Nonequilibrium Theory for Adaptive Systems in Varying Environments

TL;DR

This work applies recent results from nonequilibrium physics to show that organisms' fitness parses into a static generalist component and a nonequilibrium tracking component, providing the foundations for a generic physical theory of adaptivity.

Abstract

Biological organisms are adaptive, able to function in unpredictably changing environments. Drawing on recent nonequilibrium physics, we show that in adaptation, fitness has two components parameterized by observable coordinates: a static Generalism component characterized by state distributions, and a dynamic Tracking component sustained by nonequilibrium fluxes. Our findings: (1) General Theory: We prove that tracking gain scales strictly with environmental variability and switching time-scales; near-static or fast-switching environments are not worth tracking. (2) Optimal Strategies: We explain optimal bet-hedging and phenotypic memory as the interplay between these components. (3) Control: We demonstrate, with an example, how to suppress pathogens by independently attacking their Generalism robustness (via environmental time fractions) and Tracking capabilities (via environmental switching speed). This work provides a physical framework for understanding and controlling adaptivity.

Figures (6)

• Figure 1: The state space, observables, and types of system-environment coupling of a simple adapting individual.(a) The total state space of the two-state system $\{a,b\}$ under a Markov environmental switch $\{A,B\}$. Vertical arrows are environmental transitions and horizontal arrows are system transitions. (b) Four statistical observables that fully parameterize the system's dynamics. $\pi_a$ is the long-term time fraction that the system is in $a$ regardless of the environmental state. Traffic $\tau_E$ are the sum of fluxes in both direction under environment $E$. $J$ is the net cycle flux. (c) Three types of system-environmental coupling classified by the observables. Net flux $J$ is needed to track the environment and outperform an independent switch in the synchronization task. Fitness values shown on the right are the long-term time fraction of being at the synchronized (fit) states. Parameters used in (c) are as follows: $R_{AB}=R_{BA}=1$, $g_a(A)=g_b(B)=1$ and $g_a(B)=g_b(A)=0$ for all three cases. $R_{ab}(E)=R_{ba}(E)=1, E=A \text{ or } B,$ for the independent case; $R_{ab}(A)=R_{ba}(A)=0.1,R_{ab}(B)=R_{ba}(B)=10$ for the symmetric-coupling case; $R_{ab}(A)=0.2,R_{ba}(A)=10,R_{ab}(B)=10,R_{ba}(B)=0.2$ for the asymmetric-coupling case.
• Figure 2: The dynamics and total state space for a phenotype-switching population.(a) We consider the population dynamics of two-state individuals with Poisson birth-death and switching. (b) The population type fraction (of $a$) tracks the environmental changes and determines the instantaneous per capita growth rate. (c) The system-environment total state space consists of two line segments $f\in [0,1]$, one for each environment $A$ and $B$. The population can jump between the two segments at any $f$. (d) The (cyclic) net flux over the total state space leads to the long-term conditional probability density $\rho(f|E)$, forming a flux-tracking relation for population dynamics. Parameters used in (b) and (d) are: $R_{AB}=2$, $R_{BA}=1$, $g_a(A)=30$, $g_b(B)=10$, $g_a(B)=g_b(A)=-10$, and $w_{ab}(E)=w_{ba}(E)=1$.
• Figure 3: Fitness Parsing explains the existence of optimal nonzero stochastic switching rates. Analyzing the dependence of Long-Term (logarithmic) Growth Rate (LTGR) on the stochastic switching rates $(w_{ab},w_{ba})$. Black +" denotes the position where $(w_{ab},w_{ba})=(R_{AB},R_{BA})$ whereas red o" denotes the optimal $(w_{ab},w_{ba})$. (a) Fast growth limit. The optimal switching rates match the environment's switching rates in this limit. (b) When the Generalism strategy is optimal. The case where the growth rate is not fast. The optimum LTGR happens at the boundary, with $w_{ba}=0$. (c) Pure Tracking / Distinct Tunability. This is a case where the environmental average growth rates of the two types are the same: $\bar{g}_a=\bar{g}_b=0.5$. The Generalism term becomes a flat surface, leaving the fitness landscape to be determined solely by the Tracking term. Parameters used are the following: (a) $R_{AB}=3$, $R_{BA}=1$, $g_a(A)=1000$, $g_b(A)=-1000$, $g_a(B)=-1000$, $g_b(B)=1000$; (b) $g_a(A)=1$, $g_b(A)=-1$, $g_a(B)=-1$, $g_b(B)=1$. All other parameters are the same as (a); (c) $R_{AB}=3$, $R_{BA}=1$, $g_a(A)=5$, $g_b(A)=-1$, $g_a(B)=-1$, $g_b(B)=1.$ Parameter ranges are selected to illustrate the fast-growth limit (a) and the topological features of the fitness landscape (b,c).
• Figure 4: Fitness Parsing explains the existence of optimal phenotypic memory. From (a) to (c), we increase $R_{AB}$ from $2$ to $10/3$ to $10$. The average growth rates of the two environments change from $(\bar{g}_a,\bar{g}_b)=(0.8,0.\overline{6})$ to $\approx(0.769,0.769)$ and to $=(0.\overline{72},0.\overline{90})$. Optimal at intermediate phenotype memory cannot happen in (b) or (c) when $\bar{g}_a\le \bar{g}_b$. Note that the y-axes for the components are individually scaled to visualize the qualitative trends in each regime. Other parameters are chosen according to the phenomenological model in skanata_evolutionary_2016: fixed at $R_{BA}=1$, $h=1$, $\tau_{\text{ON}}^{-1}=1$, $g=1$, $c=0.3$. All parameter values are chosen relative to the growth parameter $g$.
• Figure 5: Fitness Parsing predicts the parameter dependence of fitness-memory relationship.(a) Using the parameter set of Fig. \ref{['fig: optimal memory']}(a) inherited from Ref. skanata_evolutionary_2016 (solid lines in this figure) as the baseline, we scale up and down the overall environmental switching rates $\theta R_{AB},\theta R_{BA}$: $\theta=0.8$ case shown in dash-dotted lines and $\theta=1.2$ case in dash lines. This varies mainly the Tracking gain. (b) We perturb the growth rate under glucose, $g+\phi$: $\phi=-0.2$ shown in dash-dotted lines, and $\phi=+0.2$ in dash lines. This shifts only the Generalism gain. Strengths of parameter perturbations are chosen phenomininological for illustration.