Transitions in optimal adaptive strategies for populations in fluctuating environments

Stochastic chemical kinetics of cell fate decision systems: From single cells to populations and back

Stochastic chemical kinetics is a widely used formalism for studying stochasticity of chemical reactions inside single cells. Experimental studies of reaction networks are generally performed with cells that are part of a growing population, yet the population context is rarely taken into account when models are developed. Models that neglect the population context lose their validity whenever the studied system influences traits of cells that can be selected in the population, a property that naturally arises in the complex interplay between single-cell and population dynamics of cell fate decision systems. Here, we represent such systems as absorbing continuous-time Markov chains. We show that conditioning on non-absorption allows one to derive a modified master equation that tracks the time evolution of the expected population composition within a growing population. This allows us to derive consistent population dynamics models from a specification of the single-cell process. We use this approach to classify cell fate decision systems into two types that lead to different characteristic phases in emerging population dynamics. Subsequently, we deploy the gained insights to experimentally study a recurrent problem in biology: how to link plasmid copy number fluctuations and plasmid loss events inside single cells to growth of cell populations in dynamically changing environments.

The chemical Langevin equation for biochemical systems in dynamic environments

Modelling and simulation of complex biochemical reaction networks form cornerstones of modern biophysics. Many of the approaches developed so far capture temporal fluctuations due to the inherent stochasticity of the biophysical processes, referred to as intrinsic noise. Stochastic fluctuations, however, predominantly stem from the interplay of the network with many other - and mostly unknown - fluctuating processes, as well as with various random signals arising from the extracellular world; these sources contribute extrinsic noise. Here we provide a computational simulation method to probe the stochastic dynamics of biochemical systems subject to both intrinsic and extrinsic noise. We develop an extrinsic chemical Langevin equation-a physically motivated extension of the chemical Langevin equation- to model intrinsically noisy reaction networks embedded in a stochastically fluctuating environment. The extrinsic CLE is a continuous approximation to the Chemical Master Equation (CME) with time-varying propensities. In our approach, noise is incorporated at the level of the CME, and can account for the full dynamics of the exogenous noise process, irrespective of timescales and their mismatches. We show that our method accurately captures the first two moments of the stationary probability density when compared with exact stochastic simulation methods, while reducing the computational runtime by several orders of magnitude. Our approach provides a method that is practical, computationally efficient and physically accurate to study systems that are simultaneously subject to a variety of noise sources.

A scaling law in CRISPR repertoire sizes arises from the avoidance of autoimmunity

Some prokaryotes possess CRISPR-Cas systems that use DNA segments called spacers, which are acquired from invading phages, to guide immune defense. Here, we propose that cross-reactive CRISPR targeting can, however, lead to "heterologous autoimmunity," whereby foreign spacers guide self-targeting in a spacer-length-dependent fashion. Balancing antiviral defense against autoimmunity predicts a scaling relation between spacer length and CRISPR repertoire size. We find evidence for this scaling through a comparative analysis of sequenced prokaryotic genomes and show that this association also holds at the level of CRISPR types. By contrast, the scaling is absent in strains with nonfunctional CRISPR loci. Finally, we demonstrate that stochastic spacer loss can explain variations around the scaling relation, even between strains of the same species. Our results suggest that heterologous autoimmunity is a selective factor shaping the evolution of CRISPR-Cas systems, analogous to the trade-offs between immune specificity, breadth, and autoimmunity that constrain the diversity of adaptive immune systems in vertebrates.

Roadmap on biology in time varying environments

Biological organisms experience constantly changing environments, from sudden changes in physiology brought about by feeding, to the regular rising and setting of the Sun, to ecological changes over evolutionary timescales. Living organisms have evolved to thrive in this changing world but the general principles by which organisms shape and are shaped by time varying environments remain elusive. Our understanding is particularly poor in the intermediate regime with no separation of timescales, where the environment changes on the same timescale as the physiological or evolutionary response. Experiments to systematically characterize the response to dynamic environments are challenging since such environments are inherently high dimensional. This roadmap deals with the unique role played by time varying environments in biological phenomena across scales, from physiology to evolution, seeking to emphasize the commonalities and the challenges faced in this emerging area of research.

Fitness value of information with delayed phenotype switching: Optimal performance with imperfect sensing

The ability of organisms to accurately sense their environment and respond accordingly is critical for evolutionary success. However, exactly how the sensory ability influences fitness is a topic of active research, while the necessity of a time delay between when unreliable environmental cues are sensed and when organisms can mount a response has yet to be explored at any length. Accounting for this delay in phenotype response in models of population growth, we find that a critical error probability can exist under certain environmental conditions: An organism with a sensory system with any error probability less than the critical value can achieve the same long-term growth rate as an organism with a perfect sensing system. We also observe a tradeoff between the evolutionary value of sensory information and robustness to error, mediated by the rate at which the phenotype distribution relaxes to steady state. The existence of the critical error probability could have several important evolutionary consequences, primarily that sensory systems operating at the nonzero critical error probability may be evolutionarily optimal.

Adapt or Perish: Evolutionary Rescue in a Gradually Deteriorating Environment

We investigate the evolutionary rescue of a microbial population in a gradually deteriorating environment, through a combination of analytical calculations and stochastic simulations. We consider a population destined for extinction in the absence of mutants, which can survive only if mutants sufficiently adapted to the new environment arise and fix. We show that mutants that appear later during the environment deterioration have a higher probability to fix. The rescue probability of the population increases with a sigmoidal shape when the product of the carrying capacity and of the mutation probability increases. Furthermore, we find that rescue becomes more likely for smaller population sizes and/or mutation probabilities if the environment degradation is slower, which illustrates the key impact of the rapidity of environment degradation on the fate of a population. We also show that our main conclusions are robust across various types of adaptive mutants, including specialist and generalist ones, as well as mutants modeling antimicrobial resistance evolution. We further express the average time of appearance of the mutants that do rescue the population and the average extinction time of those that do not. Our methods can be applied to other situations with continuously variable fitnesses and population sizes, and our analytical predictions are valid in the weak-to-moderate mutation regime.

Tuning environmental timescales to evolve and maintain generalists

Natural environments can present diverse challenges, but some genotypes remain fit across many environments. Such "generalists" can be hard to evolve, outcompeted by specialists fitter in any particular environment. Here, inspired by the search for broadly neutralizing antibodies during B cell affinity maturation, we demonstrate that environmental changes on an intermediate timescale can reliably evolve generalists, even when faster or slower environmental changes are unable to do so. We find that changing environments on timescales comparable with evolutionary transients in a population enhance the rate of evolving generalists from specialists, without enhancing the reverse process. The yield of generalists is further increased in more complex dynamic environments, such as a "chirp" of increasing frequency. Our work offers design principles for how nonequilibrium fitness "seascapes" can dynamically funnel populations to genotypes unobtainable in static environments.

Cost and benefits of clustered regularly interspaced short palindromic repeats spacer acquisition

Clustered regularly interspaced short palindromic repeats (CRISPR)-Cas-mediated immunity in bacteria allows bacterial populations to protect themselves against pathogens. However, it also exposes them to the dangers of auto-immunity by developing protection that targets its own genome. Using a simple model of the coupled dynamics of phage and bacterial populations, we explore how acquisition rates affect the probability of the bacterial colony going extinct. We find that the optimal strategy depends on the initial population sizes of both viruses and bacteria. Additionally, certain combinations of acquisition and dynamical rates and initial population sizes guarantee protection, owing to a dynamical balance between the evolving population sizes, without relying on acquisition of viral spacers. Outside this regime, the high cost of auto-immunity limits the acquisition rate. We discuss these optimal strategies that minimize the probability of the colony going extinct in terms of recent experiments.

This article is part of a discussion meeting issue ‘The ecology and evolution of prokaryotic CRISPR-Cas adaptive immune systems’.

Heterogeneous Timing of Gene Induction as a Regulation Strategy

In response to environmental changes, cells often adapt by up-regulating genes to synthesize proteins that generate a benefit in the new environment. Several such cases of gene induction have been reported where the timing was heterogeneous, with some cells responding early and others responding late, although the microbial population was genetically homogeneous and the environment was well mixed. Here, we explore under which conditions heterogeneous timing of gene induction could be advantageous for the population as a whole. We base our study on a mathematical model that accounts for the cost of protein synthesis in terms of resources, which cells must provide immediately, whereas the associated benefit accumulates only slowly over the protein lifetime. Due to this delayed benefit, gene induction can be a risky investment, if resources are scarce and the environment fluctuates rapidly and unpredictably. Unprofitable gene induction then depletes the remaining limiting resource needed for maintenance of cell viability. We show that whenever gene induction is associated with a transient risk but beneficial in the long run, the stochastic timing of gene induction maximizes the reproductive success of a population. In particular, in an environment of stochastic periods of famine and feast, an optimum emerges from a trade-off between short-term growth, favoring rapid and homogeneous responses, and long-term survival, favoring a broadly heterogeneous response. Our analysis suggests that the optimal variability of induction times is just as large as the time required for the amortization of the initial investment into protein synthesis.

Fitness Gain of Individually Sensed Information by Cells

Mutual information and its causal variant, directed information, have been widely used to quantitatively characterize the performance of biological sensing and information transduction. However, once coupled with selection in response to decision-making, the sensing signal could have more or less evolutionary value than its mutual or directed information. In this work, we show that an individually sensed signal always has a better fitness value, on average, than its mutual or directed information. The fitness gain, which satisfies fluctuation relations (FRs), is attributed to the selection of organisms in a population that obtain a better sensing signal by chance. A new quantity, similar to the coarse-grained entropy production in information thermodynamics, is introduced to quantify the total fitness gain from individual sensing, which also satisfies FRs. Using this quantity, the optimizing fitness gain of individual sensing is shown to be related to fidelity allocations for individual environmental histories. Our results are supplemented by numerical verifications of FRs, and a discussion on how this problem is linked to information encoding and decoding.

Evolving generalists in switching rugged landscapes

Evolving systems, be it an antibody repertoire in the face of mutating pathogens or a microbial population exposed to varied antibiotics, constantly search for adaptive solutions in time-varying fitness landscapes. Generalists refer to genotypes that remain fit across diverse selective pressures; while multi-drug resistant microbes are undesired yet prevalent, broadly-neutralizing antibodies are much wanted but rare. However, little is known about under what conditions such generalists with a high capacity to adapt can be efficiently discovered by evolution. In addition, can epistasis-the source of landscape ruggedness and path constraints-play a different role, if the environment varies in a non-random way? We present a generative model to estimate the propensity of evolving generalists in rugged landscapes that are tunably related and alternating relatively slowly. We find that environmental cycling can substantially facilitate the search for fit generalists by dynamically enlarging their effective basins of attraction. Importantly, these high performers are most likely to emerge at intermediate levels of ruggedness and environmental relatedness. Our approach allows one to estimate correlations across environments from the topography of experimental fitness landscapes. Our work provides a conceptual framework to study evolution in time-correlated complex environments, and offers statistical understanding that suggests general strategies for eliciting broadly neutralizing antibodies or preventing microbes from evolving multi-drug resistance.

(In)exhaustible Suppliers for Evolution? Epistatic Selection Tunes the Adaptive Potential of Nongenetic Inheritance

Nongenetic inheritance media-from methyl-accepting cytosines to culture-tend to mutate more frequently than DNA sequences. Whether this makes them inexhaustible suppliers for adaptive evolution will depend on the effect of nongenetic mutations (hereafter, epimutations) on fitness-related traits. Here we investigate how these effects might themselves evolve, specifically whether natural selection may set boundaries to the adaptive potential of nongenetic inheritance media because of their higher mutability. In our model, the genetic and epigenetic contributions to a nonneutral phenotype are controlled by an epistatic modifier locus, which evolves under the combined effects of drift and selection. We show that a pure genetic control evolves when the environment is stable-provided that the population is large-such that the phenotype becomes robust to frequent epimutations. When the environment fluctuates, however, selection on the modifier locus also fluctuates and can overall produce a large nongenetic contribution to the phenotype, especially when the epimutation rate matches the rate of environmental variation. We further show that selection on the modifier locus is generally insensitive to recombination, meaning it is mostly direct, that is, not relying on subsequent effects in future generations. These results suggest that unstable inheritance media might significantly contribute to fitness variation of traits subject to highly variable selective pressures but little to traits responding to scarcely variable aspects of the environment. More generally, our study demonstrates that the rate of mutation and the adaptive potential of any inheritance media should not be seen as independent properties.

Environment-to-phenotype mapping and adaptation strategies in varying environments

Biological organisms exhibit diverse strategies for adapting to varying environments. For example, a population of organisms may express the same phenotype in all environments ("unvarying strategy") or follow environmental cues and express alternative phenotypes to match the environment ("tracking strategy"), or diversify into coexisting phenotypes to cope with environmental uncertainty ("bet-hedging strategy"). We introduce a general framework for studying how organisms respond to environmental variations, which models an adaptation strategy by an abstract mapping from environmental cues to phenotypic traits. Depending on the accuracy of environmental cues and the strength of natural selection, we find different adaptation strategies represented by mappings that maximize the long-term growth rate of a population. The previously studied strategies emerge as special cases of our model: The tracking strategy is favorable when environmental cues are accurate, whereas when cues are noisy, organisms can either use an unvarying strategy or, remarkably, use the uninformative cue as a source of randomness to bet hedge. Our model of the environment-to-phenotype mapping is based on a network with hidden units; the performance of the strategies is shown to rely on having a high-dimensional internal representation, which can even be random.

Exploration-exploitation tradeoffs dictate the optimal distributions of phenotypes for populations subject to fitness fluctuations

We study a minimal model for the growth of a phenotypically heterogeneous population of cells subject to a fluctuating environment in which they can replicate (by exploiting available resources) and modify their phenotype within a given landscape (thereby exploring novel configurations). The model displays an exploration-exploitation trade-off whose specifics depend on the statistics of the environment. Most notably, the phenotypic distribution corresponding to maximum population fitness (i.e., growth rate) requires a nonzero exploration rate when the magnitude of environmental fluctuations changes randomly over time, while a purely exploitative strategy turns out to be optimal in two-state environments, independently of the statistics of switching times. We obtain analytical insight into the limiting cases of very fast and very slow exploration rates by directly linking population growth to the features of the environment.