Fluctuation-Response Design Rules for Nonequilibrium Flows

Biological machines like molecular motors and enzymes operate in dynamic cycles representable as stochastic flows on networks. Current stochastic dynamics describes such flows on fixed networks. Here, we develop a scalable approach to network design in which local transition rates can be systematically varied to achieve global dynamical objectives. It is based on the fluctuation-response duality in the recent Caliber Force Theory – a path-entropy variational formalism for nonequilibria. This approach scales efficiently with network complexity and gives new insights, for example revealing the transition from timing- to branching-dominated fluctuations in a kinesin motor model.

💡 Research Summary

The paper addresses a fundamental challenge in nonequilibrium network design: how to systematically adjust local transition rates in order to achieve desired global dynamical objectives such as specific flux means, variances, or higher‑order statistics. While Markov state models and master equations can predict the stochastic dynamics on a fixed network, they do not provide a principled way to “design” the network. The authors fill this gap by exploiting the fluctuation‑response duality inherent in the recently formulated Caliber Force Theory (CFT), a path‑entropy variational framework that parallels equilibrium thermodynamics.

In CFT the logarithm of the path probability ratio is expressed as a linear combination of extensive counting observables X (edge traffic, node dwell times, cycle net fluxes) and their conjugate forces F (affinities associated with edge exchange, node dwelling, and cycle completion). This conjugate pair satisfies a universal relation: the susceptibility of any observable to a change in a force equals the asymptotic covariance of the corresponding observables. By differentiating the forces with respect to the logarithm of the transition rates, the authors construct a sparse Jacobian matrix A with elements A_{ij,β}=∂F_β/∂ln k_{ij}. This matrix maps the independent kinetic noise sources λ_{ij} (defined as deviations of transition counts from their mean) onto the observable space and, crucially, its inverse A⁻¹ provides the exact response of any observable to a perturbation of any rate constant.

The central “Response‑Inverse‑Matrix” (RIM) relation, ∂⟨x_α⟩/∂ln k_{ij}=π_i k_{ij}