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Home , Natura non facit saltus, Saltation (biology)

On the feasibility of saltational

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 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 , and being fixed all together in the 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, 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 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) 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 , 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 ( 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 . In evolutionary hij L−hij qij = μ ð1 − μÞ , [1] biology, the rejection of takes the form of , 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 , it is generally assumed that evolution (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. 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

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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, ν x − x W = q i j [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 ≈ . [10] 10 ei =10 = 100

8 8 For simplicity, we start with an assumption that the values of Di and ri are independent of i. For the model 8 6 6 Z∞ iky 1 r dyGðyÞe = iD − [11] 4 k + ie k − ie . 4 −∞ 2 2 Then, from Eq. 5, the fixation rate of an h-mutation leap is equal to -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Z∞ νΔ φðhÞ = dΔ ρ ðΔÞ [12] νΔ h . Fig. 2. Rates of leaps on a landscape dominated by deleterious mutations. e − 1 μ −∞ Rates of transitions are plotted against the per-genome mutation rate (L ) and the leap length for different strengths of selection (A: νjsj = 10 and B: νjsj = 100). Contour lines indicate orders of magnitude and start from the Substituting 11 into 7, we obtain − rate of 10 5 leaps per generation. " # h+1 h h−1 h i e d k + ie −ikΔ ρhðΔÞ = − 1 − r e , Δ > 0 ðh − Þ!ð + rÞh dkh−1 k − ie Z∞ 1 1 k=−ie dk Y ie ρ ðΔÞ = e−ikΔ j [16] h π ie + k. 2 j j " # −∞ ih+1eh dh−1 k − ie h ρ ðΔÞ = − r e−ikΔ Δ < h h h− , 0. ðh − Þ!ð + rÞ dk 1 k + ie For example, in Kimura’s neutral evolution model (21), ei is a 1 1 k=ie binary random variable that takes a value of ∞ (jsij = 0, neutral [13] EVOLUTION mutation), with the probability f, and a value of 0 (jsij = ∞, lethal h mutation), with the probability 1 − f. Then, ρhðΔÞ = f δðΔÞ, Consider first the case r = 0 (all mutations are deleterious). Then, h φðhÞ = f and Lμ is replaced with Lfμ in Eq. 3, a trivial replace- ρhðΔ < 0Þ = 0. For Δ > 0, that is, decrease of the fitness, we have ment of the total genome length L with the length of the part of Lf Δh−1eh the genome where mutations are allowed, . Accordingly, −eΔ WðhÞ = P ðLμfÞ ρhðΔÞ = e . [14] h , and multimutation leaps become relevant for ðh − 1Þ! Lμf ≥ 1. Let us now estimate the probability of leaps with beneficial Then, the fixation rate 12 of an h-mutation leap is equal to mutations ðΔ < 0Þ. Assuming r 1 (rare beneficial mutations), 13 Z∞ Eq. takes the form zh the−zt φðhÞ = dt = hzhζðh + 1, z + 1Þ, [15] hre ðh − Þ! et − ρ ðΔÞ ≈ e−ejΔj [17] 1 1 h h− 0 2 1 z = e=ν ζðx yÞ and the fixation rate of a leap including beneficial mutations is where P∞ and , is the Hurwitz zeta function ζðx yÞ = 1 , ðk + yÞx. Therefore, the rate of fixation for leaps of hr k = 0 φðhÞ = zζð2, zÞ. [18] the length h is equal to WðhÞ = PhðLμÞφðhÞ. 2h−1 In one extreme, if z 1(νjsj1, neutral landscape), φðhÞ ≈ 1 and mutations are fixed at the rate they occur. In the opposite If z 1 (weak positive selection), zζð2, zÞ ≈ 1, so that the role z extreme case of strong negative selection ðz 1, νjsj 1Þ, of beneficial mutations is negligible. If 1 (strong positive h selection), φðhÞ ≈ hz ζðh + 1Þ where ζðxÞ is the Riemann zeta function. For ζðh + Þ a rough estimate, 1 can be replaced by 1, and then, hr 1 WðhÞ ≈ Lμe−LμP ðLμzÞ W h φðhÞ = . [19] h−1 . In this case, the maximum of ( )is 2h−1 z reached at h = Lμz ≅ Lμ=νjsj, which gives a nonnegligible fraction of multimutation leaps ðh > 1Þ among the fixed mutations only for Comparing Eq. 19 with the result for Δ > 0 (Eq. 18), one can see Lμ ≥ νjsj. However, in this case, the value of WðhÞ at this maximum that, in this case, beneficial mutations are predominant among is exponentially small because e−Lμ < e−νjsj. Therefore, in the regime the fixed mutations if of strong selection against deleterious mutations and at high mutations r > ð e=νÞh+1 [20] rates ðLμ ≥ νjsjÞ, multiple mutations actually dominate the mutational 2 . landscape, but their fixation rate is extremely low. Qualitatively, In this regime, multimutation leaps ðh > 4Þ occur at nonnegligible this conclusion seems obvious, but we now obtain the quantitative “ ” rates under sufficiently high (but not excessive) mutation rates criteria for what constitutes strong selection. We find that, even (Fig. 3). νjsj ∼ ðh = Þ for 10, the rate of multimutation leaps 4 can be non- The model considered above assumes independent effects of −4 negligible (>10 per generation; Fig. 2A) at the optimal Lμ values, different mutations (no epistasis, “ideal gas of mutations” model). whereas for νjsj ∼ 100, any leaps with h>1 are unfeasible (Fig. 2B). Now, let us take into account epistasis. In the case of strong epis- Under a more realistic model, all values of ei (the inverse of tasis, effects of combinations of different mutations are increasingly the fitness effect of a mutation) are different. For Δ > 0 and r = 0 strong, diverse, and, effectively, unpredictable, resulting in a rugged (no beneficial mutations), using Eq. 13, we get fitness landscape (22). In the limit of epistasis strength and

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Fig. 3. Rates of leaps on a landscape combining beneficial and deleterious mutations. Rates of leaps are plotted against the per-genome mutation rate (Lμ) and the leap length for different strengths of selection (A and C: νjsj = 10; B and D: νjsj = 100) and for different frequencies of beneficial mutations (A and B: − − − r = 10 4; C and D: r = 10 3). Contour lines indicate orders of magnitude and start from the rate of 10 5 leaps per generation.

ðLμ ≈ jsjÞ r > 2 unpredictability, epistasis creates numerous highly beneficial (23), the condition corresponds to νjsj2,thatis,the combinations that, once they occur, are highly likely to be fixed, frequency of beneficial multimutation combinations should be and a far greater number of highly deleterious combinations that unrealistically high to sustain evolution by multimutation leaps. are immediately lethal. Due to the effective randomness of ge- netic interactions, we consider the resulting landscape as essentially A Nonequilibrium Model of Stress-Induced Mutagenesis. The analysis random for h > 1, with the frequency of the beneficial combinations presented above suggests that the necessary condition for fixation r independent of h. In this case, the effective number of fixed leaps of multimutational leaps is the high-mutation regime. At low with h > 1issimply mutation rates ðLμ 1Þ, multimutation ðh > 1Þ events occur too rarely to be fixed in realistic settings even if the frequency of WðhÞ = QðhÞf ∼ PhðLμÞr. [21] beneficial combinations among them is reasonably high. However, in the high-mutation regime ðLμ 1Þ, the above analysis is ðh = Þ ðνjsj Þ If all single mutations 1 are deleterious 1 , their rate problematic for 2 reasons. First, the expression for the fixation 15 Wð Þ = P ðLμÞ 1 of fixation (Eq. ) can be approximated by 1 1 νjsj, rate (Eq. 5) is technically valid only for the case when the new whereas for all leaps of the length h > 1, effective number of fixed mutation is either fixed or lost before the emergence of the next leaps is Wðh > 1Þ = ð1 − P ðLμÞ − P ðLμÞÞf. Therefore, the condi- 0 1 one, which implies Lμ < 1=ν 1. Second, under any realistic tion for Wðh > 1Þ > Wð1Þ is model of the fitness landscape, most mutations should be dele- Lμ > Lμe−Lμ 1 terious. Thus, 1 implies that most of the progeny carries 1 or r > . [22] more mutations, and therefore suffers from these deleterious ef- ð1 − ðLμ + 1Þe−LμÞ νjsj fects. Under these conditions, the assumption of constant Ne is In the high-mutation regime ðLμ 1Þ, multiple mutations occur unrealistic, because the size of such a population will decrease orders of magnitude more frequently than single mutations, over- under the mutational load, down to an eventual crash. whelming the difference of scale between r and 1=νjsj,andmaking The complete analysis of the behavior of a variable-size pop- multimutation leaps much more likely. In the low-mutation regime ulation under the high-mutation regime and strong mutational ðLμ 1Þ, the balance between single and multiple mutations effects is currently beyond the state of the art. Therefore, here we analyze a simplified model of the short-term behavior of a 1 − ≈ 2 tends to 2 Lμ 1 Lμ and the condition for the dominance of (microbial) population after the onset of stress-induced muta- 2 1 multimutation leaps becomes r > Lμ νjsj. Around the Eigen threshold genesis ðLμ 1Þ.

4of8 | www.pnas.org/cgi/doi/10.1073/pnas.1909031116 Katsnelson et al. Downloaded by guest on September 24, 2021 N Consider a microbial population consisting of 0 individuals. N∞r e−2 ≤ E N∞ ≤ N∞r [25] Under typical conditions, the population is in an equilibrium, so 4 w 2 B 2 w 2. that approximately N =2 individuals survive the average gener- 0 r ðhÞ = r ξh−2 N ≈ N Indeed, let us consider first the simplest model h 2 ation span and produce 1 0 progeny by division (here we ð < ξ < Þ 24 consider simple asexual division as the progeny-generating pro- 0 1 . Then, Eq. takes the form cess whereby each surviving individual produces 2 offspring; other E N∞ ≈ N∞r ξ−2e−Lμ eLμξ − Lμξ − [26] demographic models can be accommodated without loss of gen- B 2 w 2 1 . erality). The typical mutation rate is low (Lμ ≈ 1=Ne ≈ 1=N , 0 0 Lμ according to refs. 24 and 25), so the population can be con- As a function of , the quantity sidered homogeneous. Upon the onset of unfavorable condi- φðLμÞ = e−Lμ eLμξ − Lμξ − [27] tions, the survival rate of the wild-type individuals drops to 1 fw = 1 2 and the mutation rate in the stressed individuals in- p creases such that Lμ > 1. reaches the maximum at Lμ = x=ξ, where x is the solution of the If fw is not too small ðfwN0 1Þ, the immediate wild-type equation survivors produce 2fwN first-generation progeny. With the 0 ex − expected number of mutations per descendant being Lμ, the 1 = 1 [28] distribution of the number of mutations in the progeny is given x 1 − ξ by the Poisson distribution with the expected number of mutants with h mutations of 2fwPhðLμÞN0. with the value at the maximum Let us consider a mutation landscape that is dominated by p 1−ξ − =ξ deleterious mutations with strong sign epistasis. All single mu- φðLμ Þ = ð1 − ξÞ ξ xξð1 − ξ + xÞ 1 . [29] tations are deleterious, so the survival of their carriers over the f f p ∞ ∞ −2 generation time is 1 w. An overwhelming majority of multi- For ξ 1 (rapid decay of rhÞ, Lμ = 2 and EðNB Þ ≈ 4Nw r2e .In p 1 mutation combinations have even stronger negative effects, so the opposite limit of slowly decaying rh, ξ → 1, Lμ ≈ ln − ξ and p 1 for h > 1, fh f1. Some small fraction rh of these combinations, φðLμ Þ ≈ 1. however, is strongly beneficial in the new conditions, conferring For a general slowly decaying function rhðhÞ, one can find that

to their carriers the survival rate of ∼ 1=2. EVOLUTION r ðhÞ r ðhÞ What would the h function look like? Intuitively, h rh h Lμp ≈ ln . [30] should decay to 0 at large , or at least not grow, as it is over- jdrhj whelmingly likely that a sufficiently large set of mutations would dh h≈Lμp contain a subset that it unconditionally lethal. Here, for sim- Lμp plicity, we consider a general form of rhðhÞ that equals 0 for h = 1 Importantly, even in this case, the optimal mutation rate and monotonically decays with h from r at an arbitrary rate. increases only logarithmically with the decay rate; furthermore, 2 r ðhÞ If the deleterious effect of mutations is strong enough the optimum value is notably robust to changes in h (Fig. 4). ð0 ≈ fh ≈ f fwÞ, then the only plausible source of beneficial The approximate condition for population survival, 1 EðN∞Þ > 23 25 mutants is the population of wild-type individuals (neither B 2, can be derived from Eqs. and and is bounded single mutants nor multiple mutants that do not carry the from below by beneficial combinations survive to the next generation). The e2 population of the wild-type individuals decays exponentially r2 > [31] through both the diminished survival and through mutations, 2N0fw k−1 reaching fwð2fwP ðLμÞÞ N at the k-th generation after the onset 0 0 Lμ of the unfavorable conditions. Ignoring stochastic fluctuations, the at the optimal value of . total number of wild-type individuals that survive until the pop- Discussion ulation collapse can be estimated as Here, we obtained analytic expressions for the probability of the ∞ fixation of multimutation leaps for deleterious and beneficial Nw ≈ fwN0 ð1 − 2fwP0ðLμÞÞ, [23] mutations depending on the parameters of the evolutionary pro- f N f P ðLμÞ = cess, namely, effective genome size (L), mutation rate (μ), effec- which is approximately equal to w 0 if w 0 1 2. ðνÞ Over the combined lifetimes of the surviving wild-type indi- tive population size , and distribution of selection coefficients of s viduals, the expected number of beneficial mutants is mutations ( ). Leaps in random fitness landscapes in the context of ! punctuated equilibrium have been previously considered for X∞ X∞ e−LμðLμÞh infinite (26, 27) or finite (28) populations. However, unlike the E N∞ ≈ N∞ ðr P ðLμÞÞ = N∞ r [24] present work, these studies have focused on the analysis of the B 2 w h h 2 w h h! , h=2 h=2 dynamics of the leaps rather than on the equilibrium distribution of their lengths. We further address the plausibility of beneficial which depends on the genome-wide mutation rate Lμ and the multimutation leaps under epistasis and outside of equilibrium, for shape of the rhðhÞ function. example in a microbial population under stress. Let us first consider the 2 extreme cases of rhðhÞ. In the limit of The principal outcomes of the present analysis are the conditions a completely flat function (rh = r2 for all h > 2), Eq. 23 gives under which multimutation leaps are fixed at a nonnegligible rate EðN∞Þ ≈ N∞r ð − P ðLμÞ − P ðLμÞÞ B 2 w 2 1 0 1 . This function asymptoti- in different evolutionary regimes (Fig. 5A). If the landscape is ∞ cally reaches the value of 2Nw r2 with Lμ → ∞. In the other ex- completely flat (strict neutrality, s = 0), the leap length is distributed treme of a rapidly decaying rhðhÞ, that is, rh = 0 for h > 2, Eq. 23 around Lμ that is , simply, the expected number of mutations per ∞ ∞ gives EðNB Þ ≈ 2Nw r2P2ðLμÞ. This function reaches its maximum genome per generation. If Lμ 1, leaps are effectively impossible, ∞ ∞ −2 at Lμ = 2 with EðNB Þ ≈ 4Nw r2e . and evolution can proceed only step by step (12). A considerable It can be shown that the estimates for all other monotoni- body of data exists on the values of each of the relevant parameters cally decaying rhðhÞ functions reach their maxima at finite values that define the probability of leaps. Generally, in the long term, the of Lμ with total expected number of mutations per genome per generation has

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Fig. 4. Abundance of beneficial multimutation combinations depending on the mutation rate. Abundance of beneficial multimutation combinations, φðLμÞ, −α h−2 given by Eq. 24, relative to r2.(A) rhðhÞ = r2ðh − 1Þ with α = 0 (blue), α = 0.5 (orange), α = 1 (green), and α = 2 (red). (B) rhðhÞ = r2ξ de with ξ = 1 (blue), ξ = 0.9 (orange), ξ = 0.5 (green), and ξ = 0.25 (red).

to be of the order of 1 or lower (Eigen threshold) because, if nonnegligible evolutionary factor. For organisms with Lμ < 1 Lμ 1, the population ultimately spirals into error catastrophe (it and larger v, the probability of leaps is substantially lower than should be emphasized that error catastrophe, i.e., the loss of high- the above estimates, so that under “normal” evolutionary re- fitness genotypes through accumulation of deleterious mutations, is gimes (at equilibrium) the contribution of leaps is negligible. distinct from catastrophe, i.e., loss of the entire population However, in some biologically relevant and common situations, caused by deleterious mutation) (15, 23, 29–32). The selection for such as stress-induced mutagenesis, which occurs in microbes in lower mutation rates is thought to be limited by the drift barrier and, response to double-stranded DNA breaks, the effective mutation accordingly, the genomic mutation rate appears to be inversely pro- rate can locally and temporarily increase by orders of magnitude portional to the effective population size, that is, Lμ ∼ 1=ν (24, 25). (33, 34) while the population is going through a severe bottleneck Thus, Lμν = const, which appears to be an important universal in (Fig. 5B). If the fraction of beneficial combinations of mutations evolution. satisfies the condition (31), even in the extreme case when the rest To estimate the leap probability, we can use Eq. 15 and the of the mutations are lethal, the population has a chance to survive characteristic values of the relevant parameters, for example when its mutation rate (Lμ) assumes a value close to the optimum those for human populations. As a crude approximation, Lμ = 1, value given by Eq. 30. This value depends on the rate of the decay − v = 104, jsj = 10 2 which, in the absence of beneficial mutations, of the fraction of beneficial combinations of mutations with the translates into the probability of a multimutation leap of about number of mutations. Specifically, the optimal value of Lμ equals − 4 × 10 5. Thus, such a leap would, on average, require over 2 for the steepest decay of r(h) and increases logarithmically slowly 23,000 generations, which is not a relevant value for the evolu- for more shallow functions. Under an extremely severe stress 9 −3 tion of mammals (given that ∼140 single mutations are expected (N0 = 10 , fw = 10 ), the survival threshold [r(h)] corresponds to − to be fixed during that time as calculated using the same for- the fraction of beneficial pairs of mutations of about 3 × 10 6.This mula). However, short leaps including beneficial mutations can means that, in the case of a typical bacterial genome of 3 × − occur with reasonable rates, such as 5 × 10 4 for h = 3, and the 106 base pairs, for each (deleterious) mutation, there is, on aver- − frequency of beneficial mutations r = 10 4, so such leaps are only age, 1 other mutation that yields a beneficial combination. This 8 times less frequent than single-mutation fixations. Conceiv- estimate pertains to the extreme case when all individual muta- ably, such leaps of beneficial mutations could be a minor but tions are highly deleterious. Under more realistic conditions, when

AB / 2 r(h) = const ≫ 0 / 2 all non-beneficial mutations are neutral ≪ first-generation survival first-generation 0 high extreme low high extreme

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Fig. 5. Summary of the modeling results: evolution by multimutation leaps depending on the evolutionary regime and fitness landscape. (A) Multimutation leaps in the equilibrium regime. (B) First-generation multimutation leaps under stress-induced mutagenesis. The hatched area denotes the of the parameter space in which saltational evolution may or may not be possible depending on the r(h) value.

6of8 | www.pnas.org/cgi/doi/10.1073/pnas.1909031116 Katsnelson et al. Downloaded by guest on September 24, 2021 many mutations are effectively neutral, and a small fraction is onset strongly and positively correlates with the number of drivers beneficial, the threshold fraction of beneficial combinations will be (53, 54). However, for a substantial fraction of tumors, no drivers considerably lower. These estimates indicate that multimutation are readily identifiable suggestive of the possibility that, in these leaps are likely to be an important factor of adaptive evolution under cases, tumor progression is driven by “epistatic drivers” (53), that is, stress. An implication of these findings is that stress-induced muta- combinations of mutations that might occur by leaps. genesiscouldbeaselectableadaptive mechanism, however contro- Another, completely different area where multimutation leaps versial an issue the evolution of evolvability might be (35–39). It could be important could be evolution of primordial replicators, should be further noted that, in this situation, large populations will in particular those in the hypothetical RNA world, that are have a higher innovation potential than small populations because thought to have had an extremely low replication fidelity, barely the former produce a greater diversity of multimutation combina- above the error catastrophe threshold (23, 55, 56). Furthermore, tions. In other terms, large populations have a greater chance to because the primordial replicators are likely to have been in- cross the entropy barrier to higher fitness genotypes (40). Thus, the completely optimized, the fraction of beneficial mutational stress-induced innovation regime is an alternative to innovation by combinations could be relatively high. Under these conditions, drift that occurs, primarily, in small populations (during population multimutational leaps could have been an important route of bottlenecks) (14, 25). This conclusion complements the previous evolutionary acceleration and thus might have contributed sub- findings that large populations can readily cross fitness valleys stantially to the most challenging evolutionary transition of all, through a series of consecutive mutations when the intermediate that from precellular to cellular forms. states are close to neutrality (41). An important caveat of the above conclusions on the bi- Remarkably, experiments on adaptive evolution of bacterial ological relevance of multimutational leaps is that the present populations revealed repeated emergence of hypermutators (i.e., analysis disregards clonal interference, that is, competition be- mutations in repair genes that greatly increase the mutation rate in tween in an evolving population, that plays a substantial – therespectiveclones)(4244) resulting, in some case, in simulta- role in the evolution of large populations under the high- “ ” neous fixation of cohorts of beneficial mutations (45). Further- mutation regime as indicated by both theory (17, 57, 58) and more, subsequent analyses have shown that mutator genotypes experiment (45, 59). Clearly, clonal interference has the poten- exist only transiently but exert long-lasting effects on the population tial to dampen the effect of multiple mutations. Nevertheless, it evolution (46). These findings seem to provide direct experimental appears likely that a clone with multiple mutations would be a validation of the multimutational leaps predicted by our model. strong competitor under strong selection pressure, for example EVOLUTION A different context in which multimutation leaps potentially in the case of stress-induced mutagenesis. might play a role is evolution of cancers. In most tumor types, Taken together, all these biological considerations suggest that mutation rate is dramatically, orders of magnitude elevated multimutation leaps with a beneficial effect, the probability of compared to normal tissues (47, 48). The effective population size which we show to be nonnegligible under conditions of elevated in tumors is difficult to estimate, and therefore there is not enough mutagenesis, could be an important mechanism of evolution that information to use the condition (31) to assess the plausibility of so far has been largely overlooked. Given that elevated mutation multimutation leaps. Nevertheless, given the extremely high values rate caused by stress is pervasive in nature, saltational evolution, of Lμ, it cannot be ruled out that the frequency of leaps is non- after all, might substantially contribute to the , in negligible. Most of the mutations in tumors are passengers that direct defiance of “.” have no effect on cancer progression or exert a deleterious effect (49, 50). Traditionally, tumorigenesis is thought to depend on ACKNOWLEDGMENTS. M.I.K. acknowledges financial support from the Dutch several driver mutations that occur consecutively (51, 52). This is Research Council via the Spinoza Prize. Y.I.W. and E.V.K. are funded through indeed likely to be the case in many tumors because the age of the Intramural Research Program of the National Institutes of Health.

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