[Paper Review] Topologically-constrained fluctuations and thermodynamics regulate nonequilibrium response

Gabriela Fernandes Martins, Jordan M. Horowitz|PubMed|May 30, 2023

Advanced Thermodynamics and Statistical Mechanics70 references4 citations

TL;DR

This paper derives fundamental bounds on nonequilibrium response in Markovian systems by combining topological constraints of the state space with thermodynamic driving forces. It shows that sensitivity in observables—such as receptor binding—can be maximized only up to a Hill coefficient enhanced by topological and thermodynamic factors, with no need for kinetic rate details beyond state-space structure.

ABSTRACT

The limits on a system's response to external perturbations inform our understanding of how physical properties can be shaped by microscopic characteristics. Here, we derive constraints on the steady-state nonequilibrium response of physical observables in terms of the topology of the microscopic state space and the strength of thermodynamic driving. Notably, evaluation of these limits requires no kinetic information beyond the state-space structure. When applied to models of receptor binding, we find that sensitivity is bounded by the steepness of a Hill function with a Hill coefficient enhanced by the chemical driving beyond the structural equilibrium limit.

Motivation & Objective

• To identify universal limits on steady-state response of physical observables in nonequilibrium systems.
• To address the gap in existing theories that neglect topological correlations between microscopic states in response bounds.
• To develop a framework that incorporates both state-space topology and thermodynamic driving to constrain response sensitivity.
• To demonstrate that these bounds are independent of kinetic rate values, relying only on state-space connectivity and cycle forces.

Proposed method

• Modeling the system as a continuous-time Markov jump process on a graph with directed transitions and reverse pairs for thermodynamic consistency.
• Using graph-theoretic methods to compute steady-state distributions and their derivatives with respect to kinetic rates, leveraging Kirchhoff’s matrix-tree theorem.
• Defining thermodynamic forces via cycle affinities, computed as log-ratios of forward and backward rate products around closed loops in the state-space graph.
• Deriving response bounds using constrained optimization over stationary distributions, where the bounds depend on observable values and topological splittings of the state space.
• Introducing a topologically consistent splitting of states into subsets to decouple response contributions and identify extremal configurations.
• Applying the formalism to receptor binding models, showing that maximum sensitivity is bounded by a Hill function with an enhanced Hill coefficient due to thermodynamic driving.

Figure 1: Representative example: (a) Five-state house graph with perturbed edge $e_{12}$ highlighted in pink. (b) Cycles associated to the thermodynamic forces $F_{C}$ . Only cycles through the perturbed edge $e_{12}$ constrain the response. (c) Topologically-consistent splittings ${\mathcal{V}}^{1

Experimental results

Research questions

• RQ1 How do topological features of the state space constrain the maximum possible response of a physical observable in a nonequilibrium steady state?
• RQ2 To what extent can thermodynamic driving enhance the sensitivity of a system beyond equilibrium limits, and how is this quantified?
• RQ3 Can response bounds be derived without knowledge of individual kinetic rate constants, relying only on state-space structure?
• RQ4 What is the role of correlations between responses at different microscopic configurations in setting fundamental limits on system sensitivity?

Key findings

• The maximum response of an observable is bounded by a function that depends on the observable’s range and the topological structure of the state space, with no dependence on individual rate values.
• The bound on sensitivity in receptor binding models is equivalent to a Hill function with a Hill coefficient enhanced by thermodynamic driving beyond the structural equilibrium limit.
• The optimal response is achieved when the stationary distribution concentrates on two states, with the maximum sensitivity occurring at a specific value of the observable’s deviation from the mean.
• The maximum attainable sensitivity is limited by the product of the observable’s minimum and maximum deviations from the mean, with the bound saturating when the domain of possible values allows for a critical ratio of these extremes.
• For a given thermodynamic driving force, the system’s sensitivity is maximized when the state-space topology allows for a balanced splitting of states that amplifies response to perturbations.
• Numerical validation using receptor binding models confirms that the derived bounds are tight and physically realizable under specific rate configurations.

Figure 2: Response ratio $\mathcal{R}_{B}$ as a function of the maximum cycle force ${\mathcal{F}}_{\rm max}$ around cycles containing edge $e_{12}$ of the house graph (Fig. 1 ) for 15000 rate matrices with the logarithmic asymmetric $\ln(W_{ij}/W_{ji})$ and symmetric $\ln(W_{ij}W_{ji})$ parts sampl