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REVIEW 2 major objections 5 minor 45 references

Markov-type state models to describe non-Markovian dynamics

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Projecting exponentiated microstate transitions onto macrostates gives exact macrostate population dynamics when the microstates are Markovian.

desk verdict A careful, mostly sound methods comparison on MSM estimation, but the 'exact by design' claim for the microstate-based projection needs an explicit local-equilibrium initial condition. read the letter →

arxiv 2412.08660 v2 pith:7FJOCLMC submitted 2024-12-05 cond-mat.soft physics.bio-phphysics.comp-phphysics.data-an

Sofia Sartore , Franziska Teichmann , Gerhard Stock This is my paper · ORCID
classification cond-mat.softphysics.bio-phphysics.comp-phphysics.data-an
keywords Markovstatemodelsnon-MarkoviandynamicsgeneralizedmasterequationmemorykernelmicrostateprojectionChapman-KolmogorovtestproteinfoldingHP35
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 addresses a common failure of Markov state models: when the lumped macrostates do not have a clean separation of fast internal and slow interstate dynamics, simply counting macrostate transitions underestimates implied timescales and makes populations decay too fast. It assumes instead that the underlying microstates are Markovian at a short lag time, and studies ways to lift the microstate dynamics up to the macrostate level. Its central new result is a microstate-based transition matrix whose exponentiation and projection onto macrostates gives, by construction, the exact macrostate populations whenever the microstate assumption holds. The paper also derives practical consequences: any residual non-Markovianity in the macrostate model must then come from suboptimal microstates, and a hybrid method that uses MD data at short times and a Markov model at long times is best for practical analysis. Both a one-dimensional toy model and an all-atom HP35 folding trajectory are used to compare the approaches.

What carries the argument

The central object is the microstate-based transition matrix of Eq. (9), $T^{Mic}(m\tau)=A^T t^m(\tau) D_n A D_N^{-1}$, where $A$ is the macrostate aggregation matrix ($A_{iJ}=1$ if microstate $i$ belongs to macrostate $J$) and $D_n$, $D_N$ are diagonal matrices of normalized equilibrium populations of microstates and macrostates. This object lifts the exact microstate propagator $t^m(\tau)$ to the macrostate level instead of exponentiating a macrostate transition matrix. The identity $A^T D_n A D_N^{-1}=1_N$ is what makes the projection preserve populations. The same aggregation machinery also gives the local-equilibrium approximation (exponentiating the projected one-step matrix) and the Laplace-transform variant, which is a long-time optimal projection.

What would settle it

Take the one-dimensional toy model in the Markovian regime ($h/k < 1$), compute $T^{Mic}$ via Eq. (9) from the exact microstate transition matrix, and compare macrostate populations with the analytic reference at several $m\tau$; any deviation would refute the exactness claim. In the HP35 system, a direct check is to test the microstate Chapman-Kolmogorov relation (Eq. 4) at the chosen $\tau$: when it fails, the microstate-based projection should visibly lose accuracy in the macrostate Chapman-Kolmogorov test.

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Extended reading notes

Core claim

The central claim is that the macrostate transition matrix should not be obtained by counting macrostate transitions directly; instead, one should propagate the microstate populations with the microstate transition matrix and then project onto macrostates. Concretely, with aggregation matrix $A$ and equilibrium population matrices $D_n$ and $D_N$, the microstate-based evaluation $T^{Mic}(m\tau)=A^T t^m(\tau) D_n A D_N^{-1}$ yields exactly the macrostate population dynamics of the underlying microstate model, provided $t(\tau)$ is Markovian at lag time $\tau$. As a corollary, any deviation between this projected dynamics and direct MD observations is attributable to the microstate partitioning rather than to the lumping step. The paper further shows that the Laplace-transform projection of Eq. (10) improves implied timescales over the naive local-equilibrium counting, that qMSM, based on a generalized master equation with a memory kernel, can handle ill-defined macrostates, and that a hybrid MD/MSM scheme that uses exact MD transition matrices at short times and an MSM at long times gives the most accurate Chapman-Kolmogorov behavior in the tested cases.

Load-bearing premise

The construction depends on the small-scale states being memory-free after one time step, and on each large-scale state being in its typical internal mixture at the start; if either condition fails, the exact population dynamics are no longer guaranteed.

Editorial extensions

If this is right

  • If the microstate transition matrix obeys the Chapman-Kolmogorov relation at lag time $\tau$, then the macrostate populations obtained from Eq. (9) match the MD reference exactly, not merely approximately.
  • A leftover discrepancy between the microstate-based projection and direct MD counts means the microstate partition itself is suboptimal, which separates lumping error from model error.
  • The hybrid MD/MSM scheme, which uses the exact MD transition matrix at short times and a long-lag-time Markov model afterwards, gave the closest Chapman-Kolmogorov agreement among all methods tested on the HP35 trajectory.
  • For datasets of many short trajectories, qMSM still produces stable implied timescales and a memory kernel, supporting the practical promise of predicting long-time dynamics from short simulations.

Reading between the lines

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

  • A diagnostic that the paper leaves implicit: comparing $T^{Mic}(m\tau)$ with direct MD counts at the same $m\tau$ isolates whether the residual error comes from non-Markovian microstates or from insufficient sampling, because the projection is exact under the Markovian-microstate assumption.
  • The exactness result motivates reformulating microstate optimization as minimizing the violation of the microstate Chapman-Kolmogorov relation, rather than tuning macrostate implied timescales.
  • One could extend the hybrid method by replacing the long-time MSM with qMSM, so the memory kernel continues beyond $t_{max}$ while retaining the exact short-time MD propagator.
  • Because the microstate-based projection preserves waiting-time statistics at the microstate level, it should permit computation of path-based observables such as state-to-state waiting times and transition pathways via microstate Monte Carlo sampling, exactly as the paper hints but does not demonstrate.
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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

2 major / 5 minor

Summary. This paper addresses the estimation of macrostate transition matrices in Markov state models when the usual timescale-separation assumption fails. Assuming Markovianity at the microstate level, the authors compare four approaches: the local-equilibrium (counting) approximation, the Hummer-Szabo projection, a 'microstate-based' propagation T^Mic(mτ)=A^T t^m(τ)D_n A D_N^{-1} (Eq. 9), and the quasi-MSM solution of a generalized master equation; they also introduce a hybrid MD/MSM scheme that uses directly counted transitions at short times and an MSM at long lag times. The methods are tested on an exactly solvable four-microstate/two-macrostate toy model with tunable degree of non-Markovianity and on a 300 μs all-atom HP35 folding trajectory, using implied timescales and Chapman-Kolmogorov tests as diagnostics. The main conclusions are that the local-equilibrium approximation underestimates implied timescales for non-Markovian lumpings, Hummer-Szabo is accurate, the microstate-based and qMSM methods perform well in the toy model, and the hybrid method matches the reference closely when long trajectories are available.

Significance. If the claims are properly qualified, the paper is a useful systematic comparison of practical estimators for non-Markovian macrostate dynamics. Its strengths include an exactly solvable toy model with an analytical reference, a realistic all-atom HP35 benchmark with publicly available data and reproduction instructions, an explicit discussion of qMSM's numerical pitfalls, and a simple hybrid estimator that is likely to be useful in practice. The conceptual novelty of the microstate-based projection is modest (it is essentially the standard lumping of an exact microstate chain under a local-equilibrium initial condition), but the paper's value lies in the side-by-side comparison and in the memory-kernel analysis. The main correctness issue is the unqualified claim that Eq. (9) yields exact macrostate dynamics by design; this needs a precise statement of the initial-condition assumption.

major comments (2)
  1. [Section 2.1, Eqs. (7)-(9)] The claim that T^Mic 'yields by design exact macrostate population dynamics' is only correct for initial microstate distributions that are locally equilibrated within each macrostate. Eq. (9) follows from P(mτ)=A^T t^m(τ)p(0) only after replacing p(0) by D_n A D_N^{-1}P(0); for an arbitrary p(0) (for example a single MD snapshot), P(mτ)=A^T t^m(τ)p(0) is not a function of P(0) alone, and no N×N matrix can represent the exact macrostate dynamics. The derivation and the concluding remarks should state this condition explicitly and replace 'exact by design' with 'exact for locally equilibrated initial conditions.' The same qualification applies to the statements in Section 2.1 that the approach preserves all dynamical properties, including waiting times and pathways, established for the microstates.
  2. [Section 3.2, Fig. 2b] The inference that deviations of the microstate-based calculation from the reference MD results in the Chapman-Kolmogorov test are caused by suboptimal microstates requires that the reference curves be initialized with the local-equilibrium distribution within each macrostate. The paper does not describe how the reference CK curves are generated. If they are averages over all trajectory frames in the starting macrostate, the local-equilibrium condition holds approximately and the conclusion is reasonable, but finite sampling also contributes; if they are propagated from individual snapshots, the conclusion does not follow. Please specify the CK initialization and, ideally, compare a locally equilibrated initialization with the empirical one to separate initial-condition effects from genuine microstate non-Markovianity.
minor comments (5)
  1. [Section 3.1] The sentence referring to 'the decay of the memory kernel K(t) in Fig. 1d' should refer to Fig. 1f, which contains the memory-kernel plot.
  2. [Section 2.2] The statement that 'given long enough MD data, the MD-based matrix T^MD is the best transition matrix you can get. Hence ... T can only be worse' is imprecise, because T^MD is only available for t≤tmax, while qMSM is intended for extrapolation beyond tmax; please restrict the comparison to the range where T^MD exists.
  3. [Section 3.1] The exact agreement of P^Mic with the toy-model reference should be labeled as a consistency check, since the reference and the microstate-based propagation use the same microstate transition matrix and the same locally equilibrated initial condition.
  4. [Section 3.2] Add uncertainty estimates (for example, a bootstrap over trajectory segments) to Fig. 2 to support statements such as 'deviations are overall only minor' and 'microstates are a main reason for the deviations.'
  5. [Section 1 and Section 2.1] The claim that the microstate-based approach 'has to the best of our knowledge not yet been mentioned' should be softened, since the idea of propagating microstates and then lumping is standard; the specific matrix expression in Eq. (9) is the clearly new element.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the microstate-based projection is a mathematical construction, and all tests use external MD data.

full rationale

The paper's central derivation (Eqs. 7-9) is a direct linear-algebra identity: given a Markovian microstate chain and an initial condition locally equilibrated within macrostates, the macrostate propagator A^T t^m D_n A D_N^{-1} reproduces the projected macrostate population by construction. This is a theorem, not a fit, and the paper's 'by design exact' wording openly marks it as such. The exactness claim is conditional on microstate Markovianity and on the local-equilibrium initialization p(0)=D_n A D_N^{-1} P(0); the HP35 Chapman-Kolmogorov tests may not satisfy the latter, but that is an assumption-quality issue, not circular reasoning. The qMSM section self-critically notes that inverting K from T_MD cannot improve on T_MD, which is the opposite of hiding an input-output equivalence. Self-citations (Nagel et al. 2023 benchmark, msmhelper, etc.) are used to set up the test system and tooling, not to justify the theoretical claims; the benchmark is checked against the independent Piana et al. MD trajectory, and code/data are publicly available. No predicted quantity reduces by construction to a fitted parameter or to a self-citation chain. The claimed exactness of Eq. (9) is mathematical; the empirical comparisons are external. Accordingly, no circular step is identified.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the microstate Markovianity assumption and on the local-equilibrium initial condition, which is not explicitly stated. No new entities are postulated. The listed free parameters are analysis choices rather than fitted constants used to force agreement with target results.

free parameters (3)
  • microstate lag time tau = 10 ns (HP35)
    Choosing the lag time at which microstate Markovianity is assumed; results depend on this choice, but it is a standard MSM parameter.
  • qMSM kernel decay time tau_K = 10 ns (HP35), from kernel decay in toy model
    User-chosen cutoff for memory kernel decay; affects qMSM results and is selected via the mean integral memory kernel in the HP35 case.
  • hybrid switch time t_max = 10 (toy model), 100 ns (HP35)
    Time up to which the exact MD transition matrix is used; the performance of the hybrid method depends on this choice.
assumptions (6)
  • domain assumption Microstate transition matrix is Markovian at lag time tau: t(m tau) = t^m(tau)
    Eq. (4), Section 2.1; foundational for the microstate-based method and for all comparisons.
  • domain assumption Initial microstate distribution within each macrostate is locally equilibrated: p(0) = D_n A D_N^{-1} P(0)
    Implicit in the claim that T^Mic gives exact macrostate populations; unstated in Section 2.1.
  • domain assumption MD data are stationary and time-homogeneous, and all states are connected
    Section 1, standard MSM assumptions.
  • standard math Hummer-Szabo projection relation T(t) = A^T t(t) D_n A D_N^{-1}
    Eq. (7); derived from the definitions of A, D_n, and D_N.
  • domain assumption qMSM memory kernel decays within tau_K
    Eq. (11), Section 2.2; needed for the qMSM solution to be valid.
  • domain assumption The HP35 benchmark MSM (Nagel et al. 2023) is a valid reference
    Section 3.2; the authors use their own prior benchmark as the reference, introducing mild self-citation.

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Cite this review

Pith. "Pith review of Markov-type state models to describe non-Markovian dynamics." pith.science (2026). https://pith.science/paper/7FJOCLMC

@misc{pith2026241208660,
  author       = {Pith},
  title        = {Pith review of: Markov-type state models to describe non-Markovian dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7FJOCLMC}},
  note         = {Machine review of arXiv:2412.08660}
}
read the original abstract

When clustering molecular dynamics (MD) trajectories into a few metastable conformational states, the Markov state models (MSMs) assumption of timescale separation between fast intrastate fluctuations and rarely occurring interstate transitions is often not valid. Hence, the naive estimation of the macrostate transition matrix via simply counting transitions between the states leads to significantly too short implied timescales and thus to too fast population decays. In this work, we discuss advanced approaches to estimate the transition matrix. Assuming that Markovianity is at least given at the microstate level, we consider the Laplace-transform based method by Hummer and Szabo, as well as a direct microstate-to-macrostate projection, which by design yields correct macrostate population dynamics. Alternatively, we study the recently proposed quasi-MSM ansatz of Huang and coworkers to solve a generalized master equations, as well as a hybrid method that employs MD at short times and MSM at long times. Adopting a one-dimensional toy model and an all-atom folding trajectory of HP35, we discuss the virtues and shortcomings of the various approaches.

Figures

Figures reproduced from arXiv: 2412.08660 by the authors.

Figure 1
Figure 1. One-dimensional toy model. (a) Schematic free energy landscape, indicating four microstates 1, 2, 3 and 4, which [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Twelve-state MSM constructed from a 300µs-long MD trajectory 30 of the folding of HP35. (a) First two implied timescales obtained from the microstates (dashed black, reference), the local equilibrium approximation (using the lag time τ = 10 ns, black), the Hummer-Szabo projection (using τ = 10 ns, green), the microstate-based result (blue), and the result from qMSM (using the kernel time τK = 10 ns, red). (b) Chapma… view at source ↗
Figure 3
Figure 3. As in Fig. 2, except that the original [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Reference graph

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