Fluctuation-Response Design Rules for Nonequilibrium Flows

Abstract

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.

Figures (4)

• Figure 1: The sparse structure of fluctuation-response duality.(a) The Jacobian matrix $\textbf{A}$ acts as the bridge between stochasticity and control: it maps physical observables $\textbf{x}$ to independent noise sources $\boldsymbol{\lambda}$ (Fluctuation Geometry), while simultaneously linking transition rates $\ln \textbf{k}$ to conjugate forces $\boldsymbol{\mathfrak{F}}$ (Response Geometry). (b) The complete observable basis: edge traffic $\phi_{ij}$, node frequency $f_n$, and cycle net flux $\psi_c$. (c) A representative 4-state network showing that every transition edge generates an intrinsic independent noise source $\lambda_{ij}$. (d) The explicit structure of the Jacobian $\textbf{A}$ for the 4-state example. The symbol $k$ (or $-k$) denotes the transition rate $k_{ij}$ corresponding to the specific row index. The prime notation $\lambda'_{mj}$ denotes the shifted noise source $\lambda'_{mj} = \lambda_{mj} + k_{mj}$ for transitions leaving the reference state $m=1$. White squares denote structural zeros.
• Figure 2: Dissecting Molecular Motor's Randomness.(a) The 6-state model for kinesin liepelt_kinesins_2007. (b) The bigger the load that the motor has to pull, the slower it runs, for given ATP energy sources liepelt_kinesins_2007. (c) Decomposition of the motor's randomness parameter $r$. The total randomness is decomposed into contributions from functional groups (aggregating forward $\mathcal{F}$ and backward $\mathcal{B}$ cycles), represented by the stacked areas. The upper boundary of the stacked areas are the randomness calculated by the kinesin model, fitted to Visscher et al. visscher_single_1999. The motor becomes very inefficient when pulling heavy loads.
• Figure 3: Breaking the Computational Bottleneck.(a) The $\Theta$-shape topology used for benchmarking, representing a generalized motor model with multiple mechanical substeps to mimic the diffusive stepping process. (b) Scaling of the computation time for the gradient of the randomness parameter (ratio of variance and average of the flux) as a function of the number of mechanical substeps (and thus the system size).
• Figure 4: Universal Kinetic Bounds. The scatter plot numerically validates the derived sensitivity hierarchy using an ensemble of random 4-state networks as an example (schematic, upper right). Each grey dot represents the normalized response of the forward flux $p_{ij}$ (x-axis) versus the induced response of the reverse flux $p_{ji}$ (y-axis) to a perturbation in the forward rate $k_{ij}$. The feasible response region is strictly bounded by three physical limits: Population Depletion, Causality, and Le Chatelier-like Compensation.