On the Feasibility of Saltational Evolution
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On the feasibility of saltational evolution Mikhail I. Katsnelsona,1, Yuri I. Wolfb, and Eugene V. Kooninb,1 aInstitute for Molecules and Materials, Radboud University, 6525AJ Nijmegen, The Netherlands; and bNational Center for Biotechnology Information, National Library of Medicine, Bethesda, MD 20894 Contributed by Eugene V. Koonin, August 28, 2019 (sent for review May 28, 2019; reviewed by Christoph Adami and Claus O. Wilke) Is evolution always gradual or can it make leaps? We examine a in the landscape, for example to the slope of a different, mathematical model of an evolutionary process on a fitness land- higher peak, via simultaneous fixation of multiple mutations scape and obtain analytic solutions for the probability of multimutation (Fig. 1). leaps, that is, several mutations occurring simultaneously, within a We sought to obtain analytically, within the population single generation in 1 genome, and being fixed all together in the genetics framework, the conditions under which multimuta- evolving population. The results indicate that, for typical, empir- tional leaps might be feasible. The results suggest that, under ically observed combinations of the parameters of the evolution- most typical parameters of the evolutionary process, leaps ary process, namely, effective population size, mutation rate, and distribution of selection coefficients of mutations, the probabil- cannot be fixed. However, taking sign epistasis into account, we ity of a multimutation leap is low, and accordingly the contribu- show that saltational evolution could become relevant under tion of such leaps is minor at best. However, we show that, conditions of elevated mutation rate under stress so that stress- taking sign epistasis into account, leaps could become an induced mutagenesis could be considered an evolvable adaptation important factor of evolution in cases of substantially elevated strategy. mutation rates, such as stress-induced mutagenesis in microbes. We hypothesize that stress-induced mutagenesis is an evolvable Results adaptive strategy. Multimutation Leaps in the Equilibrium Regime. Let us assume (binary) genomes of length L (in the context of this analysis, L fitness landscape | stress-induced mutagenesis | epistasis | should be construed as the number of evolutionarily relevant EVOLUTION purifying selection | positive selection sites, such as codons in protein-coding genes, rather than the total number of sites), the probability of single mutation μ << venerable principle of natural philosophy, most consistently 1 per site per round of replication (generation), and constant Apropounded by Leibnitz (1) and later embraced by prom- effective population size Ne >> 1. Then, the transition inent biologists, in particular Linnaeus (2), is “natura non facit i j saltus” “ ” probability from sequence to sequence is (equation 3.11 in ( nature does not make leaps ). This principle then be- ref. 15) came one of the key tenets of Darwin’s theory that was inherited by the modern synthesis of evolutionary biology. In evolutionary hij L−hij qij = μ ð1 − μÞ , [1] biology, the rejection of saltation takes the form of gradualism, that is, the notion that evolution proceeds gradually, via accu- where hij is the Hamming distance (number of different sites mulation of “infinitesimally small” heritable changes (3, 4). between the 2 sequences). The number of sequences separated However, some of the most consequential evolutionary changes, by the distance h is equal to the number of ways h sites can be such as, for example, the emergence of major taxa, seem to selected from L, that is, occur abruptly rather than gradually, prompting hypotheses on the importance of saltational evolution, for example by Goldschmidt (“hopeful monsters”) and Simpson (“quantum Significance evolution”). Subsequently, these ideas have received a more systematic, even if qualitative, treatment in the concepts of In evolutionary biology, it is generally assumed that evolution punctuated equilibrium (5, 6) and evolutionary transitions occurs in the weak mutation limit, that is, the frequency of (7, 8). multiple mutations simultaneously occurring in the same genome Within the framework of modern evolutionary biology, and the same generation is negligible. We employ mathematical gradualism corresponds to the weak-mutation limit, that is, modeling to show that, although under the typical parameter an evolutionary regime in which mutations occur one by one, values of the evolutionary process the probability of multi- consecutively, such that the first mutation is assessed by se- mutational leaps is indeed low, they might become substantially lection and either fixed or purged from the population, be- more likely under stress, when the mutation rate is dramatically fore the second mutation occurs (9). A radically different, elevated. We hypothesize that stress-induced mutagenesis in saltational mode of evolution (10, 11) is conceivable under microbes is an evolvable adaptive strategy. Multimutational the strong-mutation limit (9) whereby multiple mutations leaps might matter also in other cases of substantially increased occurring within a single generation and in the same genome mutation rate, such as growing tumors or evolution of primordial potentially could be fixed all together. Under the fitness replicators. landscape concept (12, 13), gradual or more abrupt evolu- Author contributions: M.I.K. and E.V.K. designed research; M.I.K. and Y.I.W. performed tionary processes can be depicted as distinct types of trajec- research; M.I.K., Y.I.W., and E.V.K. analyzed data; and M.I.K. and E.V.K. wrote tories on fitness landscapes (Fig. 1). The typical evolutionary the paper. paths on such landscapes are thought to be 1 step at a time, Reviewers: C.A., Michigan State University; and C.O.W., The University of Texas at Austin. uphill mutational walks (12). In small populations, where The authors declare no competing interest. genetic drift becomes an important evolutionary factor, the This open access article is distributed under Creative Commons Attribution-NonCommercial- likelihood of downhill movements becomes nonnegligible NoDerivatives License 4.0 (CC BY-NC-ND). (14). In principle, however, a different type of moves on 1To whom correspondence may be addressed. Email: [email protected] or fitness landscapes could occur, namely, leaps (or “flights”) [email protected]. across valleys when a population can move to a different area www.pnas.org/cgi/doi/10.1073/pnas.1909031116 PNAS Latest Articles | 1of8 Downloaded by guest on September 24, 2021 AB CD Fig. 1. Walks and leaps on different types of fitness landscapes. Dots show genome states; blue (shirt straight) arrows indicate consecutive moves via fixation of single mutations; red (long curved) arrows indicate multimutation leaps. (A) Nearly neutral landscape. (B) Landscape dominated by slightly deleterious mutations. (C)Kimura’s model landscape (a fraction of mutations is neutral; the rest are lethal). (D) Landscape combining beneficial and deleterious mutations. L! Lh epistasis, that is, with additive fitness effects of individual N = ≈ [2] mutations: h ðL − hÞ!h! h! , ΔðhÞ = y1 + y2 + ... + yh, [6] where the last, approximate expression is valid under the assump- L >> L >> h h tion that 1and ( can be of the order of 1). where yi are independent random variables with the distribution func- μ << Assuming also 1, we obtain a typical combinatorial prob- tions GjðyjÞ. Then, the distribution function of the fitness difference is h ability of leaps over the distance : ! Y Z X h −Lμ ðLμÞ e ρ ðΔÞ = dyjGj yj δ yj − Δ QðhÞ ∼ N qðhÞ = P ðLμÞ ≡ [3] h h h h! , j j Z∞ Z [7] dk Y which is a Poisson distribution with the expectation Lμ. = e−ikΔ dy eikyj G y π j j j , i 2 j In steady state, the probability of fixation of the state is −∞ proportional to expð−νxiÞ where which is obtained by using the standard Fourier transformation ν = N − ð Þ [4] e 1 Moran process of the delta function. Now, let us specify the distribution of the fitness effects of ν = 2ðNe − 1Þ ðhaploid Wright-Fisher processÞ mutations GiðyiÞ, assuming an exponential dependency of the probability of a mutation on its fitness effect, separately for ν = 2Ne − 1 ðdiploid Wright-Fisher processÞ beneficial and deleterious mutations: x = − f f i x D e−eiyi y > and i ln i where i is the fitness of the genotype [ i is P ðy Þ = i , i 0 [8] i i eiyi , analogous to energy in the Boltzmann distribution within the Dirie , yi < 0 analogy between population genetics and statistical physics (16)]. For other demographic structures and assumptions on the mu- where Di is the normalization factor, ri is the ratio of the probabil- tation process, the relationship between fixation probability can ities of beneficial and deleterious mutations, and ei is the inverse of quantitatively differ while retaining the same form. In particular, the characteristic fitness difference for a single mutation (discussed for a population that produces offspring by binary division (fis- below). For simplicity, we assume here the same decay rates for the sion), ν ≈ 4Ne (17, 18). probability density of the fitness effects of beneficial and deleterious mutations. Empirical data on the distributions of fitness effects of Then, the rate of the occurrence and fixation of the transition r i → j is (15) mutations (19, 20) clearly indicate that i 1. From the normali- zation condition, À Á ν x − x W = q  À i jÁà [5] ei ij ij ν x − x − . Di = ≈ ei. [9] exp i j 1 1 + ri The distribution function of the fitness differential Δij = xi − xj Note that the mean of the fitness difference (selection co- has to be specified (hereafter, we refer to x as fitness, omit- efficient) when the distribution of the fitness effects is given ting logarithm for brevity). We analyze first the case without by 8 is 2of8 | www.pnas.org/cgi/doi/10.1073/pnas.1909031116 Katsnelson et al. Downloaded by guest on September 24, 2021 Z 1 A10 B jsij = dyiGiðyiÞyi ≈ .