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REVIEW 3 major objections 5 minor 47 references

Kinesin’s six-state chemomechanical network shows the Mpemba effect, and motor velocity can reveal it without full state readout.

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2026-07-31 20:54 UTC pith:IW6WE2GP

load-bearing objection Solid application of Markovian Mpemba tools to Lipowsky’s kinesin network; velocity-as-readout is the useful hook, transfer to real motors is the soft spot. the 3 major comments →

arxiv 2607.27998 v1 pith:IW6WE2GP submitted 2026-07-30 cond-mat.stat-mech cond-mat.softphysics.bio-phphysics.chem-ph

Mpemba effect in a chemomechanical model of the Kinesin molecular motor

Karthik Cheruvary , Arnab Pal This is my paper
classification cond-mat.stat-mech cond-mat.softphysics.bio-phphysics.chem-ph PACS 05.70.Ln87.16.Nn05.40.-a
keywords Mpemba effectkinesinchemomechanical networkanomalous relaxationmolecular motorsMarkov master equationnon-equilibrium steady statemotor velocity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper asks whether the Mpemba effect—faster relaxation from a state farther from equilibrium than from one closer—can appear in a living molecular machine. Using the standard six-state Markov network for kinesin, the authors show that anomalous relaxation already exists in chemical equilibrium and that its phase boundaries track how temperature and concentrations reshape the free-energy landscape and its metastable states. Mechanical load and chemical driving away from ATP-hydrolysis equilibrium mainly stretch or compress that phase diagram rather than erase the effect in the physically relevant regime. Crucially, the probability current through the mechanical step—the motor’s velocity—shares the same slowest relaxation mode as the full state distribution, so velocity crossings after a quench are a direct experimental signature. The work positions molecular motors as a concrete baseline for looking for Mpemba-like speedups in non-equilibrium biochemical networks.

Core claim

In the six-state chemomechanical kinesin model, the Mpemba effect occurs under chemical equilibrium, persists when detailed balance is broken by load or non-equilibrium ATP hydrolysis (which mainly reshape the Mpemba phase diagram), and is faithfully mirrored by the relaxation of the mechanical current, i.e., the motor velocity, whenever the slow-mode spectral criterion holds and that current projects onto the slowest mode.

What carries the argument

The spectral a2 criterion: after a quench, long-time relaxation is dominated by the slowest non-stationary eigenmode of the rate matrix; non-monotonic dependence of its amplitude a2 on the initial control parameter (temperature or force) is necessary and sufficient for Mpemba crossings in distance measures, and the same a2 multiplies the projection of that mode onto the mechanical current J25, so velocity inherits the anomalous relaxation.

Load-bearing premise

The model puts all load dependence on a single mechanical edge and chooses intrinsic rates to explore energy landscapes rather than lock them to a fully validated kinesin parameterization; if real force reshapes the whole landscape or rates differ qualitatively, the phase diagrams and velocity signature need not carry over.

What would settle it

In a single-molecule optical-trap quench of temperature or chemical conditions on kinesin, prepare two ensembles at different initial temperatures (or concentrations), quench to the same final bath, and test whether the hotter (or farther) ensemble’s step-averaged velocity reaches the final steady velocity before the cooler one whenever the model’s a2 criterion predicts a crossing.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Motor velocity alone can serve as an experimental readout of the Mpemba effect without reconstructing microscopic state occupations.
  • Chemical and mechanical driving mainly move Mpemba phase boundaries rather than destroy the phenomenology in the explored kinesin regime.
  • Force quenches, unlike temperature quenches, do not produce a Mpemba effect in this network over the parameters surveyed.
  • The same spectral-plus-current logic suggests looking for velocity or flux Mpemba signatures in other finite-state motor and enzyme networks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If velocity inherits a2 whenever the mechanical projection is nonzero, other routinely measured motor observables (run length, waiting-time statistics) may also display Mpemba crossings when they couple to the same slow mode.
  • The reported absence of force-induced Mpemba may be specific to load acting only on one edge; full landscape force dependence could reopen a force-quench Mpemba window.
  • Extending the protocol to dynein or myosin kinetic models would test whether anomalous relaxation is generic among processive motors or kinesin-specific.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript studies anomalous relaxation (the Mpemba effect) in Lipowsky’s six-state chemomechanical Markov network for kinesin. Using the master equation, L1 distance, and the equilibrium spectral a2 criterion, the authors map Mpemba phases under chemical equilibrium as functions of bath temperature and concentration scale, interpret them via the free-energy landscape and metastability, and then show that mechanical load and chemical driving (parameter c) mainly stretch or compress those phases without destroying the phenomenology in the regimes explored. They further argue that the mechanical current J25 (motor velocity) inherits the slowest non-stationary mode whenever the projection C2,25 is nonzero, so velocity relaxation can exhibit Mpemba crossings and serve as an experimentally accessible signature. Force quenches are also examined and no force-induced Mpemba effect is reported.

Significance. The work is a clear first step in bringing the Mpemba effect into a standard, experimentally motivated biochemical network. The within-model spectral analysis is standard and correctly applied; the current expansion (Eqs. 19–21) cleanly shows why velocity can track a2; and the phase diagrams under equilibrium and broken detailed balance give a concrete baseline for anomalous relaxation in living systems. Identifying motor velocity as a probe is a useful, falsifiable experimental hook for single-molecule optical-trap work. Strengths include transparent use of literature Mpemba criteria (Lu–Raz, Klich et al.) and an honest separation of equilibrium intuition from non-equilibrium driving. The main scientific value is existence and phenomenology inside a consistent kinesin-type network, not a quantitative fit to a particular motor.

major comments (3)
  1. [Section 2; Section 5] Sec. 2 and Sec. 5: the free energies Ei and barrier heights Bij that define κ ij (Eq. 3) are never tabulated, nor is a complete numerical rate set provided. Phase diagrams (Figs. 5, 8, 10) and a2 curves are therefore not reproducible from the text alone. For a computational spectral study this is load-bearing: please supply a table (or SI) of all Ei, Bij (or all independent κ ij) used for the main figures, and state which quantities were held fixed when s, r, c, F, θ were varied.
  2. [Section 2, Eq. (4); Section 5.4] Sec. 2, Eq. (4) and Sec. 5.4: load dependence is restricted to the single mechanical edge (φ25=e^{-θF}, φ52=e^{(1-θ)F}, all other φij=1), and rates are chosen to explore landscapes rather than fixed to a validated kinesin parameterization. The algebra that velocity tracks a2 when C2,25 eq0 (Eqs. 20–21) is correct inside the model, but the claim that velocity is an “experimentally accessible signature” for real kinesin needs an explicit limitations paragraph: if force reshapes chemical barriers or empirical rates reorder λ2 / flip a2(T) / drive C2,25 through zero, the driven phases and the velocity probe need not transfer. Temper the abstract/conclusion wording from a general experimental route to a model-supported prediction under stated closures.
  3. [Section 5.4] Sec. 5.4: the statement that C2,25 “remains non-zero throughout the parameter ranges investigated” is central to the velocity–Mpemba link but is only asserted. Please add a brief systematic check (e.g., min |C2,25| on the same grids used for Figs. 5, 8, 10, or a supplementary plot) and note any loci where C2,25 is consistent with zero, where velocity would cease to report the slow mode.
minor comments (5)
  1. [Figure 3; Figure 4] Fig. 3 caption has a duplicated “(a)” label; panel (b) is mis-tagged. Clean up figure labels throughout (e.g., Fig. 4 “(b)(b)”).
  2. Typos: “sytems” (p. 2), “Mpema” (Fig. 6 caption), “nonethelss”, “Paramteres” (Fig. 9), “supresses” (Fig. 8). Standard copy-edit pass needed.
  3. [Section 3.1; Section 5.3] Sec. 3.1: condition (i) for non-temperature quenches is correctly flagged as not automatic; when discussing force quenches (Sec. 5.3), state explicitly whether L1(Fi) was verified monotonic in the plotted examples (Fig. 9b already shows a problematic case).
  4. [Section 5.1] Sec. 5.1: sampling a2 at 250 temperatures in (Tb+0.1, Tb+50) is reasonable; briefly note how non-monotonicity was decided (e.g., discrete derivative sign changes) to avoid false positives from numerical noise near a2=0 (strong Mpemba).
  5. [Section 1; Section 6] Introduction/conclusion cite a broad living-systems outlook (dynein, myosin, proofreading). One or two sentences on why the six-state kinesin topology is representative enough for that outlook—or that it is only a baseline—would tighten the framing.

Circularity Check

0 steps flagged

No significant circularity: Mpemba criteria and kinesin network are external; phase diagrams and velocity inheritance follow from spectral analysis of M, not from fitted or self-defined targets.

full rationale

The paper applies the standard Markovian Mpemba definition and a2 spectral criterion (Lu & Raz; Klich et al.) to Lipowsky’s established six-state kinesin network. Equilibrium and driven phase diagrams are obtained by sampling a2(T,Tb) from the master-equation generator M under author-chosen but explicitly stated energy-landscape and rate constructions; crossings are outputs of diagonalization, not inputs fitted to force crossings. The claim that motor velocity mirrors anomalous relaxation is a direct algebraic consequence of the eigenmode expansion of P(t) inserted into J25 (Eqs. 16, 20–21): long-time current tracks a2 whenever the projection C2,25 ≠ 0, which the authors report as nonzero on the grids checked. That is a legitimate spectral implication, not a definitional loop. Modeling closures (single-edge load factors ϕ25/ϕ52; hand-built Ei, Bij) limit experimental transfer but do not make any reported “prediction” reduce by construction to its inputs. Self-citations (e.g. related Pal/Biswas Mpemba work) are contextual, not load-bearing uniqueness theorems. No fitted-input-as-prediction, self-definitional, or ansatz-via-self-citation circularity is present.

Axiom & Free-Parameter Ledger

5 free parameters · 7 axioms · 0 invented entities

The central claim rests on (1) the standard continuous-time Markov master equation and spectral Mpemba criterion, (2) Lipowsky’s reduced six-state kinesin topology with a hand-built Arrhenius landscape, and (3) several modeling closures (force only on the mechanical edge; phosphate set by c; backward-cycle rate κ54 fixed by DB). No new physical entities are postulated. Free parameters are the landscape and bath/drive knobs used to paint phase diagrams; they are exploration parameters, not fits to a measured Mpemba curve.

free parameters (5)
  • State free energies Ei and barrier heights Bij (forward cycle)
    Set the intrinsic rates κij via Arrhenius factors; paper never tabulates numerical values, yet all phase diagrams depend on them. Chosen to explore metastability rather than fit a specific kinesin dataset.
  • Concentration scale s and ATP/ADP ratio r = example working points e.g. r=0.1, s in ~0.1–20
    Primary axes of the Mpemba phase diagrams; scanned by hand to map boundaries.
  • Chemical drive c = [P]/[P]_eq factor
    Controls departure from ATP-hydrolysis equilibrium; used to stretch/compress phase diagrams (c=0.1, 1, 5.5 shown).
  • Load F and load-sharing θ = θ≈0.6 in several figures
    Enter only through ϕ25, ϕ52; θ fixed in physical range (e.g. 0.6) while F is scanned; determine mechanical nonequilibrium and stall.
  • Bath temperature Tb and quench window for a2 sampling = 250 uniform T samples
    Tb is both control parameter and final quench target; a2 sampled at 250 points in (Tb+0.1, Tb+50) to declare Mpemba phases.
axioms (7)
  • domain assumption Continuous-time Markov master equation on a finite state space fully describes motor relaxation between transitions (separation of timescales / Kramers picture).
    Sec. 2–3: internal conformational fluctuations assumed equilibrated between interstate jumps; dynamics reduced to M acting on six occupation probabilities.
  • standard math For detailed-balance generators, non-monotonic a2(T) is necessary and sufficient for Mpemba crossings in suitable distances (Lu–Raz spectral criterion).
    Sec. 3.2, Eq. (18); used to classify all equilibrium phase diagrams.
  • standard math L1 distance satisfies the three Lu–Raz axioms (ordering of initial distances, monotonicity, convexity), so crossings in L1 define the effect.
    Sec. 3.1; L1 used throughout as the operational distance.
  • domain assumption Biochemical constraints reduce the 9 nucleotide configurations to the six-state bicyclic network of Lipowsky et al., with one mechanical edge and forward/backward ATP-consuming cycles.
    Sec. 2 and Fig. 1; topology and cycle structure taken from Refs. [30–32].
  • ad hoc to paper External load modifies only the mechanical rates R25, R52 via exponential load-sharing factors; all chemical rates are force-independent.
    Sec. 2 Eq. (4): authors explicitly retain ‘only its dominant effect on the mechanical step’ to minimize parameters; acknowledged as a simplification of the real force-tilted landscape.
  • ad hoc to paper Backward-cycle intrinsic rate κ54 is fixed by enforcing detailed balance on the backward cycle given the forward landscape (Eq. 9).
    Sec. 2: choice of which backward rate is dependent is called arbitrary; sets the equilibrium reference before c or F break DB.
  • domain assumption Steady-state occupation is a practical proxy for metastability (deeper wells ↔ larger Pss), sufficient for qualitative landscape explanations of phase boundaries.
    Sec. 4: authors note the proxy is approximate because of network coordination numbers.

pith-pipeline@v1.2.0-daily-grok45 · 18356 in / 4031 out tokens · 86871 ms · 2026-07-31T20:54:02.019127+00:00 · methodology

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read the original abstract

The Mpemba effect, wherein a system prepared farther from equilibrium relaxes faster than one initially closer to equilibrium, has been extensively investigated in a wide range of physical systems. In contrast, its role in biologically relevant non-equilibrium processes remains largely unexplored. Here, we investigate anomalous relaxation in the six-state chemomechanical network model of the Kinesin molecular motor under both equilibrium and non-equilibrium conditions. We first establish the existence of the Mpemba effect in chemical equilibrium and show that many of its qualitative features can be understood from the underlying free-energy landscape. We then examine the effects of mechanical and chemical driving, showing that breaking detailed balance primarily reshapes the Mpemba phase diagram without qualitatively altering the relaxation phenomenology over the physically relevant parameter regime. Finally, we demonstrate that the relaxation of the motor velocity also mirrors the anomalous relaxation of the underlying stochastic dynamics, thereby identifying an experimentally accessible signature of the Mpemba effect. Our results establish molecular motors as a promising baseline for studying anomalous relaxation in living systems and suggest a broader framework for exploring the Mpemba effect in non-equilibrium biochemical networks.

Figures

Figures reproduced from arXiv: 2607.27998 by Arnab Pal, Karthik Cheruvary.

Figure 1
Figure 1. Figure 1: Panel (a): The chemo mechanical network representing Kinesin motion. This [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Non-equilibrium nature of the Kinesin depicted by the velocity defined [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Examples illustrating equivalence of the spectral [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Energy landscape of the Kinesin walker that determines [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Panel (a): Mpemba phases in the plane of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Example showcasing that metastability is not sufficient to predict the Mpemba [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Variation in a2(T, Tb) with T for different parameters showing the two distinct behaviors near Mpemba phase boundaries. Curves normalized to show shape and x axis re-centered at Tb for each curve. Panel (a): |a2(T, Tb)| vs T for different Tb with s = 15, r = 0.1. Increasing Tb corresponds to moving from right to left in the phase diagram shown in [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Plots showing the effect of adding external forcing. Panel (a) Mpemba [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Examples of behavior seen after a force quench. Panel(a) Example of [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Effect of c on the Mpemba phase diagram with all other parameters as in [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Examples showing crossing over of the probability current [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗

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Reference graph

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