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Fluctuation Theorems from a Continuous-Time Markov Model of Information-Thermodynamic Capacity in Biochemical Signal Cascades

Tatsuaki Tsuruyama

TL;DR

This work develops an information-thermodynamic framework for biochemical signaling by modeling a cascade as a continuous-time Markov jump process and treating reaction durations as code lengths. It derives a capacity-like entropy-rate measure, links forward and dual trajectory probabilities to detailed and integral fluctuation theorems, and describes finite-time fluctuations via the scaled cumulant generating function and Gallavotti-Cohen symmetry. The theory connects abstract capacity to experimentally accessible MAPK/ERK time courses, providing a concrete method to estimate dissipation from activation/inactivation times and to quantify irreversibility at finite step counts. Together, the results offer a rigorous bridge between information transmission in signaling cascades and nonequilibrium thermodynamics, with potential for experimental validation and refined coarse-graining in biological systems.

Abstract

Biochemical signaling cascades transmit intracellular information while dissipating energy under nonequilibrium conditions. We model a cascade as a code string and apply information-entropy ideas to quantify an optimal transmission rate. A time-normalized entropy functional is maximized to define a capacity-like quantity governed by a conserved multiplier. To place the theory on a rigorous stochastic-thermodynamic footing, we formulate stepwise signaling as a continuous-time Markov jump process with forward and reverse competing rates. The embedded jump chain yields well-defined transition probabilities that justify time-scale-based expressions. Under local detailed balance, the log ratio of forward and reverse rates can be interpreted as entropy production per event, enabling a trajectory-level derivation of detailed and integral fluctuation theorems. We further connect the information-theoretic capacity to the mean dissipation rate and outline finite-time fluctuation structure via the scaled cumulant generating function (SCGF) and Gallavotti--Cohen symmetry, including a worked example using MAPK/ERK timescales.

Fluctuation Theorems from a Continuous-Time Markov Model of Information-Thermodynamic Capacity in Biochemical Signal Cascades

TL;DR

This work develops an information-thermodynamic framework for biochemical signaling by modeling a cascade as a continuous-time Markov jump process and treating reaction durations as code lengths. It derives a capacity-like entropy-rate measure, links forward and dual trajectory probabilities to detailed and integral fluctuation theorems, and describes finite-time fluctuations via the scaled cumulant generating function and Gallavotti-Cohen symmetry. The theory connects abstract capacity to experimentally accessible MAPK/ERK time courses, providing a concrete method to estimate dissipation from activation/inactivation times and to quantify irreversibility at finite step counts. Together, the results offer a rigorous bridge between information transmission in signaling cascades and nonequilibrium thermodynamics, with potential for experimental validation and refined coarse-graining in biological systems.

Abstract

Biochemical signaling cascades transmit intracellular information while dissipating energy under nonequilibrium conditions. We model a cascade as a code string and apply information-entropy ideas to quantify an optimal transmission rate. A time-normalized entropy functional is maximized to define a capacity-like quantity governed by a conserved multiplier. To place the theory on a rigorous stochastic-thermodynamic footing, we formulate stepwise signaling as a continuous-time Markov jump process with forward and reverse competing rates. The embedded jump chain yields well-defined transition probabilities that justify time-scale-based expressions. Under local detailed balance, the log ratio of forward and reverse rates can be interpreted as entropy production per event, enabling a trajectory-level derivation of detailed and integral fluctuation theorems. We further connect the information-theoretic capacity to the mean dissipation rate and outline finite-time fluctuation structure via the scaled cumulant generating function (SCGF) and Gallavotti--Cohen symmetry, including a worked example using MAPK/ERK timescales.
Paper Structure (21 sections, 4 theorems, 37 equations, 1 table)

This paper contains 21 sections, 4 theorems, 37 equations, 1 table.

Table of Contents

  1. Introduction
  2. Model and information-entropy formulation
  3. A chain-reaction view of information transmission
  4. Reaction durations as code lengths.
  5. Shannon entropy from symbol counts
  6. Entropy-rate maximization and capacity
  7. Continuous-time Markov formulation of stepwise signalling
  8. CTMC dynamics and embedded transition probabilities
  9. Log-ratio observable on each link
  10. Fluctuation theorems and capacity--irreversibility link
  11. Dual (adjoint) dynamics
  12. Path probability and trajectory entropy production
  13. Main results: DFT, IFT, second law
  14. Finite-time fluctuations: SCGF and Gallavotti--Cohen symmetry
  15. Practical implication / Parameterization by MAPK time courses
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\Sigma_t$ be defined by Eq. eq:Sigma_def. Denote by $P_{F,t}(\Sigma)$ the probability density of $\Sigma_t$ under the forward dynamics, and by $P^\dagger_t(\Sigma)$ the probability density of $-\Sigma_t$ under the dual dynamics. Then

Theorems & Definitions (5)

  • Remark 1
  • Theorem 1: Detailed fluctuation theorem (DFT), forward/dual form
  • Corollary 1: Integral fluctuation theorem (IFT)
  • Corollary 2: Second-law inequality
  • Proposition 1: Gallavotti--Cohen (GC) symmetry of the SCGF