Symmetry-Driven Discovery of Dynamical Variables in Molecular Simulations
Authors: Jeet Mohapatra
Nima Dehmamy
Csaba Both
Subhro Das
Tommi Jaakkola
Venue: ICML 2025 poster (2025)
Keywords: AI4Science
Molecular Dynamics
Second Order Methods
Symmetry Discovery
Primary area: applications->chemistry_physics_and_earth_sciences
TL;DR: We relate effective degrees of freedom to symmetries and use symmetry discovery find important DoF such as dihedral angles in alanine dipeptide

Abstract: We introduce a novel approach for discovering effective degrees of freedom (DOF) in molecular dynamics simulations by mapping the DOF to approximate symmetries of the energy landscape. Unlike most existing methods, we do not require trajectory data but instead rely on knowledge of the forcefield (energy function) around the initial state. We present a scalable symmetry loss function compatible with existing force-field frameworks and a Hessian-based method efficient for smaller systems. Our approach enables systematic exploration of conformational space by connecting structural dynamics to energy landscape symmetries. We apply our method to two systems, Alanine dipeptide and Chignolin, recovering their known important conformations. Our approach can prove useful for efficient exploration in molecular simulations with potential applications in protein folding and drug discovery.

Symmetry-Driven Discovery of Dynamical Variables in Molecular Simulations
Jeet Mohapatra'
Nima Dehmamy? Csaba Both Subhro Das? 2 Tommi Jaakkola
Abstract
most likely conformations correspond the minima of this energy landscape. Sampling from this low-energy sub- space is crucial for understanding the function and stability of biomolecules, but it presents significant computational challenges due to the vast number of possible configura- tions and the presence of energy barriers separating dif- ferent conformational states. Existing methods for sam- pling these low-energy conformations include enhanced sampling techniques such as metadynamics (Laio & Par- rinello 2002), umbrella sampling (Torrie & Valleau 1977).
We introduce novel approach for discovering effective degrees of freedom (DOF) in molecu- lar dynamics simulations by mapping the DOF to approximate symmetries of the energy land- scape. Unlike most existing methods, we do not require trajectory data but instead rely on knowledge of the forcefield (energy function) around the initial state. We present a scalable symmetry loss function compatible with exist- ing force-field frameworks and Hessian-based method efficient for smaller systems. Our ap- proach enables systematic exploration of confor- mational space by connecting structural dynam- ics to energy landscape symmetries. We apply our method to two systems, Alanine dipeptide and Chignolin, recovering their known important conformations. Our approach can prove useful for efficient exploration in molecular simulations with potential applications in protein folding and drug discovery.
and replica exchange molecular dynamics (REMD) Sugita & Okamoto 1999). which aim to overcome energy barri- ers and improve exploration of the conformational space. However, these methods often require careful tuning and can be computationally expensive. More recently, machine learning has shown great success in enhanced transition path sampling (Holdijk et al., 2024 Sipka et al., (023). Our work is complementary to these efforts, providing a data-free way to discover reaction coordinates or collective variables.
In this work, we present a novel approach for discover- ing degrees of freedom (DOF) that effectively move the system along the low-energy manifold, enabling more effi- cient exploration of relevant conformations in the molecu- lar landscape. Our key observation is that low-energy DOF can be related to approximate symmetries of the energy function. Recent works in machine learning have made progress in discovering symmetries in data (Benton et al.] 2020 Dehmamy et al. 202T Yang et al., 2023b a). Our setting is different in that we do not have data, but an en- ergy function. The problem of finding symmetries of the energy function overlaps with the task addressed in LieGG (Moskalev et al. 2022). which discovers infinitesimal sym- metry generators, i.e. Lie algebra elements, for a given loss function. In our case, we are interested in transformations which change the energy below certain threshold. mean- ing approxiamte symmetries of the energy. For example, we don't 't want change in energy to be so high that it would break chemical bonds.
1. Introduction
Molecular dynamics (MD) is an essential tool for a wide range of applications, including drug discovery (Jorgensen]
2004 Hollingsworth & Dror 2018), protein folding (Shaw et al. 2010; [indorff-Larsen et al. 2011), and under- standing the physics of biological systems at the molec- ular level (Karplus & Kuriyan 2005 McCammon et al.) 1977). The configuration space of molecules is very high- dimensional, with each atom contributing three degrees of freedom (xyz coordinates). However, the physically relevant conformations of molecules typically occupy much lower-dimensional subspace. This space corresponds to regions of low free energy, which can be thought of as a negative log-likelihood of the system's state. The
'Massachusetls Institute of Technology, Cambridge, MA. USA "MIT-IBM Watson AI Lab, Cambridge, MA, USA 2 ENotth University, Boston, MA, USA. Correspondence to: Ject Mohapatra <joetmo@mine mit.edu>.
We find approximate symmetries by considering small transformations of the original DOF and deriving condi- tions for the near-invariance of the energy. This process yields symmetry loss, which we then minimize. In the case of small molecules, show that the problem can be
Proceedings of the 1nnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Symmetry-Driven Discovery
formulated as finding symmetries of the Hessian. Aside from the optimization approach, we also provide an ana- lytic solution based on degenerate Hessian eigenspaces.
parametrizing the low-energy subspace can facilitate prob- lems such sampling. We will now review some of the literature on sampling and identifying DOF.
Our method connects the structural dynamics of molecules to the approximate symmetries of the energy landscape. This enables a systematic exploration of conformational space that is both computationally efficient and physically insightful. We use our method on Alanine dipeptide and Chignolin. We find that these effective DOFs can be used to efficiently explore the phase space of the molecule. In some cases, the DOF significantly overlap with chemical features such as dihedral angles. Hence, these DOFs fa- cilitate targeted exploration and sampling of biologically relevant conformations. We show that this method of sam- pling can significantly reduce the computational burden as- sociated with high-dimensional MD simulations. It also enhances the ability to explore areas of the conformational landscape that are difficult to access through conventional methods.
Space sampling. Monte Carlo methods are advantageous for sampling configurational space due to the absence of inherent timescales, but they struggle to capture transitions between conformations and can become trapped in local minima behind high-energy barriers, leaving some regions of the energy landscape poorly sampled (Heilmann et al., 2020). Umbrella sampling, introduced by (Torrie& & Valleau 1977). addresses this by replacing the standard Boltzmann weighting with a biasing potential, effectively enabling a random walk across energy barriers.
Methods such as replica exchange molecular dynam- ics (REMD) (Hansmann & Okamoto 1999] Sugita & Okamoto 1999) employ MD simulations simultaneously on series of replica systems with different conditions,e.g temperature, and randomly exchange the states of any two replicas with a regular schedule (Qi et al. 2018) to effi- ciently sample and overcome high energy barriers in the landscape. However, the need for numerous replicas can significantly increase computational demands, making the approach challenging to implement in practice (Rathore et al. 2005 Liu et al. 2005 Wang et al., 2020).
Our contributions can be summarized as follows.
1. DOF as Symmetries: We formulate identifying ef- fective DOF (EDOF) as an optimization problem aim- ing to discover approximate symmetries of the energy.
2. Symmetry discovery using the Hessian: We de- rive the relation between symmetry generators and the Hessian near critical points. We also provide a method for constructing symmetries from the spectrum of the Hessian.
Metadynamics is another approach to improve sampling of the energy landscape of system by driving it through collective variables (CV), which represent key coordinates in the landscape (Laio & Parrinello 2002] Bussi Laio 2020). In Metadynamies, Gaussian biases are periodically added to MD to prevent the system from revisiting previ- ously explored regions, facilitating the discovery of new minima. The challenge is that finding CV is non-trivia and typically relies on prior knowledge.
3. Extraction of internal coordinates for molecules: We show that our discovered DOF can overlap with well-known dihedral angles, but also include addi- tional DOF.
Identifying the DOF. Identifying suitable CV for energy landscape sampling is challenging. often introducing bi- ases and facing issues related to data quality and inter- pretability. Principal component analysis (PCA) has been employed to define CVs (Hori et al.) 2009). but its linear nature often fails to represent the complexity of protein energy landscapes (Maisuradze et al., 2009). Recent ap- proaches range from autoencoders learning nonlinear CVs from data (Chen & Ferguson| 2018), to bath-based methods like DeepLNE (Fröhlking et al. (024) or DeepLDA (Bon-) ati et al. 2020 Majumder & Straub 2024) that learn CVs from molecular dynamics trajectories. These methods have proven quite effective, but they rely on a significant amount of data and are not easily interpretable. Other including self-supervised deep neural networks, have been developed to identify slow CVs or reaction coordinates (Wehmeyer & Noč, 2018). Other CV finding methods using ML follow the coarse-grate granning approach to accelerate MD, improving sampling efficiency and expanding accessible phase space
4. Exploring conformations of Alanine dipeptide and Chignolin: We show that our DOF can be used to sample diverse conformations, particularly shallow lo- cal minima, which are difficult to sample using exist- ing methods.
2. Related Work
Exploring the conformation space of molecular systems poses significant challenges due to their numerous degrees of freedom and the highly nonconvex energy landscape. Traditional molecular dynamics and Monte Carlo meth- ods often struggle to fully map this landscape, as they tend to become trapped in local minima and fail to cap- ture rare conformations. This is largely due to the high en- ergy barriers surrounding the minima or the long timescales required for conformational changes to evolve (Bernardi jtt al., 2015). Our goal in this work is not to design bet- ter sampling methods. Instead, we want to show how
2

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Symmetry-Driven Discovery
Souza et al. 2021| Majewski et al.
2023 Noč et al. 2020).
consider the general linear group (L)) acting on the pa- rameter space X. Global translations and rotations (F) can also be included, but since MD potentials are gener- ally invariant under E(3), they lead to trivial symmetries, which we are not interested in here. Starting from a ref- erence point 2o E X, the orbit of GL(X), defined as Orbit = gan | g E GL(X)), generates a manifold of transformed configurations. This manifold effectively de- scribes the set of configurations related to To by symmetry transformations, This allows us to replace z with transfor- mations 9 that reach x from Z0. Now, if we focus on a subset of symmetries G C GL(X)) that approximately pre- serves the potential energy. we can extract DOF that corre- spond to motion along low-energy directions, thus provid- ing a natural way to explore the low-energy landscape of the system.
Our method falls in the category of finding CVs. However, the crucial difference between our method and the above is that ours does not require trajectory or simulation data.
3. Theory
We will now presents our theoretical framework for dis- covering effective DOF, based identifying approximate symmetries in the energy landscape. We introduce two ap- proaches: scalable symmetry loss function compatible with existing force-field frameworks, and a Hessian- based method effective for smaller systems. Both methods con- nect molecular structural dynamics to energy function sym- metries, enabling systematic conformational space explo- ration. We derive the mathematical foundations of these ap- proaches, showing how they lead to the discovery of physi- cally meaningful DOF, such as dihedral angles in peptides.
Group parameters as DOF In order to map g to degrees of freedom, we need parametrization for g. Since ( is a continuous group, we use the Lie algebra and the ex- ponential map to parameterize in terms of the Lie alge- bra basis La E gl(X') where the Lie algebra gi(x²) = (x)) is the tangent space at the identity of GL(X). In the case of matrix Lie groups such as GL(X), the ex- ponential map exp gl(x)) - GL(X) can be written in terms of the matrix exponential. Exponentiating an element in the Lie algebra yields a group element y = exp(0 L), where is a vector of parameters. In more general cases, where a group element requires a nontrivial path on the group manifold, the transformation may be expressed as a product of exponentials g = II, exp(0. L). In both cases, small group elements (i.e. near identity) can be expanded as g R I 0- L O where I is the identity matrix. This formulation allows us to define the group parameters @ as the new DOF, reparametrizing the system in terms of group transformations that capture the low-energy dynam- ies. Next, we define more concretely what we mean by low-energy dynamics and effective DOF.
3.1. Symmetry and DOF
While we primarily focus on physical and molecular sys- tems, our approach is general and can be formulated in broader terms by treating the energy as a general loss func- tion. Let E : Z + IR represent the potential energy of a physical system, which, analogous to a loss function, we assume to be smooth over large regions of the parameter space Z, and bounded from below. The parameters correspond to the system's 's degrees of freedom (DOF). We assume that Z is vector space. In the context of MD, the standard DOF= positions T; and momenta Pi = mmdd. all particles i E {I,.... n), where m, denotes the mass of particle i.
Z
(x.P) = (L.,P%) }2= includes the 3D
Let 7. E X. p. E P, and 2 X x P
Temperature and kinetic energy In this work, we are pri- marily interested in the static conformations of the system and therefore ignore the kinetic energy term in our formu- lation. By focusing on the potential energy, we capture the equilibrium properties of molecular systems. Although temperature induces thermal fluctuations in real systems, our current approach neglects these effects. These fluctu- ations could be incorporated into future extensions. partic- ularly when accounting for fiite--emperaauuee effects and exploring dynamic behavior. Ignoring the kinetic DOF pa by setting p = 0, our parameter space reduces to the space X' x {0} C Z of positions x = {Z.]. Hence, we redefine the energy to be just the potential energy E : X X R.
3.2. Defining Effective Degrees of Freedom
What does it mean for this DOF to be an "effective" or "low-energy" DOF? Implicitly, low-energy DOF assumes that there exists a hierarchy in the energy scales, where some barriers in the energy landscape are much smaller than some others. In MD, we can clearly see such a hi- erarchy in force-fields used F(c)), such as AMBER (Cor-) nell et al. 1995). The energy landscape comprises strong quadratic terms representing bond lengths and bond angles, along with nonlinear terms boodin"od which are much weaker. The non-bonding energy includes Lennard- Jones (E a12 - 6 producing the weak van der Waals forces and Hydrogen bonding) and Coulomb (F ~ 1 forces, both of which dominate at short distances but decay at larger separations (App. B The bonding energy
Lifting DOF to a group Our core idea is that transfor- mations acting on the DOF can be used to replace the original DOF, thereby lifting the DOF to group action on X. While not all DOF can be lifted in this manner, we demonstrate that this approach enables us to link low- energy DOF to underlying symmetries of the system. We
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Symmetry-Driven Discovery
convex and the energy barriers between local minima are mostly related to the non-bonding energies. Thus, effective DOF must satisfy two competing requirements:
subject || I 1, ensuring that L remains within bounded region of the Lie algebra. The L minimizing equation 4 define the set of effective DOF, Note that L are not necessarily exact symmetries of E and the sym- metry loss does not need to vanish (e.g. in MD oF ~ O (uonbond) is permissible). However, in problems such as MD there are important global symmetry considerations, which we discuss next.
At small scales: they must provide sufficient & to overcome local barriers and escape metastable states, with 8E Ebarrier.
At larger scales: they must preserve the molecular structure, e.g. not break bonds, óE < Eband-
3.4. Excluding global symmetries
In other words, we want the DOF to yield small óE in large range of parameters. When SE = 0 while moving along a DOF, we say the DOF symmetry of E. Let zo be a reference point at which we discover the effective DOF, and 1 x = xig + c. We want the DOF to be an approximate symmetry of E when e is large, but also want it result in relatively large 5E at small to move straight toward energy barriers ensure can explore other minima. Our strategy involves two steps:
In MD, the configuration space X' naturally isomorphic to Raxd, where n is the number of particles and is the spatial dimension. For d = the system often has a global (E(3) symmetry, corresponding to rotations and transla- tions in diree-iimessional space. However, we are not in- terested in this symmetry, as it represents trivial motions that do not affect the relative configuration of the parti- cles. Therefore, we restrict the Lie algebra element L to act on the particle indices, while being invariant under SE(3) transformations.
1. Minimization: Discover candidate set of DOF at by requiring that for large e:
Given this restriction, the action of L affects only the n- dimensional part of x E Raxd The condition for approxi- mate symmetry, (VE - Lx)², can now be written as
S = (DOFF (8E(e)) < for et > m (1)
loss:
(5)
2. Maximization: Select optimal DOF:
dE Σ 3.j.1
C(L, x) =
- Duy Lija
= (Tr (TF((V)) Lx])
DOF = arg max (e.)) for E. < Emin
(2)
LES
where i.j index the particles, and indexes the spatial components. Here, VE E Raxd is the gradient of the energy with respect to the particle positions, and the ma- trix product involves L, which acts on the particle indices, while is the current configuration of the system. We will be working with equation instead of equation 4 Addi- tionally, in small molecular systems we can use another level of simplification using the Hessian, described next.
where 17- Ebarreer is small energy scale and émin cor- responds to displacements on the order of non-bonded in- teraction distances, typically ~ 0.1 - 0.5mm in molecular systems. In the next section we will make these statements concrete, deriving the explicit optimization objectives.
3.3. Symmetry loss
Let x E X' be a reference point, corresponding to config- uration where the quadratic energy terms are minimized. The condition (8F(())) < 1) in the discovery step, means up to order 1) changes, the L are approximate symmetries of F. Note that the main difference between the two steps equation 1 and equation exponential is difficult, we consider a small but finite transformation RV I + OL, where 0 E IR is now just a magnitude and E gl(X') yielding
3.5. Hessian Approach for Symmetry Loss
Assume that x, is critical point of the energy, meaning that VE(z,) = 0. The x in equation S is not necessarily a critical point, but we can assume x is close to a E., meaning & = T. - where e ~ /N00, o²1) is a random perturbation around z.. We can Taylor expand VE(x) around x, to first order
> Eg. Since working with
VE(x) = VE(I.) + H(x)) (x - xx.) = H H(x)) e (6)
SE ~ AV E()) Lx
(3)
Here, H(x.) is the Hessian matrix at T., which has com- ponents Hý = E/Od,, day.
The optimization objective corresponding to step 1 of our method is to find the L that minimizes the symmetry loss
Expectation over Gaussian Perturbations Now, assume that we have many samples x, such that is drawn from a
Symmetry loss: 2(L,x)- (VE(x)- Lx)²
(4)
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Symmetry-Driven Discovery
Gaussian distribution. Since (0 o2), d o² 8
have
In practice, content with having small but nonzero Tr [L" H2L]. In this case, L corresponds to directions in the configuration space along which the energy is approxi- mately flat.
Substituting this expansion into the symmetry loss
[[C(L,x)] = E (Ir [c H(I)) L(x, + +)])
(8)
2. Anti-Symmetric part:
where H = (x. and we used the symmetry of the Hes- sian (H T = H).. Note that in the Minimization step equa- tion Te want the approximate symmetry condition hold for which are relatively large compared the Maximiza- tion step equation 2 To distinguish between the E of these two steps, we should choose different orders of Tay- lor expansion for them. Therefore, for the minimization step we will keep the (c²2) terms, but for the maximiza- tion we only keep (e)).
Proposition 3.2 (Anti-symmetric K). If HL is antisym- metric, the Irace loss can also be minimized. This requires
+ [5] = Tr K
=0
which implies that generates transformations that pre- serve the structure of the Hessian. One solution to this condition is if L itself is antisymmetric L -.L. In this case, the commutator between H and L vanishes
The cross-term in equation is OEe2) and vanishes because is normal. Since H and L always appear together and for ease of notation, let us denote K = HL. For the first term in equation 8 using equation for all derivations below)
LL, HJ -0
7 we get (see Appendix A.
This implies that L commutes with H, and therefore defines symmetry directions where the Hessian is invariant.
E Tr Kr. 1
Kr
If H has degenerate subspace corresponding k de- generate eigenvalues, this subspace has an inherent SO(k) symmetry. Because of this, the Lie algebra elements L E so(k) of this subspace symmetry satisfy [L, H] = 0. This
The last term, this in equation 8 yields
E Tr [E Ke 21 = o (21r [3] + Tr (Ks]")
(10)
is a special case the proposition 3.2 More formally:
where Kg = (K T + K2 is the symmetric part of K. Putting these together, the symmetry loss in this approxi- mation becomes
Proposition 3.3 (Degenerate subspace solution). let A be the diagonalized form of H, with set of k-fold degenerate eigenvalues X = 2 the corresponding eigenspace forms k-dimensional sub- space of symmetry. The action of L in this subspace can be viewed as a rotation, and can be chosen belong to the Lie algebra of rotations SO(k) restricted to the degenerate subspace:
= If A has
Hessian symmetry loss:
(11)
E [[((L, 0² 12 (2T [ + Tr [Ks] j²)
3.6. Analytical Solutions to the Trace Loss in ID
We note that there exists a simple analytical way min- imize each of the two loss terms. In the 1-D case, the of term only depends on the eigenvalues of the symmetric part of Ks HL + L H. Let λι, .2...., An be the eigen- values, then the optimization is equivalent to minimizing E.= 2X (Er= >x)² which can be minimized by mini- mizing the operator norm of Ks. The space of L minimiz- ing the operator norm of Ks can be given as combination of the following (see Appendix for details):
L E so(k),
L =
(13)
The matrix L generates rotations within the degenerate eigenspace, leaving the overall structure of H invariant.
In the three dimensional case, owing to the global spatial symmetries, we focus on L E Raxn which only act on the particle indices ji and not the spatial indices (H, Thus, we can relate this anti-symmetric one dimensional solution to the general case by having [H, L] = 0 be given as [Huw L] = for all (H,D). This not always possi- ble to do and therefore we approximate it to the condition 0 for H2 Σ, Hp.... Similarly for the sym- metric case, we replace || with || for H2 Σn, H² M. Thus, we generalize this one-dimensional an- alytic framework by looking at slow eigenspaces and de- generate eigenspaces for suitable SE(3) invariant matrices H
Symmetric L part:
Proposition 3.1 (L in "slow" subspace of the Hessian), If L has support only on the span of eigenvectors with the smallest eigenvalues of H²
H2,
|2 = Tr [T
(12)
then it minimizes the o term in equation 11 and also min- imizes Tr [Kx]² for arbitrary x.
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4. Selecting optimal effective DOF
(ø, v)) are used as coordinates to describe the states of the molecule as density plot called the Ramachandran plot. We want to investigate whether the DOF discovered using our method can be used to effectively explore the states of the system. We will examine different regions of the Ra- machandran plot reached by varying our discovered DOF and whether we can discover the known conformers of the alanine dipeptide in this way. We will also investigate whether (ø, 1/) overlap with our DOF, which are discov- ered directly from the forcefield without using any prior knowledge about the importance of the peptide bonds. In case of chignolin, there are a lot of local minima around a properly folded state and another near a misfolded state (forming two large groups of conformers). In the litera- ture (Kührová et al.) 2012). it has been shown that the two groups of conformers can be distinguished by the configu- ration of the carbon backbone of the Glycine 7 residue in chignolin. Denoting the backbone dihedral angles of Gly7 as o, v, we repeat the same analysis for chignolin. How- ever, we only examine chignolin in the presence of solvent as the 3 hairpin structure of chignolin is only formed in presence of solvent.
Using the discovery procedure, we get a set of Lie algebra elements L.. Given the set of L. we now want to find the transformations that help us navigate the energy landscape most effectively. Unlike the discovery step for L, where we wanted EE to be small, now we want to maximize it as in equation 2 Therefore, we define the most effective L as the one leading to the largest perturbations in energy, As stated above, we want the largest oE in the close vicinity of F., meaning this time we will assume e is very small and will only keep the o² term, discarding the 04. Using the result from equation 9 as a proxy of the magnitude of structural change, we can solve the following optimization problem to get the most effective DOFs among n1 discov- ered L:
L(c)=
= a=1
ca Lar
Find C. arg max [c)x.)
(14)
e.1c|,=1
But since this term is identical to the x² term we used for the discovery of L, it makes sense only minimize the a² term for discovering candidate L and use the o² term only experiments. In the end, we keep the top two L(c) as the effective DOF. When using equation 4 for discovering L, we also use the full juu in equation 14 When using the other Hessian approaches, we use H2 instead of H in equation [4
As the conformers of the alanine dipeptide as well as the ramachandran plot change with the ambient medium, we consider two MD setting for alanine dipeptide. For chignolin the 3 hairpin configuration is only observed in solvent, so we only consider chignolin in water. In to- tal, we consider three MD settings. 1) Alanine Dipepe- tide in Vacuum where we only use the amber forcefield (amher99shamr) corresponding
in the second step. This is what we do in
Optimal L without Hessian. For the optimization ver- sion of the problem, we see that as mentioned before the L are discovered at high noise levels (large o) by mini- mizing the symmetry loss over the samples. In order to find the most effective DOF, we maximize the symme- try loss at low noise levels (small σ). Given m samples Ej ~ (0, o (xxd) where der is smaller than a used for discovery. Then
interactions within the molecule and 2) Alanine Dipeptide in Water where we use the molecule forcefield (amber99sbnmr) along with the amber forcefield for the solvent (amber99_obc) modelled as implicit. 3) Chignolin in Water where we use the molecule forcefield (amber99sbnmr) along with the amber forcefield for the solvent (amber99 obc) modelled as im- plicit. Additionally, we put Hbond constraints (that keeps the length of bonds between heavy atoms and hydrogen fixed) and use heavy hydrogen in order to stabilize the in- tegration steps (as used in (Satoh et al. 2006
Tr [VE(E, e)) [())(e, + +))
c. arg max RERR GERR|I|||2_]
(15)
Methods for Extracting DOF: Based on our theoreti- cal results, we will use four methods to discover effective DOF:
5. Experiments
To evaluate our method's effectiveness, we conducted ex- periments on two well-characterized molecular systems: alanine dipeptide and the designed mini-protein chignolin. These systems serve as canonical test cases in the molecu- lar dynamics community, offering a balance between com- putational tractability and biological relevance.
1. Direct Optimization: Solving equation 5
2. Full Hessian:
Minimizing the 4f in equation
3. Slow Hessian:
Using smallest eigenvalue subspace of H2 (minimizng the 0² term of equation 11 for arbi- trary x..).
In the case of the alanine dipeptide, the two DOF that capture the most important conformations are known to be the dihedral angles @ and & over the peptide bonds.
4. Degenerate Hessian: Using a degenerate subspace of
Hz. We use the subspace with the highest dimensions
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Symmetry-Drine Driven Discovery
(a) Long Simulation
(b) Direct Optim
(c) Full Hessian
(d) Slow Hessian
(c) Degen Hessian
Figure 1: Ramachandran plot for alanine dipeptide in vacuum: a) long (500ns) simulation starting at 3, b) direct
optimization of equation 4 c) analytically solving the of term in equation 11 d) slow subspace of H2 e) fast degenerate
subspace of H2. The blue and red grid lines are traced by transforming / alanine along the discovered DOF.
(a) Long Simulation
(b) Direct Optim
(c) Full Hessian
(d) Slow Hessian
Degen Hessian Optim w Solvent
Figure 2: Ramachandran plot for alanine dipeptide in water: a) long (500ns) openMM sim with implicit solvent starting at 3 alanine dipeptide, b) direct optimization of equation 4 c) of analytical solution d) slow subspace of H2 e) fast degenerate subspace of H2 f) direct optimization over the forcefield with solvent. The blue and red grid lines are transforming B alanine along the DOF.
Note that the methods involving an explicit Hessian cannot be easily applied to settings with a stochastic solvent force. But, the direct optimization of equation 4 is applicable in all cases, including used systems with an implicit model for the solvent. We discuss the results for both settings: optimization method using just intra-molecular forces and optimization method including solvent forcefield.
Comment on sampled densities. It is important to note that our goal is not recovering the density of states found via the long simulation. In fact, our goal is to be able to visit states which are much harder to reach using the baseline method. Figures] 3 show the density of states sampled. We have also plotted the energies of the states (Appendix C). confirming that we indeed arrive low-energy states.
Procedure: Our experimental procedure is as follows:
5.1. Experiments in Vacuum
1. Start from an initial conformation Ho. the Cs con- former for alanine dipepetide and properly folded conformer in case of chignolin.
When modeling alanine dipeptide in vacuum, only sider the molecular forces between the atoms. For this setting, we see that all the Hessian-basse based methods recover all the major conformations of alanine dipeptide with rela- tively short simulation times. As the Lie algebra elements L in our problem span a R²" space, we need at least Onn²) point to avoid overfitting. For our experiments, we use 16n² samples for discovery and 16n² samples for finding the most effective degrees of freedom. Using larger values of can give more information about long-range sym- metries but using large also increases the stochasticity causing very high variance in the estimates.
2. Extract the top two DOF L1, Le near Z0-
3. Make 31 31 grid of angles (01. 02) 2
4. Generate deformed conformers 2t 021.2
5. Use openmm, mi nimi ze and run short 2ps simulations to find stable conformations from the de- formed structure.
We then compare the conformers found using the method above against conformers found using baseline method of long openMM simula-
sampling. For our baseline tions for 500ns at 300K with friction coefficient of 1ps-1 and step size of 2 amounting to 2.5e8 steps. We use the same amber forcefields both the last step of our method and the baseline simulations in order to maintain consis- tency.
5.2. Experiments with Solvent
We experimented with both alanine dipeptide (Figs 4b
and Chignolin (Figs 3 5) in water. Alanine dipeptide has different stable conformers in water than in vacuum, sug- gesting that incorporating the solvent force is essential for

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Symmetry-Driver Discovery
THAN Gara
Pold
(a) Long Sim
(b) Direct Optim 0.1am (c) Direct Optim 0.5mm
(d) Full Hessian
(c) Optim Solvent O. Inm (f) Optin Solvent 0.5nm
Figure 3: Ramachandran plot for chignolin in water a) long sim (500ns) with implicit solvent starting at folded chig-
nolin, b), c) direct optimization with DOF discovered medium (0. Inm) and long range (0.5nm), d) analytical solution of 0 term in equation 11 e).f) direct optimization of forcefield with solvent and DOF discovered at medium (0.1nm) and long range (0.5nm). The blue and red grid lines transforming chignolin along the discovered DOF.
Simulation (Spentrt))
Direct Optim txtton Metecular
Nesstan Lows Optiiaatiio
Reduced Hesslon Loss Optimization
Long [OpenMM))
Direct Optimization
Full Hesslan
Redused Optimization
Direct including Sovent Forces
Optimizatoo
Forcse
C5
B
C7 ax
a
C7
RR
(a) Alanine Dipeptide in Vacuum
(b) Alanine Dipeptide in Water
Figure 4: Closest Structure the known conformers. We only consider structure be conformer candidate if it is stable and its ()) dihedral angles close that of the corresponding conformer.
Oypine
Sbeet
Lold
Gal Lafee
based approach can be easily generalized to any forcefield. So, we also incorporate the implicit solvent forcefield in the optimization- the combined forcefields. As seen in the figures above, all the approaches proposed discover the major conformers of alanine dipeptide as well folded and misfolded conform- of chignolin. Another important fact that want to highlight is that discovered DOF are general enough to al- low us to find the conformers in settings with forcefields slightly different from the ones they were learnt on.
Folded
approach to find symmetries over
Misfolded
Figure 5: Closest Structure to the conformers of Chigno- lin discovered for simulations in water. Only the last two columns use the solvent force field to calculate the DOF.
modeling the DOF. Note that optimization approach using equation 4 can the forcefield with explicit sol- vent forces included to calculate the DOF, and can there- fore be applied to the case with solvent. However, our Hessian-based methods do not work with stochastic forces, and therefore we use DOF discovered using force- field vacuum also in the case with solvent. Our results suggest the vacuum DOF also work well the presence of solvent. Similarly, as well as the direct optimization method to calculate DOF for chignolin in vacuum and use those to navigate the free energy landscape in the presence of solvent. Moreover, we also see that although the Ressian-based approaches not easily transfer over the solvent scenario, the optimization
Discussion on speed
Although our methods use the openmm simulation in the fi- nal step, only need fraction of the total steps required by openmm to find all conformations. Furthermore, the simulations can be run for all grid points simultaneously. Thus, in principle, the effective time required for simula- tions using our method could be orders of magnitude less than the time required for the baseline long openmm simu- lations. The major bottleneck for our approach the Hes- sian computation which scales quadratically with the size of the system (2sec for alanine-dipeptide, 200sec for chig- nolin) and does not easily generalize to stochastic force- fields. While the direct optimization method overcomes
also the Hessian based method
8

---

Symmetry-Driver Discovery
0.25
0.5
2.5
5
25
50
250
500
C5
0.257
0.268
0.233
0.247
0,246
0.238
0.215
0.23
Alanine Dipeptide in Vacuum
CTeq
0.479
0.429
0.448
0.436
0.436
0.425
0.382
0.408
CTaa
0
0
0.027
0.117
0.059
B 0.411
0.424
0.416
0.404
0.367
0.39
0.39
0.39
Alanine Dipeptide in Solvent
aL 0.043
0
0
0.001
0.001
0.001
0.001
ag
0.043
0.048
0.047
0.041
0.043
0.043
0.043
folded misfolded
0.992
0.994
0.997
0.79
0.206
0.4
0.4
0.4
Chignolin in Solvent
0
0.183
0.073
0.048
0.048
0.048
Table 1: Fractional Distribution of Conformers with respect to elapsed simulation time (in ns) for standard long openmm simulation. This denotes the relative frequency and exploration of the conformal landscape as a function of the time elapsed in the simulation.
0.002
0.01
0.02
0.1
0.2
2
10
C
0.4659
0.3676
0.2463
0.2168
0.2264
0.2082
0.2232
0.2340
Alanine Dipeptide in Vacuum
CTeg
0.1818
0.2745
0.3515
0.3745
0.3708
0.3311
0.3357
0.3367
C7 ax
0
0
0
0
0.0768
0.0467
0.0226
0.9205
0.7451
0.6052
0.4808
0.4336
0.4359
0.4422
0,4505
Alanine Dipeptide in Solvent
a7
0
0
0
0
0.0017
0.0063
0.0061
OR 0.0000
0.0956
0.2166
0.3079
0.3417
0.3383
0.3238
0.3200
Fractional Distribution of Conformers with respect to elapsed simulation time (in ns) for REMD simulation with 8 parallel simulations at logarithmically spaced out temperatures between 300 K 500 K with attempted transition between adjacent temperatures every 50 steps. This denotes the relative frequency and exploration of the conformal land- scape as a function of the time elapsed in the simulation.
Table 2:
the second challenge in principle we find to be less stable than the Hessian-basee approaches in practice and it also has similar quadratic computation cost (15sec for alanine- dipeptide and 450sec for chignolin). The pairwise nature of the atomic interactions leads to quadratic cost, but it can be pruned to be almost linear by using distance cut-off. Table I compares the relative distribution of different conform- ers as a function of the simulation time. This shows that the conformational landscape poorly explored by long simulation. On the other hand, our proposed method is able to find structure close every conformer within 2 ps of simulation time leading to much more efficient and thor- ough exploration of the conformational landscape. Addi- tionally we also compare to Replica Exchange Meta Dy- namics on Alamine-dipeptide using parallel simulations logarithmically spaced out temperatures between 300 K 500 K with an attempted transition between adjacent tem- peratures every 50 steps. Table shows that under this enhanced sampling formulation, the proposed method produces orders of magnitude in speed-up (as some con- formers are only discovered after ns of simulation time).
NSF Expeditions grant (award 1918839): Collaborative Research: Understanding the World Through Code. Csaba Both's work done partly during his internship at the MIT-IBM Watson AI Lab.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
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12)9187-199, 1977.

---

Symmetry-Driven Discovery
A. Derivations Of Results
A.1. Hessian Symmetry Loss
Substituting this expansion into the symmetry loss
E x]] E (Tr (xx)) Lx. + +)]]
E Tr [e HL..] 1² + 2Tr [e' HLv.] Tr [e' HLc] + Tr [c HLc] ?
(16)
where = (x.) and we used the symmetry of the Hessian (H1 T H). For the second term we have
E Tr ( HLv.] Tr [c HL]] =0 (vanishes due to E = 0)
(17)
Since H and L always appear together and for ease of notation, let us denote K HL. For the first term in equation 16
using equation] 7 we get
Tr Kx.
Σ
E c
(Kr.)² (Kx.)
(18)
i.j.
(Kx)) (Kx,) = 0²1 Kx,
(19)
The last term, this in equation
16 yields
E Tr
Ke)
Σ k.j.a.d...........
K Le
(20)
{K
+ K
+ Ki Kji
(21)
ju
Tr K K + K]] + Tr {
0 Tr (K + K) + Tr [K]
(22)
2
Defining the symmetric part Ks (KT + K)/2, we have
E Tr [e T Ke Ke]² | 0
2Tr [ Tr {s}}}
(23)
where we used the fact that Tr [
Tr [Ks). Putting these together, the symmetry loss in this approximation becomes
Hessian symmetry loss:
E [C(,,))) x Kx.11 +o 2Tr [K'] + [K[]K)
(24)
Now, note that since K = HL the components are Kijj ku
Ex
Hikh I uu0
Let us define the trace over spatial indices, u4,
and the node (particle) indices i, as follows
Tr, [H
Σ
Hij
The [H] pv = Σ H
(25)
pgo
For the trace terms we have
Tr K Σ
Kii
Σ
Hik tk Tr, (Tr [H] L)
Tr KK = Σ
(Kij
Σ
Hik I ki Hil I' Tra L Tr, [H2]] L]
yu luu
kupn
Tr K-2 = Σ
(K(H)
pu
Σ
pu
kj H I" Tr, (rr [HLHL]
(26)
12

---

Symmetry-Driven Discovery
This provides relationship between the Hessian, the Lie algebra element L, and the effective DoF defined by L, allowing identify approximate symmetries by minimizing this loss. Note that the above still approximately holds even 2, not a critical point, but some point where the gradient is small, meaning [VE(x.) < for some small
A.2. Analytical Solutions to the Trace Loss in 1D
In the 1-D case, the o² term only depends the eigenvalues of the symmetric part of Ks HL H Let
A1, .2,...,4 An be the eigenvalues, then the optimization equivalent to minimizing Σ 22 + (E Xx) which can be minimized by minimizing the operator norm of Kg. This be equivalently written as
I H HL) x= x H + H) Lx 0
(27)
Any L that satisfies equation 27 for arbitrary r is a symmetry the Hessian. We establish the following result, that shows that the slow modes of the Hessian form and the rotation of the degenerate subspaces of the Hessian are the most dominant modes of the symmetry.
Theorem A.1. given symmetric real matrix A with eigen-decoopoositoon given as A VAVT we see that only matrices B of the form
B = VIVT with Σ trr s0,
satisfy the equation
Ve s.t ||||| 11 x² (B" A+ AB) x (())
(28)
Proof. For any real symmetric matrix A, there exists eigendecompositiinn for A such that A VAVT where VVT_ VTV and A Diag(A) is diagonal matrix of eigenvalues A.
IJ AB A+ AB)
x" VA' + +VA B)
=''IB'VAV" Is x" "VAV' BIx
= =x"VV" B" VAV" x o VV'VAV" BVVT2
BV BV) vTx
= =(V'))" B' AV' BV) (V*z)
Let = V BV which is equivalent to B = VIVT, we get that T [BT A+ AB) x (c)) for all x with ||r||2 equivalent to TT (TT + AT) x Occ) for all x with ||01|2 (as V 1 ||=||]).
This is further equivalent the condition that max |=]|_=1 xT (T AT) x = || AF O Finally, we see that || [M 2 < || I /F |M|] holds true for all r-rank matrices M. Thus, we have
+ - AT (2)
Considering the result in Theorem A.I there are many linear transformations that satisfy equation equation 27
We
restrict attention the only matrices given by the anstaz motivated by the following corollary.
Theorem A.2. For any matrix B, we can divide into two matrices: symmetric matrix Bş = 1/2 (B + B²) and anti-
symmetric matrix BA = 3/2 (B - B²). We can further restrict Bg and BA to the following family of matrices to give an ansatz for constructing an that satisfies equation equation 28
.s has dominant modes only along the slow modes of the A
BB has block diagonal structure which mixes almost- demos--degenerate eigenspaces of A.
13

---

Symmetry-Driven Discovery
Proof. Let Iss 1 I1+ and La - ))). We start by observing that Bş ( + B²)
(VTVT + +VT V (T)
VT= = VT VT. Similarly we can show that B VTAV
VT
(Ty) + (TAle X
(Ts))s (Ta))t) 2
Σ (Arr + +x
2
1/
- Σ
1
(1: )1) (L's)
+ 1 Σ
(1, - ((a)?
2
2
ij
ij
Thus it suffices to have
2)+ = =t and
[a) =
(29)
4
As a result of the first part, we get that (Ts),, = (c) if max(Ar; A²) # O(e) which shows that the dominant modes of Bs are along the slow modes of A. Similarly, the second part gives us that (T'))ss O(e) if [Λ, -Ayl * O(e) which shows that A only mixes the almost-degenerate eigenspaces (space of eigenvectors whose the eigenvalues very close together) of A.
14

---

Symmetry-Driven Discovery
B. Experimental Setup AMBER Force-field
We implement simplified force-field with implicit solvent (i.e. water molecules are not modeled and appear as hydrogen- bonding and hydrophobicity terms. In protein folding our energy function consists of five potential energies for: bond length Ebond, bond angles Bangle. Van der Waals Boaw, hydrophobic Enp and hydrogen bonding Eh (Ceci et al. 2007).
We incorporate the AMBER force field, known for its accurate representation of molecular interactions, particularly in proteins. This force field is implemented using the parameters from OpenMM (Eastman et al. 2017), and it comprehensively models the following interactions:
Protein Folding with Classical MD Using AMBER Force Field
Bond lengths Ebond and bond angles Eangle
Torsional angles Etorsion
Non-bonded interactions including van der Waals Evaw and electrostatic Estec forces
We utilize the functional forms and parameters specified in the AMBER force field:
Ebond =
Σ
krond(r - ro)²
Bangle
Σ
kangle (0 0o)
(30)
bondl
angles
Aij
Bij
Etorsion
Σ torsions
Vn [1 + cos(nw -
EndW
Σ
(31)
12
rij
Eelec
Σ KSj
9:qj
(32)
Here, r and 0 represent the bond lengths and angles, respectively, with ro and 0o as their equilibrium values. The torsional term Etorsion includes a sum over all torsion angles w, with periodicity n, amplitude V,, and phase 7y. The Lennard-Jones potential in Eeaw is characterized by parameters Aiy and Biy. and Estec is calculated using the Coulombic potential with partial charges qis qj and the relative permittivity 6r-
In this simulation, exclude the modeling of solvent effects entirely, focusing solely the protein in vacuum. This approach simplifies the computational model while emphasizing the direct interactions within the protein.
The overall energy of the system is then given by:
((X) Enond + Eangle + Exorsion + Endw + Estec
(33)
15

---

Symmetry-Driven Discovery
C. Further Experimental Results
C.1. Energy Results
In table 3 and table 4 we list for each simulation method and for each experiment, the value the lowest energy config- uration found structurally close given conformer. The closeness of the values to the ground truth found by the long simulation show that the configurations found are valid conformers. As the direct optimization method has two param- eters (units nanometers), for sampling around the starting point to estimate the Hessian (eqs and 7), and one for the maximization step to choose best L's (eq 15), we report two different experiments with varying values of the parameters.
Vacuum Results. The last three columns are for the three prominent conformations.
Method
C
C
Long Simulation
-80
-77
-83
Direct Optim (0.01,
-80
-84
-78
Direct Optim (0.01,
-80
-78
-78
Full Hessian
-80
-78
-84
H2 backbone (slow subspace)
-80
-84
-84
H2 backbone (degenerate subspace)
-80
-78
-84
Table 3: mmergy(kJJ/mol) in Vacuum for discovered Conformers of Alanine-Dipeptide
Solvent Results. These are for simulations in water. The last three columns for the three prominent conformations. "Solvent Direct Optim" means using the symmetry loss 3 and 5 with the energy function of a forcefield that includes explicit solvent terms (using openMM).
Method
8
a
Long Simulation
-120
-127
-128
Direct Optim e (0.01, 0.1)
-105
-127
-128
Direct Optim é = (0.01.0.5)
-120
-127
-128
Solvent Direct Optim = (0.01, 0.1)
-128
-127
-120
Solvent Direct Optim = 0..51,
-97
-127
-128
Full Hessian
-10
-127
-128
H2 backbone (slow subspace)
-128
-127
-128
Hy backbone (degenerate subspace)
1120
-127
-128
Table Energy(k:J/mol) in Solvent for discovered Conformers Alanine- DimineeDipepttde
16

---

Symmetry-Driven Discovery
Starting Point
Starting
C7
Cl
(a) Long Sim (500ns)
(b) Direct Optim
(c) Full Hessian
Starting Point
Starting Paint
G7
C700
Clu
(d) Slow Hessian
(e) Degenerate Hessian
Figure 6: Ramachandran plot for alanine dipeptide in Vacuum based on the a) long (500 ns) openMM simulation with implicit solvent starting at 3 alanine dipeptide, b) direct optimization based approach over the molecular forcefield, c) analytically solving the full 4 term in equation d) slow subspace H₂ e) fast degenerate subspace of H₂ f) direct optimization based approach over the solvent and molecular forcefield. The blue and red grid lines on the plots refer the grid traced by transforming B alanine using the most effective DOF discovered by algorithms. Additionally the scatter plot gives the values of the potential energy (in presence of solvent) at the points sampled.
17

---

Semmetry-Driven Discovery
Hariing Port
Startes PARK
SEAO PURE
STRONg PURZ
(a)
(b)
(c)
(d)
MANTE PAR
(c)
(f)
Figure 7: Ramachandran plot for alanine dipeptide in water based on the a) long (500 ns) openMM simulation with implicit solvent starting at 3 alanine dipeptide, b) direct optimization based approach over the molecular forcefield, c) analytically solving the full o term in equation 4 d) slow subspace of H2 e) fast degenerate subspace of H₂ f) direct optimization based approach over the solvent and the molecular forcefield. The blue and red grid lines the plots refer to the grid traced by transforming 3 alanine using the two most effective DOF discovered by our algorithms. Additionally the scatter plot gives the values of the potential energy (in presence of solvent) at the points sampled using the corresponding method.
18

---

Drrmen Discovery
Starting Point
Starting Point
Starting Point
C7
dk
C7
30
-00
-10
Starting Point
130
Starting Point
155
c7
4A
8
Starting Point
13h
Starting Point
020
Starting Point
C7
-15
-00
-20
(a)
(b)
(c)
Figure 8: The gridlines and conformers found for three independent runs (row 1, row and row 3) using the direct optimization method with E = 0.1 and E 0.01. a) give the results in vacuum b) gives the results where the simulation is conducted with solvent but the initial trajectory is derived without solvent and c) where the simulation is conducted in solvent and the initial optimization problem is also solved using the solvent forcefield.
19

---

Semmetry-Drivver Discovery
Starting Point
Starting Paint
Starting Point
677B
Starting Point
Starding Point
Starting Point
Starting Point
Starting Point
(a)
(b)
(c)
Figure 9: The gridlines and conformers found for three independent (row 1, row and row 3) using the direct optimization method with E 0.5 and E2 0.01. give the results in vacuum b) gives the results where the simulation is conducted with solvent but the initial trajectory is derived without solvent and c) where the simulation is conducted in solvent and the initial optimization problem is also solved using the solvent forcefield.
20

---

Symmetry-Driven Discovery
ralded Starting Point
Stamiig
Shaning Point
miil dided
mistoided
(a)
(b)
(c)
folded
foided
Bided Starting Point
Starting
ting Point
misfoldded
mistelead
(d)
(c)
(f)
Figure 10: Ramachandran plot for alanine dipeptide in water based on the a) long (500 ns) openMM simulation with implicit solvent starting at folded chignolin, b) direct optimization based approach over the molecular forcefield with symmetry discovered close range (0.1 nm), с) direct optimization based approach over the molccular forcefield with symmetry discovered at close range (0.1 nm), d) analytically solving the full o² term in equation e) direct optimization based approach over the solvent and the molecular forcefield with symmetry discovered long range (0.1 nm f) direct optimization based approach over the solvent and the molecular forcefield with symmetry discovered long range (0.5 nm). The blue and red grid lines on the plots refer to the grid traced by transforming folded chignolin using the two most effective DOF discovered by our algorithms.
21


Reviews:
[{"review_id": "GBCkp4tE7Q", "reviewer": "Reviewer_p5r2", "summary": "This work proposes an approach for discovering degrees of freedom in MD simulations, relying primarily on the Hessian matrix rather than large simulation data. The method was applied to two prototypical peptide systems, showing that it is capable of efficient exploration of the configurations space along discovered DOFs.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### Literature Review for Hessian-based methods ###\nWe thank the reviewer for pointing us toward this literature. We will include a detailed overview of the work in the literature and a comparison with the current paper. For geometric structure optimization, most of the methods in the literature build upon the idea of the Newton-Raphson method to accelerate gradient descent using second-order Hessian information to provide quadratic convergence near local minima by providing faster updates without calculating the full Hessian. On the other hand, transition state-finding methods utilize information about the eigenvector corresponding to a negative eigenvalue of the Hessian to find saddle points. Some synchronous transit methods like LST and QST also use Hessian information to better estimate the energy landscape near intermediate states between two final reaction states. While conformer discovery shares some of the same objectives as geometric structure optimization and transition state finding, unlike geometric structure optimization, we need to escape local energy basins to find a diverse set of conformers. Unlike the transition state finding task, we do not have a specified final reaction state that we intend to reach. Our primary objective is to discover certain local degrees of freedom to explore the energy landscape and find more conformers efficiently. In our method, we discover local DoF for a given conformer and use the discovered DoF to explore the basin near the conformer without any additional knowledge of CVs.\n\n### Better Presentation of the Methodology ###\nAs suggested by the reviewer, some of the methodology is quite hard to understand from the current presentation. We plan to reduce some of the jargon used in the paper and add a table of references in supplementary materials to help with the notation used in the paper. Additionally, we will add a section with some illustrative examples of applying our proposed methods to small systems. We present the following 1-d, 2-particle system suggested by Reviewer zCrM as an example. Due to space constraints, we only show the discovered local DoF using the Full Hessian method. We leave a more detailed discussion for larger systems in the paper. Consider the energy function: $E(x) = F(|x_1| - x_2) + \\epsilon \\sin\\left(\\frac{x_2}{\\epsilon}\\right)$ for $x \\in \\mathbb{R}^2$, where $F$ is a 1-D potential. In this case, we can use the first order optimality condition to see that at any minima, $F(|x_1| - x_2) = 0$ and $\\cos\\left(\\frac{x_2}{\\epsilon}\\right) = 0$. Then, we see that the Hessian at any such minima, $x'$, can be given as \n\n$$ H_E(x') =  H_F(|x'_1| - x'_2) \\begin{bmatrix} 1 & -\\textbf{sgn}(x'_1) \\\\\\\\ -\\textbf{sgn}(x'_1) &1 \\end{bmatrix}\n    -  \\frac{\\cos(x'_2/\\epsilon)}{\\epsilon}\\begin{bmatrix} 0 & 0 \\\\\\\\ 0 & 1\\end{bmatrix} = C \\begin{bmatrix} 1 & -\\textbf{sgn}(x'_1) \\\\\\\\ -\\textbf{sgn}(x'_1) & \\gamma \\end{bmatrix} $$\nfor some constants $C, \\gamma$. \nIn order to find a local symmetry transformation $L$ such that for any $x$ near $x'$, we need $E(e^{\\eta L} x) - E(x) \\approx 0, \\implies (\\nabla E (x) \\^\\top L x)^2 \\approx 0,\\implies ((x-x')^\\top H_E(x') L x)^2 \\approx 0$.\n\nFull Hessian method needs us to minimize $2tr(K_S^2) + tr(K_S)^2$ where $K_S = (L^\\top H_E (x') + H_E(x') L)/2$. This results in a generally complicated equation, but we get a simple solution for $\\gamma \\approx 1$. For $L^* = \\frac{1}{2} \\begin{bmatrix} 1 & \\textbf{sgn}(x'_1)  \\\\\\\\ \\textbf{sgn}(x'_1) &1 \\end{bmatrix}$ gives $K_S = a\\begin{bmatrix} 0 & -\\textbf{sgn}(x'_1)\\frac{1 - \\gamma}{2} \\\\\\\\ -\\textbf{sgn}(x'_1)\\frac{1 - \\gamma}{2} & 1-\\gamma \\end{bmatrix}$. Thus, $2tr(K_S^2) + tr(K_S)^2 = 4(1 - \\gamma)^2 \\approx 0$ for $\\gamma \\approx 1$. Thus, $L^*$ forms a local DoF for points around a minima $x'$.\n\nAs ${L^*}^2 = L^*$, exponentiating this matrix we get, $e^{\\eta L^*} = \\sum_{j=0}^\\infty \\frac{\\eta^j}{j!}{L^*}^j  = I + (\\sum_{j=1}^\\infty \\frac{\\eta^j}{j!})L^* = I + (e^{\\eta} - 1)L^*$. \nThus, we have $ e^{\\eta L^*} x = x + (e^\\eta - 1) L^* x = \\begin{bmatrix} x_1 + (e^\\eta - 1)(x_1 +  \\textbf{sgn}(x'_1) x_2) & x_2 + (e^\\eta - 1)(x_2 + \\textbf{sgn}(x'_1) x_1) \\end{bmatrix}^\\top$. So, we see that if $x'_1 > 0, x_1 - x_2 = (e^{\\eta L^*} x)_1 - (e^{\\eta L^*} x)_2$ and if $x'_1 < 0, x_1 + x_2 = (e^{\\eta L^*} x)_1 + (e^{\\eta L^*} x)_2$ giving two different local DoF (conserved $x_1 - x_2$ when $x'_1 >0$ and conserved $x_1 + x_2$ when $x'_1 < 0 $) depending on the local structure of the hessian.\n\n### Quantitative Results and Baselines ###\nPlease refer to our response to Reviewer 9HNJ for additional results. Although there are existing methods for CV discovery, our method considers the slightly orthogonal task of discovering local DoF. Unlike CVs the discovered DoF are local in nature and do not necessarily correspond to any global Collective Variable. Thus, the proposed method is not directly comparable to existing baselines.", "claims_and_evidence": "The main results are on the qualitatively side. At least I am not sure how to interpret the plots in a quantitative way. \n\nThere is a lack of quantitative metrics. This is perhaps understandable given that the purpose is to discovery collective variables, a somewhat ambiguous goal.", "methods_and_evaluation_criteria": "There is no comparison with competing existing approaches.\n\nSee also above.", "theoretical_claims": "I tried to go through the derivations but got lost half way. So this part of my review should be considered incomplete.", "experimental_designs_or_analyses": "See above in \"Claims And Evidence\"", "relation_to_broader_scientific_literature": "There was literature review of several aspects. But the paper needs more discussions about existing Hessian based methods. See below", "essential_references_not_discussed": "Key literature survey of Hessian or approximate Hessian based methods for materials or molecular modeling is missing. Hessian based methods are extensively used in structure optimization and transition state finding/optimization. The latter is especially relevant for this paper.", "other_strengths_and_weaknesses": "Strength: the method is geared towards an important scientific question of collective variable identification and conformation exploration for molecular system, with potential applications in drug design, protein folding dynamics, etc. An advantage of the method is that it does not require big MD simulation data.\n\nWeakness: the presentation of the methodology is not easy to follow. Honestly I did not quite finish it. The the different methods to find the CVs were given without immediate explanation. There were some discussions/motivations scattered at other places. But it was not clear how to interpret the results properly without clear understanding of these methods. The results are lacking quantitative metrics and comparison with existing methods. This paper sounds like it has a lot of potential. But the present presentation prevents me from fully appreciating it.", "other_comments_or_suggestions": "* About equation 4: The \"direct optimization\" uses Eq. 4, but it was stated that eq. 5 should be used instead of eq. 4. \n* In \"Full Hessian\", Eq. 11 contains both sigma^2 and sigma^4. Was the whole of Eq. 11 used, or just the sigma^4 term?\n* typo \"a stable conformations\"", "questions_for_authors": "An easier to follow presentation of the theory could be very beneficial. A lot of jargon was invoked without explanation, which does not work for everybody.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0O0wGwiL0U", "reviewer": "Reviewer_9HNJ", "summary": "This paper introduces a data-free scheme for discovering the effective degrees of freedom in a molecular simulation, which is based upon the exploration of energy landscape symmetry. The effectiveness of this framework is theoretically illustrated and experimentally validated.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their comments, and we try to address some of the concerns below:\n\n### More Quantitative Results ###\nAs pointed out by the reviewer, some of the results presented in the paper might look a bit qualitative.  Here, we provide some more quantitative statements about the results shown in the paper. We intend to add this table in the main paper in order to better illustrate the effectiveness of the discovered DOF.\n\nLong Simulation Results: The table below shows the frequency of observing the molecule in a specific conformation as a function of the simulation time ( simulation is run with a step size of 2 femtoseconds). We consider a molecule within a confirmation basin if the dihedral angles are within the known range for the specific conformer.\n\n|                              |          | 0.25 ns | 0.5 ns | 2.5 ns | 5 ns  | 25 ns | 50 ns | 250 ns | 500 ns |\n| ---------------------------- | -------- | ------- | ------ | ------ | ----- | ----- | ----- | ------ | ------ |\n|   | $C5$    | 0.257   | 0.268  | 0.233  | 0.247 | 0.246 | 0.238 | 0.215  | 0.23   |\n| Alanine Dipeptide in Vacuum |$C7_{eq}$                    | 0.479    | 0.429   | 0.448  | 0.436  | 0.436 | 0.425 | 0.382 | 0.408  |\n|  |$C7_{ax}$                    | 0        | 0       | 0      | 0      | 0     | 0.027 | 0.117 | 0.059  |\n|                              |          |         |        |        |       |       |       |        |        |\n| | $\\\\beta$ | 0.411   | 0.424  | 0.416  | 0.404 | 0.367 | 0.39  | 0.39   | 0.39   |\n| Alanine Dipeptide in Solvent  |$\\\\alpha_L$                  | 0        | 0       | 0      | 0      | 0.001 | 0.001 | 0.001 | 0.001  |\n| |$\\\\alpha_R$                  | 0.043    | 0.043   | 0.048  | 0.047  | 0.041 | 0.043 | 0.043 | 0.043  |\n|                              |          |         |        |        |       |       |       |        |        |\n| Chignolin in Solvent | folded   | 0.992   | 0.994  | 0.997  | 0.79  | 0.206 | 0.4   | 0.4    | 0.4    |\n| |misfolded                    | 0        | 0       | 0      | 0.183  | 0.073 | 0.048 | 0.048 | 0.048  |\n\nDOF-assisted Simulation Results: Under our method, we only run a simulation of 2 picosecond duration for every starting configuration provided by our DOF discovery algorithm. Thus, the conformers do not have enough time to escape from one conformational basin to another. Thus, the initial distribution of the conformers stays the same throughout the short simulation. Here, we provide the frequency of finding a molecule within a given conformational basin (within 20 ps of simulation) given the initial 1000 starting configurations.\n\n|                             |       | Full Hessian | Slow Hessian | Degenerate Hessian | Optimization eps 0.1_0.01 |\n| --------------------------- | ----- | ------- | -------- | --------- | ------------ |\n|  | $C5$ | 0.175   | 0.08     | 0.302     | 0.044        |\n| Alanine Dipeptide in Vacuum | $C7_{eq}$                   | 0.181 | 0.194   | 0.171    | 0.268     |\n|  |$C7_{ax}$                   | 0.214 | 0.194   | 0.134    | 0.231     |\n\n|                              |          | Full Hessian  | Slow Hessian | Degenerate Hessian | Optimization eps 0.1_0.01 | Optimization with solvent eps 0.1_0.01 |\n| ---------------------------- | -------- | ------- | -------- | --------- | ------------ | -------------------- |\n|  | $\\\\beta$ | 0.427   | 0.149    | 0.485     | 0.299        | 0.369                |\n| Alanine Dipeptide in Solvent | $\\\\alpha_L$                  | 0.069    | 0.144   | 0.022    | 0.207     | 0.157        |\n|  | $\\\\alpha_R$                  | 0.106    | 0.284   | 0.105    | 0.116     | 0.105        |\n\n|                      |        | Full Hessian  | Optimization eps 0.1_0.01 | Optimization eps 0.5_0.01 | Optimization with solvent eps 0.1_0.01 | Optimization with solvent eps 0.5_0.01 |\n| -------------------- | ------ | ------- | ------------ | ------------ | -------------------- | -------------------- |\n| Chignolin in Solvent | folded | 0.648   | 0.6          | 0.646        | 0.678                | 0.674                |\n| | misfolded            | 0.006  | 0.024   | 0.013        | 0.024        | 0.019                |\n\n### Simulating the algorithm on additional smaller systems ###\nAs suggested by the reviewer, we intend to add some example applications of the algorithm for smaller analytical systems. Please check bullet point 2 in our response to Reviewer p5r2 for further details and a small example system.", "claims_and_evidence": "Yes. The theoretical and experimental arguments and evidence are very convincing. I especially like the idea of using just the energy, rather than data, to learn the symmetry.", "methods_and_evaluation_criteria": "Yes. Although more thorough benchmarks using even smaller systems, such as Lenoard-Jones particles, charged particles, can be also interesting.", "theoretical_claims": "Yes. They appear correct and sound.", "experimental_designs_or_analyses": "Yes. They look reasonable. Although figures 3-5 are somewhat too qualitative and can benefit from more quantitative arguments to be more closely related to the theoretical arguments.", "relation_to_broader_scientific_literature": "The contributions are extremely relevant to the molecular simulation and drug discovery community.", "essential_references_not_discussed": "NA.", "other_strengths_and_weaknesses": "NA", "other_comments_or_suggestions": "NA", "questions_for_authors": "NA", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RUkQPAz2fh", "reviewer": "Reviewer_zCrM", "summary": "This paper proposes a method to identify effective degrees of freedom (DOF) of dynamics by connecting them with approximate symmetries of the energy function. This is done through identifying a state $x$ with a group element (here restricted to the general linear group) whose action on $x_0$ (a reference state) produces $x$. For sufficiently close $x$ to $x_0$, this can then be parameterised by a linear combination of basis vectors of the Lie algebra. These symmetry based DOFs are optimised according to two principles: the transformations should not cause large changes in energy globally (so that main conformations are still distinct under this parameterisation), but should cause enough change in energy to overcome local energy barriers. This results in loss functions that can either be directly optimised, or under a local assumption be used to derive analytical loss expressions involving the Hessian. Both approaches are tested on two relatively simple test problems (alanine dipeptide and Chignolin), where it is shown that these eDOFs are useful in exploring the conformation landscape.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thorough and thoughtful review. We address some points below:\n\n### Quantitative Metrics & Baselines ###\nPlease refer to our response to Reviewer 9HNJ for additional results. Although there are existing methods for CV discovery, our method considers the slightly orthogonal task of discovering local DoF. Unlike CVs, the discovered DoF are local in nature and do not necessarily correspond to any global Collective Variable. Moreover, we show that we can sample all the conformers for these molecules within 20 fs (10 steps) of starting the simulation from the sampled gridpoints. This setup is very different from the standard CV setup in metadynamics or replica exchange where the initial configuration is fixed.\n### Presentation ###\nThank you for the pointing out the notational inconsistencies. We will add a glossary of the relevant notation in the supplementary material. Furthermore, we will also add a section on illustrative examples applying the proposed method on small systems. Please refer to our response to Reviewer p5r2 for a discussion of a small synthetic system similar to the one you proposed. As the Hessian based method in \n### Ablation Studies ###\nWe will add the suggested ablation studies in the supplementary material. It is not always possible to plot the trajectories for the highest valued eigenvectors of the Hessian as the transformations can lead to unnatural states that lead to errors in the openmm simulation. However, we will add a more detailed discussion in the paper.\n### Local CV and Relation to other CV discovery work ###\nWe thank the reviewer for the reference [1], but the CVs discussed do not align very well with the present work. Although one could define local variables $z_i = L_i x_0$, these resulting variables behave differently from collective variables. We treat all configurations that can be reached from $x_0$ as an equivalence class; thus, all the z_i's would form a basis for this subspace. Thus, all the local $z_i$'s can be used to index the local conformation basin, but it is unclear if these can be patched together or if patching these would give a good global CV.\n### Clarifications for Typos ###\n- Page 1: \"both computationally efficient and physically insightful.\" -> what do you mean by physically insightful?\nWe mean it provides insight into the effective dynamics of the system. \n- Page 3: I find the notation for general linear group strange. … $GL(\\mathcal{X})$ doesn't make sense to me, … not consistent with the subsequent usage of (special) Euclidean groups.\nActually, $GL(V)$ is the standard way the general linear group for a vector space $V$ is denoted, though it may not look as familiar as $GL(n,R)$. We will explain and add the definition of $SE(n)$. \n- Page 3: The Orbit definition seems trivial as written … Do you mean to consider a subgroup?\nYou are correct for $\\mathcal{X} \\sim R^n$. We are indeed usually concerned with a subgroup (the approximate symmetries of $E$). Will reword it. \n- Page 3: symbols in \"Group parameters as DOF\" is quite unclear to me.\n  - Lie algebra basis $L_a$, is this a single basis element, or does $a$ run through … the Lie algebra?\n$L_a$ is a single basis element. \n  - For the notation $\\theta \\cdot L$:\nYes, it is $\\sum_a \\theta_a L_a$\n  - If θ is a vector, the Taylor expansion error term should be O(|θ|2)\nAny combination $\\theta_a \\theta_b$ is second order, but, yes, one can pull out the norm of $\\theta = |\\theta| \\hat{\\theta}$ and write the error as $O(\\theta^2)$. \n  - Since … always GL, what's the point of this abstraction to Lie groups/algebras? Why not just identify the Lie algebra with $M_n(R)$? \nThey are matrices, yes, but we want a subgroup\n  - Boldfaced $I$ vs normal. Any difference? No, will fix it.\n  - Overall, I suggest … properly define the notation. Thank you, will do. \n- Page 4: Equation 1 is not clear, is $S$ considered a subset of $GL(n,R)$ or $M_n(R)$? It should be the latter…\nWe should clarify in the paper that we are working with the canonical representation of $GL(n,R)$ which maps elements to a subset of $M_n(R)$. So, yes, all in $M_n(R)$.  \nAlso, what's the dependence of δE on DOF? Why not just use $E_{barrier}$ which is already used in the previous page and not introduce an extra η?\nEq 3, $\\delta E = \\theta \\nabla E \\cdot Lx$ (here $\\theta  \\in R$) shows dependence of $\\delta E$ on DOF $L$. We can replace $\\eta$ with $E_{barrier} in eq 1. \n- Page 4: the variables $\\theta_l,\\theta_s$ do not appear in eqs 1 and 2. \nSorry, we changed it to $\\epsilon_l, \\epsilon_s$, will fix. \n- g≈I+θL now takes θ as a number instead of a vector.\nA bit of abuse of notation here, will fix. Here $L$ denotes an arbitrary vector in the Lie algebra instead of specific basis elements $L_a$. Then, $\\theta \\in R$ is a small $\\epsilon$ making clear that $g$ is near identity. \n- P6: Eq 15, $K= n_L$ as in eq (14)?\nYes, will fix. \n\n[1] Understanding recent deep-learning techniques for identifying collective variables of molecular dynamics", "claims_and_evidence": "The two examples presented demonstrate to a certain extent the effectiveness of the method, although some quantitative evaluation of the effectiveness of the DOFs would have been better. Moreover, there lacks a general comparison with other methods which tries to construct eDOFs (such as collective variables), although such a comparison may be difficult without a well-defined evaluation criterion. At the very least, one could compare meta-dynamics driven by learned CVs and the current method on the Ramachandran plot. Note that meta-dynamics can be used to perform sampling from the correct distribution, and compute the free energy, but the current method does not seem to be able to do this. The authors may wish to comment on this, and perform some numerical comparison.", "methods_and_evaluation_criteria": "The visualisation of the Ramachandran plot is useful in demonstrating that the eDOF constructed can be used to probe the conformations, although I do not believe it is easy to perform quantitative comparisons with different methods.\n\nThere is also a lack of ablation studies. For example, can one compare the Ramachandran plots of the case which carries out steps 1-5 on page 7, except in step 2 one uses either 1) two randomly sampled $L_1, L_2$; or 2) replaces $L_1 x_0$ and $L_2 x_0$ by the top two eigenvectors of the Hessian of $E$ at $x_0$?", "theoretical_claims": "I checked the overall claims and they appear correct. I did not check every step in the derivation in the appendix.", "experimental_designs_or_analyses": "The experiments are reasonably designed, although as discussed above the evaluation and ablation can be improved.", "relation_to_broader_scientific_literature": "The related work is generally well written and relevant.", "essential_references_not_discussed": "It may be useful to discuss https://arxiv.org/pdf/2307.00365 and methods therein for learning/constructing collective variables.", "other_strengths_and_weaknesses": "trengths\n\n- The problem of finding effective ways (DOFs) to navigate conformational landscape of macromolecules is an important application area, both for qualitative analysis and quantitative computation (sampling, free energy computation, etc)\n- The method appears quite interesting and has new ideas that connect (approximate) symmetries of the energy function to effective DOFs.\n- The qualitative experiments over two simple but representative problems demonstrate the potential of the method\n\nWeaknesses\n\n- Lack of quantitive evaluations of their method and comparisons with related methods (e.g. learned/constructed CVs as effective DOFs)\n- Lack of ablation studies\n- Last but not least, while I do like the new ideas in the paper, the presentation of the paper can be *substantially* improved, there are many typos and many expressions lacking proper definition (details below in \"Other Comments or Suggestions\")", "other_comments_or_suggestions": "- Abstract: \"... we do not require data and rely on knowledge ...\" -> \"... we do not require data but rely on knowledge ...\"\n\n- Page 1: \"we don’t want to change in energy to be so high that it would break chemical bonds\" -> Grammar issue\n\n- Page 1: \"both computationally efficient and physically insightful.\" -> what do you mean by physically insightful?\n\n- Page 3: I find the notation for general linear group strange. It is usually $\\text{GL}(n)$ or  $\\text{GL}(n,F)$ if you want to specify the field, but $\\text{GL}(\\mathcal X)$ doesn't make sense to me, and moreover it is not consistent with the subsequent usage of (special) Euclidean groups (which are also not properly defined in the text).\n\n- Page 3: The Orbit defintion seems trivial as written, since its either $0$ or whole space minus 0. Do you mean to consider a subgroup?\n\n- Page 3: The symbols used in the section under \"Group parameters as DOF\" is quite unclear to me. \n\n  - For the notation of the Lie algebra basis $\\mathbf{L}_a$, is this a single basis element, or does $a$ run through the dimension of the Lie algebra?\n  - For the notation $\\theta \\cdot \\mathbf{L}$, does this mean $\\sum_a \\theta_a \\mathbf{L}_a$ or something else? It is not defined. \n  - If $\\theta$ is a vector, the Taylor expansion error term should be $O(|\\theta|^2)$.\n  - Since it appears to me that we are always considering the general linear group, what's the point of this abstraction to Lie groups/algebras? Why not just identify the Lie algebra with $M_n(\\mathbb R)$?\n  - Boldfaced $\\mathbf{I}$  is used but subsequently $I$ is used instead (e.g. page 4). Is there any meaning to the different notations?\n\n  Overall, I suggest the authors properly define the notations (even if it is placed in the appendix). These issues affect readability of the core parts of the paper.\n\n- Page 4: \"set of DOF at by\" -> Grammar issue\n\n- Page 4: Equation 1 is not clear, is $S$ considered a subset of $GL(n,\\mathbb R)$ or $M_n(\\mathbb R)$? It should be the latter given equation 2. Also, what's the dependence of $\\delta E$ on DOF? Why not just use $E_\\text{barrier}$ which is already used in the previous page and not introduce an extra $\\eta$?\n\n- Page 4: the variables $\\theta_l,\\theta_s$ do not seem to appear in equations 1 and 2.  Moreover, subsequent line with $g\\approx I + \\theta \\mathbf{L}$ now takes $\\theta$ as a number instead of a vector (as introduced on the previous page). \n\n- Page 5: \"where we wanted $\\delta E$\" -> \"where we wanted $\\delta E$ to be small\" \n\n- Page 6: \"menaing\" -> Spelling, \"oonly\" -> Spelling\n\n- Page 6: Equation 15, should $K$ be $n_L$ as in equation (14)?\n\n- Page 6: \"minimas\" -> \"minima\"\n\n- Page 6: \"presense\" -> \"presence\"\n\n- Page 7: \"to find the conformers is settings with\" -> Grammar issue", "questions_for_authors": "1. Can the authors give a minimal example to illustrate your approach and why the DOFs parameterised through symmetries make sense? For example, we can take d=1, n=2 and consider some potential like $E(x_1,x_2) = F(x_1-x_2)+\\epsilon\\sin(x_2/\\epsilon)$ where $F$ is say a double-well potential (or something to that extent, involving potentials of different scales) . We know that the effective DOF should be $x_1-x_2$. It would be helpful to illustrate that the method correctly identifies this.\n2. As I understand, since you always take a linear approximation of the exponential map of the general linear group, this is the same as finding perturbation directions as $x_0 \\mapsto x_0 + \\sum_{i} c_i L_i x_0$. This locally yields \"collective variables\" in the form of $z_i = L_i x_0$, and if one can piece this together for neighbourhoods of each $x_0$, then one should arrive at some global collective variable? Is this understanding correct? If so, there should be some discussion on this point and the connection of this approach to e.g. the collective variable discovery approaches outlined in https://arxiv.org/pdf/2307.00365.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bMlvmdYs3o", "reviewer": "Reviewer_YyW7", "summary": "The paper introduces a framework for discovering effective degrees of freedom (DOF) in molecular simulations by identifying approximate symmetries of the energy function. Instead of relying on simulation trajectories or training datasets, the authors formulate an optimization problem—called a “symmetry loss”—and, in one approach, use second-order (Hessian-based) information to find transformations that keep the energy nearly invariant. These transformations become a set of collective variables or DOF that help explore low-energy regions of the molecular landscape more efficiently. The authors demonstrate their methods on alanine dipeptide (in both vacuum and implicit solvent) and on chignolin, showing that the discovered DOF can reach known conformers and reveal states that are traditionally harder to sample with basic molecular dynamics. Overall, the paper’s main contribution is a data-free technique that systematically derives low-energy directions from the force field itself, offering a way to speed up or enrich sampling of molecular conformations.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank you for your review and we hope we can address some of your concerns below :\n\n### Quantitative Metrics and Baselines ###\nWe understand your concerns regarding the rigor of the evaluation criteria. Please refer to the response for Reviewer 9HNJ to view a table of quantitative results that might address some of the concerns you raised. As seen in the table, the grid of initial configurations given by the discovered DoF provides coverage of the entire conformational landscape. As a result, we are able to find at least some points within each conformational basin within only 10 simulation steps. \n\nTo address your second point about the lack of comparison with other enhanced sampling methods, we find other method to be orthogonal in nature to some of the other sampling methods available. Most enhanced sampling methods like umbrella sampling and metadynamics based methods require specified CVs in order to sample the space. As a result these methods, work on a reduced low-dimensional space unlike the our proposed method which directly works on the all atom model. While replica exchange methods do directly sample from the high-dimensional all-atom positions, they do not involve learning any local symmetries. Furthermore the replica exchange methods require communication between different replicas and cannot explore different parts of the energy landscape simultaneously. We provide a highly parallelizable method which uses local symmetry information to efficiently index and explore conformational basins. Using parallel exploration we only need to simulate <10 steps/starting configuration (total 1000 starting configuration) to get to all the known conformational basins for the explored systems. As a result, the method cannot be directly compared to the sequential algorithms like metadynamics and replica exchange models. \n\n### Scaling to Larger Systems ###\n\nAs mentioned in the paper, the direct optimization based methods can be applied to any system immersed in a force field. Even in the presence of explicit solvent, the procedure only needs to access the force exerted on the atoms in the molecule for given configurations. As correctly stated by the reviewer, the paper is introducing the idea of discovering local DoF, we tried to focus mostly on small systems with well-known and interpretable DoF. \n\n### Concerns Regarding Applicability to ML ###\n\nWe believe the symmetry discovery/optimization part of the procedure as well as the problem of transfer and generalizability of learnt DoF is an ML algorithm, an ML venue is much better qualified to judge its validity than a bio venue. Additionally, comp bio, drug discovery and physical simulation are among the fastest growing and most important use cases of ML.", "claims_and_evidence": "The paper’s key theoretical claim—namely, that low-energy degrees of freedom can be discovered by finding approximate symmetries of the energy landscape—receives strong support through detailed derivations and proofs. The authors demonstrate mathematically how to use the gradient or Hessian of the force field to identify transformations that leave the energy almost unchanged, then validate these ideas on small systems (alanine dipeptide and chignolin). This evidence is clear and consistent for the claim that the proposed methods can uncover physically relevant low-energy directions in the configuration space.\n\nHowever, the assertion that these methods lead to significantly more diverse sampling or noticeable speedups compared to established enhanced-sampling techniques is less rigorously substantiated. The authors do present examples where their approach discovers states that standard molecular dynamics simulations take longer to find, but there is no in-depth, head-to-head comparison of sampling efficiency (e.g., run-time or number of force evaluations) against commonly used methods. Thus, while the theoretical foundation and small-scale results strongly support the symmetry-based framework, the more expansive claims about large gains in sampling diversity or efficiency remain only partially demonstrated.", "methods_and_evaluation_criteria": "The authors focus on two well-known test systems—alanine dipeptide and chignolin—both of which are standard benchmarks in molecular dynamics. These systems provide clear baselines for verifying whether new approaches capture known conformers and relevant collective variables. Because the paper’s main claim is that approximate symmetries of the energy can yield physically meaningful DOF, using these relatively small yet widely studied molecules is a sensible choice: it makes it straightforward to check whether the discovered degrees of freedom align with known dihedral angles or populated states. That said, the evaluation consists mainly of seeing whether the method recovers recognized conformations and can occasionally reach less-populated states; it does not include more comprehensive comparisons against standard enhanced-sampling approaches.", "theoretical_claims": "The paper’s main derivations—specifically, using Taylor expansions around a reference configuration and linking approximate energy invariance to low-curvature or degenerate directions of the Hessian—align with standard linear algebra and perturbation results. The statement that degenerate eigenvalues in the Hessian induce rotation-like symmetries follows well-known principles. While minor indexing or notation issues are present, they do not invalidate the arguments, and the proofs appear consistent with conventional theoretical foundations.", "experimental_designs_or_analyses": "The experimental approach—focusing on alanine dipeptide (both vacuum and implicit solvent) and chignolin (implicit solvent)—is broadly sound as a preliminary demonstration that the discovered transformations can access known conformers. Both systems are widely recognized testbeds for validating the discovery of local minima and dihedral-angle variations. A long MD simulation serves as a reference to confirm that the proposed method recovers states consistent with standard MD. The procedure involves extracting approximate symmetries at a reference configuration, systematically applying transformations (on a grid) to generate new structures, and then minimizing and running short simulations to confirm low-energy stability. This is appropriate to gauge whether the method meaningfully covers conformational diversity; however, the study does not provide comparisons to established sampling methods. Moreover, while the results are promising, the scope is limited to relatively small systems, leaving open questions on scalability, performance with explicit solvent, and how comprehensively the new approach samples large, complex conformational landscapes.", "relation_to_broader_scientific_literature": "The paper addresses the longstanding problem of identifying collective variables for molecular simulations, a task also tackled by methods such as metadynamics, replica-exchange MD, and various data-driven approaches. Unlike those methods, which often rely on extensive trajectories or predefined biases, this work derives low-energy directions straight from the force field’s gradient and Hessian, placing it closer in spirit to normal-mode analyses used in protein dynamics.", "essential_references_not_discussed": "All the relevant references are discussed.", "other_strengths_and_weaknesses": "List of typos:\nLine 34, «approximate»\nLine 131, missed space in front of «of»\nLine 275, «assume», «meaning»\nLine 288, «only»\nLine 316, «minimizing»\nLine 551, 569, «eq 5» and similar — no reference\nLine 681, «DOF»\nLine 715, «One»\nLine 718, has to be «degenerate eigenspaces for each»", "other_comments_or_suggestions": "I don’t think that this work lies in the field of machine learning, so I would recommend that the authors submit it to some applied biology conferences or journals.\n\nI see potential in this paper, but it requires significant refinement—especially in how the methodology is presented and how it is compared with competing approaches.", "questions_for_authors": "No questions", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The authors propose a new method to discover automatically degrees of freedom (DOF) in molecular dynamics simulations. The method does not rely on data, but seeks to compute instead approximate symmetries of the energy landscape. The method was praised for its novelty by all reviewers. Both reviewers that gave it the lowest score of "1 - Reject" still maintain that `This paper sounds like it has a lot of potential. But the present presentation prevents me from fully appreciating it.` or `I see potential in this paper`. 

While I agree that the presentation is perfectible, I prefer to see the upsides in presenting a computationally novel approach. While Reviewer 9HNJ wrote something like a blank check (a strong 4 accept rating with little substance, and no follow-up post-rebuttal), I have taken into consideration the long review by Reviewer zCrM and their positive overall assessment of the paper. I see that the authors have taken steps to assuage some of the concerns expressed by Reviewer YyW7 on the presentation of experimental results. 

For these reasons, I trust that the authors will take significant care in implementing **all** of the suggested presentation improvements, and I am overall supportive of this paper.
Decision: Accept (poster)