{"id":"4b15fb29-58dc-477f-a38c-3d78e5a61efc","arxiv_id":"2509.08587","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A bond-cluster model with exponential load-sharing length predicts two coexisting rupture pathways and critical crack sizes that match simulations.","lead":"This paper introduces a one-dimensional model of transient molecular bonds in which an applied force is shared according to an exponential decay length, interpolating between global and local load sharing. The authors derive approximate rupture conditions for the intermediate regime and confirm with stochastic simulations that failure proceeds either by uniform bond opening or by a spreading crack.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unquantified log-term drop in the continuum force approximation (Eq. 12→13) can bias the crack-boundary force that feeds the analytical critical-gap predictions (Eqs. 22–26), and printed factor errors hinder independent verification.","rationale":"The reader's weakest-assumption diagnosis (symmetry/dropped log term) is a legitimate risk to the analytical apparatus, but it is not obviously fatal for the mean-field symmetric configuration used to derive Eqs. (24) and (26), where the log term vanishes exactly. The concern is more acute for the stochastic asymmetric configurations that actually initiate cracks, and the paper does not quantify the error there. I agree with the CONDITIONAL verdict: the qualitative two-pathway claim is backed by simulation, but the analytical critical-gap predictions should be verified with an exact-force computation and the printed factor errors fixed before quantitative use. The proposed concrete test directly settles whether the dropped-log-term approximation changes the predicted critical gap beyond the simulation scatter. If the test shows small errors, the CONDITIONAL verdict could be upgraded to ACCEPT; if large errors appear, the analytical support for the central claim weakens. Thus I recommend keeping the reader's CONDITIONAL verdict unchanged.","tokens_in":18420,"tokens_out":19085,"duration_ms":175006,"concrete_test":"For the parameter set of Fig. 8 (N=200, K=0.9, σ̃=0.7, ℓ̃=0.2 and 0.5), compute the exact discrete force on the boundary of the largest gap from Eq. (5) for the mean-field configuration (gap of size d̃, all other closed bonds spaced δL/K). Use this exact force in the balance condition d̃ k_on = 2 k_off,0 exp(f/f_d) to solve for the critical gap size, and compare with Eq. (24), Eq. (26), and the simulation mean in Table I. Also evaluate the dropped log term in Eq. (12) relative to the retained second term for the gap adjacent to the crack in stochastic configurations taken just before rupture; if the log term is non-negligible (e.g., >10% of the second term) and the exact-force critical gap differs from Eq. (26) by more than the simulation error bar, the approximation is load-bearing and the analytical predictions need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative results for finite ℓ—the critical gap size and its growth with ℓ—are built on Eq. (13), obtained from the exact integral expression Eq. (12) by dropping the logarithmic term under the claim that typical configurations are 'approximately symmetric' and the term is negligible (Sec. III). This step is inherited by the small-ℓ force Eq. (17), the mean-field critical-gap equation (24), and the more accurate Eq. (26). The assumption is not quantified. For the perfectly symmetric mean-field configuration (one large gap, otherwise uniform spacing) the log term is exactly zero, so it is not the weak point there. But actual near-rupture configurations, as in the Fig. 5 kymographs, are asymmetric: the bond distribution is not mirror-symmetric about the large gap's midpoint. Fig. 3 shows Eq. (13) already overestimates forces on random configurations, with increasing error at large forces. Since rupture is initiated precisely by high force on a crack-boundary bond, a systematic bias in that force would shift the predicted critical crack size and the claimed stabilization trend. The printed equations also contain apparent factor/unit inconsistencies (e.g., Eq. (24) has dimensionful imbalance and Eq. (23)'s factor 2 is missing from Eq. (24)), so the numerical values in Table I cannot be reproduced from the text alone.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a one-dimensional stochastic model of N transient bonds under constant load, in which the force from each site is distributed among closed bonds with an exponential kernel of characteristic length ℓ, recovering the previously studied global and local load-sharing limits. The paper derives a continuum approximation for the force on a closed bond and uses it to derive critical gap conditions for crack-like rupture, supplemented by bounds on the critical gap size. Gillespie simulations are used to show that the model exhibits two failure pathways—uniform bond opening and crack spreading—that coexist for intermediate ℓ, and to compare the predicted critical gap sizes. The central quantitative claim is that the critical crack size grows with ℓ, i.e., a longer force-distribution length stabilizes the cluster.","tokens_in":18892,"tokens_out":19568,"duration_ms":191530,"significance":"The conceptual contribution is genuinely useful: the model interpolates between two well-studied loading scenarios with a single, physically motivated length scale, and the two-pathway picture is clearly demonstrated in simulations. Strengths include the absence of fitted parameters in the rupture conditions, exact recovery of the global and local limits in Sec. III, and the explicit check of analytical predictions against stochastic simulations of the same model. The bounds in Sec. IV C are a non-trivial consistency check on the mean-field treatment. However, several printed algebraic steps contain factor and inversion errors that prevent reproduction of the reported numbers; until these are corrected, the quantitative predictions—especially the ℓ-dependence of the critical gap—are not verifiable from the text.","major_comments":[{"comment":"The local-limit rupture condition is internally inconsistent. The text before Eq. (20) says the on-rates of open bonds in the gap are equated with the off-rates of the two neighbouring closed bonds, which requires a factor 2 on the right-hand side; Eq. (20) has no such factor. Appendix C, Eq. (C6), contains a factor 2 but with the K factor inverted: after dividing by k_off,0, k_on/k_off,0 = K/(1-K), so the correct rearranged form is (d~_c-1) exp(-σ̃ d~_c/2) = 2(1-K)/K exp(σ̃/(2K)), not 2K/(1-K) exp(...). The Lambert-W argument in Eq. (C8) is therefore also incorrect; it should be -σ̃(1-K)/K exp(σ̃/2(1+1/K)). As printed, Eq. (21) cannot generate the local-model column of Table I. This is load-bearing because the local critical gap is the baseline for all finite-ℓ comparisons.","section":"IV A, Eqs. (20), (21), Appendix C"},{"comment":"Eq. (24) does not follow from Eq. (23). With K = k_on/(k_on+k_off,0), k_off,0/k_on = (1-K)/K. Eq. (23) therefore implies d~_c * K/[2(1-K)] = exp(...), not d~_c*(1/K-1) = exp(...). The printed left-hand side has the K dependence inverted and, for the parameters of Fig. 8 (K=0.9, σ̃=0.7, d~≈9.7), is about 1 while the exponential is of order 40; thus the reported small-ℓ estimate 9.663 is not a solution of the printed Eq. (24). Please correct the equation and regenerate Table I and Fig. 8 accordingly.","section":"IV B, Eqs. (23)-(24)"},{"comment":"The continuum approximation drops the logarithmic term in Eq. (12) based on the statement that typical configurations are approximately symmetric and the term is 'negligible compared to the second.' This is not quantified and is load-bearing: the dropped term feeds Eqs. (17), (22), (24), and (26), i.e., all the analytical critical-gap predictions. Figure 3 compares random configurations and shows systematic overestimation at large forces, but rupture is set by the force on the boundary of the largest gap, i.e., precisely the high-force tail. I ask for a quantitative estimate of the dropped term in near-rupture configurations (e.g., measured in the simulations used for Figs. 8 and 9), or for a bound that justifies the approximation.","section":"III, Eqs. (12)-(13)"},{"comment":"The procedure for extracting the critical gap from simulations excludes only the last N Gillespie steps, with the justification that rupture takes no more than N steps after initiation. This is not guaranteed when open bonds can rebind during the cascade, and the paper itself notes that bond reformation is common for larger ℓ (Fig. 5B). If the cascade takes more than N steps, unstable gaps before the excluded window are not removed; if it takes fewer, the window may include pre-rupture configurations. Since the histograms in Fig. 8B,C and the mean ± SD in Table I are the quantitative validation of Eqs. (24) and (26), this criterion should be justified or replaced (e.g., by detecting the time at which the largest gap grows monotonically).","section":"IV B, critical-gap extraction"}],"minor_comments":[{"comment":"Typo: 'force-depended' should be 'force-dependent'.","section":"Abstract"},{"comment":"The parameter string 'N=200, K=0.9, σ̃=0.7' is duplicated in the caption.","section":"Fig. 8 caption"},{"comment":"Typo: in one exponential, '\\tilde ell' should be '\\tilde\\ell'.","section":"Eq. (28)"},{"comment":"Typo in the text following Fig. 5: 'enlarge the the quick succession'.","section":"Sec. IV B"},{"comment":"The claim that rupture time saturates for ℓ̃ ≳ 2 rests on visual overlap of curves; error bars or confidence intervals would strengthen this statement.","section":"Fig. 7"}],"recommendation":"major_revision","confidential_remarks":"The algebraic errors in Eqs. (20)-(21) and (23)-(24) are extensive and sit in the quantitative core of the paper. They look like rearrangements/typos rather than a flaw in the model, since the simulation comparisons are qualitatively consistent with the intended picture. If the authors correct the equations and re-run the comparisons, the paper could be publishable. I would also encourage the editor to ask whether simulation code or data can be made available, as none is mentioned. No novelty or attribution concerns beyond this."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nThe short version: this paper gives a new load-sharing model with an exponential decay length ℓ that interpolates between the local and global force-distribution limits, and uses it to predict critical gap sizes and a transition between two rupture pathways. The qualitative story — crack-mediated vs uniform rupture, coexistence at intermediate ℓ, and stabilization with increasing ℓ — is supported by the simulations and by the correct limiting cases. The interesting new piece is the cosh kernel in Eqs. (3)–(5) and the finite-ℓ critical-gap analysis, including upper and lower bounds on the critical gap size in Sec. IV C. That is a real advance over earlier local and global models; the interpolation is not just a footnote.\n\nThe paper is also honest in its validation: no parameters are fitted. The approximate rupture conditions are compared with stochastic simulations of the same model, so it is a self-consistency check rather than external validation, but that is legitimate for a theory paper. The model is 1D and the mean-field treatment is clearly labeled as such.\n\nThe main problems are in the printed equations, and they are serious enough that I cannot reproduce the analytical results from the text alone. Eq. (24) does not follow from Eq. (23) with the relation between k_on and k_off0; the factor (1/K − 1) is off by K/(1−K). Eq. (20) and Eq. (C6) differ by a factor of 2, which propagates into the Lambert-W expression. In addition, the derivation of the continuum force Eq. (13) from Eq. (12) drops the logarithmic term under an unquantified “approximate symmetry” assumption. That step is not neutral: the force on the crack-boundary bond, which sets the critical gap size, is exactly where configuration asymmetry matters. Fig. 3 shows Eq. (13) overestimates forces on random configurations at large forces, so the bias is in the direction you’d worry about. These are not fatal to the central qualitative claims — the limiting cases and simulation comparisons are consistent — but they need to be fixed and the symmetry assumption quantified before the numbers in Table I can be used.\n\nI’d send this to peer review. It is useful for anyone working on adhesion clusters or cytoskeletal crosslinkers who needs an intermediate load-sharing model, and the analytical predictions give a handle on the intermediate regime that earlier work lacked. After the equation fixes and a check of the log-term drop, it would be a solid paper. I’d be happy to referee it myself.","headline":"A useful interpolation model for bond-cluster failure with a new length-scale force distribution; the analytical rupture conditions need equation fixes but the simulations and limiting cases hold up.","tokens_in":19214,"tokens_out":2359,"would_cite":true,"duration_ms":26639,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that adding a finite length scale to load sharing between transient bonds creates two coexisting rupture pathways—uniform opening and crack growth—and predicts that the critical crack size grows with that length scale.","keywords":["Transient bonds","Cell adhesion","Cytoskeleton","Fracture","Non-local interaction","Bond clusters","Load sharing","Rupture conditions"],"falsifier":"Seed a stochastic simulation with a deliberately asymmetric configuration—for example, all closed bonds concentrated on one side of the largest gap—at ℓ/δL = 0.5, and compare the measured force on the gap boundary with Eq. (26). If the boundary force differs systematically from the prediction, or if the measured critical gap size departs from the predicted value by more than the simulation spread, the symmetry assumption fails.","tokens_in":18363,"feed_emoji":"💥","tokens_out":6572,"duration_ms":66106,"temperature":0.7,"pith_summary":"The paper introduces a one-dimensional model of transient bond clusters in which an applied load is shared among closed bonds through an exponentially decaying kernel with a characteristic length scale ℓ, interpolating between previously studied extremes: global sharing (ℓ→∞) and nearest-neighbor local sharing (ℓ→0). The authors derive a continuum approximation for the force on each bond and use it to obtain rupture conditions, which they validate with stochastic simulations. The central finding is that failure occurs through two distinct pathways—above a critical force the cluster ruptures uniformly, while for finite ℓ a single large gap of open bonds, a crack, grows and spreads through the system; for intermediate ℓ both pathways coexist. Analytically, the critical crack size increases with ℓ, meaning that distributing force over a few neighbors stabilizes the cluster relative to purely local load sharing.","feed_headline":"Force-sharing range sets the crack size that kills bond clusters","feed_subtitle":"A model spanning local and global load sharing predicts two failure modes—uniform opening and spreading cracks—and matches simulations.","key_machinery":"The central object is the force-decay length ℓ: each bit of applied force is distributed among closed bonds with weight cosh((L/2 − d)/ℓ) on a periodic ring, so that ℓ→∞ recovers equal load sharing and ℓ→0 recovers nearest-neighbor local load sharing. The analytic engine is the continuum approximation in Eq. (13), which turns the force on a bond into a sum over gaps, each contributing half the gap's applied load attenuated by exponentials in distance; the mean-field critical-gap balance—the closing rate of open bonds inside the largest gap equals the unbinding rate of its boundary bonds—then yields the implicit rupture conditions in Eqs. (24) and (26).","core_discovery":"For a one-dimensional ring of N reversible bonds that open with rate k_off,0 exp(f_i/f_d) and close with constant k_on, the paper defines the force f_i on closed bond i as the sum over all force-attack sites of a normalized cosh((L/2 − d_ij)/ℓ) kernel. That single kernel contains the two known extremes: ℓ→∞ gives equal force on every closed bond, and ℓ→0 gives force only from the two neighboring gaps, the local rule. Replacing the discrete sum by an integral and dropping the logarithmic term in Eq. (12) under an approximate-symmetry assumption yields Eq. (13), in which the force is a sum over gaps, each contributing half the gap's applied load attenuated exponentially with distance. From tha","pith_inferences":["Beyond the paper: because the dropped logarithmic term in Eq. (12) is sizeable exactly when the closed-bond distribution is asymmetric around a gap, one can stress-test the theory by initializing simulations with one-sided bond configurations and checking whether the boundary force follows Eq. (26).","Beyond the paper: the ℓ-dependence of the critical gap size suggests a design rule—tuning the effective force-reach of a network, for example via crosslink spacing or filament stiffness, should directly set the rupture threshold; a controlled network experiment could test this.","Beyond the paper: the coexistence of uniform and crack pathways for intermediate ℓ implies that mean rupture time alone may miss the failure mode; measuring the spatial statistics of first openings would distinguish which pathway dominates."],"forward_implications":["If ℓ is small but nonzero, rupture is initiated by a critical crack, and increasing ℓ increases the critical gap size, so a broader load-sharing neighborhood stabilizes the cluster.","For intermediate ℓ both failure pathways coexist—uniform thinning and crack spreading—so no single order parameter captures rupture across the whole parameter range.","The critical gap size estimated from the fuller continuum approximation, Eq. (26), matches simulations for ℓ/δL up to about 0.5, while the simpler local-model estimate only works very close to ℓ→0.","The analytical bounds in Eqs. (28)–(30) define a stable region in gap-size/boundary-force space; simulation trajectories mostly stay inside before rupture and leave it after rupture.","The model contains the global and local load-sharing models as limiting cases, so previously derived rupture conditions are recovered as special limits."],"supporting_citations":[{"why":"Supplies the local-load-sharing limiting model and the crack-initiation rupture picture that this model generalizes.","marker":"[21]"},{"why":"Supplies the global-load-sharing rupture condition that the equal-loading limit reproduces.","marker":"[3]"},{"why":"Basis for the metastable-cluster and stochastic-rupture picture used throughout the paper.","marker":"[4]"},{"why":"Establishes the constant-force adhesion-cluster stability problem and the rupture-time observable.","marker":"[5]"},{"why":"Provides the dynamic-loading rupture-regime criterion related to the global critical-force condition.","marker":"[7]"},{"why":"Supplies the exponential force-dependent unbinding kinetics used in Eq. (1).","marker":"[2]"},{"why":"Exact stochastic simulation algorithm used for all numerical results.","marker":"[25]"}],"fun_headline_variants":["Short-range load sharing makes bond clusters crack first","Bond cluster failure: uniform rupture vs crack growth","Load-sharing length sets whether clusters crack or burst","Crack failure emerges when load sharing is local","Interpolating load-sharing model predicts two failure modes"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"All later analytical force formulas assume that bond configurations near the middle of each gap are approximately mirror-symmetric, so the logarithmic correction in Eq. (12) can be dropped; if actual crack configurations are systematically asymmetric, the predicted critical gap sizes shift.","fun_headline_variants_meta":{"raw":{"variants":["Short-range load sharing makes bond clusters crack first","Bond cluster failure: uniform rupture vs crack growth","Load-sharing length sets whether clusters crack or burst","Crack failure emerges when load sharing is local","Interpolating load-sharing model predicts two failure modes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000815,"raw_usage":{"total_tokens":3380,"prompt_tokens":685,"completion_tokens":2695,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":429,"completion_tokens_details":{"reasoning_tokens":2622}},"tokens_in":429,"tokens_out":2695,"duration_ms":23124,"temperature":1.0,"reasoning_tokens":2622,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T20:24:59.903350+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Seed a stochastic simulation with a deliberately asymmetric configuration—for example, all closed bonds concentrated on one side of the largest gap—at ℓ/δL = 0.5, and compare the measured force on the gap boundary with Eq. (26). If the boundary force differs systematically from the prediction, or if the measured critical gap size departs from the predicted value by more than the simulation spread, the symmetry assumption fails.","supporting_citations":[],"review_version":1}