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Dynamics Washington for of and Institute Planck B.X. Max A.N., research; paper.y Reviewers: the designed wrote S.L. S.L. and B.X. and and P.S., research; B.X., contributions: Author responses. phenotypic diverse develop to organisms environment model allows phenotypic the of that our of representation to internal range high-dimensional Essential wide a is conditions. a environmental explore to map- to responses “environment-to-phenotype picture us of allowed unifying which model This strength ping,” a strategies. the using possible obtained emerge and of is adaptation cues continuum of a strategies environmental from particular of selection, Depend- accuracy natural environments. of the varying on to adaptation ing for solution eral can 15). 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BIOPHYSICS AND COMPUTATIONAL BIOLOGY 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 εµ 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; εµ). 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 ; εt ) offspring, where εt 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 | εt )f (Φ(ξt ); εt ), [1]

mapping that maximizes the population growth rate in varying ξt environments. Consider a population of organisms that reproduce asexu- where P(ξt | εt ) is the probability that a cue ξt is received when ally in discrete numbers of generations. The environment they the environment is εt . 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 → ∞. This long-term T N0 mental condition is described by an n-dimensional vector ε, 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 | ε )f (Φ(ξ); ε ), [2] that the environment switches between several different condi- µ ξ tions, labeled by εµ 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 condition ε ronmental condition and can potentially be used to distinguish occurs. We use Λ as the measure of evolutionary success for a the actual environment. This environmental cue is denoted by a population. The optimal phenotypic response is determined by vector ξ, which is assumed to belong to the n-dimensional envi- the function Φ that maximizes the value of Λ. ronment space. Note that, in the same environment εµ, each For simplicity, we assume that the environmental cue ξ is ran- organism may receive a different cue ξ. domly distributed around the actual environment εµ according µ 2 Similarly, the phenotype of an organism is described by a to a Gaussian distribution, P(ξ | εµ) = 1 exp{− (ξ−ε ) }, p-dimensional vector φ, whose components represent different (2πσ2)n/2 2σ2 where σ represents the noisiness of the environmental cue. The characteristic traits, such as the shape of body parts or the speed µ of movement. The phenotype that an organism expresses may fitness is also assumed to be a Gaussian function, f (φ; ε ) = γ2(φ−ψµ)2 depend on the environmental cue ξ that it receives. We describe Fµ exp{− 2 }, where Fµ is a constant representing the such dependence by a function, φ = Φ(ξ), which represents a maximum number of offspring in the environment εµ, and ψµ is

A B

Fig. 1. Schematic illustration of our modeling framework. (A) Phenotypic responses described by a mapping from an n-dimensional environment space to a p-dimensional phenotype space. The environment can be in one of m conditions, labeled by εµ, each favoring a phenotype ψµ. In a given environmental condition (distinguished by color), each individual organism receives a noisy cue ξ (distribution represented by color shade in environment space) and expresses a phenotype according to the mapping Φ (distribution of phenotypes induced by the mapping is represented by color shade in phenotype space). The fitness of a phenotype depends on its distance to the favorable phenotype (illustrated by the fitness landscape in phenotype space). Note that the organism sees only the cue and does not know the true environment. (B) A network model with one hidden layer. The input ξ has n components, ξa. P The hidden layer has q components, given by ηα = g( a Hαaξa), where Hαa is the representation matrix and g is a sigmoid function. The output φ has p P components, determined by φi = α Giαηα, where Giα is the expression matrix.

13848 | www.pnas.org/cgi/doi/10.1073/pnas.1903232116 Xue et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 es h idnndso h ewr a ersn internal represent may network the possible of nodes all hidden capture The tems. hence and approx- can function (16)] responses. smooth phenotypic “perceptron” feed- a any multilayered as imate a [known such With network matrix.) variables, the forward of internal part as many optimized is sufficiently that term constant the is constant a e.g., represents term; that [“bias”] column additional an has matrix (Each matrix” “expression function an through variables nal a as thought be vector matrix” input can environment; the resentation external by nodes determined the are hidden of values representation their These internal 1B. an form Fig. to in illustrated as phenotype q the of has nents layer has output layer the input the The to layer. responding hidden a with network as motivated biologically is convenient. a that computationally introduce as function we well the possible following, of all the form allow In particular to responses. of general phenotypic form of sufficiently The types be numerically. should parameters function the the over optimize function can the we specify (but to general need in we explicitly solved be see cannot equation this equation nately, variational the satisfies which in described pheno- rate long-term the the growth assumptions, and those Under space respectively. environment space, type the for (see scales environments characteristic all for S3 same Fig. the be to assumed u tal. et Xue archetypes the by spanned shaded) (gray plane a on environments fall and points noise All environmental space. for phenotype the condition in environmental points actual by the represents (color space ronment 2. Fig. param- The environment. that in eter phenotype favorable most the oe,aptnilylrenme oprdwith compared number large potentially a nodes, h tutr fti oe sisie ymn ilgclsys- biological many by inspired is model this of structure The function the of model Our function ideal the in interested are We aeil n Methods and Materials tanh γ itiuino niomna cues environmental of Distribution (A) network. optimized an by produced strategy adaptation an of Example ersnstesrnt fntrlslcin hc is which selection, natural of strength the represents o ifrn ae.Nt that Note case). different a for Φ ucin h uptvector output The function. ae h form the takes ε aeil n Methods. and Materials Λ φ µ i n ewe w archetypes two between and seautdnmrclyacrigt Eq. to according numerically evaluated is Φ = P n i (ξ H a opnnso h niomna cue environmental the of components p = ) H n olna transformation nonlinear a and oe,crepnigt the to corresponding nodes, AB α h idnlyri hsnt have to chosen is layer hidden the φ; γ a X o pca ae) opoedfurther, proceed To cases). special for = ξ α a o eeto tegh hc ersn hrceitcsae htaeo h aeodra h itnebtentwo between distance the as order same the of are that scales characteristic represent which strength, selection for 1 Φ ≡ G ae h omo feed-forward a of form the takes i P Φ α g a n naprmti om othat so form, parametric a in =1

X H a φ δ ψ α Λ/δ µ a eed nteinter- the on depends H Φ respectively. , ξ σ α a ∗ a Φ(ξ G + and htmaximizes that ξ ξ a loehr the Altogether, . H ! hog “rep- a through Unfortu- 0. = ) α . n 1/γ 0 IAppendix, SI where , oe,cor- nodes, p n ev as serve compo- g and such , ε as 2, µ H [3] itiuino hntpspoue yteotmzdntok represented network, optimized the by produced phenotypes of Distribution (B) ). α Λ, p ξ 0 ; , h hntp itiuin n o tcagsudrdiffer- under corresponding changes the strategy. of characterizes it adaptation shape conditions, how The environmental and 2B. dis- ent Fig. distribution, a in phenotype generates illustrated network the as optimized phenotypes, the environment of 2A. by tribution Fig. given in given a illustrated mapping as In cues, The of 2B). distribution (Fig. a receive space organisms phenotype the in phenotype favorable condition most environmental (p each occurrence For of respectively). probabilities in and (marked 2A) space (m environment Fig. the conditions in positions three chosen arbitrarily between switches environment (p (n space space environment notype 2D a noise Consider examples. the environmental in of strategies tation selection level natural network the of optimized strength on the depend from resulting will strategy adaptation The Strategies Adaptation Different of Emergence rate of opti- growth the individuals for of look where we values and case mal matrices, same the the consider share evolu- population undergoes we the which simplicity, genotype, For organism’s tion. the in encoded is matrix matrix representation the sion by specified of is stages ping later facilitate expanding to of signals 20). role (19, input cognition the of play dimensionality may the neurons some information; of sensory layers mul- processing have intermediate for mammals, neurons and of insects layers of tiple networks, systems num- neural olfactory large biological the a as Similarly, to such (18). rise states giving possible and of other ber modification each multiple with have exam- interacting often for sites, that networks, proteins many signaling involve responses, Cellular ple, suggested. phenotypic been refined also more has produce the and perceive inter- better high-dimensional environment to organisms a allows of nodes which formation hidden representation, The as nal (17). act molecules, network promoters, gene the a of and of by group factors growth described large phenotypic as a be such plant’s where can a network, conditions example, growth-regulatory environmental For to organism. responses the of variables norntokmdl h niomn-opeoyemap- environment-to-phenotype the model, network our In Λ. G H hs arcsmyrpeetifrainthat information represent may matrices These . and 3 = ψ ,ada2Ditra pc (q space internal 20D a and ), PNAS µ G o hs lt eue aaee values parameter used we plots these For . (σ htmxmz h ogtr population long-term the maximize that , | ψ γ uy9 2019 9, July ) µ γ alda aceye hereafter, “archetype” an called , aaee pc sn numerical using space parameter eepoeterneo adap- of range the explore We . ξ ersne ypit nteenvi- the in points by represented | o.116 vol. µ 0 = H 2 = .,ad0.3, and 0.5, .2, n h expres- the and | ε µ o 28 no. easg a assign we , ,a3 phe- 3D a ), .The 20). = σ 3 = n the and ,with ), | 13849 σ ε = µ 1 ,

BIOPHYSICS AND COMPUTATIONAL BIOLOGY A prominent feature of the emerged geometric structure, Fig. S1). However, depending on the selection strength γ, the shown in Fig. 2B, is that all phenotypes lie on a flat plane phenotype distribution has very different characters. Fig. 3A spanned by the archetypes, {ψµ}. This structure can be explained shows the case of weak selection, where the characteristic scale by a “Pareto efficiency” argument as follows. Since the fitness 1/γ is much larger than the typical distance between two phe- of a phenotype depends on its distance to the archetypes, a notypes (chosen to be '1); i.e., γ  1. In this case, the pheno- ¯ P µ phenotype located off the plane will always be less fit than its types are centered near the average phenotype, ψ = µ pµψ , perpendicular projection onto the plane. Therefore, in the opti- regardless of the environmental condition. It means that the mal phenotype distribution, all phenotypes should fall on the organisms may ignore the cue when it is noisy and exhibit a plane. In general, if there are m archetypes, the optimal phe- constant phenotype. The optimal constant phenotype strikes notype distribution will be contained in a (m − 1)-dimensional a balance between all of the archetypes, similar to the result subspace spanned by those archetypes. If m is small com- in ref. 10. pared with the dimensionality of the original phenotype space, Bet-hedging strategy under high noise and strong selection. p, then the dimensionality of phenotypes will be significantly When the cue is noisy and the selection is strong (σ, γ  1), reduced. however, the unvarying strategy fails because the average phe- Such dimensional reduction, as well as the Pareto efficiency notype ψ¯ suffers from low fitness values in all environments. In argument, is similar to that found in the model in ref. 10. In that this case, surprisingly, the organisms do not ignore the uninfor- model, the archetypes represent different biological tasks that mative environmental cue, but use it in a completely different every individual organism must perform during its lifetime, with way—each organism expresses one of the archetypes according varied degrees of importance to its overall fitness. To compare to the cue, so that the population diversifies into multiple sub- with our model, we can associate the tasks with environmen- populations due to the randomness of the cue. As shown in tal conditions that individuals may encounter and need to adapt Fig. 3C, the phenotype distribution is sharply peaked around to, with varied probabilities of occurrence. From this perspec- every archetype ψµ, and the size of each peak changes little with tive, the model in ref. 10 corresponds to the situation where the the environmental condition. This bet-hedging strategy guaran- phenotype does not depend on the present environment (i.e., no tees that, in any environment εµ, a subpopulation expressing the phenotypic plasticity), and the phenotype distribution of a popu- corresponding archetype ψµ will have a high fitness value. The lation is simply localized at a given point in the phenotype space. relative size of each subpopulation depends on the probabil- This form of phenotypic response and the resulting phenotype ity pµ that each environment occurs. In the limit of extremely distribution are characteristic of the unvarying strategy, which is strong selection (γ → ∞), we expect to recover the result of discussed later. In contrast, by allowing the phenotype to depend previous bet-hedging models (e.g., ref. 21), in which the prob- on environmental cues through the environment-to-phenotype ability of expressing the archetype ψµ matches the probability of mapping, our model encompasses a wider range of adaptation encountering the environment εµ (Materials and Methods). This strategies, as we describe below. is indeed the case, as the fraction of phenotypes near each ψµ agrees well with the environment probability pµ (SI Appendix, Examples of Strategies. In the following, we examine the distribu- Fig. S1C). tion of phenotypes for different parameters σ and γ, represented Intermediate strategies. Besides the above extreme cases that by the density of points in the archetype plane, as shown in correspond to well-categorized adaptation strategies, interme- Fig. 3 (see SI Appendix, Fig. S1 for clarity). In many cases, the diate cases are also found. A combination of bet-hedging and density is high near the archetypes. We divide the plane into µ tracking strategies is seen in the case of a medium noise level regions surrounding each ψ , marked by boundary lines in Fig. 3; (σ ' 1) and strong selection (γ  1). As shown in Fig. 3F, the Insets show fraction of phenotypes lying inside each region. By phenotype distribution is peaked around the archetypes, but comparing those fractions as well as the shape of the pheno- the relative sizes of the peaks are biased toward the one that type distribution between different environmental conditions, we matches the actual environment (see also SI Appendix, Fig. S1F). identify a wide range of adaptation strategies. This case may represent the situation of bet hedging with par- Tracking strategy under low noise. Examples of low environmen- tial environmental information, in which the population uses an tal noise are shown in Fig. 3 G–I. In these cases, the width imperfect cue to moderately adjust its phenotype distribution of the noise distribution is much smaller than the typical dis- (21, 22). Similarly, we can see intermediate cases between bet- tance between two environmental conditions (chosen to be '1); hedging and unvarying strategies (high noise σ  1 and medium i.e., σ  1. Therefore, the environmental cue is very accurate selection γ ' 1, Fig. 3B), as well as between unvarying and track- about the present environmental condition. As a result, in each ing strategies (medium noise σ ' 1 and weak selection γ  1, environment εµ, the phenotype distribution is highly concen- µ Fig. 3D). trated near the corresponding archetype ψ —the surrounding The transition of adaptation strategies with the parameters region contains almost 100% of the phenotypes, so the plots in σ and γ, illustrated by the examples in Fig. 3, can also be Fig. 3 G–I, Insets look diagonal. This means that the organisms understood analytically using approximate solutions of the ideal can express the most favorable phenotype that tracks the vary- function Φ∗ for extreme parameter values (Materials and Meth- ing environmental condition. The picture hardly changes as the ods). Those approximate solutions do not rely on the para- selection strength γ is varied (compare among Fig. 3 G–I). It metric form of the function, Eq. 3, showing that our results is understandable since, without a significant cost for sensing, are more general than the numerical examples. Generally, the organisms should always use environmental cues when those are accuracy of environmental cues, measured by the noise level reliable. σ, determines the bias of the phenotype distribution toward Unvarying strategy under high noise and weak selection. The the archetype in a given environmental condition. The selection opposite case where the environmental noise level is high (σ  strength γ, on the other hand, modifies the shape of the phe- 1) is shown in Fig. 3 A–C. In these examples, the environmental notype distribution, which tends to be more clustered near the cue has a broad distribution and is largely uninformative about archetypes when the selection is strong and more scattered into the actual environment. Therefore, we expect the optimal phe- the interior space between the archetypes when the selection notype distributions to look similar in all environments. This is is weak. verified by Fig. 3 A–C, Insets, which show that there is a signifi- cant fraction of phenotypes in each region and the fractions vary Quantification of Strategies. The shape of the phenotype distri- slightly between different environments (see also SI Appendix, butions illustrated above can be characterized quantitatively.

13850 | www.pnas.org/cgi/doi/10.1073/pnas.1903232116 Xue et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 oyecnb eopsdas decomposed be can notype distribution is probability notype conditional probabilities a environment by denoted π be can tion noise how properties, environmental examine these the and describe strength with quantities To vary concentrated characteristic how archetypes. they two and the introduce how environment we near are the distributions are with phenotype they vary the they of much properties main Two conditions. environmental h nescinpitcrepnst h vrg phenotype, average the to corresponds point with plane, intersection archetype the the in plotted phenotypes represent 3. Fig. u tal. et Xue (φ pcfial,i ahenvironment each in Specifically, | ε µ (A–I sdfie nEq. in defined as ), γ π . hntp itiuin rdcdb ewrsotmzdfrdfeetvle ftenielevel noise the of values different for optimized networks by produced distributions Phenotype ) = (φ) P µ p µ π (φ p 15 µ | h vrl itiuino h phe- the of distribution overall the , ε .Gvnthe Given Methods). and (Materials µ = V[φ] h oa aineo h phe- the of variance total The ). ε µ V[ h hntp distribu- phenotype the , E[φ σ | EF DE C AB HI GH ε n h selection the and µ ]+ ]] ϕ 1 E[Vφ , ϕ ψ ¯ 2 = snwcodnts ahdlnsdvd h ln norgosta r ls oeach to close are that regions into plane the divide lines Dashed coordinates. new as P | ε µ µ p ]]. µ ψ µ . h pia hntp itiuin r otie nbetween in contained the are of distributions variance vari- phenotype the the optimal by of the a terms trace in the the normalize varies archetypes, take and phenotype we matrices sec- clarity, the ance the For much whereas environment. how environment, given characterizes the term adapta- with ond different varies characterize much phenotype how to characterizes the term terms first the two Essentially, strategies. these tion use can envi- the We to respect probabilities with expectation ronment conditional the environment of given variance a for phenotype term, first the In Insets hwtefato fpeoye nieec einudrdifferent under region each inside phenotypes of fraction the show V[ψ ] acrigt h aeoefiinyargument, efficiency Pareto the to (according E[φ PNAS | ε p µ µ σ ] n iial o h eodterm. second the for similarly and , | n h eeto strength selection the and stecniinlepcaino the of expectation conditional the is uy9 2019 9, July ε µ | and , o.116 vol. V[Eφ | γ ooe circles Colored . o 28 no. | ε µ ]] | sthe is 13851 ψ µ ;

BIOPHYSICS AND COMPUTATIONAL BIOLOGY AB

Fig. 4. Characterization of adaptation strategies using quantities VE and EV. (A) Plot of the parameter space showing how VE and EV vary with the noise level σ and the selection strength γ. Circles represent data points from numerical calculations, with values of VE and EV illustrated by grayscale; thick colored circles correspond to examples shown in Fig. 3. (B) Plot of VE−EV space showing examples from Fig. 3. Dashed line represents the bound VE + EV ≤ 1. Corners of the VE−EV space represent special limits that correspond to the tracking, unvarying, and bet-hedging strategies.

the archetypes; hence V[φ] ≤ V[ψ]). Thus, our characteristic all environments. In other words, the organism can use only the quantities are unvarying strategy, even though it is not favorable in many situ- ations. On the other hand, a large q enables organisms to form tr ( [ [φ | εµ]]) tr ( [ [φ | εµ]]) various types of adaptation strategies, as we have seen for q = 20. VE ≡ V E , EV ≡ E V . [4] tr (V[ψ]) tr (V[ψ]) The price, however, is having to tune a lot of parameters. This could mean a much longer time for a population to adapt to a Fig. 4A shows how the values of these quantities change varying environment. according to the parameters σ and γ. In our numerical computation, we found that it is much slower To see how these quantities help characterize different adap- to optimize over the representation matrix H than over the expres- tation strategies, consider the three strategies described above. sion matrix G, because the latter is directly connected to the For the tracking strategy, the phenotypes are concentrated near output phenotype being selected but the former is not. This sug- the corresponding archetype in each environment, and hence gests that it is harder for an organism to adjust the way it creates µ µ µ E[φ | ε ] ≈ ψ and V[φ | ε ] ≈ 0; therefore, VE ≈ 1 and EV ≈ 0. an internal representation of the environment than to adjust the Similarly, for the unvarying strategy, the phenotypes are always mechanism that produces the phenotype directly. It is therefore concentrated near the center of the archetypes, which means interesting to ask whether one can keep the representation matrix µ µ E[φ | ε ] ≈ ψ¯ and V[φ | ε ] ≈ 0; therefore, VE ≈ 0 and EV ≈ 0. H fixed while optimizing over the expression matrix G alone. Finally, for the bet-hedging strategy, the phenotype distributions are largely independent of the environment and are concentrated near the archetypes in proportion to the environment probabili- ties pµ; this leads to VE ≈ 0 and EV ≈ 1. Therefore, those three strategies can be clearly distinguished by different limits of the characteristic quantities, as shown in Fig. 4B. Dimensionality of Internal Representation So far we have fixed the dimensionality of the network’s hidden layer at a relatively large number, q = 20, compared with that of the environment space, n = 2. The motivation was to create an adequate expansion of dimensionality from the input layer to the hidden layer, q/n = 10, so that the network can be used to approximate well the ideal function Φ∗ in all cases. The approxi- mation is verified in the limit γ → 0, where explicit solutions can be found (Materials and Methods); the numerical solutions we obtained are very close to the ideal function Φ∗ (SI Appendix, Fig. S2 B and C). Let us now explore how the results change if we vary the dimensionality q. Fig. 5 shows how the maximum value of Λ increases with q. For a small q, the network model becomes very Fig. 5. Long-term population growth rate Λ vs. the dimensionality q of the restrictive because it does not have many parameters that can be intermediate layer of the network. Each point represents a network with tuned. In that case, the phenotype distribution that results from a random, fixed representation matrix H (entries drawn from N (0, 1) inde- optimizing the network will be deformed from that for the ideal pendently) and an optimized expression matrix G. (To aid visualization of the density of points, a small random horizontal displacement is added.) Φ∗ SI Appendix A function ( , Fig. S2 ). In particular, in the limit Dashed and solid (Inset) lines show the mean and SD of the values of Λ. q → 0, the intermediate layer of the network vanishes, so the out- Horizontal bars mark the maximum values of Λ when the matrix H is also put becomes disconnected from the input. This means that the optimized for each q; dotted line (Inset) shows the difference between the phenotype can no longer depend on the environmental cue, and maximum and the mean values. For this example the parameter values σ = 1 hence the organism is forced to express the same phenotype in and γ = 1 are used.

13852 | www.pnas.org/cgi/doi/10.1073/pnas.1903232116 Xue et al. 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BIOPHYSICS AND COMPUTATIONAL BIOLOGY studying evolution, since the same form of phenotypic responses which cannot be solved analytically. Here we derive approximate solutions may be naturally selected even if it is implemented by different for some extreme values of the parameters σ and γ. Our results in this molecular mechanisms. For instance, many bacteria can stochas- subsection do not rely on the network ansatz, Eq. 3, of the function Φ. Weak selection, γ → 0. In this limit, we can expand the integrand in Eq. 5, tically switch from a normal growth state to a dormant persister 2 state, which prevents cell death from unforeseeable antibiotic to first order in γ , yielding attack (34). Different molecular mechanisms have been found Z 2 X µ µ Λ ≈ − γ ξ ξ | ε Φ ξ − ψ
2 to underlie such bacterial persistence (35). Nevertheless, the 2 pµ d P( ) ( ) , [6] growth benefit of this particular adaptation strategy can be µ understood without using those mechanistic details (36). Such where P(ξ | εµ) is the Gaussian distribution of ξ. To maximize the value of methods have recently been applied to other types of adaptation Λ, we set its variational derivative over the function Φ(ξ) to zero, strategies (21, 37). We have used a network model as a simple example of pos- δΛ 2 X µ µ
= −γ pµP(ξ | ε ) Φ(ξ) − ψ = 0. [7] sible forms of the environment-to-phenotype mapping. In our δΦ(ξ) µ model the connections of the network store information about the environmental conditions and their statistics, as well as Solving this equation yields about the favorable phenotypes. Besides varying the dimension- ality of the internal representation or the number of intermedi- − 1 (ξ−εµ)2 P µ µ P µ 2σ2 µ pµP(ξ | ε )ψ µ pµψ e ate layers (38), a possible further generalization of our model Φ∗(ξ) = = . [8] P ν 1 ν 2 pν P(ξ | ε ) − (ξ−ε ) ν P 2σ2 would be to consider a recurrent network with evolvable internal ν pν e dynamics (39). Such a network could allow organisms to store ∗ P µ µ information about their past phenotypes and encode temporal This result can also be written succinctly as Φ (ξ) = µ P(ε | ξ)ψ , using structures of the environmental history. The environment for the Bayes’ rule. The same expression has been derived in ref. 37. In the subcase where σ is small, i.e., when the cue ξ is accurate, the prob- organisms can also include ecological interactions with individu- µ µ als of the same population or other species. Such generalizations ability P(ε | ξ) is nearly 1 for the correct environment ε , and hence the ψµ could lead to potentially more complex adaptation strategies. phenotypes are concentrated at the corresponding archetype . This yields the tracking strategy. However, when σ is large, i.e., when the cue is noisy, all environments εµ are likely; Eq. 8 becomes Φ∗(ξ) ≈ P p ψµ ≡ ψ¯, which Materials and Methods µ µ means that an average phenotype ψ¯ is produced regardless of the cue. This Numerical Methods. Our goal is to maximize the long-term growth rate corresponds to the unvarying strategy. Λ with respect to the phenotypic response function Φ. The function Φ is Low noise, σ → 0. In this limit, the Gaussian distribution of ξ in Eq. 5 is parameterized by the matrices H and G, as in Eq. 3. The value of Λ, according concentrated near its mean, εµ, so we can expand the integrand around to Eq. 2, is given by that point. This yields, to first order in σ2,

γ2 2 X X D − (Φ(ξ)−ψµ)2 E γ X h µ µ
2 2 µ µ
i Λ = pµ log Fµ + pµ log e 2 [5] Λ ≈ − 2 pµ Φ(ε ) − ψ + σ ∂aΦi(ε )∂aΦi(ε ) + ··· . [9] ξ∼N (εµ,σ) µ µ µ

where h·i represents the expectation value with respect to the Gaussian This expression depends on the local values of the function Φ and its deriva- µ µ ∗ µ µ random variable ξ. The first term does not depend on the parameters of Φ tives, Φ(ε ), ∂Φ(ε ), etc. To maximize Λ, we should have Φ (ε ) ≈ ψ and ∗ µ ∗ µ and is ignored. The optimization is done numerically by iterating over two ∂Φ (ε ) ≈ 0. It means that the ideal function Φ maps each environment ε µ steps: calculating the expectations in Eq. 5 given the current values of H and to its archetype ψ , and the mapping is locally “flat”—the function value µ G, then updating these matrices to improve the value of Λ. changes little in the neighborhood of ε . Since, for low noise, the cues ξ µ For the first step, we calculated the expectation values by numerically are close to the actual environment ε , they will all be mapped to near the µ integrating over the Gaussian distributions. We used the python package correct archetype ψ . This leads to the tracking strategy for any value of “scipy.integrate,” which calls the Fortran library QUADPACK. An alterna- the selection strength γ. tive approach to numerical integration is to generate a random sample of High noise, σ → ∞. In this limit, the cue ξ has a broad distribution that µ µ ξ from the Gaussian distribution and use it to estimate the expectation val- varies little with the environment ε , and hence P(ξ | ε ) ≈ P(ξ). As a result, ues. This approach represents a finite sampling of the environmental cues, the phenotype distribution will also be independent of the environment which allows for the analysis of the effect of finite population sizes and the and can be defined as stability of the optimal solutions. We tried both approaches and did not find Z significant differences in performance. π(φ) ≡ dξ P(ξ) δ(φ − Φ(ξ)). [10] For the second step, we searched parameters using the python package “scipy.optimize” with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algo- Λ rithm. This step involves calculating the gradient of the function Λ over the Using this phenotype distribution, the long-term growth rate can be matrices H and G and then using the gradient to update their values. One written as Z γ2 X − (φ−ψµ)2 could update the matrices simultaneously or optimize one while holding the Λ ≈ pµ log dφ π(φ) e 2 . [11] other fixed and then iterate. It turns out that optimizing the matrix G alone µ is efficient, because G is directly connected to the output without having a The distribution π∗(φ) that maximizes Λ will constrain the ideal function Φ∗ nonlinear transformation. Using this observation, we chose to optimize G through Eq. 10. at every step of updating H. In this case, the gradient of Λ(G∗(H), H) over H Let us treat the subcases of small and large γ separately. For a small γ, i.e., ∂Λ ∂Λ can be simply calculated as ∂H because ∂G = 0. 2 G∗ G∗ weak selection, we once again expand Λ to first order in γ , which yields For the examples shown in Fig. 3, the coordinates of the environments 1 2 3 Z and the archetypes are ε = [−0.1, 0.9], ε = [−0.8, −0.4], ε = [0.9, −0.5], 2 X µ Λ ≈ − γ p dφ π(φ)(φ − ψ )2 ψ1 = [−0.6, 0.5, 0.8], ψ2 = [0.4, 0.6, −0.9], ψ3 = [0.5, −0.8, 0.4]; the envi- 2 µ µ ronment probabilities are [p , p , p ] = [0.2, 0.5, 0.3]. The same values are 1 2 3 Z  used for Figs. 4 and 5. In Fig. 4, for each pair of parameter values σ and γ2 ¯
2 = − 2 dφ π(φ) φ − ψ + V[ψ] , [12] γ, we ran eight replicate optimizations starting from random initial values (every entry of H and G drawn i.i.d. from N (0, 1)); the order parameters are P µ 2 ¯2 averaged over these replicates. In Fig. 5, for each dimensionality q, we ran where V[ψ] = µ pµ(ψ ) − ψ . From this expression it is clear that the 100 examples, each having a fixed H with random entries. optimal phenotype distribution is π∗(φ) = δ(φ − ψ¯), which agrees with the unvarying strategy found above. Analytic Limits. Nonparametrically, the ideal response function Φ∗ that For a large γ, it can be seen from Eq. 11 that the distribution π(φ) should maximizes Eq. 5 should satisfy the variational equation δΛ/δΦ(ξ) = 0, become sharply peaked at points where φ = ψµ. We can use the ansatz

13854 | www.pnas.org/cgi/doi/10.1073/pnas.1903232116 Xue et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 0 .Kihauty .M emnsa,T oa .M aca,V Balasubramanian, V. Walczak, M. A. representations. Mora, T. sensory Hermundstad, M. in A. expansion Krishnamurthy, K. and 20. Sparseness complexity Sompolinsky, The Goldstein, H. B. Babadi, Perelson, S. to B. A. environment Blinov, 19. L. connects M. Faeder, perceptron R. J. plant Hlavacek, S. The W. Putten, 18. der van H. W. Scheres, B. 17. cor- a when environment varying annuals. randomly desert a in of reproduction guild Optimizing Cohen, a D. in 14. hedging Bet Venable, L. D. 13. Moczek P. A. 11. Shoval O. 10. 6 .Hrz .G amr .S Krogh, S. A. Palmer, G. R. Hertz, J. 16. Sono- in germination among-year and Within- Venable, L. D. Kimball, S. Gremer, R. J. 15. environment. varying randomly a in reproduction Optimizing Cohen, D. 12. ttearchetypes the at oyedsrbto a eudrto yaayigtegoer fthe µ of geometry the the phe- between analyzing the 40). trade-off by of (8, a understood set” shape “fitness the is be case, can there this distribution archetypes, In notype environments. the different of in one values fitness only to 1/γ scale close characteristic be the by measured as another probability total a selection, has partition Strong each that such function space the environment distribution, phenotype each a such at erate peaks probability separate the to of proportional consist will optimal tion The 21). ref. of (e.g., hedging bet values of model the recovers expression This π a by separated archetypes corresponding the distance with environments, two the strength 6. Fig. u tal. et Xue .A ae,T oa .Rvie .M aca,Tastosi pia adaptive optimal in Transitions Walczak, Grant, R. P. M. A. 9. Rivoire, O. 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Rivoire, O. 21. 8 .Fidadr .E ao .Tut,U ln vlto fbwteacietrsin architectures bow-tie of Evolution growth Alon, U. population Tlusty, T. for Mayo, plasticity E. A. phenotypic Friedlander, T. of 38. Benefits Leibler, S. Xue, K. B. 37. 0 .Lvn,Eouini hnigevrnet:Sm hoeia explorations. theoretical Some environments: changing in Evolution fluctua- Levins, environmental R. under 40. plasticity and robustness of Evolution Kaneko, K. 39. dacdSuy(rn 481.BX n ..aefne yteEi and Eric the Study. Advanced by for for funded Institute Institute the are at the P.S. Biology and in and Membership S.L.) B.X. Schmidt 345430) (to 345801). Wendy Foundation (Grant (Grant Simons Study University the Advanced Rockefeller from grants the by research through This supported discussions. partly helpful been for Young has Lai-Sang and Kaneko, Kunihiko the at ACKNOWLEDGMENTS. peaked be will distribution phenotype optimal mainly the archetypes archetypes. distribution cue, cor- phenotype the the the the of near near that less means points phenotypes This of of 1). when combinations (0, that, consists and linear seen 0) by (1, be formed at can ners It largely 6. be Fig. will in set shown is set fitness γ distance the case, a this at In be ity. to assumed are archetypes distribution phenotype the where phenotypic given a phenotypes with population all a of for function fitness response average points the such Then, set. of ness collection The space. r ie by given are (8). distribution phenotype long- the the and maximizes response Eq. that set rate, in fitness growth distribution extended term the the phenotype from within the point points by the of locating combination weighted linear set, a fitness as original considered be can set extended Φ points, those of collection The (ξ  sa xml,cnie w environments, two consider example, an As metaphor. 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