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# On the feasibility of saltational evolution

Contributed by Eugene V. Koonin, August 28, 2019 (sent for review May 28, 2019; reviewed by Christoph Adami and Claus O. Wilke)

## Significance

In evolutionary biology, it is generally assumed that evolution occurs in the weak mutation limit, that is, the frequency of multiple mutations simultaneously occurring in the same genome and the same generation is negligible. We employ mathematical modeling to show that, although under the typical parameter values of the evolutionary process the probability of multimutational leaps is indeed low, they might become substantially more likely under stress, when the mutation rate is dramatically elevated. We hypothesize that stress-induced mutagenesis in microbes is an evolvable adaptive strategy. Multimutational leaps might matter also in other cases of substantially increased mutation rate, such as growing tumors or evolution of primordial replicators.

## Abstract

Is evolution always gradual or can it make leaps? We examine a mathematical model of an evolutionary process on a fitness landscape and obtain analytic solutions for the probability of multimutation leaps, that is, several mutations occurring simultaneously, within a single generation in 1 genome, and being fixed all together in the evolving population. The results indicate that, for typical, empirically observed combinations of the parameters of the evolutionary process, namely, effective population size, mutation rate, and distribution of selection coefficients of mutations, the probability of a multimutation leap is low, and accordingly the contribution of such leaps is minor at best. However, we show that, taking sign epistasis into account, leaps could become an important factor of evolution in cases of substantially elevated mutation rates, such as stress-induced mutagenesis in microbes. We hypothesize that stress-induced mutagenesis is an evolvable adaptive strategy.

A venerable principle of natural philosophy, most consistently propounded by Leibnitz (1) and later embraced by prominent biologists, in particular Linnaeus (2), is “*natura non facit saltus*” (“nature does not make leaps”). This principle then became one of the key tenets of Darwin’s theory that was inherited by the modern synthesis of evolutionary biology. In evolutionary biology, the rejection of saltation takes the form of gradualism, that is, the notion that evolution proceeds gradually, via accumulation of “infinitesimally small” heritable changes (3, 4). However, some of the most consequential evolutionary changes, such as, for example, the emergence of major taxa, seem to occur abruptly rather than gradually, prompting hypotheses on the importance of saltational evolution, for example by Goldschmidt (“hopeful monsters”) and Simpson (“quantum evolution”). Subsequently, these ideas have received a more systematic, even if qualitative, treatment in the concepts of punctuated equilibrium (5, 6) and evolutionary transitions (7, 8).

Within the framework of modern evolutionary biology, gradualism corresponds to the weak-mutation limit, that is, an evolutionary regime in which mutations occur one by one, consecutively, such that the first mutation is assessed by selection and either fixed or purged from the population, before the second mutation occurs (9). A radically different, saltational mode of evolution (10, 11) is conceivable under the strong-mutation limit (9) whereby multiple mutations occurring within a single generation and in the same genome potentially could be fixed all together. Under the fitness landscape concept (12, 13), gradual or more abrupt evolutionary processes can be depicted as distinct types of trajectories on fitness landscapes (Fig. 1). The typical evolutionary paths on such landscapes are thought to be 1 step at a time, uphill mutational walks (12). In small populations, where genetic drift becomes an important evolutionary factor, the likelihood of downhill movements becomes nonnegligible (14). In principle, however, a different type of moves on fitness landscapes could occur, namely, leaps (or “flights”) across valleys when a population can move to a different area in the landscape, for example to the slope of a different, higher peak, via simultaneous fixation of multiple mutations (Fig. 1).

We sought to obtain analytically, within the population genetics framework, the conditions under which multimutational leaps might be feasible. The results suggest that, under most typical parameters of the evolutionary process, leaps cannot be fixed. However, taking sign epistasis into account, we show that saltational evolution could become relevant under conditions of elevated mutation rate under stress so that stress-induced mutagenesis could be considered an evolvable adaptation strategy.

## Results

### Multimutation Leaps in the Equilibrium Regime.

Let us assume (binary) genomes of length *L* (in the context of this analysis, *L* should be construed as the number of evolutionarily relevant sites, such as codons in protein-coding genes, rather than the total number of sites), the probability of single mutation *μ* << 1 per site per round of replication (generation), and constant effective population size *N*_{e} >> 1. Then, the transition probability from sequence *i* to sequence *j* is (equation 3.11 in ref. 15)*h* is equal to the number of ways *h* sites can be selected from *L*, that is,*L* >> 1 and *L* >> *h* (*h* can be of the order of 1).

Assuming also *µ* << 1, we obtain a typical combinatorial probability of leaps over the distance *h*:*Lμ*.

In steady state, the probability of fixation of the state *i* is proportional to

and *i* [

Then, the rate of the occurrence and fixation of the transition *x* as fitness, omitting logarithm for brevity). We analyze first the case without epistasis, that is, with additive fitness effects of individual mutations:

Now, let us specify the distribution of the fitness effects of mutations **8** is*i*. For the model **8****5**, the fixation rate of an *h*-mutation leap is equal to**11** into **7**, we obtain*r* = 0 (all mutations are deleterious). Then, **12** of an *h*-mutation leap is equal to*h* is equal to

In one extreme, if *W*(*h*) is reached at ^{−4} per generation; Fig. 2*A*) at the optimal *B*).

Under a more realistic model, all values of **13**, we get**3**, a trivial replacement of the total genome length L with the length of the part of the genome where mutations are allowed,

Let us now estimate the probability of leaps with beneficial mutations **13** takes the form**19** with the result for **18**), one can see that, in this case, beneficial mutations are predominant among the fixed mutations if

The model considered above assumes independent effects of different mutations (no epistasis, “ideal gas of mutations” model). Now, let us take into account epistasis. In the case of strong epistasis, effects of combinations of different mutations are increasingly strong, diverse, and, effectively, unpredictable, resulting in a rugged fitness landscape (22). In the limit of epistasis strength and unpredictability, epistasis creates numerous highly beneficial combinations that, once they occur, are highly likely to be fixed, and a far greater number of highly deleterious combinations that are immediately lethal. Due to the effective randomness of genetic interactions, we consider the resulting landscape as essentially random for **15**) can be approximated by

### A Nonequilibrium Model of Stress-Induced Mutagenesis.

The analysis presented above suggests that the necessary condition for fixation of multimutational leaps is the high-mutation regime. At low mutation rates **5**) is technically valid only for the case when the new mutation is either fixed or lost before the emergence of the next one, which implies *N*_{e} is unrealistic, because the size of such a population will decrease under the mutational load, down to an eventual crash.

The complete analysis of the behavior of a variable-size population under the high-mutation regime and strong mutational effects is currently beyond the state of the art. Therefore, here we analyze a simplified model of the short-term behavior of a (microbial) population after the onset of stress-induced mutagenesis

Consider a microbial population consisting of

If

Let us consider a mutation landscape that is dominated by deleterious mutations with strong sign epistasis. All single mutations are deleterious, so the survival of their carriers over the generation time is

What would the

If the deleterious effect of mutations is strong enough

Over the combined lifetimes of the surviving wild-type individuals, the expected number of beneficial mutants is

Let us first consider the 2 extreme cases of **23** gives **23** gives

It can be shown that the estimates for all other monotonically decaying **24** takes the form

For a general slowly decaying function

The approximate condition for population survival, **23** and **25** and is bounded from below by

## Discussion

Here, we obtained analytic expressions for the probability of the fixation of multimutation leaps for deleterious and beneficial mutations depending on the parameters of the evolutionary process, namely, effective genome size (*L*), mutation rate (*μ*), effective population size *s*). Leaps in random fitness landscapes in the context of punctuated equilibrium have been previously considered for infinite (26, 27) or finite (28) populations. However, unlike the present work, these studies have focused on the analysis of the dynamics of the leaps rather than on the equilibrium distribution of their lengths. We further address the plausibility of beneficial multimutation leaps under epistasis and outside of equilibrium, for example in a microbial population under stress.

The principal outcomes of the present analysis are the conditions under which multimutation leaps are fixed at a nonnegligible rate in different evolutionary regimes (Fig. 5*A*). If the landscape is completely flat (strict neutrality,

To estimate the leap probability, we can use Eq. **15** and the characteristic values of the relevant parameters, for example those for human populations. As a crude approximation, *Lμ =* 1, *v =* 10^{4}, *|s| =* 10^{−2} which, in the absence of beneficial mutations, translates into the probability of a multimutation leap of about 4 × 10^{−5}. Thus, such a leap would, on average, require over 23,000 generations, which is not a relevant value for the evolution of mammals (given that ∼140 single mutations are expected to be fixed during that time as calculated using the same formula). However, short leaps including beneficial mutations can occur with reasonable rates, such as 5 × 10^{−4} for *h* = 3, and the frequency of beneficial mutations *r* = 10^{−4}, so such leaps are only 8 times less frequent than single-mutation fixations. Conceivably, such leaps of beneficial mutations could be a minor but nonnegligible evolutionary factor. For organisms with *Lμ* < 1 and larger *v,* the probability of leaps is substantially lower than the above estimates, so that under “normal” evolutionary regimes (at equilibrium) the contribution of leaps is negligible.

However, in some biologically relevant and common situations, such as stress-induced mutagenesis, which occurs in microbes in response to double-stranded DNA breaks, the effective mutation rate can locally and temporarily increase by orders of magnitude (33, 34) while the population is going through a severe bottleneck (Fig. 5*B*). If the fraction of beneficial combinations of mutations satisfies the condition (31), even in the extreme case when the rest of the mutations are lethal, the population has a chance to survive when its mutation rate (*Lμ*) assumes a value close to the optimum value given by Eq. **30**. This value depends on the rate of the decay of the fraction of beneficial combinations of mutations with the number of mutations. Specifically, the optimal value of *Lμ* equals 2 for the steepest decay of *r*(*h*) and increases logarithmically slowly for more shallow functions. Under an extremely severe stress (*N*_{0} = 10^{9}, *f*_{w} = 10^{−3}), the survival threshold [*r*(*h*)] corresponds to the fraction of beneficial pairs of mutations of about 3 × 10^{−6}. This means that, in the case of a typical bacterial genome of 3 × 10^{6} base pairs, for each (deleterious) mutation, there is, on average, 1 other mutation that yields a beneficial combination. This estimate pertains to the extreme case when all individual mutations are highly deleterious. Under more realistic conditions, when many mutations are effectively neutral, and a small fraction is beneficial, the threshold fraction of beneficial combinations will be considerably lower. These estimates indicate that multimutation leaps are likely to be an important factor of adaptive evolution under stress. An implication of these findings is that stress-induced mutagenesis could be a selectable adaptive mechanism, however controversial an issue the evolution of evolvability might be (35⇓⇓⇓–39). It should be further noted that, in this situation, large populations will have a higher innovation potential than small populations because the former produce a greater diversity of multimutation combinations. In other terms, large populations have a greater chance to cross the entropy barrier to higher fitness genotypes (40). Thus, the stress-induced innovation regime is an alternative to innovation by drift that occurs, primarily, in small populations (during population bottlenecks) (14, 25). This conclusion complements the previous findings that large populations can readily cross fitness valleys through a series of consecutive mutations when the intermediate states are close to neutrality (41).

Remarkably, experiments on adaptive evolution of bacterial populations revealed repeated emergence of hypermutators (i.e., mutations in repair genes that greatly increase the mutation rate in the respective clones) (42⇓–44) resulting, in some case, in simultaneous fixation of “cohorts” of beneficial mutations (45). Furthermore, subsequent analyses have shown that mutator genotypes exist only transiently but exert long-lasting effects on the population evolution (46). These findings seem to provide direct experimental validation of the multimutational leaps predicted by our model.

A different context in which multimutation leaps potentially might play a role is evolution of cancers. In most tumor types, mutation rate is dramatically, orders of magnitude elevated compared to normal tissues (47, 48). The effective population size in tumors is difficult to estimate, and therefore there is not enough information to use the condition (31) to assess the plausibility of multimutation leaps. Nevertheless, given the extremely high values of *Lμ*, it cannot be ruled out that the frequency of leaps is nonnegligible. Most of the mutations in tumors are passengers that have no effect on cancer progression or exert a deleterious effect (49, 50). Traditionally, tumorigenesis is thought to depend on several driver mutations that occur consecutively (51, 52). This is indeed likely to be the case in many tumors because the age of onset strongly and positively correlates with the number of drivers (53, 54). However, for a substantial fraction of tumors, no drivers are readily identifiable suggestive of the possibility that, in these cases, tumor progression is driven by “epistatic drivers” (53), that is, combinations of mutations that might occur by leaps.

Another, completely different area where multimutation leaps could be important could be evolution of primordial replicators, in particular those in the hypothetical RNA world, that are thought to have had an extremely low replication fidelity, barely above the error catastrophe threshold (23, 55, 56). Furthermore, because the primordial replicators are likely to have been incompletely optimized, the fraction of beneficial mutational combinations could be relatively high. Under these conditions, multimutational leaps could have been an important route of evolutionary acceleration and thus might have contributed substantially to the most challenging evolutionary transition of all, that from precellular to cellular life forms.

An important caveat of the above conclusions on the biological relevance of multimutational leaps is that the present analysis disregards clonal interference, that is, competition between clades in an evolving population, that plays a substantial role in the evolution of large populations under the high-mutation regime as indicated by both theory (17, 57, 58) and experiment (45, 59). Clearly, clonal interference has the potential to dampen the effect of multiple mutations. Nevertheless, it appears likely that a clone with multiple mutations would be a strong competitor under strong selection pressure, for example in the case of stress-induced mutagenesis.

Taken together, all these biological considerations suggest that multimutation leaps with a beneficial effect, the probability of which we show to be nonnegligible under conditions of elevated mutagenesis, could be an important mechanism of evolution that so far has been largely overlooked. Given that elevated mutation rate caused by stress is pervasive in nature, saltational evolution, after all, might substantially contribute to the history of life, in direct defiance of “*Natura non facit saltus*.”

## Acknowledgments

M.I.K. acknowledges financial support from the Dutch Research Council via the Spinoza Prize. Y.I.W. and E.V.K. are funded through the Intramural Research Program of the National Institutes of Health.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: M.Katsnelson{at}science.ru.nl or koonin{at}ncbi.nlm.nih.gov.

Author contributions: M.I.K. and E.V.K. designed research; M.I.K. and Y.I.W. performed research; M.I.K., Y.I.W., and E.V.K. analyzed data; and M.I.K. and E.V.K. wrote the paper.

Reviewers: C.A., Michigan State University; and C.O.W., The University of Texas at Austin.

The authors declare no competing interest.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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