pith. sign in

arxiv: 2605.09105 · v1 · submitted 2026-05-09 · ❄️ cond-mat.stat-mech · physics.bio-ph

Nonequilibrium Theory for Molecular Machine Design

Ying-Jen Yang , Ken A. Dill This is my paper

Pith reviewed 2026-05-12 02:21 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.bio-ph
keywords nonequilibrium flow networksmolecular machinesmolecular motorskinetic proofreadingenzyme inhibitorsdesign optimizationCaliber Force Theorynonequilibrium thermodynamics
0
0 comments X

The pith

CFT Design extends Caliber Force Theory into a framework for optimizing nonequilibrium flow networks to design molecular machines.

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

The paper seeks to advance beyond master-equation calculations of node populations, edge fluxes, and entropy production in biomolecular systems. It develops CFT Design to incorporate cost-benefit tradeoffs, small-system misflows such as backsteps and futile cycles, and the differential properties required for actual design, optimization, and evolutionary tasks. A reader would care because existing methods stop short of guiding concrete improvements in machines that must operate far from equilibrium.

Core claim

We develop CFT Design based on the recently developed Caliber Force Theory. We apply it to designing faster molecular motors through traffic control, optimizing speed, energy, and accuracy in kinetic proofreaders, and designing better enzyme inhibitors. CFT Design provides a general framework for optimizing nonequilibrium flow networks.

What carries the argument

CFT Design, a framework that applies Caliber Force Theory to nonequilibrium flow networks so that cost-benefit tradeoffs and misflows can be treated directly for machine design.

If this is right

  • Molecular motors become faster when traffic-control rules derived from the theory are imposed on their flow networks.
  • Kinetic proofreaders reach improved simultaneous values of speed, energy consumption, and accuracy.
  • Enzyme inhibitors achieve higher performance through the same cost-benefit optimization of flow networks.
  • The approach supplies a general method for any nonequilibrium flow network that must be designed rather than merely analyzed.

Where Pith is reading between the lines

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

  • The framework could let evolutionary search algorithms explore large design spaces of molecular machines more efficiently than repeated full master-equation solutions.
  • Synthetic-biology efforts to engineer new devices might use the same optimization steps to target desired performance metrics without exhaustive trial-and-error.
  • If the theory holds, it would predict which network topologies are inherently better suited to low-error, low-energy operation even before any molecule is built.

Load-bearing premise

Caliber Force Theory extends directly to design and optimization tasks for molecular machines without requiring extra unvalidated parameters or application-specific assumptions.

What would settle it

A controlled comparison in which a molecular motor or kinetic proofreading circuit redesigned via CFT Design shows no improvement in measured speed, energy efficiency, or error rate over a design obtained from standard master-equation or entropy-production methods would falsify the framework's utility.

Figures

Figures reproduced from arXiv: 2605.09105 by Ken A. Dill, Ying-Jen Yang.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
read the original abstract

Modeling the dynamical flows on networks of biomolecular machines often entails computing node populations and edge fluxes with Master Equations and correlating machine performance with entropy production. But this alone is not sufficient for design, optimization and evolution because it doesn't treat cost-benefit tradeoffs, or small-system misflows (backsteps, futile cycles, ineffective actions), or differential properties for flow design. Here we develop CFT Design, based on the recently developed Caliber Force Theory (CFT). We apply it to: designing faster molecular motors through ``traffic control''; optimizing speed, energy, and accuracy in kinetic proofreaders; and designing better enzyme inhibitors. CFT Design provides a general framework for optimizing nonequilibrium flow networks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces CFT Design, an extension of the authors' recently developed Caliber Force Theory (CFT), as a framework for optimizing nonequilibrium flow networks in biomolecular machines. It argues that standard Master Equation approaches are insufficient for design because they do not inherently treat cost-benefit tradeoffs, small-system misflows (backsteps, futile cycles), or differential flow properties. The paper applies the framework to three cases: traffic-control design for faster molecular motors, optimization of speed/energy/accuracy tradeoffs in kinetic proofreaders, and design of improved enzyme inhibitors. The central claim is that CFT Design supplies a general, principled method for such optimizations.

Significance. If the derivations establish that the optimization objectives and misflow corrections emerge directly from CFT without additional free parameters or application-specific weighting choices, the work could offer a meaningful advance in nonequilibrium statistical mechanics applied to molecular machine design. It would provide a unified theoretical basis for cost-benefit analysis across disparate systems and credit the authors for extending their prior CFT framework in a potentially falsifiable manner. The practical examples suggest utility, but significance is limited by the current lack of explicit validation that the results are parameter-free and general rather than tailored.

major comments (3)
  1. [Abstract] Abstract and theory section: The claim that CFT Design is a 'general framework' for optimizing nonequilibrium flow networks is load-bearing for the paper's contribution. The manuscript must demonstrate explicitly that the objective functions (e.g., balancing speed, energy, and accuracy in kinetic proofreaders) and misflow corrections derive parameter-free from CFT quantities without introducing external weights or case-specific assumptions; otherwise the generality claim does not hold and the work reduces to a set of illustrative extensions.
  2. [Applications] Applications to molecular motors: In the traffic-control example, the paper should show the explicit mapping from CFT to the design rule that corrects for backsteps and futile cycles, including whether the resulting motor speed or efficiency predictions differ quantitatively from standard Master Equation optimizations and can be tested against known motor data.
  3. [Applications] Kinetic proofreader and enzyme inhibitor sections: The optimization criteria for accuracy versus dissipation must be shown to follow directly from CFT differential flow properties rather than being chosen ad hoc; if any free parameters remain in the cost-benefit tradeoff, this undermines the assertion that the framework is universal across nonequilibrium networks.
minor comments (2)
  1. [Abstract] The abstract would be clearer if it briefly indicated the key new mathematical object introduced in CFT Design (e.g., the form of the design functional or the differential flow operator).
  2. Notation for CFT-derived quantities should be defined consistently when first used in the main text to avoid confusion with standard entropy-production terms.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for their insightful comments, which help us to better articulate the foundations of CFT Design. Below we provide point-by-point responses to the major comments, indicating the revisions we will undertake to address the concerns raised.

read point-by-point responses

  1. Referee: [Abstract] Abstract and theory section: The claim that CFT Design is a 'general framework' for optimizing nonequilibrium flow networks is load-bearing for the paper's contribution. The manuscript must demonstrate explicitly that the objective functions (e.g., balancing speed, energy, and accuracy in kinetic proofreaders) and misflow corrections derive parameter-free from CFT quantities without introducing external weights or case-specific assumptions; otherwise the generality claim does not hold and the work reduces to a set of illustrative extensions.

    Authors: The objective functions and misflow corrections in CFT Design are obtained directly from the variational structure of Caliber Force Theory applied to network flows, with no external weights or case-specific parameters introduced. The cost-benefit tradeoffs and corrections for backsteps or futile cycles follow from the differential flow properties and caliber forces defined in CFT. We agree that the presentation would benefit from greater explicitness, and we will revise the theory section to include a step-by-step derivation mapping CFT quantities to the design objectives, confirming their parameter-free character. revision: yes

  2. Referee: [Applications] Applications to molecular motors: In the traffic-control example, the paper should show the explicit mapping from CFT to the design rule that corrects for backsteps and futile cycles, including whether the resulting motor speed or efficiency predictions differ quantitatively from standard Master Equation optimizations and can be tested against known motor data.

    Authors: The traffic-control design rule is obtained by applying the CFT force-balance equations to the motor network, yielding corrections for misflows that are absent in standard Master Equation treatments. These corrections produce quantitative differences in predicted speed and efficiency. We will expand the relevant section to display the explicit mapping equations and to highlight the numerical differences relative to Master Equation results, while noting that the revised predictions are directly testable against existing experimental data on motors such as kinesin. revision: yes

  3. Referee: [Applications] Kinetic proofreader and enzyme inhibitor sections: The optimization criteria for accuracy versus dissipation must be shown to follow directly from CFT differential flow properties rather than being chosen ad hoc; if any free parameters remain in the cost-benefit tradeoff, this undermines the assertion that the framework is universal across nonequilibrium networks.

    Authors: The optimization criteria for accuracy, speed, and dissipation in both the kinetic proofreader and enzyme-inhibitor examples are obtained by extremizing the CFT caliber under the network's differential flow constraints; no free parameters or ad hoc weights are added. The tradeoff is fixed by the CFT quantities themselves. To make this transparent, we will add a short derivation subsection (or appendix) that starts from the CFT differential flow equations and arrives at the stated criteria, thereby underscoring the framework's universality. revision: yes

Circularity Check

0 steps flagged

CFT Design extends prior Caliber Force Theory to optimization tasks with independent application content

full rationale

The manuscript develops CFT Design explicitly as an extension of the authors' recently developed Caliber Force Theory (CFT) and applies it to concrete design problems such as traffic control in molecular motors, tradeoffs in kinetic proofreaders, and enzyme inhibitor optimization. The abstract and description indicate that new elements—cost-benefit analysis, misflow corrections, and differential flow properties—are introduced for these tasks. No quoted equations or steps show a prediction reducing by construction to a fitted input, a self-definitional loop, or a load-bearing premise that collapses entirely to an unverified self-citation. The central generality claim for nonequilibrium flow networks therefore retains independent content beyond the self-citation of CFT, consistent with a minor, non-load-bearing self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Based solely on the abstract, the central claim rests on the validity of master-equation modeling of biomolecular networks and on the prior Caliber Force Theory. No numerical free parameters are mentioned. The new framework itself is the primary added element.

axioms (2)
  • domain assumption Dynamical flows on networks of biomolecular machines are adequately described by master equations that yield node populations and edge fluxes
    Explicitly stated as the current standard approach whose limitations the new method aims to overcome.
  • domain assumption Caliber Force Theory provides a suitable foundation for extending nonequilibrium descriptions to design and optimization
    The entire CFT Design construction is based on this recently developed theory.
invented entities (1)
  • CFT Design framework no independent evidence
    purpose: To incorporate cost-benefit tradeoffs, small-system misflows, and differential flow properties into optimization of nonequilibrium molecular-machine networks
    This is the new construct introduced in the paper.

pith-pipeline@v0.9.0 · 5408 in / 1476 out tokens · 59881 ms · 2026-05-12T02:21:31.716451+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

70 extracted references · 70 canonical work pages

  1. [1]

    evolutionary

    DECOUPLING SPEED, ACCURACY, AND COST IN KINETIC PROOFREADING Hopfield [17] and Ninio [16] were the first to explain how biomolecular processes can harness non-equilibrium driving to achieve certain biochemical process accuracies beyond the limits of thermodynamic equilibrium. For example, inkinetic proofreading(KP), a DNA polymerase enzyme copies DNA sequ...

    work page

  2. [2]

    # EI EIS E ES 𝐽$%

    STRONGER ENZYME INHIBITION: DEAD-ENDS VERSUS LEAKY LOOPS Classical pharmacology evaluates enzyme inhibitor efficacy based on binding affinities,ln(k on/koff)of the inhibitor molecule to the target enzyme [21]. By treating Michaelis-Menten (MM) inhibition as a NEQ flow-routing problem, we demonstrate that inhibitor efficacy fundamentally depends on network...

    work page doi:10.5281/zenodo.20057619

  3. [3]

    Qian, Phosphorylation Energy Hypothesis: Open Chemical Systems and Their Biological Functions, Annual Review of Physical Chemistry58, 113 (2007)

    H. Qian, Phosphorylation Energy Hypothesis: Open Chemical Systems and Their Biological Functions, Annual Review of Physical Chemistry58, 113 (2007)

    work page 2007

  4. [4]

    J. M. Horowitz and M. Esposito, Thermodynamics with Continuous Information Flow, Phys. Rev. X4, 031015 (2014)

    work page 2014

  5. [5]

    J. M. R. Parrondo, J. M. Horowitz, and T. Sagawa, Thermodynamics of information, Nature Phys11, 131 (2015)

    work page 2015

  6. [6]

    Seifert, From Stochastic Thermodynamics to Thermodynamic Inference, Annual Review of Condensed Matter Physics10, 171 (2019)

    U. Seifert, From Stochastic Thermodynamics to Thermodynamic Inference, Annual Review of Condensed Matter Physics10, 171 (2019)

    work page 2019

  7. [7]

    Peliti and S

    L. Peliti and S. Pigolotti,Stochastic Thermodynamics: An In- troduction(Princeton University Press, 2021)

    work page 2021

  8. [8]

    Cao and S

    Y . Cao and S. Liang, Stochastic thermodynamics for biological functions, Quantitative Biology13, e75 (2025), _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/qub2.75

    work page doi:10.1002/qub2.75 2025

  9. [9]

    M. P. Leighton and D. A. Sivak, Flow of Energy and Information in Molecular Machines, Annual Review of Physical Chemistry76, 379 (2025)

    work page 2025

  10. [10]

    Floyd, A

    C. Floyd, A. R. Dinner, A. Murugan, and S. Vaikuntanathan, Limits on the computational expressivity of non-equilibrium biophysical processes, Nat Commun16, 7184 (2025)

    work page 2025

  11. [11]

    Tu, Nonequilibrium Thermodynamics of Biochemical Networks: Energetics of Cellular Functions, Annual Reviews of Biophysics 10.1146/annurev-biophys-021424-112247 (2026)

    Y . Tu, Nonequilibrium Thermodynamics of Biochemical Networks: Energetics of Cellular Functions, Annual Reviews of Biophysics 10.1146/annurev-biophys-021424-112247 (2026)

    work page doi:10.1146/annurev-biophys-021424-112247 2026

  12. [12]

    Qian and D

    H. Qian and D. A. Beard, Metabolic futile cycles and their functions: a systems analysis of energy and control, Syst Biol (Stevenage)153, 192 (2006)

    work page 2006

  13. [13]

    Baiesi and C

    M. Baiesi and C. Maes, Life efficiency does not always increase with the dissipation rate, J. Phys. Commun.2, 045017 (2018)

    work page 2018

  14. [14]

    Maes, Frenesy: Time-symmetric dynamical activity in nonequilibria, Physics Reports850, 1 (2020)

    C. Maes, Frenesy: Time-symmetric dynamical activity in nonequilibria, Physics Reports850, 1 (2020)

    work page 2020

  15. [15]

    Yang and K

    Y .-J. Yang and K. A. Dill, A principled basis for nonequilibrium network flows, Nat Commun 10.1038/s41467-026-71239-9 (2026)

    work page doi:10.1038/s41467-026-71239-9 2026

  16. [16]

    Yang and K

    Y .-J. Yang and K. A. Dill, Fluctuation-Response Design Rules for Nonequilibrium Flows (2026), arXiv:2602.10957 [cond- mat]

    work page arXiv 2026

  17. [17]

    Gerritsma and P

    E. Gerritsma and P. Gaspard, Chemomechanical Coupling and Stochastic Thermodynamics of the F 1 -ATPase Molecular Motor with an applied external toque, Biophys. Rev. Lett.05, 163 (2010)

    work page 2010

  18. [18]

    Ninio, Kinetic amplification of enzyme discrimination, Biochimie57, 587 (1975)

    J. Ninio, Kinetic amplification of enzyme discrimination, Biochimie57, 587 (1975)

    work page 1975

  19. [19]

    J. J. Hopfield, Kinetic Proofreading: A New Mechanism for Reducing Errors in Biosynthetic Processes Requiring High Specificity, PNAS71, 4135 (1974)

    work page 1974

  20. [20]

    Banerjee, A

    K. Banerjee, A. B. Kolomeisky, and O. A. Igoshin, Elucidating interplay of speed and accuracy in biological error correction, Proceedings of the National Academy of Sciences114, 5183 (2017)

    work page 2017

  21. [21]

    J. D. Mallory, A. B. Kolomeisky, and O. A. Igoshin, Trade- Offs between Error, Speed, Noise, and Energy Dissipation in Biological Processes with Proofreading, J. Phys. Chem. B123, 4718 (2019)

    work page 2019

  22. [22]

    W. W. Cleland, The kinetics of enzyme-catalyzed reactions with two or more substrates or products: II. Inhibition: Nomenclature and theory, Biochimica et Biophysica Acta (BBA) - Specialized Section on Enzymological Subjects67, 173 (1963)

    work page 1963

  23. [23]

    R. A. Copeland,Evaluation of Enzyme Inhibitors in Drug Dis- covery: A Guide for Medicinal Chemists and Pharmacologists (Wiley-Interscience, Hoboken, N.J, 2005)

    work page 2005

  24. [24]

    Cornish-Bowden,Fundamentals of Enzyme Kinetics(Wiley- VCH, Weinheim, 2012)

    A. Cornish-Bowden,Fundamentals of Enzyme Kinetics(Wiley- VCH, Weinheim, 2012)

    work page 2012

  25. [25]

    Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev

    J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Mod. Phys.48, 571 (1976)

    work page 1976

  26. [26]

    T. L. Hill,Free Energy Transduction and Biochemical Cycle Kinetics(Springer-Verlag, New York, 1989)

    work page 1989

  27. [27]

    Seifert, Stochastic thermodynamics: From principles to the cost of precision, Physica A: Statistical Mechanics and its Applications504, 176 (2018)

    U. Seifert, Stochastic thermodynamics: From principles to the cost of precision, Physica A: Statistical Mechanics and its Applications504, 176 (2018)

    work page 2018

  28. [28]

    Maes, Local detailed balance, SciPost Phys

    C. Maes, Local detailed balance, SciPost Phys. Lect. Notes , 32 (2021)

    work page 2021

  29. [29]

    Falasco and M

    G. Falasco and M. Esposito, Local detailed balance across scales: From diffusions to jump processes and beyond, Phys. Rev. E103, 042114 (2021)

    work page 2021

  30. [30]

    J. A. Owen, T. R. Gingrich, and J. M. Horowitz, Universal Thermodynamic Bounds on Nonequilibrium Response with Biochemical Applications, Phys. Rev. X10, 011066 (2020)

    work page 2020

  31. [31]

    Fernandes Martins and J

    G. Fernandes Martins and J. M. Horowitz, Topologically constrained fluctuations and thermodynamics regulate nonequilibrium response, Phys. Rev. E108, 044113 (2023)

    work page 2023

  32. [32]

    Aslyamov and M

    T. Aslyamov and M. Esposito, Nonequilibrium Response for Markov Jump Processes: Exact Results and Tight Bounds, Phys. Rev. Lett.132, 037101 (2024)

    work page 2024

  33. [33]

    See Supplementary Information (SI) at [URL] for more details

    work page

  34. [34]

    E. T. Jaynes, The Minimum Entropy Production Principle, Annual Review of Physical Chemistry31, 579 (1980)

    work page 1980

  35. [35]

    Pressé, K

    S. Pressé, K. Ghosh, J. Lee, and K. A. Dill, Principles of maximum entropy and maximum caliber in statistical physics, Rev. Mod. Phys.85, 1115 (2013)

    work page 2013

  36. [36]

    J. D. Mallory, A. B. Kolomeisky, and O. A. Igoshin, Kinetic control of stationary flux ratios for a wide range of biochemical processes, Proceedings of the National Academy of Sciences 117, 8884 (2020)

    work page 2020

  37. [37]

    J. A. Wagoner and K. A. Dill, Mechanisms for achieving high speed and efficiency in biomolecular machines, PNAS116, 5902 (2019)

    work page 2019

  38. [38]

    A. I. Brown and D. A. Sivak, Theory of Nonequilibrium Free Energy Transduction by Molecular Machines, Chem. Rev.120, 434 (2020)

    work page 2020

  39. [39]

    Kacser and J

    H. Kacser and J. A. Burns, The control of flux, Biochem Soc Trans23, 341 (1995)

    work page 1995

  40. [40]

    Heinrich and T

    R. Heinrich and T. A. Rapoport, A Linear Steady- State Treatment of Enzymatic Chains, European 9 Journal of Biochemistry42, 89 (1974), _eprint: https://febs.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1432- 1033.1974.tb03318.x

    work page doi:10.1111/j.1432- 1974

  41. [41]

    Murugan, D

    A. Murugan, D. A. Huse, and S. Leibler, Speed, dissipation, and error in kinetic proofreading, Proceedings of the National Academy of Sciences109, 12034 (2012)

    work page 2012

  42. [42]

    Sartori and S

    P. Sartori and S. Pigolotti, Thermodynamics of Error Correction, Phys. Rev. X5, 041039 (2015)

    work page 2015

  43. [43]

    anti-proofreading

    We define “anti-proofreading” by the net reversal of the proofreading cycle, which operates via a different physical mechanism than the state-occupancy or speed-driven anti- proofreading regimes discussed previously by Muruganet al. [?]

    work page

  44. [44]

    Rao and L

    R. Rao and L. Peliti, Thermodynamics of accuracy in kinetic proofreading: dissipation and efficiency trade-offs, J. Stat. Mech.2015, P06001 (2015)

    work page 2015

  45. [45]

    Chiuchiu, S

    D. Chiuchiu, S. Mondal, and S. Pigolotti, Pareto optimal fronts of kinetic proofreading, New J. Phys.25, 043007 (2023)

    work page 2023

  46. [46]

    Goetz, J

    A. Goetz, J. Barrios, R. Madsen, and P. Dixit, Non-equilibrium strategies for ligand specificity in signaling networks, eLife14, 10.7554/eLife.107524.1 (2025)

    work page doi:10.7554/elife.107524.1 2025

  47. [47]

    Y .-J. Yang, C. D. Kocher, and K. A. Dill, Nonequilibrium Theory for Adaptive Systems in Varying Environments (2025), arXiv:2506.19018 [cond-mat]

    work page arXiv 2025

  48. [48]

    Chetrite and H

    R. Chetrite and H. Touchette, Nonequilibrium Microcanonical and Canonical Ensembles and Their Equivalence, Phys. Rev. Lett.111, 120601 (2013)

    work page 2013

  49. [49]

    Nonequilibrium Theory for Molecular Machine Design

    H. Touchette, The large deviation approach to statistical mechanics, Physics Reports478, 1 (2009). 1 Supplementary Information for: “Nonequilibrium Theory for Molecular Machine Design” Ying-Jen Yang1,∗ and Ken A. Dill1,2,3 1Laufer Center of Physical and Quantitative Biology, Stony Brook University 2Department of Physics and Astronomy, Stony Brook Universi...

    work page 2009

  50. [50]

    Node Energy as an Isotropic Scaler 5

    work page

  51. [51]

    invertible

    Kinetic Barrier as a Topological Router 6 II. Example 1: F 1-ATPase Molecular Rotors 6 A. The Gerritsma and Gaspard (2010) model and implementation details 6 B. Equal Traffic processivity bound and inverse design 7 C. Response Relations and Schur Complement Inversion 9 III. Example 2: Kinetic Proofreading and the Performance Envelope 10 A. Construction of...

    work page 2010

  52. [52]

    The perturbation operator is∂ En ≡P j(̸=n) knj∂knj

    Node Energy as an Isotropic Scaler Perturbing the energyE n of statenmodifies all outgoing transition rates from that state proportionally. The perturbation operator is∂ En ≡P j(̸=n) knj∂knj . Substituting this operator directly into the Node Escaping Symmetry (Eq. S19) yields: 1 πm ∂¯xα ∂Em − 1 πn ∂¯xα ∂En =δ ¯xα,πn .(S22) For any flow observable distinc...

    work page

  53. [53]

    canonical-like

    Kinetic Barrier as a Topological Router Perturbing the symmetric kinetic barrierB ij on an edge modifies both the forward and backward rates equally and oppositely in logarithmic space. The operator is∂ Bij ≡ −k ij∂kij −k ji∂kji . Applying the Edge Reciprocity symmetry (Eq. S20), we substitutek ji∂kji ¯xα = 2πjkjiδ¯xα,τij − πj kji πi ∂kij ¯xα directly int...

    work page 2010

  54. [54]

    (S56) and (S57)

    Evaluate the four fundamental cycle fluxesJ c ∈ {J R, JW, J′ R, J′ W}from the target metrics(S, ε, C, r ′)using Eqs. (S56) and (S57)

    work page

  55. [55]

    For instance, the initial cognate binding edge E⇌ER participates in both the incorporation and discard cycles, yieldingJ E→ER = JR +J ′ R

    Compute the local net fluxJ ij on every edge by superimposing the fundamental cycle fluxes traversing it. For instance, the initial cognate binding edge E⇌ER participates in both the incorporation and discard cycles, yieldingJ E→ER = JR +J ′ R. As another example mapped in Fig. S1, the activation edge ER⇌ER ∗ in the T7 DNA polymerase carries exclusively t...

    work page

  56. [56]

    (S59) to algebraically resolve the four dependent chord trafficsτ c

    With all local fluxesJ ij determined, substitute the fixed cycle forcesF cycle and the remaining tree trafficsτ rem into Eq. (S59) to algebraically resolve the four dependent chord trafficsτ c

    work page

  57. [57]

    Having completely specified all state probabilitiesπ i, edge trafficsτ ij, and local fluxesJ ij, the 16 exact microscopic transition rates are reconstructed via the fundamental CFT mapping: kij = τij +J ij 2πi , k ji = τij −J ij 2πj .(S60) This sequence perfectly isolates the physical parameters, allowing us to explicitly design the microscopic transition...

    work page

  58. [58]

    The network is connected by invertible binding transitions, but the production step is non-invertible: ES kcat − − →E (1→0)

    Model Topology and Transition Matrix:We consider a four-state enzyme system: free enzyme E (node 0), enzyme- substrate complex ES (node 1), enzyme-inhibitor-substrate complex EIS (node 2), and enzyme-inhibitor complex EI (node 3). The network is connected by invertible binding transitions, but the production step is non-invertible: ES kcat − − →E (1→0). T...

    work page

  59. [59]

    We then select a spanning tree covering the transitions1⇌0⇌3⇌2

    Observable Basis and Fundamental Cycles:To construct the linearly independent observable basis ¯x, we select node 0(E) as the reference nodem, makingπ 1, π2, π3 the independent state probabilities. We then select a spanning tree covering the transitions1⇌0⇌3⇌2. The fundamental cycles are defined by adding the remaining chords. The reversible binding edge1...

    work page

  60. [60]

    (S8a), which requires first constructing the tilted matrix ˜M

    Tilted Matrix and Conjugate Forces:To identify the rate expressions of conjugate forces, we resort to the rate parameterization in Eq. (S8a), which requires first constructing the tilted matrix ˜M. The off-diagonal elements are tilted by the forces associated with each specific transition. The diagonal elements capture node constraints defined by the out-...

    work page

  61. [61]

    The Jacobian MatrixA:The Jacobian matrixA (ij),α =k ij∂Fα/∂kij serves as the dictionary mapping local rate perturbations to global force responses. We define the rate vectork= (k 01, k10, kcat, k03, k30, k12, k21, k23, k32)T as the rows, and the conjugate force vectorF= (F edge,01,F edge,03,F edge,12,F edge,23,F node,1,F node,2,F node,3,F cat,F leak)T as ...

    work page

  62. [62]

    However, the irreversible catalytic sink at state 1 (ES→E) induces a steady-state compensatory backflow,J leak

    Flux Conservation and Force Balance:The inner loop E⇌ES⇌EIS⇌EI⇌E is futile with no net effect, requiring its cycle force to vanish (Fleak = 0). However, the irreversible catalytic sink at state 1 (ES→E) induces a steady-state compensatory backflow,J leak. Applying Kirchhoff’s Current Law at each node dictates the flow routing:

    work page

  63. [63]

    At node 1 (ES): The net flux from node 0 (J 01) must satisfyJ 01 =J cat −J leak. 16

    work page

  64. [64]

    At the sideway nodes (2 and 3): The net fluxesJ 21, J32, J03 are equal toJ leak, which strictly impliesJ 12, J23, J30 = −Jleak. Denoting the edge affinityA ij =arctanh(J ij/τij), the zero cycle force constraintP (ij)∈cycle Aij = 0along the loop0→1→ 2→3→0is expressed as: arctanh Jcat −J leak τ01 − arctanh Jleak τ12 +arctanh Jleak τ23 +arctanh Jleak τ30 = 0...

    work page

  65. [65]

    The Positivity ofJ leak:The requirement that the compensatory leak flux is strictly positive (J leak >0) is a direct consequence of probability conservation combined with the thermodynamic constraint of the futile loop (F leak = 0). Because the catalytic step is strictly non-invertible (k cat >0), it acts as a constant sink, driving a unidirectional proba...

    work page

  66. [66]

    This dictates that the partial derivative is taken while holding the kinetic trafficτ and the cycle forceF leak = 0constant

    Proof of Monotonic Coupling via Chain Rule:We evaluate the sensitivity of production to the leak,∂J cat/∂Jleak, within the mixed coordinatez= (π,τ, J leak,F leak). This dictates that the partial derivative is taken while holding the kinetic trafficτ and the cycle forceF leak = 0constant. Using the derivative rules d dx tanh(u) =sech 2(u) du dx and d dxarc...

    work page

  67. [67]

    The resistance functionf(τ) =arctanh(J leak/τ)is convex (f ′′(τ)>0) for all physical regimes whereτ > J leak

    Jensen’s Inequality and Global Bounds on Production:To identify the limits of inhibitor efficacy, we analyze how the distribution of sideway trafficτ k modulates the production fluxJ cat under a fixed total traffic budgetP τk = 3¯τ. The resistance functionf(τ) =arctanh(J leak/τ)is convex (f ′′(τ)>0) for all physical regimes whereτ > J leak. This convexity...

    work page

  68. [68]

    The local resistance of that edge diverges: arctanh(J leak/τ1)→ ∞

    work page

  69. [69]

    Consequently, the total sideway resistance diverges:A side → ∞

    work page

  70. [70]

    Substituting this limit into Eq

    Using the asymptotic property of the hyperbolic tangent,lim x→∞ tanh(x) = 1. Substituting this limit into Eq. (S77), we derive the upper bound for production: J max cat =J leak +τ 01.(S84) This upper bound reflects the state of the inhibitor loop: a bottleneck severs the compensatory leak, freeing the production flux to its maximum capacity dictated by th...

    work page