[Paper Review] A case study of thermodynamic bounds for chemical kinetics

Karel Proesmans, Luca Peliti+1 more|arXiv (Cornell University)|Apr 3, 2018

Advanced Thermodynamics and Statistical Mechanics17 references3 citations

TL;DR

This paper investigates thermodynamic uncertainty relations in chemical kinetics, demonstrating that precision in enzyme-driven reaction fluxes is fundamentally limited by energy dissipation. Using unicyclic and multi-cyclic networks—such as isomerization, Michaelis-Menten, and misfolding reactions—it derives bounds linking the Fano factor of particle production to entropy production, proving that the product of dissipation and relative variance is bounded below by 2, with saturation only in the linear regime near equilibrium.

ABSTRACT

In this chapter, we illustrate recently obtained thermodynamic bounds for a number of enzymatic networks by focusing on simple examples of unicyclic or multi-cyclic networks. We also derive complementary relations which constrain the fluctuations of first-passage times to reach a threshold current.

Motivation & Objective

• To establish thermodynamic bounds on the precision of chemical fluxes in enzymatic networks, particularly in relation to energy dissipation.
• To investigate how fluctuations in particle conversion rates are constrained by entropy production in non-equilibrium steady states.
• To extend the thermodynamic uncertainty relation to first-passage times, linking the statistics of reaching a threshold number of converted particles to thermodynamic costs.
• To validate the universal nature of the uncertainty relation across different network topologies, including unicyclic and multi-cyclic systems.
• To explore connections between current fluctuations and first-passage time statistics using large-deviation techniques in Markovian networks.

Proposed method

• Models enzymatic networks using master equations for probability distributions over particle numbers and network states.
• Employs generating functions and cumulant generating functions to compute mean fluxes and variances, particularly the Fano factor.
• Derives the thermodynamic uncertainty relation via the cumulant generating function θ(λ), showing that Cε² ≥ 2, where C is dissipation and ε is relative uncertainty.
• Applies the relation to unicyclic systems (isomerization, Michaelis-Menten) and a two-cycle misfolding network, confirming the bound holds in all cases.
• Uses Laplace transforms and recurrence relations to compute first-passage time statistics, deriving expressions for mean and variance of time to reach n converted particles.
• Verifies the first-passage time uncertainty relation ΣVar(T)/⟨T⟩ ≥ 2, where Σ is the entropy production rate, using analytical solutions for model systems.

Experimental results

Research questions

• RQ1 How do thermodynamic constraints limit the precision of chemical fluxes in unicyclic enzymatic networks?
• RQ2 What is the role of entropy production in bounding the Fano factor of particle conversion in enzyme kinetics?
• RQ3 Can the thermodynamic uncertainty relation be extended to first-passage times in chemical networks?
• RQ4 How do multi-cyclic networks with multiple affinities differ in their uncertainty bounds compared to unicyclic systems?
• RQ5 What is the connection between large-deviation functions of currents and first-passage time statistics in Markovian networks?

Key findings

• The product of the total energy cost C and the relative variance ε² of the flux is bounded below by 2, with equality only in the linear regime near equilibrium: Cε² ≥ 2.
• For the isomerization reaction, the Fano factor is given by ε² = (k⁺ + k⁻)/((k⁺ − k⁻)²t), and the uncertainty relation holds universally, even far from equilibrium.
• In the Michaelis-Menten mechanism, the uncertainty relation is confirmed via Laplace-transformed master equations, with the cumulant generating function θ(s) derived from λ(s).
• For the misfolding reaction, a two-cycle network, the same uncertainty relation holds when mapped onto an effective Michaelis-Menten system with modified rate constants.
• The first-passage time uncertainty relation ΣVar(T)/⟨T⟩ ≥ 2 is verified analytically for both isomerization and Michaelis-Menten systems, with Var(T) and ⟨T⟩ derived from the inverse of the cumulant generating function.
• The study confirms that the thermodynamic uncertainty relation is robust across different network topologies and extends to first-passage time statistics, suggesting a deep connection between current fluctuations and timing precision in non-equilibrium systems.