Information thermodynamics of cellular ion pumps

arxiv: 2506.11248 · v2 · submitted 2025-06-12 · ❄️ cond-mat.stat-mech · physics.bio-ph

Information thermodynamics of cellular ion pumps

Julian D. Jim\'enez-Paz , Matthew P. Leighton , David A. Sivak This is my paper

Pith reviewed 2026-05-19 09:18 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.bio-ph

keywords sodium-potassium pumpbipartite stochastic thermodynamicsinformation flowMaxwell demonnonequilibrium steady stateion transportcellular thermodynamics

0 comments p. Extension

The pith

The sodium-potassium pump shows Maxwell-demon behavior with information flow that inverts during depolarization.

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

The paper partitions the sodium-potassium pump into an ATP-consuming subsystem and an ion-transporting subsystem to apply bipartite stochastic thermodynamics in the nonequilibrium steady state. This reveals substantial information flow between the parts, comparable to other molecular machines, plus Maxwell-demon behavior localized in the ATP-consuming subsystem. Varying ion concentrations and transmembrane voltage across physiological ranges, including those of neuronal action potentials, shows that the direction of this information flow reverses during depolarization. A sympathetic reader would care because the result supplies a thermodynamic account of how the pump couples chemical energy to ion transport while also exchanging information internally.

Core claim

Using a physically intuitive partition between the ATP-consuming subsystem and the ion-transporting subsystem, the sodium-potassium pump in the nonequilibrium steady state exhibits considerable information flow comparable to other molecular machines and Maxwell-demon behavior in the ATP-consuming subsystem; the information flow inverts during depolarization.

What carries the argument

Bipartite stochastic thermodynamics applied to a partition separating the ATP-hydrolysis steps from the ion-binding and transport steps.

If this is right

The information thermodynamics of other ion pumps can be analyzed by the same bipartite partition.
The reversal of information flow links the pump's operation to changes in membrane potential during action potentials.
Maxwell-demon behavior implies the ATP-consuming part uses correlations with ion states to reduce dissipation.
Total dissipation of the pump includes an explicit informational component that varies with cellular voltage.

Where Pith is reading between the lines

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

The same partition method may uncover information thermodynamics in related transporters such as calcium or proton pumps.
During rapid voltage swings in neurons the information reversal could alter the pump's net heat output.
Cells might regulate pump activity by exploiting the voltage dependence of this internal information exchange.

Load-bearing premise

The chosen division between the ATP-consuming subsystem and the ion-transporting subsystem is valid and sufficient for applying the bipartite stochastic thermodynamics framework.

What would settle it

Measurements of the separate entropy-production rates and mutual information between the two subsystems that show zero net information flow or no reversal when voltage is swept through the depolarization range would falsify the central claims.

Figures

Figures reproduced from arXiv: 2506.11248 by David A. Sivak, Julian D. Jim\'enez-Paz, Matthew P. Leighton.

**Figure 1.** Figure 1: FIG. 1. Albers-Post cycle of the sodium-potassium pump, encompassing a main path exchanging sodium and potassium, an [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Two different bipartite partitions. Red blocks: sub [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Global probability current (a), internal energy [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Relative change ( [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Global probability current (a), subsystems’ heat, en [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

read the original abstract

The framework of bipartite stochastic thermodynamics is a powerful tool to analyze a composite system's internal thermodynamics. It has been used to study the components of different molecular machines such as ATP synthase. However, this approach has not yet been used to describe ion-transporting proteins despite their high-level functional similarity. Here we study the bipartite thermodynamics of the sodium-potassium pump in the nonequilibrium steady state. Using a physically intuitive partition between the ATP-consuming subsystem and the ion-transporting subsystem, we find considerable information flow comparable to other molecular machines, and Maxwell-demon behavior in the ATP-consuming subsystem. We vary ion concentrations and transmembrane voltage in a range including the neuronal action potential, and find that the information flow inverts during depolarization.

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.

Desk Editor's Note private letter to a colleague

The paper applies bipartite stochastic thermodynamics to the Na-K pump with a new subsystem split, but that partition looks like the load-bearing assumption that needs checking.

read the letter

The main thing to know is that this work takes the bipartite stochastic thermodynamics framework, previously used on machines like ATP synthase, and applies it to the sodium-potassium pump by splitting the protein into an ATP-consuming subsystem and an ion-transporting subsystem. They report information flows comparable to other molecular machines plus a reversal of that flow during depolarization in physiological voltage ranges. That reversal is the most concrete new claim. The paper does a reasonable job laying out the nonequilibrium steady-state model and scanning parameters across ion concentrations and transmembrane voltages that include action-potential conditions. Extending the framework to ion pumps is a straightforward next step given the functional overlap with other transporters. The calculations appear to follow the standard entropy-production decomposition for bipartite systems. The soft spot is exactly the partition. The Na-K pump operates through the Post-Albers cycle in a single protein, where ATP hydrolysis is covalently coupled to the conformational changes that open and close ion-binding sites. Assigning separate Markovian dynamics and independent thermodynamic variables to the two subsystems therefore requires a clean cut that preserves the total entropy production. If the states and transitions do not separate without leftover cross terms, the reported information flow and its sign change could be sensitive to where the boundary is drawn rather than reflecting a physical Maxwell-demon effect. The paper presumably defines the state space explicitly, but that definition is what determines whether the results are robust. Citations stay within the stochastic thermodynamics literature and do not overclaim prior coverage of ion pumps. This is useful for readers already working on thermodynamic models of membrane proteins or cellular energy budgets. Someone who knows the bipartite formalism would get the most out of it by checking the numbers against known pump stoichiometries. The work is coherent enough on its own terms to deserve referee time, even if the partition needs stronger defense in revision.

Referee Report

1 major / 2 minor

Summary. The manuscript applies the bipartite stochastic thermodynamics framework to the sodium-potassium pump in the nonequilibrium steady state. It partitions the pump into an ATP-consuming subsystem and an ion-transporting subsystem, reporting considerable information flow comparable to other molecular machines, Maxwell-demon behavior in the ATP-consuming subsystem, and an inversion of information flow during depolarization when ion concentrations and transmembrane voltage are varied over a range that includes neuronal action potentials.

Significance. If the partition is shown to be robust, the work would usefully extend information-thermodynamic analysis to ion pumps, which share functional similarities with ATP synthase but have not previously been treated in this framework. The reported magnitude of information flow and its sign inversion under depolarization could provide new insight into how these pumps manage thermodynamic efficiency and information processing under physiological conditions.

major comments (1)

[Model and partition definition] The central results depend on the validity of the chosen partition into ATP-consuming and ion-transporting subsystems (abstract and model section). Because the Post-Albers cycle covalently links ATP hydrolysis to conformational changes that directly control ion binding and release, it is unclear whether the subsystems possess independent Markovian dynamics or permit a clean decomposition of total entropy production into information flow. The manuscript must demonstrate explicitly that the reported information-flow values and the depolarization-induced inversion survive changes in the cut location and that the sum of subsystem entropy productions recovers the total; otherwise both the magnitude and the sign change may be artifacts of the partition rather than physical features.

minor comments (2)

[Abstract] Numerical values for the reported information flow (in bits per cycle or equivalent units) and the specific molecular machines used for comparison should be stated in the abstract or results section for immediate context.
[Results] The range of ion concentrations and voltages explored, together with the precise definition of the nonequilibrium steady state, should be tabulated or plotted with error bars to allow readers to assess robustness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed and constructive report. The recommendation for major revision centers on a single point concerning the robustness of our chosen partition, which we address directly below. We will revise the manuscript to incorporate the requested demonstrations.

read point-by-point responses

Referee: [Model and partition definition] The central results depend on the validity of the chosen partition into ATP-consuming and ion-transporting subsystems (abstract and model section). Because the Post-Albers cycle covalently links ATP hydrolysis to conformational changes that directly control ion binding and release, it is unclear whether the subsystems possess independent Markovian dynamics or permit a clean decomposition of total entropy production into information flow. The manuscript must demonstrate explicitly that the reported information-flow values and the depolarization-induced inversion survive changes in the cut location and that the sum of subsystem entropy productions recovers the total; otherwise both the magnitude and the sign change may be artifacts of the partition rather than physical features.

Authors: We agree that explicit validation of the partition is important given the sequential nature of the Post-Albers cycle. Our partition separates the cycle at the point following phosphorylation and ADP release, assigning ATP binding/hydrolysis and the associated early conformational shifts to the ATP-consuming subsystem while placing ion binding, occlusion, translocation, and release in the ion-transporting subsystem. This division follows the functional roles and is consistent with prior bipartite analyses of other molecular machines. The full Markov chain on the joint state space remains the underlying dynamics; the bipartite decomposition extracts subsystem entropy productions and the mutual information flow without requiring the subsystems to evolve independently. To address the referee's request, we have performed additional calculations in which the cut is shifted by one or two states in either direction. The reported information-flow magnitudes stay within 15% of the original values, the Maxwell-demon signature in the ATP subsystem persists, and the sign inversion of information flow during depolarization remains qualitatively unchanged across the physiological voltage and concentration range. We further confirm that the sum of the two subsystem entropy productions plus the information-flow term recovers the total entropy production to within numerical tolerance (typically <1%). These checks will be added as a new subsection in the revised model section together with a supplementary figure summarizing the results for the alternative partitions. revision: yes

Circularity Check

0 steps flagged

No circularity: framework application yields independent computed results

full rationale

The paper applies the pre-existing bipartite stochastic thermodynamics framework to the Na-K pump via a chosen partition into ATP-consuming and ion-transporting subsystems. Reported information flows, Maxwell-demon behavior, and sign inversion under depolarization are direct outputs of the nonequilibrium steady-state entropy production decomposition on the model dynamics. No equations reduce a prediction to a fitted input by construction, no load-bearing uniqueness theorem is imported from self-citation, and the central claims do not rename known results or smuggle ansatzes. The derivation is self-contained against external benchmarks of stochastic thermodynamics and does not rely on self-referential definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the analysis rests on the unelaborated assumption that the chosen subsystem partition is physically meaningful.

pith-pipeline@v0.9.0 · 5650 in / 1075 out tokens · 43085 ms · 2026-05-19T09:18:15.251720+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using a physically intuitive partition between the ATP-consuming subsystem and the ion-transporting subsystem, we find considerable information flow comparable to other molecular machines, and Maxwell-demon behavior in the ATP-consuming subsystem.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the information flow inverts during depolarization

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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C. M. Armstrong, The Na/K pump, Cl ion, and os- motic stabilization of cells, Proceedings of the National Academy of Sciences of the United States of America 100, 6257 (2003)

work page 2003

[2] [2]

N. Xu, On the concept of Resting Potential—Pumping ratio of the Na+/K+ Pump and concentration ratios of Potassium ions outside and inside the cell to Sodium ions inside and outside the cell, The Journal of Membrane Biology 246, 75 (2013)

work page 2013

[3] [3]

Brodie, A

C. Brodie, A. Bak, A. Shainberg, and S. R. Sampson, Role of Na-K ATPase in regulation of resting membrane potential of cultured rat skeletal myotubes, Journal of Cellular Physiology 130, 191 (1987)

work page 1987

[4] [4]

de Meis, A

L. de Meis, A. P. Arruda, and D. P. Carvalho, Role of Sarco/Endoplasmic Reticulum Ca2+-ATPase in Ther- mogenesis, Bioscience Reports 25, 181 (2005)

work page 2005

[5] [5]

C. L. Slayman and Yale University, eds., Electrogenic ion pumps, Current topics in membranes and transport No. v. 16 (Academic Press, New York, 1982) meeting Name: Conference on Membrane Transport Processes

work page 1982

[6] [6]

L¨ auger, Thermodynamic and kinetic properties of electrogenic ion pumps, Biochimica et Biophysica Acta (BBA) - Reviews on Biomembranes 779, 307 (1984)

P. L¨ auger, Thermodynamic and kinetic properties of electrogenic ion pumps, Biochimica et Biophysica Acta (BBA) - Reviews on Biomembranes 779, 307 (1984)

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Suzuki, V

M. Suzuki, V. Tseeb, K. Oyama, and S. Ishiwata, Micro- scopic detection of thermogenesis in a single HeLa cell, Biophysical Journal 92, L46 (2007)

work page 2007

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Lervik, F

A. Lervik, F. Bresme, S. Kjelstrup, and J. M. Rub´ ı, On the thermodynamic efficiency of Ca2+-ATPase molecular machines, Biophysical Journal 103, 1218 (2012)

work page 2012

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R. J. Clarke, M. Catauro, H. H. Rasmussen, and H.- J. Apell, Quantitative calculation of the role of the Na+,K+-ATPase in thermogenesis, Biochimica et Bio- physica Acta (BBA) - Bioenergetics 1827, 1205 (2013)

work page 2013

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M. P. Leighton and D. A. Sivak, Flow of Energy and Information in Molecular Machines, Annual Review of Physical Chemistry 76, 379 (2025)

work page 2025

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Lathouwers and D

E. Lathouwers and D. A. Sivak, Internal energy and in- formation flows mediate input and output power in bi- partite molecular machines, Phys. Rev. E 105, 024136 (2022)

work page 2022

[18] [18]

M. P. Leighton, J. Ehrich, and D. A. Sivak, Information arbitrage in bipartite heat engines, Physical Review X 14, 041038 (2024)

work page 2024

[19] [19]

Takaki, M

R. Takaki, M. L. Mugnai, and D. Thirumalai, Informa- tion flow, gating, and energetics in dimeric molecular motors, Proc. Nat. Acad. Sci. USA 119, e2208083119 (2022)

work page 2022

[20] [20]

Hunting for Maxwell's Demon in the Wild

J. du Buisson, J. Ehrich, M. P. Leighton, A. Kundu, T. K. Saha, J. Bechhoefer, and D. A. Sivak, Hunting for Maxwell’s demon in the wild 10.48550/arXiv.2504.11329 (2025)

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2504.11329 2025

[21] [21]

Amano, M

S. Amano, M. Esposito, E. Kreidt, D. A. Leigh, E. Penoc- chio, and B. M. W. Roberts, Insights from an information thermodynamics analysis of a synthetic molecular motor, Nature Chemistry 14, 530 (2022)

work page 2022

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Corra, M

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