{"id":"5beace6a-2cb9-484d-aeaa-6193ccc82910","arxiv_id":"2501.08754","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"All-atom simulations of two proteins show small ergodicity-breaking parameters on picosecond-to-nanosecond timescales, suggesting prior reports of non-ergodicity in this window stem from incomplete convergence.","lead":"Using 100 independent computer simulations of two proteins, this paper tests whether protein motion is ergodic, the idea that long time averages match averages over many identical copies. It reports that in the picosecond-to-nanosecond window the simulated motion looks ergodic, contradicting recent claims that protein dynamics is non-ergodic.","discovery_kind":"replication","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claim of complete ergodicity rests on extrapolating a monotonic decrease of the EB parameter to zero without demonstrating that the EB plateau is zero; the paper's own multi-funnel discussion admits global non-ergodicity.","rationale":"The paper's headline claim is that proteins exhibit 'completely ergodic dynamic' in the pico-to-nanosecond window regardless of size. The only quantitative evidence for this is the behavior of the ergodicity-breaking parameter and the related ϕ distribution. I agree with the reader's weakest assumption: the inference from a finite-time monotonic decrease of EB to a zero asymptotic limit is unjustified. A non-ergodic system with a separation of timescales would produce exactly the same finite-time trend if the plateau timescale exceeds the 100-ns sampling window. The paper provides no extrapolation, error bars, or longer-time checks to rule out this alternative. Additionally, the Discussion's own multi-funnel argument states that the protein is 'by definition non-ergodic' globally, which directly contradicts the abstract's unqualified 'completely ergodic' phrasing. The most defensible conclusion would be local ergodicity within a single native-state valley over the sampled window, which is a much weaker claim than the one made. The lack of simulation details (force field, package, replica generation) in the present paper further undermines reproducibility, though the primary logical flaw is the finite-time extrapolation. For these reasons, the central claim is not supported, and the paper should be rejected in its current form. If the authors revise to a claim of local ergodicity and add a proper scaling analysis, a conditional acceptance might be possible, but as written the rejection stands.","tokens_in":6953,"tokens_out":5755,"duration_ms":60008,"concrete_test":"Re-analyze the EB(T,t) data from Fig. 3 for a fixed lag time t = 1 ns by fitting the model EB(T) = a T^{-b} + c to the observed values (e.g., T = 10, 20, 50, 100 ns). Obtain a 95% confidence interval for the plateau parameter c. If the interval excludes zero and c is comparable to the observed EB values, the non-ergodic plateau is supported; if the interval includes zero and the fit shows clean power-law decay (b > 0), the ergodic interpretation is strengthened but still requires longer simulations for confirmation. This single check, using the paper's existing data, would settle whether the monotonic decrease is compatible with a nonzero asymptotic EB.","verdict_should_be":"REJECT","load_bearing_attack":"The central evidence for ergodicity is the monotonic decrease of the EB parameter (Eqs. 6–7) below 10^-2 as T increases and the narrow ϕ distribution centered at 1. But for a non-ergodic system with slow inter-valley transitions, trajectories on timescales shorter than the transition time remain confined within a single valley, so EB initially decreases with T/t and only later plateaus at a nonzero value reflecting static heterogeneity. The paper samples only T up to 100 ns and lag times t up to 1 ns, so T/t ≤ 100. An observed decreasing trend over this limited range cannot distinguish ergodic convergence to zero from a transient approach to a plateau positioned beyond the sampled range. The statement that Var is 'at least one tenth' of the squared average (EB < 10^-2) is a finite-time bound, not evidence of a zero limit. Furthermore, the Discussion explicitly concedes that if multiple native valleys exist, a single protein molecule is 'by definition non-ergodic' because it cannot visit another valley; this contradicts the abstract's 'completely ergodic dynamic ... irrespective of their size.' The defensible conclusion is 'locally ergodic within the sampled window,' not 'completely ergodic.'","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper uses 100 independent 100-ns all-atom MD simulations of two proteins, the 35-residue Villin headpiece and the ~400-residue CAP homodimer, to test whether protein dynamics is ergodic in the picosecond-to-nanosecond window. It computes the ensemble-averaged mean-square displacement and the time-averaged MSD of individual replicas, the ergodicity-breaking parameter EB (Eq. 6), and the distribution phi of normalized time-averaged MSDs. The authors report that EB decreases below 10^-2 with increasing T/t and that phi is a narrow distribution centered at 1, and they conclude that protein dynamics is \"completely ergodic\" in this window irrespective of protein size, with deviations from ergodicity attributed to incomplete convergence. The Discussion, however, concedes that if multiple native valleys exist, a single protein molecule is \"by definition non-ergodic\" because it cannot visit another valley, a statement that conflicts with the abstract's absolute wording.","tokens_in":7153,"tokens_out":4996,"duration_ms":52108,"significance":"If established, the result would challenge the interpretation of recent single-molecule experiments (Li et al., Ye et al.) as evidence of non-ergodicity in the short-time regime and would support the common practice of comparing short MD trajectories with ensemble-averaged experimental observables. The paper's strengths are that EB and phi are computed directly from trajectories using standard definitions with no fitted parameters entering the central comparison, and the two-protein design with very different sizes directly addresses the previously proposed size dependence. The main weakness is that the finite-time data cannot distinguish ergodic convergence to zero from a transient approach to a nonzero plateau, and the headline claim of complete ergodicity is internally contradicted by the paper's own multi-valley discussion.","major_comments":[{"comment":"The abstract and conclusion state that protein exhibits a \"completely ergodic dynamic within a time window ranging from pico-to-nanoseconds, irrespective of their size,\" but the Discussion explicitly says that if multiple native valleys exist, a single protein molecule is \"by definition non-ergodic\" because it cannot change its native state by visiting another valley. Since the simulations reported here are, on the paper's own model, confined to a single valley, they cannot support the word \"completely.\" The defensible conclusion is that the dynamics is locally ergodic within the sampled valley and time window, not globally ergodic; the overclaim should be removed or substantially qualified.","section":"Discussion (final paragraph) and Abstract"},{"comment":"The central evidence for ergodicity is the monotonic decrease of EB with T/t, with T up to 100 ns and lag times t up to 1 ns, so T/t is at most 100. This finite-time trend is equally consistent with an ergodic system converging to zero and with a non-ergodic system whose EB would eventually plateau at a nonzero value because trajectories remain trapped in one valley on the sampled timescale. The paper does not demonstrate that the EB plateau is zero; it only shows that EB is below 10^-2 over the sampled range. Therefore the claim that \"deviations from ergodic behavior are due to incomplete convergence\" is not established by the presented data. A quantitative test, such as an extrapolation of EB to the T/t -> infinity limit, a fit to a known functional form, or a comparison with a non-ergodic model over the same range, is needed.","section":"Methods and Results, Fig. 3 and Eqs. 6-7"},{"comment":"The entire MD simulation protocol is deferred to reference 12. The manuscript does not state the force field, simulation package, water model, thermostating/barostating scheme, equilibration procedure, or how the 100 independent replica starting configurations were extracted from the 1-microsecond simulation. Because the central conclusion depends on these trajectories adequately sampling the native-state dynamics of both proteins, the results are not reproducible from the manuscript as written, and the applicability of the previous protocol to these two systems cannot be assessed. At minimum, a summary of the protocol and a statement of replica-generation details should be included.","section":"Methods, first paragraph"}],"minor_comments":[{"comment":"The text contains numerous ligature and OCR artifacts, such as \"signi5cant,\" \"5ndings,\" and \"simualted,\" which should be corrected in a polished manuscript.","section":"Throughout"},{"comment":"The caption contains typographical errors, including \"segement\" and \"beetween,\" and the phrase \"distance beetween the arginines\" should be reworded for clarity.","section":"Figure 1 caption"},{"comment":"The statement that EB below 10^-2 means the variance of the TA-MSD is \"at least one tenth\" of the squared average is backwards: from Eq. 6, EB < 10^-2 implies Var < 10^-2 <delta^2>^2, i.e., the variance is less than one hundredth of the squared average, which would actually strengthen the authors' qualitative claim.","section":"Methods and Results, paragraph after Eq. 6"},{"comment":"No error bars or confidence intervals are reported for the EB estimates; with 100 replicas, the sampling uncertainty of EB should be quantified so that the reader can judge whether the decrease with T/t is significant.","section":"Fig. 3"},{"comment":"The normalized root-mean-square errors of 15% and 11% are reported without any null model or confidence interval, so the reader cannot assess whether these values are small enough to support the claim of equality between the two MSD estimates.","section":"Fig. 5 and surrounding text"}],"recommendation":"reject","confidential_remarks":"The manuscript is a short report with a potentially interesting negative result about short-time ergodicity, but as submitted the central claim overreaches the evidence. The internal contradiction between the abstract and the Discussion, together with the inability of the finite-time EB data to distinguish ergodic convergence from a nonzero plateau, are load-bearing problems that would require a substantial reframing and additional analysis to address. If the authors revise the claim to \"locally ergodic within the sampled window,\" the paper could be reconsidered as a more modest contribution, but in its current form I cannot recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper applies the ergodicity-breaking parameter and phi distribution from the anomalous-diffusion toolbox to 100 independent 100 ns all-atom MD replicas of two proteins of very different size, Villin and CAP. That is a new and useful application, and the result directly counters Li et al.'s recent claim that non-ergodicity in the picosecond-to-nanosecond window depends on protein size. The EB values decrease below 10^-2 and the phi distributions are narrow and centered at 1, which is at least consistent with ergodic behavior on this timescale. The authors also correctly note that variability among TA-MSDs can reflect incomplete convergence rather than non-ergodicity.\n\nThe central soft spot is the inference from a finite-time decreasing EB to convergence to zero. For a non-ergodic process with slow inter-valley transitions, the same decreasing trend would appear on timescales shorter than the transition time, followed by a plateau at a nonzero value. The paper samples T up to 100 ns and T/t up to 100, so it cannot distinguish those two cases. The statement that EB is below 10^-2 is a finite-time bound, not a limit. This is a load-bearing gap, not a minor quibble.\n\nThe second issue is the internal contradiction between the abstract's \"completely ergodic dynamic ... irrespective of their size\" and the Discussion's admission that a protein with multiple native valleys is \"by definition non-ergodic.\" The defensible conclusion is \"locally ergodic within the sampled window,\" which is still a meaningful statement. The paper's own multi-funnel discussion implicitly concedes the abstract overshoots.\n\nThe simulation protocol is deferred to reference 12, the author's own prior paper. That is not circular by itself, but it makes this paper unverifiable in isolation: no force field, package, or replica-generation details are given. The comparison between the ensemble MSD and the averaged TA-MSD is quantified by 15% and 11% normalized root mean square error, which is not particularly small and deserves discussion.\n\nOverall, the paper is a reasonable check of a published claim, with standard tools and a clear negative result for the sampled window. But the headline claim as written is not supported. With a revised abstract and a careful statement of what the EB trend can and cannot prove, it would be a useful contribution.\n\nI would not cite it as it stands, but I would bring it to a reading group to discuss the interpretation of EB in finite-time MD. Send it to a good referee for the question and the data; it warrants a serious look, with a request for major revision.","headline":"Useful finite-time ergodicity check with overreach in the headline claim; EB decrease alone cannot prove convergence to zero.","tokens_in":7711,"tokens_out":2253,"would_cite":false,"duration_ms":21168,"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":"Using 100 replicas of two proteins, this paper argues that protein dynamics is completely ergodic in the picosecond-to-nanosecond window, and that reported non-ergodicity reflects incomplete convergence.","keywords":["protein dynamics","ergodicity","molecular dynamics simulation","ergodicity breaking parameter","time-averaged mean square displacement","villin headpiece","CAP protein","conformational sampling"],"falsifier":"Compute the ergodicity-breaking parameter for the same trajectories at lag times approaching the trajectory length, for instance T/t = 2 and 5; if EB stops decreasing and levels off at a nonzero value, the conclusion that the dynamics is completely ergodic in this window would be wrong.","tokens_in":6681,"feed_emoji":"🧬","tokens_out":6210,"duration_ms":54989,"temperature":0.7,"pith_summary":"This paper tries to settle whether protein dynamics is non-ergodic in the picosecond-to-nanosecond window, as recent single-molecule experiments and simulations have suggested. Using 100 independent 100-ns all-atom molecular dynamics trajectories of two proteins of very different size, it computes the ergodicity breaking parameter and the distribution of time-averaged mean square displacements. The author argues that these quantities converge to zero and to a narrow Gaussian centered at unity, respectively, indicating that dynamics is completely ergodic in this window regardless of protein size. If correct, the result would restore the formal basis for comparing simulation time averages with experimental ensemble averages, at least on these timescales.","feed_headline":"Protein motion is fully ergodic on pico-to-nanosecond timescales","feed_subtitle":"Two proteins of very different size show matching time and ensemble averages, challenging earlier non-ergodicity claims.","key_machinery":"The central objects are the ergodicity breaking parameter $EB(T,t)=\\mathrm{Var}[\\delta_k^2(T,t)]/\\langle\\delta_k^2(T,t)\\rangle^2$ and the distribution $\\phi(\\xi)$ of the replica-normalized time-averaged mean square displacement $\\xi_k(T,t)=\\delta_k^2(T,t)/\\langle\\delta^2(T,t)\\rangle$. The argument runs by showing that $EB$ decreases monotonically toward zero as $T/t$ grows (with $T=100$ ns and $t\\le 1$ ns), and that $\\phi$ narrows to a Gaussian centered at 1 without weight at zero; the paper takes these as the standard signatures of ergodicity, and uses the collapse of ensemble-averaged MSD onto the ensemble-averaged TA-MSD as a direct check.","core_discovery":"The paper's central claim is that, over timescales from picoseconds to nanoseconds, a folded protein explores its available native-state conformations in a fully ergodic manner, and that apparent non-ergodicity reported by earlier work (notably Li et al., ref 24) is an artifact of incomplete convergence of the measured quantities. The evidence is drawn from 100 independent 100-ns replica trajectories of villin headpiece (35 residues) and the CAP-cAMP homodimer (~400 residues). For each replica the paper computes a time-averaged mean square displacement (TA-MSD) and compares it with the ensemble average over replicas. The ergodicity breaking parameter $EB$ — the variance of TA-MSD divided by its squared ensemble average — falls monotonically below $10^{-2}$ as $T/t$ grows, and the distribution of the ratio $\\xi = \\delta_k^2 / \\langle \\delta^2 \\rangle$ is a narrow Gaussian centered at 1 with no weight at zero. The paper concludes these are the signatures of an ergodic system and that the size dependence suggested by Li et al. does not appear in this time window.","pith_inferences":["If the ergodicity claim holds for these two proteins, it likely extends to other globular proteins in the same timescale, but only for descriptors that equilibrate quickly; slow collective variables may still show replica-to-replica spread.","A sharper test of the paper's conclusion would be to compute EB for the same trajectories using lag times approaching T (e.g., T/t = 2), where a non-ergodic system would reveal a plateau.","The multi-funnel picture proposed in the discussion predicts a bimodal distribution of escape times — fast intrabasin and slow interbasin — which could be tested by long single-molecule trajectories or by enhanced-sampling simulations.","The paper's reliance on a single MD protocol leaves open the question of force-field sensitivity; rerunning the analysis with different force fields would clarify whether ergodicity in this window is a generic property or protocol-dependent."],"forward_implications":["Simulation time averages and experimental ensemble averages can be compared directly for folded proteins within the pico-to-nanosecond window.","The size-dependent non-ergodicity reported by Li et al. does not hold for these two systems in this window; the apparent effect likely comes from incomplete statistical convergence.","The multi-funnel interpretation of static heterogeneity implies that ergodicity within a native-state valley is compatible with non-ergodicity between valleys.","The low EB values justify using ensemble-averaged TA-MSD rather than single-trajectory TA-MSD in analyses of subdiffusion in this timescale.","The conclusion is limited to the sampled native states; potential non-ergodicity over larger timescales remains open."],"supporting_citations":[{"why":"The non-ergodicity and size-dependence claim this paper sets out to test and challenge.","marker":"[24]"},{"why":"Supplies the molecular dynamics simulation protocol and the definition of the high T/t regime used throughout.","marker":"[12]"},{"why":"Defines the ergodicity breaking parameter in rescaled form and the criterion T/t > 100 for the high-value regime.","marker":"[30]"},{"why":"Introduces the finite-time ergodicity breaking parameter used here.","marker":"[31]"},{"why":"Provides an additional source for the EB parameter and weak ergodicity breaking.","marker":"[32]"},{"why":"Supports the use of alpha-carbon RMSD descriptors for villin headpiece dynamics.","marker":"[29]"}],"fun_headline_variants":["Protein dynamics truly ergodic on pico-to-nanosecond timescales","Earlier non-ergodicity claim overturned by replicate simulations","Proteins explore all states in nanoseconds, simulations show","Ergodicity restored: protein motion matches ensemble averages","Fast protein dynamics are ergodic, contrary to prior reports"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument hinges on reading the monotonic decrease of the ergodicity-breaking parameter below $10^{-2}$, within trajectories of 100 ns and lag times no larger than 1 ns, as evidence that it would converge to zero rather than to a small nonzero plateau outside the sampled range.","fun_headline_variants_meta":{"raw":{"variants":["Protein dynamics truly ergodic on pico-to-nanosecond timescales","Earlier non-ergodicity claim overturned by replicate simulations","Proteins explore all states in nanoseconds, simulations show","Ergodicity restored: protein motion matches ensemble averages","Fast protein dynamics are ergodic, contrary to prior reports"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000218,"raw_usage":{"total_tokens":1433,"prompt_tokens":929,"completion_tokens":504,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":421}},"tokens_in":545,"tokens_out":504,"duration_ms":5076,"temperature":1.0,"reasoning_tokens":421,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:18:22.876190+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the ergodicity-breaking parameter for the same trajectories at lag times approaching the trajectory length, for instance T/t = 2 and 5; if EB stops decreasing and levels off at a nonzero value, the conclusion that the dynamics is completely ergodic in this window would be wrong.","supporting_citations":[],"review_version":1}