DSR: Dynamical Surface Representation as Implicit Neural Networks for Protein
Authors: Daiwen Sun
He Huang
Yao Li
Xinqi Gong
Qiwei Ye
Venue: NeurIPS 2023 poster (2023)
Keywords: Protein molecular dynamics
Protein surface representation
Implicit neural representation
Signed distance function
Continuous time modeling

Abstract: We propose a novel neural network-based approach to modeling protein dynamics using an implicit representation of a protein’s surface in 3D and time. Our method utilizes the zero-level set of signed distance functions (SDFs) to represent protein surfaces, enabling temporally and spatially continuous representations of protein dynamics. Our experimental results demonstrate that our model accurately captures protein dynamic trajectories and can interpolate and extrapolate in 3D and time. Importantly, this is the first study to introduce this method and successfully model large-scale protein dynamics. This approach offers a promising alternative to current methods, overcoming the limitations of first-principles-based and deep learning methods, and provides a more scalable and efficient approach to modeling protein dynamics. Additionally, our surface representation approach simplifies calculations and allows identifying movement trends and amplitudes of protein domains, making it a useful tool for protein dynamics research. Codes are available at https://github.com/Sundw-818/DSR, and we have a project webpage that shows some video results, https://sundw-818.github.io/DSR/.

DSR: Dynamical Surface Representation as Implicit Neural Networks for Protein
Daiwen Sun', He Huang?.), Yao Li Xinqi Gong Qiwei Ye
Institute for Mathematical Sciences, School of Mathematics, Renmin University of China, Beijing. China
2 Beijing Academy of Artificial Intelligence, Beijing, China
School of Life Sciences, Tsinghua University, Beijing, China
sundaivenOruc..ed en, huanghe@baai.ac.com 1iyao arraol7@tsinnhua.oom org.
xinqigongèruc. eew.onngrueeeu..... cn, qwye@baai. ac.
Abstract
We propose a novel neural network-based approach to modeling protein dynamics using implicit representation of protein's 's surface in 3D and time. Our method utilizes the zero-level set of signed distance functions (SDFs) to represent protein surfaces, enabling temporally and spatially continuous representations of protein dynamics. Our experimental results demonstrate that our model accurately captures protein dynamic trajectories and can interpolate and extrapolate in 3D and time. Importantly, this is the first study to introduce this method and successfully model large-scale protein dynamics. This approach offers a promising alternative current methods, overcoming the limitations of first-principles plerriiipeesbbaedd and deep learning methods, and provides a more scalable and efficient approach to modeling protein dynamics. Additionally, our surface representation approach simplifies calculations and allows identifying movement trends and amplitudes of protein domains, making it useful tool for protein dynamics research. Codes are available at hhtpps//github.b com/Sundw-$ Sunddw&18/DSRR and we have a project webpage that shows some video results, https://sundw-88 Sttps/uuddww-t---iiiiiig 8.t...ttub..o/DSSSL
Introduction
Proteins, which are made up of chains of amino acids, are essential molecules in living organisms. Protein with specific three-dimensional structures responsible for a wide range of biological processes such as enzymatic reactions, transport of molecules, and structural support. However, proteins are not static objects in vivo, but constantly in motion, with individual amino acids vibrating. rotating, and shifting in response to the change of environment. The biological functions of proteins fulfilled by their dynamical behaviors. and researchers study the dynamics and functions of proteins by molecular dynamics(MD) simulations I 2
First-principle molecular dynamics the forces calculated from the electronic states by solving the Schrödinger equation (namely, density functional theory) to compute the coordinate trajectories with programs such VASP[4] and GPAW[5) However, the computational expense of MD typically increases exponentially in relation to the number of electronic degrees of freedom. In order to reduce the computational cost, empirical force field (classical force field) models with simplified functional forms are employed in MD simulation, such as 6MBER 71. CHARMM 9. and GROMOS 12]. which are designed to capture the bonded and non-bonded interactions between atoms. Besides, machine learning techniques can be employed to fit force fields or potential energy surfaces surfaces[]3]
These force field-based methods are much faster than ab initio methods without exhaustive QM
"Correspondence to: Xinqi Gongexiiggignnggruuc edu, en>, Qiwei Ye <quyedbaai. awweeebbaii.c.ccmm ac.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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calculations. However, the integration time step in MD cannot be too large to resolve the fastest motion in the system. for example the vibration of hydrogen-inven involvedd bonds whose timescales are as short as a few femtoseconds. This restricts MD from having more speed advantages. To address this problem, coarse-grained methods are being developed to perform molecular dynamies simulations faster and on larger systems by eliminating some unimportant degrees of freedom [17220]. Fast as it is, coarse-grained simulations are only able to reproduce molecular properties at low resolution, such as Rg", sacrificing the all-atom details. Given this perspective, existing molecular dynamics approaches suffer from either unaffordable computational costs or insufficient precision. Moreover, inherent limitations such the considerable random fluctuation of simulated trajectories and the consequential low signal-to- coonal-to-noise noise ratio pose significant challenges in detecting crucial protein functional motion. Even for the current most advanced Anton 3. under the parallel processing of 512 nodes, for large system of million atoms, only 100 microseconds of MD simulation can be performed each day (2])
These aforementioned methods give thorough under- standing of the natures of proteins by modeling pro- tein molecules as discrete particles. But when study- ing protein-protein interaction or protein-ligand bind- ing, knowledge about surfaces of proteins is what re- searchers need. To fulfill such demands, a continuous, high-level representation of molecular shape known as the molecular surface is proposed, as shown in Figure Based on the the concept of "key-lock pairs" empha- sizing complementarity of molecular surface the continuous shape with geometric and chemical features. One repre- sentative method, molecular surface interaction fingerprinting (MaSIF), a geometric deep learning method that captures fingerprints important for specific biomolecular interactions, enabling accurate and efficient analysis of protein-ligand complexes and potential applications in drug discovery and molecular design 23 244. Furthermore, some works investigated the application of end-to-end learning techniques for the analysis of protein surfaces and the identification of potential functional sites, such as drug-target binding sites 25).
The protein surface representa- tion. Showing the protein surface (PDB ID: IPGA) at different time steps in molecular dynamics trajectory.
Figure 1:
molecular surface models a protein
Previously, surface representation was mainly used for studying inter-molecular interactions or assisting molecular docking, and it had not been applied to analysing molecular dynamics simulation data. Inspired by the above works, propose Dynamical Surface Representation (abbreviated as DSR), a method for modeling protein dynamics by representing protein surfaces with implicit neural networks. Implicit neural representations [66] 27) are class of deep learning models that can represent complex geometric shapes using a continuous function without requiring an explicit surface representation. It's well-suited for modeling the dynamic shapes of proteins which have complex and varied conformations. Currently, deep learning applications in molecular dynamics (MD) mainly target small molecules, while our method can be applied to very large proteins.
In this paper, we explore the use of implicit neural representations for modeling the dynamic shapes of proteins. To be specifie, protein surfaces are represented as zero-level set of signed distance function (SDF) in DSR. The value of SDF is defined as the minimum distance from a point in Euclidean space to the surface of the object, where the positive value indicates that the point is outside the object, otherwise, it is inside. In addition, we introduce a time variable in DSR model to simulate the change of the protein surface over time, which is the dynamic simulation of the protein. In summary, we use the idea of implicit neural representation and use the neural network as the implicit neural representation of points on the 3D and time region, so as to achieve the purpose of modeling protein dynamic changes in continuous space and time domain.
In summary, our paper's main contribution is two-fold:
We build an implicit neural network that learns SDF from raw point cloud data 3D + time domain, which is a both temporally and spatially continuous protein dynamic representation.
We have achieved dynamic modeling of large proteins by using surface representations, which overcomes the limitations of previous atomic models or coarse-grained models that were restricted to small molecules or proteins.

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2 DSR
In this section, we present DSR, dynamical surface representation implicit neural networks for protein. We first introduce the definition of SDF, and then describe how we use SDF to model proteins. The architecture of the whole method is shown in Figure 2
2.1 SDF Representation
Let Ω = |-1, 1 be spatial domain, T = [-1,1] a temporal domain, and Mr be a 3D manifold embedded in S2 at time & E T. For any point (z, y, 2) I at time t, the fM. 22 - R is define
as
d(x, Me)
if outside Mr
(x)
if x belonging to Mr
(I)
Mc) x inside M₁
where d(x, M,) == yeM, d(x; y). d(x, v) = || yll2-
The zero-level set, which is the surface of the protein at time t, is presented by the points where [M, (.) 0. That is, the points on the protein surface at time t can be represented as set {ø [M, (x) = 0, æ 2].
2.2 Modeling Protein Dynamical Surface with SDFs
The protein surface representation is in the form of 3D shape. In order utilize the implicit function form of SDF, a typical approach is to partition the space into a grid, and then compute the SDF value for each grid point. However, for irregular objects. the accurate SDF values cannot be computed and thus approximation algorithms, such as the 8SSEDT algorithm, are usually employed. This method is associated with two limitations: firstly, protein surfaces are often intricate and rugged, and estimation errors can result in larger biases after modeling; secondly, the SDF calculation for dynamic models linearly dependent on time and cubically dependent on resolution, and pre-computed SDFs will incur high computational costs. Hence, an alternative strategy is used herein, whereby SDF is learned directly from the raw point clouds, rather than utilizing pre-computed SDF for supervised learning. This completely circumvents the need for pre-computing SDF, significantly reduces computational costs, and enhances computational efficiency. In the following, we will elaborate on how learn SDF directly from the raw point clouds.
For given input point cloud X' fer R time 1,with point normal vector data (optional), N = nn hie C R3, our objective is to find the optimal parameters & of an MLP f(z, t; f). where f: TR l+d > IR, that can accurately estimate a signed distance function to surface M, defined by the point cloud X' and normals N at time f.
The form of our loss function is as follows:
{()) = (x) + AE ([, -1) + || where x > 0 parameter, || || = || ||2 is the Euclidean 2-norm, and
(2)
(x(
1
Σ
+l (z., n2,l) -
(3)
7 GET
promotes the vanishing of f on X, and in the presence of normal data (i.e., T == 1), Va/ to the given normals N.
The first term in the summation part of Eq. 3 indicates that the SDF value of the surface points should be as small as possible. The second term represents the loss between the point normal vector and the ground truth.
The second term in Eq. called the Eikonal term promotes the gradients V Va to possess 2-norm of unity, i.e. |V 1, which is the gradient property SDFs. Note that this property is crucial to the loss design, allowing us to avoid using pre-computed SDF values for supervised learning. While its sufficiency has not been fully established in previous literature, we provide more comprehensive proof in the Appendix B
3

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The third term in Eq. 22 is latent code that we introduce into the model to specify different protein types, and its parameters are learnable. We claim that with sufficient data, can explore the latent space for greater generalization.
The global minimum of the loss in Eq. will be the solution of the Eikonal partial differential equation, i.c.
[>/(=))= 1,
which will also be a signed distance function, where f approaches 0 on X. with gradients N. The calculation of the expectation is based on some probability distribution D in 12 which will be introduced in Section 3.2
Throughout the optimization process, we did not the pre-computed SDF value supervise the training, which is an end-to-end learning architecture from raw point clouds data to SDF value. the next section, we will introduce the experiment setup in detail, including the datasets, implementation details, and model evaluation metrics.
manifold SDF
menifold loss
menifold points
Gradient Module
points
space
predicted SDF
difierant leval sets
points
DSR
Model
Figure 2: Overview of DSR. The upper part is the training process. First, the manifold and space points are sampled different times, added the time dimension and input into the DSR Model to obtain SDF, And then the Gradient Module is used to calculate the derivative of SDF with respect to (xx,) The lower part is the inference process. We first divide the space into grid based on the desired resolution, then adds the time dimension and inputs them into the DSR Model to predict the SDF. Finally, the Marching Cube algorithm is utilized to generate different level sets, with the zero-level set representing the protein dynamical surface.
3
DSR on large-scale proteins simulation data
3.1 Dataset.
500ns Trajectory Data. This data set currently contains two 100ns atomistic molecular dy- namies trajectories Abeta (40 residues), one wild type and one E22G, following the protocol the 500ns trajectories published in (29]. The simulation was performed at 310K in generalized Born implicit solvent. In this work, we used the first 2000 frames of wild type trajectories.
MDAnalysisData. MDAnalysisDataa collects set of data resources pertaining to computational biophysies, primarily focusing on molecular dynamics (MD) simulations and the structural and dynamic attributes of biomolecules. We used five of these datasets,
(1) AdK equilibrium dataset 30 of 4187 frames,
(2) AdK conformation transition 31 sampled with methods:
Dynamic importance sampling molecular dynamics (DIMS MD) of 102 frames 32
Framework Rigidity Optimized Dynamics Algorithm (FRODA) of 142 frames 33

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(3) A short MD trajectory of I-FABP (intestinal fatty acid binding protein) of 500 frames 34
(4) A trajectory of the NhaA membrane protein of 5000 frames (35]
(5) Two MD trajectories of YiiP membrane protein, the short one has 900 frames, and the long one has 9000 frames[36]. We opted for the longer one here.
For these 6 trajectories data, we use a fixed size 2000 to crop the frames due to the computational resources, and we keep the full length for those shorter than the cropping size.
GPCRmd. The GPCRmd (http://gpermd.org) [37] is an online platform that incorporates web- based visualization capabilities and shares data. The GPCRmd database includes at least one representative structure from each of the four structurally characterized GPCR classes. The GPCRmd platform holds more than 600 GPCR MD simulations from GPCRmd community and individual contributions. Each system was simulated for 500 ns in three replicates (total time 1.5 jus). We selected one system per GPCR family, and total of 4 trajectories were used to train the model, of which trajectory ids are 10792_trj jrrrr____. 81, 10840 _rj__r________ 87, 10912 [0912_trj__0) 95 and 15711 1_tr____r______ 791 respectively.
DRYAD _DD This dataset contains the trajectory data of 23 proteins simulated by Jumper 38] It is worth noting that the protein trajectories within this dataset exhibit significant oscillations, posing challenge for model training and resulting in high failure rate. For this reason, we screened 14 proteins with mean RMSD less than 1.0 for our experiments.
3.2 Implementation Details.
Problem setup. In this paper, we use PyMol to obtain the protein surface representation, the points and normal vectors, to train the model. The goal to obtain a protein surface dynamic model with respect to continuous space and time.
Architecture. For representing shapes we used level sets of MI.P [(x, 1; 0) = Rd x Rdd - R. called DSR Model, with 8 layers, each containing 512 hidden units, and a single skip connection from the input to the middle layer. We set the loss parameters to À 0.1, r = 1,a = 1e 3. And the model architecture is Figure 2 The activation function between fully connected layers is softplus activation: x + / In (1 + 8z where 3 = 100. The initial latent code vector z of size 192, were sampled from (O, 1.0 The Gradient Module was implemented by PyTorch Autograd to calculate (z)())
Distribution D. For all experiments, we utilize the distribution D as defined in the IGR for the expectation in Eq. (2). This distribution determined by taking the average of uniform distribution and sum of Gaussians that are centered at X' with a standard deviation equal to the distance to the k-th nearest neighbor (where we set 50).
Level set extraction. We extract the zero (or any other) level set of a trained model f(x, f; 0) using the Marching Cubes algorithm [I implemented in the python scikit-image package, which can use any large-size size grid to achieve any high resolution.
3.3 Evaluation Metrics.
We use three evaluation metries commonly used in 3D modeling to evaluate the similarity between two 3D shapes from three aspects: volume, distance, and normal vectors, which are Volumetric Intersection over Union (IoU), Chamfer distance and Normal Consistency (NC). These three metrics are normalized to a range of 0-1, providing a comprehensive evaluation of the model's 's performance from different perspectives.
IoU. IoU compares the reconstructed volume with the ground truth shape (higher better). For two arbitrary shapes A, B C R is attained by:
A B
IoU
AUB B
'The RMSD here was calculated between consecutive frames to reflect the smoothness of the protein trajectory.
5

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Chamfer distance. Chamfer distance is a standard metrics to evaluate the distance between two point sets X R (lower is better).
1 do (21,
d (21, + 2z (12)
(6)
2
where
da (Ki,
1
Σ
min #2ER2
|| 22
X1
NC NC evaluate estimated surface normals (higher is better). Normal consistency between two normalized unit vectors n, and n; is defined as the dot product between the two vectors. For evaluating the surface normals, given the object surface points and normal vectors: Xpred {(2,, n2.) ). and the ground truth surface points and normal vectors: Xgt = {(リ mÿ) }. the surface normal consistency between Xpred and gtt, denoted as I, is defined as:
T (X gt: Xpred)
mnj th o Xmas)
(8)
|x
jE|Xuul
where
(yjo Xpred = {(x n2))3) = arg min y 2z
iE Xpod
Results and Analysis
We design experiments from three aspects verify the ability to learn SDF from the point clouds, the ability to reconstruct the protein dynamical surface, and the generalization of the model in terms of temporal interpolation and extrapolation. To knowledge, dynamics of large proteins through surface representation, with no other methods currently available for comparison. Therefore, we conduct the following analysis to verify the learning ability of the DSR model and the effectiveness of protein surface representation. Additionally, to demonstrate that our model is doing something trivial, we compare our method with linear interpolation on interpolation tasks.
are the first model long-term
4.1 The Ability Learn SDF
We show the reconstruction results of training on Abeta protein in Figure which demonstrates the ability of our model to learn the signed distance field. The first row in the figure the reconstruction results of our model, and the second row is the ground truth. It can be seen that our model has the ability to reconstruct the dynamic shape of the protein at different moments, and can restore a clearer and more delicate shape than the ground truth. Furthermore, we show some videos of this protein on the webpage and discuss the effect of normal vectors on learning process in the Appendix D
t t= t= 5
t 10
t= 15
t 20
25
t 30
35
Figure 3: Given different times as input, the zero-level set visualization of the SDF value predicted by the model for protein abeta. The first row the reconstruction result of our model, and the second row is the ground truth.
Note that SDF represents the shortest distance (signed) from the surface of the object, which means that different level sets form surfaces certain distances from the manifold surface. As shown

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in Figure (a), the visualization of different level sets of SDF values predicted by our model is consistent with the meaning expressed by SDF. In addition, our model is continuous with respect to spatial and can use to reconstruct and predict for arbitrary space resolutions. Figure shows the reconstruction of protein different resolutions certain time. We also calculated three evaluation metrics, namely IoU, Chamfer distance and NC, which have values of 0.9361 #0.0265, 0.0009#0.0008 and 0.9967#0.001 respectively.
Resolation
5121
Figure 4: (a) The visualization of different level-set of the SDF predicted by our model; positive level sets are in red; negative ones are in blue; the zero level set, represents the approximated surface in white. (b) The reconstruction of a protein at different resolutions.
4.2 Reconstruction across Time
In this experiment, we validate the ability of our trained model to reconstruct the protein shapes in training frames (see T.. For data AdK equilibrium, NhaA, YiiP, and DRYAD_MD, we used the first 1500 frames of each trajectory and diluted them by ten times. That is, selecting 150 frames for each trajectory for training. Similarly, every second frame was selected within the first 80 frames for DIMS, every third frame was chosen within the first 120 frames for FRODA, and every fifth frame was selected within the first 400 frames for I-FABP.
As from the Table 1 the model can reconstruct well in many For example, DIMS, the loU value as high as 0.9316, which means that the shape reconstructed by the model is almost completely consistent with the ground truth the entire timeline.
Table I: Evaluation on the surface reconstruction across time.
loUt
_ist
NC1
loUt
Chamfer_dian
NC1
DIMS
0.9316+0.00 0070
0.0003+0.000
0.9671=0.005D
I-FABP
0.8907#0.00 0071
0.0005:0 0001
0.9382=0.0072
YliP
0.8425
0.000340.0001
0.873720.0164
FRODA
0.8318+0.000 0286
0.0004 0001
0.9098:0.0175
NahA
0.8265+0.0202
0.0005=0.0001
0.8372=0.0245
adk_equi
0.7526+0.0396
0.0020=0.000
0.7914=0.0308
protG
0.6580=0.0699
0.003640.000 0017
0.8287=0.0595
nt1
0.6342+0.124 1244
0.0066+0.000
0.8165=0.0911
bba
0.616340.
0.0057=0.0073
0.8158=0.1044
espa
0.6136:00 0.68
0.0074:0. 0036
0.8047=0.0527
T0765
0.5997#0.0977 0923
0.0039H0.0022
0.7991=0.0757
0.5920#0. 1056
0.0088+0.0055
0.8196=000691
T0855
0.5585+0.0899
0.0117=0.009
0.7804=0.0647
T0773
0.5289+0.0833
0.0123+0.009
0.7404=0.0632
T0816
0.528440. 1366
0.0083+0.0071
0.7368=0.0979
gpW
0.4836+0. 1253
0.0093+0.000 0072
0.7176=0.0905
0.460340. 1422
0.0094#0.006
0.6968=0.00 0896
0.98
hyp
0,4578#0
0.0099:0
0.7104=0.0873
T0769
0.4535+0.0871
0.0191+0.0101
0.7102=0.0738
T0771
0.4219H0.0711
0.011140.0073
0.6537+0.0491
We analyze the effect of RMSD model training quality, indicating that data containing high levels of noise and significant vibration pose challenges to model training. As illustrated in the Figure 5 model shows superior performance smoother and less noisy trajectory data.
the
4.3 Generalization Ability
Our model has generative capabilities to predict missing or future trajectories of a protein. Here we obtain different samples in two aspects, interpolation and extrapolation of time.
4.3.1 Interpolation
Our model enable the prediction of protein shape at any given time due to its temporal continuity. This is very useful when we only know the structures of two moments but not what the intermediate process During training sample selection, we extract one frame at intervals of several frames to evaluate the model's interpolation capability in the temporal space. To demonstrate that model
7

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68-
4a
Chertker daace
-08
0.8
006
08
9
15%
60
4.0
124
0.0 0.
-00
Chamioo dolance
Figure 5: The correlation between RMSD and three metrics was evaluated across various protein trajectories, revealing significant relationship between the metrics and RMSD, highlighting that smaller RMSD and system noise levels correspond to improved model performance and casier modeling of protein surface dynamics.
learns the protein's inherent dynamic variations and is not just performing trivial tasks, we compared it with linear interpolation, as shown in Figure even though there were no previous works for direct comparison. The evaluation results of our model and linear interpolation are shown in Appendix Table 4 3 respectively and the performance of interpolation has reached a level comparable to that of reconstruction.
03-
20-
derr
122
0.6
14-
0.4-
10*
0.2
Liver
Figure 6: The three evaluation metries for the interpolation task are shown in the plot, with the red line representing our method and the blue line representing linear interpolation.
4.3.2 Future Prediction
Our model can not only interpolate missing frames in the middle, but also predict future dynamics by extrapolating over time. To validate this, we use models trained on the MDAnalysisData and DRYAD MD datasets to predict future frames. Table jablws shows the prediction results of the model for the next 10 frames and the next 11 ~ 20 frames some proteins. Due to space limitations of the main text, the full results are presented in the Appendix E.2
Table 2: Evaluation the future 10 frames and 11 20 frames
Future 10 frames
Future 11~20
LoU't
Chamfer dist.
NCT
IoL't
Chamfer dist.)
NC1
DIMS
0.846740.0423
0.0010+0.00 0005
0.866620.00 0504
0.6835=0.0551
0.0055 0020
0.6932+0.0474
IFABP
0.8878:0.0142
0.0006+0.000
0.9271+0.015
0.844420.0153
0.001220.000
0.8797=0.0164
FRODA
0.7660+0.0502
2.000200
0.826740.0490
0.6043=0.0474
0.004410.0012
0.6783=0.0384
4.3.3 A Case Study on GPCRmd
Furthermore, we have explored the model's simultancous simulation of multiple protein trajectories. Here we use the four protein trajectories data in GPCRmd dataset. The model trains different latent vectors to represent different proteins. In addition, in selecting training frames, we have employed another strategy, wherein the interval between the frames utilized for training is gradually increased with the sequence.
As shown in Figure 7 our model can learn shapes of multiple proteins simultaneously, and can recover the fine details of the protein surfaces well despite their ruggedness. An interesting finding that the trajectories obtained from molecular dynamics simulations often have lot of random jittering or noise, while the trajectories interpolated by our model can well fit the main functional

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ground truth
reconstruction
A 90"
(b)
(c)
(d)
Figure 7: Comparison of ground truth and reconstruction for GPCRmd data at / 2000. From (a) to
(d) are 10792_trj trj__rt1 81, 10912_tr 10_rtj__5,, 15711 15711__rr_770 [rj_791 and 16105_ __r_r_______ _88 respectively, ground truth and
reconstruction (top row) after rotating 90 degrees (bottom row) clockwise along the horizontal axis.
motion of the protein without noise. We put the detailed results and discussion in the Appendix video demonstration on the webpage.
5
Discussions
Molecular surface Representation The molecular surface representation is commonly adopted for tasks involving molecular interfaces where non-covalent interactions (e.g.. hydrophobic interactions) play a decisive role (3). Non-Euclidean convolutional neural networks [44] and point cloud-based learning models 45 have been applied encode the molecular surface for downstream applications, e.g., protein binding site prediction[45, [46]. However, it has not been applied to molecular dynamic simulation, especially protein. In fact, this work explored the more essential application scenarios of molecular surface representation, that is, we believe that chemical molecules are more suitable be regarded as electron clouds that can be represented by 3D shapes.
Deep learning molecular dynamics Deep learning is being used more molecular dynamics simulations, which has shown promising results. There have been several approaches, such Behler- Parrinello network(A) (47). DTNN[48], and Sch that focus on predicting molecular properties and potential energy. However, these methods are limited small molecules and may not work for proteins, which have many atoms. To address this issue, several coarse-grained models have been proposed, such as CGnet 17). 55)) (RCG and a CG auto-encoder(52) Recent works, flow-matching (53) and DFF (F], have used normalizing flows and diffusion models to model coarse-grained force fields for dynamic simulation small proteins. Another approach, DiffMD (55), can predict simulation trajectories without energy or forces. However, these methods are currently limited to simulating small proteins due to computational constraints and are not yet practical for real-world applications, while the method in this paper provides new idea for long term modeling of large proteins.
Implicit neural representations SIREN 56) and 2cRF[26) have led to the popularity of implicit neural representations (INR) for tasks like new view synthesis and geometry reconstruction. INR involves creating a continuous function that can map coordinates to high-frequency information. NeRF's highly expressive nature has allowed it to be effectively employed in numerous fields, such as image processing(57) (e.g., compression, denoising, super-resolution, inpainting). video processing(58) medical imaging[59) etc. Signed distance function (SDF) is type of INR that characterizes object's shape well. DeepSDF uses INR to model SDF, and IGR [40 simplifies the modeling of SDF by using its gradient property. Drawing inspiration from the aforementioned studies, our work learns a unified implicit representation of proteins that will facilitate future large- scale modeling of protein dynamics like NeRF's ' contribution in vision.
Conclusions and Future work
We introduced a method for learning implicit neural representations of protein surface dynamics in continuous spatial and temporal space. Experimental results show that our method is very effective in reconstructing proteins with smooth dynamic changes, and has certain ability to interpolate in time.

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In addition, we found that our model tends to model major conformation changes of proteins rather than those noise-like random vibrations. The case study of GPCRmd shows that the model has the potential to learn the dynamics of a class of similar proteins, such as protein family.
The main limitation of this method is that it is sensitive noise in protein dynamic trajectories, as discussed in Section 4.2 In addition, the generalization of the model needs to be further improved. The first the temporal generalization, especially future prediction. So far, there is effective method that can better predict the future in various motion prediction tasks. We believe that modeling motion or velocity fields can help future predictions for longer time.
We anticipate DSR as a powerful tool for building purpose protein surface models, and hope that our work helps shed some light more efficient and generalizable protein representation.
Acknowledgments
This work supported by Mathematical Intelligence Application Laboratory. MIALAB, Institute for Mathematical Sciences, Renmin University of China, Beijing Academy of Artificial Intelligence and Public Computing Cloud of Renmin University of China.
References
[1] Yao Li, Tong Wang, Juanrong Zhang, Bin Shao, Haipeng Gong, Yusong Wang, Xinheng He, Siyuan Liu, and Tie-Yan Liu. Exploring the regulatory function of the n-terminal domain of sars-cov-2 spike protein through molecular dynamics simulation. Advanced theory and simulations, 2(10):210012,, 2021.
[2] Maxwell Zimmerman, Justin R Porter, Michael Ward, Sukrit Singh, Neha Vithani, Artur Meller, Upasana L Mallimadugula, Catherine E Kuhn, Jonathan H Borowsky, Rafal P Wiewiora, al. Citizen scientists create an exascale computer combat covid-19. BioRxiv, pages 2020-06 06:00-06222 2020.
[3] Richard Car and Mark Parrinello. Unified approach for molecular dynamics and density- functional theory. Physical review letters, 55(22):2471, 1985.
[4] Kresse Georg. Vasp group, theoretical physics departments, vienna. Retrieved February, 21, 2011.
[5] Jussi Enkovaara, Nichols A Romero, Sameer Shende, and Jens J Mortensen. Gpaw-massively parallel electronic structure calculations with python- software. Procedia Computer Science, 4:17-25, 2011.
[6] Wendy D Cornell, Piotr Cieplak, Christopher I Bayly, Ian Gould, Kenneth Merz, David M Ferguson, David Spellmeyer, Thomas Fox, James W Caldwell, and Peter A Kollman. A second generation force field for the simulation of proteins, nucleic acids, and organic molecules. Journal of the American Chemical Society, 117(19):517--519) 1995.
[7] Lijiang Yang. Chun-hu Tan, Meng-Juei Hsieh, Junmei Wang, Yong Duan, Piotr Cieplak, James Caldwell, Peter A Kollman, and Ray Luo. New-generatee amber united-ator force field. The journal of physical chemistry B, 110(26):13166 131661177, 2006.
[8] Sandeep Patel and Charles L Brooks III. Charmm fluctuating charge force field for proteins: I parameterization and application to bulk organic liquid simulations. Journal of computational chemistry, 25(1):1-16, 2004.
[9] Sandeep Patel, Alexander D Mackerell Jr, and Charles Brooks III. Charmm fluctuating charge force field for proteins: li protein/solvent properties from molecular dynamics simulations using a nonadditive electrostatic model. Journal of computational chemistry, 25(42):1504-1514, 2004.
[10] Wilfred F van Gunsteren and Herman Berendsen. Groningen molecular simulation (gromos) library manual. Biomos, Groningen, 24(682704):13, 1987.
10

---

[11] Lukas D Schuler, Xavier Daura, and Wilfred F Van Gunsteren. An improved gromos96 force field for aliphatic hydrocarbons the condensed phase. Journal of computational chemistry. 22(1 (/111):122 (2(11):1205-1216 2001.
[12] Markus Christen, Philippe Hünenberger. Dirk Bakowies, Riccardo Baron, Roland Bürgi. Daan P Geerke. Tim N Heinz, Mika A Kastenholz, Vincent Kräutler, Chris Oostenbrink, et al. The gromos software for biomolecular simulation: Gromos05. Journal of computational chemistry, 26(16):1719-177 2005.
[13] Stefan Chmicla, Alexandre Tkatchenko, Huziel Sauceda, Igor Poltavsky, Kristof T Schütt, and Klaus-Robent Müller. Machine learning of accurate energy-conserving molecular force fields. Science advances, 3(5):e660301,, 2017.
[14] Stefan Chmiela, Huziel E Sauceda, Klaus-Rober Müller, and Alexandre Tkatchenko. Towards exact molecular dynamics simulations with machine-learned force fields. Nature communica- tions, 91):3887, 2018.
[15] Huziel Sauceda, Stefan Chmicla, Igor Poltavsky, Klaus-Robert Müller, and Alexandre Tkatchenko. Molecular force fields with gradient-domain machine learning: Construction and application to dynamies of small molecules with coupled cluster forces. The Journal of chemical physics, 150(1 I):1 14102, 2019.
[16] Stefan Chmiela, Huziel Sauceda, Igor Poltavsky, Klaus-Robert Müller, and Alexandre Tkatchenko. sgdml: Constructing accurate and data efficient molecular force fields using machine learning. Computer Physics Communications, 240:38-45, 2019.
[17] Jiang Wang, Simon Olsson, Christoph Wehmeyer, Adrià Pérez, Nicholas E Charron, Gianni De Fabritüs, Frank Noé, and Cecilia Clementi. Machine learning of coarse-grained molecular dynamics force fields. ACS central science, 5(5):755-767, 2019.
[18] Feliks Nüske, Lorenzo Boninsegna, and Cecilia Clementi. Coarse Coarse-graining molecular systems by spectral matching. The Journal of chemical physics, 15 (41(4)044411, 2019.
[19] Jiang Wang, Stefan Chmiela, Klaus-Robert Müller, Frank Noé, and Cecilia Clementi. Ensemble learning of coarse -oannee molecular dynamics force fields with a kernel approach. The Journal of chemical physics, 152(19):194106, 2020.
[20] Xiang Fu, Tian Xie, Nathan J Rebello, Bradley D Olsen, and Tommi Jaakkola. Simulate time-integrated coarse coarse-grained molecular dynamics with geometric machine learning. arXiv preprint 2204 10348, 2022.
[21] David E Shaw, Peter J Adams, Asaph Azaria, Joseph A Bank, Brannon Batson, Alistair Bell, Michael Bergdorf, Jhanvi Bhatt, Adam Butts, Timothy Correia, et al. Anton 3: twenty mi- croseconds of molecular dynamics simulation before lunch. In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, pages 1-11, 2021.
[22] Ye Li, Xianren Zhang, and Dapeng Cao. The role of shape complementarity in the protein- protein interactions. Scientific reports, 3(1):3271, 2013.
[23] Pablo Gainza, Freyr Sverrisson, Frederico Monti, Emanuele Rodola, D Boscaini, MM Bronstein, and BE Correia. Deciphering interaction fingerprints from protein molecular surfaces using geometric deep learning. Nature Methods, 17(2):84-192, 2020.
[24] Pablo Gainza, Sarah Wehrle, Alexandra Van Hall-Beauvais. Anthony Marchand, Andreas Scheck, Zander Harteveld, Stephen Buckley, Dongchun Ni, Shuguang Tan, Freyr Sverrisson, et al. De novo design of site-specific protein interactions with learned surface fingerprints. bioRxiv, pages 2022-06. 2022.
[25] Freyr Sverrisson, Jean Feydy, Bruno E Correia, and Michael M Bronstein. Fast end-to-end learning on protein surfaces. In Proceedings of the IEFF/CVE Conference on Computer Vision and Pattern Recognition, pages 15272-15281 2021.

---

[26] Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoor- thi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. Communications of the ACM, 65(1)999-106, 2021.
[27] Chengan He, Jun Saito, James Zachary, Holly Rushmeier, and Zhou. Nemf: Neural motion fields for kinematic animation. arXiv preprint arXiv:226..03887. 2022.
[28] Carter Butts, 500ns Trajectory Data for "Comparative Exploratory Analysis of Intrinsically Disordered Protein Dynamics Using Machine Learning and Network Analytic Methods," (Grazi- oli.ct.al. 2019), 2022. (ttL:htss:/// :/tp::://ooi.. org/10 (ttg/1/.791////////./ 7910/DV//ERY0ZS
[29] Gianmare Grazioli, Rachel W Martin, and Carter T Butts. Comparative exploratory analysis of intrinsically disordered protein dynamics using machine learning and network analytic methods. Frontiers in molecular biosciences, 6:42, 2019.
[30] Sean Seyler and Oliver Beckstein. Molecular dynamics trajectory for benchmarking mdanalysis, 6 2017. URL: https:/fiishhae. com/aricces//oolecalar: _dyaamicrd_dmdsis____mma trajec- doi, 10:m9.
[31] Sean L Seyler, Avishek Kumar, Michael F Thorpe, and Oliver Beckstein. Path similarity analysis: a method for quantifying macromolecular pathways. PLoS computational biology, 11 (10):e004456, 2015.
[32] Juan R Perilla, Oliver Beckstein, Elizabeth Denning, and Thomas Woolf. Computing ensem- bles of transitions from stable states: Dynamic importance sampling. Journal of computational chemistry, 12(2):196-209 2011.
[33] Daniel Farrell, Kirill Speranskiy, and MF Thorpe. Generating stereochemically acceptable protein pathways. Proteins: Structure, Function, and Bioinformaties, 78(14):2908-22) 2921, 2010.
[34] Oliver Beckstein. Molecular dynamics trajectory of i-fabp for testing and benchmarking solvent dynamics analysis, 2018.
[35] Ian M. Kenney, Shujie Fan, and Oliver Beckstein. Molecular dynamics trajectory of membrane protein nhaa, 2018.
htt:://oi..or111...81.0. figshare. 7185203. v2
[36] Shujie Fan and Oliver Beckstein. Molecular dynamics trajectories of membrane protein yiip, 2019. URL htapl//ttss//ab..or//00000/////1 http://00..004/9..1m www.sharr..222214...o
[37] Ismael Rodríguez- Srariguez-Esigggrres, Mariona Forrens-Fontanals, Johanna KS Tiemann, David Aranda- García, Juan Manuel Ramírez-Anguita, Tomasz Maciej Stepniewski. Nathalie Worp, Alejandro Varela-Rial, Adríán Morales- Pastor, Brian Medel-Lacruz, et al. Gpermd uncovers the dynamics of the 3d-gpcrome. Nature Methods, 1778):777-777, 2020.
[38] John M Jumper, Nabil F Faruk, Karl F Freed, and Tobin R Sosnick. Data from: Trajectory- based training enables protein simulations with accurate folding and boltzmann ensembles in cpu-hours, 2018 URL https://ooo:: wttp://./oo..//////////.... /061/dryad......oo/ h9f8sb7
[39] Schrödinger, LLC. The PyMOL molecular graphics system, version 1.8. November 2015.
[40] Amos Gropp, Lior Yariv, Niv Haim, Matan Atzmon, and Yaron Lipman. Implicit geometric regularization for learning shapes. arXiv preprint arXiv:2002.10094 2020.
[41] William E Lorensen and Harvey E Cline. Marching cubes: A high resolution 3d surface construction algorithm. ACM siggraph computer graphics. 21(4):163-16) 1987.
[42] Dina Duhovny, Ruth Nussinov, and Haim J Wolfson. Efficient unbound docking of rigid molecules. Algorithms Bioinformatics: Second International Workshop, WABI 2002 Rome, Italy, September 17-21, 2002 Proceedings 2, pages 185-200. Springer, 2002.
[43] Kim Sharp. Electrostatic interactions macromolecules. Current Opinion in Structural Biology, 4(2):234-23) 1994.
12

---

[44] Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodola, Jan Svoboda, and Michael M Bronstein. Geometric deep learning on graphs and mamifolds using mixture model cnns. In Proceedings of the IEEF conference computer vision and pattern recognition, pages 5115-5124, 2017.
[45] Freyr Sverrisson, Jean Feydy, Joshua Southern, Michael M Bronstein, and Bruno E Correia. Physics-informed deep neural network for rigid-body protein docking. In MLDD workshop of ICLR 2022, 2022.
[46] Stelios K Mylonas, Apostolos Axenopoulos, and Petros Daras. Deepsurf: surface-based deep learning approach for the prediction of ligand binding sites on proteins. Bioinformaties, 37(12): 1681-1690, 2021.
[47] Jörg Behler and Michele Parrinello. Generalized neural-network representation of high- dimensional potential-energy surfaces. Physical review letters, 98(14):146401, 20077
[48] Kristof T Schütt, Farhad Arbabzadah, Stefan Chmiela, Klaus R Müller, and Alexandre Tkatchenko. Quantum-chemiert insights from deep tensor neural networks. Nature com- munications, 8(1):13890, 2017.
[49] Kristof Schütt, Huziel Sauceda, P-J Kindermans, Alexandre Tkatchenko, and K-R Müller. Schnet-a deep learning architecture for molecules and materials. The Journal of Chemical Physics, 144(22) 2018.
[50] Linfeng Zhang, Jiequn Han, Han Wang, Roberto Car, and Weinan Deepeg: Constructing coarse-grained models via deep neural networks. The Journal of chemical physics, 149(3): 034101, 2018.
[51] Aleksander EP Durumeric and Gregory A Voth. Adversarria-ressullllcoossss machine learning theory to systematic molecular coarse saanse-rrainng. The Journal of chemical physics, (11()2):124110, 2019.
Applying
[52] Wujie Wang and Rafael Gómez-Bombarelli. Coarse-graining auto-encoders for molecular dynamics. npj Computational Materials, 5(1):125, 2019.
[53] Jonas Kohler, Yaoyi Chen, Andreas Kramer, Cecilia Clementi, and Frank Noé. Flow-matching: Efficient coarse coarse-graining of molecular dynamics without forces. Journal of Chemical Theory and Computation, 19(3):942-952 2023.
[54] Marloes Arts, Victor Garcia Satorras, Chin-Wei Huang, Daniel Zuegner, Marco Federici, Cecilia Clementi, Frank Noé, Robert Pinsler, and Rianne van den Berg. Two for one: Diffusion models and force fields for coarse boarse-grained molecular dynamics. arXiv preprint arXiv:2302.00600 2023.
[55] Fang Wu13 and Stan Z Li. Diffmd: A geometric diffusion model for molecular dynamics simulations. 2023.
[56] Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein. Im- plicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems, 33:7462-7473, 2020.
[57] Mikolaj Czerkawski, Javier Cardona, Robert Atkinson, Craig Michie, Ivan Andonovic, Carmine Clemente, and Christos Tachtatzis. Neural knitworks: Patched neural implicit representation networks. arXiv preprini arXiv:2219. 14406, 2021.
[58] Yoni Kasten, Dolev Ofri, Oliver Wang, and Tali Dekel. Layered neural atlases for consistent video editing. ACM Transactions Graphics (TOG), 40(6): 2021.
[59] Liyue Shen, John Pauly, and Lei Xing. Nerp: implicit neural representation learning with prior embedding for sparsely sampled image reconstruction. IEEE Transactions on Neural Networks and Learning Systems, 2022.
13

---

[60] Jeong Joon Park, Peter Florence, Julian Straub, Richard Newcombe, and Steven Lovegrove. Deepsdf: Learning continuous signed distance functions for shape representation. In Proceed- ings of the IEEF/CVF conference on computer vision and pattern recognition, pages 165-174, 2019.
[61] Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang. Zachary Vitito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. 2017.
[62] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6988) 2014.
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Reviews:
[{"review_id": "9jQmY2A2d0", "reviewer": "Reviewer_2SKi", "summary": "The paper proposes to learn an implicit neural representation to model protein surface across continuous spatial and temporal space.", "questions": "I think the paper can be greatly improved if the author can address the weaknesses section and I am happy to consider raising my score then.\n\nIn addition, the author should consider spending more text to describe the evaluation setup, specifically around how train/test data is split and what has been done to prevent data leakage.", "limitations": "The author has sufficiently and fairly discussed the limitation of their methods.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "This is a great application of implicit neural networks for modeling protein surfaces, and the results seem promising.", "weaknesses": "The evaluation of the method can be improved:\n- Even though there is no prior work as a baseline, simple methods should still be used to confirm the learned model is not doing something trivial. For instance, what happens if the author just linearly interpolate the 3D coordinates for the experiment in 4.3?\n- The reconstructions metrics do not really reflect the usability of the model predicted trajectory. Can we see the correlation for docking on ground truth and generated surface? What about the distance between ground truth and the generated surface in terms of their MaSIF distance?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your review comments. \n\n**Q1**: Comparing the learned model with simple methods, such as linearly interpolate?\n\n**A1**: Following the reviewer's suggestion, we conduct additional experiments comparing our methods to linear interpolation that generates intermediate structures between two endpoints. \n\nA partial summary of the results is shown in the table below. The complete results table (Table 1. and Table 2. ) and comparative charts (Figure 1. ) can be found in the attachment PDF in the global rebuttal, and we have included the results and a comparison line chart in the revised manuscript. As shown in the table below, our method outperforms linear interpolation in terms of surface quality. However, we would like to point out that this comparison is not entirely fair to our method, as linear interpolation skips the difficulty of generating reasonable surface details. In contrast, our model is capable of learning the underlying structure of the data and generating high-quality surfaces that capture the subtle details of the input data. We believe that these additional experiments and comparisons demonstrate the strengths of our method and its ability to generate high-quality surfaces that capture the complexity and nuances of the input data.\n\n**Table: Comparison of evaluation on the surface interpolation between trained frames by our model and linear interpolation**\n\n|  |  | Ours |  |  |  | Linear |  |\n|:---:|:---:|:---:|:---:|---|:---:|:---:|:---:|\n|  | IoU | Chamfer_dist | NC |  | IoU | Chamfer_dist | NC |\n| DIMS | 0.9225±0.0035 | 0.0003±0.0001 | 0.9527±0.0034 |  | 0.9084±0.0071 | 0.0003±0.0001 | 0.9266±0.0079 |\n| I-FABP | 0.8833±0.0078 | 0.0006±0.0001 | 0.9254±0.0066 |  | 0.8770±0.0076 | 0.0006±0.0001 | 0.9053±0.0082 |\n| adk_equi | 0.7459±0.0436 | 0.0021±0.0012 | 0.7816±0.0327 |  | 0.7181±0.0440 | 0.0020±0.0010 | 0.7320±0.0352 |\n| bba | 0.6204±0.1296 | 0.0052±0.0064 | 0.8074±0.0972 |  | 0.5249±0.1506 | 0.0076±0.0079 | 0.7193±0.1056 |\n| cspa | 0.5957±0.0714 | 0.0080±0.0039 | 0.7812±0.0574 |  | 0.5189±0.0728 | 0.0070±0.0032 | 0.6994±0.0556 |\n| T0816 | 0.5245±0.1289 | 0.0087±0.0088 | 0.7221±0.0951 |  | 0.3994±0.1344 | 0.0097±0.0092 | 0.6259±0.0845 |\n| gpW | 0.4769±0.1262 | 0.0097±0.0086 | 0.7008±0.0967 |  | 0.3846±0.1411 | 0.0109±0.0092 | 0.6228±0.0912 |\n| bbl | 0.4380±0.1372 | 0.0108±0.0088 | 0.6605±0.0895 |  | 0.3304±0.1558 | 0.0130±0.0122 | 0.5964±0.0881 |\n\nWe believe that our model is effective in handling protein dynamic changes and has practical applications. We will continue to improve our research to meet the expectations of both reviewers and readers.\n\n**Q2**: Can we see the correlation for docking on ground truth and generated surface? What about the distance between ground truth and the generated surface in terms of their MaSIF distance?\n\n**A2**: We employ three evaluation metrics: the Intersection over Union (IoU) metric, which quantifies the overlap between the predicted and ground truth structures; the Chamfer distance, which measures the distance between the predicted and ground truth structures in terms of their surface normals; and Normal Consistency (NC), which assesses the consistency of surface normals between the predicted and ground truth structures. These three metrics collectively provide a comprehensive evaluation of the model's predictive capability from various perspectives.\n\nIn addition to the evaluation metrics, we have provided some comparison figures between the ground truth and generated surfaces in the paper, such as Figure 3 and Figure 6. Furthermore, in the video available at https://anonymouswk.github.io/DSR, we showcase examples of the ground truth and reconstructed surfaces overlaid for comparison. These examples demonstrate the correlation between the ground truth and generated surfaces. \n\nFurthermore, we would like to address your concern about the MaSIF distance. While it is true that the MaSIF distance requires the full atomic coordinates, we have found that it is not directly applicable to our case. However, based on its calculation principle, the MaSIF distance is equivalent to our Chamfer distance. Therefore, we believe that our Chamfer distance metric is a suitable substitute for the MaSIF distance in this case.\n\n**Q3**: In addition, the author should consider spending more text to describe the evaluation setup, specifically around how train/test data is split and what has been done to prevent data leakage.\n\n**A3**: Thank you for your suggestion. We have incorporated additional details regarding the evaluation, including the splitting of datasets, in the manuscript."}, {"review_id": "5hmkMAtMNg", "reviewer": "Reviewer_ZJGr", "summary": "The paper proposes a framework for predicting protein dynamics through learning the time-dependent SDF. The authors showcase that the learned SDF can reconstruct/interpolate protein surfaces and predict the dynamics, thus bypassing the computationally expensive molecular dynamics simulation.", "questions": "* The units for RMSD and CD in the result section are missing (I assume it is Angstrom).\n*  The latent code \"z\" in the equation 2 is not defined and discussed in the method section.", "limitations": "The authors have discussed the limitation of their work in the conclusion section.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "* The paper validates the model by conducting extensive numerical experiments on various protein systems.\n* Learning efficient surrogate models for protein dynamics prediction is important due to the computational cost associated with simulating protein dynamics using MD. The proposed method can potentially benefit both computational biology and drug discovery efforts.", "weaknesses": "The presentation of the current manuscript can make it difficult to grasp its technical contribution. From the perspective of developing new deep learning models, the paper has combined many existing techniques. For example, parametrizing SDF with neural networks have extensively studied in many neural field works like SIREN/DeepSDF, and the Eikonal term has also been explored in prior work IGR (Gropp, Amos, et al. ). The experiments also do not show how it addresses or improves upon issues in existing deep learning models. For example, what is the advantage of the proposed method over a point-cloud GNN that learns to predict atom-wise displacement? \n\nOn the other hand, if the authors' primary objective is to introduce a new way to predict the protein dynamics for the simulation community, it is also not very clear to what extent the model can be used to replace classical MD. To use the model for prediction, whenever there is a new protein system, one needs to first train the model on its existing trajectory (needs to run MD to get it) and then extrapolate from it. The model cannot do any prediction from scratch on new configuration of molecular system without re-training. \n\nIn addition, the model is neither energy conserved nor equivariant, this might not be a serious issue in fitting and reconstructing a single protein trajectory but can cause instability for the long-term prediction.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your review comments.\n\n**Q1**: What is the advantage of the proposed method over a point-cloud GNN that learns to predict atom-wise displacement?\n\n**A1**: Thank you for your question regarding the advantages of our proposed method over a point-cloud GNN that learns to predict atom-wise displacement. While our work is inspired by previous works such as DeepSDF and IGR, our approach is based on the claim that a protein's surface is crucial for understanding its properties.\n\nTraditional biochemists tend to focus on atomic representations of proteins. However, from a quantum mechanics perspective, atoms and electrons have inherent uncertainties. Therefore, representing proteins using electron density maps to study their properties may be more appropriate, which is why we chose to focus on protein surface representations.\n\nBased on our work, we are also conducting further research, including issues related to the generalization of larger protein spaces, modeling protein surface force fields based on vector fields, and assisting protein simulation from scratch.\n\nRegarding the advantages over point-cloud GNN, we claim that using a continuous surface representation can better cover uncertainties compared to a discrete atomic representation.\n\n**Q2**: It is also not very clear to what extent the model can be used to replace classical MD. \n\n**A2**: Thank you for your feedback regarding the extent to which our model can replace classical molecular dynamics (MD) for predicting protein dynamics. We understand the need for step-by-step progress towards achieving the goal of replacing MD. To address this limitation and pave the way for future advancements, we have introduced a latent code in our model, which allows for greater generalization across various protein types. This latent code serves as a starting point for training the model on existing trajectory data obtained from MD simulations. This paper is just the beginning of our work to validate the feasibility of using a protein surface's implicit neural representation. We are currently developing a follow-up study that incorporates and learns a vector field to guide proteins in simulating from scratch. We would like to emphasize that our paper represents an initial exploration into the feasibility of using a protein surface's implicit neural representation. We recognize the need for further research and development to bridge the gap between our current approach and a fully independent prediction model.\n\n**Q3**: The units for RMSD and CD in the result section are missing (I assume it is Angstrom).\n\n**A3**: Thank you for your feedback regarding the missing units for RMSD and CD in the results section. We confirm that the unit for RMSD is indeed Angstrom, which has been added to the manuscript. For Chamfer Distance (CD), this is a standardized unitless quantity with values ranging from 0 to 1. We have added an explanation of this to the manuscript. Thank you for bringing these issues to our attention, and we appreciate your helpful feedback. \n\n**Q4**: The latent code $z$ in the equation 2 is not defined and discussed in the method section.\n\n**A4**: Thank you for your feedback regarding the latent code $z$ in Equation 2. Here, we explain the latent code and we have added the definition and discussion in the manuscript. As previously mentioned, the latent code is for the model's generalization, and its parameters are learned during training. In Section 4.3.3 of the manuscript, we demonstrate the use of different latent codes for each protein in our multiple protein dynamics. \n\nActually, we claim that with sufficient data, we can explore the latent space to achieve greater generalization across various protein types. We apologize for any confusion caused and will revise the manuscript accordingly to provide a more detailed explanation. "}, {"review_id": "Y2jHu61Qz1", "reviewer": "Reviewer_EXzZ", "summary": "The author puts forwards the DSR model, which is an implicit neural network, uses this model to accurately capture the protein dynamic trajectories and interpolate and extrapolate in 3D and time. The DSR is a scalable and efficient method to modeling protein dynamics, overcomes the limitations of first-principles-based and deep learning methods. As the most important contribution, it is successful for the large protein dynamic study.", "questions": "1. In the background, there are some traditional method which is costly and low speed. Is there some reason or explanation that this method is fast and cheap, other than the application of neural network?\n2. In the reconstruction task, there are three evaluation metrics, in what range do the values of these metrics indicate that the model is performing well?\n", "limitations": "1. The DSR model is sensitive to noise in protein dynamic trajectories, which is a challenge while faced with data with little smoothness or great noise.\n2. As the analysis for the GPCRmd, the DSR model has the potential to learn the dynamics of a class of similar proteins, such as a protein family.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The method learns SDF well and achieves dynamic modeling for large proteins for the first time. This is an excellent strength.\n2. The surface representation approach is useful in protein dynamic field. It simplifies calculations and allows identifying movement trends and amplitudes of protein domains.", "weaknesses": "1. In the reconstruction task, there are little comparison between the DSR and other methods.\n2. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your review comments.\n\n**Q1**: In the reconstruction task, there are little comparison between the DSR and other methods.\n\n**A1**: Thank you for your feedback regarding the comparison between our DSR method and other methods in the reconstruction task. As we are the first to propose and model this issue, there are no previous works to compare our method with. However, at the suggestion of another reviewer, we have added a comparison with a linear interpolation to our manuscript. A partial summary of the results is shown in the table below. The complete results table (Table 1. and Table 2. ) and comparative charts (Figure 1. ) can be found in the attachment PDF in the global rebuttal, and we have included the results and a comparison line chart in the revised manuscript. \n\n**Table: Comparison of evaluation on the surface interpolation between trained frames by our model and linear interpolation**\n\n|||Ours||||Linear||\n|:-:|:-:|:-:|:-:|-|:-:|:-:|:-:|\n||IoU|Chamfer_dist|NC||IoU|Chamfer_dist|NC|\n|DIMS|0.9225±0.0035|0.0003±0.0001|0.9527±0.0034||0.9084±0.0071|0.0003±0.0001|0.9266±0.0079|\n|I-FABP|0.8833±0.0078|0.0006±0.0001|0.9254±0.0066||0.8770±0.0076|0.0006±0.0001|0.9053±0.0082|\n|adk_equi|0.7459±0.0436|0.0021±0.0012|0.7816±0.0327||0.7181±0.0440|0.0020±0.0010|0.7320±0.0352|\n|bba|0.6204±0.1296|0.0052±0.0064|0.8074±0.0972||0.5249±0.1506|0.0076±0.0079|0.7193±0.1056|\n|cspa|0.5957±0.0714|0.0080±0.0039|0.7812±0.0574||0.5189±0.0728|0.0070±0.0032|0.6994±0.0556|\n|T0816|0.5245±0.1289|0.0087±0.0088|0.7221±0.0951||0.3994±0.1344|0.0097±0.0092|0.6259±0.0845|\n|gpW|0.4769±0.1262|0.0097±0.0086|0.7008±0.0967||0.3846±0.1411|0.0109±0.0092|0.6228±0.0912|\n|bbl|0.4380±0.1372|0.0108±0.0088|0.6605±0.0895||0.3304±0.1558|0.0130±0.0122|0.5964±0.0881|\n\nThe comparison results demonstrate that our method outperforms the linear interpolation method, indicating that our model is capable of learning the inherent dynamic changes of proteins. We have updated the manuscript to reflect this comparison.\n\n**Q2**: In the background, there are some traditional method which is costly and low speed. Is there some reason or explanation that this method is fast and cheap, other than the application of neural network?\n\n**A2**: Thank you for your feedback The traditional methods for molecular dynamics simulation, such as solving the Schrödinger equation, the Born-Oppenheimer approximation method, and Density Functional Theory (DFT), are computationally expensive and time-consuming. To provide a more detailed explanation, we can give specific numbers to illustrate the computational complexity of these methods. For instance, solving the Schrödinger equation directly for a system of 100 atoms would require solving a matrix with approximately $1.26 \\times 10^6$ elements, which is computationally intractable even with supercomputers. The Born-Oppenheimer approximation, which separates electronic and nuclear motion, is also computationally expensive. For example, a single energy minimization calculation using this approximation for a protein containing 10,000 atoms can take several hours on a high-performance computing cluster. DFT, which is a widely used approximation for solving the Schrödinger equation, has a time complexity of $O(n^3)$, where $n$ is the number of atoms in the system. For example, a DFT calculation for a system of 1000 atoms would require approximately 1 billion calculations. \n\nSince traditional biochemists tend to focus on atomic representations of proteins, the computational complexity is closely related to the number of atoms in the system. However, we have converted the representation to a continuous function of the protein surface, which can significantly reduce the computational complexity. \n\n**Q3**: In the reconstruction task, there are three evaluation metrics, in what range do the values of these metrics indicate that the model is performing well?\n\n**A3**: In general, for the IoU metric, a value of 0.5 or above is commonly considered as a good performance indicator. This signifies a significant overlap between the predicted and ground truth structures. However, it's worth noting that specific assessment criteria may vary depending on the application domain and the particular task at hand.\n\nSince there is no established performance score for the problem we are studying, we have provided a comparison table (Table 2) of evaluation metrics for a 3D reconstruction task as a reference for reviewers to understand[1]. Additionally, we have included visualizations in the figures and videos in the provided link https://anonymouswk.github.io/DSR to showcase the excellent modeling of protein surface dynamics achieved by our method.\n\n**Table2: Single view reconstruction results on Pix3D Chairs.**\n\n|Metric|Pix3D|AtlasNet|Mesh R-CNN|Pixel2Mesh|DISN|MeshSDF (raw)|MeshSDF|\n|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|\n|IoU|0.282|-|0.240|0.254|0.333|0.337|0.407|\n|EMD|0.118|0.128|0.125|0.115|0.117|0.119|0.098|\n|CD-$\\sqrt{l_2}$|0.119|0.125|0.110|0.104|0.104|0.102|0.089|\n\nIt is important to emphasize that determining whether a representation is good or not ultimately depends on its impact on downstream tasks. Further extending such representations to downstream work is an ongoing effort for us. We are actively working on integrating our approach into downstream tasks to assess its effectiveness and applicability.\n\n[1] Remelli E, Lukoianov A, Richter S, et al. Meshsdf: Differentiable iso-surface extraction[J]. Advances in Neural Information Processing Systems, 2020, 33: 22468-22478.\n\n"}, {"review_id": "KQoKuIEbq1", "reviewer": "Reviewer_MFaL", "summary": "This paper proposes to learn an implicit neural network to model protein surface and dynamics. A protein surface is represented as a signed distance function (SDF), which can be modeled by implicit neural networks based on MLP. The training objective follows standard practice: the difference between true and predicted SDF values at each point (both interior and exterior) and an Eikonal term at the surface points to ensure the learned function is a valid SDF. DSR is evaluated on multiple protein dynamic simulation data such as GPCRmd. The model is able to reconstruct the surface with different levels of accuracy, depending on the types of proteins. DSR is also able to predict the future frames given partial protein trajectory.", "questions": "In section 4.3.3, when you model multiple proteins, do you provide protein type as an input to DSR model? If not, how do you train the model so that DSR knows there are different protein types?", "limitations": "The paper has included a short discussion on limitations and societal impact.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* This paper presents an interesting direction to protein surface. Modeling protein dynamics at the surface level is a novel contribution to the field.\n* Using implicit neural networks to model protein surface is a novel application of existing surface modeling methods.\n* This paper presents a comprehensive analysis over a collection of 20 protein dynamic datasets, which can be a valuable resource to the protein dynamic modeling field.", "weaknesses": "There are two limitation of the method.\n1. Currently, DSR cannot generalize to a new protein. In the current set of results, DSR is trained and tested on the same protein, i.e., it is trained on a partial trajectory of one protein and used to predict the future of that protein. In practice, we usually want to simulate a protein that we haven't simulated before.\n2. DSR only models the surface of the protein. Although the shape of a protein contains useful information, it is not all. We also need to know the chemical composition over the protein surface, i.e., which part of the surface is more hydrophobic/charged/etc. Predicting the surface itself is not sufficient for downstream analysis. For example, MaSIF (Nature Method 2020) used protein surface to predict protein-protein interaction. They find that their model performs poorly when using only the geometric shape as input. When they add chemical features of the protein surface patches (hydrophobicity, charge, etc.), the performance improved substantially. Therefore, it is necessary to predict not only the geometric shape but also the chemical composition of the protein surface.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your review comments.\n\n**Q1**: Currently, DSR cannot generalize to a new protein. \n\n**A1**: We appreciate your valuable feedback regarding the generalizability of our DSR method to new proteins. We acknowledge that the current set of results demonstrates a limitation in the ability of DSR to generalize to new proteins, which is a concern for practical applications.\n\nHowever, it is crucial to emphasize the significance of our work in establishing the foundation for continuous modeling of protein surfaces based on implicit neural representations. Without achieving continuous modeling of protein surfaces, it would be challenging to make progress in this field. By demonstrating the effectiveness of DSR in predicting the future dynamics of a protein based on a partial trajectory of the same protein, we provide a starting point for further investigations.\n\nIt is important to note that our research is still in the exploratory stage, and we are actively working on improving the practicality of our method. In our upcoming work, we plan to address the issue of generalizability by introducing and learning a vector field (force field) that can simulate the dynamics of new proteins given an initial structure. This extension will enable our model to simulate the dynamics of new proteins based on the learned force field.\n\nWe appreciate the reviewer's insightful comments, and we are dedicated to addressing these challenges in our ongoing research.\n\n**Q2**: DSR only models the surface of the protein. Although the shape of a protein contains useful information, it is not all. We also need to know the chemical composition over the protein surface, i.e., which part of the surface is more hydrophobic/charged/etc. Predicting the surface itself is not sufficient for downstream analysis.\n\n**A2**: Indeed, it is an active area of research in the community to explore the physical and chemical properties of protein surfaces for downstream tasks such as protein-protein interactions (PPI) and docking. However, our work focuses on addressing some of the underlying and underappreciated aspects of protein representation.\n\nOur motivation stems from the fact that traditional molecular simulations are based on quantum mechanics, where atoms and electrons have inherent uncertainties. In this regard, we believe that representing proteins using electron density maps, as opposed to atomic representations, may be more suitable. Therefore, we adopted a surface-based representation of proteins to capture their shape. While geometric information is the initial step, we fully recognize the importance of incorporating chemical and biological information to enhance our model and expand its applicability.\n\nIn our ongoing research, we are actively working on incorporating chemical and biological features to enrich our model. We aim to explore and integrate additional information such as hydrophobicity, charge, and other relevant properties of the protein surface. By doing so, we intend to strengthen our model's capability and enable it to be applied to a wider range of downstream tasks.\n\nWe appreciate the reviewer's insightful comment and will continue to improve our approach by incorporating chemical composition and exploring various downstream applications.\n\n**Q3**: In section 4.3.3, when you model multiple proteins, do you provide protein type as an input to DSR model? If not, how do you train the model so that DSR knows there are different protein types?\n\n**A3**: In our model, we introduce a latent code, which is a vector with learnable parameters. By specifying an index for the latent code, we inform the model about the specific protein type. This allows the DSR model to differentiate between different protein types during training and prediction.\n\nFurthermore, we acknowledge the potential for further exploration of the latent space. We aim to collect additional data in the future to investigate the generalization capabilities of our model across a broader range of protein types. By expanding the dataset, we hope to enhance the model's ability to generalize to different protein types. Thank you for raising this important question, and we appreciate the opportunity to clarify the mechanism by which we incorporate protein type information into our DSR model."}, {"review_id": "2ZQEblpMhi", "reviewer": "Reviewer_Dvrt", "summary": "The work proposes to model the surface of proteins using signed distance field (SDF) expressed by implicit neural representations. Previously, protein surfaces were studied to characterize protein interactions and ligand binding. In this work, surface representation learning is for analyzing MD simulation data. The network takes in 3D coordinates, as well as time, and returns SDF and is optimized in the typical fashion by minimizing the distance of 3D points on the surface, matching normals, plus Eikonal term regularization. The experiments on curated datasets show that network is able to provide dynamic reconstructions, while the performance degrades on structures that exhibit rapid movements. The networks is able to interpolate between moments and predict future dynamics.", "questions": "1. I suggest that you provide more context on why those rapid movements in the structure is noise and not part of real dynamics. I think that it is not fully noise but a blend of noise and true dynamics which is completely eliminated by smoothness inductive bias in the network.\n2. In the interpolation experiments, how do you extract one frame out of several frames? How much is the RMSD between two states that are used as endpoints of interpolation? ", "limitations": "Please address the weaknesses and questions above.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The work is very-well written. In the introduction section, the relation with previous work and their issues are clarified and the task addressed in the work is well-motivated. Datasets used for evaluation are clearly described improving reproducibility. \n2. Additional visualizations provide in the supplement and main text helps understanding results on modeling dynamics with implicit neural representations. For example, both quantitative and qualitative results shows that with increase of RMSD the reconstructions do not match well with the ground-truth.\n3. The proposed method is evaluated in multiple contexts. For example, its ability to interpolate or extrapolate in time is assessed. Also, a case study shows that one can learn a latent space encode dynamics of similar proteins.\n ", "weaknesses": "1. According to the provided visualizations (e.g. Figure 3), the proposed method could add some non-existent tiny areas close to the structure but disconnected. I believe these artifacts are due to implicitly representing surfaces.\n2. In L211, it is mentioned that the reconstructions are clearer than GT. This is a weird statement as one expect that GT be the most faithful representation of the surface. It is not clear if small rapid movements are due to noise or part of the true dynamics; If the latter, then the inductive bias in the network to output a smooth function is not really a merit of the work. On the other hand, if the GT is noisy, then it is not clear how to quantitatively evaluate the reconstructions; thus such results become less valid and hard to rely on.\n3. It is claimed that, unlike previous work, the proposed method is able to resolve dynamics of large proteins. However, there is no quantitative comparison with previous work to validate this.\n4. It is not clearly stated why implicit representation for surfaces is used, rather than other representations (such as algebraic ones). There is a lack of experimental validation on this, too.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1**: According to the provided visualizations (e.g. Figure 3), the proposed method could add some non-existent tiny areas close to the structure but disconnected. I believe these artifacts are due to implicitly representing surfaces.\n\n**A1**: We acknowledge the complexity of protein surfaces, which pose significant challenges in characterizing them using a continuous function model. Our study focuses on a continuous function model in a 3D+time space, which increases the learning difficulty. In Figure 3, the protein exhibits substantial dynamic variations, including transitions between different folding patterns. This adds to the complexity of learning across both spatial and temporal dimensions. In comparison, proteins with smaller motion amplitudes or more regular oscillations tend to exhibit fewer of these tiny areas.\n\n**Q2**: The statement that the reconstructions are clearer than GT may not be appropriate.\n\n**A2**: Here, we would like to clarify that the surface representation of proteins that we have access to is in the form of discrete point clouds, which inherently have limited resolution. We aim to use a continuous model to capture a more smooth representation. Perhaps using the term \"GT\" might not be the most appropriate in this context, and \"reference\" would be a more suitable term.\n\n**Q3**: It is claimed that, unlike previous work, the proposed method is able to resolve dynamics of large proteins. However, there is no quantitative comparison with previous work to validate this.\n\n**A3**: In response to this concern, we would like to provide a comprehensive explanation of the computational complexity of traditional methods for molecular dynamics simulation. For instance, solving the Schrödinger equation directly for a system of 100 atoms would require solving a matrix with approximately $1.26 \\times 10^6$ elements. As the number of atoms increases, the computational intractability of this method becomes evident, even with the aid of supercomputers. Additionally, Density Functional Theory (DFT), a widely used approximation for solving the Schrödinger equation, has a time complexity of $O(n^3)$, where n represents the number of atoms. Consequently, a DFT calculation for a system of 1000 atoms would cost around 1 billion calculations. In contrast, our method, which converts the protein representation to a continuous function of the protein surface, significantly reduces the computational complexity. Unlike traditional methods that heavily rely on atomic representations of proteins, our approach for generating the dynamic representation of proteins is independent of the number of atoms. With a well-trained model, the generation of each time step's surface structure, on average, only requires approximately 30 seconds.\n\n**Q4**: It is not clearly stated why implicit representation for surfaces is used, rather than other representations (such as algebraic ones). There is a lack of experimental validation on this, too.\n\n**A4**: We appreciate your concerns. In our response, we would like to emphasize the importance of the protein surface and why we chose the implicit neural representation method. Traditionally, biochemists have used atomic representations to describe proteins. However, the most accurate methods for molecular dynamics simulations rely on quantum mechanics. From a quantum mechanics perspective, both atoms and electrons have inherent uncertainties. Therefore, using atomic representations to calculate atomic forces may not be accurate, and representing proteins through electron density maps might be a better choice.\n\nHence, in order to model the dynamics of proteins from this perspective, we initially chose to represent the protein surface. Implicit neural representation has proven to be an excellent method for modeling 3D objects. However, we acknowledge that we did not provide experimental validation for this specific choice.\n\nRegarding the comparison with other methods, as we are the first to use this approach to model such problems, there is no previous work to compare against. However, based on the suggestions from other reviewers, we have conducted additional comparison experiments with linear interpolation methods to demonstrate that our approach can indeed capture the dynamic changes of protein surfaces. The complete results table (Table 1. and Table 2. ) and comparative charts (Figure 1. ) can be found in the attachment PDF in the global rebuttal, and we have included the results and a comparison line chart in the revised manuscript. \n\n**Q5**: Why are those rapid movements in the structure noise and not part of real dynamics?\n\n**A5**: Thank you for your suggestion. Indeed, the rapid movements in the protein structure contain both noise and true dynamics. However, what I intended to convey was the smoothness of the trajectory. In other words, it refers to the ease or difficulty of modeling that particular case.\n\nThe continuous inductive bias of the model does help to eliminate noise to some extent. However, our evaluation of the model is based on the data we have access to. The application of smoothing techniques may lead to a decrease in evaluation metrics. At present, it is challenging to determine which representation is more appropriate. In our future work, we plan to quantitatively evaluate the performance of the representations through downstream tasks.\n\n**Q6**: In the interpolation experiments, how do you extract one frame out of several frames? How much is the RMSD between two states that are used as endpoints of interpolation?\n\n**A6**: In our modeling approach, we extracted frames at fixed time intervals, specifically selecting every 10th frame to ensure equal time intervals between the frames. We did not extract frames based on the RMSD between states. This decision was made because our aim was to model a continuous function of 3D+time, and thus, we provided the model with a time-related input to facilitate the learning of temporal dynamics."}]

Author rebuttal:
I would like to extend my sincere appreciation for the reviewers’ efforts in coordinating the review process for our submission. In response to the reviewers' comments, we have carefully analyzed and compiled their concerns and suggestions. I would like to summarize the key points raised by the reviewers and assure you that we have taken them into serious consideration:

1. The reviewers mentioned the need for comparative experiments since we were the first to propose using surface representation to model protein dynamics, without any previous work for comparison. In response to the suggestion from Reviewer 2SKi, "Even though there is no prior work as a baseline, simple methods should still be used to confirm the learned model is not doing something trivial. For instance, what happens if the author just linearly interpolates the 3D coordinates for the experiment in 4.3?" we conducted additional experiments comparing our model with linear interpolation, and the results indeed demonstrated the effectiveness of our model.
2. Another concern raised by the reviewers was the generalization of our method and how it applies to simulating new proteins. Regarding this, I would like to clarify that it is indeed the focus of our future work. First and foremost, the reason we proposed using an implicit neural representation model to capture the dynamics of protein surfaces is because atoms and electrons have inherent uncertainties from the perspective of quantum mechanics. Representing proteins using atomic coordinates to calculate interactions may not always be accurate, while using electron density maps could be more appropriate. Therefore, we introduced surface representation. Our ultimate goal is to gradually achieve an alternative to traditional Molecular Dynamics (MD) simulations. However, the first step is to validate the feasibility of dynamic surface representation, which is the focus of our current study. In our model, we have also incorporated a latent code. Through this latent space, we aim to explore the representation space of proteins and enable the switching between different protein types by manipulating the latent code. Moving forward, our considerations include introducing vector field modeling and leveraging methods such as NeuralODE to achieve more realistic simulations in the following works.

In addition, the reviewers also pointed out several strengths and expressed a strong appreciation for our work. Specifically, they can be summarized as follows:

1. Application and novelty: Multiple reviewers mentioned that this paper presents an interesting direction in the field of protein surface modeling. Simulating protein dynamics using surface representation is considered a novel contribution, and the application of latent neural networks for modeling protein surfaces is seen as a novel approach.
2. Computational efficiency and practicality: The reviewers acknowledged the emphasis placed on the computational efficiency and practicality of our proposed method. They recognized the importance of learning efficient surrogate models for protein dynamics prediction, considering the computational cost associated with traditional molecular dynamics simulations. The reviewers also acknowledged the potential benefits of our approach for computational biology and drug discovery efforts, validating the significance of our work in their assessment.
3. Comprehensive analysis and presentation: The reviewers also commented on the quality of the writing in our paper. They noted that the introduction section effectively establishes the relationship with previous work and highlights the motivation behind the addressed task. Additionally, they appreciated the clear description of the datasets used for evaluation, which enhances reproducibility. Furthermore, they commended the extensive use of visuals, including images and videos in the supplementary materials, for effective visualization of the presented work.

In summary, the ultimate goal of our series of work is to model protein molecular dynamics using surface representation, which can also be applicable to downstream tasks. This paper serves as a starting point, aiming to validate the effectiveness of surface representation in modeling dynamics. This step is crucial and necessary for further progress. Furthermore, we have addressed all the reviewers' concerns and made corresponding modifications in the manuscript.

Meta-review: This work proposes a new way to model protein surface dynamics through implicit neural networks. Reviewers agreed that the ideas are interesting and novel. Two key identified methodological limitations entail limited generalizability to new proteins and the current inability of the method to model the important surface biophysical properties. The paper is also lacking in terms of baselines. It would be valuable to complement the existing results with an investigation of how the proposed method compares to alternative approaches for predicting protein surface dynamics, e.g., by predicting how the entire structure evolves and retaining the surface. Additional ablations would also be insightful on which components of the method affect performance. Despite these limitations, the innovativeness of the method and problem make this paper worth presentation. The approach will hopefully inspire follow-up work from the authors and the community.
Decision: Accept (poster)