Somatic Selection and Adaptive Evolution: On the Inheritance of Acquired Characters

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However, an absence of evolution does not necessarily imply an absence of diversity. When but , the population is phenotypically diverse, although rather than transmitted from one generation to the next, the same diversity is reproduced at each generation. Moreover, a population may also be adapted to a fluctuating environment without showing any diversity: for small environment fluctuations, we find indeed that and Fig. The diversity of the population, either genotypic or phenotypic, can be more precisely quantified within the model: we thus have , for the genotypic diversity, and, for the phenotypic diversity.

Although new variations are beneficial when the fluctuations of the environment are large enough, our model predicts that natural selection should favor their introduction at different levels, depending on the statistical structure of these fluctuations. As indicated in Fig. Genotypic variations are suppressed , however, when the environment is not strongly correlated a small. This may be rationalized by noticing that nontrivial inheritance is relevant only when successive generations share correlated selective pressures. The study can be extended to an environment undergoing directed changes, with SI Appendix.

Acquired characteristics

In this case, we find that nonzero genotypic variations are always needed to keep up with the environmental changes, but a transition between phenotypic canalization and phenotypic plasticity is still observed as the environmental fluctuations increase, or as the speed c of the environmental changes decreases, as summarized in Fig. Living organisms do not harbor a single trait, but many, each potentially subject to a selective pressure with a different statistical structure.

For instance, in bacteria, the strength of selection may be very different between central metabolism 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 transmitted. However, our model is obviously extremely schematic and does not account for a number of features that affect the evolution of mechanisms of inheritance.

In particular, it does not consider the cost of these mechanisms, which may strongly limit their actual diversity: suppressing any variation by error corrections, checkpoints, canalization, etc. This separation, however, cannot be taken for granted, and is in fact absent in many if not most living organisms, including notably plants However, mammals do seem to possess specific mechanisms to enforce a separation; for instance, murine primordial germ cells undergo resetting and erasing of maternal and paternal imprints, genome-wide DNA methylation, extensive histone modifications, and inactive X-chromosome reactivation We follow here this approach by examining within model the selective value of a feedback from phenotype to transmitted genotype.

To analyze their relative adaptive value, we study the model depicted by Fig. S2 , for an alternative analysis where the optimization is performed over discrete values, ]. When to separate phenotype and transmitted genotype? This question is addressed within a model where the heredity kernel is optimized. They show that a feedback from phenotype to transmitted genotype is prevented in two limits: the limit of uncorrelated environments, , and the limit of deterministic environments.

In the first limit, vanishes as well; hence the absence of feedback does not imply an isolated germ line, but simply an absence of nontrivial heredity for small a but large , the solution , , indicates that only noise is transmitted to the offsprings. B Same results presented as a function of a for three fixed values of.

In the two previous models, the role of the environment is confined to selection, and any new variation is introduced independently of the environmental state. Examples, however, abound of living organisms generating new traits that are correlated with the environment.

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Several 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 provided, for instance, by the bacterial immune system 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.

The constraints to which the evolution of such mechanisms are subject are determining but potentially nongeneric, and, in any case, difficult to model. Formally, model P is described by the following: thus corresponding to the general model with and , whereas model L is described by the following: thus corresponding to the general model with and.

Where to acquire information? This question is addressed by comparing two models in presence of the same information with. A Model P, described by Eq. B Model L, described by Eq. The results of a comparison between the two models are shown in Fig. The main controlling parameter appears to be the correlation a or equivalently of the environmental fluctuations, with the Lamarckian modality systematically becoming more favorable when this correlation is large, in line with the intuition that transmitting acquired information is beneficial when the selective pressure experienced by the offspring is sufficiently similar to that experienced by the parents.

Note that this simple conclusion conceals in fact a much richer diversity of strategies, revealed by considering the values of the parameters optimizing the two models SI Appendix , Fig.

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We discuss here the relations between our model and these two lines of work. Price equation is a general formula, applicable to any model of population dynamics, which uses a covariance formalism to express the change in the mean value of a trait between successive generations Being a mathematical identity, it necessarily holds true.

Applied in many other contexts, it has also been used as a general mathematical framework for studying the different possible modes of inheritance 39 , As for any other model of population dynamics, a Price equation can be written for our model SI Appendix. Price equation is, however, limited to a short-term description of the dynamics: as it considers only the mean values of traits, it cannot be iterated to describe changes in subsequent generations; this indeed requires the full distribution. As illustrated in the previous sections, systems of inheritance may, however, have qualitatively different implications in environments with different statistical structures.

Only a long-term analysis of the dynamics of a population can thus fully reveal their evolutionary properties. In our approach, this is achieved by coupling Price equation, the recursion over the mean of the trait, Eq. Within the Gaussian assumptions that define our model, these two quantities are sufficient to fully capture the population dynamics.

Because this quantity decides the eventual fate of two competing populations, it allows us to associate with each scheme of inheritance an adaptive value, and thus to derive the conditions under which a given scheme confers a selective advantage over the others. At the core of adaptation is the problem of anticipating the next state of the environment. In a stochastically fluctuating environment, no system of inheritance can, however, perform better than direct sensing of the present environment by the individual that experiences it.

Sensors, when not altogether absent, are generically imperfect. Interestingly, the very same dilemma is encountered in various problems of engineering, such as for instance the automatic guidance of aircrafts, where decisions must also be made based on two potentially conflicting sources: the past states of the system, and the signals from the sensors. A classical algorithm for solving this problem is the Kalman filter Maybe not surprisingly, it involves the same essential mathematical ingredients that make our model solvable: linearity and Gaussianity.

We present here a limit case of our model where the two approaches formally coincide, thus revealing that the scope of the concepts of inheritance extends beyond the study of biological organisms. Control in engineering typically involves a single system, rather than a population of diverse individuals. This corresponds in our model to a limit where no developmental noise is present, i. First, assuming that no external information is available, with a model described by where note the slight difference with Eq.

Assuming now that some information is available, which is derived from as with , we can extend this result to a model with. We introduce here only to make the correspondence with the Kalman filter where this rescaling is generally not assumed. This model, which corresponds to a continuous version of the model studied in ref. With a Gaussian environment and Gaussian noisy channel, and , this yields for : In this case, it is thus proved that the optimal form of the heritability kernel H is Gaussian. The formal correspondence with the solution to the Kalman filter holds only in the limit of infinite selectivity , where the population is perfectly homogeneous at every time.

This limit, where the optimal scheme for processing information follows the Bayesian principles, is also the limit in which the value of the information can be quantified by the usual concepts of information theory We may thus view our model as a generalization 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 Kalman filter has been extended along several lines since its original formulation 46 , and the mathematical formalisms thus developed may suggest ways along which generalizations of our model could be analyzed.

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Classical models of population genetics take the mechanisms for generating and transmitting new traits as given. Several previous studies have extended these models to analyze how the mechanisms 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 inheritance 48 , plasticity and memory 49 , and relationship quantitative trait loci Here, we proposed a simple model to compare the adaptive value of different schemes for generating and transmitting variations in populations.

Its analysis indicates that different modalities of inheritance are favored depending on the statistical structure of the fluctuations of the environment. For an organism with various traits, each potentially subject to a different selective pressure, this analysis suggests that multiple inheritance systems operating in parallel may be selected for, consistently with observations. Our model is schematic but captures a key feature of evolutionary dynamics: information can be transmitted between generations along different canals, and not only does the topology of these canals affect the dynamics, but this topology can be itself subject to selection.

Certainly, the model does not encompass the full diversity of possible modes of inheritance, but it can still be extended along several lines while retaining its analytical tractability. For instance, rather than combining the inherited and the acquired information into a single attribute, it could include two channels of transmission, one for the germ line and another for somatic elements, as for instance in ref.

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It can similarly be extended to account for multiple timescales, for instance by introducing a temporal delay between the developmental stage and the time of reproduction formally , for. Different environmental processes can also be analyzed, which do not need to be Gaussian for the model to be solvable; e. The model is abstracted from material implementations and, in particular, does not refer to the genetic or nongenetic nature of what is transmitted. It cannot, therefore, account for the constraints and costs that the evolution of any specific mechanism for generating and transmitting variations must face.

The model can certainly be extended to include such costs, both constitutive attached to the mechanisms or inductive stemming from their use , but only at the price of introducing new, ad hoc parameters. More generally, the model does not account for the fact that a reliable hereditary mechanism must precede the evolution of a Lamarckian mechanism, if this mechanism is to be faithfully transmitted. Despite these limitations, we hope that our approach may be of value for providing theoretical limits to the evolution of systems of inheritance, in the spirit of the theoretical limits that Shannon derived for the communication of signals over noisy channels, after similarly abstracting from practical costs and constraints As shown in our previous work 23 , the similarity between the two problems extends beyond the mere analogy: the fundamental quantities of information theory are recovered as a limit of our model.


We exposed here, in the same limit, another formal analogy, with the solution to the Kalman filter used in stochastic control The model presented in this paper thus provides a versatile, analytically tractable framework for clarifying and unifying common issues and concepts in population genetics, information theory, and stochastic control, which may contribute to stimulate further crossbreeding between these disciplines.

We thank B. Chazelle, B.