Optimal phenotypic adaptation in fluctuating environments

Abstract

Phenotypic adaptation is a universal feature of biological systems navigating highly variable environments. Recent empirical data support the role of memory-driven decision making in cellular systems navigating uncertain future nutrient landscapes, wherein a distinct growth phenotype emerges in fluctuating conditions. We develop a simple stochastic mathematical model to describe memory-driven cellular adaptation required for systems to optimally navigate such uncertainty. In this framework, adaptive populations traverse dynamic environments by inferring future variation from a memory of prior states, and memory capacity imposes a fundamental trade-off between the speed and accuracy of adaptation to new fluctuating environments. Our results suggest that the observed growth reductions that occur in fluctuating environments are a direct consequence of optimal decision making and result from bet hedging and occasional phenotypic-environmental mismatch. We anticipate that this modeling framework will be useful for studying the role of memory in phenotypic adaptation, including in the design of temporally varying therapies against adaptive systems.

Figure 1

Illustration of memory-driven phenotypic switching. (A) The model consists of cells capable of stochastic transition between two states, $S_A$ and $S_B$, where state $S_i$ has an advantage in the $i$-type environment. (B) The general model is applied to study cases where the environment may become hostile or may represent low nutrient availability. (C) Cellular decision making uses a memory of prior environments to optimally respond to future ones by selecting the phenotype that maximizes expected future growth.

Figure 2

Maximal attainable growth potential in fluctuating environments. The value function for cells optimally selecting their phenotypic state based on a history of fluctuating environmental states is given as a function of the underlying environmental parameter $p$ for (A) systems with infinite-memory and variable $\alpha$ together with limiting behavior ($\alpha =1$ and $\alpha \to \infty$). (B) For systems with finite memory capacity, the value functions are plotted for $\alpha =2$ over increasing memory size, along with the infinite-memory limit (in all cases, $r\nu =2$, $c\lambda =1$, $\delta =0.9$).

Figure 3

Growth in fluctuating environments. Representative stochastic trajectories of (A) cumulative growth potential and (B) averaged cumulative growth potential are depicted for cells navigating $A$-predominant (left), neutral (middle), and $B$-predominant (right) environments. Adaptive systems (blue) capable of phenotypic switching have lower growth potential than static phenotypes (red, yellow) matched to the proper environment, but outperform those phenotypes whenever the environment is mismatched. Growth dynamics for all strategies collapse to a common process in the neutral environment (in all cases, $r\nu =2$, $c\lambda =1$, $\alpha =2$, $\delta =0.9$, $N=20$).

Figure 4

Dynamics of fixed and adaptive-memory systems navigating constant-to-fluctuating nutrient environments. (A) Fixed-memory cells experiencing low-constant to high-fluctuating nutrient environments undergo phenotypic switching. The maximal growth potential and switching times both vary directly with memory capacity. (B) Illustration of the probability distribution for the estimate ${\pi }_n$ of $p$ for varying memory sizes as in (A). Successful estimation occurs whenever ${\pi }_n>p_I$ (green arrow; right of dashed line). The probability of environmental mis-estimation occurs whenever ${\pi }_n<p_I$ (red arrow; left of dashed line). Cumulative averaged growth potential and memory sizes are plotted in time for adaptive cells undergoing (C) constant-high to fluctuating-low and (D) constant-low to fluctuating-high nutrient environments (in all cases, $r_L=0.05$, $r_H=1$ giving ${\beta }_C=1+r_H/r_L$. $\beta ={\beta }_C/2$ giving $p_I=0.32$. All initial memory capacities tend toward a unique long-term limit (dashed horizontal lines). For (B) and (C) $N_{\min }=3$, $N_{\max }=20$, high-fluctuating environments were given by $p=0.60$, low-fluctuating by $p=0.20$, and memory was updated based on detected distance to environmental indifference $d_n=\left|{\pi }_n-p_I\right|$. A total of ${10}^4$ stochastic simulations were averaged to generate growth and memory curves. Two stochastic realizations are depicted for initial memory sizes $N_0=N_c$ selected in via linear interpolation between $N_{\min }$ and $N_{\max }$ of the initial distance between $p_I$ and the high- (resp. low-) constant environments, modeled by $p_0=1$ (resp. 0).

Figure 5

Phenotypic adaptation in oscillating environments. Adaptation dynamics are studied by considering the fluctuation experiment in (22) and imputing estimates of the corresponding growth rates for each phenotype-environment pair. (A) The average growth rate distribution of 100 adaptive cells in an oscillating-nutrient environment with period $T=60$ minutes is depicted as a function of time assuming intrinsic noise in each cell’s ability to identify the past environmental encounters. Model application for a variety of memory sizes $N$ recapitulates empirically observed timescales of adaptation memory sizes that are in excess of environmental oscillations. (B) Focusing on growth trajectories with memory size $N=3T$, the average growth rates over the nutrient-high $\left(R_{high}\right)$, nutrient-low $\left(R_{low}\right)$, and total $\left(R_{avg}\right)$ time intervals are plotted (error bars depict standard deviation) and recover the dynamics observed in (22) ($p_{noise}=0.4$ as in see supporting material, Section S7.3).

Figure 6

Growth deficits due to phenotypic switching in stochastic environments. $C_{\text{avg}}$ describes the average environment given by the convex combination $p{C}_{\text{high}}+\left(1-p\right)C_{\text{low}}$ against which the rapidly fluctuating $p$ environment can be compared. (A) The dynamics of cumulative averaged growth potential for fixed- and adaptive-memory cells navigating fluctuating and comparable constant environments ($p=0.4$, $p_I=0.2$, $\beta ={\beta }_{\text{Crit}}/4$). Across all environmental parameters, $p$, and allowable growth coefficients, $\beta$, simulated long-run growth potential for (B) $p$-random and (C) $p$-averaged constant environments reveal a universal growth deficit (D) of the $p$-fluctuating case relative to $C_{\text{avg}}$, which is particularly pronounced in the $p=p_I$ environment. (E and F) The total expected growth deficit of fluctuating environments consists of an intrinsic contribution maximized at $p=p_I$ and an extrinsic contribution (illustrated for adaptive cells with limited $N=3$ memory capacity) owing to the risk of environmental phenotypic mismatch. In all cases, $r_L=0.01$, $r_H=1$, ${\beta }_{\text{Crit}}=1+r_H/r_L$, and ${10}^3$ stochastic simulations were evaluated for each simulation in (A) over time and at each parameter value in (B)–(F).