{"id":"e99f375a-6b51-4757-ad3e-63c68d86d98f","arxiv_id":"2507.05912","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"The variance of the mean back relaxation estimator is derived analytically for Gaussian processes in terms of the mean squared displacement, with a minimum at length l ≈ sqrt(Δx²(τ)), and cell data deviate from this prediction, indicating non-Gaussian motion.","lead":"The authors study the mean back relaxation, a three-time-point measure of trajectory memory, and show how its statistical error depends on the length and time parameters used to compute it, in living cells and in a minimal model. For Gaussian processes the error formula reduces to a closed expression in the mean squared displacement, yielding a practical rule for choosing the length cutoff and exposing that cell motion is non-Gaussian.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The cell-data non-Gaussianity test and the l-selection rule presuppose stationary increments; drift or slow remodeling could mimic non-Gaussian VBR, so the Gaussian reference needs a stationarity control.","rationale":"The reader's weakest_assumption correctly flags stationarity, but the precise requirement is stationary increments, not absolute stationarity of x(t). The RHC model has a free-diffusing trap q, so ⟨x⟩ is not well-defined and x is non-stationary, yet Eq. (40) is confirmed by simulation because the increment process is stationary. The paper's concession about ⟨x⟩ is therefore not by itself grounds for doubt. The load-bearing gap is the lack of any test for non-stationary increments in the cell data: drift or remodeling on the ~1 s window would make the MSD time-average path-dependent and invalidate Eq. (40) as a Gaussian reference. A detrending or split-half control would settle whether the Fig. 10a upward deviation is genuine non-Gaussianity or merely non-stationarity. The Gaussian VBR formula itself is well derived and checked in the RHC simulation, so the analytic contribution is secure. Because the cell conclusions are already conditional in the reader's verdict, and the concern is exactly the stationarity premise, no verdict change is needed; the paper should add the proposed control before stronger claims are made.","tokens_in":25690,"tokens_out":15329,"duration_ms":158851,"concrete_test":"Detrend each A549 trajectory by subtracting a linear least-squares fit over the analysis window (or a moving average with a timescale ~10×τ), then recompute MSD, MBR, and VBR; if the detrended VBR curve collapses onto the Gaussian prediction of Eq. (40), the reported non-Gaussianity is an artifact of non-stationary drift rather than intrinsic non-Gaussian dynamics. A second, complementary check is to compute MBR(τ,t,l) and VBR separately on the first and second halves of each trajectory; a significant difference indicates non-stationary increments and invalidates the Gaussian reference.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Both central cell conclusions—the non-Gaussianity marker from Fig. 10a and the practical rule l≈√Δx²(τ)—rest on the premise that the cell trajectories are realizations of a Gaussian process with stationary increments, so that the three-time W3 in Eq. (3), the MSD time-average, and the VBR formula Eq. (40) are valid. Section II concedes that no mean ⟨x⟩ exists for cell probes, but the RHC model (free-diffusing trap q) shows absolute stationarity is not required; only the increments must be time-translation invariant. The true threat is slow cell drift, cytoskeletal remodeling, or active transport over the ~1 s windows, which breaks increment stationarity. Under non-stationary Gaussian increments, the empirical MSD is not the ensemble MSD of a stationary-increment process, so Eq. (40) is no longer a valid reference, and a false positive for non-Gaussianity can occur. The paper performs no stationarity control (no detrending, no sub-window MBR/VBR comparison, no check of time-reversal symmetry of increments). Thus both the non-Gaussianity claim and the l-optimality transfer from Gaussian theory to cells are unsecured.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper analyzes the mean back relaxation (MBR) observable for stochastic trajectories, focusing on its dependence on the conditioning time τ, observation time t, and length cutoff l. The authors introduce a modified \"Random Horse and Cart\" (RHC) model with memory and show it reproduces qualitative features of MBR in cell data, including non-monotonic time dependence and a τ-dependent long-time plateau. They extend the phenomenological relation between the effective energy amplitude E0 and the long-time MBR to finite τ, for both cells and the model. The central new result is the variance of back relaxation (VBR): for stationary Gaussian processes, the authors derive a closed-form expression, Eq. (40), linking VBR to the MSD and a dimensionless length η, and identify the optimal length l ≈ 0.925√(Δx²(τ)) that minimizes VBR. They verify this formula in simulations of the RHC model and then compare cell data to the Gaussian prediction, finding an excess in VBR that they interpret as evidence of non-Gaussian dynamics.","tokens_in":25962,"tokens_out":11354,"duration_ms":106987,"significance":"The Gaussian VBR result is a substantive and useful contribution: it gives a parameter-free (up to the MSD) expression for the statistical error of MBR, provides a concrete rule for choosing the length cutoff, and yields a potential non-Gaussianity diagnostic. The derivation in Appendix A1 is internally consistent and is corroborated by the RHC simulations in Fig. 10b, which strengthens confidence in the algebra. The RHC model, though simple, captures several non-trivial features of the cell MBR phenomenology, and the extension of the E0–MBR linear relation to larger τ is of practical value for lower-resolution experiments. The main caveat is that the two cell-data conclusions—non-Gaussianity and the optimal-l rule—assume stationary increments, and this assumption is not tested in the manuscript.","major_comments":[{"comment":"The conclusion that cell trajectories are non-Gaussian rests on comparing VBR computed from the cell data with the prediction of Eq. (40), which is derived in Appendix A1 under the explicit assumption of a stationary Gaussian process. For the prediction to be a valid reference, the cell increments must be time-translation invariant so that the empirical MSD equals the ensemble MSD and the three-time distribution W3 in Eq. (3) is stationary. The manuscript performs no stationarity control: there is no detrending, no sub-window comparison of MBR or VBR, and no check of increment time-reversal symmetry. Since Sec. II concedes that a mean position ⟨x⟩ cannot be claimed to exist for probe particles in cells, slow drift, cytoskeletal remodeling, or active transport over the roughly one-second windows could break increment stationarity and generate an excess VBR even for a Gaussian process. This would make the non-Gaussianity claim a false positive. Please add explicit stationarity checks (e.g., comparing MBR/VBR from disjoint sub-windows, detrending, or testing time-reversal symmetry of increments) or substantially weaken the claim to report a deviation from the stationary Gaussian reference rather than a demonstration of non-Gaussianity.","section":"Sec. V D, Fig. 10(a)"},{"comment":"The practical recommendation l ≈ √(Δx²(τ)) is derived from the Gaussian minimum of VBR, η_min ≤ 0.654 (Eq. (44) and Fig. 9). The transfer of this rule to cell data is justified by the qualitative flatness of the cell VBR curve around the minimum, but the location of the minimum depends on α(τ,t), which is itself computed from the same empirical MSD. If the cell process has non-stationary increments, the empirical MSD is not the stationary MSD, so both the reference VBR curve and the estimated optimal l are shifted. The rule of thumb is therefore only as reliable as the stationarity assumption. A stationarity control, or at least an explicit discussion of the sensitivity of η_min to drift, is needed before recommending l ≈ √(Δx²(τ)) for cell experiments.","section":"Sec. V D, Discussion"}],"minor_comments":[{"comment":"The phrase \"we determine its absolute minimum as a function of the length and time parameters\" overstates the result: the minimum is computed with respect to η (or l) for fixed τ and t, while the dependence on t and τ is analyzed asymptotically but not jointly minimized. Please rephrase to avoid ambiguity.","section":"Abstract and Sec. V B"},{"comment":"The sentence \"the statement hols true for τ and t exchanged\" contains a typo; it should read \"holds true.\"","section":"Sec. V B, after Eq. (45)"},{"comment":"The displayed formula for MBR(τ → ∞, t → ∞) is garbled in the text; please rewrite the expression with clear notation for the prefactor and the τ dependence.","section":"Eq. (20)"},{"comment":"The red data points for cell VBR are shown without error bars or an uncertainty estimate; adding them, even as approximate bootstrap intervals, would strengthen the comparison with the Gaussian prediction.","section":"Fig. 10(a)"},{"comment":"The sentence \"limiting ourselves to τ ≤ 0.01 s; In Fig. 2(a), we see that, in this regime, MBR decreases with τ\" uses a semicolon and capitalization awkwardly; also clarify that this restriction is chosen to avoid the MBR minimum.","section":"Sec. IV C"},{"comment":"The caption contains \"fir τ → 0\" which should be \"for τ → 0.\"","section":"Fig. 4 caption"},{"comment":"The word \"ration\" in the sentence about the critical activity should be \"ratio.\"","section":"Appendix A2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central theoretical result—the VBR formula for stationary Gaussian processes—is sound and well verified by simulation. The main risk is the unguarded transfer of this result to cell data without a stationarity control, which threatens both the non-Gaussianity claim and the practical l-selection rule. The authors appear to be aware of the stationarity limitation (Sec. II), so adding a direct test or softening the claims is feasible within the scope of a major revision. The heavy reliance on the authors' own prior work (Refs. 33, 34) is appropriate given the topic, but the new contribution would be more convincing if the cell-data analysis included at least one independent stationarity check."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing to know: the paper's central new result, the closed-form variance of back relaxation (VBR) for Gaussian processes, is real and useful. Eq. (40) expresses the estimator variance in terms of the MSD and MBR, has a unique minimum at η_min ≤ 0.654, and yields the practical rule l ≈ sqrt(Δx²(τ)). The derivation in Appendix A1 is straightforward and internally consistent, and the authors check it against their own RHC simulations in Fig. 10b with good agreement. That is a genuine advance over the earlier MBR work.\n\nThe RHC model is a modest but sensible extension of their prior model. Adding the bath particle introduces memory, and the model reproduces the non-monotonic MBR(t) seen in cells without being oversold.\n\nThe soft spots are mostly at the level of the cell-data interpretation. The non-Gaussianity conclusion for cells rests on comparing VBR from cell trajectories to a Gaussian reference built from the same trajectories' MSD. The paper itself concedes that no mean position exists for cell probes, and there is no stationarity control—no detrending, no sub-window comparison, no check that increments are time-translation invariant over the ~1 s windows. Slow drift or cytoskeletal remodeling would break increment stationarity, and then both the non-Gaussianity marker and the transfer of the l-rule to cells are not secured. This is not a fatal flaw in the formal result, but it is a real gap in the experimental claim. The plotted experimental data also lack error bars, which is ironic for a paper whose point is a variance estimate. Data and code are 'available upon request,' which is weaker than it should be.\n\nThe E0–MBR extension to finite τ is interesting but limited: the model shows the linear relation is only valid near MBR = 1/2 for larger τ, and the experimental analysis restricts to τ values where the relation is expected to hold. That is honest, but it narrows the scope.\n\nOverall: the Gaussian VBR result deserves a serious referee and will be useful to anyone analyzing MBR from particle tracking data. The cell-level conclusions need revision—at minimum a stationarity control and public data—before I would rely on them.\n\nRecommendation: send to peer review, with careful attention to the experimental sections.","headline":"The Gaussian VBR formula is a genuine, useful analytical result, but the cell-level conclusions need a stationarity control and public data before they convince.","tokens_in":26556,"tokens_out":2189,"would_cite":true,"duration_ms":24238,"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":"For Gaussian trajectories, the variance of mean back relaxation is set exactly by the mean square displacement, and the statistical error is minimized at l ≈ sqrt(MSD(τ)); cell data deviate, marking non-Gaussianity.","keywords":["mean back relaxation","variance of back relaxation","effective energy","Gaussian process","non-Gaussianity","microrheology","detailed balance","active matter"],"falsifier":"Compute VBR at fixed τ and t from long simulated trajectories of a stationary Gaussian process (e.g., Brownian motion in a harmonic trap) and check that the l-dependence follows the closed-form formula with its minimum at η_min ≤ 0.654; any systematic deviation would falsify the 'any Gaussian process' claim. For the non-Gaussianity conclusion, repeat the VBR comparison on an equilibrium viscoelastic sample with the same MSD as the cells; if the excess over the Gaussian curve also appears there, the marker is not specific to active non-Gaussian intracellular motion.","tokens_in":25471,"feed_emoji":"📏","tokens_out":11738,"duration_ms":107569,"temperature":0.7,"pith_summary":"Mean back relaxation (MBR) compares a particle's future displacement with its past displacement and has been proposed as a probe of broken detailed balance. This paper asks how the two time parameters (conditioning time $\\tau$ and observation time $t$) and one length cutoff $l$ control MBR and its statistical error. The central result is an exact formula for the variance of back relaxation (VBR) valid for any stationary Gaussian process, expressed purely through the mean squared displacement (MSD). The formula shows VBR has a unique minimum in $l$, located at $l \\approx 0.925\\sqrt{\\Delta x^2(\\tau)}$, so the rule of thumb $l \\approx \\sqrt{\\Delta x^2(\\tau)}$ yields the smallest error bar in MBR measurements. Comparing VBR from probe particles in A549 cells with the Gaussian prediction reveals a clear excess, serving as a non-Gaussianity marker for intracellular motion.","feed_headline":"Best length for back relaxation is set by the mean square displacement","feed_subtitle":"A closed-form variance formula fixes the optimal length scale and exposes cell non-Gaussianity.","key_machinery":"The central object is the variance of back relaxation (VBR), the mean-square fluctuation of the ratio $\\delta x(t)/\\delta x(-\\tau)$ weighted by the cutoff $\\vartheta_l$. For Gaussian processes the paper derives the closed-form identity $$\\mathrm{VBR} = \\mathrm{MBR}^2\\left[(\\$\\alpha$-1)g(\\eta)+h(\\eta)\\right],$$ where $g$ and $h$ are explicit functions of $\\eta = l/\\sqrt{2\\Delta x^2(\\tau)}$ built from error functions, and $\\alpha$ is a dimensionless ratio of MSDs. This identity does three pieces of work: it proves VBR diverges at small and large $l$, it locates the minimum of VBR in $l$ at $\\eta_{\\min}\\le 0.654$ (hence $l\\approx 0.925\\sqrt{\\Delta x^2(\\tau)}$), and it supplies a data-only null model—since all inputs are the measured MSD and MBR—against which cell data can be tested for Gaussianity.","core_discovery":"The authors establish that for any stationary Gaussian process the variance of the mean back relaxation is exactly $$\\mathrm{VBR}(\\tau,t,l) = \\mathrm{MBR}(\\tau,t)^2\\left[(\\$\\alpha$(\\tau,t)-1)\\,g(\\eta)+h(\\eta)\\right],$$ with $\\eta = l/\\sqrt{2\\Delta x^2(\\tau)}$, $\\alpha = \\frac{1}{\\mathrm{MBR}^2}\\frac{\\Delta x^2(t)}{\\Delta x^2(\\tau)}$, and $g,h$ given in terms of complementary error functions. MBR itself is independent of $l$ for Gaussian processes, but VBR depends on $l$ and diverges for both $l\\to 0$ and $l\\to\\infty$; its single minimum lies at $\\eta_{\\min}\\le 0.654$, i.e., at $l\\approx 0.925\\sqrt{\\Delta x^2(\\tau)}$. The authors further show that the 'Random Horse and Cart' model, a linear Gaussian model with a nonreciprocally driven trap and a memory-adding bath particle, reproduces the characteristic non-monotonic dependence of MBR on $t$ and its dependence on $\\tau$ found in living cells. They extend the phenomenological linear relation between effective energy amplitude $E_0$ and the long-time MBR value to a larger range of $\\tau$ values. Finally, they compare VBR of the cell data with the Gaussian prediction and find that the cell VBR lies above it, demonstrating that the intracellular process is non-Gaussian.","pith_inferences":["Because Eq. (40) is purely a function of MSD and MBR, it can be evaluated in any experiment that already measures trajectories; the same formula could be used to retroactively re-analyze existing passive microrheology data sets and report error bars on previously published MBR values.","The VBR excess could be turned into a quantitative non-Gaussianity index, e.g., the ratio of measured VBR to the Gaussian prediction at a fixed η; this index might correlate with the activity parameter Dq or with the effective energy amplitude E0 across cell types.","The η_min ≤ 0.654 bound is derived for Gaussian processes; for genuinely non-Gaussian cell dynamics, the optimal l for MBR evaluation may differ, but the Gaussian rule of thumb remains a safe starting point because the VBR curve is flat around the minimum.","A testable extension would be to measure VBR in reconstituted active networks with controlled activity; if the VBR excess scales with ATP-driven activity, VBR becomes a simple optical-microscopy readout of nonequilibrium activity."],"forward_implications":["For any stationary Gaussian process, the VBR formula gives the statistical error of MBR as a function of the length cutoff, so experiments should set l ≈ √Δx²(τ) to minimize the error bar.","Because MBR is l-independent but VBR is not, the length parameter can be tuned to reduce noise without biasing the MBR value, for Gaussian data.","The cell-data VBR lying above the Gaussian prediction provides a practical non-Gaussianity test that only requires passive trajectory data and the MSD.","The E0–MBR linear relation holds for a wider range of conditioning times τ (up to 0.01 s in the cell data), making effective-energy estimates accessible to setups with lower temporal resolution.","The Random Horse and Cart model with a bath particle reproduces the non-monotonic MBR(t) curves seen in cells, indicating that memory is the source of this shape."],"supporting_citations":[{"why":"Supplies the original cell data, the MBR–effective energy linear relation, and the optical-tweezer microrheology protocol that this work extends.","marker":"33"},{"why":"Established the MBR definition, its long-time value 1/2 under detailed balance, and the Gaussian MBR–MSD identity on which the VBR derivation rests.","marker":"34"},{"why":"Provides the intracellular mechanical fingerprint data and the effective-energy power law with amplitude E0 used in the correlation analysis.","marker":"13"},{"why":"Introduced the effective energy as the ratio of correlation to response in active gels, grounding the E0 definition.","marker":"29"},{"why":"Supplies the analytic MSD solution for the Random Horse and Cart model used to compute MBR and effective energy in the model system.","marker":"45"},{"why":"Provides the Gaussian conditional distribution used to integrate the VBR formula.","marker":"54"},{"why":"Numerical simulation library used to verify the Gaussian VBR prediction in the model system.","marker":"49"},{"why":"Curve-fitting routine used to obtain R-squared values for the linear E0–MBR fits.","marker":"56"}],"fun_headline_variants":["Mean back relaxation variance minimum fixes optimal length scale","Cell trajectories reveal non-Gaussian back relaxation variance","Analytical variance of back relaxation gives optimal length scale","Back relaxation variance minimum reveals non-Gaussian cell motion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The cell trajectories are treated as a stationary, time-translation-invariant process with a well-defined mean, although the paper concedes that no mean position can be claimed for probe particles in cells; if slow drift or active remodeling occurs on the experimental window, the Gaussian reference and the MBR–E0 correlation are contaminated.","fun_headline_variants_meta":{"raw":{"variants":["Mean back relaxation variance minimum fixes optimal length scale","Cell trajectories reveal non-Gaussian back relaxation variance","Analytical variance of back relaxation gives optimal length scale","Back relaxation variance minimum reveals non-Gaussian cell motion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000912,"raw_usage":{"total_tokens":3975,"prompt_tokens":1062,"completion_tokens":2913,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":678,"completion_tokens_details":{"reasoning_tokens":2852}},"tokens_in":678,"tokens_out":2913,"duration_ms":20263,"temperature":1.0,"reasoning_tokens":2852,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:16:16.634007+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute VBR at fixed τ and t from long simulated trajectories of a stationary Gaussian process (e.g., Brownian motion in a harmonic trap) and check that the l-dependence follows the closed-form formula with its minimum at η_min ≤ 0.654; any systematic deviation would falsify the 'any Gaussian process' claim. For the non-Gaussianity conclusion, repeat the VBR comparison on an equilibrium viscoelastic sample with the same MSD as the cells; if the excess over the Gaussian curve also appears there, the marker is not specific to active non-Gaussian intracellular motion.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the original cell data, the MBR–effective energy linear relation, and the optical-tweezer microrheology protocol that this work extends."},{"cited_title":"Knotz and M","cited_arxiv_id":null,"evidence_quote":"Established the MBR definition, its long-time value 1/2 under detailed balance, and the Gaussian MBR–MSD identity on which the VBR derivation rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the intracellular mechanical fingerprint data and the effective-energy power law with amplitude E0 used in the correlation analysis."},{"cited_title":"Mizuno, C","cited_arxiv_id":null,"evidence_quote":"Introduced the effective energy as the ratio of correlation to response in active gels, grounding the E0 definition."},{"cited_title":"Höper, ``Mean back relaxation for a driven viscoelastic model system,'' 2023","cited_arxiv_id":null,"evidence_quote":"Supplies the analytic MSD solution for the Random Horse and Cart model used to compute MBR and effective energy in the model system."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Gaussian conditional distribution used to integrate the VBR formula."},{"cited_title":"Rackauckas and Q","cited_arxiv_id":null,"evidence_quote":"Numerical simulation library used to verify the Gaussian VBR prediction in the model system."},{"cited_title":"Virtanen, R","cited_arxiv_id":null,"evidence_quote":"Curve-fitting routine used to obtain R-squared values for the linear E0–MBR fits."}],"review_version":1}