The Maximum Caliber Variational Principle for Nonequilibria

Abstract

Ever since Clausius in 1865 and Boltzmann in 1877, the concepts of entropy and of its maximization have been the foundations for predicting how material equilibria derive from microscopic properties. But, despite much work, there has been no equally satisfactory general variational principle for nonequilibrium situations. However, in 1980, a new avenue was opened by E.T. Jaynes and by Shore and Johnson. We review here maximum caliber, which is a maximum-entropy-like principle that can infer distributions of flows over pathways, given dynamical constraints. This approach is providing new insights, particularly into few-particle complex systems, such as gene circuits, protein conformational reaction coordinates, network traffic, bird flocking, cell motility, and neuronal firing.

Keywords: biological dynamics; caliber; entropy; inference; network models; nonequilibrium; variational principles.

Figure 1

Timeline of key developments in nonequilibrium physics: (top row) developers, (middle row) nonequilibrium processes and models, and (bottom row) equilibria. The blue boxes denote principles explored and realized.

Figure 2

Path entropy measures the uniformity of the traffic distribution through different routes. Line thickness indicates the traffic density, i.e., pathway probability.

Figure 3

The use of maximum caliber plus a Markov model to represent two-state dynamics, A ↔ B. (a) Different trajectories are represented by different landscapes. Maximum caliber enumerates trajectories, weighted by path weights that are unknown at the start. (b) Measured averages, like the average number of times a particle remained in state A (⟨N_aa⟩), then determine those weights and thus the relative probabilities of all the paths. (c) The predicted variances, given the mean, agree with experiments. Panel c adapted with permission from Reference .

Figure 4

Diffusion-equation modeling treats concentration c(x, t) as continuous and differentiable—for example, in Fick’s law, ⟨J⟩ = −D∂c/∂x.

Figure 5

Diffusion at the microscopic level of colloidal particles on a microscopic slide (18). (a) The microfluidic apparatus used to measure the free diffusion of a few colloidal beads to quantify fluctuations. (b) A snapshot from the video used to track the movements of the beads over time. (c) Three typical concentration profiles measured using the apparatus. These profiles indicate that few-particle flows such as the one measured in this experiment entail large fluctuations. Phenomenology like Fick’s law just describes averages for large numbers of particles, and not the fluctuations indicated here. Figure adapted from Reference with permission from the American Association of Physics Teachers.

Figure 6

(a) Concentration gradient c(x) showing bins i and i + 1, illustrating the different numbers N of particles in each one. (b) One possible trajectory, labeled with the statistical weights of the steps.

Figure 7

Simple diffusion, from maximum caliber with the dog-flea model, successfully predicts few-particle experiments (19). First, it derives Fick’s first law, ⟨J⟩ = −D∂c/∂x, from a variational principle. Second, it shows that Fick’s law holds down to the few-particle limit. Third, it correctly predicts the full rate distribution. Fourth, it computes a Maxwell’s demon–like quantity of wrong-way flows (quantified by ϕ_badactor), showing that they become negligible as the net flux gets larger. And fifth, it accurately gives a flux fluctuation relationship. All these predictions result when given only one quantity, equivalent to knowing the diffusion constant, D (see Reference for more information). Figure adapted with permission from Reference .

Figure 8

Maximum caliber derives Kirchoff’s current principle—namely, that current divides at junctions in proportion to the relative flow resistances of the channels.

Figure 9

Maximum caliber (Max Cal) predicts the dynamics of an autoactivation gene circuit. (a) Gene α produces protein A. When A₂ (dimer of A) binds to the promoter, production of protein A speeds up (20). Note that negative feedback can be readily treated similarly, except that the light blue region here called the promoter is replaced by a region called the repressor, and the effect of repression is to slow down, rather than speed up, production of A. (b) The experiment measures this stochastic switchlike trajectory. (c) Using this as input, Max Cal predicts protein production rates in the normal (g) and accelerated (g*) states, as well as degradation rate d.

Figure 10

Maximum caliber (Max Cal) gives rate distributions in a toggle-switch gene circuit. (a) Gene α produces protein A. Gene β produces protein B. The binding of A represses production of B, and the binding of B represses production of A. (b) The effect is bistability (winner-take-all): When either species, A or B, enters into small excess, it then grows further to fully dominate the population. The stochastic trajectory is experimentally measured. Bottom part of panel b adapted with permission from Reference . (c) In the absence of anything else, Max Cal uses the stochastic trajectory for both proteins to infer different rate parameters g, g*, d, etc.

Figure 11

Maximum caliber (Max Cal) gives the few-particle dynamics of the repressilator gene circuit. (a) Genes α, β, and γ produce proteins A, B, and C, respectively. The binding of A represses production of B, the binding of B represses production of C, and the binding of C represses production of A. (b) The effect is an oscillatory time trace. The distribution of populations of A, B, and C contains less information than the stochastic time trajectory, typically measured in experiments. (c) In the absence of anything else, Max Cal uses the stochastic trajectory for all three proteins to infer different rate parameters g, g∗, and d and feedback strength K. Panel c adapted with permission from Reference .

Figure 12

Traditional dynamical models of gene networks. (a) Mass-action (MA) model describes averages (A) with arbitrary nonlinear function f to model feedback (k_d is the degradation rate). (b) MA + random noise model adds random noise around the MA equation, yielding a Langevin-type equation. (c) Chemical master equation (CME) model describes time evolution of probability distributions (P) in terms of transition probabilities (W) determined by invoking a set of auxiliary species ({Y}) that are often not seen in experiments. (d) CME + MA is a coarse-grained model for the time evolution of probability when substituting the phenomenological f functions used in MA for W.

Figure 13

Maximum caliber gives a fast estimate of traffic flows on networks. If we know the steady-state populations of the mobile elements that flow through a network, then the maximum caliber equation (Equation 25) estimates the full transition-rate matrix, as that which maximizes the path entropy.

Figure 14

Maximum caliber can identify good RCs. A complex potential energy landscape (panel a) can be projected onto any RC; ΔG is free energy (panels b and c). Maximum caliber allows us to quickly estimate the approximate kinetics along any RC and identify the RC that leads to the highest separation in timescales (panels d and e). In this simple example, RC₁ is a better reaction coordinate than RC₂. Abbreviations: CV, collective variable; RC, reaction coordinate.

Figure 15

Model of the growth factor–activated pathway. The four states (①–④) of the receptor are as follows. (①) Ligand-free receptors (green) on the cell surface can be (②) bound by a ligand (yellow) and then (③) phosphorylated. (④) All receptors can be internalized and degraded, albeit at different rates. The arrows indicate transition rates, and the most altered transition rate, as predicted by maximum caliber, is shown in (a) gray and (b) pink.