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REVIEW 3 major objections 5 minor 65 references

Reaction-Coordinate-Dependent Non-Markovian Friction Governs Protein-Folding Dynamics

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The paper shows that the memory friction governing protein folding depends strongly on the reaction coordinate and is largest in the folded, compact state, and that including this dependence in generalized Langevin equation simulations mark

desk verdict Genuinely new data-driven extraction of coordinate-dependent memory friction for protein folding, with a plausible qualitative result and a real but addressable weakness in the diagonal approximation. read the letter →

arxiv 2607.20504 v1 pith:YODNIBYW submitted 2026-06-30 physics.bio-ph

Lucas Tepper , Benjamin J. A. Héry , Cihan Ayaz , Florian N. Brünig , Benjamin Dalton , Anton Klimek , Roland R. Netz This is my paper · ORCID
classification physics.bio-ph PACS 87.15.hm05.40.-a
keywords proteinfoldingmemoryfrictiongeneralizedLangevinequationreactioncoordinatenon-MarkoviandynamicsconditionalVolterrainternalmeanfirst-passagetime
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

The paper sets out to show that the friction memory function in a generalized Langevin equation description of protein folding is strongly coordinate-dependent, not a single function of time. Using a conditional Volterra equation applied to long all-atom molecular dynamics trajectories of six fast-folding proteins, the authors extract a coordinate-dependent memory kernel and an effective coordinate-dependent mass. They find that friction is markedly larger in the folded state than in the unfolded state, consistent with internal friction in compact conformations. When the coordinate-dependent kernel is fed into a Markovian-embedded GLE simulation, the mean first-passage times between folded and unfolded states match the molecular-dynamics reference for most proteins, and for proteins with pronounced coordinate dependence they match substantially better than the standard coordinate-independent GLE. The claim is that coordinate-dependent non-Markovian friction is a fundamental feature of protein-folding dynamics, not a minor correction.

What carries the argument

The central object is the conditional Volterra equation (Eq. 10), which relates the running integral G(t,x) of the memory kernel to single-position-conditioned velocity autocorrelation functions C_vv(t,x) and velocity-force correlations C_vF(t,x). Its practical feasibility rests on the diagonal approximation that replaces the double-conditional velocity correlation C_vv(s,x,x_s) by C_vv(s,x)δ(x−x_s), decoupling the equations per position x. The extracted kernels are fitted to factorized multi-exponentials and simulated via a Markovian embedding that samples the correct stationary distribution, allowing direct comparison of mean first-passage times with MD.

What would settle it

Run a controlled test with a known position-dependent memory kernel: simulate long trajectories from the GLE, then apply the paper's diagonal conditional-Volterra extraction and compare the recovered kernel with the true one, especially in the barrier region where the reaction coordinate changes rapidly. A systematic mismatch there would falsify the method; alternatively, compute the full double-conditional correlation in the MD data and show that the off-diagonal terms are non-negligible.

Watch

Extended reading notes

Core claim

The central discovery is that the memory kernel extracted from MD trajectories of six fast-folding proteins varies significantly with the reaction coordinate, rising towards the folded state, with the slow friction component dominating barrier-crossing kinetics. The paper further shows that a GLE with both coordinate-dependent mass and coordinate-dependent memory friction (Eq. 3) reproduces the MD mean first-passage times at least as well as, and for several proteins better than, the commonly used coordinate-independent GLE (Eq. 1). For one protein (λ-repressor), which has a largely coordinate-independent friction profile, the simpler model performs comparably or slightly better, indicating

Load-bearing premise

The extracted friction profile rests on the assumption that, when the trajectory is conditioned on a starting position, velocity correlations at two different later positions are negligible; if those off-diagonal correlations matter, the position-dependent kernel and the reported folded-state friction enhancement would be biased.

Editorial extensions

If this is right

  • For proteins with strong coordinate dependence, coordinate-independent GLE simulations fail to reproduce MD folding and unfolding kinetics; including the coordinate-dependent kernel restores agreement.
  • The slow memory component, with timescales from about 10 ns to 60 ns, dominates barrier crossing and is largest in the folded state, locating the kinetic effect of internal friction in the folded basin.
  • The coordinate-dependent mass also rises toward the folded state, and the effective potential used in the GLE produces the correct stationary distribution in position and velocity.
  • The authors state that the framework should extend naturally to larger proteins and other macromolecular systems.

Reading between the lines

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

  • Because the diagonal approximation is validated only indirectly through mean first-passage times, a direct check on the magnitude of off-diagonal correlations in the double-conditional velocity correlation could strengthen or revise the extracted friction profile; if those terms are non-negligible near the barrier, the reported folded-state enhancement could be biased upward.
  • The method should be portable to single-molecule force spectroscopy experiments where position-dependent diffusion coefficients have been inferred; a position-dependent memory kernel would predict refolding time distributions different from the Markovian models currently fitted to such data.
  • If the slow folded-state friction component is the dominant kinetic control, mutations that alter the compactness of the folded state should measurably change folding rates through the friction term, a testable prediction.
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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 / 5 minor

Summary. The paper introduces an extraction method for reaction-coordinate-dependent non-Markovian friction in a generalized Langevin equation (GLE) with position-dependent mass and memory kernel. Starting from a conditional Volterra equation derived from an exact GLE, the authors approximate the double-conditional velocity autocorrelation as diagonal in x, extract Γ(t,x) from all-atom MD trajectories of six fast-folding proteins, and report that the memory friction increases strongly toward the folded state. They fit the kernels to two-exponential forms, embed them in a Markovian scheme, and compare simulated mean first-passage times with MD. The central claim is that including RC-dependent memory friction improves the low-dimensional description of protein folding kinetics and provides evidence for internal friction in compact states.

Significance. If the extraction is valid, this is a useful contribution: it gives a data-driven route to spatially resolved non-Markovian friction, connects to internal-friction interpretations, and supplies open-source code. The paper also derives a Fokker–Planck stationary distribution for the embedding and checks marginal distributions analytically. The multi-protein comparison is a strength, and the raw observation that the memory kernel rises in the folded basin is physically plausible. However, the quantitative claims rest on an unvalidated diagonal approximation, a two-exponential fit whose slow time constants exceed the extraction window for several proteins, and an embedding whose correction factor is not negligible. These issues need to be resolved before the headline conclusions are fully supported.

major comments (3)
  1. [Main text, Eq. (8) to Eq. (9)] The load-bearing step is the replacement C_vv(s,x,x_s) ≈ C_vv(s,x)δ(x−x_s). The text says 'the validity of this approximation will be checked below,' but the subsequent check is the joint τ_MFP comparison in Fig. 4, which tests the diagonal ansatz together with the two-exponential fit, the mass fit, and the embedding. It cannot localize errors from omitted off-diagonal correlations. In barrier bins, x(t) changes rapidly, so C_vv(s,x,x_s) will have substantial weight at x_s≠x. If those terms contribute, the extracted Γ(t,x), especially the folded-state peak in γ2(x), is biased. Please provide a direct test of the off-diagonal terms, e.g., evaluate the omitted integral in Eq. (8) or compare the diagonal-extracted Γ with a solution of the full Volterra equation in a simplified model.
  2. [SI §III, Fig. S1 and Fig. 3G–L] The slow memory times τ2 exceed the maximum extraction window for several proteins: Villin τ2=46.7 ns with t_max≤16 ns, and λ-repressor τ2=64.3 ns with t_max≤16 ns. Protein G and α3D have τ2≈14–15 ns with t_max≈16 ns, which is marginal. A two-exponential fit whose slow time constant is several times longer than the fitted time window is not identified by the data; the reported rise of γ2(x) toward the folded state may be an extrapolation artifact of the fit, not a feature of the MD data. The SI short-vs-long window comparison (Table S4) is for coordinate-independent kernels only and does not test the RC-dependent fits. Please show that the γ2(x) trends are robust to the fitting window, e.g., by fixing τ2 to values above and below the extraction window and rerunning the τ_MFP comparison.
  3. [SI §IV.B, Eq. (S30), Fig. S8] The Markovian embedding is claimed to reproduce the target kernel, but Eq. (S30) contains an exponential correction factor exp(∫ γ'_i/(2γ_i) v ds'), which is assumed to be ≈1 because ⟨v⟩=0. Figure S8 shows that this factor ranges between about 0.5 and 3 in the Villin simulation, and Fig. S9 shows deviations E that are non-negligible for some frames. Since the GLE simulations are used as validation of the extracted parameters, this embedding error propagates into Fig. 4 and weakens the quantitative comparison with MD. Please quantify the effect of this factor on the memory kernel and on the computed τ_MFP values, or modify the embedding to remove the approximation.
minor comments (5)
  1. [Fig. 2 caption] The color coding of the x-dependent curves is described only as 'blue unfolded and yellow folded state'; please add a color bar or explicit legend so the reader can map colors to x values.
  2. [Main text after Eq. (3)] The statement 'the GLEs in Eqs. 2 and 3 are exact' may be misinterpreted: Eq. (3) is an exact form for the projected dynamics only with the appropriate projection operator, and the extraction in this paper uses an additional approximation. Please qualify the wording to distinguish the exact GLE form from the approximate extraction scheme.
  3. [SI Table S2] Some fitted mass parameters are extremely small or large (e.g., θ1=1.724e-96 for Villin), suggesting poor conditioning of the fit parameterization. A more stable basis for m(x) would improve reproducibility.
  4. [Fig. 4] The figure compares simulated and MD τ_MFP curves without error bars or quantitative error metrics. Given that the central claim is that RC-dependent friction improves the description, please report a quantitative measure (e.g., log-mean-squared error) and, where feasible, bootstrap uncertainties.
  5. [References] Reference [42] is an arXiv preprint; if the derivation of Eqs. (2) and (3) relies on it, please ensure the reference is to a published version or peer-reviewed source, and state the extent of the reliance.

Circularity Check

1 steps flagged · score 4.0 of 10

The extraction itself is data-driven and not forced, but the exactness of the starting RC-dependent GLE is imported from same-author citations, and the main validation is an in-sample consistency check rather than an independent prediction.

  1. self citation load bearing [Main text, Eqs. 2-3 discussion (p.2)]
    "Note that the GLEs in Eqs. 2 and 3 are exact, while the GLE in Eq. 1 only holds for m(x) = m0, which is not a good approximation for proteins."

    The whole extraction scheme starts from the claimed-exact RC-dependent GLE, Eq. 3. Exactness is not demonstrated here; it is attributed to refs. [42] and [45], both by the present group (Héry/Tepper/Netz; Ayaz/Tepper/Netz). If that imported exactness were wrong, the extracted Γ(t,x) would be an effective fitting object rather than the true memory kernel. The empirical folded-state friction peak is therefore built on a load-bearing self-citation, though the profile itself could in principle have been flat or decreasing, so the central claim is not forced by construction.

full rationale

The central derivation, Eqs. 6-10, is a genuine Volterra inversion: G(t,x) is solved from conditional velocity correlation functions, so the reported increase of γ2(x) toward the folded state is an empirical outcome that could have been flat or decreasing; it is not an identity. I do not raise the score to 6 because no τ_MFP value is a fitted parameter: the GLE simulations are emergent and could have disagreed with the MD reference. Two weaknesses are noted but not counted as circularity. First, the diagonal approximation C_vv(s,x,x_s) ≈ C_vv(s,x)δ(x−x_s) that converts Eq. 8 into Eq. 9 is announced with 'the validity of this approximation will be checked below', but the subsequent check is the global τ_MFP comparison, a joint test of the diagonal ansatz, the two-exponential fit, the Markovian embedding, and the m(x) fits; it cannot isolate errors from omitted off-diagonal correlations. Second, the Fig. 4 comparison uses the same MD trajectories from which Γ(t,x) and m(x) were extracted, making the agreement a consistency check of the inversion/fitting/embedding chain rather than an out-of-sample prediction. These limit the strength of the empirical claim but do not make the derivation circular by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

Every dynamical quantity used in the final simulations (m(x), γ_i(x), τ_i) is fitted to the same MD data used for validation. The method itself adds no new physical entity; the main unproven ingredient is the diagonal approximation. The exact GLE background is pulled from the authors' own prior papers.

free parameters (5)
  • τ₁, fast memory time per protein = e.g. 384.7 ps (Villin)
    Fitted to extracted Γ(t,x) via Eq. S11; shared across x; controls memory decay in simulations.
  • τ₂, slow memory time per protein = e.g. 46.7 ns (Villin)
    Fitted via Eq. S11; for Villin and λ-repressor exceeds the extraction window.
  • γ₁(x), γ₂(x) friction profiles = 5 polynomial coefficients per component per protein, SI Table S3
    Fitted per x-bin; central empirical outputs and inputs to Eq. S12.
  • m(x) mass profile fit parameters θ_j = 6 parameters per protein, SI Table S2
    Mass computed from MD Eq. 11 then fit to exponential-plus-polynomial for simulation (Eq. S37); affects U_eff and dynamics.
  • RC parameters β, γ = β=30 nm⁻¹, γ=1.6
    Taken from prior work [27], not fitted here, but they define the reaction coordinate and thus influence all extracted profiles.
assumptions (6)
  • domain assumption GLE Eq. 3 with RC-dependent mass and memory kernel is an exact projection of the atomic dynamics
    Used as starting point; the paper does not rederive it, citing Refs. [42,44,45].
  • domain assumption Orthogonality ⟨v(0)δ(x(0)−x)F_R(t)⟩=0
    Needed to pass from Eq. 6 to Eq. 7; quoted from Ref. [42].
  • ad hoc to paper Diagonal approximation C_vv(s,x,x_s)≈C_vv(s,x)δ(x−x_s)
    Transforms exact Eq. 8 into tractable Eq. 9; validity asserted but not directly tested.
  • domain assumption Markovian embedding Eq. S12 reproduces the GLE Eq. 3, with correction factor C≈1
    Needed to simulate the GLE; tested numerically for Villin in SI Figs. S8–S9 with error small for most frames.
  • domain assumption MD trajectories are long enough and equilibrium-sampled for correlation estimates
    Trajectories 125–1154 µs from Ref. [52]; no convergence diagnostics provided.
  • domain assumption Velocity via central difference at 200 ps sampling is adequate
    Velocities computed from x(t) with Eq. S62; discretization effects not quantified.

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Pith. "Pith review of Reaction-Coordinate-Dependent Non-Markovian Friction Governs Protein-Folding Dynamics." pith.science (2026). https://pith.science/paper/YODNIBYW

@misc{pith2026260720504,
  author       = {Pith},
  title        = {Pith review of: Reaction-Coordinate-Dependent Non-Markovian Friction Governs Protein-Folding Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YODNIBYW}},
  note         = {Machine review of arXiv:2607.20504}
}
read the original abstract

It is common to project the full atomic-resolution representation of a protein onto a one-dimensionalreaction coordinate (RC) to capture the protein-folding kinetics. As a direct consequence ofthis dimensionality reduction, non-Markovian friction emerges in the framework of the general-ized Langevin equation (GLE). All previous applications of GLEs to protein folding employed anRC-independent friction memory function and therefore did not account for the different frictionin the folded and unfolded states. Using a recently derived GLE with RC-dependent mass andfriction memory function, we introduce a novel method to extract memory functions from timeseries data via a conditional Volterra equation. When applied to molecular dynamics (MD) data ofsix fast-folding proteins, we find strongly RC-dependent memory friction in line with the intuitiveexpectation that friction is higher in the folded than in the unfolded state due to internal proteinfriction. Our numerically efficient method to simulate the GLE confirms the accuracy of the GLEparameter extraction by comparison with the MD data. We show that RC-dependent memoryfriction not only adds physical insight into the folding process but also significantly improves thedescription of protein folding kinetics using low-dimensional RCs.

Figures

Figures reproduced from arXiv: 2607.20504 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Friction profiles [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Mean first-passage-time profiles [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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