diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzeziz" "b/data_all_eng_slimpj/shuffled/split2/finalzzeziz" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzeziz" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:Introduction}\n\\noindent Biological eyes are one of the most efficient and sophisticated neural systems. As a vital part of eyes, the retina can precisely and efficiently capture motion information \\cite{murphy2018old}, especially for motions caused by moving objects \\cite{olveczky2003segregation}, in natural scenes. Compared with the retina, most state-of-the-art motion estimation approaches are still limited by challenging conditions such as motion blur and high dynamic range (HDR) illuminations. There have been several attempts (e.g., \\cite{mcintosh2016deep}) trying to imitate the retina by using artificial neural networks (ANNs). However, it is difficult for ANNs to imitate the asynchronous nature of the retina. In contrast, event cameras (e.g., DAVIS \\cite{brandli2014240}) are asynchronous visual sensors with very high dynamic range and temporal resolution ($>$120 dB, $<$1 ms). These bio-inspired sensors help event-based methods to perform better in many computer vision and artificial intelligence tasks. In particular, event cameras can filter out non-motion information from the visual input, under stable illumination conditions or infrequent light variations: thus saving a lot of computation power, and giving clear clues about where object movement occurred.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\resizebox{0.84\\columnwidth}{!}{\n\t\\includegraphics{figure1}\n }\n\t\\caption{An illustration of the encoding process of the proposed TSLTD frames. The left and right sub-figures in the first row show two snapshots, while the middle one shows the retinal events between the two snapshots. Note that the two snapshots are just for reference to show what a conventional camera would have seen before and after the retinal events occur. The second row shows a sequence of the encoded TSLTD frames of the retinal events.}\n\t\\label{fig:retinalEvents}\n\\end{figure} \n\nSeveral 3D 6-DoF ego-motion estimation methods for event-based input, such as \\cite{mueggler2014event,kim2016real,gallego2017event}, have been proposed during recent years. They have shown the superiority of event cameras on the motion estimation task. However, there are only a few studies devoted to analyzing object-level motion, and most of these studies are designed for some special scenarios (e.g., \\cite{pikatkowska2012spatiotemporal} is for the pedestrian tracking scenario). Moreover, none of these methods are based on the regression methodology, which gives an explicit motion model for retinal events. As shown in Fig. \\ref{fig:retinalEvents}, the retinal events, collected between the previous snapshot and the current snapshot, show clear visual patterns about object motions. With this intuition, we present a 5-DoF object-level motion estimation method based on the event camera inputs. \n\nIn this study, we propose a novel deep neural network (called Retinal Motion Regression Network, abbreviated as RMRNet) to regress the corresponding 5-DoF object-level motion for moving objects. Here, the 5-DoF object-level motion is a 2D transform between the object bounding box in the previous frame and the estimated object bounding box in the current frame. The proposed RMRNet has a lightweight network structure, and it is end-to-end trainable. In order to leverage the proposed network, we encode asynchronous retinal events into a sequence of synchronous Time-Surface with Linear Time Decay (TSLTD) frames, as shown in the second row of Fig. \\ref{fig:retinalEvents}. The TSLTD representation, based on the Time-Surface representation \\cite{lagorce2017hots}, contains clear spatio-temporal motion patterns corresponding to the original retinal events, which is convenient for extracting motion information using RMRNet. Overall, this study makes the following contributions:\n\n\\begin{itemize}\n\\item We present the TSLTD representation that is amenable for preserving the spatio-temporal information of retinal events, and for training motion regression networks.\n\\item We introduce a 5-DoF object-level motion model to explicitly regress object motions for visual tracking.\n\\item We propose a Retinal Motion Regression Network (RMRNet) that allows one to end-to-end estimate the 5-DoF object-level motion, directly from TSLTD frames.\n\\end{itemize}\n\nWe evaluate our method on an event camera dataset and an extreme event dataset. The results demonstrate the superiority of our method when it is compared with several state-of-the-art object tracking methods.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\resizebox{1.75\\columnwidth}{!}{\n\t\\includegraphics{figure2}\n }\n\t\\caption{The full pipeline of the proposed method. Initially, retinal events are integrated and formatted as a sequence of TSLTD frames. The input of RMRNet is the object patches cropped from the TSLTD frames for each object. These object patches are sent to a set of convolutional layers (marked in blue) to extract deep features. Then, these features are stacked or they are in one-by-one style, to pass through an LSTM module (marked in orange), for feature compression. Finally, the network is divided into five branches of fully connected layers (marked in green) to predict 5-DoF object-level motions separately.}\n\t\\label{fig:pipeline}\n\\end{figure*}\n\n\\section{Related Work}\n\\label{sec:RelatedWork}\nEvent cameras have achieved great success on a variety of computer vision and artificial intelligence tasks \\cite{gallego2019event}. Event-based motion estimation is a popular topic in these tasks. For 2D motion estimation, event-based optical flow can be calculated by using a sliding window variational optimization algorithm \\cite{bardow2016simultaneous}, or the best point trajectories of the event data \\cite{gallego2018unifying}, by using a self-supervised neural network \\cite{Zhu_2019_CVPR}, or a time-slice block-matching method\\cite{liu2018adaptive}. For 3D motion estimation, the event-based 6-DoF motion can be predicted by using a line constraint \\cite{mueggler2014event}, using interleaved probabilistic filters \\cite{kim2016real}, or using photometric depth maps \\cite{gallego2017event}. For event-based visual-inertial data, the 3D motion information can also be estimated by using an extended Kalman filter on image features \\cite{zihao2017event}, using a keyframe-based nonlinear optimization algorithm \\cite{rebecq2017real} or using a continuous-time representation \\cite{mueggler2018continuous}. As described in these works, event cameras have shown a unique and superior ability for the motion estimation task. From their results, we can see that event cameras usually outperform conventional cameras, especially when coping with some harsh conditions such as fast motion and HDR scenes.\n\nDespite the fact that event-based object tracking methods can benefit a lot from the high spatio-temporal resolution and high HDR features of event cameras (when compared with conventional tracking methods e.g., \\cite{fan2017robust,Yun_2017_CVPR,lan2018robust,Li_Zhu_Hoi_Song_Wang_Liu_2019,Qi_Zhang_Zhang_Su_Huang_Yang_2019,Huang_Zhou_2019}), there are only a few works done in this area. These works can be roughly divided into two categories. The works in the first category need a clustering process to group events into clusters. The works in the second category do not need the clustering process. \n\nIn the first category, \\cite{pikatkowska2012spatiotemporal} proposes a method for tracking multiple pedestrians with occlusions. They use a Gaussian mixture model for clustering. Similarly, \\cite{camunas2017event} also rely on a clustering algorithm to track objects with occlusions using a stereo system of two event cameras. In \\cite{glover2017robust}, they propose a variant of particle filter to track the cluster centers grouped by a Hough transform algorithm. \n\nIn the second category, \\cite{mitrokhin2018event} proposes an object detection and tracking method, which is built on top of the motion compensation concept, and they use the Kalman filter tracker for tracking. \\cite{ramesh2018long} also proposes a detection and tracking method for long-term tracking using a local sliding window approach. \\cite{barranco2018real} presents a real-time method for multi-target tracking based on both mean-shift clustering and Kalman tracking. \n\nFor the above-mentioned studies, we have found that all of them involve handcrafted strategies. However, in our study, we prefer to learn an explicit 2D object motion model from original retinal events, with minimal human intervention, in an end-to-end manner.\n\n\\section{Proposed Method}\n\\label{sec:Methods}\nOur method can directly predict frame-wise 5-DoF in-plane object-level motion (i.e., a 2D transform for bounding boxes) from retinal events. The full pipeline of our method is shown in Fig. \\ref{fig:pipeline}. The retinal events are initially created by an event camera (we use event channel data from a DAVIS sensor \\cite{brandli2014240}, as the input in this work), and the events are converted into a series of synchronous Time-Surface with Linear Time Decay (TSLTD) frames. Then the TSLTD frames are fed to our new Retinal Motion Regression Network (RMRNet) to regress the corresponding 5-DoF in-plane motions. In the remainder of this section, we introduce the proposed method in detail.\n\n\\subsection{Time-Surface with Linear Time Decay}\n\\label{subsec:RetinalEventFlow}\nThe $k$-th event $e_{k}$ of retinal events $\\mathcal{E}$ can be represented as a quadruple:\n\\begin{equation}\ne_{k} \\doteq (u_{k}, v_{k}, p_{k}, t_{k}),\n\\end{equation}\nwhere $u_{k}$ and $v_{k}$ are the horizontal and vertical coordinates of $e_{k}$; $p_{k}$ indicates the polarity (\\emph{On} or \\emph{Off}) of $e_{k}$, and $t_{k}$ is the timestamp of $e_{k}$. Retinal events can occur independently in an asynchronous manner, which makes it difficult for conventional computer vision algorithms to directly process the raw data. There are several attempts \\cite{lagorce2017hots,sironi2018hats,maqueda2018event} that try to convert asynchronous retinal events to synchronous frames. For example, \\cite{lagorce2017hots} adopts a hierarchical model, which is beneficial to the recognition task but containing less motion information. \\cite{sironi2018hats,maqueda2018event} use specially designed histograms to format retinal events, which cut off the continuity of motion patterns in the temporal domain. In contrast, we prefer to create a clear and lightweight motion pattern for training our network with the help of event cameras, which allows our method to use a much smaller network structure, and maintain high precision estimation and real-time performance simultaneously.\n\nIn this work, we propose a synchronous representation of retinal events named Time-Surface with Linear Time Decay (TSLTD), as shown in Fig. \\ref{fig:retinalEvents}. The TSLTD representation is based on the Time-Surface representation in \\cite{lagorce2017hots}, but with two major differences: (1) the Time-Surface representation, which is designed for object recognition, consists of clear object contours with little motion information. We replace the exponential time decay kernel in Time-Surface with a linear time decay kernel to efficiently create effective motion patterns; (2) our TSLTD representation does not need the two hyper-parameters (i.e., the size of neighborhood and the time constant of exponential kernel) in Time-Surface. Thus, TSLTD can be effectively generalized to various target objects for motion estimation. \n\nIn TSLTD, motion information, captured by an event camera, is encoded and represented as a set of TSLTD frames over every time window \\emph{T}. The time window \\emph{T} will be discussed later. Each of the TSLTD frames is initialized to a three-dimensional zero matrix $\\mathcal{M} \\in {\\mathcal{N}^{h \\times w \\times 2}}$. Here $h$ and $w$ are the height and width of the event camera resolution, the third dimension indicates the polarity of events (and the information from \\emph{On} or \\emph{Off} events will be stored separately). Then asynchronous retinal events within the current time window are used to update the matrix in an ascending order in terms of their timestamps.\n\nMore specifically, supposing that we are processing a retinal event set $\\mathcal{E}_{t_{s},t_{e}}$ which is collected between the start timestamp $t_s$ and the end timestamp $t_{e}=t_{s}+T$ of the current time window, to yield a new TSLTD frame $\\mathcal{F}_{t_{s},t_{e}}$. $\\mathcal{F}_{t_{s},t_{e}}$ is initialized to a 3D zero matrix $\\mathcal{M}_{t_{s},t_{e}}$. During the updating process, we process ${{\\cal E}_{t_{s},t_{e}}} = \\left\\{ {{e_i},{e_{i + 1}}, \\ldots ,{e_j}} \\right\\}$ from ${e_i}$ to ${e_j}$, where ${e_i}$ is the first event and ${e_j}$ is the last event. Each of the events in $\\mathcal{E}_{t_{s},t_{e}}$ triggers an update, which assigns a value ${g}$ to $\\mathcal{M}_{t_{s},t_{e}}$ at the coordinates ${(u, v)}$ corresponding to the triggering event. The value ${g_k}$ for the $k$-th update caused by ${e_k}$ is calculated using the following equation:\n\\begin{equation}\n{g_k} = round(255*\\left( {{t_k} - {t_s}} \\right)\/T),\n\\end{equation}\nwhere ${t_{k}}$ is the timestamp of ${e_{k}}$. So the assigned value for each update is proportional to the timestamp of the triggered event. ${t_k}-{t_s}$ is the linear time decay, and we use $255\/T$ to normalize the decay. When a pixel of an object moves to a new coordinate, an event will occur at that coordinate and the TSLTD frame records a higher value of $g$ than the previous one at that coordinate. As a result, the time-motion variation information is naturally embedded in the TSLTD frames to form intensity gradients as shown in Fig. \\ref{fig:pipeline}, which have shown a clear pattern of the direction and magnitude of the corresponding object motion. Therefore, the TSLTD format facilitates our network to extract the motion information through intensity gradient patterns that are embedded in the TSLTD frames.\n\nThere are two main problems that are related to the time window \\emph{T} value, used in generating TSLTD frames. The first problem is that if \\emph{T} is set to a large interval (e.g., more than 16 $ms$), there are two consequences for fast moving objects. Since these objects move fast, they can return to the previous position or move far from the previous position, during a large time interval. As a result, new motion patterns overlap and contaminate the previous patterns, or become too large to recognize. On the contrary, if \\emph{T} is set to a small interval (e.g., less than 3 $ms$), TSLTD can only capture a very small movement, which may not be distinguished from sensor noises, and thus it may cause an ambiguous motion pattern (especially for a low-resolution event camera). After testing \\emph{T} with different objects and motions, we experimentally set \\emph{T} to be $6.6$ $ms$ (according to the sampling frequency of 150 Hz) for good generalization performance.\n\n\\subsection{Network Structure}\n\\label{subsec:NetworkStructure}\nNowadays, pre-trained deep models, such as VGGNet \\cite{simonyan2014very} and ResNet \\cite{he2016deep}, are very popular among many computer vision tasks. But most of these pre-trained models were trained using three-channel RGB images, which is not the optimal option for two-channel TSLTD frames. In addition, since TSLTD frames have clear motion patterns, we do not need very deep and complex networks to extract very high-level features for general pattern recognition.\n\nHere we design a lightweight network, named RMRNet, to learn object motions directly from TSLTD in an end-to-end manner. Between every two adjacent video frames, there are five TSLTD frames, which contain multiple objects. For individual object motion estimation, we crop object patches from the TSLTD frames. During the training stage, we crop the object patches from adjacent five TSLTD frames according to $\\tau$ times of their axis-aligned bounding boxes, to preform a joint training. Here $\\tau$ is a parameter that renders the cropped region slightly larger than the previous bounding box to capture the full pattern of object motion. During the test stage, we crop object patches frame-by-frame according to $\\tau$ times of their axis-aligned bounding boxes. Finally, these object patches are resized to 64 $\\times$ 64, and sent to the proposed RMRNet as the input. \n\nAs shown in Fig. \\ref{fig:pipeline}, the first part of RMRNet contains four convolutional layers for feature extraction. The initial three layers share the same kernel size of 3 $\\times$ 3 with stride of 2, which is similar to the VGG-Network \\cite{simonyan2014very}. The kernel size of the final layer, which is used to reduce the feature dimensions, is 1 $\\times$ 1 with stride of 1. The filter numbers of the four layers are 32, 64, 128 and 32, respectively. The four convolutional layers are followed by a batch normalization layer. A dropout layer is added in the end during the training stage. Finally, the output feature is flattened and sent to the next part. The second part of RMRNet is an LSTM module. This module contains three layers with 1568 channels in each layer. By adding the LSTM module, we can stack object patches in one regression process. Then the LSTM module can fuse the stacked motion features from the CNN part to regress an accumulated motion, which is the motion between the first object patch and the final object patch. The final part of RMRNet is a set of fully connected layers, which are used to predict 5-DoF motions. The first fully connected layer has 1568 channels. Then the following layers are divided into five branches for different components of the 5-DoF motion. Each branch contains two layers, which respectively have 512 and 128 channels. This network structure is chosen due to its desirable performance on balancing both precision and speed. The output of RMRNet is a 5-DoF transform ($e_1$ to $e_5$), as described next.\n\n\\begin{table*}[t]\n\t\\caption{The details of the ECD and EED datasets. The FM, BC, SO, HDR and OC, in the Challenges column, are fast motion,\n\t\tbackground clutter, small object, HDR scene and occlusion, respectively.}\\smallskip\n\t\\centering\n\t\\resizebox{2\\columnwidth}{!}{\n\t\t\\begin{tabular}{|l|l|c|c|}\n\t\t\t\\hline\n\t\t\tDataset & Sequence names & Feature & Challenges\\\\\n\t\t\t\\hline\\hline\n\t\t\tECD & shapes\\_translation & B\\&W shape objects mainly with translations & FM \\\\\n\t\t\tECD & shapes\\_6dof & B\\&W shape objects with various 6-DoF motions & FM \\\\\n\t\t\tECD & poster\\_6dof & Natural textures with cluttered background and various 6-DoF motions & FM+BC \\\\\n\t\t\tECD & slider\\_depth & Various artifacts at different depths with only translations & BC \\\\ \\hline\n\t\t\tEED & fast\\_drone & A fast moving drone under a very low illumination condition & FM+SO+HDR \\\\\n\t\t\tEED & light\\_variations & Same with the upper one with extra periodical abrupt flash lights & FM+SO+HDR \\\\\n\t\t\tEED & what\\_is\\_background & A thrown ball with a dense net as foreground & FM+OC \\\\\n\t\t\tEED & occlusions & A thrown ball with a short occlusion under a dark environment & FM+OC+HDR \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t}\n \\label{tab:dataset}\n\\end{table*}\n\nA 5-DoF transform ${\\mathcal{T}_{i,j}^{o}}$ between a frame ${i}$ and the next frame ${j}$ for an object ${o}$ can be defined as a subset of the 2D affine transform on an object bounding box in the ${i}$-th frame:\n\\begin{equation}\n{\\mathcal{T}_{i,j}^{o}} \\doteq (d_{x}, d_{y}, \\theta, s_{x}, s_{y}),\n\\end{equation}\nwhere ${\\mathcal{T}_{i,j}^{o}}$ is represented as a quintet, $d_{x}$ and $d_{y}$ are respectively the horizontal and vertical displacement factors, $s_{x}$ and $s_{y}$ are respectively the horizontal and vertical scale factors, and $\\theta$ is the rotation factor. Note that the rotation factor $\\theta$ and the scaling factors $s_{x}$ and $s_{y}$ in ${\\mathcal{T}_{i,j}^{o}}$ are ``in-place operations'', which means that we will keep center alignment before and after these two operations. The resulting coordinate transform is as follows:\n\\begin{equation}\n\\begin{bmatrix} {x}' \\\\ {y}' \\end{bmatrix} =\n\\begin{bmatrix} s_{x} & 0 \\\\ 0 & s_{y} \\end{bmatrix} \\otimes\n\\begin{bmatrix} cos\\theta & -sin\\theta \\\\ sin\\theta & cos\\theta \\end{bmatrix} \\otimes\n\\begin{bmatrix} x + d_{x} \\\\ y + d_{y} \\end{bmatrix}.\n\\end{equation}\nHere the original coordinates $(x,y)$ of the bounding box of object ${o}$ in previous frame ${i}$ are transformed into the new coordinates $({x}',{y}')$ in current frame ${j}$ through the transform ${\\mathcal{T}_{i,j}^{o}}$. The operator $\\otimes$ indicates an in-place operation. Note that the five parameters $e_1$ to $e_5$ predicted by RMRNet are normalized to a range of $[-1.0,1.0]$ using the Tanh activation function. Then we set five boundary parameters $p_1$ to $p_5$ for $e_1$ to $e_5$ to constrain the range of object movements. Finally, the transform ${\\mathcal{T}_{i,j}^{o}}$ is calculated as follows:\n\\begin{equation}\n\\label{eq:5dof}\n{\\cal T}_{i,j}^o = \\left\\{ {\\begin{array}{*{20}{l}}\n\t{{d_x} = {e_1}*{p_1}}\\\\\n\t{{d_y} = {e_2}*{p_2}}\\\\\n\t{\\theta = ({e_3}*{p_3})*\\pi \/180}\\\\\n\t{{s_x} = 1 + {e_4}*{p_4}}\\\\\n\t{{s_y} = 1 + {e_5}*{p_5}}\\\\\n\t\\end{array}} \\right..\n\\end{equation}\nIn this paper, we respectively fix $p_1$ to $p_5$ to 72, 54, 30, 0.2 and 0.2, and fix $\\tau$ to 1.2 according to the 240 $\\times$ 180 resolution of the DAVIS camera for RMRNet. This parameter setting will allow a relatively large range for object movements. Thus, the setting is suitable for estimating the majority of object motions, which includes most of the fast movements.\n\n\\subsection{Learning Approach}\n\\label{subsec:LearningApproach}\nThere are two key points that should be mentioned in relation to the training stage. The first one is that if we only use B\\&W object samples (i.e., black objects with a white background) as the training data, our network can only learn some relatively simple motion patterns for the corresponding object motions. Thus, we follow the standard five-fold cross-validation protocol and use the object pairs (refer to the next section) from the \\emph{shapes\\_6dof}, \\emph{poster\\_6dof} and \\emph{light\\_variations} sequences as the training and validation data (while the other five sequences are unseen during the training stage) to train and validate the proposed RMRNet. The second point is that it is difficult to learn an object motion model from a single TSLTD frame. There are five TSLTD frames for every object pair. A single TSLTD frame usually includes only a small movement and it has only a weak motion pattern. After extensive experiments, we find that stacking five TSLTD frames in one prediction has gained the optimal performance during the training stage.\n\nThe proposed network is trained using the ADAM solver with a mini-batch size of 16. We use a fixed learning rate of 0.0001, and our loss function is the MSE loss:\n\\begin{equation}\nMSE_{loss}\\left( {\\hat {\\cal T},{\\cal T}} \\right) = \\frac{1}{{{N_{train}}}}\\sum\\limits_{l = 1}^{{N_{train}}} {{{\\left\\| {{{\\hat {\\cal T}}_l} - {{\\cal T}_l}} \\right\\|}^2}},\n\\end{equation}\nwhere $\\hat{\\cal{T}}$ is the estimated 5-DoF motion, $\\cal{T}$ is the corresponding ground truth, $N_{train}$ is the number of training samples, and the subscript $l$ indicates the $l$-th sample.\n\n\\section{Experiments}\n\\label{sec:Experiments}\n\n\\subsection{Pre-processing}\n\\label{subsec:Pre-processing}\nFor our evaluation, we use a challenging mixed event dataset including a part of the popular Event Camera Dataset (ECD) \\cite{mueggler2017event} and the Extreme Event Dataset (EED) \\cite{mitrokhin2018event}, which were recorded using a DAVIS event camera in real-world scenes. The details of the dataset can be found in Table \\ref{tab:dataset}. Note that the mixed dataset contains both the event data sequences and the corresponding video sequences for every sequence. Since the ECD dataset does not provide ground truth bounding boxes for object tracking, we labeled a rotated or an axis-aligned rectangle bounding box as the ground truth for each object in the dataset to evaluate all the competing methods.\n\nWe choose five state-of-the-art object tracking methods, including SiamFC \\cite{bertinetto2016fully}, ECO \\cite{danelljan2017eco}, SiamRPN++ \\cite{Li_2019_CVPR}, ATOM \\cite{Danelljan_2019_CVPR} and E-MS \\cite{barranco2018real}, as our competitors. About these competitors: SiamFC and SiamRPN++ are fast and accurate methods, which are based on Siamese networks. ECO and ATOM are state-of-the-art methods that have achieved great performance on various datasets. E-MS \\cite{barranco2018real} is recently proposed for event-based target tracking based on mean-shift clustering and Kalman filter. We extend E-MS to support bounding box-based tracking, by employing the minimum enclosing rectangle of those events that belong to an identical cluster center as its estimated bounding box. \n\nTo perform ablation studies, we also compare our RMRNet with an event-based variant of ECO (called as ECO-E) and a variant of RMRNet (called as RMRNet-TS). ECO-E, which uses our proposed TSLTD frames as its inputs, is an event-based variant of ECO \\cite{danelljan2017eco}. ECO-E is used to evaluate the performance of ECO on the event data sequences. RMRNet-TS uses the classical Time-Surface frames in Hots \\cite{lagorce2017hots} instead of using the proposed TSLTD frames as its inputs. RMRNet-TS is used to evaluate the performance of the classical Time-Surface representation and our TSLTD representation. For SiamFC, ECO, SiamRPN++ and ATOM, we use the video sequences as their inputs. For ECO-E, E-MS, RMRNet-TS and the proposed RMRNet, we use the event data sequences (in quadruple format) as their inputs. For all the five competitors, we use their released codes, default parameters and best pre-trained models.\n\n\\begin{table*}[t]\n\t\\caption{Results obtained by the competitors and our method on the ECD dataset. The best results are in \\textbf{bold}.}\\smallskip\n\t\\label{tab:ecd_results}\n\t\\centering\n\t\\resizebox{2\\columnwidth}{!}{\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t\\multirow{2}{*}{Method} & \\multicolumn{2}{c|}{shapes\\_translation} & \\multicolumn{2}{c|}{shapes\\_6dof} & \\multicolumn{2}{c|}{poster\\_6dof} & \\multicolumn{2}{c|}{slider\\_depth} \\\\ \\cline{2-9}\n\t\t\t& AOR & AR & AOR & AR & AOR & AR & AOR & AR \\\\ \\hline\\hline\n\t\t\tSiamFC\\cite{bertinetto2016fully} & 0.812 & 0.940 & 0.835 & 0.968 & 0.830 & 0.956 & 0.909 & \\textbf{1.000} \\\\ \\hline\n\t\t\tECO\\cite{danelljan2017eco} & 0.823 & 0.943 & 0.847 & 0.969 & 0.846 & 0.960 & \\textbf{0.947} & \\textbf{1.000} \\\\ \\hline\n\t\t\tSiamRPN++\\cite{Li_2019_CVPR} & 0.790 & 0.942 & 0.779 & 0.972 & 0.753 & 0.899 & 0.907 & \\textbf{1.000} \\\\ \\hline\n\t\t\tATOM\\cite{Danelljan_2019_CVPR} & 0.815 & 0.945 & 0.803 & 0.974 & 0.835 & 0.961 & 0.897 & \\textbf{1.000} \\\\ \\hline\n\t\t\tECO-E\\cite{danelljan2017eco} & 0.821 & 0.941 & 0.834 & 0.960 & 0.783 & 0.878 & 0.771 & 0.993 \\\\ \\hline\n\t\t\tE-MS\\cite{barranco2018real} & 0.675 & 0.768 & 0.612 & 0.668 & 0.417 & 0.373 & 0.447 & 0.350 \\\\ \\hline\n\t\t\tRMRNet-TS & 0.491 & 0.564 & 0.467 & 0.509 & 0.504 & 0.558 & 0.814 & 0.993 \\\\ \\hline\n\t\t\tRMRNet & \\textbf{0.836} & \\textbf{0.951} & \\textbf{0.866} & \\textbf{0.980} & \\textbf{0.859} & \\textbf{0.962} & 0.915 & \\textbf{1.000} \\\\ \\hline\n\t\t\\end{tabular}\n }\n\\end{table*}\n\n\\begin{table*}[t]\n\t\\caption{Results obtained by the competitors and our method on the EED dataset. The best results are in \\textbf{bold}.}\\smallskip\n\t\\label{tab:eed_results}\n\t\\centering\n\t\\resizebox{2\\columnwidth}{!}{\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t\\multirow{2}{*}{Method} & \\multicolumn{2}{c|}{fast\\_drone} & \\multicolumn{2}{c|}{light\\_variations} & \\multicolumn{2}{c|}{what\\_is\\_background} & \\multicolumn{2}{c|}{occlusions} \\\\ \\cline{2-9}\n\t\t\t& AOR & AR & AOR & AR & AOR & AR & AOR & AR \\\\ \\hline\\hline\n\t\t\tSiamFC\\cite{bertinetto2016fully} & 0.766 & \\textbf{1.000} & 0.772 & \\textbf{0.947} & 0.712 & 0.833 & 0.148 & 0.000 \\\\ \\hline\n\t\t\tECO\\cite{danelljan2017eco} & 0.830 & \\textbf{1.000} & 0.782 & 0.934 & 0.675 & 0.750 & 0.209 & 0.333 \\\\ \\hline\n\t\t\tSiamRPN++\\cite{Li_2019_CVPR} & 0.717 & 0.941 & 0.497 & 0.500 & 0.653 & 0.833 & 0.096 & 0.167 \\\\ \\hline\n\t\t\tATOM\\cite{Danelljan_2019_CVPR} & 0.763 & \\textbf{1.000} & 0.652 & 0.921 & \\textbf{0.725} & \\textbf{0.917} & 0.387 & 0.500 \\\\ \\hline\n\t\t\tECO-E\\cite{danelljan2017eco} & 0.728 & 0.882 & 0.685 & 0.803 & 0.099 & 0.000 & 0.308 & 0.333 \\\\ \\hline\n\t\t\tE-MS\\cite{barranco2018real} & 0.313 & 0.307 & 0.325 & 0.321 & 0.362 & 0.360 & 0.356 & 0.353 \\\\ \\hline\n\t\t\tRMRNet-TS & 0.199 & 0.118 & 0.096 & 0.066 & 0.108 & 0.000 & 0.000 & 0.000 \\\\ \\hline\n\t\t\tRMRNet & \\textbf{0.892} & \\textbf{1.000} & \\textbf{0.802} & \\textbf{0.947} & 0.202 & 0.083 & \\textbf{0.716} & \\textbf{0.833} \\\\ \\hline\n\t\t\\end{tabular}\n }\n\\end{table*}\n\nThe output of our RMRNet is a 2D 5-DoF frame-wise bounding box transform model, estimated between two adjacent frames. In order to evaluate the quality of the estimated frame-wise bounding box transform model, we compare the proposed RMRNet with all the competitors on frame-wise object tracking, that is, our evaluation on these competing methods is based on object pairs, each of which includes two object regions on two adjacent frames corresponding to an identical object. During the evaluation, we treat each of the object pairs as a tracking instance in the corresponding frame. In addition, all the competitors only estimate axis-aligned bounding boxes, whereas the proposed RMRNet can estimate both rotated and axis-aligned bounding boxes (by using the 5-DoF motion model). Therefore, we evaluate all the methods on the axis-aligned ground truth bounding boxes for a fair precision comparison.\n\n\\subsection{Evaluation Metrics}\n\\label{subsec:EvaluationMetrics}\nFor evaluating the precision of all the methods, we calculate the Average Overlap Rate (AOR) as follow:\n\\begin{equation}\n\\label{eq:aor}\n{AOR} = \\frac{1}{{{N_{rep}}}}\\frac{1}{{{N_{pair}}}}\\sum\\limits_{u = 1}^{{N_{rep}}} {\\sum\\limits_{v = 1}^{{N_{pair}}} {\\frac{{O_{u,v}^E \\cap O_{u,v}^G}}{{O_{u,v}^E \\cup O_{u,v}^G}}}},\n\\end{equation}\nwhere $O_{u,v}^{E}$ is the estimated bounding box in the $u$-th round of the cross-validation for the $v$-th object pair, and $O_{u,v}^{G}$ is the corresponding ground truth. Eq. (\\ref{eq:aor}) shows that the AOR measure is related to the Intersection over Union (IoU). ${N_{rep}}$ is the repeat times of the cross-validation, and ${N_{pair}}$ is the number of object pairs in the current sequence. We set ${N_{rep}}$ to 5 for all the following experiments. \n\nWe also calculate the Average Robustness (AR) to measure the robustness of all the competing methods as follow:\n\\begin{equation}\n{AR} = \\frac{1}{{{N_{rep}}}}\\frac{1}{{{N_{pair}}}}\\sum\\limits_{u = 1}^{{N_{rep}}} {\\sum\\limits_{v = 1}^{{N_{pair}}} {succes{s_{u,v}}}},\n\\end{equation}\nwhere $succes{s_{u,v}}$ indicates that whether the tracking in the $u$-th round for the $v$-th pair is successful or not (0 means failure and 1 means success). If the AOR value obtained by a method for one object pair is under 0.5, we will consider it as a tracking failure case.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\resizebox{1.53\\columnwidth}{!}{\n\t\\includegraphics{figure3}\n }\n\t\\caption{Tracking results obtained by SiamFC, ECO, SiamRPN++, ATOM, E-MS and our method. Each row represents a sequence of the two datasets. From top to bottom, the corresponding sequences are \\emph{shape\\_6dof}, \\emph{poster\\_6dof}, \\emph{slider\\_depth}, \\emph{light\\_variations}, \\emph{what\\_is\\_background} and \\emph{occlusions}, respectively. From left to right, the first, third and fifth columns show the results of the competing methods with the axis-aligned GT. The second, fourth and sixth columns show the results of our method with the rotated GT. The seventh column show the actual TSLTD frames of the second column. Best viewed in color.}\n\t\\label{fig:results}\n\\end{figure*}\n\n\\subsection{Evaluation on the Event Dataset}\n\\label{subsec:Evaluation}\nWe use the mixed event dataset to evaluate the eight competing methods. We choose the \\emph{shapes\\_translation}, \\emph{shapes\\_6dof}, \\emph{poster\\_6dof} and \\emph{slider\\_depth} sequences from the ECD dataset \\cite{mueggler2017event} as the representative sequences for comparison. The first three sequences have increasing motion speeds. The fourth sequence has a constant motion speed. The object textures of these sequences vary from simple B\\&W shapes to complicated artifacts. For these sequences, we are mainly concerned with the performance of all methods for various motions, especially for fast 6-DoF motion, and for different object shapes and textures.\n\nFor comparison, the quantitative results are reported in Table \\ref{tab:ecd_results}. We also provide some representative qualitative results obtained by SiamFC, ECO, SiamRPN++, ATOM, E-MS and our method in the top three rows of Fig. \\ref{fig:results}. From Table \\ref{tab:ecd_results}, we can see that our method achieves the best performance on the first three sequences and it achieves the second best performance on the fourth sequence. SiamFC, ECO, SiamRPN++ and ATOM also achieve competitive results. However, as we can see in Fig. \\ref{fig:results}, our method has achieved better performance in estimating fast motion. In comparison, SiamFC, ECO, SiamRPN++ and ATOM usually lose the tracked objects, due to the influence of motion blur. Comparing with the original ECO, ECO-E has achieved inferior performance, which shows that state-of-the-art object tracking methods, like ECO, are not suitable to be directly applied to event data. The classical Time-Surface frames are designed for object recognition and detection, which contain less motion information for motion estimation and object tracking. Thus, RMRNet-TS shows much inferior results in this evaluation. By leveraging the high temporal resolution feature of the event data, E-MS can also effectively handle most fast motions. However, E-MS is less effective to handle complicated object textures and cluttered backgrounds (e.g., for the \\emph{poster\\_6dof} and \\emph{slider\\_depth} sequences). In contrast, the proposed RMRNet outperforms E-MS by a large margin, which shows the superiority of our method in handling various object textures and cluttered backgrounds.\n\nMoreover, we also choose the EED dataset to evaluate the eight competing methods. The EED dataset \\cite{mitrokhin2018event} contains four challenging sequences: \\emph{fast\\_drone}, \\emph{light\\_variations}, \\emph{occlusions} and \\emph{what\\_is\\_background}. The first three sequences respectively record a fast moving drone under low illumination environments. The fourth sequence records a moving ball with a net as foreground. Using this dataset, we want to evaluate the competing methods in low illumination conditions and in occlusion situations.\n\nThe quantitative results and some representative qualitative results are shown in Table \\ref{tab:eed_results} and Fig. \\ref{fig:results}, respectively. From the results, we can see that our method achieves the best performance on most sequences except for the \\emph{what\\_is\\_background} sequence, on which our method has obtained low AOR and AR. This is because that the foreground net in the \\emph{what\\_is\\_background} sequence covers the ball, which destroys the corresponding motion patterns. In contrast, our method has achieved the highest AOR and AR on the \\emph{occlusions} sequence. This is because that in this sequence, the occlusion time is short and only one object pair involves the occlusion. Our method fails at tracking that object pair but it successfully estimates the other object pairs. As the competitors, SiamFC, ECO, SiamRPN++ and ATOM show their low effectiveness with fast motion and low illumination conditions. Moreover, ECO-E and RMRNet-TS are respectively inferior to ECO and RMRNet. Although SiamFC and ECO achieve relatively good results for the first two sequences, the sequences include relatively clean backgrounds, which helps the two methods to achieve the performance. However, if we add a small amount of noise around the objects, the performance of the two methods will be significantly reduced. In contrast, our method can maintain its performance even with the severe sensor noises of an event camera, as shown in the seventh column of Fig. \\ref{fig:results}. Meanwhile, the performance of E-MS is highly affected by the sensor noises in the HDR environments. Thus, E-MS show unsatisfied results on the EED dataset.\n\n\\subsection{Time Cost}\n\\label{subsec:timecost}\nSince the proposed RMRNet is a relatively shallow network, our method has an advantage of relatively high efficiency. The proposed RMRNet is implemented using PyTorch on a PC with an Intel i7 CPU and an NVIDIA GTX 1080 GPU. For the mixed dataset, our method achieves real-time performance and the average computational time for each object pair (including five TSLTD frames) is 38.57 ms. \n\n\\section{Conclusion}\n\\label{sec:Conclusion}\nIn this paper, we demonstrate the great potential of event cameras for object tracking under severe conditions such as fast motion and low illumination scenes. By using the event camera, our method only extracts motion related information from the inputs. Then we present the TSLTD representation to encode input retinal events into a sequence of synchronous TSLTD frames. TSLTD can represent the spatio-temporal information with clear motion patterns. Finally, to leverage the motion clues contained in TSLTD, we propose the RMRNet to regress 5-DoF motion information in an end-to-end manner. Extensive experiments demonstrate the superiority of our method over several other state-of-the-art object tracking methods.\n\n\\section{Acknowledgments}\nThis work is supported by the National Natural Science Foundation of China under Grants U1605252 and 61872307. Haosheng Chen and David Suter acknowledge funding from ECU that enabled the first author to visit ECU and undertake a small portion of this work.\n\n\\bibliographystyle{aaai}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe sensitivity dependence on initial conditions (SDIC) of trajectory of a chaotic dynamical system was first discovered by Poincar\\'{e} \\cite{poincare1890probleme}, and then rediscovered by Lorenz \\cite{lorenz1963deterministic} with a more popular name ``butterfly-effect''. Besides, Lorenz \\cite{lorenz1989computational,lorenz2006computational} further discovered that trajectories of a chaotic dynamical system have the sensitivity dependence {\\em not only} on initial conditions (SDIC) {\\em but also} on numerical algorithms (SDNA), because numerical noises (i.e. truncation error and round-off error) are {\\em unavoidable} for all numerical algorithms. All of these phenomena are based on the exponential increase of noises (or small disturbances) of chaotic dynamical system. Naturally, this kind of non-replicability\/unreliability of chaotic trajectory brought some heated debates on the credence of numerical simulation of chaotic system, and some even made a rather pessimistic conclusion that ``for chaotic systems, numerical convergence cannot be guaranteed {\\em forever}'' \\cite{Teixeira2007Time}. Besides, it is currently reported that ``shadowing solutions can be almost surely nonphysical'', which ``invalidates the argument that small perturbations in a chaotic system can only have a small impact on its statistical behavior'' \\cite{Chandramoorthy2021JCP}.\n\n\nIn order to gain a {\\em reproducible\/reliable} numerical simulation of chaos, Liao \\cite{Liao2009} suggested a numerical strategy, namely the ``Clean Numerical Simulation'' (CNS) \\cite{Liao2009,Liao2013,Liao2014,LIAO2014On}, to greatly reduce the background numerical noises (i.e. truncation error and round-off error) in a finite interval of time that is long {\\em enough} for statistics. In the frame of the CNS \\cite{Liao2009, Liao2013, Liao2014, LIAO2014On}, the spatial and temporal truncation errors are reduced to a {\\em required} tiny level by means of a fine {\\em enough} spatial discretization (such as the spatial Fourier expansion) and a high {\\em enough} order of Taylor expansion in the temporal dimension, respectively. Especially, by means of a large {\\em enough} number of significant digits to represent all physical and numerical variables\/parameters in the multiple precision \\cite{oyanarte1990mp}, the round-off error can be reduced to a {\\em required} tiny level. Furthermore, an additional simulation with the even smaller background numerical noises is performed so as to determine the so-called ``critical predictable time'' $T_c$ by comparing such two simulations that the numerical noises (caused by truncation and round-off errors) can be negligible, i.e. several orders of magnitude smaller than the physical solution, and thus the computer-generated result of chaos is reproducible\/reliable within the whole spatial domain in the time interval $t\\in[0,T_c]$. In this way, the CNS can give the reproducible\/reliable trajectories of a chaotic dynamical system in an interval of time $[0,T_{c}]$ that is long enough for statistics. \n\nThe CNS provides us a useful tool to gain {\\em reproducible\/reliable} simulations of chaos in a long enough interval of time. Up to now, the CNS has been successfully applied to solve many chaotic systems, such as the Lorenz equations \\cite{LIAO2014On}, the two-dimensional Rayleigh-B\\'{e}nard turbulence \\cite{lin2017origin}, the chaotic motion of a free fall desk \\cite{xu2021accurate} and some spatiotemporal chaotic systems such as the complex Ginzburg-Landau equation \\cite{hu2020risks}, the damped driven sine-Gordon equation \\cite{qin2020influence} and so on. Especially, by means of the CNS, more than $2000$ new families of periodic orbits of the three-body system \\cite{Li2017More,li2018over,li2019collisionless} have been found, which were reported twice by the popular magazine {\\em New Scientist} \\cite{NewScientist2017,NewScientist2018}, because only three families of periodic orbits of the three-body problem were reported in 300 years after Newton mentioned this famous problem in $1687$. All of these illustrate the novelty, great potential and validity of the CNS for chaotic dynamic systems.\n\nObviously, a chaotic numerical simulation given by the CNS can be used as a benchmark solution to study the influence of numerical noise to chaos. Using the CNS as a tool, it is found that, for some chaotic systems, such as the Lorenz equations \\cite{lorenz1963deterministic}, which has one positive Lyapunov exponent, and the so-called hyper-chaotic Rossler system \\cite{Stankevich2020Chaos}, which has two positive Lyapunov exponents, their statistics always keep the same, although their trajectories are rather sensitive to small disturbances. We call them the normal-chaos \\cite{Liao2022AAMM}. However, it was found that the statistical properties (such as the probability density function) of some chaos are extremely sensitive to the tiny noise\/disturbance \\cite{hu2020risks, qin2020influence, xu2021accurate}, which is called ultra-chaos \\cite{Liao2022AAMM}. \n In this letter, we use the Arnold-Beltrami-Childress (ABC) flow as an example to illustrate that ultra-chaos indeed widely exists and is in a higher disorder than a normal-chaos. Besides, our results highly suggest that turbulence should have a close relationship with ultra-chaos. \n\n\\section{Ultra-chaos in the ABC flows}\n\nThe Arnold-Beltrami-Childress (ABC) flow\n\\begin{eqnarray}\n&& \\mathbf{u}_{ABC}(x,y,z) \\nonumber \\\\\n&=&[A\\,sin(z)+C\\,cos(y)]\\,\\mathbf{e}_x \n+\\;[B\\,sin(x)+A\\,cos(z)]\\,\\mathbf{e}_y \n+\\;[C\\,sin(y)+B\\,cos(x)]\\,\\mathbf{e}_z \\label{ABC-0}\n\\end{eqnarray}\ndescribes a kind of stationary flow of the incompressible fluid with periodic boundary conditions, where $\\mathbf{u}_{ABC}$ is the velocity vector field, $A$, $B$ and $C$ are arbitrary constants, $x$, $y$ and $z$ are Cartesian coordinates, $\\mathbf{e}_x$, $\\mathbf{e}_y$ and $\\mathbf{e}_z$ are the direction vectors of Cartesian coordinate system, respectively. It was first discovered as a class of analytical solutions of the Euler\/Navier-Stokes equations by Arnold \\cite{arnold1965}, and since then the Lagrangian chaotic property \\cite{arnold1965,henon1966topologie,dombre1986chaotic,galloway1986dynamo,galloway1987note} as well as the so-called Beltrami property, i.e. substantial helicity $\\mathbf{u}_{ABC}\\times(\\nabla\\times\\mathbf{u}_{ABC})=0$, of this kind of flow have aroused wide concern in nonlinear dynamics, hydrodynamics and magnetohydrodynamics. The property of exponential deviation of a fluid particle (i.e. Lagrangian chaos) in the ABC flow is typical for chaotic dynamical systems \\cite{arnold1965,henon1966topologie,dombre1986chaotic,maciejewski2002non,galloway2012abc,blazevski2014hyperbolic,didov2018analysis,didov2018nonlinear} and essential for the development of turbulent flows \\cite{dombre1986chaotic,galloway1987note,podvigina1994non}, and this feature as well as the above-mentioned substantial helicity is also essential for the fast dynamo action (i.e. fast generation of magnetic field in conducting fluids) and for the origin of magnetic field of large astrophysical objects \\cite{childress1970new, moffatt1985topological, galloway1986dynamo, finn1988chaotic, zienicke1998variable, archontis2003numerical, teyssier2006kinematic, alexakis2011searching, galloway2012abc, bouya2013revisiting}.\n\n\nLet $x(t)$, $y(t)$ and $z(t)$ represent the location coordinates of a fluid particle, $\\dot{x}(t)$, $\\dot{y}(t)$ and $\\dot{z}(t)$ denote their temporal derivatives, respectively. Thus, in the Lagrangian sense, the motion of a fluid particle of the ABC flow, which is reported to be a typical chaotic dynamical system in many cases, is governed by \n\\begin{equation}\n\\left\\{\n\\begin{array}{l}\n\\dot{x}(t)=A\\, \\sin[z(t)]+C\\,\\cos[y(t)], \\\\\n\\dot{y}(t)=B\\,\\sin[x(t)]+A\\,\\cos[z(t)], \\\\\n\\dot{z}(t)=C\\,\\sin[y(t)]+B\\,\\cos[x(t)], \n\\end{array}\n\\right. \\label{ABC}\n\\end{equation}\n with the initial condition\n\\begin{equation}\n(x(0), y(0), z(0) ) = {\\bf r}_0, \\label{ABC-ini}\n\\end{equation}\nwhere ${\\bf r}_0$ denotes a starting point of the fluid particle. Without loss of generality, let us consider the case of $A=1$ and different values of $B$ and $C$. It should be emphasized here that, by means of the CNS, we can always gain a reproducible\/reliable trajectory of the chaotic motion of a fluid particle of the ABC flow in a long enough interval of time. \nTo investigate the influence of small disturbance on trajectory of the fluid-particle in the ABC flow (\\ref{ABC-0}) starting from $\\mathbf{r}_{0}= (x(0), y(0), z(0))$, we compare the trajectories of two close fluid-particles of the ABC flow, starting from the initial positions ${\\bf r}_0 $ and ${\\bf r}_{0}'={\\bf r}_{0}+\\hspace{0.2mm}(0,0,1) \\times \\delta$, respectively, where $\\delta = |{\\bf r}_{0} - {\\bf r}_{0}' |$ is a tiny constant. Note that $\\delta = 0$ when ${\\bf r}_{0} = {\\bf r}_{0}'$. \n\n\nFor example, without loss of generality, let us consider the motion of a fluid-particle of the ABC flow (in the Lagrangian sense) starting from the point ${\\bf r}_{0} = (0,0,0)$ in the case of $A=1$ and different values of $B$ and $C$. To investigate its chaotic property, we compare its trajectory with that starting from a very close one ${\\bf r}_{0}' ={\\bf r}_{0}+\\hspace{0.2mm}(0,0,1)\\times \\delta$, where we choose either $\\delta = 10^{-5}$ or $10^{-10}$, respectively. \n It is found that in each case we can always gain a reproducible chaotic simulation in a quite long interval $t\\in[0,10000]$ by means of a parallel algorithm of the CNS using the $200$th-order Taylor expansion with the time-step $\\Delta t = 0.01$ and representing all data in $500$-digit multiple-precision (MP), whose replicability\/reliability is guaranteed via another CNS result with even smaller background numerical noises, given by the $205$th-order Taylor expansion (with the same time-step) and the $520$-digit multiple-precision. \n\n\n\\begin{figure}\n \\begin{center}\n \\subfigure[]{\\includegraphics[width=2in]{Fig1a.pdf}}\n \\subfigure[]{\\includegraphics[width=2in]{Fig1b.pdf}}\\\\\n \\subfigure[]{\\includegraphics[width=2in]{Fig1c.pdf}}\n \\subfigure[]{\\includegraphics[width=2in]{Fig1d.pdf}}\n \\caption{ {\\bf Influences of the tiny disturbances on the phase plot $x-z$ and the probability density function (PDF) of a normal-chaotic motion of the fluid-particle in the ABC flow.} The curves are based on the CNS results in $t\\in[0,10000]$ of the normal-chaotic motion of the fluid-particle, governed by the ABC flow (\\ref{ABC}) with (\\ref{ABC-ini}) in the case of $A=1$, $B=0.7$, and $C=0.42$ (with the maximum Lyapunov exponent $\\lambda_{max}=0.01$), from the starting point ${\\bf r}_{0}'=(0,0,0)+\\hspace{0.2mm}(0,0,1) \\times \\delta$ when $\\delta=0$ (red), or $\\delta=10^{-5}$ (black), or $\\delta=10^{-10}$ (blue), respectively. (a) The phase plot $(x,z)$ when $\\delta=0$; (b) The phase plot $(x,z)$ when $\\delta=10^{-5}$; (c) The phase plot $(x,z)$ when $\\delta=10^{-10}$; (d) The PDFs of $z(t)$. } \\label{C042}\n \\end{center}\\vspace{-0.6cm} \n\\end{figure}\n\n\nFor example, in the case of $A=1$, $B=0.7$ and $C=0.42$, the fluid-particle starting from ${\\bf r}_{0}=(0,0,0)$ has a chaotic motion (with the maximum Lyapunov exponent $\\lambda_{max}=0.01$) in a {\\em restricted} spatial domain, as shown in Fig.~\\ref{C042}(a) for its phase plot $(x,z)$. In the case of $\\delta=10^{-5}$ and $\\delta=10^{-10}$, although the chaotic trajectories of the two fluid-particles, starting from the points very close to ${\\bf r}_{0}=(0,0,0)$, are rather sensitive to the starting point, their attractors and statistical properties such as the probability density function (PDF) are almost the same as those given by the chaotic trajectory starting from ${\\bf r}_{0}=(0,0,0)$ that corresponds to $\\delta=0$, as shown in Fig.~\\ref{C042} (b), (c) and (d), respectively. Thus, in the case of $A=1$, $B=0.7$ and $C=0.42$, the motion of the fluid-particle starting from ${\\bf r}_{0}=(0,0,0)$ is a {\\em normal-chaos}, since its statistical properties such as the PDF of $z(t)$ are {\\em not} sensitive to the small disturbances of the starting point. \n \n\n\\begin{figure}\n \\begin{center}\n \\subfigure[]{\\includegraphics[width=2in]{Fig2a.pdf}}\n \\subfigure[]{\\includegraphics[width=2in]{Fig2b.pdf}}\n \\caption{ {\\bf Influences of tiny disturbances on the phase plot $x-z$ and the probability density function (PDF) of an ultra-chaotic motion of the fluid-particle in the ABC flow.} The curves are based on the CNS results in $t\\in[0,10000]$ of the ultra-chaotic motion of the fluid-particle, governed by the ABC flow (\\ref{ABC}) with (\\ref{ABC-ini}) in the case of $A=1.0$, $B=0.7$, and $C=0.43$ (with the maximum Lyapunov exponent $\\lambda_{max}=0.06$) from the starting point ${\\bf r}_{0}'=(0,0,0)+\\hspace{0.2mm}(0,0,1) \\times \\delta $ when $\\delta=0$ (red), or $\\delta=10^{-5}$ (black), or $\\delta=10^{-10}$ (blue), respectively. (a) The phase plots $(x,z)$; (b) The PDFs of the normalized results $z'(t)$. } \\label{C043}\n \\end{center}\\vspace{-0.6cm} \n\\end{figure}\n\n \nHowever, in the case of $A=1$, $B=0.7$ and $C=0.43$, i.e. with a small change of $C$, the chaotic motion (with the maximum Lyapunov exponent $\\lambda_{max}=0.06$) of the fluid particle of the ABC flow starting from ${\\bf r}_{0} = (0,0,0)$ becomes quite different from that in the case of $A=1$, $B=0.7$ and $C=0.42$ mentioned above: the fluid-particle moves {\\em far and far} away from ${\\bf r}_{0}$ and besides its phase plot $(x,z)$ becomes very sensitive to the small disturbance of the starting position, as shown in Fig.~\\ref{C043}(a). These are quite different from the results in the case of $A=1$, $B=0.7$ and $C=0.42$. Since the ABC flow is periodic, we \n normalize the values of $z(t)$ to $[-\\,\\pi,\\pi)$, i.e.\n\\begin{align}\nz'(t)& = z(t)+2\\pi\\hspace{0.2mm} n_z,\n\\end{align}\nwhere the values of $n_z$ are integers, and $-\\pi \\leq z' < +\\pi $. Note that, as illustrated in Fig.~\\ref{C043}(b), the tiny disturbances of the starting position can lead to huge deviations of the PDFs of the normalized chaotic simulations $z'(t)$ in $t\\in[0,10000]$. In other words, in the case of $A=1$, $B=0.7$ and $C=0.43$ of the ABC flow (\\ref{ABC}), even statistical properties of the chaotic motion of the fluid particle starting from ${\\bf r}_{0} = (0,0,0)$ are very sensitive to the initial position, and thus the corresponding motion of the particle is a kind of {\\em ultra-chaos}. \nObviously, this kind of ultra-chaos is at a higher-level of disorder than that normal-chaos, as shown in Fig.~\\ref{C042} and Fig.~\\ref{C043}. This example illustrates that ultra-chaos indeed exists in the famous ABC flow.\n\n\n\\begin{figure}\n \\begin{center}\n \\subfigure[]{\\includegraphics[width=2in]{Fig3a.pdf}}\n \\subfigure[]{\\includegraphics[width=2in]{Fig3b.pdf}}\\\\\n \\subfigure[]{\\includegraphics[width=4in]{Fig3c.pdf}}\n \\caption{ {\\bf Influences of the tiny disturbances on the $x-z$ phase plots of the ensemble-averaged trajectories of the normal-chaotic or ultra-chaotic fluid-particle in the ABC flow.} The $x$-$z$ phase plots of the ensemble-averaged trajectories are based on the CNS results in $t\\in[0,10000]$ of the normal-chaotic or the ultra-chaotic fluid-particle of the ABC flow (\\ref{ABC}) in the case of $A=1$, $B=0.7$ and either $C=0.42$ or $ C=0.43$ from the starting point ${\\bf r}_{0}=(0,0,0)+(0,0,1)\\times \\delta_i $, $1\\leq i\\leq1000$, with either $\\sigma_{d}=\\sqrt{\\langle\\delta^{2}_{i}\\rangle}=10^{-5}$ (black) or $\\sigma_{d} = 10^{-10}$ (blue), respectively. (a) The $x$-$z$ phase-plot of the normal-chaotic fluid-particle when $C=0.42$ with $\\sigma_{d}=10^{-5}$; (b) The $x$-$z$ phase-plot of the normal-chaotic fluid-particle when $C=0.42$ with $\\sigma_{d}=10^{-10}$; (c) The $x$-$z$ phase-plot of the ultra-chaotic fluid-particle when $C=0.43$ with either $\\sigma_{d}=10^{-5}$ or $10^{-10}$. } \\label{xz_ea}\n \\end{center}\\vspace{-0.6cm} \n\\end{figure}\n\n\\begin{figure}\n \\begin{center}\n \\subfigure[]{\\includegraphics[width=2in]{Fig4a.pdf}}\n \\subfigure[]{\\includegraphics[width=2in]{Fig4b.pdf}}\n \\caption{ {\\bf Influences of the tiny disturbances on the PDFs of the ensemble-averaged trajectories of the normal-chaotic or ultra-chaotic fluid-particles in the ABC flow.} The PDFs of the ensemble-averaged trajectories are based on the CNS results in $t\\in[0,10000]$ of the normal-chaotic or the ultra-chaotic fluid-particle of the ABC flow (\\ref{ABC}) with (\\ref{ABC-ini}) in the case of $A=1$, $B=0.7$ and either $C=0.42$ or $ C=0.43$ from the starting point ${\\bf r}_{0}=(0,0,0)+(0,0,1) \\times \\delta_i $, $1\\leq i\\leq 1000$, with either $\\sigma_{d}=\\sqrt{\\langle\\delta^{2}_{i}\\rangle}=10^{-5}$ (black) or $\\sigma_{d} = 10^{-10}$ (blue), respectively. (a) The PDFs of $z(t)$ of the normal-chaotic fluid-particle when $C=0.42$; (b) The PDFs of the normalized results $z'(t)$ of the ultra-chaotic fluid-particle when $C=0.43$.} \\label{PDF_ea}\n \\end{center}\\vspace{-0.6cm} \n\\end{figure}\n\n\nBesides the PDF, let us further investigate other statistics such as the ensemble average to demonstrate the higher-level of disorder given by the ultra-chaos than the normal-chaos mentioned above. Here we consider the ensemble average of the chaotic trajectories of a fluid-particle starting from the point ${\\bf r}_{0}' = {\\bf r}_{0}+ (0,0,1) \\times \\delta_{i}$ with 1000 different tiny initial disturbances $\\delta_{i}$ ($i=1,2,3, ...,1000$), which are given by the Gaussian random number generator in the case of the standard deviation $\\sigma_{d}=\\sqrt{\\langle\\delta^{2}_{i}\\rangle}$ as well as zero mean, i.e. $\\mu_d =\\langle\\delta_{i}\\rangle= 0$, where $\\langle\\;\\rangle$ denotes the average operator. It is found that, in the case of $A=1$, $B=0.7$, $C=0.42$ and ${\\bf r}_{0} = (0,0,0)$, corresponding to the normal-chaotic motion of the fluid particle, the ensemble averages of the phase plots $x-z$, which are given respectively either by $\\sigma_{d}=10^{-5}$ or $\\sigma_{d}=10^{-10}$, are almost the same, as shown in Fig.~\\ref{xz_ea}(a) and (b). \nOn the contrary, in the case of $A=1$, $B=0.7$ and $C=0.43$, the ensemble averages of the phase plots $x-z$ (of the ultra-chaotic motions of the fluid particle), which are given respectively either by $\\sigma_{d}=10^{-5}$ or $\\sigma_{d}=10^{-10}$, are totally different, as shown in Fig.~\\ref{xz_ea}(c). \nFurthermore, the PDFs of the ensemble-averaged trajectories of the ultra-chaotic fluid-particles starting from ${\\bf r}_{0} = (0,0,0)$ is also very sensitive to the starting position, which are completely different from those given by the normal-chaotic fluid-particles, as illustrated in Fig.~\\ref{PDF_ea}.\nAll of these results illustrate that, unlike the normal-chaos, even the ensemble-averaged quantities as well as their corresponding PDFs of the ultra-chaos in the ABC flow are rather sensitive to the tiny disturbances. It further illustrates that this kind of ultra-chaotic motion in the ABC flow is indeed at a higher-level of disorder than the normal-chaos.\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=2.5in]{Fig5.pdf}\n \\caption{ {\\bf Chaotic states of the fluid-particle in the ABC flow (\\ref{ABC}) starting from ${\\bf r}_{0} = (0,0,0)$ for different values of $B$ and $C$ when $A=1$.} Red domain: ultra-chaos; Blue domain: normal-chaos; Gray domain: non-chaos. } \\label{ultra}\n \\end{center}\\vspace{-0.6cm} \n\\end{figure}\n\n\nUsing $A=1$ but various values of $B$ and $C$, it is found that there exist the non-chaos, normal-chaos, and the ultra-chaos for the motion of the fluid particle starting from ${\\bf r}_{0} = (0,0,0)$, as shown in Fig.~\\ref{ultra}. For a non-chaotic motion, its trajectory is not sensitive to the tiny disturbances of starting position. For a normal-chaotic motion, although its trajectory is rather sensitive to the tiny disturbances of starting position, its attractor and especially its statistical properties are {\\em not} sensitive to the tiny disturbances. However, for an ultra-chaotic motion, even its statistical properties are sensitive to the tiny disturbances of starting position. Note that, for a normal-chaotic motion, the fluid particle starting from ${\\bf r}_{0}=(0,0,0)$ always moves in a {\\em restricted} spatial domain (i.e. its position is in a restricted domain of the phase plot $x-z$). However, for an ultra-chaotic motion, the fluid-particle starting from ${\\bf r}_{0}=(0,0,0)$ departs from its starting point far and far away. This further illustrates that an ultra-chaotic motion of the fluid-particle in the ABC flow has a higher disorder than a normal-chaotic motion, although the velocity field of the ABC flow itself as a whole is periodic and steady-state. \n\n\n\\begin{figure}\n \\begin{center}\n \\subfigure[]{\\includegraphics[width=2in]{Fig6a.pdf}}\n \\subfigure[]{\\includegraphics[width=2in]{Fig6b.pdf}}\n \\subfigure[]{\\includegraphics[width=2in]{Fig6c.pdf}}\\\\\n \\subfigure[]{\\includegraphics[width=2in]{Fig6d.pdf}}\n \\subfigure[]{\\includegraphics[width=2in]{Fig6e.pdf}}\n \\subfigure[]{\\includegraphics[width=2in]{Fig6f.pdf}}\n \\caption{ {\\bf States of chaos for the motions of the fluid-particles starting from different points ${\\bf r}_{0}=(x(0),y(0),z(0))$ in the ABC flow (\\ref{ABC}) in the case of $A=1$, $B=0.7$ and $C=0.43$.} (a) on $z(0)=0$; (b) on $z(0)=\\pi\/8$; (c) on $z(0)=\\pi\/4$; (d) on $z(0)=3\\,\\pi\/8$; (e) on $z(0)=7\\,\\pi\/16$; (f) on $z(0)=\\pi\/2$. Red points: ultra-chaos; Blue points: normal-chaos. } \\label{r0_1}\n \\end{center}\\vspace{-0.6cm} \n\\end{figure}\n\n\n\\begin{table}\n\\tabcolsep 0pt\n\\caption{ {\\bf Statistical values of the maximum Lyapunov exponents $\\lambda_{max}$ given by the normal-chaos and ultra-chaos for the motions of the fluid particles in the ABC flow.} These results are obtained via solving the ABC flow (\\ref{ABC}) in the case of $A=1$, $B=0.7$ and $C=0.43$, using various starting points ${\\bf r}_{0}=(x(0),y(0),z(0))$ of the fluid particles, where $-\\,\\pi \\leq x(0),y(0),z(0)\\leq +\\,\\pi$. }\n\\vspace*{-2pt}\\label{Lye}\n\\begin{center}\n\\def0.5\\textwidth{0.75\\textwidth}\n{\\rule{0.5\\textwidth}{1pt}}\n\\begin{tabular*}{0.5\\textwidth}{@{\\extracolsep{\\fill}}ccc}\n~ & \\hspace{1.0cm} Normal-chaos \\hspace{1.0cm} & \\hspace{1.0cm} Ultra-chaos \\hspace{1.0cm} \\\\\n\\hline\n~Maximum value of $\\lambda_{max}$ & $1.3\\times 10^{-2}$ & $8.7\\times 10^{-2}$~ \\\\\n~Minimum value of $\\lambda_{max}$ & $8.5\\times 10^{-5}$ & $4.3\\times 10^{-2}$~ \\\\\n~Mean & $9.7\\times 10^{-4}$ & $6.9\\times 10^{-2}$~ \\\\\n~Standard deviation & $7.5\\times 10^{-4}$ & $1.0\\times 10^{-2}$~ \\\\\n\\end{tabular*}\n{\\rule{0.5\\textwidth}{1pt}}\n\\end{center}\n\\end{table}\n\n\nOn the other hand, keeping $A=1$, $B=0.7$, $C=0.43$ and using various positions of the starting point ${\\bf r}_{0}=(x(0),y(0),z(0))$ in the ABC flow, where $-\\,\\pi \\leq x(0),y(0),z(0)\\leq +\\,\\pi$, it is found that both the normal-chaos and ultra-chaos (for the motions of the fluid particle starting from ${\\bf r}_{0}$) widely exist, and these two states of chaos co-exist at the same time in the ABC flow, as shown in Fig.~\\ref{r0_1}. \nThe statistical values of their maximum Lyapunov exponents $\\lambda_{max}$ are given in Table~\\ref{Lye}. Statistically speaking, the maximum Lyapunov exponents $\\lambda_{max}$ of an ultra-chaotic motions of the fluid particle in the ABC flow is about two-order magnitude larger than that of a normal-chaos. \n\nNote that, when $z(0)$ increases from $0$ to $\\pi\/2$, there exists a kind of structure constituted by the starting positions $(x(0),y(0))$ of the fluid particles with the normal-chaotic motions (corresponding to blue points) as well as the ultra-chaotic motions (corresponding to red points), and this kind of structure has continuous deformations, as shown in Fig.~\\ref{r0_1}. \nLet $\\alpha(x(0),y(0),z(0))=0$ or $1$ denote the normal-chaotic motion or the ultra-chaotic motion of fluid particle starting from the point ${\\bf r}_{0}=(x(0),y(0),z(0))$, respectively. It is found that, for $-\\,\\pi \\leq x(0),y(0) \\leq +\\,\\pi$, there exist the following relationships (symmetries): \n\\begin{equation}\n\\alpha(x(0),y(0),z(0))=\\alpha(-\\,x(0),y(0),\\pi-z(0)),\n\\end{equation}\nwhere $z(0)\\in [\\pi\/2, \\pi]$, \n\\begin{equation}\n\\alpha(x(0),y(0),z(0))=\\alpha(x(0),\\pi-y(0),-\\,z(0)),\n\\end{equation}\nwhere $y(0)\\in[0,\\pi]$, $z(0)\\in[-\\,\\pi,0]$, and\n\\begin{equation}\n\\alpha(x(0),y(0),z(0))=\\alpha(x(0),-\\,\\pi-y(0),-\\,z(0)),\n\\end{equation}\nwhere $y(0)\\in[-\\,\\pi,0]$, $z(0)\\in[-\\,\\pi,0]$.\n\n\n\\renewcommand\\arraystretch{1.2}\n\\begin{table}\n\\tabcolsep 0pt\n\\caption{Values of the parameter $C$ versus $N_{ultra}\/N_{all}$, where $N_{ultra}$ denotes the number of the starting points corresponding to the ultra-chaos (for the motion of a particle in the ABC flow) and $N_{all}=20^3$ denotes the total number of the equidistant starting points, respectively. \nThese results are obtained via solving the chaotic dynamical system (\\ref{ABC}) in $t\\in[0,10000]$ by means of the CNS, in the case of $A=1.0$, $B=0.7$, $0 \\leq C \\leq 0.43$, using various starting points ${\\bf r}_{0}=(x(0),y(0),z(0))$ of the fluid particles, where $-\\,\\pi \\leq x(0),y(0),z(0)\\leq +\\,\\pi$.\n}\n\\vspace*{-2pt}\\label{C_ultra-T}\n\\begin{center}\n\\def0.5\\textwidth{0.5\\textwidth}\n{\\rule{0.5\\textwidth}{1pt}}\n\\begin{tabular*}{0.5\\textwidth}{@{\\extracolsep{\\fill}}cc}\n \\hspace{1.0cm} $C$ \\hspace{1.0cm} & \\hspace{1.0cm} $N_{ultra}\/N_{all}$ \\hspace{1.0cm} \\\\\n\\hline\n~~$0.43$ & $49\\%$~~ \\\\\n~~$0.2$ & $47\\%$~~ \\\\\n~~$0.1$ & $43\\%$~~ \\\\\n~~$0.01$ & $20\\%$~~ \\\\\n~~$0.001$ & $6\\%$~~ \\\\\n~~$0.0001$ & $2\\%$~~ \\\\\n~~$0.0$ & $0\\%$~~ \\\\\n\\end{tabular*}\n{\\rule{0.5\\textwidth}{1pt}}\n\\end{center}\n\\end{table}\n\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=2.5in]{Fig7.pdf}\n \\caption{$N_{ultra}\/N_{all}$ versus $C$ in the case of $A=1$, $B=0.7$ and $C\\leq0.1$.} \\label{C_ultra-F}\n \\end{center}\\vspace{-0.6cm}\n\\end{figure}\n\nLet $\\beta$ denote the ratio of the numbers of the starting fluid-particles with ultra-chaotic trajectories to the whole particle numbers in $-\\pi\\leq x,y,z \\leq +\\,\\pi$. In theory it holds that\n\\begin{equation}\n\\beta = \\frac{1}{(2\\pi)^{3}}\\int_{-\\pi}^{+\\pi} \\int_{-\\pi}^{+\\pi}\\int_{-\\pi}^{+\\pi} \\alpha(x,y,z)\\; dx dy dz,\n\\end{equation}\nsince $\\alpha=1$ and $\\alpha=0$ correspond to an ultra-chaos and a normal-chaos, respectively. In practice, we use the Monte Carlo method to approximately calculate the ratio\n\\begin{equation}\n\\beta \\approx N_{ultra}\/N_{all},\n\\end{equation}\nwhere $N_{all}$ denotes the number of whole randomly chosen starting fluid-particles ${\\bf r}_{0}\\in\\Omega=\\left\\{(x,y,z): -\\pi \\leq x,y,z \\leq +\\,\\pi\\right\\}$, and $N_{ultra}$ is the number of the starting fluid-particles with an ultra-chaotic trajectory. Obviously, the larger $N_{all}$, the more accurate the result $\\beta$ given by the Monte Carlo method. \n In the case of $A=1$, $B=0.7$ and $0 \\leq C \\leq 0.43$, it is found using $N_{all}=8000$ that the ratio $\\beta \\approx N_{ultra}\/N_{all}$ is dependent upon the value of $C$, as shown in Table~\\ref{C_ultra-T}. Especially, when $C\\leq0.1$, there is a power-law relationship between $\\beta \\approx N_{ultra}\/N_{all}$ and $C$, i.e. \n\\begin{equation}\n\\beta\\approx N_{ultra}\/N_{all}\\approx C^{\\hspace{0.2mm}0.4}, \\label{relation_C_ultra}\n\\end{equation}\nas illustrated in Fig.~\\ref{C_ultra-F}. Thus, when the parameter $C$ decreases, the value of $N_{ultra}$, i.e. the number of the starting fluid-particles with an ultra-chaotic trajectory, decreases until $N_{ultra} = 0$ when $C = 0$. This is reasonable since it is well-known that the ABC flow in the case of $C=0$ is stable and thus chaos does not exist in $C=0$. \n\n\\section{Relationship between ultra-chaos and turbulence}\n\n\nThe velocity $\\mathbf{u}_{ABC}$ of the famous Arnold-Beltrami-Childress (ABC) flow (\\ref{ABC-0})\n was first discovered by Arnold \\cite{arnold1965} as a steady-state solution of the dimensionless Navier-Stokes equations\n\\begin{equation}\n\\frac{\\partial\\mathbf{u}}{\\partial t}+(\\mathbf{u}\\cdot\\nabla)\\mathbf{u}=-\\nabla p +\\frac{1}{Re}\\Delta \\mathbf{u}+\\mathbf{f}, \\label{NS-1}\n\\end{equation}\n\\begin{equation}\n\\nabla\\cdot \\mathbf{u}=0, \\label{NS-2}\n\\end{equation}\nwhere $t\\geq0$ denotes the time, $\\nabla$ is the Hamilton operator, $\\Delta$ is the Laplace operator, $Re$ is the Reynolds number, $p$ is the pressure and $\\mathbf{f}=\\mathbf{u}_{ABC}(x,y,z)\/Re$ is the given external force per unit mass, respectively, with the periodic boundary conditions at $x =\\pm \\pi, y =\\pm \\pi, z =\\pm \\pi$.\n\n\n\\begin{figure}\n \\begin{center}\n \\subfigure[]{\\includegraphics[width=2in]{Fig8a.pdf}}\n \\subfigure[]{\\includegraphics[width=2in]{Fig8b.pdf}}\n \\subfigure[]{\\includegraphics[width=2in]{Fig8c.pdf}} \\\\\n \\subfigure[]{\\includegraphics[width=2in]{Fig8d.pdf}}\n \\subfigure[]{\\includegraphics[width=2in]{Fig8e.pdf}}\n \\subfigure[]{\\includegraphics[width=2in]{Fig8f.pdf}}\n \\caption{(a) Total kinetic energy; (b)-(f) Module values $|\\bm{\\omega}|$ of instantaneous vorticity fields at $t=30$, $t=50$, $t=60$, $t=100$, and $t=300$, respectively, in the case of $A=1.0$, $B=0.7$ and $C=0.43$.} \\label{C043-results}\n \\end{center}\\vspace{-0.6cm}\n\\end{figure}\n\n\nWithout loss of generality, let us consider the case of $A=1.0$, $B=0.7$ and $0 \\leq C \\leq 0.43$. As reported by \\citet{podvigina1994non}, the Reynolds number $Re=50$ corresponds to a turbulent flow if the initial velocity field $\\mathbf{u}_{ABC}$ is under a small disturbance at order of magnitude $10^{-3}$. \nSuch kind of turbulent flow is solved numerically in $t\\in[0,500]$: the spatial domain $[-\\,\\pi,+\\pi]^3$ is discretized by a uniform mesh with $128^3$ points for the spatial Fourier expansion, where the maximum grid spacing is less than the minimum Kolmogorov scale \\cite{pope2001turbulent}, and the $3\/2$ rule for dealiasing \\cite{pope2001turbulent} is used, with the time-step $\\triangle t=10^{-3}$. \n\n\n\\renewcommand\\arraystretch{1.2}\n\\begin{table}\n\\tabcolsep 0pt\n\\caption{$T_{tran}$ versus $C$ by means of using the ABC flow in the case of $A=1$, $B=0.7$ and $C\\leq0.1$ as the initial condition under a small disturbance at order of magnitude $10^{-3}$ for the NS equations (\\ref{NS-1}) and (\\ref{NS-2}), where $T_{tran}$ denotes the time of the transition occurrence. }\n\\vspace*{-2pt}\\label{C_Tran-T}\n\\begin{center}\n\\def0.5\\textwidth{0.5\\textwidth}\n{\\rule{0.5\\textwidth}{1pt}}\n\\begin{tabular*}{0.5\\textwidth}{@{\\extracolsep{\\fill}}cc}\n~~$C$ & $T_{tran}$~~ \\\\\n\\hline\n~~$0.1$ & $50.5$~~ \\\\\n~~$0.01$ & $59.5$~~ \\\\\n~~$0.001$ & $70.0$~~ \\\\\n~~$0.0001$ & $80.0$~~ \\\\\n~~$0.0$ & $-$~~ \\\\\n\\end{tabular*}\n{\\rule{0.5\\textwidth}{1pt}}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[t]\n \\begin{center}\n \\subfigure[]{\\includegraphics[width=2.5in]{Fig9.pdf}}\n \\caption{Relationship between $C$ and $T_{tran}$ by means of using the ABC flow in the case of $A=1$, $B=0.7$ and $C\\leq0.1$ as the initial condition under a small disturbance at order of magnitude $10^{-3}$ for the NS equations (\\ref{NS-1}) and (\\ref{NS-2}), where $T_{trans}$ denotes the time of transition occurrence.} \\label{C_Tran}\n \\end{center}\\vspace{-0.6cm}\n\\end{figure}\n\nLet us first use the {\\em unstable} ABC flow in the case of $A=1.0$, $B=0.7$ and $C=0.43$ as the initial condition of the NS equations (\\ref{NS-1}) and (\\ref{NS-2}). It is found that, at the beginning, since the time is not long enough for the tiny velocity disturbances to transfer into the macro-level, it is very close to the ABC flow, i.e. about 49\\% starting fluid-particles are ultra-chaotic, according to Table~\\ref{C_ultra-T}. The transition from laminar flow to turbulence occurs approximately at $t\\approx 50.0 = T_{trans}$, as shown in Fig.~\\ref{C043-results}, where $T_{trans}$ denotes the time of the transition occurrence. Knowing the velocity field $\\mathbf{u}$ of the NS equations (\\ref{NS-1}) and (\\ref{NS-2}), we can investigate the chaotic property of trajectory of the fluid-particle starting from $\\mathbf{r}_{0}$ in a similar way as mentioned in \\S~2. \nWhen $t=50$, we use the Monte Carlo method to randomly choose 10000 starting fluid-particles in $-\\pi\\leq x,y,z < +\\pi$ and found that {\\em all} trajectories of fluid-particles starting from them are {\\em ultra-chaotic}. This strongly suggest that ultra-chaotic trajectories of {\\em all} fluid-particles should be a necessary condition of turbulence. \n\n\nSimilarly, let us consider the {\\em stable} ABC flow in the case of $A=1$, $B=0.7$ and $C=0$. Using the ABC flow $\\mathbf{u}_{ABC}$ under a small perturbation at order of magnitude $10^{-3}$ as the initial condition, we numerically solve the NS equations(\\ref{NS-1}) and (\\ref{NS-2}) in $t\\in[0,2000]$ and investigate the chaotic property of trajectory of the fluid-particles. It is found that in the case of $C=0$ the transition from laminar to turbulence {\\em never} occurs and besides there is {\\em no} ultra-chaotic motion of fluid-particles in the whole $t\\in[0,2000]$. This suggests from another side that ultra-chaotic trajectories of fluid-particle should have a close relationship with turbulence. \n\n Let us further consider the ABC flows in the case of $A=1$, $B=0.7$ and $0\\leq C < 0.43$. It is found that the time $T_{trans}$ of transition from laminar flow to turbulence increases as $C$ decreases from $0.1$ to $0.0001$, as shown in Table~\\ref{C_Tran-T}. When $0\\leq C\\leq0.1$, there is a linear relationship:\n\\begin{equation}\nT_{tran}\\approx-10\\log_{10}(C)+40, \\label{relation_C_Tran}\n\\end{equation}\nas illustrated in Fig.~\\ref{C_Tran}, indicating that $T_{tran} \\to +\\infty$ as $C\\to 0$, say, the transition from laminar flow to turbulence should never occur when $C=0$, which agrees with our numerical simulation in the case of $C=0$ mentioned above. In {\\em all} cases of $A=1$, $B=0.7$ and $0 \\leq C \\leq 0.43$ under consideration, we use the Monte Carlo method to randomly choose 10000 starting points in $-\\pi\\leq x,y,z < +\\pi$ and found that {\\em all} trajectories of fluid-particles starting from these randomly chosen 10000 fluid-particles are {\\em ultra-chaotic}. Thus, when $A=1$, $B=0.7$ and $0 < C \\leq 0.43$, the less the number of ultra-chaotic fluid-particles at beginning, corresponding to an unstable ABC flow with smaller $C$, the larger the transition time $T_{trans}$, say, more time is needed for {\\em all} fluid-particles to become ultra-chaotic at $t =T_{trans}$. All of these highly suggest that ultra-chaotic trajectories of nearly all fluid-particles should be a necessary condition of transition occurrence from laminar to turbulence for the viscous flow governed by the NS equations (\\ref{NS-1}) and (\\ref{NS-2}) considered here. \n \n\n\\section{Concluding remarks}\n\nIn this paper, we illustrate that trajectories of many fluid-particles in the unstable steady-state ABC flow (\\ref{ABC-0}) are ultra-chaotic, say, their statistical properties are sensitive to tiny disturbances of their starting position. Obviously, such kind of ultra-chaotic motions of fluid-particles represent a higher disorder than the normal-chaotic ones. Besides, using the ABC flow as an initial condition of the NS equations (\\ref{NS-1}) and (\\ref{NS-2}) with a small disturbance at the order $10^{-3}$ of magnitude, it is found that trajectories of {\\em all} fluid-particles become {\\em ultra-chaotic} after the transition from laminar to turbulent flow occurs. Thus, ultra-chaos should have a close relationship with turbulence. Our results highly suggest that trajectories of nearly all fluid-particles become ultra-chaotic should be a necessary condition of transition occurrence from laminar to turbulence, at least for the viscous flows considered in this paper. \n\n\nNote that the chaotic property of the ABC flow is essential for the development of turbulence \\cite{dombre1986chaotic, galloway1987note, podvigina1994non}. Hopefully, the ultra-chaos as a new concept \\cite{Liao2022AAMM} might open a brand-new door to study the chaos theory, \n turbulence and especially their relationships. \n\n\t\t\nThis work is partly supported by the National Natural Science Foundation of China (No. 91752104).\n\t\n\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLocal DNA flexibility is crucial for its biological function.\nThe deformation of the double helix is achieved through three types of mobility in the DNA backbone:\nconcerted rotations around $\\alpha$ and $\\gamma$ \\cite{2002-alpha-gamma-Lavery}, $\\zeta$ and $\\varepsilon$ \\cite{1993-BI-BII-Lavery} torsions,\nand ribose flexibility. The last mentioned type makes the main contribution into providing the observed DNA flexibility.\n\nThe local sugar repuckerings are very common in physiological saline. In vivo, they take place in binding of proteins (such as TBP, SRY, LEF-1, PurR) to the minor groove of B-DNA.\nDuring this process the minor groove widens through several sugar repuckerings into A-like conformations \\cite{1999-MinorGrooveBindingProteins}. Many local B to A conversions have been observed in protein and drug-bound DNA crystal complexes \\cite{2000-Olson-A-DNA-with-proteins}. Particularly, such conversion takes place when an enzyme interacts with the atoms ordinarily buried within the backbone (O3', for example). A-DNA is more resistant to ultraviolet radiation, therefore DNA undergoes a transition from B to A in Gram-positive bacteria during sporulation \\cite{1991-A-DNAInSpores}.\n\nOne can obtain DNA crystals from a solution in both the A- and B-forms depending on salt concentration, relative humidity and base pairs sequence (for a review, see, for example, \\cite{2006-Historical-DNA-polymorphism} and references therein). In a solution, one can induce a B to A transition by increasing salt concentration and\/or adding ethanol to the solvent \\cite{Ivanov1973,Nishimura1986}. The ethanol concentration, at which the B to A transition occurs, depends on the nucleotide sequence \\cite{1992-Ivanov-A-in-ethanol}: more C:G pairs shift the transition to smaller ethanol concentrations. It should be noted that other characteristics of a DNA nucleotide (both structural and dynamical ones) also depend on its base and a few neighboring bases.\n\nTherefore, the geometry (and, as a consequence, mechanical and dynamical properties) of both the A-DNA and B-DNA forms is a result of a complex balance of interdependent factors: torsion and valence angles in the backbone; sugar puckers; sequence dependent base pair stacking; electrostatic interactions of DNA with solvent molecules and with salt ions.\nFrom the physical point of view, the understanding of this balance is equivalent to the construction of a coarse-grained (CG) DNA model able to reproduce key features of the DNA behavior.\n\nThe last ten years witnessed the extensive development of CG models for different substances \\cite{2012-Coarse-grained-for-macrochemistry}, particularly for large biomolecules \\cite{2012-Coarse-grained-biomolecules}. Regular methods for obtaining CG force fields from the all-atom ones have been developed. However, all the regular methods imply that all potentials are pairwise and have a certain simple form. As we will show later, if the objective is to model the effect of ribose flexibility on DNA geometry, one has to use at least one three-particle potential.\n\nIn many works, DNA behaviour is simulated within the framework of CG models with phenomenologically chosen parameters (for a review, see, for example, \\cite{2013-Papoian-review-CG-DNA}). The model with the coarsest grain (consisting of two complementary nucleotides), the worm-like chain model, can reproduce the simplest macro-mechanical properties which are\ncommon for DNA and the other semiflexible polymers. A modification of this model \\cite{2009-Mazur-CG-bending-torsion} allows one to obtain both bending and torsional persistent lengths (consistent with experiments). However, the modified model does not provide the correct value for the relaxation rate of bending fluctuations, because the DNA bending proves to be highly anharmonic. Another approach with similar \"coarseness\" is the representation of DNA as a stack of interacting plane bases \\cite{2007-CG-DNA-rigid-bases}. To simulate DNA melting, one needs a model with finer grain, at least one grain for every nucleotide \\cite{2008-DNA-CG-melting-2grains}. However, this scale still does not allow to take into account even the simplest geometrical nonlinearity of DNA strands. The next step is to divide a nucleotide into 3 grains: phosphate, ribose and nucleic base \\cite{2007-CG-DNA-3-grains}. On this scale, it is already possible to fit helical parameters either for B-form or for A-form (not for both) by tuning the lengths, angles and rigidities \\cite{2009-Kovaleva-DNA-3-grains}. The next approximation to the double helix structure is a pair of zigzag chains with interacting plane nucleobases (that is two grains and one solid body or, equivalently, five grains per nucleotide) \\cite{2010-DNA-CG-2grains-plus-ellipsoid}. However, this model still can not reproduce the ribose flexibility. We will show that for this purpose one needs at least 3 grains and one solid body (or, equivalently, six grains) per nucleotide, one nonlinear potential, and, in addition, explicit modeling of ions.\n\nHere, we present a minimal CG model of the DNA molecule which can reproduce ribose flexibility, and, correspondingly, both A-DNA and B-DNA forms. On the basis of experimental and theoretical (MD simulations within the framework of an all atom model) data, we analyze the mobility of the DNA double helix, unite atomic groups into grains and choose the potentials of interaction. Besides backbone interactions in strands, we define the minimal adequate solvent model which enables to simulate A-DNA.\n\nThe approach we use in this work has been originally formulated in \\cite{2010-first-full-publication}. There, the choice of grains and potentials for simulating ribose flexibility has been described. In work \\cite{2011-Biofizika} the B-DNA form has been modeled in the framework of this approach, but without implementation of ribose flexibility and with implicit solvent (generalized Born approximation with the Debye-Huckel correction for salt effects \\cite{Case1999}). In the framework of this version, DNA heat conductivity was estimated \\cite{Savin2011} and DNA stretching was investigated \\cite{Savin2013}. However, this version is admissible only when one may neglect the ribose flexibility (under low temperatures or when the molecule can be extended but not bent and not compressed). At temperature 300~K, the behaviour of this CG B-DNA and the all atom AMBER model significantly differ \\cite{2011-Biofizika}. In the present work we suggest a realization of the model (hereinafter referred to as the \"sugar\" CG DNA model) which adequately describes ribose flexibility and interactions between negative charges on DNA and explicit ions in implicitly treated solvent. This realization of the model is capable of simulating both A- and B-DNA forms under corresponding conditions.\n\\section{Construction of the sugar CG DNA model}\nAs a basis for the CG model construction, we accept the AMBER force field \\cite{2000-Parm99} for base pairs interactions.\nThis choice is justifiable, considering that the empirical force fields reproduce the interactions between bases best \\cite{2006-QM-and-AMBER-base-stacking}.\n We analyze the DNA dynamics in solvents at 300~K in the framework of the all-atom AMBER model, as well as the geometric changes of the DNA helix\nin A-B transition and base-pair opening. On the basis of these data, we divide the DNA chains into grains\nand find the potentials for interactions between the grains. Finally, we choose the description of the medium:\nthe models for the ions and the solvent.\n\\subsection{Base stacking and base pairing}\nWe model a base as three rigidly bound grains which can rotate\naround an axis coinciding with the real rotation axis of the base: the glycosidic bond $\\chi$.\nWe placed the grains on some (heavy) atoms of the bases (see fig.~\\ref{DNA-bases}).\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth] {fig01.eps}\n\\caption{Coarse-graining of the natural base pairs (A:T and G:C). We show the locations and and the charges of three rigidly bound grains modeling every base.\nThe rotation axes (glycosidic bonds $\\chi$ with the sugar rings) are marked.\nDNA atoms are designated according to \\cite{2008-Neidle-Principles-of-DNA-structure}, and we depict their van-der-Waals radii.\nThe black arrows present electric dipole moments of the bases (according to the charge distribution in AMBER),\nand the circles near atoms of the bases - zones around A-DNA where\none can find ions Na$^+$ or Cl$^-$ with maximal probability \\cite{1999-ions-around-A-DNA-AMBER-CHARMM}.}\n\\label{DNA-bases}\n\\end{center}\n\\end{figure}\nOne of the atoms was chosen on one side of the rotation axis, the two others -\non the other side, at maximum distance from the center of the base. The objective of this choice was to best approximate the real moments of inertia of the base.\n\nFor base A (Adenine) we placed the grains on atoms C8, N6, C2; for base T (Thymine) - on atoms\nC7, O4, O2; for base G (Guanine) - on atoms C8, O6, N2; and for base C (Cytosine) - on atoms C6, N4, O2 (see fig.~\\ref{DNA-bases}).\n\nMasses of grains $m_1$, $m_2$, $m_3$ on a base X (X = A, T, G, C) were found from the two conditions: (1) equality of the\ntotal mass of the grains to the mass of the base ($m_1+m_2+m_3=m_X$) and (2) coincidence of the mass centers of the\nthree grains and the base. Numerical values for masses of grains and moments of inertia of the CG and real bases are\ngiven in table~\\ref{tab1}.\n\n\\begin{table}\n\\caption{Masses of grains $m_1$, $m_2$, $m_3$ and moments of inertia of the real i$_{xx}$ and i$_{yy}$ and the CG I$_{xx}$ and\nI$_{yy}$ bases A, T, G, C. Masses of the grains are given in a.e.m., the moments of inertia - in a.e.m.$\\cdot \\mbox{\\AA}^2$.\n}\n\\label{tab1}\n\\begin{tabular}{cccccccc}\n\\hline\nX & $m_1$ & $m_2$ & $m_3$ & i$_{xx}$ & I$_{xx}$ & i$_{yy}$ & I$_{yy}$\\\\\n\\hline\nA & $52.230$ & $28.139$ & $53.632$ & $690$ & $475$ & $1704$ & $1712$\\\\\nT & $51.822$ & $16.204$ & $56.974$ & $584$ & $256$ & $1636$ & $1543$\\\\\nG & $61.731$ & $34.357$ & $53.912$ & $1302$ & $800$ & $1885$ & $1858$\\\\\nC & $39.254$ & $35.492$ & $35.254$ & $233$ & $164$ & $1344$ & $1231$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nKnowing the coordinates of the three grains, one can find the coordinates of all the base atoms and compute the energy of base pairing and base stacking as the sum of pairwise\natomic electrostatic and van-der-Waals interactions. For them, we used the AMBER force field \\cite{2000-Parm99}.\nIn articles \\cite{Savin2011} and \\cite{Savin2013}, where only the B-DNA was simulated, the partial interaction between bases was used to decrease the computation time. In the present realization of the model every atom of a base interacts with\nevery atom of the complementary base and with all the atoms of two neighboring base pairs. It allows to simulate the A-DNA form and other deformed structures (for example, base pair opening).\n\nThe accepted interactions between bases are computationally expensive (as compared to the other interactions in the CG model). We used them only as a foundation to construct the adequate CG backbone and test the obtained structure. If one uses the sugar CG model in long simulations of biological processes in the future, the described scheme\nis the first thing one would want to change.\n\\subsection{Choice of grains for backbone: C1', P, C3'}\n\\label{choice-of-grains}\nWe choose the locations of all the grains on (heavy) atoms. The main principle was:\none may distort the mass distribution of the molecule, but one preserves its key geometric nodes.\nLet us show that we will need three grains per nucleotide for the backbone\n(see fig.~\\ref{DNA-strand}).\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth] {fig02.eps}\n\\caption{Coarse-graining of the sugar-phosphate backbone. We show locations of the grains and (for the backbone)\ngroups of atoms united into the grains. Notations for atoms and torsion angles are from \\cite{2008-Neidle-Principles-of-DNA-structure}.\nA nucleobase rotates about the glycosidic bond $\\chi$.}\n\\label{DNA-strand}\n\\end{center}\n\\end{figure}\n\nThe first grain was chosen on atom C1' because it is the point of suspension of the base, and the rotation axis of the base (glycosidic bond)\npasses through this atom. Crystallographic data \\cite{1996-rotation-around-glicosidic-bond} show that bases rotate around this axis considerably,\nwhile valence angles O4'C1'N1(N9) and C2'C1'N1(N9) practically do not change and fix the direction of the glycosidic bond relative to the ribose ring.\nThis is the key geometric peculiarity of the connection between a base and the backbone, and we keep this peculiarity in our model.\n\nThe second grain was chosen on phosphorus atom, and united phosphate group and three atoms (C5', H5'1, H5'2) which normally move\ntogether with the phosphate \\cite{1999-PO4CH2-together}. As we want to minimize the number of grains per nucleotide, we would tend\nto restrict the backbone to the chain (...-P-C1'-P-C1'-...). However, all atom MD simulations show that the presence\nof the very flexible sugar rings between the atoms P and C1' allows the \"torsion\" angles in this chain to vary over a very wide range,\nwhile the \"torsion\" angles in the chain (...-P-P-P-...) keep their mean values quite satisfactorily. To preserve this feature,\nwe need at least one more grain per nucleotide.\n\nBesides, we would like to build a backbone which can serve as a support for the moving bases at their opening and in A-B transition\n(which is the case for the real sugar-phosphate backbone of DNA). This purpose is also not achieved for the chain (...-P-C1'-P-C1'-...)\nas the atoms C1' always move together with the bases.\n\nWhen a sugar repuckers, the torsion angles around the bonds adjoining the sugar ring also change, and so does the position of the ring\nrelative to the two neighboring phosphate groups as well as the distance between the groups. The adjacent base pairs move relative\nto each other, and the local geometry changes from B-like to A-like (see fig.\\ref{B-A-rings}).\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.98\\linewidth] {fig03.eps}\n\\caption{Position of a sugar ring relative to the chain of phosphorus atoms in B- and A-DNA forms (GLACTONE \\cite{1998-GlactoneGeorgia}).\nThe valence bonds and angles, on the one hand, and the van-der-Waals interactions, on the other hand, provide two stable positions\nof the ring relative to the phosphates exactly\nwhen the ring has C2'-{\\it endo} (B-DNA) and C3'-{\\it endo} (A-DNA) puckers: between two phosphates and under the chain of phosphates.\nIntermediate positions suffer strong steric hindrance, which results in a barrier between these two energy minima. When the ring\nchanges its conformation, it has to change its position relative to the chain of phosphates (distance C1'P2). This, in turn, has to change\nthe distance between the phosphates. So we have correlations: (C2'-{\\it endo} $\\rightleftharpoons$ small $\\mid$C1'P2$\\mid$ $\\rightleftharpoons$\nlarge $\\mid$P1 P2$\\mid$ $\\rightleftharpoons$ B-DNA) and\n(C3'-{\\it endo} $\\rightleftharpoons$ large $\\mid$C1'P2$\\mid$ $\\rightleftharpoons$ small $\\mid$P1 P2$\\mid$ $\\rightleftharpoons$ A-DNA).\n}\n\\label{B-A-rings}\n\\end{center}\n\\end{figure}\nOne can see that the atom C4' moves with the sugar ring in sugar repuckering. On the contrary, the displacement\nof the atom C3' is mostly caused by the necessity to change the geometric form of the chain of phosphates.\nTherefore, we choose the third grain on the atom C3'.\n\nAs a result, to hold the needed form of the DNA strand, we use \"valence\" bonds, and \"valence\" and \"torsion\" angles in the chain of grains\n(...-P-C3'-P-C3'-...). To model the ribose flexibility, we add grains C1' (connected to bases)\nattached to the C3' grains of this chain. The position of the bond C3'-C1' relative to the chain (...-P-C3'-P-C3'-...)\nshould have two locations divided by a barrier - which reflects two main states of the ribose.\n\\subsection{CG simulation of ribose flexibility: choice of potentials for pyramid \\{P1 P2 C3' C1'\\}}\n\\label{choice-of-pyramid}\nRibose flexibility causes deformations of the CG pyramid \\{P1 P2 C3' C1'\\} (see fig.~\\ref{pyramid})\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.98\\linewidth] {fig04.eps}\n\\caption{CG model of sugar repuckering from C2'-{\\it endo} (B-DNA) to C3'-{\\it endo} (A-DNA).\nWe show the pyramid \\{P1 P2 C3' C1'\\} composed of grains of the sugar CG model in B-DNA and A-DNA forms (GLACTONE \\cite{1998-GlactoneGeorgia}).\nFor the bonds C3'-P2, P1-C3' and C3'-C1' we choose sufficiently rigid harmonic potentials. For the bond C1'-P2, we introduce a\ndouble-well potential. The two wells correspond to two main sugar puckers (see fig.~\\ref{B-A-rings}). The correlation of the distance\n$\\mid$P1 P2$\\mid$ with $\\mid$C1' P2$\\mid$ is described by potential (\\ref{equation-3-point-potential}). This potential can be symbolically depicted as\none spring thrown from grain P1 to grain C1' over grain P2. For the long bond P1-C1', we use a soft spring.}\n\\label{pyramid}\n\\end{center}\n\\end{figure}\nand is modeled through double-well potential for bond C1'-P2. As we saw (fig.~\\ref{B-A-rings}),\none also needs correlation between the distances $\\mid$P1 P2$\\mid$ and $\\mid$C1' P2$\\mid$, which we introduce through three-particle potential\n\\begin{equation}\n\\label{equation-3-point-potential}\nU=\\frac{1}{2} k_P (\\left|P(1)P(2)\\right|+t_P\\left|C1'P(2)\\right|-l_{P0})^2,\n\\end{equation}\nwhere $t_P>0.$\n\nFrom direct geometric considerations, one could also expect a (positive) correlation between distances $\\mid$P1C1'$\\mid$ and $\\mid$C1'P2$\\mid$.\nHowever, as we will see from the all-atom simulations (chapter \\ref{MD-pyramid}), one can use a soft harmonic potential for this bond:\nthe distance $\\mid$P1 C1'$\\mid$ does not correlate with the sugar pucker. Indeed, there are five valence bonds and three torsion angles between\natoms P1 and C1', and these degrees of freedom are only weakly connected with sugar repuckering.\n\\subsection{Obtaining parameters of CG potentials from the all-atom AMBER model}\n\\label{section-AMBER}\nIn spite of the problems in simulation with large ethanol or salt concentrations, one can consider empirical force fields AMBER and CHARMM\nas providing sufficiently reliable balance of interactions in the DNA molecule in water with both small (\\cite{1998-A-to-B-Beveridge}, B-DNA)\nand high (\\cite{2003-Mazur-A-B-in-drop} and \\cite{1996-B-to-A-Pettitt-Sharmm}, B-A transition) salt concentrations, which allows to lean\nupon these all-atom models in construction of CG models.\n\nFor coarse-graining, we used two methods. First, to obtain rigidities of some \"valence\" bonds, \"valence\" and \"torsion\" angles in our CG model,\nwe used the simplest Boltzmann inversion method \\cite{1998-Boltzmann-inversion} for all-atom MD trajectories of Dickerson-Drew dodecamer\n(the sequence 5'-CGCGAATTCGCG-3' on the first chain and, correspondingly, 3'-GCGCTTAAGCGC-5' on the other; in the following, the number\nof a nucleotide is its serial number in counting first along the first chain from 5' to 3' and then along the second chain from 5' to 3').\nMore exactly,\nwe analyzed the behavior of a B-DNA in water (AMBER, parmbsc0 \\cite{2007-parmbsc0}) and an\nA-DNA in the mixture of ethanol and water (85:15) (AMBER, parm99 \\cite{2000-Parm99}). We supposed that\nthe bonds and angles under consideration have Boltzmann distribution $p(l) \\sim exp (-U(l)\/kT)$, where $U$ is a harmonic function:\n$U(l)=\\frac{1}{2}K_l(l-l_0)^2$ ($U(\\theta)=\\frac{1}{2}K_{\\theta}(\\theta-\\theta_0)^2$). For every nucleotide, we made a frequency histogram\nfor every relevant distance and angle. Then we approximated the histogram by Gaussian distribution or by the sum of two Gaussian distributions,\nand determined their means (centers) and standard deviations. Finally, we took the averages of the centers and of the deviations over the nucleotides.\nWe believed that so obtained distances $l_0$ (angles $\\theta_0$) and rigidities $K_l$ ($K_{\\theta}$) can be regarded as estimates of mean values\nand effective rigidities for bonds and angles in our CG model.\n\nLet us notice that, in this approach, the obtained rigidities take into account not only the (valence, torsion and van-der-Waals) interactions\nbetween DNA atoms, together with DNA solvation. In addition, these rigidities partly \"include\" several interactions which we plan\nto introduce separately: base stacking; electrostatic interactions between charges on DNA, and between charges on DNA and ions.\nTherefore, to verify the obtained rigidities and, in some cases, to evaluate the potentials which can not be derived in such a way\n (for example, in the case of the double-well potential for the bond C1'-P2), we used an another method (method of \"relaxation\").\n\nNamely, to obtain the energy of interaction between two grains at a given distance, we took an all-atom fragment of one DNA strand\n(without charges, in vacuum) between the atoms corresponding to these grains, and located these atoms at the needed distance\none from another. Then we minimized the energy of the system (in the framework of the AMBER force field) as a function of coordinates\nof all the rest atoms (and so we carried out the \"relaxation\" of the system). The obtained value of energy was regarded as the energy\nof interaction between the grains at this distance. Changing the distance between the grains, we got the dependence of the energy\non the distance. In this way one can evaluate potentials of \"valence\" bonds in a CG model.\n\nThe details of the estimations of the mean values and rigidities for the bonds and angles of the CG model are collected in Appendix A (section\n\\ref{AppendixA}). In the rest of this section, we shortly list the most important results.\n\\subsubsection{\"Valence\" bonds C3'-C1', C3'-P(2) and P(1)-C3'}\n\\label{subsection-skeleton}\nThe rigidities of the bonds C3'-C1' and C3'-P(2) (see fig.~\\ref{valence-bonds-angles}) are provided by valence bonds and angles (under tension)\nand by van-der-Waals forces (under compression), therefore the bonds C3'-C1' and C3'-P(2) are rigid. The bond P(1)-C3', consisting\nof four valence bonds, is softer.\n\\begin{figure}\n\\includegraphics[width=0.75\\linewidth] {fig05.eps}\n\\caption {\"Valence\" bonds and angles of the sugar CG model. The double-well bond C1'-P(2) models ribose flexibility, the length $\\mid$P(1)P(2)$\\mid$\ncorrelates with the length $\\mid$C1'P(2)$\\mid$ (potential (\\ref{equation-3-point-potential}) is symbolically depicted as the polyline P(1)-P(2)-C1').\nAtom N (N1 or N9) does not belong to the grains of the CG model, it is shown merely to determine the direction of the glycosidic bond,\nwhich the base rotates around.}\n\\label{valence-bonds-angles}\n\\end{figure}\nThe frequency histograms obtained from MD trajectories for these bonds have one narrow peak for all the nucleotides.\nThe estimates of the lengths and the rigidities made by different methods are listed in table~\\ref{table-valence-bonds} of Appendix A (section~\\ref{AppendixA}).\n\\subsubsection{Bonds P(1)-C1', C1'-P(2) and P(1)-P(2) in pyramid \\{P(1)P(2)C3'C1'\\} modeling sugar ring}\n\\label{MD-pyramid}\nIn pyramid \\{P(1)P(2)C3'C1'\\} three bonds C3'-C1', C3'-P(2) and P(1)-C3' are rigid and we use for them harmonic potentials\n(see \\ref{subsection-skeleton}), while the rest three bonds P(1)-C1', C1'-P(2), P(1)-P(2) are connected with the flexible sugar\nring. As we expected from geometric considerations (see \\ref{choice-of-pyramid}), only the length\nC1'-P(2) unambiguously correlates with pseudorotation angle $\\tau$ (see fig.~\\ref{A-pyramid-correlations} in Appendix A (section~\\ref{AppendixA})).\nThe length P(1)-P(2) demonstrates broad double-humped distribution not always connected\nwith the sugar repuckering, and the length P(1)-C1' has large additional noise.\n\nTo estimate the double-well potential of the bond C1'-P(2), we used the method of \"relaxation\" (see the beginning of section~\\ref{section-AMBER}),\nbecause the depths of the wells and the height of the barrier can not be obtained from MD simulations.\nThe derived potential proved to be double-well as we had expected, only the depths of the wells are different: B-minimum is lower\n(see fig.~\\ref{fig-2wells-relax} in Appendix A (section~\\ref{AppendixA})). The lengths and rigidities in both wells well agree\nwith the estimates from the MD trajectory for A-DNA.\n\nIt is known \\cite{2001-QuantChem-DNA-conformations-review} that the sugar pucker typical of B-DNA is energetically\nmore favorable for all bases except cytosine. Estimates made by different methods give the energy difference in the interval\n0.2-0.9 kcal\/mol. For cytosine, the energy minimum corresponding to A-form is 0.3-0.5 kcal\/mol lower than that of B-form.\nThe energy barrier between these two main sugar puckers is about 2-4 kcal\/mol (at temperature 25$^o$C, kT$\\approx$0.6 kcal\/mol).\n\nBecause the fragment of chain used in \"relaxation\" method did not have any base attached to it, one would expect the depths of the two\nwells to be equal one to another. But we see that the B-minimum is lower. In this connection, one can remember work \\cite{1997-Cheatham-MD-AB},\nwhere, to provide in AMBER force field a spontaneous B to A transition in d[CCAACGTTGG]$_2$ sequence in 85\\% ethanol solution,\nthe authors had to make \"reduction of the V2 term in the O-C-C-O torsions from 1.0 to 0.30 kcal\/mol to better stabilize\nthe C3'-endo sugar pucker\". We also found that the higher A-minimum obstructs formation of the A-DNA, requiring unrealistic\npotentials for interactions between ions. Therefore we chose the double-well potential for the bond C1'-P(2) with equal minima.\n\nWe calculated parameters $t_P$ and $l_{P0}$ of the three-particle potential (\\ref{equation-3-point-potential}) from conditions\nthat the characteristic values for lengths $\\mid$P(1)P(2)$\\mid_A$ and\n$\\mid$C1'P(2)$\\mid_A$ obtained from MD simulation of A-DNA (see table~\\ref{table-pyramid}) have to correspond to zero of the potential;\nas well as the uncharacteristic values for these lengths (only for $\\mid$C1'P(2)$\\mid_B$ we chose, instead of the value 4.3$\\mbox{\\AA}$,\nthe lesser amount 4.2$\\mbox{\\AA}$). The two equations give $t_P=1.27$ and $l_{P0}=12.235\\mbox{\\AA}$.\nDeviations of the values $\\left|P(1)P(2)\\right| + t_P \\left|C1'P(2)\\right| - l_{P0}$ from zero depend on the pseudorotational angle: in A-DNA, they\nare less in the \"north\"(A) region. Therefore the rigidity $k_P$ in (\\ref{equation-3-point-potential}) can not be chosen uniquely.\nIt may be estimated as 10-40~kcal\/(mol$\\cdot$\\AA$^2$) in A-DNA, while in \"soft\" B-DNA it is around 5~kcal\/(mol$\\cdot$\\AA$^2$).\nWe used the amount\n39~kcal\/(mol$\\cdot$\\AA$^2$) to provide better correlation between the lengths $\\mid$P(1)P(2)$\\mid$ and $\\mid$C1'P(2)$\\mid$.\n\nFor the bond P(1)-C1', one can observe two peaks in histograms in both A- and B-DNA forms, but this is not connected with\nthe ribose flexibility: the populations of the peaks are the same in both forms (see table~\\ref{table-pyramid} in Appendix A, section~\\ref{AppendixA}),\nand from fig.~\\ref{A-pyramid-correlations} one can see that the correlation of this length with pseudorotational angle is small.\nEven where this correlation does exist, the changes of this length are comparable with its thermal fluctuations.\nThis fact allows one not to introduce a three-particle potential of type (\\ref{equation-3-point-potential}) for this bond,\nbut to restrict oneself to a soft harmonic potential with the rigidity close to the rigidity obtained from all-atom simulation.\n\\subsubsection{\"Valence\" and \"torsion\" angles}\nIn our CG model, every one of the two DNA strands is a sequence of pyramids \\{P(1)P(2)C3'C1'\\} surrounding riboses.\nEvery pyramid has two common vertices with two neighboring pyramids, and every pyramid can change its shape,\nwhich imitates ribose flexibility.\nIn the all atom models the zigzag (...-C3'-P-C3'-P-...) keeps its shape sufficiently well. To govern the shape of this backbone, one has to introduce\n\"valence\" and \"torsion\" angles into it. These angles do\nnot differ appreciably in the all atom MD simulations of A- and B-DNA forms (see table~\\ref{table-skeletal-angles} in Appendix A, section~\\ref{AppendixA}), which confirms\nthe adequacy of our choice of grains. So, we will consider the shape of the zigzag (...-C3'-P-C3'-P-...) to be universal: not depending\non sugar puckers.\n\nContrary to the angles on the backbone, the \"valence\" angles responsible for the direction of the glycosidic bond relative to the backbone,\nas well as \"torsion\" angles for base pair opening and for rotation of base around the glycosidic bond differ considerably\nin crystallographic (GLACTONE \\cite{1998-GlactoneGeorgia}) A- and B-DNA forms (see table~\\ref{table-differing-angles} in Appendix A, section~\\ref{AppendixA}).\nHowever, in MD trajectories, \"torsion\" angle C1'C3'P(2)C3' for base pair opening around the \"short\" bond C3'P(2) and the angle of rotation of base around\nthe glycosidic bond C3'C1'N(1,9)C(6,8) do not differ at all. In other cases we always chose the values corresponding to A-DNA.\n\nOne should also note, that the rigidity of the angle C3'C1'N(1,9)C(6,8), obtained from the all atom trajectories, has nothing to do with\nthe real rigidity of this angle. Indeed, in one nucleotide, there in no steric hindrance for the rotation of a base,\nwhich means very small own rigidity of this angle. The obstacles to the rotation create the atoms of the neighboring bases.\nTherefore, in our CG model, the rigidity of this angle is chosen to be very small.\n\\subsection{Modeling of DNA environment}\nIt is clearly seen from the all atom modeling (see fig.1 in \\cite{1998-A-to-B-Beveridge-2} and fig.7 in \\cite{1998-A-to-B-Beveridge}) that most counter-ions are situated near the surface of A-DNA, and almost in one layer. Closer examination shows that the counter-ions are located mostly in the major groove of A-DNA both in ethanol-water mixture \\cite{1997-Cheatham-MD-AB} and in a small water drop \\cite{2003-Mazur-A-B-in-drop}. It allows the phosphates on the opposite sides of the major groove to approach each other, and thus the characteristic cavity of A-DNA forms. It is obvious that such ion distribution can not be described in the framework of any implicit approach.\nAnd, indeed, in work \\cite{2006-Chocholousova-A-DNA-Implicit-Solvent}, where the generalized Born approximation with Debye-Huckel correction for salt effects was used, only some shift of the DNA form from B to A was reported when 1M of salt was added, while in work \\cite{1996-B-to-A-Pettitt-Sharmm} with the explicit media modeling the transition to A-DNA was registered in 1.5 nanoseconds with only 0.45M of salt added. Therefore we introduce ions explicitly.\n\nExplicit modeling of solvent takes lion's share of computational resources, and we know of no evidence that interaction of DNA\nwith solvent molecules should be treated explicitly. Therefore, all the known effects of the medium (electrostatics and solvation),\nwhich may affect the balance of interactions in the system, are represented implicitly, through effective potentials of interaction between the grains of the CG model,\nbetween the ions, and between the ions and the grains of the CG model.\n\\subsubsection{Electrostatic interaction between phosphate grains: distance dependent permittivity $\\varepsilon(r)$}\nThe simplest way to model electrostatic forces between phosphates is to put negative charges (-$e$) on phosphate grains and to introduce the Coulomb potential of interaction between these charges. Because of the small distances between phosphates and because of the appreciable changes in the distances in B$\\leftrightarrow$A transition one has to use a distance dependent permittivity $\\varepsilon(r)$ in this potential. Indeed, the dielectric constant $\\varepsilon$ is close to vacuum at small distances between the charges because there are not enough solvent molecules between the phosphates for screening their charges. Therefore, $\\varepsilon$ is normally taken equal to 2-3 at small distances. With increasing distance $r$, $\\varepsilon(r)$ is expected to reach its macroscopic value. Starting value and slope of this curve depend on size and dipole moment of solvent molecules, as well as on the location of the charges on the DNA molecule.\n\nOne may use different analytical forms for the dependence $\\varepsilon(r)$ (see a review in \\cite{2002-DNA-explicit-ions-implicit-water-Broyde} and \\cite{1991-Jernigan-eps(r)-review}).\nWe adopted the simplest representation already used in one CG DNA model \\cite{2011-CG-DNA-Freeman-de-Pablo}:\n\\begin{equation}\n\\label{formula-eps(r)_new}\n\\varepsilon(r)=\\varepsilon{_0}+\\varepsilon{_1} \\tanh [\\exp \\left( \\frac{ \\alpha} {2} (r-r_{0}) \\right)] ,\n\\end{equation}\nwith differing parameters: $\\varepsilon_{0}=58$ and $\\varepsilon_1=22$, $\\alpha = 12\\mbox{\\AA}^{-1}$, $r_0=8.5\\mbox{\\AA}$.\nThis function is shown in fig.~\\ref{n1-eps(r)}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.95\\linewidth] {fig06.eps}\n\\caption{ Screening of phosphate charges: distance dependence of permittivity $\\varepsilon(r)$ in the Coulomb potential of repulsion between negative charges on phosphate grains.\nWe compare the dependence adopted in our model with the dependencies offered or used in other works:\nby Hingerty et al \\cite{1985-Hingerty-eps(r)} for charges on biopolymers, by J.Mazur at al \\cite{1991-Jernigan-eps(r)-review} and\nby Wang et al \\cite{2002-DNA-explicit-ions-implicit-water-Broyde} for all-atom B-DNA modeling, and\nby Freeman et al \\cite{2011-CG-DNA-Freeman-de-Pablo} for a CG B-DNA.\nThe points show the dielectric constants on MD trajectory of our sugar CG model for the nearest grains\nalong the strand in A-DNA and B-DNA: $\\varepsilon(6.29\\mbox{\\AA})=36$ and $\\varepsilon(6.7\\mbox{\\AA})=36$,\nand for the nearest grains located across the major groove: $\\varepsilon(7.3\\mbox{\\AA})=36$ (A-DNA), $\\varepsilon(16.2\\mbox{\\AA})=80$ (B-DNA).\n}\n\\label{n1-eps(r)}\n\\end{center}\n\\end{figure}\n\\subsubsection{Interactions between ions, and between ions and DNA grains: solvation effects and sequence dependence}\nA-DNA can not be simulated in water with sodium counterions, even in a small box. The only exception we know of is in work \\cite{2003-Mazur-A-B-in-drop} where\na B to A transition was observed in a tiny water drop in which the surface tension contributed to the formation of the compact A-DNA. Normally, one needs to add salt to DNA with\ncounterions to observe an A-DNA (see, for example, \\cite{1996-B-to-A-Pettitt-Sharmm}). Therefore, in the CG model,\nwe included explicit ions Na$^+$ and Cl$^-$ and interactions between them, and between the ions and the charges on DNA.\n\nWe also found that the charges (-e) on phosphates are not sufficient for the formation of A-DNA,\none needs to put partial charges on bases (on grains $B_1$, $B_2$, $B_3$ of all the bases), and so to introduce the sequence dependence. More exactly, the A-DNA conglomerate\nis not stable if there are no charges on bases keeping the sodium ions inside the major groove. Therefore, we distributed partial charges on grains\nso that the dipole moment of every (neutral) base coincided with the moment in the AMBER force field\n(table~\\ref{table-base-charge}).\n\\begin{table}\n\\caption{Charges (in units of the elementary charge e) of base grains interacting with ions in solution.}\n\\label{table-base-charge}\n\\begin{tabular}{c c c c}\n\\hline\ntype of base & B$_1$ & B$_2$ & B$_3$ \\\\\n\\hline\nAdenine & -0.048 & 0.109 & -0.061 \\\\\n\\hline\nThymine & 0.390 & -0.240 & -0.150 \\\\\n\\hline\nGuanine & -0.496 & 0.134 & 0.362 \\\\\n\\hline\n Cytosine & 0.433 & 0.061 & -0.494 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nOne can obtain effective potential of interaction between ions in a solvent from a radial distribution function (rdf) in an all atom\nsimulation by several different methods \\cite{1999-Lyubartsev-DNA-ions-potentials-inverseMC, 2002-Lyubartsev-DNA-ions-potentials-HNC, \n2006-Vegt-Na-Cl-interaction, 2009-Papoian-CG-ions-renormgroup}. These methods yield the potentials of close shapes. We chose the \nanalytical representation of the potential function proposed in \\cite{2009-Papoian-CG-ions-renormgroup}:\n\\begin{equation}\n\\label{equation-ions-interaction}\n\\mathcal{V}_{ij} = \\frac{A}{{r_{ij}}^{12}}\n+ \\sum_{k=1}^{5} D_{k} \\exp^{ - C_{k} \\left[r_{ij}-R_{k} \\right]^2 } +\n\\frac{q_i q_j}{4\\pi \\varepsilon_0 \\varepsilon r_{ij}} ,\n\\end{equation}\nwhere $r_{ij}$ is the distance between $i$th and $j$th particles (between ions or between an ion and a DNA grain).\nIn this expression, the first term introduces excluded volume, the second term describes the shape of peaks\nand minima of the potential due to solvation, and the last term is the long-distance asymptotics: electrostatic (Coulombic) interaction between the charges.\n\nWe adopted Cl$^-$-Na$^+$ and Cl$^-$-Cl$^-$ potentials obtained in work \\cite{2002-Lyubartsev-DNA-ions-potentials-HNC}\nfrom all atom simulation of 0.5M NaCl electrolyte. In the simulation the authors used the Smith-Dang's ion model \\cite{1994-Smith-Dang-ions}.\nBecause we have few Cl$^-$ ions interacting mostly with Na$^+$ ions and one with another,\nwe assumed that we may use these potentials, even with a DNA molecule added in solution. For the interactions of Cl$^-$ ions with phosphate grains\nCl$^-$-P$^-$, we adopted the potential obtained in \\cite{1999-Lyubartsev-DNA-ions-potentials-inverseMC}, again from all atom simulations\nwith the Smith-Dang's ion model. We also used Na$^+$-Na$^+$, Na$^+$-Cl$^-$ and Cl$^-$-Cl$^-$ potentials from \\cite{2002-Lyubartsev-DNA-ions-potentials-HNC}\nas templates for potentials of interaction between ions and grains on bases. The details of the derivation of these potentials are in Appendix B (section~\\ref{AppendixB}).\n\nPotentials of interaction between sodium ions and between sodium ions and phosphate grains were chosen so that there were both A-DNA and B-DNA,\nas well as both A$\\rightarrow$B and B$\\rightarrow$A transitions in the model.\nIn figures \\ref{n3-Na} and \\ref{n3-Na-P} we compare these potentials with the ones obtained by different methods or exploited in some works.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.99\\linewidth] {fig07.eps}\n\\caption{Comparison of effective potentials of sodium ions interaction. The solid line corresponds to the potential used in the sugar CG DNA model. Dotted, dashed and dash-dotted lines\nrepresent the potentials obtained by different methods from all-atom simulations of ions in aqueous solution of salt in works by Lyubartsev et al\n\\cite{2002-Lyubartsev-DNA-ions-potentials-HNC}, Hess et al \\cite{2006-Vegt-Na-Cl-interaction}, Savelyev et al \\cite{2009-Papoian-CG-ions-renormgroup}. We also show the potential\nused in CG modeling of B-DNA by Freeman et al \\cite{2011-CG-DNA-Freeman-de-Pablo} - dash-dot-dot line. Thin dashed curve corresponds\nto the Coulombic interaction between the charges (the last term in equation (\\ref{equation-ions-interaction})).}\n\\label{n3-Na}\n\\end{center}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.99\\linewidth] {fig08.eps}\n\\caption{Comparison of effective potentials of interaction between phosphate grain and sodium ion. The potential used in the sugar CG model - solid line, the one obtained by Lyubartsev et al \\cite{1999-Lyubartsev-DNA-ions-potentials-inverseMC} from all-atom simulation of B-DNA - dashed line; the one used in CG modeling of B-DNA\nby Freeman et al \\cite{2011-CG-DNA-Freeman-de-Pablo} - dash-dot-dot line. Thin dashed curve corresponds to the Coulombic interaction between the charges (the last term in equation (\\ref{equation-ions-interaction})).}\n\\label{n3-Na-P}\n\\end{center}\n\\end{figure}\n\nParameters $\\{A, D_{k}, C_{k}, R_{k} \\}$ of the potentials used in our CG model are listed in tables \\ref{table-Na-Cl-P-interactions}-\\ref{table-Cl-base-interactions2} in\nAppendix B (section~\\ref{AppendixB}). We showed some of the potential curves in figures \\ref{n2-Na-Cl-P-attraction} and \\ref{n2-Na-Cl-P-repulsion}.\n\\subsubsection{Interaction of ions with grains C1', C3' and grains of bases: excluded volume}\nWe did not put charges on the grains C1' and C3', and the ions interact with them only by excluded volume potential.\nWe estimated the value of this volume from the rdf-s between sodium ions and atoms belonging to these grains.\nThese functions had been obtained in papers \\cite{1998-Lyubartsev-ions-atoms-on-bases-rdfs, 1999-Lyubartsev-DNA-ions-potentials-inverseMC}\nfor some atoms on bases and atoms O4' (belonging to the grain C1') and C4' (belonging to the grain C3') from all atom MD trajectories of B-DNA.\nFor the atom O4', solvation peaks are very low, and the rdf approaches unity at 4.5\\AA\\ (and this is true for all the uncharges grains).\nWe neglect a very low peak between 2.3 and 3.5\\AA, and accept that the rise of the rdf begins at 3.5\\AA. We take this value as the distance\nof closest approach between a Na$^+$ ion and the grain C1'. Similarly, for the grain C3', excluded volume distance is equal to 3.2\\AA.\n\nLeaning upon these estimates, we accepted the following potential functions for the excluded volume interactions of sodium ions with the uncharged grains C1' and C3':\n\\begin{eqnarray}\n\\label{Na-C1-C3-bases}\nE_{Na^+-C1'}= & \\epsilon \\left(\\sigma_{C1'} \/ r\\right)^{16}, & \\nonumber \\\\\nE_{Na^+-C3'}= & \\epsilon \\left(\\sigma_{C3'} \/ r\\right)^{12}, &\n\\end{eqnarray}\nwith $\\sigma_{Na^+-C1'}$=3.5\\AA, $\\sigma_{Na^+-C3'}$=3.2\\AA\\ and the same $\\epsilon$=~0.369~kcal\/mol ($\\approx$0.6kT at T=25$^o$C).\n\nFor the potentials for the excluded volume interactions of chlorine ions with the uncharged grains C1' and C3' we used the same functions, only with\n$\\sigma_{Cl^--C1'}$=$\\sigma_{Cl^--C3'}$=3.3\\AA.\n\\subsubsection{Interaction of ions and grains of DNA with implicit water: coefficient of friction}\nInfluence of water molecules on solute molecules within implicit solvent representation is normally simulated by Langevin equation. It provides both thermostat and viscosity. If one takes damping (friction) coefficient $\\gamma$=70 ps$^{-1}$ for sodium ions in implicit water, their diffusion coefficient proves to be equal to the experimental value. In work \\cite{2008-Sharp-explicit-ions-implicit-water}, DNA atoms, fully exposed to water, had the same damping coefficient $\\gamma$=70 ps$^{-1}$, while completely buried DNA atoms had a zero coefficient (no Langevin term). Partially exposed atoms had a damping coefficient proportional to the fraction of its solvent exposed surface area. In work \\cite{2006-Chocholousova-A-DNA-Implicit-Solvent}, $\\gamma$ was equal to 50 ps$^{-1}$ for all DNA atoms, while in work \\cite{2011-Case-Implicit-water-B-review} - to 5 ps$^{-1}$.\n\nThe value of the damping coefficient influences the rate of relaxation processes which depend on the solvent. This value does not seem to affect the balance\nof interactions in DNA molecule. We made simulations with small friction $\\gamma$=5 ps$^{-1}$ for both DNA grains and ions\nto make rapid system relaxation or to follow the behavior of the system for effectively longer time periods. To observe the behavior\ncomparable with the all atom simulation on its timescale, we used big friction $\\gamma_1=$50$ps^{-1}$ for DNA grains and $\\gamma_2=$70$ps^{-1}$ for ions.\n\\section{Description of the sugar CG DNA model}\nIn the sugar CG DNA model (see fig.\\ref{valence-bonds-angles}), every one of the two DNA strands is modeled by a zigzag of alternating grains P and C3': ...-P-C3'-P-C3'-... These grains are connected by \"valence\" bonds. A grain C1' is linked to each C3' grain by another \"valence\" bond. This \"comb\" is a skeleton of the strand. The grain C1' and the grains on the base B1, B2, B3 are connected by very rigid \"valence\" bonds C1'-B1, C1'-B3, B1-B2, B2-B3, B2-B3. We keep grains C1', B1, B2 and B3 in one plane by means of rigid \"torsion\" angle C1'-B1-B3-B2. The three rigidly bound grains (B1, B2, B3) almost freely rotate around glycosidic bond C1'-N(1,9) (position of the atom N(1,9) is calculated on each step from coordinates of the grains B1, B2, B3).\n\nTo maintain the shape of the helix ...-P-C3'-P-C3'-..., we introduce, besides the \"valence\" bonds, the \"valence\" angle C3'-P-C3' and two \"torsion\" angles\nC3'-P(2)-C3'-P(3) and P(1)-C3'-P(2)-C3'. The position of the glycosidic bond C1'-N(1,9) relative to the \"skeleton\" helix is supported by two \"valence\"\nangles P(1)-C1'-N(1,9) and C3'-C1'-N(1,9). Another \"torsion\" angle C1'-C3'-P(2)-C3' provides base pair opening.\n\nRibose flexibility is modeled by deformation of the pyramid \\{P(1)P(2)C1'C3'\\}. The possibility of the sugar repuckering is provided by a double-well\npotential for the \"valence\" bond C1'-P(2). The length of the edge P(1)-P(2) correlates with length of the \"double-well\" bond.\nThe grains P(1) and C1' are connected by a soft \"valence\" bond.\n\nThe model system consists of a DNA double helix and explicit sodium and chlorine ions. The potential energy of the system includes ten contributions:\n\\begin{eqnarray}\nH = &E_{base} + E_{hydr-bonds} + E_{stacking} + & \\nonumber \\\\\n & + E_{val-bonds} + E_{val-angles} + E_{tors-angles} + & \\nonumber \\\\\n & +E_{el} + E_{vdW} + E_{ion-DNA} + E_{ion-ion} &\n\\label{f12}\n\\end{eqnarray}\nThe corresponding potential functions and the used parameters are collected in table~\\ref{constants-and-formulas}.\n\\begin{table*}\n\\caption{A summary of the potential functions and parameters of the sugar CG DNA model. The order of grains in the notation of bonds and angles is their order along the chain direction (see fig.~\\ref{valence-bonds-angles}). The letter $N$ stands for atom $N1$ (or $N9$), and $C$ - for atom $C6$ (or $C8$) on bases.}\n\\label{constants-and-formulas}\n\\begin{tabular}{ccccc}\n\\hline\n \\bf{Interaction} & \\bf{Potential} & \\multicolumn{3} {c}{\\bf{Constants} } \\\\\n\\hline\n\\hline\n\\multicolumn{2} {c}{\"valence\" bonds } & $r_0$ , \\AA & $k_r$, kcal\/(mol $\\cdot $\\AA$^2$) & \\\\\n\\hline\nP-C3' & & 4.52 & 35 & \\\\\nC3'-C1' & $\\frac{1}{2}k_r(r-r_0)^2$ & 2.4 & 192 &\\\\\nC3'-P & & 2.645 & 201 &\\\\\nP-C1' & & 5.4 & 28 &\\\\\n \\hline\n\\hline\n\\multicolumn{2} {c}{double-well \"valence\" bond (imitating ribose flexibility)} & parameter & value & dimension \\\\\n\\hline\n\t& & $r_A$ & 4.8 & \\AA \\\\\n & & $r_B$ &4.2 & \\AA \\\\ \n & $U(r)=U_B(r - r_B) f(r) +$ & $r_C$ &4.584 &\\AA \\\\\nC1'-P & $+ [U_A(r - r_A) + \\epsilon_0] [1-f(r)] +$ & $\\epsilon_0$ & 0 & kcal\/mol \\\\\n & $+ \\epsilon_{barrier} e^{-\\mu_0(r - r_C)^2}$ & $\\epsilon_{barrier} $ & -0.46 & kcal\/mol \\\\\n& $U_j(r)=\\frac{1}{2}K_j r^2, j=A, B$ & $K_A $ & 63 &kcal\/(mol $\\cdot$ \\AA$^2$) \\\\\n& $ f(r)=\\frac{1}{1+e^{2\\mu(r-r_C)}}$ & $K_B $ & 25 &kcal\/(mol $\\cdot$ \\AA$^2$) \\\\\n& & $\\mu $ & 20 & \\AA$^{-1}$ \\\\\n& & $\\mu_0 $ & 300 & \\AA$^{-1}$ \\\\\n\\hline\n \\hline\n\\multicolumn{2} {c}{\"valence\" bond correlated with ribose conformation} & parameter & value & dimension \\\\\n\\hline\n & $U(r_{C1'P},r_{P(1)P(2)})=$ & $ k_P$ & 39 & kcal\/mol \\\\\n P(1)-P(2) & $\\frac{1}{2}k_P(r_{P(1)P(2)}+t_Pr_{C1'P}-l_{P0})^2$ & $ l_{P0}$ & 12.235 & \\AA \\\\\n & & $t_P $ & 1.27 & \\\\\n\\hline\n \\hline\n\\multicolumn{2} {c}{ \"valence\" angles } & $\\theta_0$, deg & $k_{\\theta}$, kcal\/(mol $\\cdot$ deg$^2)$ & \\\\\n \\hline\nC3'-P-C3' & & 110 & 0.017 & \\\\\nP-C1'-N & $\\frac{1}{2}k_{\\theta}(\\theta-\\theta_0)^2$ & 84 & 0.026 & \\\\\nC3'-C1'-N & & 112 & 0.032 & \\\\\n\\hline\n \\hline\n\\multicolumn{2} {c}{ \"torsion\" angles } & $\\delta_0$, deg & $\\epsilon_{\\delta}$, kcal\/mol & comment\\\\\n\\hline\nC3'-P-C3'-P & & 188 & 4.6 & long P-C3' bond \\\\\nP-C3'-P-C3' & $\\epsilon_{\\delta}(1-\\cos (\\delta-\\delta_0))$ & 194 & 4.6 & short C3'-P bond \\\\\nC1'-C3'-P-C3' & & 13 & 3.0 & base-pair opening \\\\\nC3'-C1'-N-C & & -32 & 0.03 & glycosidic bond\\\\\n\\hline\n\\hline\n\\multicolumn{2} {c}{ interactions in rigid bases } &\\multicolumn{3} {c} {} \\\\\n\\hline\n bonds & $\\frac{1}{2}k_r(r-r_0)^2$ &\\multicolumn{3} {c} { see formula (B4) } \\\\\ntorsion angle & $\\epsilon(1+\\cos \\delta)$ &\\multicolumn{3} {c} {and table~III in \\cite{Savin2011} } \\\\\n\\hline\n\\hline\n\\multicolumn{2} {c}{hydrogen bonds and stacking interactions} &\\multicolumn{3} {c} {from AMBER}\\\\\n\\hline\n\\hline\n\\multicolumn{2} {c}{ electrostatic interactions between phosphate grains} & parameter & value & dimension \\\\\n\\hline\n &\t & $\\varepsilon_{0}$ & 22 & \\\\\n$P-P$ & $\\frac{q_i q_j}{4\\pi \\varepsilon_0 \\varepsilon (r) r_{ij}} $ &$\\varepsilon_1$ & 58 & \\\\\n & $\\varepsilon(r)=\\varepsilon{_0}+\\varepsilon{_1} \\tanh [\\exp \\left( \\frac{ \\alpha} {2} (r-r_{0}) \\right)] $ & $\\alpha $ & 0.3 & $\\mbox{\\AA}^{-1}$ \\\\\n & & $r_0$ & 8.5 & $\\mbox{\\AA}$ \\\\\n\\hline\n\\hline\n \\multicolumn{2} {c}{ van der Waals interactions between skeleton grains} & $\\sigma_i$, \\AA & $ \\epsilon_i$ , kcal\/mol & \\\\\n\\hline\n $P$ & $4\\epsilon_{ij} \\left[ \\left(\\frac{\\sigma_{ij}}{r}\\right)^{12}- \\left(\\frac{\\sigma_{ij}}{r}\\right)^6 \\right] $ & 2.18 & 0.23 & \\\\\n $C3'$ & $ \\sigma_{ij}=(\\sigma_i+\\sigma_j)\/2$, $\\epsilon_{ij}=\\sqrt{\\epsilon_{i}\\epsilon_{j}} $ & 2.0 & 0.115 & \\\\\n\\hline\n \\hline\n\\multicolumn{2} {c}{interaction of Na$^+$ and Cl$^-$ ions with charged grains } & \\multicolumn{3} {c}{ } \\\\\n\\multicolumn{2} {c}{of DNA and one with another } & \\multicolumn{3} {c}{ } \\\\\n\\hline\n& & \\multicolumn{3} {c} { $q_{Na^+}$=+$e$, $q_{Cl^-}$=-$e$, $q_{P}$=-$e$,} \\\\\n& $\\frac{A}{{r_{ij}}^{12}}+ \\sum_{k=1}^{5} D_{k} \\exp^{ - C_{k} \\left[r_{ij}-R_{k} \\right]^2 } +\n\\frac{q_i q_j} {4\\pi \\varepsilon_0 \\varepsilon r_{ij}} $ & \\multicolumn{3} {c} { charges of grains of bases see in table~\\ref{table-base-charge};} \\\\\n& & \\multicolumn{3} {c} { A, D$_k$, C$_k$, R$_k$, $\\varepsilon$ are in tables \\ref{table-Na-Cl-P-interactions},\\ref{table-Na-Cl-interactions},\\ref{table-Na-base-interactions1},\\ref{table-Na-base-interactions2},\\ref{table-Cl-base-interactions1},\\ref{table-Cl-base-interactions2}} \\\\\n\\hline\n\\hline\n\\multicolumn{2} {c}{ interaction of ions with uncharged grains } & $\\sigma$, \\AA & $ \\epsilon$ , kcal\/mol & \\\\\n\\hline\nNa$^+$ with C1' & $ \\epsilon \\left(\\sigma \/ r\\right)^{16}$ & 3.5 & 0.369 & \\\\\nNa$^+$ with C3' & $ \\epsilon \\left(\\sigma \/ r\\right)^{12} $ & 3.2 & 0.369 & \\\\\nCl$^-$ with C1' & $ \\epsilon \\left(\\sigma \/ r\\right)^{16}$ & 3.3 & 0.369 & \\\\\nCl$^-$ with C3' & $ \\epsilon \\left(\\sigma \/ r\\right)^{12} $ & 3.3 & 0.369 & \\\\\n\n\\hline\n\n\\end{tabular}\n\\end{table*}\n\nThe term $E_{base}$ describes the energy of deformation of rigid bases.\nThe terms $E_{hydr-bonds}$ and $ E_{stacking}$ stand for energy of hydrogen bonds between\ncomplementary bases and for base pairs stacking, correspondingly. We recalculate the coordinates of all nucleobase atoms on each step and compute these terms\nusing the all atom force field AMBER.\n\nThe terms $E_{val-bonds}, E_{val-angles}, E_{tors-angles}$ describe energy of deformation of \"valence\" bonds, \"valence\" angles and \"torsion\" angles on the strands of CG DNA.\nEquilibrium values of the angles and the bonds, not pertaining to the ribose flexibility, were chosen equal to the values in A-DNA. For the rigidities were chosen\nthe maximal values. Two wells in\nthe double-well potential of the bond C1'-P(2) were made of equal depth, contrary to the AMBER force field (see fig.~\\ref{fig-2wells-relax}). We discuss the\nreason for these choices in section~\\ref{Discussion and conclusion}.\n\nCoulombic interactions $E_{el}$ between charged phosphate grains have distance dependent permittivity (see formula (\\ref{formula-eps(r)_new})).\nWe introduce van der Waals interactions $E_{vdW}$ for the grains $P$ and $C3'$ not connected through \"valence\" bonds, \"valence\" angles or \"torsion\" angles.\n\nInteraction of ions with DNA $E_{ion-DNA}$ includes interactions with charged phosphate grains $P$ and grains on bases and with uncharged grains (C1', C3').\nIn the present realization of the model, we introduce sequence dependence: the charges on grains of a base depend on the type of this base.\nInteractions of ions one with another $E_{ion-ion}$ and with charges on DNA (phosphate grains and grains on bases) $E_{ion-DNA}$\ntake into account solvation effects (besides direct Coulomb force).\n\nThe influence of water on DNA and on ions is described implicitly, by Langevin equation. Damping constant (friction) is taken to be equal to 5ps$^{-1}$ for system relaxation and\ntest calculations. Productive runs were made with $\\gamma$=5ps$^{-1}$, as well as with $\\gamma_1$=50ps$^{-1}$ for the DNA grains and $\\gamma_2$=70ps$^{-1}$\nfor the ions.\n\\section{Model testing}\nWe found two equilibrium states (A-DNA and B-DNA) of the system by its energy minimization (from different initial states at corresponding boundary and initial conditions).\nAfter this, we compared CG MD trajectories with the ones obtained in all atom simulations.\n\\subsection{Energy minimization: A-DNA and B-DNA.}\n\\label{energy-minimization}\nTo obtain the ground states of B- and A-DNA, we started from all atom MD configurations of DNA in water and in mixed ethanol\/water\n(85\/15) solution correspondingly. We put the grains and ions of our CG model on the places of the corresponding atoms and ions of the all atom system.\nWe also added 16 Na$^+$ and 16 Cl$^-$ ions for A-DNA. For more adequate comparison with A-DNA, we studied B-DNA not only in combination with\ncounterions (which is common), but also with the same amount of additional salt as for A-DNA. The additional ions were placed randomly in the unoccupied area of the\ncomputational cell. We modeled B-DNA in a large reservoir: a cube 60x60x60\\AA.\nIn this volume, the 32 salt ions give the molar concentration 0.12M, very close to the one of physiological saline. For A-DNA, we chose a small volume so\nthat, on the one hand, the ions could not go too far from the molecule, and, on the other hand, the energy of interaction between the chlorine ions and\nthe phosphate grains was not too high. The optimal reservoir proved to be a cylinder with diameter 18.5\\AA\\ and height 30\\AA. In it, the salt concentration\nwas 0.8M. The energy of the CG system was minimized by the method of conjugate gradients: first the ions, and then the whole system.\nAs a result, we obtained both CG A- and B-DNA conformations (in proper computational cells) which we used as initial configurations for MD simulations.\n\\subsection{MD simulations of sugar CG A-DNA and B-DNA.}\n\\label{section-CG-MD}\nFor MD simulations, we exploited the Runge-Kutta method, with a Langevin thermostat. As a boundary condition, we used the condition\nof reflection of ions from the sides of the computational cell. The initial configurations of the system were the ones obtained by energy minimization\ndescribed in section~\\ref{energy-minimization}.\n\nThe initial system relaxation was being done in two stages. First, during one nanosecond, we carried out a relaxation of the ion atmosphere\nwith the DNA molecule kept immobile.\nThen, the whole system was relaxing during the next 0.5 nanoseconds. During all the relaxation process, the friction for all the grains and ions was\n$\\gamma=$5ps$^{-1}$. The productive runs were carried out both with the small friction $\\gamma=$5ps$^{-1}$, and with the large friction\n$\\gamma_1=$50ps$^{-1}$ for the DNA grains and $\\gamma_2=$70ps$^{-1}$ for the ions.\n\nWe followed the dynamics of the both forms, A- and B-DNA, at temperature 300~K up to 8 nanoseconds with small friction and up to 18 nanoseconds with large\nfriction, which allows to consider the forms stable at the corresponding conditions. B-DNA is stable in both the simulations: with counterions and in physiological saline.\nThe obtained stable configurations are shown in figure \\ref{ABpictures}. The balance of interactions in A-DNA and B-DNA is presented in histogram in fig.~\\ref{fig-energy-balance}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth] {fig09.eps}\n\\caption{\nFrames from the trajectories of sugar CG A-DNA (on the right) and B-DNA (on the left). Temperature is 300~K. Sodium ions are yellow, chlorine - cyan.\n}\n\\label{ABpictures}\n\\end{center}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.95\\linewidth] {fig10.eps}\n\\caption {\nComparison of the average contributions to the full potential energy of the CG A-DNA and B-DNA. Both the systems consist of a DNA molecule,\ncounterions and additional salt (16 sodium and 16 chlorine ions). The temperature is 300~K. B-DNA is in a cube 60x60x60\\AA,\nA-DNA is in a cylinder with diameter 18.5\\AA\\ and height 30\\AA.\n}\n\\label{fig-energy-balance}\n\\end{center}\n\\end{figure}\n\nTables \\ref{table-A-DNA} and \\ref{table-B-DNA} present the lengths and the angles in the sugar CG model in comparison with\nthe all atom AMBER model.\n\\begin{table}\n\\caption{\nComparison of lengths and angles in the sugar CG A-DNA (with small friction $\\gamma$=5ps$^{-1}$) in concentrated\nsalt solution and in the all atom AMBER A-DNA in mixed ethanol\/water (85\/15) solution.\nThe letter $N$ stands for atom $N1$ (or $N9$), and $C$ - for atom $C6$ (or $C8$) on bases. The averages were taken over a trajectory of 8 nanoseconds long for\nthe CG model, and of 4.5 nanoseconds long for the all atom model.\n}\n\\label{table-A-DNA}\n\\begin{tabular}{cccc}\n\\hline\nvalue &dimension &sugar-CG model & AMBER \\\\\n\\hline\n$\\left|PC3'\\right|$ & & 4.5 $\\pm$0.1 & 4.5 \\\\\n$\\left|C3'C1'\\right|$ & & 2.40 $\\pm$0.05 & 2.4 \\\\\n$\\left|C3'P\\right|$ & \\AA & 2.65 $\\pm$0.05 & 2.6\\\\\n$\\left|PC1'\\right|$ & & 5.4 $\\pm$0.1 & 5.4$\\pm$0.3 \\\\\n$\\left|C1'P\\right|$ & & 4.7$\\pm$0.1 & 4.8$\\pm$ 0.2\\\\\n$\\left|P(1)P(2)\\right|$ & & 6.3$\\pm$0.2 & 6.1$\\pm$ 0.8\\\\\n\\hline\n$\\angle$C3'PC3' & & 109$\\pm$4 & 110$\\pm$14 \\\\\n$\\angle$PC1'N & deg & 83$\\pm$3 & 84$\\pm$10\\\\\n$\\angle$C3'C1'N & & 112 $\\pm$3 &112 $\\pm$ 3\\\\\n\\hline\n$\\angle$C3'PC3'P & & 173 $\\pm$14 & 173$\\pm$ 32\\\\\n$\\angle$PC3'PC3' & deg & 213$\\pm$11 & 217$\\pm$ 30\\\\\n$\\angle$C3'C1'NC & & -50$\\pm$11 & -32$\\pm$9\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table}\n\\caption{\nComparison of lengths and angles in the sugar CG B-DNA (with small friction $\\gamma$=5ps$^{-1}$)\nand in the all atom AMBER B-DNA.\nThe letter $N$ stands for atom $N1$ (or $N9$), and $C$ - for atom $C6$ (or $C8$) on bases. The averages were taken over a trajectory of 8 nanoseconds long for\nthe CG model, and of 20 nanoseconds long for the all atom model.\n}\n\\label{table-B-DNA}\n\\begin{tabular}{cccc}\n\\hline\nvalue &dimension &sugar-CG model & AMBER \\\\\n\\hline\n$\\left|PC3'\\right|$ & & 4.5 $\\pm$0.1 & 4.53 \\\\\n$\\left|C3'C1'\\right|$ & & 2.40$\\pm$0.05 & 2.368\\\\\n$\\left|C3'P\\right|$ & \\AA & 2.65 $\\pm$0.05 & 2.66\\\\\n$\\left|PC1'\\right|$ & & 5.4$\\pm$0.1 & 5.35$\\pm$ 0.4 \\\\\n$\\left|C1'P\\right|$ & & 4.3 $\\pm$0.1 & 4.4 $\\pm$ 0.4\\\\\n$\\left|P(1)P(2)\\right|$ & & 6.8 $\\pm$0.2 & 6.8$\\pm$0.6 \\\\\n\\hline\n$\\angle$C3'PC3' & & 110$\\pm$4 & 104$\\pm$ 13 \\\\\n$\\angle$PC1'N & deg & 83$\\pm$3 & 105$\\pm$14\\\\\n$\\angle$C3'C1'N & & 113$\\pm$3 & 146$\\pm$13 \\\\\n\\hline\n$\\angle$C3'PC3'P & & 186 $\\pm$16 & 170 $\\pm$31 \\\\\n$\\angle$PC3'PC3' & deg & 198$\\pm$12 & 211 $\\pm$44 \\\\\n$\\angle$C3'C1'NC & & -40$\\pm$10 & -27$\\pm$34\\\\\n\\hline\n\\end{tabular}\n\\end{table}\nAs one should expect, our model proved to be stiffer than the all atom one (we accepted the maximal rigidities observed in all atom simulations), and the angles\nin both A- and B-DNA are closer to their values in all-atom A-DNA (so we set them). The only exception is torsion $\\angle$C3'C1'NC, for which the prescibed magnitude\nwas (-32$^0$), while the observed one - (-50$^0$) in A-DNA and (-40$^0$) in B-DNA. Interestingly, that the value for A-DNA is in excellent agreement with\nthe crystallographic value (GLACTONE \\cite{1998-GlactoneGeorgia}): see table~\\ref{table-differing-angles}.\n\nFigures \\ref{fig-gist-C1P-A} and \\ref{fig-gist-C1P-BDNA} give a visualization of the ribose flexibility in CG A- and B-DNA.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.98\\linewidth] {fig11.eps}\n\\caption {\nRibose flexibility of A-DNA in the sugar CG model with both small ($\\gamma$=5ps$^{-1}$) and large ($\\gamma_1$=50ps$^{-1}$) friction in concentrated\nsalt solution {\\it vs.} ribose flexibility of A-DNA in the all atom AMBER model in mixed ethanol\/water (85\/15) solution.\nWe show histograms of the length of the bond C1'P which directly correlates with the pseudorotation angle.\nThe histograms were plotted for the counts at every 1ps over the trajectories 4.5 nanoseconds long.\n}\n\\label{fig-gist-C1P-A}\n\\end{center}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.98\\linewidth] {fig12.eps}\n\\caption {\nRibose flexibility of B-DNA with counterions in the sugar CG model with both small ($\\gamma$=5ps$^{-1}$) and large ($\\gamma_1$=50ps$^{-1}$)\nfriction {\\it vs.} ribose flexibility of B-DNA in the all atom AMBER model.\nWe show histograms of the length of the bond C1'P which directly correlates with the pseudorotation angle.\nThe histograms were plotted for the counts at every 1ps over the trajectories 20 nanoseconds long.\n}\n\\label{fig-gist-C1P-BDNA}\n\\end{center}\n\\end{figure}\nOne can see that the CG model imitates the AMBER A-DNA very closely, including sequence dependence. Sugar CG B-DNA is stiffer and has lower population\nin the area between C2'-endo and C3'-endo.\n\nA-DNA can exist only if sodium ions can assemble in the major groove so that their electrostatic interaction with the phosphate grains (and with the nearest\nchlorine ions) will give the gain in free energy greater than the loss in entropic contribution because of the ion clustering\n(see the balance of interactions in CG DNA forms in fig.~\\ref{fig-energy-balance}).\nFor the stabilization of this positively charged cluster between two rows of\nnegative charges, the presence of a solvent in the major groove is crucial.\n\nTo adequately model this balance, one should very precisely choose\nthe positions and the widths of the minima of the effective solvent-mediated potentials. We built our potentials for the Na$^+$-P$^-$ and Na$^+$-Na$^+$\npairs so that the CG radial distribution functions g(r) were as close as possible to the all atom (AMBER) ones, especially in what regards the positions of the minima.\nThe agreement between the CG rdf and the all atom one for the pair Na$^+$-P$^-$ is almost ideal (see fig.\\ref{fig-rdf-Na-P-ADNA}).\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.98\\linewidth] {fig13.eps}\n\\caption {\nRadial distribution function g(r) of Na$^+$ ions around phosphate grains P$^-$ in the sugar CG A-DNA\nwith both small (short dashed line) and large (dashed line) friction of grains $\\gamma$.\nFor comparison, we show Na$^+$-P$^-$ distribution function in the all-atom AMBER model of A-DNA in mixed ethanol\/water\nsolution (solid line).\n}\n\\label{fig-rdf-Na-P-ADNA}\n\\end{center}\n\\end{figure}\nFor the pair Na$^+$-Na$^+$ it was impossible because we simulated A-DNA in water, and not in in mixed ethanol\/water\nsolution. A-DNA with counterions in water does not exist without additional salt. Therefore we had many more sodium ions\nin the computational cell, and, correspondingly, in the major groove. It had to lead to the substantial rise of the first peak as compared\nwith the second one (see fig.\\ref{fig-rdf-Na-Na-ADNA}),\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.98\\linewidth] {fig14.eps}\n\\caption {\nRadial distribution function g(r) of the pair Na$^+$-Na$^+$ in the sugar CG A-DNA\nwith both small (short dashed line) and large (dashed line) friction of grains $\\gamma$.\nFor comparison, we show this distribution function in the all-atom AMBER model of A-DNA in mixed ethanol\/water\nsolution (solid line).\n}\n\\label{fig-rdf-Na-Na-ADNA}\n\\end{center}\n\\end{figure}\nwhich usually takes place with increase of the salt concentration (see, for example, \\cite{2011-Shen-Vegt-rdf-ions}).\nHowever, our Na$^+$-P$^-$ and Na$^+$-Na$^+$ pairs are more sticky than the default AMBER ions. One can\nsee that from rdfs for B-DNA shown in fig.~\\ref{fig-rdf-Na-P-BDNA} and \\ref{fig-rdf-Na-Na-BDNA}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.99\\linewidth] {fig15.eps}\n\\caption {\nRadial distribution function g(r) of Na$^+$ ions around phosphate grains P$^-$ in the sugar CG B-DNA (without additional salt)\nwith both small (short dashed line) and large (dashed line) friction of grains $\\gamma$.\nFor comparison, we show Na$^+$-P$^-$ distribution functions in the all-atom AMBER model of B-DNA with different\nparameters for ions: default Aqvist's (solid line), Cheatham's (dash-dot line)\\cite{2009-Noy-Orozco-rdf-ions}\nand Jorgensen's (dash-dot-dot line)\\cite{2009-Noy-Orozco-rdf-ions}.\n}\n\\label{fig-rdf-Na-P-BDNA}\n\\end{center}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.99\\linewidth] {fig16.eps}\n\\caption {\nRadial distribution function g(r) of the pair Na$^+$-Na$^+$ in the sugar CG B-DNA (without additional salt)\nwith both small (short dashed line) and large (dashed line) friction of grains $\\gamma$.\nFor comparison, we show this distribution function in the all-atom AMBER model of B-DNA (solid line).\n}\n\\label{fig-rdf-Na-Na-BDNA}\n\\end{center}\n\\end{figure}\n\nBecause of the problems of choice of ion parameters in the framework of additive, nonpolarizable and pairwise potentials,\nthere are several different sets of ion parameters. Aqvist's cations and Dang's Cl$^-$, which were used in AMBER by default,\ngive the artefact of formation of stable ion pairs and even salt crystals at moderately low concentrations (below their solubility\nlimit). Other sets of the parameters result in rdfs greatly differing in shape (see fig.~6 in \\cite{2009-Noy-Orozco-rdf-ions}).\n\nAs compared with default AMBER ions, our CG model gives for the pair Na$^+$-P$^-$ a very high first peak (see fig.~\\ref{fig-rdf-Na-P-BDNA}),\nwhich means that our Na$^+$ ions are more often located near phosphates, and, consequently, one to another (see fig.~\\ref{fig-rdf-Na-Na-BDNA}).\nRdf for the Na$^+$-P$^-$ pair for B-DNA in our model mostly resembles the corresponding rdf for Cheatham's ions\n\\cite{2008-Cheatham-ions-parameters}. When compared to the others, Cheatham's sodium ions are much more often near phospates,\navoiding chlorine ions and each other (see fig.~11 and 6 in \\cite{2009-Noy-Orozco-rdf-ions}). This feature seems to be\na drawback leading to over-neutralization of the DNA. However, it proved \\cite{2015-Case-Cheatham-ions-around-DNA} to be that\nthe number of Cheatham's sodium ions well agrees with ion counting experiments at low salt concentrations,\nand at high concentrations ($>$0.7M) is even less than in experiment. Therefore, we can regard our effective potentials for\nion interactions as trustworthy.\n\\section{Discussion and conclusion}\n\\label{Discussion and conclusion}\nWe built our CG model on the base of the all atom force field AMBER, and we still keep the all-atom base pairing and stacking\n(it should be changed to a CG version).\nStarting from AMBER, we faced the problem that our CG DNA can assume a B-form structure at almost every reasonable set of parameters,\nwhile balancing interactions in the A-DNA required some efforts.\n\nFirst, we supposed that an A-DNA can exist without placing partial charges on the bases, i.e. without introduction of\nsequence dependence. But that proved to be impossible. At temperature only as high as 300~K, the conglomerate of A-DNA proved\nto be unstable, the charges on the borders of the major groove were insufficient to keep the ions inside this groove.\n\nSecondly, the potentials and the constants, derived from the AMBER force field, required corrections. Namely, for all the bonds and the angles\n(except for C1'P and P(1)P(2) connected with ribose flexibility) we used equilibrium values of the all-atom A-DNA and maximal rigidities observed\nin the all-atom simulations. For the double-well potential of the bond C1'P we lower the A-minimum to the level of the B-minimum\n(see fig.~\\ref{fig-2wells-relax}). In this connection, one can remember work \\cite{1997-Cheatham-MD-AB},\nwhere, to provide in AMBER force field a spontaneous B to A transition of d[CCAACGTTGG]$_2$ sequence in 85\\% ethanol solution,\nthe authors had to make \"reduction of the V2 term in the O-C-C-O torsions from 1.0 to 0.30 kcal\/mol to better stabilize\nthe C3'-endo sugar pucker\".\n\nFinally, we had to exploit such effective potentials between sodium ions and phosphate grains and between\nsodium ions one with another that resulted in a rdf for the pair Na$^+$-P$^-$ very close to the rdf\nfor Cheatham's ions (see fig.~\\ref{fig-rdf-Na-P-BDNA}), and not for default AMBER ions (for more details,\nsee the discussion at the end of section~\\ref{section-CG-MD}).\n\nEvidently, the necessity of these corrections is a result of the long known B-philia of the AMBER force field\n(see, for example, \\cite{1997-Feig-Experiment-vs-Force-Fields}). Indeed, the B$\\rightarrow$A\ntransition at high salt concentration has been demonstrated for the A-philic CHARMM force field in 1996, while\nfor the AMBER force field this transition in water takes place only in a tiny drop of water \\cite{2003-Mazur-A-B-in-drop}.\nIn it, the compact A-DNA is stabilized by surface tension. The additional salt results only in salt crystallization,\ninstead of causing the transition.\n\nAfter the described fitting, we have obtained both an A-DNA and a B-DNA at the corresponding conditions,\nas well as both A$\\rightarrow$B and B$\\rightarrow$A transitions. The chosen set of parameters,\nproviding the needed balance, is, of course, not unique and, may be, not yet quite adequate.\nIt will probably be corrected in the process of application of the CG model to different physical situations.\n\nThe offered sugar CG model is adequate, physically clear, computationally cheap, allows to promptly check physical\nhypotheses, and so can be employed for the study of many interesting problems, including the cases in which\nall atom DNA models can not be used. For example, the CG model can be applied for investigation of large mechanical\ndeformations of long DNA molecules (when they package into chromosomes). The introduced sequence dependence\nallows to use the model for studying of DNA-protein interactions, including the interactions with CG proteins.\nA small change of the potentials enables base opening, and offers the possibility to simulate DNA melting,\nand investigate transcription and replication. Another large area of applications - modeling of electrostatic\ninteractions of DNA with different types of ions in different kinds of solutions.\n\\section{Acknowledgements}\nWe are sincerely grateful to our colleagues, prof.~Alexey Onufriev (Virginia, USA) who obtained a trajectory of B-DNA in water for us; prof. Modesto Orozco (Barcelona) and Agnes Noy who kindly granted us a trajectory of A-DNA obtained in work \\cite{2007-A-B-Noy-Orozco}; and U. Deva Priyakumar who sent us .pdb-files of DNA molecule with an opening base from work \\cite{2006-MD-base-flipping}. For the realization of the sugar CG DNA model we used the (properly modified) program written by Dr. A.V. Savin. We take the opportunity to thank A.V. Savin for his help in the modification of the program. The simulations were partly carried out in the Joint Supercomputer Center of Russian Academy of Sciences. We appreciate financial support of RFBR (grants 12-03-31810 mol-a and 08-04-91118-a) and CRDF (award RUB2-2920-MO-07).\n\\cleardoublepage\n\\section{Appendix A: details of obtaining sugar CG DNA model parameters from all-atom MD simulations (AMBER)}\n\\label{AppendixA}\n\"Relaxation\" method shows that, under deformation of the bond P(1)-C3', the torsion angles change first: $\\beta$, then $\\alpha$\nand $\\gamma$, and one can observe {\\it trans-gauche} transitions in these angles, and several different minima of the potential. If we\nneglect the $\\alpha$\/$\\gamma$-mobility, the potentials for this bond may be approximated by a parabola.\nIn MD simulations, the lengths $\\mid$C3'C1'$\\mid$, $\\mid$C3'P(2)$\\mid$ and $\\mid$P(1)C3'$\\mid$ (see fig.~\\ref{valence-bonds-angles})\nonly sligthly fluctuate for all the nucleotides.\nThe estimates of the lengths and the rigidities made by different methods (see section~\\ref{section-AMBER})\nare listed in table~\\ref{table-valence-bonds}.\n\\begin{table}[b]\n\\caption{Lengths of \"valence\" bonds not connected with sugar repuckering (see fig.~\\ref{valence-bonds-angles}) and\ncorresponding rigidities. We list (1) distances between the corresponding\natoms in crystallographic (GLACTONE \\cite{1998-GlactoneGeorgia}) A-DNA and B-DNA forms ($l_{A-Gl}$ and $l_{B-Gl}$); (2) mean distances and\nrigidities $l_{A-MD}$, $K_{A-MD}$ and $l_{B-MD}$, $K_{B-MD}$, obtained by Boltzmann inversion method\nfrom MD trajectories; (3) equilibrium lengths\n$l_{relax}$ and rigidities $K_{relax}$ of potentials obtained by method of \"relaxation\" (see \\ref{section-AMBER}). The distances have\ndimension of Angstroms, the rigidities - of kcal\/(mol$\\cdot$\\AA$^2$).}\n\\label{table-valence-bonds}\n\\begin{tabular}{cccc}\n\\hline\n & $\\mid$C3'C1'$\\mid$ & $\\mid$C3'P(2)$\\mid$ & $\\mid$P(1)C3'$\\mid$ \\\\\n\\hline\n$l_{A-Gl}$ & 2.361 & 2.605 & 4.162 \\\\\n$l_{A-MD}$ & 2.40 & 2.645 & 4.52 \\\\\n$K_{A-MD}$ & 192 & 201 & 35 \\\\\n$l_{B-Gl}$ & 2.368 & 2.609 & 4.072 \\\\\n$l_{B-MD}$ & - & 2.66 & 4.53 \\\\\n$K_{B-MD}$ & - & 222 & 19 \\\\\n$l_{A-relax}$ & - & 2.65 & - \\\\\n$K_{A-relax}$ & - & 230 & - \\\\\n$l_{B-relax}$ & - & 2.76 & 4.6 \\\\\n$K_{B-relax}$ & - & 288 & 16 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nFor B-DNA in water, there are few nucleotides for which pseudorotation angle $\\tau$ is localized in classical\nsouth region. For most nucleotides $\\tau$ changes considerably, and in the process the pucker often is in the north\nregion (characteristic for A-DNA) or, even more commonly, in the intermediate area.\nIn contrast, the angle $\\tau$ for A-DNA in the mixture of ethanol and water is well localized in the canonical north region.\nOne can observe jumps to the south region, but the pucker is practically never in the intermediate area.\nOn the other hand, the correlation between the distance $\\mid$C1'P(2)$\\mid$ and the pseudorotation angle is approximately the same.\nFor the convenience of the analysis of correlations between the pseudorotational angle and the lengths $\\mid$P(1)C1'$\\mid$, $\\mid$C1'P(2)$\\mid$\nand $\\mid$P(1)P(2)$\\mid$ we used the MD trajectory of A-DNA where two main ribose conformations are divided more distinctly.\nFigure \\ref{A-pyramid-correlations} shows the behaviour of the bonds in pyramid \\{P(1)P(2)C3'C1'\\} containing\nthe sugar ring. As we expected (see \\ref{choice-of-pyramid}), only the length\n$\\mid$C1'P(2)$\\mid$ unambiguously correlates with $\\tau$, therefore we assigned a double-well potential to it.\n\\begin{figure}\n\\includegraphics[width=0.75\\linewidth] {fig17.eps}\n\\caption {Time dependence of the distances $\\mid$C1'P(2)$\\mid$, $\\mid$P(1)P(2)$\\mid$ and $\\mid$P(1)C1'$\\mid$ (see fig.~\\ref{valence-bonds-angles}) in comparison\nwith the pseudorotational angle $\\tau$ for the 4th nucleotide of the Dickerson-Drew dodecamer in A form in the mixture of ethanol\nand water (85:15) (parm99). Only the length $\\mid$C1'P(2)$\\mid$ correlates with the pseudorotation angle $\\tau$ sufficiently strongly.\n}\n\\label{A-pyramid-correlations}\n\\end{figure}\n\nIn table~\\ref{table-pyramid} we list the mean lengths and the corresponding rigidities for the distances $\\mid$P(1)C1'$\\mid$, $\\mid$C1'P(2)$\\mid$ and $\\mid$P(1)P(2)$\\mid$,\nobtained from MD trajectories of A- and B-DNA forms.\n\n\\begin{table}\n\\caption{\nDistances $\\mid$P(1)C1'$\\mid$, $\\mid$C1'P(2)$\\mid$ and $\\mid$P(1)P(2)$\\mid$ (see fig.~\\ref{valence-bonds-angles}) connected with sugar repuckering.\nWe list characteristic and uncharacteristic values for these bonds and their rigidities obtained by Boltzmann inversion\nmethod from MD trajectories of A-DNA and B-DNA. In parentheses, we give the part\nof population (in per cent) near the characteristic and uncharacteristic values. In the cases when the division\nof the histogram into two peaks was difficult (the distances $\\mid$C1'P(2)$\\mid$ and $\\mid$P(1)P(2)$\\mid$ in B-DNA demonstrate one broad\noften non-symmetrical peak), we list simple means over nucleotides.}\n\\label{table-pyramid}\n\\begin{tabular}{ccc}\n\\hline\n & $l_{A-Gl}$ & A \\\\\n\\hline\n$\\mid$P(1)C1'$\\mid$, & & \\\\\n$l$, $\\mbox{\\AA}$ & 5.42 & 5.4$\\pm$0.34 (85\\%) \/\/ 5.0$\\pm$0.3 (15\\%) \\\\\n$K_l$, $\\frac{kcal}{mol \\cdot \\mbox{\\AA}^2}$ & & 19$\\pm$9 \/\/ 28 $\\pm$6 \\\\\n\n\\hline\n$\\mid$C1'P(2)$\\mid$, & & \\\\\n$l$, $\\mbox{\\AA}$ & 4.59 & 4.8$\\pm$0.2 (64\\%) \/\/ 4.3$\\pm$0.3 (36\\%) \\\\\n$K_l$, $\\frac{kcal}{mol \\cdot \\mbox{\\AA}^2}$ & & 55$\\pm$17 \/\/ 27.4 $\\pm$8 \\\\\n\n\\hline\n$\\mid$P(1)P(2)$\\mid$, & & \\\\\n$l$, $\\mbox{\\AA}$ & 5.63 & 6.14$\\pm$0.8 (75\\%) \/\/ 6.9$\\pm$0.4 (25\\%) \\\\\n$K_l$, $\\frac{kcal}{mol \\cdot \\mbox{\\AA}^2}$ & & 3.8$\\pm$1 \/\/ 18.4$\\pm$8 \\\\\n\n\\hline \\hline\n\n & $l_{B-Gl}$ & B \\\\\n\\hline\n$\\mid$P(1)C1'$\\mid$, & & \\\\\n$l$, $\\mbox{\\AA}$ & 4.89 & 5.35$\\pm$0.4 (84\\%) \/\/ 5.0$\\pm$0.35 (16\\%) \\\\\n$K_l$, $\\frac{kcal}{mol \\cdot \\mbox{\\AA}^2}$ & & 15$\\pm$5 \/\/ 19.5 $\\pm$5 \\\\\n\n\\hline\n$\\mid$C1'P(2)$\\mid$, & & \\\\\n$l$, $\\mbox{\\AA}$ & 3.67 & 4.4$\\pm$0.4 \\\\\n$K_l$, $\\frac{kcal}{mol \\cdot \\mbox{\\AA}^2}$ & & 15 $\\pm$3 \\\\\n\n\\hline\n$\\mid$P(1)P(2)$\\mid$, & & \\\\\n$l$, $\\mbox{\\AA}$ & 6.54 & 6.8$\\pm$0.6 \\\\\n$K_l$, $\\frac{kcal}{mol \\cdot \\mbox{\\AA}^2}$ & & 7 $\\pm$ 2 \\\\\n\n\\hline\n\n\\end{tabular}\n\\end{table}\n\nWe calculated the double-well potential of the bond C1'-P(2) by the method of \"relaxation\" (see the beginning of section~\\ref{section-AMBER}).\nWe used a fragment of a DNA strand between two atoms C5' (including both of them and their hydrogen atoms); from the base we kept only one\nnitrogen atom, the nearest to the backbone. The obtained potential for $\\mid$C1'P(2)$\\mid$ is shown in fig.~\\ref{fig-2wells-relax}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.85\\linewidth] {fig18.eps}\n\\caption {\nPotential of interaction between the grains C1' and P(2), obtained by method of \"relaxation\" (see the beginning of\nsection~\\ref{section-AMBER}). On the top: dependence of the energy of the \"relaxed\" fragment on the distance $\\mid$C1'P(2)$\\mid$.\nThe solid smooth curve is the potential used in the sugar CG model.\nOn the bottom: dependence of the pseudorotational angle $\\tau$ on $\\mid$C1'P(2)$\\mid$. At the distances in the area of the left\nminimum of the potential the ribose is in the south region (B-DNA), and in the area of the right minimum - in the north region \n(A-DNA). In the transition through the barrier, the pucker goes through the west region.\n}\n\\label{fig-2wells-relax}\n\\end{center}\n\\end{figure}\nThe discontinuity of the\nfunction at distance around 5\\AA\\ is because the torsion angles of the fragment pass into the region\ninaccessible for DNA double helix. The two minima of the potential are 4.184\\AA\\ and 4.75\\AA, the difference\nof energies in the minima - 0.7 kcal\/mol, the barrier height - 1.9 kcal\/mol, rigidities in the wells - 25 and 61 kcal\/mol correspondingly.\nComparing with the table~\\ref{table-pyramid}, we see that the analysis\nof the MD trajectory for A-DNA gives very good estimates for lengths and rigidities in both wells, while in B-DNA the position of the left minimum\nis overestimated, and rigidity - underestimated (exactly as one would expect).\n\nFrom table~\\ref{table-skeletal-angles} one can see that the \"valence\" and \"torsion\" angles on the backbone (...-C3'-P-C3'-P-...) do not depend\non sugar puckers.\n\n\\begin{table}\n\\caption{\"Valence\" and two \"torsion\" angles in the backbone (...-C3'-P-C3'-P-...) of a DNA strand.\nIn crystallographic (GLACTONE \\cite{1998-GlactoneGeorgia}) A- and B-DNA forms $A-Gl$ \u0438 $B-Gl$, as well as in MD simulations\nthe values of these angles are close. The order of atoms in the notations\nof the angles is their order along the chain direction (see fig.~\\ref{valence-bonds-angles}).}\n\\label{table-skeletal-angles}\n\\begin{tabular}{ccc}\n\\hline\n & $\\phi_{A-Gl}$ & A \\\\\n\\hline\n$\\angle$C3'PC3', & & \\\\\n$\\phi$, $^0$ & 110.6 & 110 $\\pm$14 \\\\\n$K_\\phi$, & & 1.2 $\\pm$0.5 \\\\\n 10$^{-2}$$\\frac{kcal}{mol\\cdot deg^2}$ & & \\\\\n\n\\hline\n$\\angle$C3'PC3'P, & & \\\\\n$\\phi$, $^0$ & 168 & 173 $\\pm$32 \\\\\n$K_{\\phi}$, & & 0.23 $\\pm$0.05 \\\\\n 10$^{-2}$$\\frac{kcal}{mol\\cdot deg^2}$ & & \\\\\n\n\\hline\n$\\angle$PC3'PC3', & & \\\\\n$\\phi$, $^0$ & 215 & 217$\\pm$30 \\\\\n$K_{\\phi}$, & & 0.28$\\pm$0.15 \\\\\n 10$^{-2}$$\\frac{kcal}{mol\\cdot deg^2}$ & & \\\\\n\n\\hline \\hline\n & $\\phi_{B-Gl}$ & B \\\\\n\\hline\n$\\angle$C3'PC3', & & \\\\\n$\\phi$, $^0$ & 130.2$^0$ & 104 $\\pm$ 13 \\\\\n$K_\\phi$, & & 1.4 $\\pm$0.6 \\\\\n 10$^{-2}$$\\frac{kcal}{mol\\cdot deg^2}$ & & \\\\\n\n\\hline\n$\\angle$C3'PC3'P, & & \\\\\n$\\phi$, $^0$ & 158 & 179 $\\pm$31 \\\\\n$K_{\\phi}$, & & 0.25$\\pm$0.09 \\\\\n 10$^{-2}$$\\frac{kcal}{mol\\cdot deg^2}$ & & \\\\\n\n\\hline\n$\\angle$PC3'PC3', & & \\\\\n$\\phi$, $^0$ & 224 & 211$\\pm$44 (90\\%) \/\/ 135$\\pm$27 (10\\%) \\\\\n$K_{\\phi}$, & & 0.12$\\pm$0.04 \/\/ 0.33$\\pm$0.11 \\\\\n 10$^{-2}$$\\frac{kcal}{mol\\cdot deg^2}$ & & \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nThe \"valence\" angles responsible for direction of the glycosidic bond relative to the backbone,\nas well as \"torsion\" angles for base pair opening and for rotation of base around the glycosidic bond are presented in table~\\ref{table-differing-angles}.\n\\begin{table}\n\\caption{\n\"Valence\" angles determining the direction of the glycosidic bond relative to the sugar-phosphate backbone\nin the sugar CG model C3'-C1'-N and P-C1'-N;\n\"torsion\" angles for base pairs openings C3'-P-C3'-C1' and C1'-C3'-P-C3';\nand the angle of rotation of base around the glycosidic bond in crystallographic (GLACTONE \\cite{1998-GlactoneGeorgia}) A- and B-DNA forms {\\it A-Gl} and {\\it B-Gl},\nas well as in MD simulations. The order of atoms in the notation of the angles is their order along the chain direction\n(see fig.~\\ref{valence-bonds-angles}).\n}\n\\label{table-differing-angles}\n\\begin{tabular}{ccc}\n\\hline\n & $\\phi_{A-Gl}$ & A \\\\\n\\hline\n$\\angle$C3'C1'N(1,9), & & \\\\\n $\\phi$, $^0$ & 110.7 & 112$\\pm$3(65\\%)\/\/145$\\pm$4(35\\%) \\\\\n$K_{\\phi}$, 10$^{-2} \\frac{kcal}{mol \\cdot deg^2}$ & & 2.5$\\pm$0.7 \/\/ 2$\\pm$1 \\\\\n\n\\hline\n$\\angle$PC1'N(1,9), & & \\\\\n$\\phi$, $^0$ & 83.6 & 84$\\pm$10(77\\%)\/\/104$\\pm$15(23\\%) \\\\\n$K_{\\phi}$, 10$^{-2} \\frac{kcal}{mol \\cdot deg^2}$ & & 2.4$\\pm$0.24 \/\/ 1.1 $\\pm$0.1 \\\\\n\n\\hline\n$\\angle$C3'PC3'C1', & & \\\\\n$\\phi$, $^0$ & 8.9 & 4$\\pm$23 (91\\%)\/\/31$\\pm$13(9\\%) \\\\\n$K_{\\phi}$, 10$^{-2} \\frac{kcal}{mol \\cdot deg^2}$ & & 0.44$\\pm$0.2 \/\/ 1.4$\\pm$1 \\\\\n\n\\hline\n$\\angle$C1'C3'PC3', & & \\\\\n$\\phi$, $^0$ & 7 & 13$\\pm$34 \\\\\n\n$K_{\\phi}$, 10$^{-2} \\frac{kcal}{mol \\cdot deg^2}$ & & 0.2$\\pm$0.1 \\\\\n\n\\hline\n\n$\\angle$C3'C1'- & & \\\\\n-N(1,9)C(6,8), & & \\\\\n$\\phi$, $^0$ & -51.4 & -32$\\pm$9 \\\\\n$K_{\\phi}$, 10$^{-2} \\frac{kcal}{mol \\cdot deg^2}$ & & 3.2$\\pm$0.1 \\\\\n\n\\hline\\hline\n & $\\phi_{B-Gl}$ & B \\\\\n\\hline\n$\\angle$C3'C1'N(1,9), & & \\\\\n $\\phi$, $^0$ & 141.5 & 146$\\pm$13 \\\\\n$K_{\\phi}$, 10$^{-2} \\frac{kcal}{mol \\cdot deg^2}$ & & 1.46$\\pm$0.6 \\\\\n\n\\hline\n$\\angle$PC1'N(1,9), & & \\\\\n$\\phi$, $^0$ & 87 &88$\\pm$12(32\\%)\/\/105$\\pm$14(68\\%) \\\\\n$K_{\\phi}$, 10$^{-2} \\frac{kcal}{mol \\cdot deg^2}$ & &1.7$\\pm$0.6 \/\/ 1.2$\\pm$0.35 \\\\\n\n\\hline\n$\\angle$C3'PC3'C1', & & \\\\\n$\\phi$, $^0$ & 49 & 3$\\pm$28(80\\%)\/\/22$\\pm$19(20\\%) \\\\\n$K_{\\phi}$, 10$^{-2} \\frac{kcal}{mol \\cdot deg^2}$ & & 0.3$\\pm$0.1\/\/0.64 $\\pm$0.2 \\\\\n\n\\hline\n$\\angle$C1'C3'PC3', & & \\\\\n$\\phi$, $^0$ & -26 & 11$\\pm$33 \\\\\n\n$K_{\\phi}$, 10$^{-2} \\frac{kcal}{mol \\cdot deg^2}$ & & 0.2$\\pm$ 0.1\\\\\n\n\\hline\n\n$\\angle$C3'C1'- & & \\\\\n-N(1,9)C(6,8), & & \\\\\n\n$\\phi$, $^0$ & -1.5 & -27 $\\pm$34 \\\\\n$K_{\\phi}$, 10$^{-2} \\frac{kcal}{mol \\cdot deg^2}$ & & 0.22$\\pm$ 0.06 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\section{Appendix B: Potentials of interactions between ions and between ions and DNA grains: derivation and parameters}\n\\label{AppendixB}\nThe parameters of the effective (solvent-mediated) potentials of interaction (\\ref{equation-ions-interaction}) for the pairs Cl$^-$-Na$^+$ and Cl$^-$-Cl$^-$\n(adopted from work \\cite{2002-Lyubartsev-DNA-ions-potentials-HNC}) are listed in table~\\ref{table-Na-Cl-interactions}.\nThe parameters of the potential of interaction of Cl$^-$ ions with phosphate grains\nCl$^-$-P$^-$ (from article \\cite{1999-Lyubartsev-DNA-ions-potentials-inverseMC}) are given in table~\\ref{table-Na-Cl-P-interactions},\ntogether with the parameters for Na$^+$-P$^-$ interactions.\n\\begin{table}\n\\caption{Parameters A, D$_{k}$, C$_{k}$, R$_{k}$ for potentials of interaction (\\ref{equation-ions-interaction}) between a sodium ion\nand a phosphate grain Na$^+$-P$^-$ and between a chlorine ion and a phosphate grain Cl$^-$-P$^-$ in the sugar CG DNA model.}\n\\label{table-Na-Cl-P-interactions}\n\\begin{tabular}{cccc}\n\\hline\nparameter &dimension & Na$^+$-P$^-$ & Cl$^-$-P$^-$ \\\\\n\\hline\n & & & \\\\\n ~A~ & kcal\/mol$\\cdot \\mbox{\\AA}^{12}$ & $8.50 \\cdot 10^{5}$ & $10.2 \\cdot 10^{7}$\\\\\n\\hline\n~$D_1$~ & & -1.25 & -0.05 \\\\\n~$D_2$~ & & 1.54 & 0.54 \\\\\n~$D_3$~ & kcal\/mol & -0.73 & -0.02\\\\\n~$D_4$~ & & 0.53 & 0.25\\\\\n~$D_5$~ & & -0.38 & 0\\\\\n\\hline\n~$C_1$~ & & 0.92 & 2.9 \\\\\n~$C_2$~ & & 5.00 & 2.0\\\\\n~$C_3$~ & $\\mbox{\\AA}^{-2}$ & 0.75 & 0.5\\\\\n~$C_4$~ & & 5.50 & 0.7\\\\\n~$C_5$~ & & 0.90 & 0\\\\\n\\hline\n~ R$_1$ ~ & & 3.65 & 5.4\\\\\n~ R$_2$ ~ & & 4.18 & 6.7\\\\\n~ R$_3$ ~ & \\mbox{\\AA} & 5.86 & 8.6\\\\\n~ R$_4$ ~ & & 6.70 & 8.9\\\\\n~ R$_5$ ~ & & 7.97 & 0\\\\\n\\hline\n ~$\\varepsilon$ ~ & & 80 & 80 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table}\n\\caption{Parameters A, D$_{k}$, C$_{k}$, R$_{k}$ for potentials of interaction (\\ref{equation-ions-interaction}) between two sodium ions Na$^+$-Na$^+$ and between a sodium ion and a chlorine ion Na$^+$-Cl$^-$ and between two chlorine ions Cl$^-$-Cl$^-$ in the sugar CG DNA model.}\n\\label{table-Na-Cl-interactions}\n\\begin{tabular}{ccccc}\n\\hline\n &dimension & Na$^+$-Na$^+$ & Na$^+$-Cl$^-$ & Cl$^-$-Cl$^-$\\\\\n\\hline\n & & & & \\\\\n ~A~ & $\\frac{kcal}{mol} \\cdot \\mbox{\\AA}^{12}$ & $15.0 \\cdot 10^{5}$ & $2.2 \\cdot 10^{5}$ & $4.0 \\cdot 10^{7}$\\\\\n\\hline\n~$D_1$~ & & -0.62 & -1.36 & -0.52\\\\\n~$D_2$~ & & 0.29 & 1.87 & 0.29\\\\\n~$D_3$~ & kcal\/mol & -0.55 & -0.27 & -0.07\\\\\n~$D_4$~ & & 0.16 & 0.276 & 0.06\\\\\n~$D_5$~ & & -0.25 & -0.073 & -0.022 \\\\\n\\hline\n~$C_1$~ & & 1.4 & 5.0 & 3.0 \\\\\n~$C_2$~ & & 4.0 & 2.0 & 4.0 \\\\\n~$C_3$~ & $\\mbox{\\AA}^{-2}$ & 0.7 & 1.5 & 3.5 \\\\\n~$C_4$~ & & 5.5 & 5.5 & 3.0 \\\\\n~$C_5$~ & & 1.5 & 5.0 & 3.0 \\\\\n\\hline\n~ R$_1$ ~ & & 3.40 & 2.80 & 5.00 \\\\\n~ R$_2$ ~ & & 4.75 & 3.62 & 6.35 \\\\\n~ R$_3$ ~ & \\mbox{\\AA} & 6.20 & 5.20 & 7.60 \\\\\n~ R$_4$ ~ & & 7.15 & 6.00 & 8.50 \\\\\n~ R$_5$ ~ & & 8.47 & 6.80 & 9.70 \\\\\n\\hline\n ~$\\varepsilon$ ~ & & 80 & 80 & 80 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nWe derived the potentials of interaction between ions and grains on bases B$_j$ from Na$^+$-Na$^+$,\nNa$^+$-Cl$^-$ and Cl$^-$-Cl$^-$ potentials obtained in \\cite{2002-Lyubartsev-DNA-ions-potentials-HNC}.\nThe procedure, which we assume to give qualitatively correct shape of the potential curves, was as follows.\nRadial distribution function (rdf) for any two charged particles in a solvent has two or three pronounced local maxima.\nThe first maximum corresponds to the direct contact of ions. It is absent for the like-charged particles.\nThe second maximum is at the distance between the particles across one solvent shell, the third maximum is behind the second solvent shell.\nWe supposed that the position of the first maximum mainly depends on the radii of interacting particles; while the position of the second\nand the third maxima - on the properties of solvent molecules (size, shape and dipole moment).\n\nTherefore, for every unlike-charged pair (Cl$^-$-B$_j^+$ or Na$^+$-B$_j^-$) we set the location of the first minimum of the effective potential\nas the sum of the radius of an ion (1.8\\AA\\ for Cl$^-$ and 1\\AA\\ for Na$^+$) and the radius of the grain B$_j$.\nFor like-charged pairs (Cl$^-$-B$_j^-$ or Na$^+$-B$_j^+$), one should add the thickness of solvation shell to this value.\nFor Na$^+$-Na$^+$ and Cl$^-$-Cl$^-$ interactions it equals about 1.6\\AA\\ \\cite{2002-Lyubartsev-DNA-ions-potentials-HNC},\nand we used this value for all our effective potentials of interactions between ions and grains of bases.\nThe second and the third minima of the potentials we put at 2.3\\AA\\ and 4.3\\AA\\ from the first minimum: at\nthe distances at which they are in all three potentials of interactions from \\cite{2002-Lyubartsev-DNA-ions-potentials-HNC}.\nFor interactions Na$^+$-B$_j^-$ we took the heights of the maxima and the depths of the minima relative to the Coulomb asymptotics\nthe same as in the potential for Na$^+$-Cl$^-$, and for interactions Na$^+$-B$_j^+$ - the same as in the potential for Na$^+$-Na$^+$.\nWe also accepted that the energy of interaction between the particles at the distance less by 0.5\\AA\\ than the position of the first minimum is 3kT.\nAnalogously, the potentials for the pairs Cl$^-$-B$_j^-$ and Cl$^-$-B$_j^+$ were patterned after the potentials between the pairs\nCl$^-$-Cl$^-$ and Cl$^-$-Na$^+$.\n\nAll the grains put on the groups NH$_2$ (B$_2$ on C, B$_2$ on A, B$_3$ on G) are positively charged (see table~\\ref{table-base-charge}).\nWe accepted for their radii the value $r_N$=1.6\\AA, $r_N=r_{N-H}+r_H$, where $r_{N-H}=1\\mbox{\\AA}$ - the distance between the centers\nof nitrogen and hydrogen, and $r_H=0.6\\mbox{\\AA}$ - van der Waals radius of hydrogen in the AMBER force field.\n\nThree of four grains put on the oxygen atoms (B$_3$ on C, B$_2$ and B$_3$ on T) are negatively charged, and we accepted for their radii\nthe value 1.66\\AA\\ - van der Waals radius of oxygen in the AMBER force field. The fourth grain on an oxygen, B$_2$ (G), carries a small\npositive charge because the dipole moment of base G has component along the axis C6-C8. Actually, the partial charge on atom O6 is negative,\n(-0.57e) in AMBER force field.\n\nAs a matter of fact, the main difference between the base pairs AT and GC is that the atoms of G looking into the major groove carry large negative charge, and the atoms of C - large positive charge. Namely (see fig.~\\ref{DNA-bases}), the atoms N$_7$ and O$_6$ of G attract Na$^+$ ions into the major groove in A-DNA, and stabilize them in this groove. Correspondingly, chlorine ions near DNA can be found beside atom N$_4$ on C (more exactly, beside hydrogen atoms connected with atoms N$_4$ and C$_5$). Contrary to this picture, bases A and T have almost zero component of electric dipole moment orthogonal to the axis C6-C8.\n\nTo model the above described feature of base G, we set the location of the first minimum of the potential for Na$^+$-B$_2$(O6) on G\nat 2.5\\AA, which corresponds to the radius of the grain 1.5\\AA, and to the absence of the solvation shell (sodium ions can come up close to\nthese grains as if they were negatively charged). The radius of the negatively charged grain B$_1$(C8) on G for interaction with sodium ions\nwas taken to be 1.3\\AA, the depth of the first minimum was shifted down by 1.3kT, and the depth of the second minimum - by 0.4 kT.\nTo not\nprevent the sodium ions from dwelling in the major groove, the radii of the grains B$_1$ on A (C8) and B$_1$ on T (C7) were chosen to\nbe 1.7\\AA\\ and 1.5\\AA\\ correspondingly. The above mentioned small radii of these three grains are maximum values at which sodium ions keep inside\nthe major groove in A-DNA.\n\nFor the grain B$_1$(C6) on C with large positive charge we set the radius $r_C=2.5\\mbox{\\AA}=r_{C-H}+r_{H4}$, where $r_{C-H}=1\\mbox{\\AA}$\n- the distance between the centers of carbon and hydrogen, and $r_{H4}=1.4\\mbox{\\AA}$ - van der Waals radius of hydrogen in the AMBER force field.\nThe same value, 2.5\\AA, was taken for the radius of the grain B$_3$(C2) on A, as well as for all the grains put on carbon atoms\n(B$_1$ on A, T, C, G and B$_3$ on A) in their potentials of interactions with chlorine ions.\n\nParameters $\\{A, D_{k}, C_{k}, R_{k} \\}$ of so obtained potentials (\\ref{equation-ions-interaction}) are listed in tables \\ref{table-Na-base-interactions1},\n\\ref{table-Na-base-interactions2}, \\ref{table-Cl-base-interactions1}, \\ref{table-Cl-base-interactions2}. The plots of some effective potentials used in\nthe CG DNA model are shown in figures \\ref{n2-Na-Cl-P-attraction} and \\ref{n2-Na-Cl-P-repulsion}.\n\\begin{table}\n\\caption{Parameters A, D$_{k}$, C$_{k}$, R$_{k}$ for potentials of interaction (\\ref{equation-ions-interaction}) between a sodium ion and\nbase grains Na$^+$-B$_j$ in the sugar CG DNA model. The case of Adenine and Thymine.}\n\\label{table-Na-base-interactions1}\n\\begin{tabular}{ccccc}\n\\hline\nparameter &dimension & Na$^+$-B$_1$ & Na$^+$-B$_2$ & Na$^+$-B$_3$ \\\\\n\\hline\n \\multicolumn{5} {c}{Adenine} \\\\\n\\hline\n$\\sqrt[12]{A}$ & $\\sqrt[12]{\\mathrm{kcal\/mol}} \\mbox{\\AA}$ & 2.43 & 4.0 & 3.256\\\\\n\\hline\n~$D_1$~ & & -1.0 & -1.237 & 1.11 \\\\\n~$D_2$~ & & 1.6 & 0.307 & 2.1\\\\\n~$D_3$~ & kcal\/mol & -0.4 & -0.3 & -0.298\\\\\n~$D_4$~ & & 0.2 & 0.115 & 0.13\\\\\n~$D_5$~ & & -0.1 & -0.08 & -0.0895\\\\\n\\hline\n~$C_1$~ & & 2 & 2 & 2.8 \\\\\n~$C_2$~ & & 4 & 4.5 & 4\\\\\n~$C_3$~ & $\\mbox{\\AA}^{-2}$ & 2 & 2 & 2 \\\\\n~$C_4$~ & & 7 & 7 & 7 \\\\\n~$C_5$~ & & 2 & 2 & 2 \\\\\n\\hline\n~ R$_1$ ~ & & 2.52 & 4.05 & 3.5\\\\\n~ R$_2$ ~ & & 3.6 & 5.27 & 4.7\\\\\n~ R$_3$ ~ & \\mbox{\\AA} & 4.9 & 6.53 & 5.8\\\\\n~ R$_4$ ~ & & 6.2 & 7.53 & 6.9\\\\\n~ R$_5$ ~ & & 7.4 & 8.53 & 7.8\\\\\n\\hline\n\\hline\n\\multicolumn{5} {c}{Thymine} \\\\\n\\hline\n$\\sqrt[12]{A}$ & $\\sqrt[12]{\\mathrm{kcal\/mol}} \\mbox{\\AA} $ & 3.86 & 2.366 & 2.366\\\\\n\\hline\n~$D_1$~ & & -1.4 & -1.0 & -1.0 \\\\\n~$D_2$~ & & 0.483 & 2.19 & 2.19\\\\\n~$D_3$~ & kcal\/mol & -0.5 & -0.298 & -0.298\\\\\n~$D_4$~ & & 0.104 & 0.132 & 0.132\\\\\n~$D_5$~ & & -0.073 & -0.0895 & -0.0895\\\\\n\\hline\n~$C_1$~ & & 3 & 3.42 & 3.42\\\\\n~$C_2$~ & & 5 & 4.5 & 4.5\\\\\n~$C_3$~ & $\\mbox{\\AA}^{-2}$ & 1.5 & 2.0 & 2.0\\\\\n~$C_4$~ & & 5 & 7 & 7\\\\\n~$C_5$~ & & 3 & 2 & 2\\\\\n\\hline\n~ R$_1$ ~ & & 3.9 & 2.66 & 2.66\\\\\n~ R$_2$ ~ & & 4.8 & 3.45 & 3.45\\\\\n~ R$_3$ ~ & \\mbox{\\AA} & 5.2 & 4.96 & 4.96\\\\\n~ R$_4$ ~ & & 6.46 & 6.1 & 6.1\\\\\n~ R$_5$ ~ & & 7.4 & 6.96 & 6.96\\\\\n\\hline\n\\hline\n ~$\\varepsilon$ ~ & & 80 & 80 &80\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table}\n\\caption{Parameters A, D$_{k}$, C$_{k}$, R$_{k}$ for potentials of interaction (\\ref{equation-ions-interaction}) between a sodium ion and\nbase grains Na$^+$-B$_j$ in the sugar CG DNA model. The case of Guanine and Cytosine.}\n\\label{table-Na-base-interactions2}\n\\begin{tabular}{ccccc}\n\\hline\nparameter &dimension & Na$^+$-B$_1$ & Na$^+$-B$_2$ & Na$^+$-B$_3$ \\\\\n\\hline\n \\multicolumn{5} {c}{Guanine} \\\\\n\\hline\n$\\sqrt[12]{A}$ & $\\sqrt[12]{\\mathrm{kcal\/mol}} \\mbox{\\AA}$ & 1.78 & 2.187 & 3.97\\\\\n\\hline\n~$D_1$~ & & -0.85 & -0.75 & -1.18 \\\\\n~$D_2$~ & & 2.149 & 1.5 & 0.304\\\\\n~$D_3$~ & kcal\/mol & -0.55 & -0.298 & -0.243\\\\\n~$D_4$~ & & 0.2 & 0.132 & 0.114\\\\\n~$D_5$~ & & -0.3 & -0.22 & -0.073\\\\\n\\hline\n~$C_1$~ & & 3 & 3.4 & 3.6 \\\\\n~$C_2$~ & & 4.5 & 4.5 & 4.5\\\\\n~$C_3$~ & $\\mbox{\\AA}^{-2}$ & 2 & 2 & 2 \\\\\n~$C_4$~ & & 7 & 7 & 5 \\\\\n~$C_5$~ & & 2 & 2 & 3 \\\\\n\\hline\n~ R$_1$ ~ & & 2.4 & 2.5 & 4.06\\\\\n~ R$_2$ ~ & & 3.29 & 3.29 & 5.27\\\\\n~ R$_3$ ~ & \\mbox{\\AA} & 4.8 & 4.6 & 6.53\\\\\n~ R$_4$ ~ & & 5.94 & 5.94 & 7.53\\\\\n~ R$_5$ ~ & & 6.9 & 6.8 & 8.53\\\\\n\\hline\n\\hline\n \\multicolumn{5} {c}{Cytosine} \\\\\n\\hline\n$\\sqrt[12]{A}$ & $\\sqrt[12]{\\mathrm{kcal\/mol}} \\mbox{\\AA} $ & 4.92 & 4.0 & 2.366\\\\\n\\hline\n~$D_1$~ & & -1.7 & -1.4 & -0.6 \\\\\n~$D_2$~ & & 0.283 & 0.31 & 2.147\\\\\n~$D_3$~ & kcal\/mol & -0.5 & -0.41 & -0.298\\\\\n~$D_4$~ & & 0.104 & 0.104 & 0.132\\\\\n~$D_5$~ & & -0.21 & -0.17 & -0.0895\\\\\n\\hline\n~$C_1$~ & & 1.5 & 1.5 & 3.42\\\\\n~$C_2$~ & & 5.5 & 4.5 & 4.5\\\\\n~$C_3$~ & $\\mbox{\\AA}^{-2}$ & 2.0 & 1.6 & 2.0\\\\\n~$C_4$~ & & 5.0 & 5.0 & 7.0\\\\\n~$C_5$~ & & 3.0 & 3.0 & 2.0\\\\\n\\hline\n~ R$_1$ ~ & & 4.94 & 4.03 & 2.66\\\\\n~ R$_2$ ~ & & 6.2 & 5.07 & 3.45\\\\\n~ R$_3$ ~ & \\mbox{\\AA} & 7.4 & 6.23 & 4.96\\\\\n~ R$_4$ ~ & & 8.46 & 7.53 & 6.1\\\\\n~ R$_5$ ~ & & 9.4 & 8.53 & 6.96\\\\\n\\hline\n\\hline\n ~$\\varepsilon$ ~ & & 80 & 80 &80\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table}\n\\caption{Parameters A, D$_{k}$, C$_{k}$, R$_{k}$ for potentials of interaction (\\ref{equation-ions-interaction}) between a chlorine ion and\nbase grains Cl$^-$-B$_j$ in the sugar CG DNA model. The case of Adenine and Thymine.}\n\\label{table-Cl-base-interactions1}\n\\begin{tabular}{ccccc}\n\\hline\nparameter &dimension & Cl$^-$-B$_1$ & Cl$^-$-B$_2$ & Cl$^-$-B$_3$ \\\\\n\\hline\n \\multicolumn{5} {c}{Adenine} \\\\\n\\hline\n$\\sqrt[12]{A}$ & $\\sqrt[12]{\\mathrm{kcal\/mol}} \\mbox{\\AA}$ & 5.7 & 3.2 & 5.7 \\\\\n\\hline\n~$D_1$~ & & -1.5 & -1.11 & -1.5 \\\\\n~$D_2$~ & & 0.29 & 2.1 & 0.29 \\\\\n~$D_3$~ & kcal\/mol & -0.245 & -0.298 & -0.245 \\\\\n~$D_4$~ & & 0.105 & 0.13 & 0.105 \\\\\n~$D_5$~ & & -0.07 & -0.0895 & -0.073 \\\\\n\\hline\n~$C_1$~ & & 3 & 2.8 & 3 \\\\\n~$C_2$~ & & 4 & 4 & 4 \\\\\n~$C_3$~ & $\\mbox{\\AA}^{-2}$ & 2 & 2 & 2 \\\\\n~$C_4$~ & & 4 & 7 & 4 \\\\\n~$C_5$~ & & 3 & 2 & 3 \\\\\n\\hline\n~ R$_1$ ~ & & 5.65 & 3.3 & 5.81 \\\\\n~ R$_2$ ~ & & 6.96 & 4.59 & 6.96\\\\\n~ R$_3$ ~ & \\mbox{\\AA} & 8.11 & 5.74 & 8.11\\\\\n~ R$_4$ ~ & & 9.11 & 6.74 & 9.11\\\\\n~ R$_5$ ~ & & 10.11 & 7.74 & 10.11\\\\\n\\hline\n\\hline\n\\multicolumn{5} {c}{Thymine} \\\\\n\\hline\n$\\sqrt[12]{A}$ & $\\sqrt[12]{\\mathrm{kcal\/mol}} \\mbox{\\AA} $ & 4.1 & 4.79 & 4.79\\\\\n\\hline\n~$D_1$~ & & -1.3 & -1.39 & -1.39 \\\\\n~$D_2$~ & & 2.1 & 0.317 & 0.317\\\\\n~$D_3$~ & kcal\/mol & -0.298 & -0.245 & -0.245\\\\\n~$D_4$~ & & 0.13 & 0.105 & 0.105\\\\\n~$D_5$~ & & -0.0895 & -0.070 & -0.070\\\\\n\\hline\n~$C_1$~ & & 2.8 & 3 & 3\\\\\n~$C_2$~ & & 4 & 4 & 4\\\\\n~$C_3$~ & $\\mbox{\\AA}^{-2}$ & 2 & 2 & 2\\\\\n~$C_4$~ & & 7 & 4 & 4\\\\\n~$C_5$~ & & 2 & 3 & 3\\\\\n\\hline\n~ R$_1$ ~ & & 4.12 & 4.76 & 4.76 \\\\\n~ R$_2$ ~ & & 5.46 & 6.12 & 6.12 \\\\\n~ R$_3$ ~ & \\mbox{\\AA} & 6.61 & 7.27 & 7.27 \\\\\n~ R$_4$ ~ & & 7.61 & 8.27 & 8.27 \\\\\n~ R$_5$ ~ & & 8.61 & 9.27 & 9.27 \\\\\n\\hline\n\\hline\n ~$\\varepsilon$ ~ & & 80 & 80 &80\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table}\n\\caption{Parameters A, D$_{k}$, C$_{k}$, R$_{k}$ for potentials of interaction (\\ref{equation-ions-interaction}) between\na chlorine ion and base grains Cl$^-$-B$_j$ in the sugar CG DNA model. The case of Guanine and Cytosine.}\n\\label{table-Cl-base-interactions2}\n\\begin{tabular}{ccccc}\n\\hline\nparameter &dimension & Cl$^-$-B$_1$ & Cl$^-$-B$_2$ & Cl$^-$-B$_3$ \\\\\n\\hline\n \\multicolumn{5} {c}{Guanine} \\\\\n\\hline\n$\\sqrt[12]{A}$ & $\\sqrt[12]{\\mathrm{kcal\/mol}} \\mbox{\\AA}$ & 4.6 & 3.2 & 3.2\\\\\n\\hline\n~$D_1$~ & & -1.36 & -1.11 & -1.11 \\\\\n~$D_2$~ & & 0.317 & 2.1 & 2.10\\\\\n~$D_3$~ & kcal\/mol & -0.245 & -0.298 & -0.298 \\\\\n~$D_4$~ & & 0.105 & 0.13 & 0.130\\\\\n~$D_5$~ & & -0.070 & -0.0895 & -0.0895\\\\\n\\hline\n~$C_1$~ & & 3 & 2.8 & 2.8 \\\\\n~$C_2$~ & & 4 & 4 & 4\\\\\n~$C_3$~ & $\\mbox{\\AA}^{-2}$ & 2 & 2 & 2 \\\\\n~$C_4$~ & & 4 & 7 & 7 \\\\\n~$C_5$~ & & 3 & 2 & 2 \\\\\n\\hline\n~ R$_1$ ~ & & 4.6 & 3.3 & 3.3\\\\\n~ R$_2$ ~ & & 5.96 & 4.62 & 4.59\\\\\n~ R$_3$ ~ & \\mbox{\\AA} & 7.11 & 5.77 & 5.74\\\\\n~ R$_4$ ~ & & 8.11 & 6.9 & 6.74\\\\\n~ R$_5$ ~ & & 9.11 & 7.77 & 7.74\\\\\n\\hline\n\\hline\n \\multicolumn{5} {c}{Cytosine} \\\\\n\\hline\n$\\sqrt[12]{A}$ & $\\sqrt[12]{\\mathrm{kcal\/mol}} \\mbox{\\AA} $ & 4.1 & 3.2 & 4.79\\\\\n\\hline\n~$D_1$~ & & -1.3 & -1.11 & -1.39 \\\\\n~$D_2$~ & & 2.1 & 2.1 & 0.317\\\\\n~$D_3$~ & kcal\/mol & -0.298 & -0.298 & -0.245\\\\\n~$D_4$~ & & 0.13 & 0.13 & 0.105\\\\\n~$D_5$~ & & -0.0895 & -0.0895 & -0.070\\\\\n\\hline\n~$C_1$~ & & 2.8 & 2.8 & 3\\\\\n~$C_2$~ & & 4 & 4 & 4\\\\\n~$C_3$~ & $\\mbox{\\AA}^{-2}$ & 2 & 2 & 2\\\\\n~$C_4$~ & & 7 & 7 & 4\\\\\n~$C_5$~ & & 2 & 2 & 3\\\\\n\\hline\n~ R$_1$ ~ & & 4.12 & 3.3 & 4.76\\\\\n~ R$_2$ ~ & & 5.46 & 4.59 & 6.12\\\\\n~ R$_3$ ~ & \\mbox{\\AA} & 6.61 & 5.74 & 7.27\\\\\n~ R$_4$ ~ & & 7.61 & 6.74 & 8.27\\\\\n~ R$_5$ ~ & & 8.61 & 7.74 & 9.27\\\\\n\\hline\n\\hline\n ~$\\varepsilon$ ~ & & 80 & 80 &80\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.99\\linewidth] {fig19.eps}\n\\caption{Effective potentials of attraction between ions and charged grains in the sugar CG DNA model.\nThe parameters of the potentials are listed in tables \\ref{table-Na-Cl-P-interactions}\nand \\ref{table-Na-base-interactions1}.\nThin dashed and dotted curves correspond to the Coulombic interaction between the charges (the last term in equation (\\ref{equation-ions-interaction})).}\n\\label{n2-Na-Cl-P-attraction}\n\\end{center}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.95\\linewidth] {fig20.eps}\n\\caption{ Effective potentials of repulsion between ions and charged grains in the sugar CG DNA model.\nThe parameters of the potentials are listed in tables \\ref{table-Na-Cl-P-interactions} and \\ref{table-Na-base-interactions1}.\nThin dashed and dotted curves correspond\nto the Coulombic interaction between the charges (the last term in equation (\\ref{equation-ions-interaction})).}\n\\label{n2-Na-Cl-P-repulsion}\n\\end{center}\n\\end{figure}\n\\cleardoublepage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nRecently, a new type of BPS vortex was found in $U(N)$ \ngauge theories \\cite{Hanany:2003hp,Auzzi:2003fs}. This is called non-Abelian vortex and\ncarries the non-Abelian charge\n${\\bf C}P^{N-1}=\\frac{SU(N)_{{\\rm C}+{\\rm F}}}{SU(N-1)_{{\\rm C}+{\\rm F}}\\times U(1)_{{\\rm C}+{\\rm F}}}$.\nReaders can find \ngood reviews in \\cite{reviews,Eto:2006pg} and references of related works therein.\nIn this talk\nwe are interested in studying interactions between {\\it non-Abelian} vortices which are non-BPS. \nThe non-BPS vortices are more natural than BPS ones in a sense that the BPS always requires\na fine tuning or supersymmetry. \nIt is well known that ANO vortices \\cite{Abrikosov:1956sx,Nielsen} in the\ntype I system feel an attractive force while those in the type II model feel a repulsive force\n\\cite{Gustafson:2000, Jacobs:1978ch,Bettencourt:1994kf,Speight:1996px}.\nSpecifically we are interested in the interactions between\nvortices with different internal orientations, which is the distinct feature from the ANO case \\cite{1stpaper}. \n\nThis talk is based on \\cite{Auzzi:2007wj} in collaboration with R.Auzzi and W.Vince.\n\n\n\\section{The model}\n\n\\subsection{A fine-tuned model}\nWe start with non-Abelian, $U(N)$, extension of the Abelian-Higgs model\n\\beq\n{\\cal L} =\n{\\rm Tr}\\left[\n- \\frac{1}{2e^2} F_{\\mu\\nu}F^{\\mu\\nu}\n+ \\mathcal{D}_\\mu H (\\mathcal{D}^\\mu H)^\\dagger\n- { \\frac{\\lambda^2 \\, e^2}{4}} \\left(v^2 {\\bf 1}_{N} - HH^\\dagger \\right)^2\n\\right].\n\\label{eq:Lag_NAH}\n\\eeq\nHere, for simplicity we take the same gauge coupling $e$ for both the $U(1)$ and $SU(N)$ groups, while $\\lambda^2 \\,\ne^2\/4$ is a scalar coupling and $v$ ($>0$) determines the Higgs VEV.\n$H$ is $N$ Higgs fields in the fundamental representation of $U(N)$.\nThe Higgs vacuum of the model is given\nby $HH^\\dagger = v^2 {\\bf 1}_N$.\nIt breaks completely the gauge symmetry, although a global color-flavor locking symmetry\n$SU(N)_{\\rm C+F}$ is preserved\n\\beq\nH \\to U_{\\rm G} H U_{\\rm F},\\quad U_{\\rm G} = U_{\\rm F}^\\dagger,\\quad\nU_{\\rm G} \\in SU(N)_{\\rm G},\\ U_{\\rm F} \\in SU(N)_{\\rm F}.\n\\eeq\nThe trace part ${\\rm Tr} H$ is a singlet under the color-flavor group and the traceless parts are\nin the adjoint representation.\nThe $U(1)$ and the $SU(N)$ gauge vector bosons\nhave both the same mass\n$M_{U(1)}=M_{SU(N)}=e \\, v$.\nThe $N^2$ real scalar fields in\n$H$ are eaten by the gauge bosons and the other $N^2$ (one singlet and\nthe rest adjoint) have same masses\n$M_{\\rm s} = M_{\\rm ad} = \\lambda \\, e \\, v$.\nThe critical coupling $\\lambda=1$ (BPS) allows an $\\mathcal{N}=2$ supersymmetric extension.\n\n\n\\subsection{Models with general couplings}\n\nA generalization of (\\ref{eq:Lag_NAH}) is to consider\ndifferent gauge couplings, $e$ for the $U(1)$ part and $g$ for the $SU(N)$ part,\nand a general quartic scalar potential\n\\beq\n{\\cal L}\n= {\\rm Tr} \\left[ - \\frac{1}{2g^2} \\hat F_{\\mu\\nu} \\hat F^{\\mu\\nu} - \\frac{1}{2e^2} f_{\\mu\\nu} f^{\\mu\\nu} + \\mathcal{D}_\\mu H (\\mathcal{D}^\\mu\nH)^\\dagger \\right] - V, \\label{flum}\n\\eeq\nwhere we have defined $\\hat F_{\\mu\\nu} = \\sum_{A=1}^{N^2-1} F_{\\mu\\nu}^A T_A$ and $f_{\\mu\\nu} = F_{\\mu\\nu}^0 T^0$ with\n${\\rm Tr}(T^AT^B) = \\delta^{AB}\/2$ and $T^0 = {\\bf 1}\/\\sqrt{2N}$\nThe scalar potential is:\n\\beq\nV \n=\\frac{\\lambda_g^2g^2}{4} {\\rm Tr} \\hat X^2 + \\frac{\\lambda_e^2e^2}{4}{\\rm Tr} \\left(X^0T^0 - v^2{\\bf 1}_N\\right)^2,\n\\label{eq:flum_pot}\n\\eeq\nwhere\n$HH^\\dagger = X^0T^0 + \\hat X$ and $\\hat X = 2\n\\sum_{A=1}^{N^2-1} \\left(H^{i\\dagger}T^AH_i\\right)T^A$.\nThe symmetries is same as the previous fine-tuned model~(\\ref{eq:Lag_NAH}). \nIn this model, the $U(1)$ and the $SU(N)$ vector bosons have different masses\n$M_{U(1)}=e \\,v, \\ M_{SU(N)}=g \\, v.$\nMoreover, the singlet part of $H$ has a mass $M_{\\rm s}$ different from that of the adjoint part\n$M_{\\rm ad}$ as\n$M_{\\rm s}=\\lambda_e \\, e \\, v, \\ M_{\\rm ad}=\\lambda_g \\, g \\, v$.\nFor the critical values $\\lambda_e=\\lambda_g=1$,\nthe Lagrangian again allows an $\\mathcal{N}=2$ susy extension.\n\n\n\n\\subsection{Vortex equations in the fine-tuned model}\n\nLet us make the\nfollowing rescaling of fields and coordinates:\n\\beq\nH \\rightarrow v H,\\quad\nW_\\mu \\rightarrow ev W_\\mu,\\quad\nx_\\mu \\rightarrow \\frac{x_\\mu}{ev}.\n\\label{eq:rescale}\n\\eeq\nThe masses of vector and scalar bosons are\nrescaled to\n\\beq\nM_{U(1)} = M_{SU(N)} = 1,\\qquad\nM_{\\rm s} = M_{\\rm ad} = \\lambda.\n\\eeq\n\nIn order to construct non-BPS non-Abelian vortex solutions,\nwe have to solve the equation of motion\nderived from the Lagrangian (\\ref{eq:Lag_NAH}),\n\\beq\n\\mathcal{D}_\\mu F^{\\mu\\nu} - \\frac{i}{2}\\left[H (\\mathcal{D}^\\nu H)^\\dagger - (\\mathcal{D}^\\nu H) H^\\dagger\\right] = 0,\n\\label{eq:eom1}\\\\\n\\mathcal{D}_\\mu\\mathcal{D}^\\mu H + \\frac{\\lambda^2}{4}\\left(1-HH^\\dagger\\right)H = 0.\n\\label{eq:eom2}\n\\eeq\nFrom now on, we restrict ourselves to static configurations\ndepending only on the coordinates $x^1,x^2$. Here we introduce a\ncomplex notation\n$z = x^1 + i x^2,\n\\partial = \\frac{\\partial_1-i\\partial_2}{2},\\ \nW = \\frac{W_1 - iW_2}2,\\ \n\\mathcal{D} = \\frac{\\mathcal{D}_1 - i\\mathcal{D}_2}2 = \\partial + iW$.\nInstead of the equation of motions itself, it might be better to study gauge invariant\nquantities. For that purpose let us define\n\\beq\n\\bar W (z,\\bar z) = - i S^{-1}(z,\\bar z)\\bar\\partial S(z,\\bar z),\\quad\nH(z,\\bar z) = S^{-1}(z,\\bar z) \\tilde H(z,\\bar z),\n\\label{eq:decomposition}\n\\eeq\nwhere $S$ takes values in $GL(N,{\\bf C})$ and it is in the fundamental representation of $U(N)$ while the gauge singlet\n$\\tilde H$ is an $N \\times N$ complex matrix. There is an equivalence relation $(S,\\tilde H) \\sim (V(z)S,V(z)\\tilde\nH)$, where $V(z)$ is a holomorphic $GL(N,{\\bf C})$ matrix with respect to $z$.\nThe gauge group $U(N)$ and the flavor symmetry act as follows\n\\beq\nS(z,\\bar z) \\to U_{\\rm G} S(z,\\bar z),\\quad\nH_0(z) \\to H_0(z) U_{\\rm F}.\n\\eeq\nAn important gauge invariant quantity is now constructed as\n\\beq\n\\Omega (z,\\bar z) \\equiv S(z,\\bar z) S(z,\\bar z)^\\dagger.\n\\eeq\nWith respect to the gauge invariant objects, the equations of motion are\n\\beq\n4 \\bar \\partial^2 \\left( \\Omega \\partial \\Omega^{-1} \\right) - \\tilde H \\bar\\partial \\left( \\tilde H^\\dagger \\Omega^{-1} \\right)\n+ \\bar\\partial \\tilde H \\tilde H^\\dagger \\Omega^{-1} = 0 ,\\qquad\\quad\n\\label{eq:eom_omega1}\\\\\n\\Omega \\partial \\left( \\Omega^{-1} \\bar \\partial \\tilde H \\right)\n+ \\bar \\partial \\left( \\Omega \\partial \\left( \\Omega^{-1} \\tilde H \\right)\\right)\n+ \\frac{\\lambda^2}{4} \\left( \\Omega - \\tilde H \\tilde H^\\dagger \\right) \\Omega^{-1} \\tilde H = 0.\n\\label{eq:eom_omega2}\n\\eeq\nThese equations must be solved with the\nboundary conditions for $k$ vortices\n$\\det \\tilde H \\rightarrow z^k,\\ \\Omega \\rightarrow \\tilde H \\tilde H^\\dagger$ as $z \\rightarrow \\infty$.\n\n\n\\subsection{BPS Limit}\n\nFor the later convenience, let us \nsee the BPS limit $\\lambda \\to 1$. \nIt can be done by just taking a holomorphic function $\\tilde H$ with respect to $z$ as\n\\beq\n\\tilde H = H_0(z).\n\\label{eq:aho}\n\\eeq\nThen the equations (\\ref{eq:eom_omega1}) and (\\ref{eq:eom_omega2}) reduce to the \nsingle matrix equation\n\\beq\n\\bar \\partial \\left( \\Omega \\partial \\Omega^{-1} \\right)\n+ \\frac{1}{4} \\left( {\\bf 1} - H_0 H_0^\\dagger \\Omega^{-1} \\right) = 0.\n\\label{eq:master}\n\\eeq\nThis is the master\nequation for the BPS non-Abelian vortex and the holomorphic matrix $H_0(z)$ is called the moduli matrix\n\\cite{Eto:2005yh,Eto:2006pg}. All the complex\nparameters contained in the moduli matrix are moduli of the BPS vortices. For example, the position of the vortices can\nbe read from the moduli matrix as zeros of its determinant $\\det H_0(z_i) = 0$. Furthermore, the number of vortices\n(the units of magnetic flux of the configuration) corresponds to the degree of $\\det H_0(z)$ as a polynomial with\nrespect to $z$. The classification of the moduli matrix for the BPS vortices is given in\nRef.~\\cite{Eto:2005yh,Eto:2006pg}.\n\nConsider $U(2)$ gauge theory. The\nminimal vortex is generated by\n\\beq\nH_0^{(1,0)} =\n\\left(\n\\begin{array}{cc}\nz-z_0 & 0 \\\\\n-b' & 1\n\\end{array}\n\\right),\\qquad\nH_0^{(0,1)} =\n\\left(\n\\begin{array}{cc}\n1 & -b \\\\\n0 & z-z_0\n\\end{array}\n\\right).\n\\label{eq:mm_single}\n\\eeq\n$z_0$ corresponds to the position of the vortex and\n$b$ and $b'$\nare the internal orientation. \nOne can extract the\norientation as the null eigenvector of $H_0(z)$ at the vortex position\n$z=z_0$ as\n\\beq\n\\vec\\phi^{~(1,0)} =\n\\left(\n\\begin{array}{c}\n1\\\\\nb'\n\\end{array}\n\\right)\n\\quad\\sim\\quad\n\\vec \\phi^{~(0,1)}\n=\n\\left(\n\\begin{array}{c}\nb\\\\\n1\n\\end{array}\n\\right).\\label{fundorient}\n\\eeq\nHere ``$\\sim$\" stands for an identification up to complex non zero factors: $\\vec \\phi \\sim \\lambda \\vec\\phi$,\n$\\lambda \\in {\\bf C}^*$, so that we have found\n${\\bf C}P^1$~\\cite{Eto:2005yh,Eto:2006pg}. We call\ntwo non-Abelian vortices with equal orientational vectors {\\it parallel}, while orthogonal orientational\nvectors {\\it anti-parallel}.\n\nArbitrary two vortices (the center of mass is fixed to be zero and the overall orientaion is fixed) \nis given by\n\\beq\nH_{0 \\ {\\rm red}}^{(1,1)} \\equiv \\left(\n\\begin{array}{cc}\nz-z_0 & -\\eta\\\\\n0 & z + z_0\n\\end{array}\n\\right).\\label{reducedpatch}\n\\eeq\nThe orientational vectors are then of the form\n\\beq\n\\vec\\phi^{~(1,1)}_1\\big|_{z=z_0} =\n\\left(\n\\begin{array}{c}\n1\\\\\n0\n\\end{array}\n\\right),\\qquad\n\\vec\\phi^{~(1,1)}_2\\big|_{z=-z_0} =\n\\left(\n\\begin{array}{c}\n\\eta\\\\\n-2 z_0\n\\end{array}\n\\right).\n\\label{eq:oris}\n\\eeq\n\n\\section{Vortex interaction in the fine-tuned model}\n\n\n\\subsection{$(k_1,k_2)$ coincident vortices}\n\nThe minimal winding solution in the non-Abelian gauge theory is a mere embedding of the ANO solution into the\nnon-Abelian theory. Embedding is also useful for another simple non-BPS configurations.\nLet us start with the moduli\nmatrix for a configuration of $k$ coincident vortices. The axial symmetry\nallows a reasonable ansatz for $\\Omega$ and $\\tilde H$\n\\beq\n\\Omega^{(0,1)} =\n\\left(\n\\begin{array}{cc}\n1 & 0\\\\\n0 & w(r)\n\\end{array}\n\\right),\\qquad\n\\tilde H^{(0,1)} =\n\\left(\n\\begin{array}{cc}\n1 & 0\\\\\n0 & f(r) z^k\n\\end{array}\n\\right).\n\\label{eq:mm_para}\n\\eeq\nWe call this ``$(0,k)$-vortex''.\nWhen $k \\ge 2$, it is possible that the ansatz (\\ref{eq:mm_para}) does not give the true solution (minimum of the energy) of\nthe equations of motion (\\ref{eq:eom_omega1}) and (\\ref{eq:eom_omega2}). This is because there could be repulsive\nforces between the vortices. With ansatz (\\ref{eq:mm_para}) we fix the positions of all the vortices at the origin by\nhand. The master equation (\\ref{eq:master}) is nevertheless still useful to investigate the\ninteractions between two vortices. The results are listed in Table \\ref{fig:spectrum}.\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{cc}\n\\begin{tabular}{c|cc}\n$\\lambda$ & $k=1$ & $k=2$\\\\\n\\hline\n0.8 & 0.91231 & 1.77407 \\\\\n0.9 & 0.95737 & 1.88936 \\\\\n1 & 1.00000 & 2.00000 \\\\\n1.1 & 1.04053 & 2.10655 \\\\\n1.2 & 1.07922 & 2.20944 \\\\\n\\end{tabular}\n&\\includegraphics[width=7cm]{spectrum2.eps}\n\\end{tabular}\n\\caption{{\\small Spectrum of the $(0,2)$ and $(1,1)$ coincident vortices.}}\n\\label{fig:spectrum}\n\\end{center}\n\\end{figure}\nFor $\\lambda = 1$, the masses are identical to integer values, up to $10^{-5}$ order, which are nothing but the winding\nnumber of the vortices.\n\n\nThere is another type of composite configuration which can easily be analyzed numerically\n\\beq\n\\Omega^{(1,1)} =\n\\left(\n\\begin{array}{cc}\nw_1(r) & 0\\\\\n0 & w_2(r)\n\\end{array}\n\\right),\\qquad\n\\tilde H^{(1,1)} =\n\\left(\n\\begin{array}{cc}\nf_1(r) z^{k_1} & 0\\\\\n0 & f_2(r) z^{k_2}\n\\end{array}\n\\right).\n\\label{eq:mm_anti_para}\n\\eeq\nThis ansatz corresponds to a configuration with $k_1$ composite vortices which wind in the first diagonal $U(1)$ subgroup\nof $U(2)$ and with $k_2$ coincident vortices that wind the second diagonal $U(1)$ subgroup.\nWe refer to these as a ``$(k_1,k_2)$-vortex''. The mass of a $(k_1,k_2)$-vortex is thus the\nsum of the mass of the $(k_1,0)$-vortex and that of the $(0,k_2)$-vortex.\n\nWe call the non-Abelian vortices in the fine-tuned model for\n$\\lambda < 1$ type I, while they will be called type II for $\\lambda>1$. From Fig.~\\ref{fig:spectrum}, we can see that \nin the type I case, the $(0,2)$-vortex is energetically preferred to the $(1,1)$-vortex, while in type II case the\n$(1,1)$-vortex is preferred. If the two vortices are separated sufficiently,\nregardless of their orientations, the mass of two well separated vortices is twice that of the single vortex. This mass\nis equal to the mass of the $(1,1)$-vortex. \n\n\n\\subsection{Effective potential for coincident vortices \\label{sect:eff_coinc}}\n\nThe dynamics of BPS solitons can be investigated by the so-called moduli approximation~\\cite{Manton:1981mp}. The\neffective action is a massless non-linear sigma model whose target space is the moduli space. \nIf the coupling constant $\\lambda$ is close to the BPS limit $\\lambda = 1$, we can still use the moduli\napproximation, to investigate dynamics of the non-BPS non-Abelian vortices \nby adding a potential of order $|1-\\lambda^2|\n\\ll 1$.\nTo this end, we write the Lagrangian\n\\beq\n\\tilde {\\cal L} = \\tilde {\\cal L}_{\\rm BPS}+\\frac{(\\lambda^2 - 1)}{4} \\left( {\\bf 1}_{N} - HH^\\dagger \\right)^2.\n\\label{eq:Lag_Hind}\n\\eeq\nWe get non-BPS corrections of order $O(\\lambda^2 - 1)$ by putting BPS solutions\ninto Eq.~(\\ref{eq:Lag_Hind}). The energy functional thus takes the following form\n\\beq\n{\\cal E}= 2 + (\\lambda^2 - 1) {\\cal V},\\quad\n {\\cal V} = \\frac{1}{8 \\pi} \\int dx^1dx^2 \\ {\\rm Tr} \\left({\\bf 1} - \\left|H_{\\rm\nBPS}(\\varphi_i)\\right|^2 \\right)^2\\label{eff gen}\n\\eeq\nwhere $H_{\\rm BPS}(\\varphi_i)$ stands for the BPS solution. \nWe have defined a reduced effective potential ${\\cal V}$ which is independent of $\\lambda$.\nThe first term corresponds to the mass of two BPS vortices and the second term is the deviation\nfrom the BPS solutions which is nothing but the effective potential we want.\n\nTo have the effective potential on the moduli space of coincident vortices, \nit suffices\nto consider only the matrix (\\ref{reducedpatch}) with turning off the relative distance $z_0$.\nIn order to evaluate it, we need to solve the BPS equations with\nan intermediate value of $\\eta$.\nBecause of the axial symmetry and the\nboundary condition at infinity\n$\\Omega \\rightarrow H_0(z)H_0^\\dagger(\\bar z)$,\n we can make an ansatz\n\\beq\n\\Omega^{(1,1)} = \\left(\n\\begin{array}{cc}\nw_1(r) & -\\eta e^{-i\\theta}w_2(r) \\\\\n-\\eta e^{i\\theta}w_2(r) & w_3(r)\n\\end{array}\n\\right).\n\\label{eq:omega_11}\n\\eeq\nThe advantage of the moduli matrix formalism is that only three functions $w_i(r)$ \nare needed and the formalism itself is gauge\ninvariant. \nThe effective potential can be obtained by plugging numerical solutions into Eq.~(\\ref{eff gen}). \nThe result is shown\nin Fig.~\\ref{fig:num_V}.\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=4cm]{V_eta.eps}\n\\caption{{\\small Numerical plots of the effective reduced potential ${\\cal V}(|\\eta|)$.}} \\label{fig:num_V}\n\\end{center}\n\\end{figure}\n\n\nThe type II effective\npotential has the same qualitative behavior as showed in the figure. It has a minimum at $|\\eta|=0$. This\nmatches the previous result that the $(1,1)$-vortex is energetically preferred to the $(2,0)$-vortex. The\ntype I effective potential can be obtained just by flipping the overall sign of\nthat of the type II case. Then the effective potential always takes a negative value, which is consistent\nwith the fact that the masses of the type I vortices are less than that of the BPS vortices. \nContrary to the type II case, the type I potential has a minimum at $|\\eta| = \\infty$, \nso that the $(2,0)$-vortex is preferred to the $(1,1)$ vortex.\n\n\n\\subsection{Interaction at generic vortex separation}\n\nNext we go on investigating the interactions of non-Abelian vortices in the $U(2)$ gauge group at generic\ndistances. We will again use the moduli space approximation.\nThe generic configurations are described by the moduli matrices in Eq.~(\\ref{reducedpatch}). \nBy putting the two vortices on the real axis, we can\nreduce $z_0$ to a real parameter $d$.\nSo $2d$ is the relative distance and $\\eta$ the relative orientation.\nNow let us study the effective potential as function of $\\eta$ and $d$.\nAs before, we first need the numerical solution to the BPS master\nequation. \nDespite the great complexity by broken axial symmetry, the moduli matrix formalism is a powerful tool\nand the relaxation method is very effective\nto solve the problem. \nOnce we get the numerical solution, the effective potential is\nobtained by plugging them into Eq.~(\\ref{eff gen}), see Fig.~\\ref{fig:effv_sep}. It for\nthe type II has the same shape, up to a small positive factor ($\\lambda^2-1$).\nThe potential forms a hill whose top is at $(d,|\\eta|)= (0,\\infty)$. It clearly shows that two vortices feel\nrepulsive forces, in both the real and internal space, for every distance and relative orientation. The minima of the\npotential has a flat direction along the $d$-axis where the orientations are anti-parallel $(\\eta=0)$ and along the $\\eta$\naxis at infinite distance $(d=\\infty)$.\nTherefore the anti-parallel vortices\ndo not interact.\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\includegraphics[width=3.5cm]{effv_sep02.eps}&\n\\includegraphics[width=3.5cm]{effv3d_ab.eps} &\n\\includegraphics[width=3.5cm]{effv3d_nab.eps}\n\\end{tabular}\n\\caption{{\\small \nLeft panel is the effective potential ${\\cal V}(\\eta,d)$.\nThe Abelian potential ${\\cal V}_e$ (middle) and the non-Abelian potential ${\\cal V}_g$ (right) for\n$\\gamma=1$.}}\\label{fig:effv_sep}\n\\end{center}\n\\end{figure}\nIn the type I case ($\\lambda < 1$) the effective potential is upside-down of that of the type II case. There is unique\nminimum of the potential at $(d,|\\eta|)= (0,\\infty)$. This means that attractive force works not only for the distance\nin real space but also among the internal orientations.\n\n\n\n\\section{Vortices with generic couplings}\n\nIn this section we sutudy the general model\ndefined in Eqs.~(\\ref{flum}) and (\\ref{eq:flum_pot}). \nWe have three effective couplings $\\gamma = g\/e,\\lambda_e,\\lambda_g$ \nafter the rescaling (\\ref{eq:rescale}). The masses of particles are rescaled as\n\\beq\nM_{U(1)} = 1,\\quad M_{SU(N)} = \\gamma,\\quad M_{\\rm s} = \\lambda_e,\\quad M_{\\rm ad} = \\gamma \\lambda_g.\\label{masses}\n\\eeq\nIn order to find the effective potential on the moduli space as before, we\nneed to clarify BPS configurations.\nThe moduli matrix in (\\ref{eq:aho}) is still valid, while\nthe master equation (\\ref{eq:master}) get a modification\n\\beq\n4\\bar\\partial \\left(\\Omega \\partial \\Omega^{-1} \\right) = \\Omega_0 \\Omega^{-1} - {\\bf 1}_N +\n(\\gamma^2 - 1) \\left( \\Omega_0 \\Omega^{-1} - \\frac{{\\rm Tr} \\left(\\Omega_0 \\Omega^{-1}\\right)}{N} {\\bf 1}_N \\right)\n\\label{eq:gene_master}\n\\eeq\nwhere $\\Omega = SS^\\dagger$ is same as before and $\\Omega_0 \\equiv H_0H_0^\\dagger$.\nIt turns out that the effective potential consists of the Abelian and the non-Abelian potentials\n\\beq\n{\\cal V}_{e}(\\eta,d;\\gamma) = \\int d\\tilde x^2\\ {\\rm Tr} (F_{12}^0T^0)^2,\\quad\n{\\cal V}_{g}(\\eta,d;\\gamma) = \\int d\\tilde\nx^2\\ {\\rm Tr} (\\hat F_{12})^2. \\label{potentials}\n\\eeq\nThe true potential is a linear combination of them\n\\beq\nV(\\eta,d;\\gamma,\\lambda_e,\\lambda_g) = (\\lambda_e^2 - 1) {\\cal V}_e(\\eta,d;\\gamma) + \\frac{\\lambda_g^2\n-1}{\\gamma^2} {\\cal V}_g(\\eta,d;\\gamma). \\label{eq:gene_effpot}\n\\eeq\n\n\n\\subsection{Equal gauge coupling $\\gamma=1$ revisited}\n\n\nThe effective potential with $\\gamma=1$ and $\\lambda = \\lambda_g=\\lambda_e$ in the left panel of\nFig.~\\ref{fig:effv_sep} should be now decomposed in the two potentials, see the middle and the right panels\nin Fig.~\\ref{fig:effv_sep}.\nIn the case with $\\lambda_e^2-1 > 0$ and $\\lambda_g^2-1>0$, the effective potential will have\nthe same qualitative behaviors like the reduced potentials in the Figs.~\\ref{fig:effv_sep}. \nThe figures shows how ${\\cal V}_e$ and ${\\cal V}_g$ behaves very differently. In particular, the\nAbelian potential is always repulsive, both in the real and internal space.\nThe non-Abelian potential is on the contrary sensitive on the\norientations.\nFig.~\\ref{fig:effv_sep} shows that it is repulsive for parallel vortices while it is\nattractive for anti-parallel ones. When the two scalar\ncouplings are equal, $\\lambda_e^2=\\lambda_g^2$, the two\npotentials exactly cancel for anti-parallel vortices.\n\nOf course, the true effective potential depends on $\\lambda_e$ and $\\lambda_g$ through the combination in\nEq.~(\\ref{eq:gene_effpot}). This indicates the interaction between non-Abelian vortices is quite rich in comparison\nwith that of the ANO vortices.\n\n\n\n\\subsection{Different gauge coupling $\\gamma \\neq 1$}\n\n\nWe now consider interactions between non-Abelian vortices with different gauge coupling $e\\neq g$ ($\\gamma \\neq 1$).\nIn Figs.~\\ref{fig:effv_e=2g} and \\ref{fig:effv_g13} we show two numerical examples for the reduced effective potentials\n${\\cal V}_e$, ${\\cal V}_g$\ngiven in Eq.~(\\ref{potentials}).\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\includegraphics[width=3cm]{effv_b0_g05.eps}\n&&\n\\includegraphics[width=3cm]{effv_b4_g05.eps}\n&&\n\\includegraphics[width=3cm]{effv_binf_g05.eps}\\\\\nanti-parallel ($\\eta=0$) && intermediate ($\\eta=4$) && parallel ($\\eta=\\infty$)\n\\end{tabular}\n\\caption{{\\small Effective potential with $\\gamma=1\/2$ vs. separation. (red, blue) = (${\\cal V}_e$, ${\\cal V}_g$). }}\n\\label{fig:effv_e=2g}\n\\ \\\\\n\\begin{tabular}{ccccc}\n\\includegraphics[width=3cm]{effv_b0_g13.eps}\n&&\n\\includegraphics[width=3cm]{effv_b04_g13.eps}\n&&\n\\includegraphics[width=3cm]{effv_binf_g13.eps}\\\\\nanti-parallel ($\\eta=0$) && intermediate ($\\eta=4$) && parallel ($\\eta=\\infty$)\n\\end{tabular}\n\\caption{{\\small Effective potential with $\\gamma=1.3$ vs. separation. (red, blue) = (${\\cal V}_e$, ${\\cal V}_g$). }}\n\\label{fig:effv_g13}\n\\end{center}\n\\end{figure}\nThese show that the qualitative features of ${\\cal V}_e$ and ${\\cal V}_g$ are basically the same as\nwhat is discussed\nin the equal gauge coupling case $(\\gamma=1)$.\nThe true effective potential in Eq.~(\\ref{eq:gene_effpot}) \ndepends on the three parameters $\\gamma$, $\\lambda_e$ and $\\lambda_g$. \nWe can have potentials which develop a global minimum at some finite non zero\ndistance, see\nFig.~\\ref{bound}\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=4.5cm]{effv_ghalf_minima.eps}\n\\caption{{\\small $\\gamma=1\/2$, $\\lambda_e=1.2$, $\\lambda_g=1.06$: From $\\eta=0$ (green) to $\\eta=7$ (blue) with\n$d=0\\sim5$ for each $\\eta$.}}\\label{bound}\n\\end{center}\n\\end{figure}\nThe figure shows the presence of a minimum around $d\\sim\n2.$ This kind of behavior\nhave not been found for the ANO type I\/II vortices and the possibility of bounded vortices really results from the\nnon-Abelian symmetry.\n\n\n\\section{Interaction at large vortex separation}\n\n\\subsection{Vortices in fine-tuned models $e=g$ and $\\lambda_e=\\lambda_g$ \\label{sec:ana_int_fine}}\n\n\nWe study an asymptotic forces between vortices\nat large separation, following Refs. \\cite{Speight:1996px}.\nWe need to find asymptotic behaviors around $(1,0)$-vortex\n\\beq\nH_0(z)^{(1,0)} =\n\\left(\n\\begin{array}{cc}\nz & 0\\\\\n0 & 1\n\\end{array}\n\\right),\\qquad\n\\vec\\phi_1^{(1,0)} =\n\\left(\n\\begin{array}{cc}\n1\\\\\n0\n\\end{array}\n\\right).\n\\eeq\nWe are\nlead to the well known asymptotic behavior of the\nANO vortex\n\\beq\nH_{[1,1]} = \\left( 1 + \\frac{q}{2\\pi}K_0(\\lambda r) \\right) e^{i\\theta},\\quad\n\\bar W_{[1,1]} = - \\frac{i}{2} \\left(\\frac{1}{r} - \\frac{m}{2\\pi}K_1(r) \\right) e^{i\\theta}, \\label{eq:asymW}\n\\eeq\nwhere $K_1 \\equiv - K'_0$ and\nwe have defined $H_{[1,1]}$ and $\\bar W_{[1,1]}$ as $[1,1]$ elements of\n$H$ and $\\bar W$ in Eq.~(\\ref{eq:decomposition}) with the $k=1$ ansatz (\\ref{eq:mm_para}).\n\nNext we treat the vortices as point particles in a linear field theory\ncoupled with a scalar source $\\rho$ and a vector current $j_\\mu$.\nTo linearize the Yang-Mills-Higgs Lagrangian, we choose a gauge such that\nthe Higgs fields is given by hermitian matrix\n$H =\n{\\bf 1}_2\n+ \\frac{1}{ 2}h^i\\sigma_i,\\ W_\\mu = \\frac{1}{ 2}w_\\mu^i\\sigma_i\n$ with $\\sigma=({\\bf 1}_2,\\vec\\sigma)$.\nwith all $h^a,w_\\mu^a$ are real.\nThen the quadratic part of the Lagrangian is\n\\beq\n{\\cal L}^{(2)}_{\\rm free} = \\sum_{a=0}^3\\left[\n- \\frac{1}{4} f_{\\mu\\nu}^a f^{a\\mu\\nu} + \\frac{1}{2} w^a_\\mu w^{a\\mu}\n+ \\frac{1}{2}\\partial_\\mu h^a \\partial^\\mu h^a - \\frac{\\lambda^2}{2}(h^a)^2\n\\right]\n\\eeq\nwith $f_{\\mu\\nu}^a \\equiv \\partial_\\mu w^a_\\nu - \\partial_\\nu w^a_\\mu$. We also take into\naccount the external source terms to realize the point vortex\n\\beq\n{\\cal L}_{\\rm source} = \\sum_{a=0}^3 \\left[ \\rho^a h^a - j_\\mu^a w^{a\\mu} \\right]. \\label{sources}\n\\eeq\nThe scalar and the vector sources should be determined so that the asymptotic behavior of the fields in\nEq.~(\\ref{eq:asymW}) are replicated. The solution of the equation of motion is\n\\beq\nh^0=h^3 = \\frac{q}{2\\pi} K_0(\\lambda r),&&\\\n\\nonumber \\rho^0=\\rho^3 = q \\delta(r),\\\\\n{\\bf w}^0={\\bf w}^3 = - \\frac{m}{2\\pi} \\hat{\\bf k} \\times \\nabla K_0(r),&&\\ {\\bf j}^0 = {\\bf j}^3= - m \\hat{\\bf k}\n\\times\\nabla \\delta(r)\\label{tobedoubled}\n\\eeq\nwhere $\\hat{\\bf k}$ is a spatial fictitious unit vector along the vortex world-volume. The vortex configuration with\ngeneral orientation is also treated easily, since the origin of the orientation is the Nambu-Goldstone mode associated\nwith the broken $SU(2)$ color-flavor symmetry\n$\nH_0 \\to H_0(z)^{(1,0)} U_{\\rm F},\\ \\vec\\phi_2 = U_{\\rm F}^\\dagger \\vec\\phi_1^{(1,0)}$.\nThe interaction between a vortex at ${\\bf x} = {\\bf x}_1$ with the orientation $\\vec\\phi_1$ and another vortex at\n${\\bf x} = {\\bf x}_2$ with the orientation $\\vec\\phi_2$ is given through the\nsource term and is summarized as\n\\beq\nV_{\\rm int} = - \\frac{\\left|{\\vec \\phi}^\\dagger_1{\\vec \\phi}_2\\right|^2}{\\left|{\\vec \\phi}_1\\right|^2\\left|{\\vec\n\\phi}_2\\right|^2} \\left( \\frac{q^2}{2\\pi} K_0(\\lambda r) -\\frac{m^2}{2\\pi} K_0(r) \\right), \\label{eq:eff_e=g}\n\\eeq\nwhere $r \\equiv |{\\bf x}_1 - {\\bf x}_2| \\gg 1$.\nWhen two vortices have parallel orientations, this potential becomes that of two ANO\nvortices \\cite{Speight:1996px}. On the other hand, the potential vanishes when their orientations are\nanti-parallel. This agrees with the numerical result found in the previous sections. In the BPS limit\n$\\lambda=1$ ($q=m$), the interaction becomes precisely zero.\n\n\n\\subsection{Vortices with general couplings}\n\nIt is quite straightforward to generalize the results\nof the previous section to the case of generic couplings.\nWe find\nthe total potential $V_{\\rm int}$\n\\beq\nV_{\\rm int} &=& \\frac{1}{2}\\left(\n-\\frac{(q^0)^2}{2\\pi} K_0(\\lambda_e r)\n+\\frac{(m^0)^2}{2\\pi} K_0(r) \\right)\\nonumber\\\\\n&+& \\left(\\frac{\\left|{\\vec \\phi}^\\dagger_1{\\vec \\phi}_2\\right|^2}{\\left|{\\vec \\phi}_1\\right|^2\\left|{\\vec\n\\phi}_2\\right|^2} - \\frac{1}{2}\\right) \\left( -\\frac{(q^3)^2}{2\\pi} K_0(\\lambda_g \\gamma r)\n+\\frac{(m^3)^2}{2\\pi} K_0(\\gamma r) \\right).\n\\eeq\nAt large distance, the interactions between vortices are dominated by the particles with the lowest mass $M_{\\rm low}$.\nThere are four possible regimes $V_{\\rm int} =$\n\\beq\n\\left\\{\n\\begin{array}{ccll}\n- \\frac{(q^0)^2}{4\\pi} \\sqrt{\\frac{\\pi}{2 \\lambda_e r}}\n e^{- \\lambda_e r} & {\\rm for} & M_{\\rm low}=M_{\\rm s},& {\\rm Type} \\, {\\rm I}\\\\\n - \\left(\\frac{\\left|{\\vec \\phi}^\\dagger_1{\\vec \\phi}_2\\right|^2}{\\left|{\\vec \\phi}_1\\right|^2\\left|{\\vec\n\\phi}_2\\right|^2} - \\frac{1}{2}\\right)\n \\frac{(q^3)^2}{2\\pi} \\sqrt{\\frac{\\pi}{2\\lambda_g \\gamma r}}\n e^{- \\lambda_g \\gamma r} & {\\rm for} & M_{\\rm low}=M_{\\rm ad},& {\\rm Type} \\, {\\rm I}^* \\\\\n \\frac{(m^0)^2}{4\\pi} \\sqrt{\\frac{\\pi}{2 r}}\n e^{- r} & {\\rm for} & M_{\\rm low}=M_{U(1)},& {\\rm Type} \\, {\\rm II}\\\\\n \\left(\\frac{\\left|{\\vec \\phi}^\\dagger_1{\\vec \\phi}_2\\right|^2}{\\left|{\\vec \\phi}_1\\right|^2\\left|{\\vec\n\\phi}_2\\right|^2} - \\frac{1}{2}\\right)\n \\frac{(m^3)^2}{2\\pi} \\sqrt{\\frac{\\pi}{2 \\gamma r}}\n e^{- \\gamma r} & {\\rm for} & M_{\\rm low}=M_{SU(2)},&{\\rm Type} \\, {\\rm II}^*\n\\end{array}\n\\right. ,\\label{classif2}\n\\eeq\nbecause of $K_0(\\lambda r)\n\\sim \\sqrt{\\pi\/2\\lambda r} e^{-\\lambda r}$.\nThis generalizes the type I\/II\nclassification of Abelian superconductors. We have found two new categories, called type I$^*$ and type II$^*$, in\nwhich the force can be attractive or repulsive depending on the relative orientation. \nIn the type I$^*$ case the forces between parallel vortices are\nattractive while anti-parallel vortices repel each other. The type II$^*$ vortices feel opposite forces to the type\nI$^*$. \nThe result in Eq.~(\\ref{classif2}) is easily extended to the general case of $U(1) \\times SU(N)$.\nThis can be done by just thinking of the orientation vectors $\\vec \\phi$ as taking values\nin ${\\bf C}P^{N-1}$.\n\nIt may be interesting to compare these results with the recently\nstudied asymptotic interactions between non-BPS non-Abelian global vortices \\cite{global}.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nTogether with transformers, masked language modeling (MLM) has revolutionized the field of self-supervised learning (SSL) in natural language processing (NLP), which enables training of generalizable NLP models containing over one hundred billion parameters~\\cite{GPT-3}. The concept of MLM is quite intuitive, i.e., a portion of the data is removed and a model is trained to predict the removed content. Recently, significant progress has been made in masked image modeling to catch up to masked language modeling, where the masking mechanism is a key factor. Context encoder~\\cite{pathak2016context}, an inpainting-based masked image modeling (MIM) pioneer, proposes to use a random and fix-shaped mask; SiT~\\cite{sit} and BEiT~\\cite{BeiT} use random ``blockwise'' masks, where patches in the local neighbourhood are masked together (also called GMML: group mask model learning); MAE~\\cite{MAE} randomly masks out 75\\% patches of an image. Actually, masking mechanisms define the specific pretext task, i.e., what kind of information is to be exploited and what kind of information is to be predicted. Thus AttMask~\\cite{AttMask} studies the problem of which tokens to mask and proposes an attention-guided mask strategy to make informed decisions. ADIOS~\\cite{shi2022adversarial} takes one step further to ``learn to mask'' by adversarial training. \n\nAlthough promising performance has been achieved, there is still a large gap for masked autoencoding (MAE) between vision and language due to different signal natures. A sentence can be semantically decomposed into words, while the semantic decomposition of an image is not trivial to be obtained. To find a visual analogue of words, we investigate part-based image representation. Specifically, the real world is composed of objects, which consist of different parts. Therefore, part-based image representation is a fundamental image representation method that fits the inherent properties of objects~\\cite{DBLP:conf\/cvpr\/ChenMLFUY14, felzenszwalb2009object, fischler1973representation, he2021partimagenet, hinton2021represent}. For example, part-based Pictorial Stracture~\\cite{fischler1973representation} dominated the image representation field for several years in the early days of computer vision, and Deformable Part Model (DPM)~\\cite{felzenszwalb2009object} was also a milestone in image recognition and detection. Moreover, GLOM~\\cite{hinton2021represent} argues that the hierarchical representation with five levels (i.e., the lowest level, sub-part level, part level, object level, and scene level) would be a powerful image representation method in the future. To this end, we argue that \\textit{semantic parts would be a potential visual analogue of words}. With such visual analogue, \\textit{more controllable hints} can be built up to guide the learning of MAE, thus high-level visual representations can be well learned.\n\n\n\nIn this paper, we first propose a self-supervised semantic part learning method to obtain semantic parts for each image. Our insight is that the spatial information to reconstruct an image is highly correlated to the position of semantic parts. In particular, our part learning model consists of a ViT-based encoder together with an attention module that generates a class token and multiple attention maps, and a StyleGAN-based decoder that reconstructs the original image. The attention maps are optimized to provide spatial information, and the class token is integrated into the decoder via AdaIN to provide texture information. We find that the optimized attention maps can indicate part positions, and we conduct an argmax operation to obtain part segmentation maps. After that, we study how semantic parts can facilitate the learning of MAE. We design a masking strategy that varies from masking a portion of patches in each part to masking a portion of (whole) parts in an image. Such a design can gradually guide the network to learn various information, i.e., from intra-part patterns to inter-part relations. Extensive experiments on various vision tasks (e.g., linear probing, fine-tuning, semantic segmentation, and fine-grained recognition) show that SemMAE can learn better image representation by integrating semantics. \n\n\nOur contributions include 1) designing a self-supervised semantic part learning method that can generate promising semantic parts on multi-class datasets, i.e., ImageNet, and 2) verifying that semantic parts can facilitate the learning of MAE by proposing a semantic-guided masking strategy. While more importantly, we hope our attempts can provide insights for the community to study the visual analogue of words and unified vision and language modeling.\n\n\n\n\\section{Related work}\n\\label{sec:rw}\n\n\n\\textbf{Semantic part learning.} Part-based image representation is a fundamental image representation method that fits the inherent properties of objects~\\cite{DBLP:conf\/cvpr\/ChenMLFUY14, felzenszwalb2009object,fischler1973representation, he2021partimagenet,hinton2021represent}. However, due to the tremendous cost of labeling parts, there are still no large-scale datasets containing part labels. Thus previous works are mainly two-fold, i.e., unsupervised\/weakly-supervised part learning and few-shot part segmentation. Unsupervised\/weakly-supervised part learning methods~\\cite{choudhury2021unsupervised,krause2015fine,MACNN} propose to mine part information by leveraging spatial priors, the semantics of convolutional channels, or designing contrastive proxy tasks. Few-shot part segmentation methods~\\cite{baranchuk2021label,saha2021ganorcon,zhang2021datasetgan} mainly learn an additional classifier over pre-trained features that are trained by GAN, self-supervised contrastive learning, or denoising diffusion probabilistic modeling. Although promising results have been obtained, these models are designed to deal with fine-grained datasets, where all images belong to a single super-class (e.g., birds, cars, or human faces). It is much more challenging to solve the problem of unsupervised part learning on multi-class datasets such as ImageNet. With the development of ViT and self-supervised learning (SSL), some recent works show a potential solution. In particular, DINO~\\cite{DINO} and iBOT~\\cite{iBoT} have observed intuitive semantics in the ViT trained by their SSL methods, where the multi-head attention maps can somehow indicate different semantic parts of an object. Inspired by these works, we design a reconstruction-based method to further refine the attention maps learned by iBOT to obtain semantic parts on the ImageNet dataset.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{imgs\/mask1.pdf}\n\n\\caption{Comparison of different masking strategies. Detailed information for each compared model can be found in Section~\\ref{sec:rw} Masked Image Modeling.}\n\\label{fig:mask}\n\n\\end{figure}\n\n\n\\textbf{Masked image modeling.}\nInspired by the success of Masked Language Modeling (MLM)~\\cite{GPT-3,bert} in pre-training of the NLP field, Masked Image Modeling (MIM) has been proposed recently and exhibits promising potential for visual pre-training~\\cite{BeiT,iGPT,MAE,shi2022adversarial}. Existing works mainly study the problem of MIM from two directions, i.e., regression targets and masking strategies. In terms of regression targets, BeiT~\\cite{BeiT}, mc-BEiT~\\cite{mc-BEiT}, and PeCo~\\cite{PeCo} adopt tokens produced by VQ-VAE~\\cite{van2017neural} or its variants. MaskFeat~\\cite{MaskFeat} studies a broad spectrum of feature types and proposes to regress Histograms of Oriented Gradients (HOG) features of the masked content. MAE~\\cite{MAE} and SimMIM~\\cite{Simmim} argue that predicting RGB values of raw pixels by direct regression performs no worse than the patch classification approaches with complex designs. In this paper, we follow MAE~\\cite{MAE} to adopt the most simple and intuitive raw pixels regression. In terms of masking strategies, \nSiT~\\cite{sit}, MC-SSL0.0~\\cite{MC-SSL0.0} and BeiT~\\cite{BeiT} use a block-wise masking strategy, where a block of neighbouring tokens arranged spatially are masked. MAE~\\cite{MAE} and SimMIM~\\cite{Simmim} use random masking with a large masked patch size or a large proportion of masked patches. MST~\\cite{MST} and AttMask~\\cite{AttMask} propose to use attention maps to guide the masking strategy, where the former proposes to mask the nonessential regions to preserve crucial patches while the latter proposes to learn image representations with challenging tasks by masking the most attended tokens. Moreover, ADIOS~\\cite{shi2022adversarial} proposes to learn an optimal mask by adversarial training. Compared to these works, our SemMAE takes one step further and explicitly learn semantic parts to build reasonable hints for masked image modeling. Figure~\\ref{fig:mask} is an illustration of different masking strategies.\n \n\n\n\\section{Semantic-guided masked autoencoders}\n\nWe propose a Semantic-guided Masked Autoencoder (SemMAE) for self-supervised image representation learning with mask image modeling. The framework of SemMAE is shown in Figure~\\ref{fig:frame}, which consists of two key components, i.e., Semantic Part Learning (A) and Semantic-Guided Masking (B). First, given an image in Figure~\\ref{fig:frame} (a), we extract the class token in Figure~\\ref{fig:frame} (b) and patch tokens in Figure~\\ref{fig:frame} (c) by an iBOT-pretrained ViT. After that, we learn an embedding over the class token to obtain part tokens in Figure~\\ref{fig:frame} (d). We calculate the correlation of each part token to patch tokens to obtain attention maps in Figure~\\ref{fig:frame} (e), whose texture information is further removed by a large-kernel blur operation. The attention maps are optimized by a diversity constraint and a reconstruction task where the attention maps and the class token are fed into a StyleGAN-based decoder to control the spatial and texture information of the reconstructed image, respectively. Finally, we conduct argmax over the attention maps to obtain part segmentation maps in Figure~\\ref{fig:frame} (f) and used them to guide the mask generation for MAE. Specifically, we design a masking strategy that varies from masking a portion of patches in each part to masking a portion of (whole) parts in an image. Such a design can gradually guide the network to learn various information, i.e., from intra-part patterns to inter-part relations. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{imgs\/framework.pdf}\n\n\\caption{An illustration of the proposed SemMAE. (A) Semantic Part Learning. A ViT-based encoder takes as input an image in (a) and produces a class token in (b) and patch tokens in (c). Our attention module first learns to embed the class token into part tokens in (d) and then generates an attention map for each part token by calculating the correlation between the part token and patch tokens. As an objective function of the attention maps, our StyleGAN-based decoder learns to reconstruct the original image from attention maps with texture information from the class token. (B) Semantic-Guided Masking. We conduct argmax over the attention maps to obtain part segmentations in (f), which are used to guide the mask generation. During the training of the MAE, the masks vary from a portion of patches in each part to a portion of (whole) parts in an image.}\n\\label{fig:frame}\n\n\\end{figure}\n\n\\subsection{Semantic part learning}\n\\label{subsec:SemanticPartLearning}\nIn this subsection, we introduce our self-supervised semantic part learning method. Previous unsupervised\/weakly-supervised part learning methods are mainly designed to deal with single-class datasets (i.e., fine-grained datasets where images belong to the same superclass). Few methods are able to solve this problem under a multi-class scenario (e.g., ImageNet). While some recent works (i.e., DINO~\\cite{DINO} and iBOT~\\cite{iBoT}) on ViT-based self-supervised learning show that the multi-head attention maps in their model can somehow indicate different semantic parts of an object. In this work, we take advantage of semantics learned in iBOT and design a reconstruction task together with a diversity constraint to refine and obtain semantic parts. \n\nIn particular, given an image $\\mathbf{I}$, we first use an iBOT-pretrained ViT to extract its features, i.e., a class token $\\mathbf{F}_\\mathrm{c} \\in \\mathbb{R}^{C \\times 1}$ and patch tokens $\\mathbf{F} \\in \\mathbb{R}^{C \\times HW}$. Then, we embed the class token into $N$ part tokens $\\mathbf{F}_\\mathrm{p} \\in \\mathbb{R}^{C \\times N}$. The main idea of such embedding is to re-weight feature channels of the class token. As shown in previous methods~\\cite{MACNN}, feature channels may be corresponding to specific semantics and channel re-weighting can group channels with similar semantics together to obtain semantic part features. Thus we can obtain part tokens by:\n\\begin{equation} \\label{eqn:emb}\n\\mathbf{F}_\\mathrm{p}^{(i)} =\\mathbf{F}_\\mathrm{c} \\circ \\mathrm{sigmoid}(\\mathbf{W}_\\mathrm{c2}^{(i)}\\tanh(\\mathbf{W}_\\mathrm{c1}^{(i)} \\mathbf{F}_\\mathrm{c})),\n\\end{equation}\nwhere $i \\in [1,2,...,N]$, $\\mathbf{F}_\\mathrm{p}^{(i)}\\in \\mathbb{R}^{C \\times 1}$ is the $i^{th}$ column vector of $\\mathbf{F}_\\mathrm{p} \\in \\mathbb{R}^{C \\times N}$, $\\circ$ indicates hadamard product, $\\mathbf{W}_\\mathrm{c1}^{(i)}$ and $\\mathbf{W}_\\mathrm{c2}^{(i)}$ are embedding weights, $\\tanh(\\cdot)$ and $\\mathrm{sigmoid}(\\cdot)$ are activation functions. \n\nAfter that, we calculate the correlation of each part token to the patch token in each position, thus we can obtain attention maps, i.e., the possibility of a semantic part to appear in each position:\n\\begin{equation} \\label{eqn:att}\n\\mathbf{M} = \\mathbf{F}_\\mathrm{p} \\otimes \\mathbf{F} \\coloneqq \\mathrm{softmax}(\\mathbf{F}_\\mathrm{p}^{T}\\mathbf{W}_\\mathrm{p}^{T}\\mathbf{W}\\mathbf{F}),\n\\end{equation}\nwhere $\\mathbf{M} \\in \\mathbb{R}^{N \\times HW}$ denotes $N$ attention maps, $\\otimes$ indicates correlation function, which is implemented by $\\mathrm{softmax}(\\mathbf{F}_\\mathrm{p}^{T}\\mathbf{W}_\\mathrm{p}^{T}\\mathbf{W}\\mathbf{F})$ in our work. $\\mathbf{W}_\\mathrm{p}$ and $\\mathbf{W}$ are embedding matrixes.\n\nTo learn such multi-attention maps (i.e., to optimize the parameters in Equation~\\ref{eqn:emb} and Equation~\\ref{eqn:att}), we propose a reconstruction task. Our insight is that the spatial information to reconstruct an image is highly correlated to the position of semantic parts. Thus, we adopt a StyleGAN-based decoder to reconstruct the original image based on the spatial information from the attention maps and the texture information from the class token. To ensure the attention maps learn spatial information, we 1) remove texture information from the attention maps by conducting a large-kernel blur operation and 2) further feed the blurred attention maps to stacked convolutional layers. To integrate the texture information from the class token into the decoder, we use Adaptive Instance Normalization (AdaIN) operation, which is widely used to integrate texture\/style information:\n\n\\begin{equation} \\label{eqn:adain}\n[\\mathbf{F}_\\mathrm{d}]_i = \\mathrm{AdaIN}([\\mathrm{conv}(\\mathbf{M})]_i,\\mathbf{F}_\\mathrm{c})\\coloneqq [\\mathbf{W}_\\mathrm{s}\\mathbf{F}_\\mathrm{c}]_i\\frac{[\\mathrm{conv}(\\mathbf{M})]_i-\\mu([\\mathrm{conv}(\\mathbf{M})]_i)}{\\sigma([\\mathrm{conv}(\\mathbf{M})]_i)}+[\\mathbf{W}_\\mathrm{b}\\mathbf{F}_\\mathrm{c}]_i,\n\\end{equation}\nwhere each feature channel $[\\mathrm{conv}(\\mathbf{M})]_i$ is normalized separately, and then scaled and biased using the corresponding scalar components from the embedded class token $\\mathbf{F}_\\mathrm{c}$. $\\mathbf{F}_\\mathrm{d}$ denotes the convolutional feature in the decoder, $\\mathbf{W}_\\mathrm{s}$ and $\\mathbf{W}_\\mathrm{b}$ are embedding weights, $[\\cdot]_i$ denotes the $i^{th}$ feature channel, $\\mathrm{conv}(\\cdot)$ denotes convolutional layers, $\\mu(\\cdot)$ and $\\sigma(\\cdot)$ calculate the mean and variance values, respectively. The reconstructed image $\\hat{\\mathbf{I}}$ can be obtained by stacking convolutional can AdaIN layers:\n\\begin{equation} \\label{eqn:img}\n\\hat{\\mathbf{I}} = \\mathrm{conv}(\\mathrm{AdaIN}(\\mathrm{conv}(\\mathbf{F}_\\mathrm{d}),\\mathbf{F}_\\mathrm{c})).\n\\end{equation}\nWe use the Mean squared error (MSE) loss function to optimize such reconstruction task:\n\\begin{equation} \\label{eqn:loss1}\n\\mathcal{L}_{rec}(\\mathbf{I}, \\hat{\\mathbf{I}}) = \\frac{1}{HW}\\sum_{i, j}^{HW}(\\mathbf{I}(i,j) - \\hat{\\mathbf{I}}(i,j))^2.\n\\end{equation}\nMoreover, to obtain diverse multiple attention maps, we follow previous work~\\cite{zheng2019learning} and add a diversity constraint over attention maps:\n\\begin{equation} \\label{eqn:loss2}\n\\mathcal{L}_{div}(\\mathbf{M}) = \\frac{1}{N^2}(\\sum_{i \\neq j}(0 - \\frac{\\mathbf{m}_i\\mathbf{m}_j^T}{\\norm{\\mathbf{m}_i}_2\\norm{\\mathbf{m}_j}_2})^2 + \\sum_{i = j}(1 - \\frac{\\mathbf{m}_i\\mathbf{m}_j^T}{\\norm{\\mathbf{m}_i}_2\\norm{\\mathbf{m}_j}_2})^2),\n\\end{equation}\nwhere attention maps are optimized to be different from each other, $\\mathbf{m}_i$ and $\\mathbf{m}_j$ denotes the $i^{th}$ and $j^{th}$ attention map, respectively. The overall objective function can be denoted by:\n\\begin{equation} \\label{eqn:loss}\n\\mathcal{L} = \\mathcal{L}_{rec}(\\mathbf{I}, \\hat{\\mathbf{I}}) + \\lambda \\mathcal{L}_{div}(\\mathbf{M}),\n\\end{equation}\nwhere $\\lambda$ is the loss weight.\n \n\n \n\n \n\\subsection{Semantic-guided masking}\n\\label{subsec:SematicGuidedMasking}\n\n\nAfter finished semantic part learning, we move to the next stage, i.e., semantic-guided MAE training. Our informed masking strategy is based on the part information learned in Subsection~\\ref{subsec:SemanticPartLearning}. Specifically, we can obtain multiple attention maps by Equation~\\ref{eqn:att}, where each attention map $\\mathbf{m}\\in \\mathbb{R}^{H \\times W}$ indicates the possibility of the corresponding semantic part appearing in $H \\times W$ positions. Thus we conduct $\\mathrm{argmax}(\\cdot)$ operation over attention maps to obtain part segmentation, where each patch is classified into a particular semantic part. The patches in the same semantic part compose a visual analogue of words, which are semantically meaningful. To leverage such visual analogue of words for MAE training, a most intuitive way is to mask a portion of semantic parts and learn to predict the masked semantic parts by other parts. However, due to the learned semantic parts being coarse-grained (e.g., 6 parts for each image), we experimentally find that such a masking strategy makes the task too hard to effectively learn image representations.\n\nTo this end, we propose an easy-to-hard reconstruction task, which can provide reasonable hints (i.e., visible patches) for the model to predict the masked patches during the training process of the MAE. Specifically, at the beginning of the training process, we mask a portion of patches in each part, thus the masked patches can be predicted based on the visual patches that belong to the same semantic part. Such a design can facilitate the models to learn intra-part patterns. After that, we gradually mask all patches belonging to some parts and a portion of patches belong to the remaining parts. Finally, we mask all patches belonging to a portion of parts and predict the remaining patches belong to the other parts, where inter-part relations or visual reasoning ability can be learned.\n\nAlgorithm~\\ref{alg:SemMasking} shows the details to obtain the number of masked patches for each semantic part. First, we define two masking settings, i.e., 1) mask a portion of patches in each part and 2) random select some parts to mask (the whole part). The number of masked patches for each semantic part can be calculated for these two settings. After that, we introduce an interpolation hyper-parameter $\\alpha$. A small $\\alpha$ means the first setting dominates the masking strategy, and vice versa. $\\alpha$ is adjusted based on training iterations and keeps increasing during the training process. Finally, we random mask a certain number of patches based on the calculated masking number. \n\\setlength{\\textfloatsep}{5pt}\n\\begin{algorithm} [t]\n \\caption{Algorithm of Semantic-Guided Masking in a PyTorch-like style.}\n \\label{alg:SemMasking}\n \\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n \n \\begin{algorithmic}[1]\n \n \\REQUIRE $L$, $x$, $num\\_patches$, $mask\\_ratio$, $total\\_epoches$, $epoch$\n \n \\color{CadetBlue}\n \\# $L$: the number of patches per image.\n \n \\# $x \\in \\mathbb{R}^{L \\times C}$: the token embeddings of image patches.\n \n \\# $num\\_patches \\in \\mathbb{R}^{N \\times 1}$: the patch number of each part, where $N$ is the number of parts.\n \n \\# $mask\\_ratio$: the ratio of masked patches.\n \n \\# $total\\_epoches$: the number of pre-training epochs.\n \n \\# $epoch$: current epoch number.\n \n \\color{black}\n \n \n \\hspace*{\\fill}\n \n \n \n \n \n \n \n\n {\\color{CadetBlue}\\# mask a portion of patches in each part}\n \n \n \n \\STATE $num\\_mask1$ = $mask\\_ratio$ * $num\\_patches$ \\hspace{8ex} \n \n \n {\\color{CadetBlue}\\# randomly select some parts to mask (with tricks to ensure a fixed mask ratio)}\n \n \\STATE $shuffle\\_num\\_patches = shuffle\\_parts(num\\_patches)$\n \n \\STATE $marks$ = L * $mask\\_ratio$-cumsum($shuffle\\_num\\_patches$)+$shuffle\\_num\\_patches$\n \n \\STATE $marks\\_remains$ = where($marks$ < 0, 0, $marks$)\n \n \\STATE $num\\_mask2$ = where($marks\\_remains$ < $shuffle\\_num\\_patches$, $marks\\_remains$,\n $shuffle\\_num\\_patches$) \n \\hspace{8ex} \n \n \\STATE $num\\_mask2 = unshuffle\\_parts(num\\_mask2)$\n \n \n {\\color{CadetBlue}\\# adaptive masking by interpolating between num\\_mask1 and num\\_mask2}\n \n \\STATE $\\alpha = (\\frac{epoch}{total\\_epoches})^\\gamma$\n \n \\STATE $num\\_mask = (1-\\alpha) * num\\_mask1 + \\alpha * num\\_mask2$\n\n \\RETURN $num\\_mask$ {\\color{CadetBlue}\\# $num\\_mask \\in \\mathbb{R}^{N \\times 1}$: the number of patches to be masked in each part.}\n \\end{algorithmic}\n \n\\end{algorithm}\n\n\n\\section{Experiments}\n\n\\label{sec:Exp}\n\\subsection{Experiment setup}\n\n\n\\textbf{Semantic part learning.} As introduced in Section~\\ref{subsec:SemanticPartLearning}, we use ViT-small~\\cite{ViT} as our part learning encoder, which is pre-trained by a self-supervised method iBOT~\\cite{iBoT}. We follow iBOT~\\cite{iBoT} to learn 6 semantic parts for each image, as the head number of the multi-head attention in ViT-Small is 6. The size of the blur kernel is experimentally set to be 7, and the loss weight $\\lambda$ in Equation~\\ref{eqn:loss} is set to be 0.03. The experiment is performed on ImageNet-1k~\\cite{Imagenet} dataset. The parameters of the ViT-based encoder are fixed, and we only optimize the attention module and the StyleGAN-based decoder. Our model converges fast, which only takes 2 hours on one A100 GPU card.\n\n\\textbf{Semantic-guided MAE training.} We follow MAE~\\cite{MAE} and adopt an encoder-decoder structure to perform MIM. Our method is general for ViT backbones, while most experiments are conducted with a relatively small version, i.e., the original ViT-Base~\\cite{ViT}, due to the limitation of computation resources. We follow the most comment setting to optimize our model by AdamW~\\cite{AdamW} with a learning rate of 2.4e-3. The batch size is set to be 4096, and the weight decay is set to be 0.05. We use a cosine learning rate strategy~\\cite{loshchilov2016sgdr} with warmup~\\cite{goyal2017accurate}. The warmup number is set to be 40 epochs, and we pre-train our model for 800 epochs. For data augmentation, we only employ random horizontal flipping in our pre-training stage. The hyper-parameter $\\gamma$ in Algorithm~\\ref{alg:SemMasking} is experimentally set to be 2. Our model is trained on 16 A-100 GPUs for 3 days, and more details can be found in our code, which is in the supplemental material and will be made publicly released.\n\n\n\n\n\\begin{table}[!h]\n\\caption{Quantitative evaluation of the effectiveness of integrating semantic information for MAE.} \n\n\\renewcommand{\\arraystretch}{1.2}\n\\centering\n\\setlength{\\tabcolsep}{1.5mm}{ \n\\begin{tabular}{c|cc|cc}\n\\hline\n\\multirow{2}{*}{Setting} & \\multicolumn{2}{c|}{16$\\times$16 patch size} & \\multicolumn{2}{c}{8 $\\times$ 8 patch size} \\\\ \\cline{2-5} \n & \\multicolumn{1}{c|}{MAE~\\cite{MAE}} & SemMAE & \\multicolumn{1}{c|}{MAE~\\cite{MAE}} & SemMAE \\\\ \\hline\nLinear probing & \\multicolumn{1}{c|}{63.7} & \\textbf{65.0} & \\multicolumn{1}{c|}{66.8} & \\textbf{68.7} \\\\ \\hline\n\\end{tabular}}\n\n\\label{table:semantic}\n\\end{table}\n\n\n\\subsection{Semantic-guided MAE}\n\\textbf{The effectiveness of integrating semantic information.} We conduct experiments under two different settings (i.e., with a patch size of 16$\\times$16 and 8$\\times$8) to verify the effectiveness of integrating semantic information for training MAE. The results in Table~\\ref{table:semantic} show that integrating semantic information can bring 1.3\\% and 1.9\\% accuracy gains for linear probing, respectively. As we use 8$\\times$8 patch size to learn semantic parts, the coarse-grained patch (i.e., large patch size) in the pre-training stage would cause imprecise part segment and suppresses the benefits of semantic parts. To further study the impact of patch size for masked image modeling, we conduct fine-tuning experiments in Table~\\ref{tab:ft}. It can be observed that in SimMIM and original MAE, a larger patch size performs better; while in our SemMAE, more precise semantic parts with 8$\\times$8 patch size can significantly improve the performance. Thus in the following experiments, we adopt 8$\\times$8 patch size for SemMAE. It is notable that although using a smaller patch size, our parameters and computational cost do not increase during pre-training and linear probing as only 1\/4 patches in each image are used. Specifically, we leverage the learned attentions maps to remove 3\/4 patches that are most likely to be the background.\n\n\n\n\n\n\n\n\\begin{table}[t]\n\n\\renewcommand{\\arraystretch}{1.2}\n\\centering\n\n\\caption{The optimal patch size for different models.} \n\\vspace{0.1cm}\n\\setlength{\\tabcolsep}{1.5mm}{ \n\\begin{tabular}{c|c|ccc|c|c}\n\\cline{1-3} \\cline{5-7}\nModel & Patch size & Fine-tuning Acc.(\\%) & & Model & Patch size & Fine-tuning Acc.(\\%) \\\\ \\cline{1-3} \\cline{5-7} \n\\multirow{4}{*}{SimMIM~\\cite{Simmim}} & 32x32 & \\textbf{82.8} & & \\multirow{2}{*}{MAE~\\cite{MAE}} & 16x16 & \\textbf{83.26} \\\\ \\cline{2-3} \\cline{6-7} \n & 16x16 & 82.7 & & & 8x8 & 83.10 \\\\ \\cline{2-3} \\cline{5-7} \n & 8x8 & 82.1 & & \\multirow{2}{*}{SemMAE} & 16x16 & 83.34 \\\\ \\cline{2-3} \\cline{6-7} \n & 4x4 & 82.0 & & & 8x8 & \\textbf{84.50} \\\\ \\cline{1-3} \\cline{5-7} \n\\end{tabular}}\n\n\\label{tab:ft}\n\\end{table}\n\n\n\\textbf{A detailed study on masking strategies.} Once obtained semantic parts, a most intuitive way to leverage such visual analogue of words for MAE training is to mask a portion of semantic parts and make the model to predict the removed content. However, due to the learned semantic parts being coarse-grained (e.g., 6 parts for each image), we experimentally find that such a masking strategy makes the task too hard to effectively learn image representations. The results can be found in Table~\\ref{tab:mask}, where masking 75\\% parts cause 13.9\\% performance drops compared to random masking. Moreover, it can be observed that masking 75\\% patches per part achieves comparable results with random masking. The self-supervised learning task of masking 75\\% patches per part would encourage the model to learn local contexts\/intra-part patterns, and masking 75\\% parts would encourage the model to learn inter-part relations. Interestingly, we find that the former task can enable the model to further learn better image representation in the latter task. The results in Table~\\ref{tab:mask} show that our proposed adaptive masking strategy (i.e., varying from masking 75\\% patches per part to masking 75\\% parts gradually) with $\\gamma=2$ yields the best performance.\n\n\n\n\n \\begin{minipage}{\\textwidth}\n \\vspace{3mm}\n \\begin{minipage}[b]{0.34\\textwidth}\n \\centering\n \\includegraphics[width=0.86\\linewidth]{imgs\/ratio.pdf}\n \n \\captionof{figure}{The curves of $\\alpha$ and $\\gamma$.}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{0.7\\textwidth}\n \\centering\n \\begin{tabular}{c|c|c|c}\n\\hline\nMask strategy & $\\alpha$ & $\\gamma$ & Linear probing \\\\ \\hline\nRandom masking & -- & -- & 66.8 \\\\ \\hline\nMask 75\\% patches & 0 & -- & 66.5 \\\\ \\hline\nMask 75\\% parts & 1 & -- & 52.9 \\\\ \\hline\n\\multirow{6}{*}{Adaptive masking} & 1 $\\to$ 0 & 1 & 63.3 \\\\ \\cline{2-4} \n & \\multirow{5}{*}{0 $\\to$ 1} & 1\/3 & 66.2 \\\\ \\cline{3-4} \n & & 1\/2 & 67.3 \\\\ \\cline{3-4} \n & & 1 & 67.9 \\\\ \\cline{3-4} \n & & 2 & \\textbf{68.7} \\\\ \\cline{3-4} \n & & 3 & 68.6 \\\\ \\hline\n\\end{tabular}\n \\captionof{table}{Quantitative evaluation of different masking strategies.}\n\\label{tab:mask}\n \\end{minipage}\n \\end{minipage}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{imgs\/part.pdf}\n\\caption{Qualitative comparison of semantic part learning. Different color indicates different semantic parts, and it can be observed that our model can better separate different parts and the background with less noise. }\n\\label{fig:part}\n\\end{figure}\n\n\n\\subsection{Semantic part learning}\nWe first evaluate the effectiveness of our proposed semantic part learning method. Both qualitative and quantitative experiments are conducted. Note that most of the previous part learning models are designed for single-class datasets and cannot be effectively applied to ImageNet. iBOT~\\cite{iBoT} is not proposed for part learning, while the multi-head attention in their model achieves the state-of-the-art part learning performance on ImageNet. Figure~\\ref{fig:part} shows the qualitative comparison of our model and iBOT~\\cite{iBoT}, and it can be observed that our model can generate more complete semantic part segmentation maps where different parts and the background are better separated with less noise. Moreover, as there is no part segmentation ground truth, we conduct quantitatively evaluation in an indirect way by training SemMAE and analyzing ImageNet classification performance. The results in Table~\\ref{table:part} show that the semantic parts obtained by iBOT are not able to benefit the learning of MAE, while our semantic part learning methods can generate more precise part segmentation maps, which are vital to learning a better image representation.\n\n\\begin{table}[!h]\n\\renewcommand{\\arraystretch}{1.2}\n\\centering\n\n\\caption{Quantitative evaluation of semantic part learning in terms of classification accuracy(\\%).} \n\\vspace{0.1cm}\n\\setlength{\\tabcolsep}{1.5mm}{ \n\\begin{tabular}{cccc}\n\\hline\n Semantic parts for masking & Baseline (w\/o parts) & iBOT-initialized parts & Our learned parts \\\\ \\hline\n Linear probing Acc. (\\%) & 63.7 & 63.6 & \\textbf{65.0} \\\\ \\hline\n\\end{tabular}}\n\n\\label{table:part}\n\\end{table}\n\n\n\n\\begin{table}[!h]\n\\caption{System-level comparison on ImageNet-1k in terms of classification accuracy using ViT-Base as the encoder. Note that we list the best performance in previous papers with $224\\times224$ inputs, and some experiment settings (e.g., training epochs and patch size) may be different.} \n\\vspace{0.1cm}\n\\renewcommand{\\arraystretch}{1.2}\n\\centering\n\\setlength{\\tabcolsep}{1.5mm}{ \n\\begin{tabular}{lcccc}\n\\hline\n Method & Pre-train dataset & Pre-train epoches & Linear probing & Fintuning \\\\ \\hline\n \\multicolumn{5}{l}{ \\emph{Traning from scratch}} \\\\\n ViT$_{384}~$\\cite{ViT} & - & - & - & 77.9 \\\\ \n DeiT~\\cite{DeiT} & - & - & - & 81.8 \\\\ \n ViT~\\cite{MAE} & - & - & - & 82.3 \\\\ \\hline\n \\multicolumn{5}{l}{ \\emph{Contrastive-based SSL Pre-Training}} \\\\\nAttMask~\\cite{iBoT} & ImageNet-1K & 100 & 75.7 & -\\\\\nDINO~\\cite{DINO} & ImageNet-1K & 300 & 78.2 & 82.8 \\\\\nMoCo v3~\\cite{Mocov3} & ImageNet-1K & 300 & 76.5 & 83.2\\\\\niBOT~\\cite{iBoT} & ImageNet-1K & 1600 & 79.5 & 84.0\\\\ \\hline\n\\multicolumn{5}{l}{\\emph{MIM-based SSL Pre-Training}} \\\\\nBeiT~\\cite{BeiT} & ImageNet-1K & 800 & 56.7 & 83.2\\\\\nMAE~\\cite{MAE} & ImageNet-1K & 1600 & 68.0 & 83.6 \\\\\nSimMIM~\\cite{Simmim} & ImageNet-1K & 800 & 56.7 & 83.8 \\\\\n\\rowcolor{gray!20} SemMAE & ImageNet-1K & 800 & \\textbf{68.7} & \\textbf{84.5} \\\\ \\hline\n\\end{tabular}}\n\\label{table:ImageClassificationCompareSOTA}\n\\vspace{4mm}\n\\end{table}\n\n\n\\subsection{Compared with other methods on ImageNet} \nLinear probing and fine-tuning on ImageNet-1K classification dataset is the most common setting to evaluate SSL methods. We collect all competitive methods that report their results on ImageNet-1K dataset. For example, we do not include the related work MST~\\cite{MST} and ADIOS~\\cite{shi2022adversarial} as they evaluate their model on other benchmarks. Table ~\\ref{table:ImageClassificationCompareSOTA} shows the comparison of our model and previous models in terms of linear probing and fine-tuning. For a fair comparison, all experiments adopt the same input size, i.e., 224 $\\times$ 224 unless specified otherwise. Compared with ``training from scratch'', our SemMAE can significantly improve the performance for both linear probing and fine-tuning. For linear probing, our SemMAE outperforms the most competitive MIM-based methods by 0.8\\% even with fewer training epochs. For fine-tuning, our SemMAE achieves 84.5\\% top-1 classification accuracy, outperforming SimMIM\\cite{Simmim} and MAE\\cite{MAE} by 0.9\\% and 0.7\\% respectively. Moreover, our SemMAE can also surpass previous contrastive learning-based methods\\cite{DINO,Mocov3} for fine-tuning.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Downstream tasks}\n\n\\textbf{Fine-grained image classification.} Table ~\\ref{table:Fine-tuning results on fine-grained datasets} shows our results of transfer learning on fine-grained datasets. Our model can surpass the most competitive MAE~\\cite{MAE} with a clear margin, i.e., 0.3\\%, 0.6\\%, and 0.2\\% on the iNaturalists\\cite{inaturalist}, CUB-Bird\\cite{CUB}, and Stanford-Cars\\cite{StanfordCars} dataset, respectively. These results show the promising transfer ability of our SemMAE for downstream classification tasks. \n\n\\begin{table}[!h]\n\\caption{ Fine-tuning results on fine-grained datasets.} \n\\vspace{0.1cm}\n\\renewcommand{\\arraystretch}{1.2}\n\\centering\n\\setlength{\\tabcolsep}{6.0mm}{ \n\\begin{tabular}{lcccc}\n\\hline\n Method & iNa$_{19}$ & CUB & Cars \\\\ \\hline\nBeiT~\\cite{BeiT} & 79.2 & - & 94.2\\\\\nDINO~\\cite{BeiT} & 78.6 & - &93.0\\\\\niBoT~\\cite{MAE} & 79.6 & -& 94.3 \\\\\nMAE~\\cite{MAE} & 81.8 & 86.5 & 94.2 \\\\\n\\rowcolor{gray!20} SemMAE & \\textbf{82.1} & \\textbf{87.1}\\ & \\textbf{94.4}\\\\ \\hline\n\\end{tabular}}\n\\label{table:Fine-tuning results on fine-grained datasets}\n\\end{table}\n\n\\textbf{Semantic segmentation.}\nSemantic segmentation aims to assign a label to each pixel of the input image. We evaluate our SemMAE on the widely used semantic segmentation dataset ADE20K~\\cite{ADE}, which contains 25K images and 150 semantic categories. We follow the most common setting to use the task layer in UPerNet~\\cite{UperNet} and fine-tune the pre-trained ViT-Base model. We use the standard setting that pre-train a ViT-Base model with a patch size of $16\\times16$ and fine-tunes 160K iterations with a batch size of 16. Such an experiment can validate the transfer ability of our SemMAE for semantic segmentation. As shown in Table ~\\ref{table:semanticSegmentation}, SeMAE surpasses MAE by 0.2 (46.3 vs. 46.1) mIoU and outperforms the supervised pre-train model by 1.0 mIoU. These results show the promising transfer ability of our SemMAE for dense prediction visual tasks.\n\n\n\n\\begin{table}[!h]\n\\caption{ Semantic segmentation results on ADE-20K.} \n\\vspace{0.1cm}\n\\renewcommand{\\arraystretch}{1.2}\n\\centering\n\\setlength{\\tabcolsep}{6.0mm}{ \n\\begin{tabular}{lc}\n\\hline\n Method & mIoU \\\\ \\hline\n Supervised Pre-Training & 45.3 \\\\ \\hline\n \\multicolumn{2}{l}{ \\emph{Self-Supervised Pre-Training}} \\\\\nBeiT & 45.8 \\\\ \nMAE(800 epochs) & 46.1 \\\\ \n\\rowcolor{gray!20} SemMAE (Ours) & \\textbf{46.3} \\\\ \\hline\n\\end{tabular}}\n\\label{table:semanticSegmentation}\n\\end{table}\n\n\n\n\n\n\n\n\n\n \n \n \n \n\n\n\\section{Conclusion}\n\\label{sec:Con}\nIn this paper, we study the visual analogue of words and propose a semantic-guided masked autoencoder model to reduce the gap between masked language modeling and masked image modeling. Our proposed self-supervised semantic part learning method can generate promising semantic parts on ImageNet and we show that the learned semantic parts can facilitate the learning of MAE. Unlike the main-stream random masking strategy, our semantic-guided mask strategy can effectively integrate semantic information in the pre-training process. Extensive experiments with superior results show the effectiveness of our SemMAE. \n\n{\\bf Limitations}: due to the lack of part segmentation labels, the semantic part in our work is kind of coarse (e.g., 6 parts per image), making it not an ideal visual analogue of words yet. Moreover, using a small patch size increases the computational cost in the fine-tuning stage. In the future, we will 1) investigate finer-grained semantic parts (e.g., 20-30 parts per image) by few-shot part segmentation and 2) replace the widely obtained patch-based tokenization with part-based tokenization to further reduce the gap between vision and language modeling. \n\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}