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Environment-To-Phenotype Mapping and Adaptation Strategies in Varying Environment-to-phenotype mapping and adaptation strategies in varying environments BingKan Xuea,b,c,1, Pablo Sartoria,b,c, and Stanislas Leiblera,b,c,1 aThe Simons Center for Systems Biology, Institute for Advanced Study, Princeton, NJ 08540; bLaboratory of Living Matter, The Rockefeller University, New York, NY 10065; and cCenter for Studies in Physics and Biology, The Rockefeller University, New York, NY 10065 Contributed by Stanislas Leibler, May 14, 2019 (sent for review February 26, 2019; reviewed by Armita Nourmohammad and Mikhail Tikhonov) Biological organisms exhibit diverse strategies for adapting to phenotype distribution would thus exhibit a narrow peak that varying environments. For example, a population of organisms tracks the environmental variation. Examples of this “tracking may express the same phenotype in all environments (“unvarying strategy” are seasonal changes of the butterfly’s wing patterns strategy”) or follow environmental cues and express alterna- and the mammal’s coat colors, which are induced by weather tive phenotypes to match the environment (“tracking strategy”), conditions and provide suitable camouflage (11). or diversify into coexisting phenotypes to cope with environ- A third strategy is such that individual organisms of the same mental uncertainty (“bet-hedging strategy”). We introduce a population express different phenotypes, so that the phenotype general framework for studying how organisms respond to envi- distribution is broad or has multiple peaks. Such diversification ronmental variations, which models an adaptation strategy by is useful in stochastically changing environments, since there will an abstract mapping from environmental cues to phenotypic always be some individuals in the population that have the right traits. Depending on the accuracy of environmental cues and phenotype to survive. A classical example of this “bet-hedging the strength of natural selection, we find different adaptation strategy” is the seed bank: To cope with unpredictable inclement strategies represented by mappings that maximize the long-term weather, some seeds quickly germinate after being dispersed growth rate of a population. The previously studied strategies while others remain in the soil for a prolonged period (12, 13). emerge as special cases of our model: The tracking strategy is Bet hedging can also be combined with cue tracking, such that favorable when environmental cues are accurate, whereas when the distribution of phenotypes varies according to the environ- cues are noisy, organisms can either use an unvarying strat- ment. For example, the fraction of seeds that germinate can BIOPHYSICS AND egy or, remarkably, use the uninformative cue as a source of depend on environmental factors such as temperature, moisture, COMPUTATIONAL BIOLOGY randomness to bet hedge. Our model of the environment-to- and the presence of other seeds (14, 15). phenotype mapping is based on a network with hidden units; We show that the above strategies are special limits of a gen- the performance of the strategies is shown to rely on having eral solution for adaptation to varying environments. Depend- a high-dimensional internal representation, which can even be ing on the accuracy of environmental cues and the strength random. of natural selection, particular strategies of adaptation emerge from a continuum of possible strategies. This unifying picture evolutionary theory j fluctuating environments j phenotypic plasticity j is obtained using a model of “environment-to-phenotype map- population dynamics j survival strategies ping,” which allowed us to explore a wide range of phenotypic responses to environmental conditions. Essential to our model is a high-dimensional internal representation of the environment o study the properties of a physical system, a phenomenolog- that allows organisms to develop diverse phenotypic responses. Tical approach is to characterize how it responds to external conditions. For instance, materials show particular patterns of Significance deformation under external forces, which reveals their elas- tic properties. Biological organisms exhibit far more complex A fundamental difference between living and nonliving sys- responses to environmental conditions. As the environment tems is that organisms can evolve responsive adaptation varies, organisms adapt by changing their phenotypes, including to external conditions. We present a theoretical framework, morphological and behavioral traits. Such phenotypic responses which unifies different adaptation strategies encountered in to the environment are modified through the process of evolu- biology. Central to our approach is the introduction of an tion, which gives rise to different forms of adaptation. Several environment-to-phenotype mapping describing how organ- adaptation strategies, as described below, have been studied both isms’ traits or behavior depend on the environment. In con- experimentally and theoretically (1–8). In this paper, we adopt trast to commonly considered genotype-to-phenotype map- the phenomenological approach to study biological adaptation ping, our approach emphasizes an evolutionary rather than by modeling general forms of phenotypic responses to environ- mechanistic understanding of organisms. Our phenomenolog- mental conditions. This approach enables us to reveal underlying ical model, inspired by artificial neural networks, also allows connections between different adaptation strategies. us to study the importance of the dimensionality of internal The simplest adaptation strategy is one in which organisms representations for the adaptation strategies. express the same phenotype in all environments. A population using this strategy has a narrow distribution of phenotypes that Author contributions: B.X., P.S., and S.L. designed research; B.X. and P.S. performed does not vary with the environment. In such an “unvarying strat- research; and B.X. and S.L. wrote the paper.y egy,” the typical phenotype is often fit for most environmental Reviewers: A.N., Max Planck Institute for Dynamics and Self Organization and University conditions. For example, birds that feed on a variety of food of Washington in Seattle; and M.T., Washington University in St. Louis.y sources (“generalists”) have a midsized beak, which is slender The authors declare no conflict of interest.y enough for catching insects and conical enough for cracking Published under the PNAS license.y seeds (9, 10). 1 To whom correspondence may be addressed. Email: [email protected] or Another strategy is for organisms to follow environmental [email protected] cues and express alternative phenotypes to match the environ- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. ment. Provided that the cues are accurate, individual organisms 1073/pnas.1903232116/-/DCSupplemental.y of a population may all express the appropriate phenotype. The Published online June 20, 2019. www.pnas.org/cgi/doi/10.1073/pnas.1903232116 PNAS j July 9, 2019 j vol. 116 j no. 28 j 13847–13855 Downloaded by guest on September 24, 2021 Our results suggest ways to experimentally evolve and identify mapping from the n-dimensional environment space to the p- different adaptation strategies. dimensional phenotype space, as illustrated in Fig. 1A. Different forms of the mapping will correspond to different adaptation Model of Environment-to-Phenotype Mapping strategies. The phenotypic responses of an organism to environmental The fitness of an organism in a given environment "µ is mea- conditions can be conceptualized as a mapping from the envi- sured by how many offspring it produces. This depends on its ronment space to the phenotype space. A certain environmental phenotype φ and is described by a function f (φ; "µ). Thus, in stimulus that the organism experiences may induce a partic- each generation, labeled by a number t, an individual organism ular phenotype. Such an environment-to-phenotype mapping that receives an environmental cue ξt will express a phenotype may represent, for example, how the development of organisms φt = Φ(ξt ) and produce as many as f (φt ; "t ) offspring, where "t is affected by the environment (known as “phenotypic plas- is the environmental condition. Let Nt be the population size in ticity”). A mapping that allows a population to survive better the tth generation; then in the next generation it will be and reach greater abundance in the long term will generally be X favored by natural selection. We study the optimal form of the Nt+1 = Nt P(ξt j "t )f (Φ(ξt ); "t ), [1] mapping that maximizes the population growth rate in varying ξt environments. Consider a population of organisms that reproduce asexu- where P(ξt j "t ) is the probability that a cue ξt is received when ally in discrete numbers of generations. The environment they the environment is "t . In the long term, the growth rate of the live in may vary from generation to generation. An environ- population is given by Λ ≡ 1 log NT for T ! 1. This long-term T N0 mental condition is described by an n-dimensional vector ", growth rate can be calculated as whose components represent different environmental factors, X X µ µ such as temperature, light, and amount of food. We assume Λ = pµ log P(ξ j " )f (Φ(ξ); " ), [2] that the environment switches between several different condi- µ ξ tions, labeled by "µ for µ = 1, ::: , m. Each individual organism µ receives an environmental cue, which is correlated with the envi- where pµ is the probability that each environmental
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