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REVIEW 3 major objections 4 minor 11 references

Learning collective variables that preserve transition rates

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper argues that enforcing the orthogonality condition $D\xi\nabla V_1 = 0$ on a learned surrogate manifold yields collective variables that preserve transition rates, and demonstrates it on butane with under ten percent rate error.

desk verdict A clean equivalence proof and a well-described CV discovery pipeline, but the surrogate-space OC is not the advertised theorem and the headline rate is 10.6%, not <10%. read the letter →

arxiv 2506.01222 v2 pith:UONIDIJE submitted 2025-06-02 math.NA cs.NAphysics.chem-phstat.ML

Shashank Sule , Arnav Mehta , Maria K. Cameron This is my paper
classification math.NAcs.NAphysics.chem-phstat.ML MSC 65C3060J60
keywords collectivevariablestransitionratesorthogonalityconditioneffectivedynamicsdiffusionmapsmanifoldlearningbutaneoverdampedLangevin
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to show that one geometric condition from effective-dynamics theory—the orthogonality condition, which asks that the collective variable’s level sets run along the slow directions of a scale-separated potential $V = V_0 + \epsilon^{-1} V_1$—is a practical criterion for learning reaction coordinates that preserve kinetics, not just metastable state separation. It proves that this condition is equivalent to the projected orthogonality condition used for pathwise error estimates, so the same constraint controls both relative entropy and pathwise distance between the projected and effective dynamics. The paper turns the condition into a numerical recipe: learn a surrogate hypersurface for the residence manifold from simulation data, learn a surrogate confining potential whose gradient is normal to that surface, and train a neural network collective variable whose gradient is orthogonal to that normal field. On butane, the learned collective variable reproduces the anti-gauche transition rate at $1.25\times 10^{-2}\,\mathrm{ps}^{-1}$ against a reference of $1.13\pm 0.08\times 10^{-2}\,\mathrm{ps}^{-1}$ (under ten percent relative error), while the dihedral angle gives 24 percent error. The paper also gives evidence that the diffusion tensor need not be uniformly positive definite for the rate to come out right.

What carries the argument

The load-bearing object is the orthogonality condition (OC), $D\xi\nabla V_1 = 0$: the gradient of the collective variable is perpendicular to the gradient of the stiff confining potential, so level sets of $\xi$ lie along the slow directions of $V = V_0 + \epsilon^{-1}V_1$. Proposition 2 identifies (OC) with the projected orthogonality condition (POC), $(I-\Pi)^{\top}\nabla V_1 = 0$, where $\Pi$ is the projection induced by $\xi$, making the same condition serve both the relative-entropy estimate and the pathwise-distance estimate. Computationally, the condition is enforced in a latent space rather than all-atom space: group-invariant features are embedded with diffusion maps, independent eigencoordinate selection and a hypersurface search choose coordinates so the embedded residence manifold is a hypersurface, diffusion nets extend the embedding out of sample, and a Laplacian conformal autoencoder removes spurious self-intersections; a surrogate potential is learned as a signed distance whose gradient is the surface normal, and the collective variable is trained so its gradient is orthogonal to that normal.

What would settle it

Use the GramMatrixCarbon feature map in the same pipeline: its embedding is parameterized by $\cos\theta$, the collective variable that overestimates the butane rate at $1.52\times 10^{-2}\,\mathrm{ps}^{-1}$. If an orthogonality-respecting collective variable learned from that embedding also lands near $1.52\times 10^{-2}$ rather than near the reference $1.13\times 10^{-2}$, then enforcing (OC) on a surrogate that misrepresents the manifold does not preserve kinetics.

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Extended reading notes

Core claim

The central claim is that a collective variable satisfying the orthogonality condition $D\xi\nabla V_1 = 0$ reproduces the statistical properties of the original overdamped Langevin dynamics under scale separation, and that this condition is equivalent to the projected orthogonality condition $(I-\Pi)^{\top}\nabla V_1 = 0$. The equivalence means one geometric constraint controls both relative entropy and pathwise error estimates in coarse graining. The paper implements this by learning a surrogate manifold from featurized simulation data, learning a signed-distance surrogate potential on it, and training an encoder whose gradient is orthogonal to the surrogate potential’s gradient; the resulting collective variable is the composition of that encoder with the manifold embedding. In the butane case study, the learned variable separates the anti and gauche states and reproduces the anti-gauche transition rate within ten percent relative error, whereas the conventional dihedral angle overestimates it by 24 percent. The paper further claims that a rank-deficient diffusion tensor—as in the $(\sin\theta, \cos\theta)$ variable—does not prevent faithful transition rates.

Load-bearing premise

The construction assumes the learned surrogate manifold faithfully represents the true residence manifold, with its normal directions matching the true stiff fast directions; if the diffusion-map embedding is distorted, self-intersecting, or mis-dimensioned, the orthogonality condition enforced in latent space need not correspond to the fast subspace of the original dynamics.

Editorial extensions

If this is right

  • Transition rates for rare conformational changes can be computed from the low-dimensional effective dynamics once the learned collective variable satisfies the orthogonality condition, avoiding brute-force all-atom simulation.
  • The equivalence of (OC) and (POC) means a single constraint improves both relative-entropy and pathwise estimates of coarse-graining error.
  • The butane experiments suggest that requiring $D\xi D\xi^{\top}$ to be uniformly positive definite is too strong; rank-deficient diffusion tensors still yield correct rates as long as the collective variable does not collapse distinct metastable states.
  • Group-invariant featurization that retains hydrogen coordinates can change the learned manifold enough to make the difference between a collective variable that separates metastable states and one that does not.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same pipeline should carry over to any molecule whose stiff degrees of freedom produce a codimension-one residence manifold, but only if the feature map yields a topologically faithful embedding; the paper’s own HyperSurface score gives false negatives, so this is the main transfer risk.
  • The rank-deficiency evidence invites a theoretical extension: error estimates for effective dynamics with degenerate diffusion tensors, replacing uniform positive definiteness with a weaker condition that the diffusion tensor does not vanish on its support.
  • The Laplacian conformal autoencoder’s ability to undo self-intersections is a standalone manifold-learning contribution that could be tested on other spectral embeddings independent of collective-variable construction.
  • One testable extension is to compare the feature-map route to Haar-averaged group-invariant diffusion-map kernels on the same butane data, since the paper notes the two routes have not been compared.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper revisits quantitative coarse-graining for overdamped Langevin dynamics, proves the equivalence of the orthogonality condition (OC) and the projected orthogonality condition (POC), and proposes a data-driven pipeline (Algorithm 1) for learning collective variables: featurize MD data, embed the residence manifold as a hypersurface via diffusion maps/diffusion nets/LAPCAE, learn a surrogate signed-distance potential, and enforce orthogonality in the latent space. The method is tested on butane, where the PlaneAlign-based CV is reported to reproduce the anti-gauche transition rate at 1.25×10−2 ps−1 versus the reference 1.13±0.08×10−2 ps−1. The paper also provides empirical evidence that a rank-deficient diffusion tensor can still reproduce transition rates and emphasizes the role of hydrogen atoms in featurization.

Significance. If the central claim were fully supported, the paper would make a useful contribution: Proposition 2 gives a clean identification of two conditions from different error estimates, the surrogate-manifold framework is a plausible route to imposing geometric conditions without knowing V1, the butane study is detailed, and the code is public. The empirical demonstration with (sin θ, cos θ) showing that a rank-deficient diffusion tensor can reproduce rates is interesting and relevant. However, the quantitative headline claim is contradicted by the authors' own table, and the connection between the latent-space condition actually imposed and the theoretical condition used in the error estimates is not established. The approach is defensible, but the paper currently overstates what is proven and what is demonstrated.

major comments (3)
  1. [Abstract; §5.4, Table 2] The paper repeatedly claims a 'less than ten percent' relative error for the PlaneAlign CV, but Table 2 gives 1.25×10−2 ps−1 against the reference 1.13±0.08×10−2 ps−1. Relative to the point estimate this is (1.25−1.13)/1.13 ≈ 10.6%, slightly above ten percent. The Introduction's phrase 'nearly 10%' is accurate, but the Abstract and §5.4 are not. This is a concrete, checkable numerical claim and should be corrected or carefully qualified (e.g., 'within 1.5 standard deviations of the reference').
  2. [Algorithm 1; §5.2; §6 (Limitations)] Algorithm 1 enforces the condition ∇bξ·∇bΦ = 0 in latent coordinates, but the theoretical estimates (32) and (40), as well as Proposition 2, concern the true condition Dξ∇V1 = 0 in the all-atom space R^N. For ξ = bξ ∘ Ψ, the chain rule gives Dξ = (Dbξ ∘ Ψ)DΨ, so latent orthogonality controls only the component of DΨ∇V1 aligned with ∇bΦ; no quantitative bound is supplied that transfers the surrogate condition to the true coarse-graining error. The acknowledgment in §6 that (OC) is imposed on the surrogate rather than in R^N does not by itself close this gap. Without such a transfer estimate, the reported PlaneAlign rate improvement cannot be attributed to satisfaction of the advertised orthogonality condition rather than to a favorable reparameterization of the dihedral angle.
  3. [§4.4; §5.2] The successful case study is selected by visual inspection, and the successful CV is then also chosen by inspection. Section 4.4 states that PlaneAlign 'clearly visually' gives the best 2D embedding despite its low HyperSurface score, and §5.2 sets bξ(x) = arctan2(x2, x1) after observing that the embedding is a circle parameterized by the dihedral angle. This weakens the demonstration that the proposed algorithm automatically discovers a good CV: the favorable result could be obtained by any smooth reparameterization of the dihedral angle. The paper should either present these two choices as human-guided and moderate the claims accordingly, or provide an automated selection criterion that recovers PlaneAlign without visual inspection.
minor comments (4)
  1. [§3.2.1] The transition rate for ξ3 = (sin θ, cos θ) is printed as '1.19 ps−1' in the text; it should read 1.19×10−2 ps−1 to match Table 1.
  2. [§2.2.1] The inequality is attributed to 'Cziszar-Kullback'; the standard spelling is Csiszár-Kullback. The same spelling appears in the proposition label and should be corrected.
  3. [§4.5; §5.1] The notation 'Leikonal' is used for the eikonal loss in equation (71); consider renaming to 'L_eikonal' for consistency with the rest of the text.
  4. [Table 3] The header 'F eature map' contains a stray space; also, the table would be easier to read if the hyperparameter column distinguished αZero, αnormals, and αOC explicitly, since αnormals is reported as 0.0 for both rows.

Circularity Check

2 steps flagged · score 4.0 of 10

Latent OC is built into the network by construction, and the PlaneAlign CV is read off from an embedding already parameterized by the dihedral angle; the headline rate is therefore a smooth reparameterization of a known CV.

  1. self definitional [Section 3.3 (Algorithm 1) and Section 6 (Limitations)]
    "Finally, we learn bξ which satisfies the (OC) ∇bξ · ∇bΦ = 0. Notably, this bξ can be lifted to an overall collective variable via composition with the parameterization, given by ξ := bξ ◦ Ψ. Therefore, our method can be described as recovering a CV by enforcing the orthogonality condition in a latent space RD+1 instead of the ambient space RN . ... in our framework, the orthogonality condition (OC) is imposed on the surrogate manifold rather than directly in the all-atom space RN ."

    The theoretical estimates (32) and (40) require Dξ∇V1 = 0 in the all-atom space RN. Algorithm 1 instead trains ∇bξ·∇bΦ=0, where bΦ is defined as the signed-distance surrogate potential whose gradient is, by construction, the normal to the learned manifold cM. Any bξ orthogonal to that normal satisfies the latent condition automatically, so the statement that the learned CV satisfies (OC) is built into the loss rather than derived from the true fast subspace. The paper explicitly concedes that OC is imposed on the surrogate manifold, not in RN, and provides no estimate transferring latent orthogonality to D(bξ∘Ψ)∇V1.

  2. renaming known result [Section 4.4 and Section 5.2]
    "PlaneAlign provides a better embedding than GramMatrixCarbon because the PlaneAlign embedding is a circle parameterized by the dihedral angle ... For PlaneAlign, we find that bξ(x) = arctan2(x2, x1) ... is orthogonal to ∇bΦ and therefore satisfies (OC). Thus, given an interpretable enough visualization, CVs may be devised by inspection. Interestingly, the CV learned via PlaneAlign correlates significantly with the dihedral angle."

    The headline CV is selected by inspecting the PlaneAlign diffusion-map embedding after seeing that it is a circle parameterized by the known good CV, the dihedral angle. Setting bξ=arctan2(x2,x1) is reading off the angular coordinate of that circle; Section 5.4 then reports that this CV 'corresponds nearly exactly to the dihedral angle.' The resulting rate 1.25e-2 is therefore a smooth reparameterization of the dihedral-angle CV, whose two-dimensional (sinθ,cosθ) form already gave 1.19e-2 in Table 1. Presenting this as a machine-learned CV that 'reproduces the anti-gauche transition rate' renames a known empirical CV in the new latent coordinates rather than deriving a new prediction from the orthogonality condition.

full rationale

The analytical core is not circular: Proposition 2 proves (OC) ⇔ (POC) algebraically from the SVD of Dξ, and the error estimates (30), (32), (39), (40) are quoted from external prior work. The transition-rate numbers are also benchmarked against an independent brute-force all-atom reference (1.13±0.08×10^-2 ps^-1) and computed by solving the low-dimensional committor BVP with estimated free energies and diffusion tensors, not by fitting to that reference. So this is not a case of the whole derivation collapsing into its inputs. The circularity is partial and localized to the experimental claim. First, the algorithm's central condition is enforced as ∇bξ·∇bΦ=0 on a learned surrogate signed-distance function, making the latent 'OC' true by construction while the paper admits it is not imposed in RN; no transfer bound connects this to the true condition that drives the theory. Second, the PlaneAlign CV used for the headline rate is chosen by inspection after observing that the embedding is parameterized by the known dihedral angle, and arctan2 is essentially that angle, so the rate is a validation of a reparameterized known CV rather than an independent discovery. The paper's own limitation that HyperSurface gives false negatives for the very PlaneAlign embedding later used for the headline rate reinforces that the rate result rests on post-hoc visual selection rather than on the proposed selection heuristic. Note also that the claimed 'less than ten percent relative error' is not met by the reported numbers: 1.25×10^-2 is 10.6% above 1.13×10^-2. No load-bearing self-citation chain was found; the cited same-author items [Cam13], [ECT22], [SEC23] are used for standard rate/diffusion-map numerics and are not invoked to force the CV choice.

Assumptions & free parameters 6 free parameters · 5 assumptions · 2 invented entities

The central rate result depends on a chain of fitted or hand-chosen quantities: diffusion-map bandwidth, embedding dimension, number of eigenfunctions, loss weights, and metastable-set boundaries. The axioms are imported theory assumptions plus the key domain assumptions that the featurized data lie on an immersed hypersurface and that the diffusion net and learned potential faithfully represent it. The two invented entities (surrogate manifold/potential and LAPCAE) lack independent falsifiable evidence outside the paper.

free parameters (6)
  • diffusion map bandwidth epsilon = tuned via kernel sum test; exact value not reported
    Bandwidth controls the diffusion map embedding and therefore the surrogate manifold. The kernel sum test determines it, but the chosen value is not stated, so the embedding is not fully reproducible.
  • embedding dimension D+1 = 2 or 3 depending on feature map
    The surrogate manifold dimension is chosen using the HyperSurface heuristic, which the paper itself concedes has false negatives. This is a hand-chosen structural parameter on which the central result depends.
  • number of Laplacian eigenfunctions m = 25
    Free hyperparameter of the diffusion map step of Algorithm 2.
  • regularization weight zeta = 0.01
    Free hyperparameter in the IES and HyperSearch scores (64), set by hand.
  • loss weights alpha_DNet, alpha_LAPCAE, alpha_Zero, alpha_normals, alpha_OC = 1.0, 0.5, 1.0, 0.0, 1.0 (per Table 3)
    All loss terms in the neural network training use hand-chosen weights. These affect the learned surrogate manifold, the surrogate potential, and the learned CV.
  • transition rate boundary widths = A = {|θ − π| ≤ 0.2}, B = {|θ − π/3| ≤ 0.1} ∪ {|θ − 5π/3| ≤ 0.1}
    The definition of the metastable sets A and B is a modelling choice that affects the computed transition rate. The paper does not provide sensitivity with respect to these widths.
assumptions (5)
  • domain assumption The featurized trajectory samples a well-defined immersed hypersurface M̂ in RD+1.
    Algorithm 1 requires M̂ to be representable as a hypersurface so that normals and a vanishing potential can be learned. Section 4 admits this is only achieved locally, with self-intersections possible.
  • domain assumption The machine-learned diffusion net Ψ approximates the diffusion map embedding faithfully on the full manifold, not only on training points.
    Out-of-sample extension is assumed in Section 4.5: the diffusion net is trained to match ψF and Laplacian eigenfunctions on training points, but the central rate computation relies on CV values at points generated during the string method and metadynamics runs.
  • ad hoc to paper The surrogate potential Φ̂ learned from the eikonal loss has gradients that approximate normals to the true residence manifold M.
    The paper replaces the unknown confining potential V1 by a learned signed-distance-like Φ̂. The normals from tangent-plane fitting are used to improve the learned gradients, but no error estimate connects ∇Φ̂ to the true normal field of M.
  • domain assumption Langevin dynamics at friction 10 ps−1 is well approximated by the overdamped dynamics (1), and the anisotropic mass rescaling in Appendix A correctly converts rates to physical time units.
    The rate formula (48) is for overdamped dynamics, but the MD simulation is Langevin dynamics. Appendix A supplies the mass-rescaled diffusion tensor and the factor γ−1. This scaling heuristic enters the reported numerical rate.
  • standard math Standard assumptions of the effective dynamics theory: log-Sobolev inequality, Lipschitz drift and diffusion, finite κ, λ.
    These are stated as Assumptions 1, 3, 4, 5 in Section 2 and are imported from [LL10; Duo+18]. They are not verified for butane.
invented entities (2)
  • Surrogate manifold M̂ with surrogate potential Φ̂
    purpose: A learned stand-in for the unknown residence manifold M and the unknown confining potential V1, used to enforce the orthogonality condition in latent space.
    The surrogate manifold is constructed from data via diffusion maps and is validated only by visual inspection, the HyperSurface heuristic, and the final rate match. There is no independent measurement of M̂ or Φ̂.
  • LAPCAE (Laplacian Conformal Autoencoder)
    purpose: A neural network embedding that removes spurious self-intersections in the diffusion-map embedding by promoting functional independence of eigencoordinates.
    The self-intersection removal is shown in Figure 10 and described in Section 4.6, but the claim that LAPCAE undoes spurious topology is supported only by visual inspection and the energy (68), not by a quantitative topology-preservation certificate.

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Pith. "Pith review of Learning collective variables that preserve transition rates." pith.science (2026). https://pith.science/paper/UONIDIJE

@misc{pith2026250601222,
  author       = {Pith},
  title        = {Pith review of: Learning collective variables that preserve transition rates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UONIDIJE}},
  note         = {Machine review of arXiv:2506.01222}
}
read the original abstract

Collective variables (CVs) play a crucial role in capturing rare events in high-dimensional systems, motivating the continual search for principled approaches to their design. In this work, we revisit the framework of quantitative coarse graining and identify the orthogonality condition from Legoll and Lelievre (2010) as a key criterion for constructing CVs that accurately preserve the statistical properties of the original process. We establish that satisfaction of the orthogonality condition enables error estimates for both relative entropy and pathwise distance to scale proportionally with the degree of scale separation. Building on this foundation, we introduce a general numerical method for designing neural network-based CVs that integrates tools from manifold learning with group-invariant featurization. To demonstrate the efficacy of our approach, we construct CVs for butane and achieve a CV that reproduces the anti-gauche transition rate with less than ten percent relative error. Additionally, we provide empirical evidence challenging the necessity of uniform positive definiteness in diffusion tensors for transition rate reproduction and highlight the critical role of light atoms in CV design for molecular dynamics.

Figures

Figures reproduced from arXiv: 2506.01222 by the authors.

Figure 1
Figure 1. The butane molecule C4H10 can be described in atomic coordinates in R 42 . butane can be coarse-grained along the dihedral angle θ in its carbon backbone, where it exhibits metastability. We simulate butane at 300K under Langevin dynamics with high friction, resulting in dynamics well-approximated by the overdamped regime (1). We provide additional details on this approximation in Appendix A where we also describe h… view at source ↗
Figure 2
Figure 2. The dynamics of butane can be effectively coarse-grained by the dihedral angle [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Free energy profiles (top) and diffusion tensors (bottom) for the collective variables [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: In BondAlign(1,2), the C1 − C2 bond is aligned along the x-axis. 4.2 Diffusion maps We pass the data X through F to get a featurized pointcloud XF = {pi} where pi = F(xi), which we posit is sampled from the featurized manifold MF := F(M). To embed the featurized manifo…
Figure 5
Figure 5. Figure 5: In BondAlign(2,3), the C2 − C3 bond is aligned along the x-axis [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: In PlaneAlign the basis is changed such that the first three carbon atoms are in the xy-plane. motivated by the fact that if ∆MF is the Laplace-Beltrami operator on MF and ψ˜ i are the Laplacian eigenfunctions satisfying: −∆MF ψ˜ i = λiψ˜ i , (54) then there exists an …
Figure 7
Figure 7. Figure 7: Plotting HyperSurface(D + 1) against D + 1 for different feature maps. Notably, GramMatrixCarbon gives the largest value of HyperSurface(·) at D +1 = 2 which corresponds to the residence manifold of dimension D = 1. Moreover PlaneAlign seemingly gives the best hypersur…
Figure 8
Figure 8. Figure 8: Visualizing the first two selected Laplacian eigenvectors for [PITH_FULL_IMAGE:figures/full_fig_p028_8.png]
Figure 9
Figure 9. Figure 9: Left to right, top to bottom: The diffusion map embeddings [PITH_FULL_IMAGE:figures/full_fig_p029_9.png]
Figure 10
Figure 10. Figure 10: Top left: The diffusion map embedding after featurization by recentering the [PITH_FULL_IMAGE:figures/full_fig_p030_10.png]
Figure 11
Figure 11. Figure 11: Gradients of potential ∇Φ learned for b BondAlign(1,2), BondAlign(2,3) and PlaneAlign. Here Φ is learned by minimizing b Lpotential. Lpotential in (72). In this case Φb ≈ ΦbSDF. However, HΦbSDF∇ΦbSDF = 0, (73) where HΦbSDF is the Hessian of the signed distance functio…
Figure 12
Figure 12. Figure 12: Gradients of CVs for BondAlign(2,3) and PlaneAlign [PITH_FULL_IMAGE:figures/full_fig_p034_12.png]
Figure 13
Figure 13. Figure 13: We visualize (θ(xi), ξ(xi)) for xi ∈ X , the training data used for learning ξ. Note that ξ is a nonlinear function of θ–the single discontinuity is due to the use of the arctan2 function which requires a branch cut at θ = 2π. 34 [PITH_FULL_IMAGE:figures/full_fig_p03…
Figure 14
Figure 14. Figure 14: When visualizing data in the CV space learned with the [PITH_FULL_IMAGE:figures/full_fig_p042_14.png]
Figure 15
Figure 15. Figure 15: Free energy and diffusion tensor for BondAlign(2,3) with LAPCAE as the global coordinate chart parameterizing the resident manifold. References [ABG06] Hernan Alonso, Andrey A Bliznyuk, and Jill E Gready. “Combining docking and molecular dynamic simulations in drug de…
Figure 16
Figure 16. Figure 16: Free energy and diffusion tensor for PlaneAlign. 44 [PITH_FULL_IMAGE:figures/full_fig_p044_16.png]

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