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Home , Adaptive value, Lamarckism

A model for the generation and transmission of PNAS PLUS variations in

Olivier Rivoirea,b,1 and Stanislas Leiblerc,d

aLaboratoire Interdisciplinaire de Physique, Université Grenoble Alpes, F-38000 Grenoble, France; bLaboratoire Interdisciplinaire de Physique, Centre National de la Recherche Scientifique, F-38000 Grenoble, France; cLaboratory of Living Matter, The Rockefeller University, New York, NY 10065; and dThe Simons Center for Systems , Institute for Advanced Study, Princeton, NJ 08540

Edited by Joshua B. Plotkin, University of Pennsylvania, Philadelphia, PA, and accepted by the Editorial Board March 24, 2014 (received for review January 8, 2014)

The inheritance of characteristics induced by the environment has Concurrent views, emphasizing the role of environmentally in- often been opposed to the theory of evolution by . duced variations in evolution, have had several insightful propo- However, although evolution by natural selection requires new nents (10–13), but were also endorsed by dubious yet influential heritable traits to be produced and transmitted, it does not pre- supporters (14). Examples of inherited acquired characteristics scribe, per se, the mechanisms by which this is operated. The have, however, been long known, from the transmission of culture in mechanisms of inheritance are not, however, unconstrained, be- humans to the uptake of extracellular DNA by bacteria. However, cause they are themselves subject to natural selection. We introduce only recently have we gained a fuller recognition of the diversity of a schematic, analytically solvable mathematical model to compare mechanisms for generating and transmitting variations (15). In ad- the of different schemes of inheritance. Our model dition to the well-recognized roles of and recombinations allows for variations to be inherited, randomly produced, or of chromosomal DNA, a nonexhaustive list would include the environmentally induced, and, irrespectively, to be either trans- transmission of acquired marks such as DNA methyla- mitted or not during reproduction. The of the different tion, the transmission of small interfering , the transmission of schemes for processing variations is quantified for a range of conformational states of molecules such as prions, or, at the cellular fluctuating environments, following an approach that links quan- level, the transmission of self-sustaining states of regulation, and, at the organismal level, so-called parental effects (16).

titative with stochastic control theory. BIOPHYSICS AND Inheritance, long treated as an autonomous and universal mechanism to be experimentally characterized and then integrated COMPUTATIONAL BIOLOGY Darwinian evolution | | | acquired characteristics | to evolutionary theory, thus appears to consist of multiple and parallel systems whose origins and implications are to be explained within an evolutionary framework. The problem of a synthesis of hree principles underlie the explanation of by inheritance with evolution is thus now doubled by the problem of Tnatural selection: (i) individuals in a population have varied the synthesis of inheritance by evolution. With this problem in characteristics; (ii) their reproductive success correlates with these view, we propose here a mathematical model where the adaptive PHYSICS characteristics; (iii) the characteristics are inherited. The last values of different schemes for generating and transmitting var- principle, of inheritance, has always been the most contentious. At iations can be compared. The model treats inheritance as a trait on the time of Darwin and Wallace, its mechanisms were unknown, which selection can act, although not in a direct way: systems of and fundamental questions, such as the role of the environment in inheritance indeed pertain to the transmission of traits between the production of new, adaptive traits, were unsettled. Adaptation individuals, and estimating their adaptive value therefore requires by natural selection does not, indeed, require any causal relation analyzing the dynamics of a population over several generations. between the environment and newly generated traits, but neither In this sense, the adaptation of a mode of inheritance is necessarily does it exclude it; Darwin, for instance, included as potential of “second order”—aformof“.” sources of variations the direct and indirect effects of the envi- Our model thus quantifies the adaptation of different modes ronment, as well as the use and disuse of organs, in line with ideas of inheritance by considering the long-term growth rate of previously propounded by Lamarck (1, 2). Prominent followers of Darwin, however, came to exclude the possibility of inheritance of acquired characteristics. This view- Significance point was notably formulated by Weismann in his theory of continuity of the germplasm (3). Experiments of amputations, Few things are excluded in biology if they are not physically which showed no incidence on the progeny, supported it. At the impossible, but some have been advocated to be, for instance, end of the nineteen century, it had became a central tenet of the absence of reverse flow of information from to “neo-” (4). Half a century later, the “Modern Syn- genotype, often assimilated to Lamarckism. Regardless of the thesis,” which produced a synthesis between evolution theory question of whether this prohibition is universally true or not, and Mendel’s laws of inheritance (5), reached the same con- could any such prohibition logically result from natural selec- clusion: it promoted a clear distinction between genotypes, tion? We introduce a mathematical model to examine this inherited but only subject to random variations, and , question and, more generally, the conditions under which affected by the environment but not directly transmitted. These various mechanisms for generating and transmitting variations conclusions were based on studies in multicellular , but in evolving populations may be favored or suppressed by subsequent experiments with , which found that natural selection. adaptive variations can precede changes of environmental con- ditions (6), further reinforced the conviction that biological Author contributions: O.R. and S.L. designed research; O.R. performed research; and O.R. evolution is mainly fueled by random variations. At a molecular and S.L. wrote the paper. level, finally, once prevalent instructional theories of enzymatic The authors declare no conflict of interest. adaptation or formation also came to be discarded in This article is a PNAS Direct Submission. J.B.P. is a guest editor invited by the Editorial the 1950s and 1960s (7, 8). At this time, the successes of molec- Board. ular biology in unraveling the mechanisms of heredity elevated 1To whom correspondence should be addressed. E-mail: [email protected]. a molecular refutation of Lamarckism, the unidirectional flow of This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. information from DNA to proteins, as its “central dogma” (9). 1073/pnas.1323901111/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1323901111 PNAS Early Edition | 1of10 Downloaded by guest on September 26, 2021 populations. The model is schematic: it relies on an abstraction Model from physical implementations, along the example of Shannon’s We provide in this section a general presentation of the model model of communication (17). The model thus defines the “ge- and derive its solution in a simple case. The general solution follows notype” as what is transmitted between successive generations, the same principles and its details are included in SI Appendix. and the “phenotype” as what determines the survival and re- production of an individual, with no reference to their material Definition. The model considers a population of asexually repro- support. As a consequence, our distinction between genotypic ducing individuals where each genotype is characterized by an “at- and phenotypic variations is not equivalent to the often-made tribute” γ. The genetic or epigenetic of this attribute is distinction between genetic and epigenetic inheritance; any irrelevant: we are only concerned with the origins of the transmitted transmitted character, whether DNA encoded or not, will be- information, either inherited, randomly produced, or environmen- long, from the standpoint of our model, to the genotype. tally induced, irrespective of its material support. In the parlance of In general, few things are excluded in biology if they are not Weismann, γ would be termed the “germplasm,” and, in the par- physically impossible—but some have been proposed to be, for lance of , the “breeding value.” At each time instance, the reverse flow of information from phenotype to step, corresponding to a generation, each individual with attribute γ genotype. Regardless of the question of whether this pro- reproduces and is replaced by ξ offsprings sharing as common at- hibition is indeed universally true or not, it is interesting to tribute γ′,withpossiblyξ = 0, in which case we conventionally define consider whether any such prohibition could logically result γ′ = γ. (The model could be generalized to produce offsprings with from natural selection. More generally, under what conditions different attributes, but we purposely ignore this unessential com- various mechanisms for generating and transmitting variations plication.) The individuals are noninteracting and the generations may be favored or suppressed by natural selection itself? We nonoverlapping. The values of ξ and γ′ can depend on γ andonthe illustrate the versatility of our model by examining this question current environmental state, which fluctuates independently of the in the context of three biological phenomena that are often population and is characterized by a variable xt. This dependency considered to be either irrelevant to evolution, or absent be- can be stochastic and is generally given by a stochastic kernel cause “forbidden”: APðξ;Rγ′jγ; xtÞ with the following properties: Aðξ; γ′jγ; xtÞ ≥ 0and dγ′ Aðξ; γ′jγ; x Þ = 1forallγ; x . i) Noninherited variations, sometimes also referred to as phe- ξ t t “ ” notypic noise, and commonly thought to have no evolution- Population Dynamics. For a given series of environmental states ary implications; for instance, in the first chapter of The ðx ; ...; x Þ, we define n ðγÞ as the expected probability density “ 1 T t Origin of (1), Darwin states: Any variation which is function of the attributeR γ in the population at time (generation) ” not inherited is unimportant for us. Considering that new t, normalized to dγ ntðγÞ = 1 at any t. It satisfies a recursive variations may generally be introduced at the genotypic and/or equation which, more formally, is the recursion for the first at the phenotypic level, and may thus be transmitted to future moment of the branching process: generations either totally, in part or not at all, what scheme is Z X most conductive to adaptation? Does natural selection gener- γ′ = −1 γ ξ ξ; γ′ γ; γ ; [1] “ ” nt+1 Wt d A j xt ntð Þ ically favor developmental canalization, i.e., a reduction of ξ phenotypic differences between individuals inheriting a com- R mon genotype? Our model will highlight how the answer where Wt is a normalization ensuring dγ′ nt+1ðγ′Þ = 1, depends on the statistical structure of the environment, and Z Z how nontransmitted variations may under some conditions be X = γ′ γ ξ ξ; γ′ γ; γ : [2] more beneficial than transmitted variations. Wt d d A j xt ntð Þ ii) The absence of reverse flow of information from phenotype to ξ genotype, advocated by Weismann, but now challenged even in the species where it is best established (18). Given that Wt represents the factor by which the population increases, on + isolating the transmitted genotype from the phenotype may average, between generations t and t 1. = involve dedicated mechanisms, can we characterize the con- Starting from a large number N0 of individuals at time t 0, ditions under which natural selection favors their presence? the expected total number NT of individuals at time T is thus the iii) The nondirected nature of new adaptive variations, associ- following: ated with the refutation of any “Lamarckian” mechanism. T Nothing in principle prevents the environment from inducing NT = ∏ Wt N0: [3] the generation of new traits, either at the phenotypic level, or t=1 at the genotypic level: the first effect, known as “plasticity,” has long been recognized (19), and the second one, long Over T generations, this results in the following growth rate: thought to be forbidden, is also observed (20). Are there XT nevertheless conditions under which direct integration of in- 1 N 1 formation into the transmitted genotype is logically excluded Λ ≡ ln T = ln W : [4] T N T t as a consequence of natural selection? 0 t=1

Despite the fundamental nature of these questions, no pre- We are interested here in the long-term limit T → ∞, under the vious formal model exists, to our knowledge, that addresses them assumption that the population does not go extinct. (We assume in a common and analytically tractable framework. Our model, that the branching process is supercritical and ignore the fluctu- however, is not without precedents: it is in line with traditional ations associated with small populations, which is justified in the models of (21) and relates to models of sto- large t limit when the population is exponentially growing.) This chastic control in engineering (22). Processing unreliable infor- limit is mathematically well defined when the environment fol- mations from the past and present to confront an uncertain future, lows an ergodic process, in which case we have the following: which may be seen as the fundamental “function” of systems of inheritance, is indeed at its core a question of control. We thus Λ = lim E½ln Wt; [5] proposed previously that evolution in fluctuating environments t→∞ could be viewed as a problem of stochastic control (23); this view is supported here by a formal analogy between our model and a where E indicates an expectation with respect to the environ- basic algorithm in stochastic control theory, the Kalman filter (24). mental fluctuations (25) (Λ is also known as the quenched Lyapunov

2of10 | www.pnas.org/cgi/doi/10.1073/pnas.1323901111 Rivoire and Leibler Downloaded by guest on September 26, 2021 exponent for the underlying branching process in a random A PNAS PLUS environment). The long-term growth rate Λ is a “group-level” property, at- tached to the population as a whole rather than to any particular individual. It is relevant for the comparison of different schemes of inheritance because of the following property (23): given two populations, characterized by kernels A1 and A2, and given an ergodic environmental process, if ΛðA1Þ > ΛðA2Þ > 0, then, al- 2= 1 = 1 2 B most surely, limt→∞ ln Nt Nt 0, where Nt and Nt represent the respective sizes of the two populations at time t.Inother words, Λ predicts the long-term outcome of a competition be- tween populations characterized by different kernels A.Thisar- gument assumes an exponentially growing population, but its conclusions are equally valid in presence of a constraint on the total population size, in which case the population with smallest growth rate almost surely becomes extinct. Fig. 1. (A) Simple model. In this model, no distinction is made between inherited genotype and phenotype: both are characterized by a single Long-Term Growth Rate. Different systems of inheritance are rep- attribute γ. The number ξ of offsprings and the attribute γ′ that they resented in the model by different kernels A. In the simplest case, transmit are independently specified by two stochastic kernels, the “re- ” ξ γ “ ” γ′ γ γ schematically represented in Fig. 1A, the environment affects re- production kernel Rð j ,xt Þ and the kernel Hð j Þ,where production but not transmission, and the kernel A is factorized into represents the attribute inherited by the individual and xt the current the product of a reproduction kernel R and an heredity kernel H, state of the environment. (B) General model. This model includes explicitly a phenotype ϕ derived from the inherited genotype γ through a stochastic kernel Dðϕjγ,y Þ representing “development.” This kernel, and the kernels A ξ; γ′jγ; x = Rðξjγ; x ÞH γ′jγ : [6] t t t Rðξjγ,xt Þ and Hðγ′jγ,ϕ,zt Þ, can depend on external, environmental factors, xt , yt , zt . The three red arrows represent respectively developmental From now on, for simplicity, we also assume a single continuous plasticity (P), phenotype-to-genotype feedback (F), and direct Lamarckian attribute γ ∈ R, although the model remains analytically solvable effects (L).

in the multidimensional case. BIOPHYSICS AND We take the heredity kernel to be Gaussian, for all t; more generally, starting from any distribution n ðγ Þ, COMPUTATIONAL BIOLOGY ! 0 0 γ′ − γ 2 the distribution of the trait in the population will converge to a γ′ γ = 1 − : [7] H j = exp Gaussian. Assuming that it does not go extinct, the long-term πσ2 1 2 2σ2 2 H H evolution of the population can thus be described in terms of just ≡ γ σ2 ≡ γ − 2 two parameters, the mean mt h it and variance t hð mtÞ it This corresponds to a standard assumption of additivity in γ of at time t. PHYSICS γ′ = γ + ν ν population genetics: H , with H a normally distributed From Eq. 1, it follows that these two parameters satisfy the σ2 ν ∼ ; σ2 random variable with variance H , H Nð0 H Þ (this relation following: is extended below to include a possible reversion toward a mean, γ′ = λγ + ν λ < i.e., H with 1). 2 mt+1 = mt + h ðxt − mtÞ; [9] To compute ntðγÞ, as in Eq. 1, we only need to specify the first t moment of the selection kernel Rðξjγ; xtÞ. We assume here that σ2 = 2σ2 + σ2 ; [10] the expected number of offsprings of an individual with attribute t+1 ht S H γ in environment xt is the following: 2 = σ2= σ2 + σ2 ! where the so-called heritability ht t ð S t Þ represents the X 2 ðγ − xtÞ contribution of the genotype to the total phenotypic variance hξiγ; ≡ ξ Rðξjγ; xtÞ = exp rmax − : [8] xt σ2 (26) and satisfies here the following recursion: ξ 2 S 1 γ − “ ” h2 = 1 − : [11] Here, the difference xt captures the fitness of an individ- t+1 + 2 + σ2 σ2 1 ht H S ual with attribute γ to the current criterion of selection xt.Be- cause in the simplest version of the model, depicted in Fig. 1A, Eq. 1 also yields in terms of these variables Wt, the factor by there is no distinction between inherited genotype and pheno- + type, the attribute γ can be thought as a phenotypic trait and the which the population size increases between times t and t 1, “state of the environment” xt is thus being defined relative to 1 1 ðm − x Þ2 the population, as the value of the trait that is optimal in this ln W = r + ln 1 − h2 − 1 − h2 t t : [12] t max t t σ2 environment (for instance, in a classical simplistic picture of 2 2 S adaptation, xt would be associated with the height of acacias γ σ2 and with the length of the neck). The variance S Stochastically Fluctuating Environments. Up to here, the equations σ2 = describes the selectivity of the environment; S 0 means that are valid for any kind of environmental process. If we now as- σ2 only one phenotype can survive at any given time, whereas S sume an ergodic environment, we obtain, by taking the t → ∞ large means that many different phenotypes can survive. Finally, limit, a formal expression for the long-term growth rate, rmax is the maximal reproductive rate per generation for the species; h i in particular, r > 0 is a necessary condition for the population not 2 max E ðmt − xtÞ Λ 1 2 1 2 to go extinct. [rmax affects the growth rate only through an addi- Λ = rmax + ln 1 − h∞ − 1 − h∞ lim ; [13] t→∞ σ2 tive constant, i.e., Λðrmax Þ = Λðrmax = 0Þ + rmax , and therefore plays 2 2 S no role when comparing different inheritance schemes.] Starting at t = 0 from a large population with a normally where h∞ represents the fixed point of ht in Eq. 11. Λ is the sum “ ” distributed attribute, n0ðγ Þ = Gσ2 ðγ − m0Þ, the distribution of γ of two terms: the first can be interpreted as the genetic load 0 0 remains normally distributed at all times, i.e., ntðγÞ = Gσ2 ðγ − mtÞ due to stabilizing selection and the second as the “evolutionary t

Rivoire and Leibler PNAS Early Edition | 3of10 Downloaded by guest on September 26, 2021 load” due to the lag between the mean phenotype mt and the lution of rates: what is optimal degree of variation for optimal phenotype xt. The dynamics of mt has generally no fixed a population adapting to a varying environment? This problem point (unless the environment is constant), but, if the environ- hasbeenexaminedinmanyprevioustheoreticalstudies(27– ment is ergodic, it has a stationary distribution and hence 30) and fueled by observations and experiments with microbial 2 E½ðmt − xtÞ has a limit for t → ∞. populations showing strikingly variable mutation rates (31). Λ may be explicitly computed for several stationary processes. Within our model, the problem amounts to determining the σ^2 σ2 Λ Two aspects of the environment are particularly relevant when value H of H that optimizes the growth rate for given ; σ2 studying the adaptive value of mechanisms of heredity: the am- environmental parameters a E. The results of this optimiza- σ2 τ plitude E of the environmental fluctuations, and the scale E of tion are shown in Fig. 2A: whereas larger environmental fluc- their temporal correlations. As a simple dynamical process that tuations are conducive to larger mutations rates as may be encapsulates these two elements, we consider the following: expected, the dependence into the temporal correlation is less σ2 < intuitive, with a nonmonotonic dependence on a for E 2. A = + ; ∼ ; σ2 : [14] σ2 = − = + xt axt−1 bt with bt N 0 X transition line, given by E 2ð1 aÞ ð1 aÞ and represented in Fig. 2B, separates environmental situations where variations σ^2 = It corresponds to fxtgt being generated by a stationary Gaussian tend to be suppressed, H 0, and environmental situations = − σ^2 > Markov process with transition kernel Pðxtjxt−1Þ Gσ2 ðxt axt−1Þ. where they are favored, H 0. The parameter σ2 controls the degree of stochasticityX and the The approach of optimizing the growth rate Λ with respect X σ2 parameter a the degree of correlation between successive environ- to one of its parameters, here H ,ispredicatedontheas- ments; assuming a ≤ 0 < 1, the process followed by x has a station- sumption that a population for which this parameter mutates − t ary distribution, namely Nð0; σ2 ð1 − a2Þ 1Þ. The amplitude σ2 at a slow rate will eventually evolve toward the value of this X E Λ and relaxation time τE of this process are thus defined by the parameter optimizing .Fig.2C shows that this assumption is following: indeed verified in numerical simulations of the population σ2 dynamics where H is taken to be part of the genotype and σ2 σ′ = ; σ + η η ∼ ; σ2 2 X 1 transmitted as H maxð0 H Þ with Nð0 M Þ (the max σ ≡ ; τE ≡− ; [15] σ ≥ E 1 − a2 ln a is here to ensure that H 0). We performed these simulations by imposing a maximum total population size (SI Appendix), and such that the results of Fig. 2C therefore also demonstrate that the as- ymptotic growth rate Λ can describe well the behavior of finite − =τ 2 t E populations. Finally, the same simulations repeated with non- E½xt+t′xt′ = σ e : [16] E Gaussian distributions for the selection, the mutations or the environmental fluctuations lead to similar results (SI Appendix, This follows from the definition xt+1 = axt + bt with E½bt = 0and E 2 = σ2 Fig. S1), indicating that our conclusions are not crucially de- ½bt X , and the assumption that xt has a stationary distribution: σ2 ≡ E 2 = E 2 = 2E 2 + E 2 = 2σ2 + σ2 pendent on the Gaussian assumptions that we make for com- E ½xt ½xt + 1 a ½xt ½bt a E X , and therefore σ2 = σ2 = − 2 E = tE 2 putational convenience. X ð1 a Þ. Similarly, from ½xt+t′xt′ a ½xt′, it follows that E − =τ − E = σ2 t E τ = − 1 ½xt+t′xt′ Ee with E ðln aÞ . This stochastic process is Generalizations known as an autoregressive AR(1) model in signal process- – The previous formulae solve the simple model schematically ing and corresponds in physics to a discrete-time Ornstein represented in Fig. 1A. A generalization of this model, depicted Uhlenbeck process. in Fig. 1B, explicitly distinguishes between the inherited geno- With this choice for the environmental temporal dynamics, we type γ and the phenotype ϕ of an individual. The phenotype can show that mt itself is normally distributed and we can com- “ E − 2 arises from the inherited genotype through a process of de- pute limt→∞ ½ðmt xtÞ (SI Appendix). The calculation leads to ” Λ velopment, represented by a stochastic kernel D, which can a long-term growth rate of the following form: show some dependence on the external environment, i.e., de- σ2 σ2 velopmental plasticity (arrow P in Fig. 1B). The generalized Λ = r + Λ a; H ; E : [17] model also incorporates the possibility of acquiring informa- max 0 σ2 σ2 S S tion from the environment and directly modifying the heri- tability kernel H (arrow L). The latter can also be influenced σ2 = Without loss of generality, we can therefore assume that S 1. internally by a feedback from the phenotype to the transmitted Under this assumption, the expression for Λ0 is given by the genotype (arrow F). This more general model, several limits of following: which we analyze in the following sections, can also be solved analytically. 1 ð1 − aÞα More precisely, the general model is characterized by the Λ a; σ2 ; σ2 = ln α − σ2 ; 0 H E 2 ð1 + αÞð1 − aαÞ E following: [18] 2 γ′ = λγ + κ + ωϕ + ν ; ν ∼ ; σ2 ; α = ; zt H H N 0 with = H + σ2 + σ2 σ2 + 1 2 ϕ = θγ + ρy + ν ; ν ∼ N 0; σ2 ; 2 H H H 4 h t D D . i D [19] 2 2 hξiγ; = exp rmax − ðϕ − xtÞ 2σ : α σ2 xt S where the relation between and H can also be inverted to give σ2 = ð1 − αÞ2=α. H The first equation specifies how the transmitted genotype γ′ Not considering the additive parameter rmax, only three σ2 σ2 depends on the inherited genotype γ, on some external informa- parameters, a, E, H , are needed to characterize this simple model. The first two parameters pertain to the environmental tion zt coming from the environment and on the current pheno- − =τ ϕ process, with a representing temporal correlations ða = e 1 E Þ type . The second equation defines how this current phenotype σ2 σ2 σ2 depends on the inherited genotype γ and on some possibly avail- and E its stationary variance (in units of S), whereas H (in the same units) describes the stochasticity of inheritance (akin to able external information yt. The third equation, finally, gives the a ). expected number of offsprings for individuals with phenotype ϕ in environment xt. How Much Variation to Introduce? This model allows us to address The description of the model is completed by the equations a first question pertaining to the evolution of heredity, the evo- governing the dynamics of the environment. As before, it is sup-

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σ2 Fig. 2. Howmuchvariationstointroduce?(A) For the simple model described by Eq. 18 anddepictedinFig.1A, optimal value of the degree H of σ2 τ = − −1 genotypic variations as a function of E , the amplitude of the environmental fluctuations and a, representing the characteristic timescale E ðln aÞ of “ ” σ2 σ2 σ2 their relaxation. (B) Phase diagram for the nonzero optimal values of H as a function of ða, E Þ.(C) In red, optimal values of H as a function of a for σ2 = σ2 σ′ = σ + η E 1. In blue, results of numerical simulations of the dynamics of a population in which H varies between parent and offspring as H maxð0, H Þ with η ∼ σ2 σ2 = 4 ≤ 3 σ2 = −5 Nð0, MÞ; the mean and SD of H are computed over T 10 generations of a population of size N 10 ,with M 10 (see also SI Appendix, Fig. S1,for σ2 other values of M).

posed to be fixed (quenched) independently of the dynamics of the next generation but not directly affecting survival and re- the population: production. How different are these two (nonexclusive) types of variation from an evolutionary standpoint? Is one type of = + ; ∼ ; σ2 ; xt axt−1 bt bt N 0 X variations more advantageous than the other? Or, does each have its own optimal degree? Here again, several observations y = x + b′; b′ ∼ N 0; σ2 ; [20] t t t t Ip support the biological relevance of these questions. Both types = + ″; ″ ∼ ; σ2 : zt xt b t b t N 0 Iℓ of variations are indeed found simultaneously in every living . An example of universally shared genotypic varia- The signals yt and zt are thus assumed to derive from xt via tions is provided by DNA mutations, whereas the so-called

additive white Gaussian noise channels, one of the simplest mod- nongenetic individuality of bacteria (34) is an example of BIOPHYSICS AND

els of signal transmission. phenotypic variations. The latter, although often assimilated to COMPUTATIONAL BIOLOGY Although involving more parameters, the derivation of the noise, may in fact confer a selective advantage to the organisms (35). solution for this general model follows the same principles as Addressing questions about the degree and nature of new var- for the simple model presented previously (SI Appendix). We iations requires a model that is both quantitative and rich enough take advantage of an analytical formula for its growth rate Λ to allow for nontrivial selective pressures. Our model meets these to analyze three particular variants of the model, which ad- criteria; in fact, it is even sufficient to consider its simplified ver-

dress the three specific questions raised in the introduction. sion, in which all three mechanisms depicted by red arrows in Fig. PHYSICS In each case, we characterize the most adaptive scheme for 1B, namely plasticity, feedback, and Lamarckian effects, are ab- generating and transmitting variations by considering the Λ— sent. This limiting case, represented in Fig. 3, is described by the value of the parameters that optimize the expected out- following equations: come of an evolutionary dynamics where these parameters evolve on a timescale longer than the characteristics timescale γ′ = γ + ν ; ν ∼ ; σ2 ; τ of the environment. [It can be shown that Λ is concave with H H N0 H [21] E ϕ = γ + ν ; ν ∼ ; σ2 : respect to its parameters (32), implying an absence of local D D N 0 D maxima into which the evolutionary dynamics could otherwise be trapped.] In this particular instance of the model (obtained from the general model by taking λ = 1, θ = 1, κ = 0, ω = 0, ρ = 0), new Where to Introduce Variation? The introduction of new variations variations are generated independently of the state of the envi- is a requirement for sustained evolution: a population into which ronment and are introduced at two levels: at the genotypic level, ν σ2 no new variations are introduced will eventually become mono- through the random variable H with variance H , and at the morphic even in absence of selection, simply as a consequence of phenotypic level, through the random variable νD with variance “random drift.” However, if the characters of the individuals are so σ2 D. The two possible types of new variations are thus tunable at new as to be uncorrelated with those of their parents, inheritance various degrees. By studying how the long-term growth Λ may is negated, and adaptation by natural selection impossible. be optimized with respect to σ2 and σ2 , we can thus analyze “ ” H D Therefore, an appropriate degree of new variations must lie quantitatively the adaptive value of these two sources of varia- between these two extremes. This logical conclusion raises a first tion as a function of the characteristics of the environment, its class of questions, which we started to address with the model of σ2 temporal correlation a and its stochasticity E. Fig. 2: What is this intermediate degree of variations? What sets σ^2 σ^2 Specifically, we analyze here the values ( H , D) of the variables its value? In addition, if an optimal degree exists, can it be selected σ2 and σ2 that optimize the long-term growth Λ, for given values for? Several biological observations hint at a positive answer to H D of ða; σ2 Þ (fixing without loss of generality σ2 = 1); this corre- this last point: a complex molecular machinery has for instance E S “ ” sponds to the expected outcome of a competition between pop- evolved to ensure a faithful replication of DNA, and thus to set σ2 σ2 its mutation rate; moreover, not all are treated equally: in ulations characterized by different values of H and D,or, bacteria, for example, the positioning of genes on the leading or equivalently, to the expected outcome of an evolutionary dynamics σ2 σ2 lagging strand of the , where they are subject to where H and D are themselves slowly varying. The results, different mutation rates, is correlated to the nature of the se- presented in Fig. 4 A and B, show that the nature of the most lective pressure that they experience (33). adaptive variations indeed depends on the statistics of the envi- σ2 > A second class of questions follows from noticing that new ronment. For instance, phenotypic noise ð D 0Þ is preferred over σ2 > variations may not only vary in degree but also in “nature.” In genotypic noise ð H 0Þ for weakly correlated environments (low particular, new variations may be phenotypic, thus affecting a). The optimization can be performed analytically to derive a survival and reproduction but not being transmitted to the phase diagram with distinct phases, defined by the presence or next generation, or/and genotypic, thus being transmitted to absence of phenotypic or genotypic stochasticity in the optimal

Rivoire and Leibler PNAS Early Edition | 5of10 Downloaded by guest on September 26, 2021 structure. For instance, in bacteria, the strength of selection may be very different between central and mechanisms of resistance to antibiotics. Our model suggests that this diversity of selective pressures may be responsible for the evolution of the diversity of ways in which new traits are generated and trans- mitted. However, our model is obviously extremely schematic and does not account for a number of features that affect the evolution Fig. 3. Where to introduce variation? This question is addressed within ϕ γ of mechanisms of inheritance. In particular, it does not consider a model where the two sources of noise, the developmental kernel Dð j Þ the cost of these mechanisms, which may strongly limit their actual and heredity kernel Hðγ′jγÞ, are jointly optimized to yield the largest long- term growth rate. The optimization is performed over the two parameters diversity: suppressing any variation by error corrections, check- σ2 σ2 ϕ γ γ′ γ ϕ = γ + ν points, canalization, etc., may be prohibitively expensive, and D and H, which define Dð j Þ and Hð j Þ by the relations D and γ′ = γ + ν ν ∼ σ2 ν ∼ σ2 evolving a different system of inheritance for every trait simply H, with D Nð0, DÞ and H Nð0, HÞ. impossible (any mechanism for introducing variations in a trait defines a new trait into which variations may be introduced). solution. The different boundaries, shown in Fig. 4E, are given by the following: When to Separate Phenotype and Transmitted Genotype? The pre- vious version of the model assumes an independent germ line, 1 1 + a 2 with a transmitted genotype γ′ that is not influenced by the phe- a = 1=3 ðblackÞ; σ2 = ðblueÞ; ϕ γ “ ” E 4 1 − a notype . Such a separation between a germplasm and a soma [22] ϕ is central to the view that new traits are exclusively generated by 1 − a random mutations in the , independently of any event σ2 = 2 ðredÞ; σ2 = 1 ðgreenÞ; E 1 + a E occurring during the lifetime of the individual. This separation, however, cannot be taken for granted, and is in fact absent in = = σ2 = σ2 < many if not most living organisms, including notably (36). with the central point at a 1 3 and E 1. When E 1, as in σ2 > However, mammals do seem to possess specific mechanisms to Fig. 4C, we thus encounter two phases as a is varied; when E 1, as in Fig. 4D, three phases are traversed. enforce a separation; for instance, murine primordial germ cells A first conclusion from these results is that a population can be undergo resetting and erasing of maternal and paternal imprints, σ2 genome-wide DNA , extensive modifications, adapted to a fluctuating environment (here have optimal H and σ2 and inactive X-chromosome reactivation (18). As a very first step D) even though its individuals are not undergoing any change: this σ^2 = in trying to understand the origin of such mechanisms, it is in- is the case for the regions of the phase diagram where H 0, corresponding to a population with a homogeneous and constant structive to abstract from the many constraints that may limit the genotype. In this sense, adaptation of a population to a fluctuating evolution of systems of inheritance, and look for the way in which environment does not require variations in the attributes of the genotypic and phenotypic features should ideally be combined to individuals. However, an absence of evolution does not necessarily ensure a maximal growth rate of the population; if a “Weismann’s σ^2 = σ^2 > ” imply an absence of diversity. When H 0but D 0, the pop- barrier segregating a germ line from the soma is never found in ulation is phenotypically diverse, although rather than transmitted such conditions, this implies that its origin must reside elsewhere. from one generation to the next, the same diversity is reproduced at We follow here this approach by examining within model the se- each generation. Moreover, a population may also be adapted to lective value of a feedback from phenotype to transmitted genotype. a fluctuating environment without showing any diversity: for small Three factors potentially contribute to the genotype γ′ transmitted σ^2 = σ^2 = environment fluctuations, we find indeed that H 0and D 0 to the offsprings: the genotype γ inherited from the parent, the (Fig. 4E); in this case, natural selection favors the suppression of phenotype ϕ of the individual, and random variations νH .Toanalyze any variation. The diversity of the population, either genotypic or their relative adaptive value, we study the model depicted by Fig. 5, phenotypic, can be more precisely quantified within the model: which allows for different combinations of these three elements: ς ≡ E γ − 2 =σ2 = − α =α we thus have γ ½ð mtÞ S ð1 Þ ,forthegenotypic ς ≡ E ϕ − 2 =σ2 = − α =α + σ2 γ′ = λγ + ωϕ + ν ; ν ∼ ; σ2 ; diversity, and, ϕ ½ð mtÞ S ð1 Þ D for the pheno- H H N 0 H [23] ϕ = γ + ν ; ν ∼ ; σ2 : typic diversity. D D N 0 D Although new variations are beneficial when the fluctuations of the environment are large enough, our model predicts that natural λ γ ω ϕ σ2 Each factor is controlled by a parameter, for , for , and H selection should favor their introduction at different levels, for νH . We thus consider optimizing the long-term growth rate Λ depending on the statistical structure of these fluctuations. As over ðσ2 ; λ; ωÞ, for various values of ða; σ2 ; σ2 Þ, using the expres- σ^2 = H E D indicated in Fig. 4E, phenotypic variations are suppressed ð D 0Þ sion for Λ of the general model, with θ = 1, κ = 0, ρ = 0 [we con- when the environmental stochasticity is small enough or the en- sider λ ≥ 0 and ω ≥ 0; see also SI Appendix, Fig. S2, for an vironmental correlation large enough: this may be interpreted as alternative analysis where the optimization is performed over a selection for “canalization,” i.e., reduction of the phenotypic 2 discrete values, ðλ; ωÞ ∈ f0; 1g ]. diversity of genotypically identical individuals, a phenomenon in- The result of a numerical optimization of Λ are presented in deed observed in biological organisms (13). Genotypic variations σ^2 = Fig. 6. They show that a feedback from phenotype to trans- are suppressed ð H 0Þ, however, when the environment is not ω^ = strongly correlated (a small). This may be rationalized by noticing mitted genotype is prevented ð 0Þ in two limits: the limit of uncorrelated environments, τE → 0 ða → 0Þ, and the limit of that nontrivial inheritance is relevant only when successive gen- σ2 → ^λ erations share correlated selective pressures. deterministic environments E 0. In the first limit, vanishes The study can be extended to an environment undergoing as well; hence the absence of feedback does not imply an iso- = + ∼ ; σ2 lated germ line, but simply an absence of nontrivial heredity directed changes, xt ct bt with bt Nð0 EÞ (SI Appendix). In 2 ^ σ^2 > (for small a but large σ2 ,thesolutionσ^ > 0, λ = 0, ω^ = 0 indi- this case, we find that nonzero genotypic variations H 0 are E H always needed to keep up with the environmental changes, but cates that only noise is transmitted to the offsprings). The system σ^2 = of inheritance most reminiscent to Weismann’sscenariois a transition between phenotypic canalization ð D 0Þ and phe- σ^2 > obtained in the limit of constant environments τ ; σ2 → ∞; 0 notypic plasticity ð D 0Þ is still observed as the environmental ð E EÞ ð Þ σ2 ; σ2 → ; fluctuations E increase, or as the speed c of the environmental ½ða EÞ ð1 0Þ, suggesting that a separation of germplasm changes decreases, as summarized in Fig. 4F. from soma is beneficial only for those aspects of the phenotype Living organisms do not harbor a single trait, but many, each subject to nonfluctuating selective pressures (“housekeeping” potentially subject to a selective pressure with a different statistical genes, for instance).

6of10 | www.pnas.org/cgi/doi/10.1073/pnas.1323901111 Rivoire and Leibler Downloaded by guest on September 26, 2021 AB where we assume a mechanism for incorporating external signals, PNAS PLUS but question the way in which it is optimally “plugged in.” This approach allows us, without discussing the mechanisms them- selves, to compare the Darwinian and Lamarckian “modalities,” and test the conjecture that each of them is tuned to a different type of selective pressure (38). We thus compare two models, where the same information = + ′ ′ ∼ ; σ2 yt xt bt with bt Nð0 I Þ is available, but where it is either processed at the phenotypic level (model P for “plasticity,” for σ2 = σ2 which I Ip; Fig. 7A), or at the genotypic level (model L for “ ” σ2 = σ2 Lamarckism, for which I Iℓ; Fig. 7B). Formally, model P is C D described by the following: γ′ = λγ + ν ; ν ∼ ; σ2 ; H H N 0 H [24] ϕ = θγ + ρ + ν ; ν ∼ ; σ2 ; yt D D N 0 D

thus corresponding to the general model with ω = 0 and κ = 0, whereas model L is described by the following: γ′ = λγ + κ + ν ; ν ∼ ; σ2 ; yt H H N 0 H [25] ϕ = θγ + ν ; ν ∼ ; σ2 ; E F D D N 0 D thus corresponding to the general model with ω = 0 and ρ = 0. We consider optimizing the long-term growth rate Λ over the parameters that control the contributions of each factor: ðθ; λ; ρÞ for model P, and ðθ; λ; κÞ for model L (considering here again only positive values of these parameters).

The results of a comparison between the two models are BIOPHYSICS AND ; σ2 shown in Fig. 8, where, as a function of ða EÞ and for different COMPUTATIONAL BIOLOGY σ2; σ2 ; σ2 values of the fixed parameters I H D, we present which of the two models, P or L, yields the highest growth rate. The main Fig. 4. Where to introduce variation? (A) For the model described by Eq. 21 σ2 controlling parameter appears to be the correlation a (or and depicted in Fig. 3, optimal values of the degree H of genotypic variations τ σ2 equivalently E) of the environmental fluctuations, with the as a function of the environmental variables ða, E Þ when optimizing jointly σ2 σ2 σ2 Lamarckian modality systematically becoming more favorable over ð H, DÞ.(B) Optimal values of the degree D of phenotypic variations as

σ2 σ2 σ2 when this correlation is large, in line with the intuition that PHYSICS a function of ða, E Þ when optimizing jointly over ð H, DÞ.(C) Optimal values σ2 σ2 σ2 = = σ2 σ2 transmitting acquired information is beneficial when the se- of ð H, DÞ as a function of a for E 1 2. (D) Optimal values of ð H, DÞ as σ2 = lective pressure experienced by the offspring is sufficiently a function of a for E 2. (E) Phase diagram for the optimal nonzero values of σ2 σ2 σ2 ð H, DÞ as a function of ða, E Þ.(F) Same phase diagram for an environment similar to that experienced by the parents. Note that this = + ∼ σ2 undergoing directed changes, xt ct bt with bt Nð0, EÞ, in which case the simple conclusion conceals in fact a much richer diversity of σ2 environmental parameters are ðc, E Þ. strategies, revealed by considering the values of the parame- ters optimizing the two models (SI Appendix, Fig. S3).

Where to Acquire Information? In the two previous models, the Connections role of the environment is confined to selection, and any new Of the many studies that share part of our intents or methods, variation is introduced independently of the environmental two lines of work stand out: (i) a different approach, based on state. Examples, however, abound of living organisms generat- Price’s equation, has been proposed with the same goal of ing new traits that are correlated with the environment. It is thus uniting the various forms of inheritance within a common well recognized that the current phenotype of an individual is mathematical framework (39, 40); (ii) a different problem, affected not only by the genotype received from its parents but pertaining to control in engineering, has been solved using also by the environment in which it develops; such a variation, a closely related mathematical framework (24). We discuss which may be adaptive, is generally referred to as “plasticity.” here the relations between our model and these two lines As already mentioned in the introduction, the question of of work. whether environmentally induced variations can be transmitted to the progeny was, and still remains, a subject of hot debates. Link to Price Equation. Price equation is a general formula, appli- This possibility was central to Lamarck’s theory of evolution, cable to any model of population dynamics, which uses a covariance and is often referred to as “Lamarckism.” Evolution by natural formalism to express the change in the mean value of a trait be- selection does not require it, but it does not exclude it. Several tween successive generations (41). Being a mathematical identity, it examples of environmentally induced traits have indeed been observed. Proving that such traits confer a selective advantage is generally delicate, but a particularly striking example is pro- vided, for instance, by the bacterial called CRISPR (37); this system relies on the insertion of phage-specific sequences into bacterial genomes and has been shown to protect against phage infection the bacteria that inherit them from their parent. This example implies a specific mechanism for incor- porating and exploiting the environmental “signal” (here, the presence of a particular strain of phage). The constraints to which Fig. 5. When to separate phenotype and transmitted genotype? This the evolution of such mechanisms are subject are determining but question is addressed within a model where the heredity kernel Hðγ′jγ,ϕÞ is λ ω σ2 potentially nongeneric, and, in any case, difficult to model. Here, optimized. The optimization is made over the three parameters , , H, γ′ γ ϕ γ′ = λγ + ωϕ + ν ν ∼ σ2 we consider a thought experiment (or Gedankenexperiment), which define Hð j , Þ by the relation H with H Nð0, HÞ.

Rivoire and Leibler PNAS Early Edition | 7of10 Downloaded by guest on September 26, 2021 A

B

Fig. 6. When to separate phenotype and trans- mitted genotype? (A) For the model described by Eq. σ2 λ ω 23 and depicted in Fig. 5, the values of ð H, , Þ that jointly optimize Λ are represented as a function of σ2 the environmental parameters ða, E Þ for a fixed σ2 = developmental noise D 1. (B) Same results pre- σ2 sented as a function of a for three fixed values of E .

necessarily holds true. Its virtue is to provide a decomposition of be made based on two potentially conflicting sources: the past evolutionary change that can illuminate its origins; initially derived states of the system, and the signals from the sensors. A classical for models of cooperation, it has contributed to clarify the nature of algorithm for solving this problem is the Kalman filter (24). Maybe group/kin selection (42). Applied in many other contexts, it has also not surprisingly, it involves the same essential mathematical been used as a general mathematical framework for studying the ingredients that make our model solvable: linearity and Gaus- different possible modes of inheritance (39, 40). As for any other sianity. We present here a limit case of our model where the two model of population dynamics, a Price equation can be written for approaches formally coincide, thus revealing that the scope of the our model (SI Appendix). concepts of inheritance extends beyond the study of biological Price equation is, however, limited to a short-term description organisms. of the dynamics: as it considers only the mean values of traits, Control in engineering typically involves a single system, it cannot be iterated to describe changes in subsequent genera- γ rather than a population of diverse individuals. We thus obtain tions; this indeed requires the full distribution ntð Þ. As illus- a formal analogy with the Kalman filter only when the pop- trated in the previous sections, systems of inheritance may, ulation is perfectly homogeneous, and described by a single however, have qualitatively different implications in environ- “state” γ. This corresponds in our model to a limit where no ments with different statistical structures. Only a long-term developmental noise is present, ϕ = γ (i.e., θ = 1, ρ = 0, σ2 = 0), analysis of the dynamics of a population can thus fully reveal D and where the environment is perfectly selective, σ2 = 0, so their evolutionary properties. In our approach, this is achieved S that each and every individual has a common γ = xt.Inthis by coupling Price equation, the recursion over the mean mt of ^ 9 σ2 limit case, the heredity kernel H that optimizes Λ can be the trait, Eq. , with a recursion over the variance t of the trait, Eq. 10, and by considering their asymptotic properties, which computed exactly for any stationary Markovian environmental σ2 process (not necessarily Gaussian). consist of a fixed point for t and a stationary distribution for mt. Within the Gaussian assumptions that define our model, these First, assuming that no external information is available, ξ; γ′ γ; = ξ γ′; γ′ γ two quantities are sufficient to fully capture the population dy- Pwith a model described by Að j xtÞ Rð j xtÞHð j Þ where ξ ξ γ′; = rmax δ γ′ − namics. More importantly, our formalism involves a central quantity ξ Rð j xtÞ e ð xtÞ (note the slight difference with 6 not present in covariance formalisms, the Lyapunov exponent Λ. Eq. ), and denoting by Pðxt+1jxtÞ the stochastic kernel defining Because this quantity decides the eventual fate of two com- the environmental process, we have the following: peting populations, it allows us to associate with each scheme 1 of inheritance an adaptive value, and thus to derive the con- Λ = rmax + lim E½Pðxt+1jxtÞln Hðxt+1jxtÞ ditions under which a given scheme confers a selective advantage t→∞ t [26] over the others. = rmax − HðPÞ − DðPkHÞ;

Link to the Kalman Filter. At the core of adaptation is the problem where HðPÞ denotes the entropy rate of the process P,and of anticipating the next state of the environment. From this per- DðPkHÞ the rate of relative entropy of P with respect to H (23). spective, different systems of inheritance can be considered as [They are defined by HðPÞ = −lim →∞ E½Pðx + jx Þln Pðx + jx Þ=t different ways to share with the current generation the “knowl- t t 1 t t 1 t P H = →∞ E P x + x P x + x =H x + x =t edge” accumulated, through natural selection, by the previous and Dð k Þ limt ½ ð t 1j tÞln ð t 1j tÞ ð t 1j tÞ (43).] The later verifies DðPkHÞ≥0, with DðPkHÞ = 0 if and only if generations. In a stochastically fluctuating environment, no system = Λ ^ = of inheritance can, however, perform better than direct sensing P H (43). The growth rate is therefore minimal for H P, of the present environment by the individual that experiences it. an instance of the so-called proportional betting strategy (44), Sensors, when not altogether absent, are generically imperfect. which consists in matching the stochasticity of the environ- The individual with such an imperfect access to its current envi- ment. In particular, when taking a Gaussian environment with = − ^ γ′ γ = γ′ − γ ronment thus faces a dilemma: it has to arbitrate between two Pðxt+1jxtÞ Gσ2 ðxt+1 axtÞ,weobtainHð j Þ Gσ2 ð a Þ,cor- X ^λ = σ^2 = σ2 X valuable but unreliable sources of information, the germplasm γ responding to a and H X [Gσ2 (x) denotes a Gaussian σ2 inherited from the parent, and the cues yt or zt that it gained from function with zero mean and variance ]. its own perception. Interestingly, the very same dilemma is en- Assuming now that some information yt is available, which is = + ′ ′ ∼ ; σ2 countered in various problems of engineering, such as for instance derived from xt as yt cI xt bt with bt Nð0 I Þ, we can extend the automatic guidance of aircrafts, where decisions must also this result to a model with Aðξ; γ′jγ; xtÞ = Rðξjγ′; xtÞHðγ′jγ; ytÞ.

8of10 | www.pnas.org/cgi/doi/10.1073/pnas.1323901111 Rivoire and Leibler Downloaded by guest on September 26, 2021 γ PNAS PLUS A of the system and the newly acquired information yt must be linearly combined to optimally define the subsequent state γ′ of the system. The formal correspondence with the solution to the Kalman σ2 → filter holds only in the limit of infinite selectivity S 0, where the population is perfectly homogeneous at every time. This B limit, where the optimal scheme for processing information fol- lows the Bayesian principles, is also the limit in which the value of the information yt can be quantified by the usual concepts of information theory (23). We may thus view our model as a gen- eralization of the problem of stochastic control encountered in engineering by incorporating biological features that are absent in this context, notably a diverse and growing population, and a distinction between genotype and phenotype. Reciprocally, the Fig. 7. Where to acquire information? This question is addressed by com- = + ′ Kalman filter has been extended along several lines since its paring two models in presence of the same information yt xt bt with ′ ∼ σ2 original formulation (46), and the mathematical formalisms thus bt Nð0, I Þ.(A) Model P, described by Eq. 24, where the information yt is incorporated to the phenotype and where the growth rate Λ is optimized developed may suggest ways along which generalizations of our with respect to the three parameters θ, λ, ρ, which define Dðϕjγ,yt Þ and model could be analyzed. γ′ γ ϕ = θγ + ρ + ν γ′ = λγ + ν ν ∼ σ2 Hð j Þ by the relations yt D and H with D Nð0, DÞ and ν ∼ σ2 σ2 σ2 Conclusion H Nð0, HÞ, and fixed D, H.(B) Model L, described by Eq. 25, where the information yt is incorporated to the transmitted genotype and where the Classical models of population genetics take the mechanisms for growth rate Λ is optimized with respect to the three parameters θ, λ, κ, generating and transmitting new traits as given. Several previous ϕ γ γ′ γ ϕ = θγ + ν γ′ = which define Dð j Þ and Hð j ,yt Þ by the relations D and studies have extended these models to analyze how the mechanisms λγ + κ + ν ν ∼ σ2 ν ∼ σ2 σ2 σ2 yt H with D Nð0, DÞ and H Nð0, HÞ, and fixed D, H. of inheritance may themselves evolve, starting from works on the evolution of mutation rates (27) and including, among several other examples, studies of maternal effects (47), nongenetic = κ σ2 (We previously assumed cI 1; and I can indeed always be inheritance (48), plasticity and memory (49), and relationship rescaled to be in this case. We introduce here cI only to make quantitative trait loci (50). Here, we proposed a simple model the correspondence with the Kalman filter where this rescaling to compare the adaptive value of different schemes for gen- BIOPHYSICS AND

is generally not assumed.) This model, which corresponds to a erating and transmitting variations in populations. Its analysis COMPUTATIONAL BIOLOGY continuous version of the model studied in ref. 23, was pre- indicates that different modalities of inheritance are favored P depending on the statistical structure of the fluctuations of the ξ R ξ γ′; x = viously analyzed in ref. 45. In the limit where ξ ð j tÞ environment. For an organism with various traits, each po- rmax δ γ′ − Λ e ð xtÞ, the growth rate is optimized by following a tentially subject to a different selective pressure, this analysis Bayesian strategy: suggests that multiple inheritance systems operating in par-

^ allel may be selected for, consistently with observations. PHYSICS H γ′jγ; y = P ; γ′jγ; y ; [27] t Xt + 1jXt Yt t Our model is schematic but captures a key feature of evolutionary dynamics: information can be transmitted between generations along where P ; ðx + jx ; y Þ denotes the conditional probability of Xt + 1jXt Yt t 1 t t different canals, and not only does the topology of these canals affect having the environmental random variable Xt+1 taking the value + the dynamics, but this topology can be itself subject to selection. xt+1 at time t 1 given that it took the value xt at time t,andthat Certainly, the model does not encompass the full diversity of possible the population observed yt at this time. With a Gaussian environ- modes of inheritance, but it can still be extended along several lines P + = 2 + − ment and Gaussian noisy channel, Xt + 1jXt ðxt 1jxtÞ Gσ ðxt 1 axtÞ while retaining its analytical tractability. For instance, rather than ^ X P = 2 − γ′ γ; and YtjXt ðytjxtÞ Gσ ðyt cI xtÞ, this yields for Hð j ytÞ: combining the inherited and the acquired information into a single I σ2 σ2 attribute, it could include two channels of transmission, one for the a I cI X − −1 γ′ − γ − : [28] germ line and another for somatic elements, as for instance in ref. 47; G σ 2 + σ−2 yt ðð I cI Þ Þ σ2 + 2σ2 σ2 + 2σ2 X I cI X I cI X because Gaussian formulae extend to multidimensional variables, the model remains indeed solvable when considering the generation and In this case, it is thus proved that the optimal form of the heritability transmission of multiple traits. It can similarly be extended to account ^λ = σ2= σ2 + 2σ2 kernel H is Gaussian. The two coefficients a I ð I cI X Þ and for multiple timescales, for instance by introducing a temporal delay ^κ = σ2 = σ2 + 2σ2 cI X ð I cI X Þ correspond exactly to the gains for the Kal- between the developmental stage and the time of reproduction man filter (24); in this context, they prescribe how the previous state (formally yt = xt−τ,for0< τ < 1). Different environmental processes

Fig. 8. Where to acquire information? Boundary ^ ^ between Λðκ = 0Þ > Λðρ = 0Þ, when acquiring infor- mation at the phenotypic level is more beneficial, “ ” Λ^ κ = < Λ^ ρ = 2 indicated by P, and ð 0Þ ð 0Þ, when ac- quiring information at the genotypic level is more beneficial, indicated by “L,” for all eight combina- σ2 σ2 σ2 ∈ 3 σ2 σ2 tions of ð H, D, I Þ f1, 10g . I represents Ip for σ2 model P and Iℓ for model L.

Rivoire and Leibler PNAS Early Edition | 9of10 Downloaded by guest on September 26, 2021 can also be analyzed, which do not need to be Gaussian for the model Despite these limitations, we hope that our approach may be to be solvable; e.g., adding a linearly varying component to the en- of value for providing theoretical limits to the evolution of vironment (as in Fig. 4F) provides a framework for studying the systems of inheritance, in the spirit of the theoretical limits that implications of different modes of adaptations to the survival of Shannon derived for the communication of signals over noisy a population facing a directed change of selective pressure (51). channels, after similarly abstracting from practical costs and The model is abstracted from material implementations and, constraints (17). As shown in our previous work (23), the sim- in particular, does not refer to the genetic or nongenetic nature ilarity between the two problems extends beyond the mere of what is transmitted. It cannot, therefore, account for the con- straints and costs that the evolution of any specific mechanism analogy: the fundamental quantities of information theory are for generating and transmitting variations must face. The model recovered as a limit of our model. We exposed here, in the same can certainly be extended to include such costs, both constitutive limit, another formal analogy, with the solution to the Kalman (attached to the mechanisms) or inductive (stemming from their filter used in stochastic control (24). The model presented in use), but only at the price of introducing new, ad hoc parameters. this paper thus provides a versatile, analytically tractable frame- This limitation for example prevents us from making a meaningful work for clarifying and unifying common issues and concepts in comparison of the relative benefit of “selection” versus “instruc- population genetics, information theory, and stochastic control, tion,” because selection results automatically from the interaction which may contribute to stimulate further crossbreeding between with the environment, whereas a specific mechanism is needed to these disciplines. acquire and incorporate environmental signals. More generally, the model does not account for the fact that a reliable hereditary ACKNOWLEDGMENTS. We thank B. Chazelle, B. Houchmandzadeh, D. Huse, mechanism must precede the evolution of a Lamarckian mecha- D. Jordan, and S. Kuehn for comments. This work was supported by Agence nism, if this mechanism is to be faithfully transmitted. Nationale de la Recherche, Grant CoevolInterProt (to O.R.).

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