{"id":"a8f79139-6669-45bf-be8f-e090e9e7384a","arxiv_id":"2412.08660","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A microstate-based projection and a hybrid MD/MSM approach estimate macrostate transition matrices more accurately than direct counting when the macrostate dynamics are non-Markovian.","lead":"This paper tests several ways to build state models that describe non-Markovian dynamics in molecular simulations, where the usual Markov assumption fails. Two new simple approaches, one that projects microstate dynamics and one that mixes short-time simulation data with a long-time Markov model, are shown to work well on a toy model and on a protein folding trajectory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (9)'s 'exact by design' macrostate populations require p(0)=D_n A D_N^{-1} P(0); the HP35 CK tests do not enforce this, so the exactness claim is overbroad and the test cannot separate initial-condition mismatch from microstate non-Markovianity.","rationale":"The reader's weakest_assumption correctly identifies both microstate Markovianity and local-equilibrium initialization as assumptions underlying Eq. (9). My stress-test focuses specifically on the initialization condition, because it is the component that makes the paper's central 'by design exact' claim overbroad as stated. The HP35 CK test uses a fixed initial MD frame, so it cannot validate exactness without either enforcing local equilibrium at initialization or accounting for the actual microstate distribution. The manuscript's own discussion in Section 3.2 blames residual deviations on suboptimal microstates, but does not isolate the initialization effect. This is a real, testable limitation rather than a fatal flaw, so the reader's CONDITIONAL verdict remains appropriate.","tokens_in":13719,"tokens_out":4642,"duration_ms":48417,"concrete_test":"Recompute the HP35 Chapman-Kolmogorov test (Fig. 2b) under two initializations: (i) set the microstate subpopulation of each macrostate J to π_i/Π_J (local equilibrium), and (ii) use the raw microstate occupancy of the initial frame. Compare T^Mic(mτ) predictions to the reference for both. If (i) matches the reference but (ii) does not, the exactness claim is initialization-dependent; if (i) also deviates, the residual is microstate non-Markovianity. Also report error bars from block or bootstrap resampling.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (9), T^Mic(mτ)=A^T t^m(τ)D_n A D_N^{-1}, is exact only when the microstate distribution at time 0 is locally equilibrated within each macrostate, p(0)=D_n A D_N^{-1}P(0). The derivation in Section 2.1 starts from P(mτ)=A^T p(mτ) and p(mτ)=t^m(τ)p(0); replacing p(0) by the equilibrium-conditional projection is an additional assumption, not a consequence of microstate Markovianity. In the HP35 Chapman-Kolmogorov test, the initial frame is a fixed MD snapshot whose microstate occupancy generally is not the local-equilibrium distribution. Hence even with perfectly Markovian microstates, T^Mic will not reproduce the reference macrostate populations unless the initialization is equilibrated. The paper attributes the HP35 deviations solely to 'suboptimal microstates' (Section 3.2) and never tests the initialization protocol, so the 'by design exact' claim is not supported by the reported experiments. The toy model avoids the issue only because the initial condition is chosen consistently; the CK test in Fig. 2b cannot distinguish initial-condition mismatch from genuine microstate non-Markovianity.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14012,"tokens_out":15832,"duration_ms":145062,"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":[{"comment":"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.","section":"Section 2.1, Eqs. (7)-(9)"},{"comment":"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.","section":"Section 3.2, Fig. 2b"}],"minor_comments":[{"comment":"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.","section":"Section 3.1"},{"comment":"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.","section":"Section 2.2"},{"comment":"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.","section":"Section 3.1"},{"comment":"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.'","section":"Section 3.2"},{"comment":"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.","section":"Section 1 and Section 2.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid methods comparison and is likely suitable for the journal after revision. The main technical issue is the unqualified exactness claim in Eq. (9), which must be fixed. The authors should also clarify the CK-test initialization in the HP35 section. The novelty of the microstate-based and hybrid methods is modest, but the systematic comparison and the discussion of qMSM's numerical behavior are valuable. I would support publication after major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know this paper is a mostly sound methods comparison with two simple but useful tricks, and one overstated exactness claim. Eq. (9) — project the microstate transition matrix after exponentiating — is exactly what it looks like: a direct corollary of the Hummer-Szabo projector in Eq. (7). The hybrid MD/MSM Eq. (14) is also straightforward: use exact transition counts at short times, then switch to a long-lag MSM. Neither is deep, but both are clearly explained and benchmarked on a solvable toy model and an all-atom HP35 trajectory.\n\nWhat the paper does well: the toy model is exactly solvable and allows tuning non-Markovianity, and all methods are compared against the exact microstate reference. The critique of qMSM is genuinely useful — the iteration fixed point and the cancellation of \\dot T(0) with K_0 are observations I had not seen stated explicitly. They also ship the HP35 data and analysis scripts on GitHub, which is reproducible work and should count.\n\nThe soft spot is the claim that Eq. (9) 'yields by design exact macrostate population dynamics.' That is only true when the initial microstate distribution within each macrostate is the local equilibrium one, i.e., p(0)=D_n A D_N^{-1} P(0). The derivation of Eq. (7) assumes that; for an arbitrary initial condition the projected dynamics does not reproduce the exact macrostate populations. The paper never states this assumption in Section 2.1. On the toy model the initial condition is chosen consistently, so the claim is not tested there. In the HP35 Chapman-Kolmogorov test, the reference is a real MD trajectory whose initial microstate occupancy is generally not locally equilibrated, so the observed deviations cannot be attributed solely to 'suboptimal microstates' — initial-condition mismatch and microstate non-Markovianity are confounded. That does not sink the paper, but the authors should either rephrase the claim or test the initialization protocol.\n\nMinor issues: the HP35 results have no error bars (single trajectory, though they split it into 10000×30ns pieces but report no spread), and the statement that qMSM 'can only be worse' than T^MD is trivially true on the time range where T^MD is known; the method's purpose is extrapolation beyond the sampled window.\n\nWho should read it: anyone building MSMs from poorly separated states, and people who want a critical look at qMSM. It deserves a serious referee — the comparison is careful and the qMSM observations are worth publishing even if the exactness claim needs a caveat.","headline":"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.","tokens_in":14527,"tokens_out":3211,"would_cite":true,"duration_ms":29736,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Projecting exponentiated microstate transitions onto macrostates gives exact macrostate population dynamics when the microstates are Markovian.","keywords":["Markov state models","non-Markovian dynamics","generalized master equation","memory kernel","microstate projection","Chapman-Kolmogorov test","protein folding","HP35"],"falsifier":"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.","tokens_in":13539,"feed_emoji":"🧬","tokens_out":10770,"duration_ms":90928,"temperature":0.7,"pith_summary":"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.","feed_headline":"Exact macrostate populations come from microstate MSM projection","feed_subtitle":"Projecting exponentiated microstate transitions onto macrostates removes lumping-induced memory loss by construction.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the equilibrium-normalized aggregation relation (Eq. 7) and the Laplace-transform optimal projection that serves as one comparison method.","marker":"[13]"},{"why":"provides the generalized master equation and the qMSM memory-kernel reconstruction scheme used in Section 2.2.","marker":"[17]"},{"why":"supplies the all-atom HP35 folding trajectory used for the benchmark tests.","marker":"[30]"},{"why":"defines the benchmark HP35 MSM construction workflow and the reference model whose macrostates are used here.","marker":"[33]"},{"why":"gives the implementation recipe for the qMSM calculations used in the paper.","marker":"[32]"}],"fun_headline_variants":["Exact macrostate populations via microstate MSM projection","Microstate MSM projection removes lumping artifacts","Skip macrostate counting: project microstate MSM","Non-Markovian dynamics? Microstate projection is exact"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exact macrostate populations via microstate MSM projection","Microstate MSM projection removes lumping artifacts","Skip macrostate counting: project microstate MSM","Non-Markovian dynamics? Microstate projection is exact"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0003,"raw_usage":{"total_tokens":1747,"prompt_tokens":974,"completion_tokens":773,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":590,"completion_tokens_details":{"reasoning_tokens":708}},"tokens_in":590,"tokens_out":773,"duration_ms":6829,"temperature":1.0,"reasoning_tokens":708,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:49:41.421029+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Optimal Dimensionality Reduction of Multistate Kinetic and Markov -State Models","cited_arxiv_id":null,"evidence_quote":"supplies the equilibrium-normalized aggregation relation (Eq. 7) and the Laplace-transform optimal projection that serves as one comparison method."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the generalized master equation and the qMSM memory-kernel reconstruction scheme used in Section 2.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the all-atom HP35 folding trajectory used for the benchmark tests."},{"cited_title":"Toward a Benchmark for Markov State Models: The Folding of HP35","cited_arxiv_id":null,"evidence_quote":"defines the benchmark HP35 MSM construction workflow and the reference model whose macrostates are used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the implementation recipe for the qMSM calculations used in the paper."}],"review_version":1}