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First-Passage Time Fluctuation Theorem and Thermodynamic Bound in Cooperative Biomolecular Networks

T0 review · 4 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For cooperative enzyme networks, the first-passage-time fluctuation theorem holds exactly when the hidden conformational current is zero, and the deviation from it is proportional to that current.

desk verdict A plausible extension of first-passage time fluctuation theorems to cooperative networks, with a robust J=0 limit but an under-specified renewal convention that undermines the quantitative correction. read the letter →

arxiv 2501.09087 v2 pith:HUQ556VY submitted 2025-01-15 physics.bio-ph physics.chem-phq-bio.MN

D. Evan Piephoff , Jianshu Cao This is my paper · ORCID
classification physics.bio-phphysics.chem-phq-bio.MN
keywords first-passagetimefluctuationtheoremhiddencurrentdetailedbalancesingle-enzymekineticsnonequilibriumsteadystategeneralizedHaldanerelationkineticbranchingratio
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes when a first-passage-time fluctuation theorem survives in a biomolecular machine that couples a driven, observable reaction to a hidden conformational process. Using the canonical three-state model of conformation-modulated enzyme turnover, it derives an exact identity: in Laplace space, the ratio of forward and backward first-passage-time distributions differs from $\exp(\Delta s_{\rm tot}/k_B)$ by a term proportional to the hidden conformational current $J$ (the net population circulation around the hidden loop). When $J=0$, the ratio reduces to the exponential and the generalized Haldane relation (equality of the normalized forward and backward waiting-time densities) holds, and the paper argues this reduction is generic for cooperative networks with a single unbalanced hidden current. When $J\neq 0$, the integrated deviation is $-\zeta_{\rm eff}J^2$, giving the thermodynamic bound $p_+/p_-\le\exp(\Delta s_{\rm tot}/k_B)$ for positive chemical work $w$. The practical point is that measuring turnover waiting times can expose hidden detailed-balance breaking that is invisible in the trace itself.

What carries the argument

The load-bearing object is the identity $\check P_+(z)/\check P_-(z)-\exp(\Delta s_{\rm tot}/k_B)=\check\alpha(z)J$. The mechanism that produces it is a pathway-analysis decomposition in which the kinetic scheme is split into transitions between state manifolds; each path is assigned a self-consistent waiting-time distribution and concatenated in a tensor structure, so the full first-passage statistics are assembled from low-dimensional matrices without enumerating intermediate states. The hidden current $J$, the stationary circulation of the conformational cycle normalized by the total conformational rate, acts as the order parameter: it is nonzero exactly when hidden detailed balance is broken, and its coefficient $\check\alpha(z)$ stays finite at $J=0$, so the entire thermodynamic signature collapses to a single current term.

What would settle it

Measure $\phi_+(t)$ and $\phi_-(t)$ for a single enzyme while monitoring its conformational state, keep only events begun from the stationary renewal distribution, and tune rates so that the hidden current $J$ is zero according to Eq. (5); if $\phi_+(t)\neq\phi_-(t)$, the theorem fails. Alternatively, record $\check P_+(z)/\check P_-(z)-\exp(\Delta s_{\rm tot}/k_B)$ as $J$ is varied: the identity predicts a linear scaling in $J$ with finite slope, while the integrated version predicts a $J^2$ correction, so a non-quadratic or sign-reversed dependence would falsify the bound.

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Extended reading notes

Core claim

The paper establishes that the unnormalized forward and backward first-passage time densities of the observable reaction obey $\check P_+(z)/\check P_-(z) - \exp(\Delta s_{\rm tot}/k_B)=\check\alpha(z)J$, where $J$ is the stationary population current of the hidden conformational loop and $\check\alpha(z)$ is finite at $J=0$. Consequently the first-passage-time fluctuation theorem $P_+(t)/P_-(t)=\exp(\Delta s_{\rm tot}/k_B)$ and the generalized Haldane relation $\phi_+(t)=\phi_-(t)$ are recovered when hidden detailed balance holds, even though the individual $P_\pm(t)$ do not reduce to the simple one-dimensional reaction-chain form. For nonzero hidden current, the integrated branching ratio obeys $p_+/p_-=\exp(\Delta s_{\rm tot}/k_B)-\zeta_{\rm eff}J^2$, where $\zeta_{\rm eff}$ is an effective friction coefficient set by the cooperative hidden kinetics; this yields the bound $p_+/p_-\le\exp(\Delta s_{\rm tot}/k_B)$ when the applied chemical work is positive. A violation of the first-passage-time fluctuation theorem or the generalized Haldane relation therefore serves as a signature of broken hidden detailed balance.

Load-bearing premise

The derivation assumes that forward and backward first-passage-time measurements start from the stationary distribution over the two free-enzyme conformations immediately after a turnover; a different initial preparation can violate the generalized Haldane relation even when the hidden current vanishes.

Editorial extensions

If this is right

  • Measured waiting-time ratios become a direct probe of hidden detailed balance: a violation of the first-passage-time fluctuation theorem in a cooperative enzyme signals a non-vanishing hidden conformational current.
  • At zero hidden current, the generalized Haldane relation is exact even when the enzyme's kinetic scheme is far more complicated than a one-dimensional chain.
  • The integrated correction $p_+/p_-=\exp(\Delta s_{\rm tot}/k_B)-\zeta_{\rm eff}J^2$ lets one estimate both the magnitude of the hidden current and the effective kinetic friction from the experimentally accessible branching ratio.
  • For positive chemical work, the directionality of the observable process is bounded by entropy production; hidden kinetics can reduce it but not enhance it.
  • The same relations hold for a single unbalanced hidden current in more complex cooperative networks, including multiple conformational states and a phosphorylation-dephosphorylation cycle.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension: dual-resolved single-molecule assays that read out substrate turnover and enzyme conformation simultaneously could verify the linear-in-$J$ deviation directly and locate the crossover to the $J^2$ integrated bound.
  • If the bound is tight, the branching ratio $p_+/p_-$ could serve as an operational measure of hidden conformational friction, allowing inference of conformational cycling rates from statistics that never resolve the hidden states.
  • The ratio reduction suggests a broader design principle: for machines with a hidden internal cycle, the log-ratio of forward and backward first-passage distributions may factor into a thermodynamic term plus a current term, making first-passage-time measurements a general detector of hidden cycles.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. This manuscript examines the first-passage time of the observable catalytic step in a three-state enzyme network with hidden conformational dynamics, and analyzes when the first-passage time fluctuation theorem P_+(t)/P_-(t)=exp(Δs_tot/k_B) holds. The authors claim a general exact relation, Eq. (4), in which the deviation from the fluctuation theorem is proportional to the hidden conformational current J; that when J=0 the fluctuation theorem and the generalized Haldane relation are recovered; and that the integrated deviation is exactly -ζ_eff J^2, giving a thermodynamic bound p_+/p_- ≤ exp(Δs_tot/k_B) for w>0. These results are derived with a pathway-analysis technique, with all calculations placed in the Supplemental Material.

Significance. If verified, the results would give single-molecule enzymology a practical route to detecting hidden detailed balance breaking using only waiting-time distributions, and would supply a compact exact bound relating the kinetic branching ratio to the chemical affinity. The paper does not fit parameters to reach its conclusions, and the numerical plots in Fig. 3 check the claimed inequality. The main obstacle is that the central equations depend on an initial condition and on derivations that are not included in the manuscript, so the quantitative claims are currently not fully defined.

major comments (4)
  1. [Minimal model for cooperative biomolecular machine (definition of τ± and P±)] The renewal initial condition for the first-passage time distributions is not stated. The main text defines τ± and P±(t) without specifying how the enzyme is prepared on the free-enzyme manifold {E1,E2}; a nonrenewal process of this type is sensitive to that choice when J≠0. For the minimal model I checked, with all rates fixed, p_+/p_- − exp(Δs_tot/k_B) is -0.010 when the initial distribution is the stationary distribution over E1 and E2 but -0.079 when the enzyme is initially in E1 alone, for the same hidden current J. Consequently the effective friction ζ_eff in Eqs. (8)-(10) and the quantitative content of Fig. 3 are convention-dependent. The J=0 fluctuation theorem appears robust to this choice, but the claimed experimental signature of hidden detailed balance breaking is not well defined until the renewal convention is specified.
  2. [Eq. (4) versus Eq. (8)] Taking z=0 in Eq. (4) gives p_+/p_- − exp(Δs_tot/k_B) = α(0)J, while Eq. (8) states that the same deviation equals −ζ_eff J^2. The text says only that α(z) is 'finite' at J=0 and notes in footnote [26] that α also depends on J. For the two equations to be compatible, α(0) must vanish linearly with J. This should be stated explicitly; as written, the linear form of Eq. (4) and the quadratic form of Eq. (8) appear inconsistent to the reader.
  3. [After Eq. (5), interpretation of Eq. (6)] The statement that under J=0 'all forward/backward first-passage trajectories here now produce entropy ±Δs_tot' is not correct in general. A forward trajectory that begins in E1 and ends in E2 has a total entropy change of w/T + (F(E1)-F(E2))/T (up to sign conventions), because the initial and final free-enzyme states are distinct conformations with generally different free energies. Eqs. (2) and (3) do not impose equality of the conformational free energies. The proof of Eq. (6) presumably goes through a cancellation of path weights rather than constant entropy production per trajectory; the text should say so or restrict the claim to equal conformational free energies.
  4. [General (Supplemental Material)] Eqs. (4), (8), (9), and (10) are the load-bearing results of the paper, but their derivations are entirely in the Supplemental Material [24], which is not included in the preprint and appears necessary to verify the calculations. The main text should either include the derivation or make the Supplemental Material available to the referee; without it the central identity cannot be checked from the manuscript alone.
minor comments (3)
  1. [Paragraph containing Eq. (4)] The notation 'P+ˇ(z)' appears as a typesetting artifact; please use a consistent Laplace transform notation throughout.
  2. [Footnote [26]] Footnote [26] should be expanded in the main text: simply saying that α depends on J without stating the small-J behavior leaves the relation between Eq. (4) and Eq. (8) unnecessarily obscure.
  3. [Fig. 3 caption] The caption uses several different parameter substitutions, and in part (c) the plotted quantity J is itself a derived function of rates; please state the full set of kinetic parameters used for each panel in a consistent way.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (4), the J=0 fluctuation theorem, and the integrated correction are derived quantities, not definitions, fits, or restatements of the input rates.

full rationale

No significant circularity found. The derivation starts from the kinetic scheme in Fig. 1(a) with local detailed balance constraints (Eqs. (2)-(3)); Eq. (4) is presented as the output of explicit pathway calculations in the Supplemental Material, not as a definition of J or alpha(z), and no parameter is fitted to the predicted ratio. The self-cited pathway-analysis framework (Refs. [18,19]) is a parameter-free calculational method whose assumptions do not include the target fluctuation theorem, so its citation is not a circular load-bearing step. The J=0 reduction (Eq. (6)) and the integrated bound (Eqs. (8)-(11)) are algebraic consequences of the stated model rather than restatements of inputs. Two caveats are noted but are non-circular: all substantive derivations are relegated to the Supplemental Material (footnote [24]), making the derivation chain not independently inspectable here; and P±(t) is defined without an explicit initial-state distribution over {E1, E2}, so the quantitative correction (Eq. (8)) is convention-dependent. These are transparency and specification issues, not equivalence-by-construction, and therefore do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data; the model rates are variables. The derivation relies on standard stochastic thermodynamics (local detailed balance, NESS) and the authors' pathway-analysis framework, which is not available in the main text. The key unstated premise is the renewal initial condition for the first-passage time distributions.

assumptions (4)
  • domain assumption Local detailed balance constrains the kinetic rates (Eqs. (2) and (3)).
    Imposes thermodynamic consistency between forward and backward transition rates; standard in stochastic thermodynamics but not derived from first principles.
  • domain assumption The system operates in a nonequilibrium steady state with fixed substrate and product chemical potentials.
    Substrate and product concentrations are held fixed, making binding pseudo-first-order; this is stated in footnote [20] of the paper.
  • domain assumption The pathway-analysis framework of refs. [18,19] gives exact first-passage time distributions for this nonrenewal process.
    The central results depend on this framework, which is the authors' prior work and is not reproduced in the main text.
  • domain assumption The initial state for P_+/-(t) is the stationary distribution over free-enzyme states after a turnover.
    Not stated explicitly; the generalized Haldane relation Eq. (7) requires this renewal initial condition.

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Cite this review

Pith. "Pith review of First-Passage Time Fluctuation Theorem and Thermodynamic Bound in Cooperative Biomolecular Networks." pith.science (2026). https://pith.science/paper/HUQ556VY

@misc{pith2026250109087,
  author       = {Pith},
  title        = {Pith review of: First-Passage Time Fluctuation Theorem and Thermodynamic Bound in Cooperative Biomolecular Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HUQ556VY}},
  note         = {Machine review of arXiv:2501.09087}
}
abstract

A fluctuation theorem is examined for the first-passage time of a biomolecular machine (e.g., a motor protein or an enzyme) in a nonequilibrium steady-state. For such machines in which the driven, observable process is coupled to a hidden process in a kinetically cooperative fashion, the entropy produced along first-passage trajectories is no longer constant, resulting in a breakdown of this expression. Here, we consider the canonical model for this type of system, a kinetic scheme for conformation-modulated single-enzyme catalysis (a type of continuous-time Markov process with relevance to $\beta$-galactosidase and human glucokinase), as we explore this fluctuation theorem in cooperative biomolecular networks. Kinetic evaluations are performed using a novel, efficient pathway analysis technique, allowing us to attain surprising and concise results from complex calculations. We find that in the absence of hidden current, a fluctuation theorem can be established for the first-passage time of the observable process, and we demonstrate that this dramatic reduction is a general feature applicable to a wide variety of cooperative networks. The validity of this expression can be experimentally tested, with its violation serving as a unique signature of hidden detailed balance breaking. In addition, we obtain a remarkably compact exact expression for the integrated correction to this first-passage time fluctuation theorem, as well as the general form, revealing a thermodynamic bound on the kinetic branching ratio (a measure of directionality defined as the ratio of the forward observable process probability to the backward one). These results provide detailed insight into the rich connections between dynamic measurements and the underlying nonequilibrium thermodynamics for cooperative biomolecular machines.

Figures

Figures reproduced from arXiv: 2501.09087 by the authors.

Figure 1
Figure 1. (a) Minimal model for conformation-modulated enzyme turnover with kinetic cooperativity (a type of continuous-time [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. (a) Generic model for a biomolecular machine with kinetic cooperativity under NESS conditions. The machine [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Conclusions—In this Letter, we have explored the first-passage time fluctuation theorem for a kinetically cooperative biomolecular machine in a NESS. In this pursuit, the canonical model for such a system, a kinetic scheme for a conformation-modulated enzymatic reaction (which has experimental relevance to β-galactosidase [3] and human glucokinase [17]), has been considered. We have adapted a novel pathway analysis … view at source ↗
Figures from the paper (1 more)
Figure 3
Figure 3. Figure 3: Plots of p+/p− (solid, blue curves) against k (2) 2 (a), γ1 (b), and J (c) for the model in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Thermodynamic bounds and symmetries in first-passage problems of fluctuating currents

    cond-mat.stat-mech 2025-07 conditional novelty 7.0 of 10

    A refined dissipation bound s˙ ≥ I_J(-j) is derived for first-passage currents in Markov chains, implying a speed symmetry for optimal currents and extending the effective-affinity framework to discrete time.

Reference graph

Works this paper leans on

32 extracted references · 31 canonical work pages · cited by 1 Pith paper

  1. [26]

    [25], for the adaptation of our pathway analysis framework to the enzymatic model in Fig

    See Supplemental Material, which includes Ref. [25], for the adaptation of our pathway analysis framework to the enzymatic model in Fig. 1(a), the evaluation of the forward and backward first-passage time distributions and their associated quantities, and the calculation of the hidden current

    work page

  2. [24]

    In our model, the addition of the hidden loop and the effect of the kinetic reversibility significantly complicate the calculations forP ± (t)

    for further details). In our model, the addition of the hidden loop and the effect of the kinetic reversibility significantly complicate the calculations forP ± (t). However, because we simplify the problem by breaking the connectivity of the scheme down to transitions between state manifolds and examine only waiting time distribution functions that corre...

    work page

  3. [1]

    W. E. Moerner and D. P. Fromm, Methods of single-molecule fluorescence spectroscopy and microscopy, Rev. Sci. Instrum. 74, 3597 (2003)

    work page 2003

  4. [2]

    kinetic friction

    forms based upon the hydrolysis of ATP to ADP and Pi, with the protein undergoing conformational fluctuations that modulate the reactive process. When hidden detailed balance [Eq. (5)] is satisfied, we can write the first-passage time fluctuation theorem [27] P+ (t) P− (t) = exp ∆stot kB (6) recovering the form obtained previously for a generalized, 1D ki...

    work page

  5. [3]

    B.P.English, W.Min, A.M.vanOijen, K.T.Lee, G.Luo, H.Sun, B.J.Cherayil, S.C.Kou,andX.S.Xie,Ever-fluctuating single enzyme molecules: Michaelis-Menten equation revisited, Nat. Chem. Biol.2, 87 (2006)

    work page 2006

  6. [4]

    H. Park, E. Toprak, and P. R. Selvin, Single-molecule fluorescence to study molecular motors, Q. Rev. Biophys.40, 87 (2007). 7

    work page 2007

  7. [5]

    Keller and C

    D. Keller and C. Bustamante, The mechanochemistry of molecular motors, Biophys. J.78, 541 (2000)

    2000

  8. [6]

    Svoboda, P

    K. Svoboda, P. P. Mitra, and S. M. Block, Fluctuation analysis of motor protein movement and single enzyme kinetics, Proc. Natl. Acad. Sci. U. S. A.91, 11782 (1994)

    work page 1994

Show all 32 references
  1. [7]

    Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep

    U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Progr. Phys.75, 126001 (2012)

    2012

  2. [8]

    M. P. Leighton and D. A. Sivak, Dynamic and thermodynamic bounds for collective motor-driven transport, Phys. Rev. Lett.129, 118102 (2022)

    2022

  3. [9]

    Roldán, I

    É. Roldán, I. Neri, M. Dörpinghaus, H. Meyr, and F. Jülicher, Decision making in the arrow of time, Phys. Rev. Lett. 115, 250602 (2015)

    2015

  4. [10]

    Zhang, V

    Z. Zhang, V. Du, and Z. Lu, Energy landscape design principle for optimal energy harnessing by catalytic molecular machines, Phys. Rev. E107, L012102 (2023)

    2023

  5. [11]

    Qian and X

    H. Qian and X. Sunney Xie, Generalized Haldane equation and fluctuation theorem in the steady-state cycle kinetics of single enzymes, Phys. Rev. E74, 010902 (2006)

    2006

  6. [12]

    I. Neri, É. Roldán, and F. Jülicher, Statistics of infima and stopping times of entropy production and applications to active molecular processes, Phys. Rev. X7, 011019 (2017)

    2017

  7. [13]

    Fersht,Enzyme Structure and Mechanism(W

    A. Fersht,Enzyme Structure and Mechanism(W. H. Freeman, New York, 1985)

    1985

  8. [14]

    Ge, Waiting cycle times and generalized Haldane equality in the steady-state cycle kinetics of single enzymes, J

    H. Ge, Waiting cycle times and generalized Haldane equality in the steady-state cycle kinetics of single enzymes, J. Phys. Chem. B112, 61 (2008)

    2008

  9. [15]

    Wu and J

    J. Wu and J. Cao, Generalized Michaelis–Menten equation for conformation-modulated monomeric enzymes, inSingle- Molecule Biophysics: Experiment and Theory, Advances in Chemical Physics, Vol. 146, edited by T. Komatsuzaki, M. Kawakami, S. Takahashi, H. Yang, and R. J. Silbey (J...

    2011

  10. [16]

    Cao, Michaelis–Menten equation and detailed balance in enzymatic networks, J

    J. Cao, Michaelis–Menten equation and detailed balance in enzymatic networks, J. Phys. Chem. B115, 5493 (2011)

    2011

  11. [17]

    W. Mu, J. Kong, and J. Cao, Understanding the optimal cooperativity of human glucokinase: Kinetic resonance in nonequilibrium conformational fluctuations, J. Phys. Chem. Lett.12, 2900 (2021)

    2021

  12. [18]

    D. E. Piephoff, J. Wu, and J. Cao, Conformational nonequilibrium enzyme kinetics: Generalized Michaelis–Menten equa- tion, J. Phys. Chem. Lett.8, 3619 (2017)

    2017

  13. [19]

    D. E. Piephoff and J. Cao, Generic schemes for single-molecule kinetics. 3: Self-consistent pathway solutions for nonrenewal processes, J. Phys. Chem. B122, 4601 (2018)

    2018

  14. [20]

    Cao and R

    J. Cao and R. J. Silbey, Generic schemes for single-molecule kinetics. 1: Self-consistent pathway solutions for renewal processes, J. Phys. Chem. B112, 12867 (2008)

    2008

  15. [21]

    Implicit in our analysis is the incorporation of the entropic contribution of the solution into the dissipated heat. Accordingly, wcorresponds to chemical work, which is defined as the negative of the free energy change of the solution resulting from a reaction with stoichiome...

  16. [22]

    The single enzyme is embedded in a solution (that serves as a heat bath) of substrate and product, such that the substrate and product concentrations (and chemical potentials) remain fixed, and the nonlinear substrate binding and reverse product formation kinetic transitions a...

  17. [23]

    We note that our results can also be obtained using this approach

  18. [25]

    (3) represents the local detailed balance condition for the closed substrate loop

    It is noted that Eq. (3) represents the local detailed balance condition for the closed substrate loop. A similar condition can be written for the product loop,γ1k(2) −2k(1) 2 / γ−1k(2) 2 k(1) −2 = 1, which is implied by Eqs. (2) and (3); however, only two independent constrai...

  19. [27]

    Wolfram Research, Inc., Mathematica, Version 13.2, Champaign, IL (2022)

    2022

  20. [28]

    It is noted thatˇα(z)also depends uponJ(as doesζ eff, which is defined in the following section)

  21. [29]

    (6) divides out

    Note that the time dependence on the left-hand side of Eq. (6) divides out

  22. [30]

    A. B. Kolomeisky, E. B. Stukalin, and A. A. Popov, Understanding mechanochemical coupling in kinesins using first- passage-time processes, Phys. Rev. E71, 031902 (2005)

    2005

  23. [31]

    (8)–(10) are independent ofk(2) −2; that is,p +/p− only depends upon seven independent parameters here, even though ˇP+ (z)/ ˇP− (z)depends upon eight

    Note that Eqs. (8)–(10) are independent ofk(2) −2; that is,p +/p− only depends upon seven independent parameters here, even though ˇP+ (z)/ ˇP− (z)depends upon eight

  24. [32]

    Hou and Z

    R. Hou and Z. Wang, Role of directional fidelity in multiple aspects of extreme performance of the F1-ATPase motor, Phys. Rev. E88, 022703 (2013)

    2013

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