A model for the generation and transmission of PNAS PLUS variations in evolution 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 Biology, 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 natural selection. 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 adaptive value of different schemes of inheritance. Our model dition to the well-recognized roles of mutations 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 chromatin marks such as DNA methyla- mitted or not during reproduction. The adaptation of the different tion, the transmission of small interfering RNAs, 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 gene regulation, and, at the organismal level, so-called parental effects (16). titative genetics 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 | Lamarckism | heredity | acquired characteristics | to evolutionary theory, thus appears to consist of multiple and epigenetics 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 adaptations 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“evolvability.” 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 phenotype to “neo-Darwinism” (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 phenotypes, 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 organisms, but in evolving populations may be favored or suppressed by subsequent experiments with microorganisms, 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 antibody 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 nature 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 population genetics, 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