The expected time to cross extended fitness plateaus

Theoretical Population Biology 129 (2019) 54–67

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The expected time to cross extended fitness plateaus ∗

Mahan Ghafari a,b ,1 , Daniel B. Weissman a , a b

Department of Physics, Emory University, Atlanta, GA 30322, USA Department of Genetics, University of Cambridge, UK

article

info

Article history: Received 27 May 2018 Available online 25 September 2019 Keywords: Fitness plateau Recombination Adaptation Mutation Complexity

a b s t r a c t For a population to acquire a complex adaptation requiring multiple individually neutral mutations, it must cross a plateau in the fitness landscape. We consider plateaus involving three mutations, and show that large populations can cross them rapidly via lineages that acquire multiple mutations while remaining at low frequency, much faster than the ∝ µ3 rate for simultaneous triple mutations. Plateau-crossing is fastest for very large populations. At intermediate population sizes, recombination can greatly accelerate adaptation by combining independent mutant lineages to form triple-mutants. For more frequent recombination, such that the population is kept near linkage equilibrium, we extend our analysis to find simple expressions for the expected time to cross plateaus of arbitrary width. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Most mutations in most natural populations are effectively neutral. Considered in isolation, these are irrelevant for adaptation. But the fitness effect of a mutation generally depends on the genetic background on which it occurs, a phenomenon known as epistasis. Thus, there are likely to be combinations of these neutral mutations that interact epistatically to have an effect on fitness. If this effect is positive for a given combination, then that combination forms a complex adaptation, separated from the wild type by a fitness plateau. How frequently do we expect populations to acquire such adaptations? On one hand, a given complex adaptation should typically be harder for a population to find than a simple adaptation requiring only a single beneficial mutation. On the other hand, if a genome of length L has O(L) possible neutral mutations, then there are O(LK ) genotypes that could potentially be a complex adaptation involving K mutations. So if even a modest fraction of these genotypes are indeed adaptive, the number of possible complex adaptations could far exceed the number of available beneficial mutations, and it could be that they are collectively a frequent form of adaptation (Fisher, 2007; Weissman et al., 2009; Trotter et al., 2014). To evaluate their importance, we must know more about how rapidly populations explore fitness plateaus. Populations can cross fitness plateaus via a sequence of neutral mutations fixing by drift until only one more mutation is needed for the (formerly complex) adaptation. But this process is slow ∗ Corresponding author. E-mail address: [email protected] (D.B. Weissman). 1 Current address: Interdisciplinary Bioscience DTP, Rex Richards Building, University of Oxford, UK https://doi.org/10.1016/j.tpb.2019.03.008 0040-5809/© 2019 Elsevier Inc. All rights reserved.

and inefficient; in a high-dimensional fitness plateau, the population will be much more likely to drift away from a complex adaptation than towards it. Large, asexual populations can cross plateaus and even fitness ‘‘valleys’’ (in which adaptation requires individually deleterious mutations) much more rapidly (e.g., van Nimwegen and Crutchfield, 2000; Komarova et al., 2003; Iwasa et al., 2004; Weinreich and Chao, 2005; Durrett and Schmidt, 2008; Weissman et al., 2009). They can do this because many mutations will be present in the population at low frequency. If the population is sufficiently large, even these low-frequency mutations will be present in a large absolute number of individuals, some of which will happen to also carry additional mutations. Thus, genotypes that are multiple mutations away from the wildtype will already be present in the population and exposed to natural selection, allowing the population to effectively ‘‘see’’ several steps away in the fitness landscape, and ‘‘tunnel’’ directly to the adaptive genotypes (Jain and Krug, 2006). Recombination changes these dynamics in two ways. First, by combining mutations that occur in different lineages, it accelerates the population’s exploration of the plateau (Christiansen et al., 1998). On the other hand, recombination breaks up the beneficial combination once it is formed (Eshel and Feldman, 1970; Feldman, 1971; Karlin and McGregor, 1971), slowing adaptation (Takahata, 1982; Michalakis and Slatkin, 1996). The latter effect has a simple quantitative description: rare recombination can be approximated as reducing the effective selective advantage of the mutant combination. The former effect, on the other hand, depends on the spectrum of mutant lineages that coexist in the population and has only been fully understood in the simplest case of two-locus plateaus (Weissman et al., 2010). Here we extend this analysis to the three-locus case, considering the full

M. Ghafari and D.B. Weissman / Theoretical Population Biology 129 (2019) 54–67

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spectrum of possible population sizes, recombination rates, mutation rates, and selective advantages of the adaptive genotype. When recombination is frequent relative to selection, the population tends to be in quasi-linkage equilibrium, greatly simplifying the analysis and allowing us to describe the dynamics of crossing arbitrary-width plateaus in this regime. 2. Model We consider a haploid Wright–Fisher population of size N. The genome consists of K loci each of which has two possible alleles, 0 and 1; for much of the analysis, we will focus on the case K = 3. Initially, all individuals have the all-0 genotype. All genotypes have the same fitness except the all-1 genotype, which has a strong selective advantage s ≫ 1/N; see Fig. 1. In all mathematical expressions we will assume that s ≪ 1, but the qualitative results are unchanged for s ≳ 1 (Figs. 7 and 8). Individuals mutate (in both directions) at a rate µ per locus per generation. Each generation, each offspring is produced asexually (with possible mutations) with probability 1 − f ; with probability f , it is the product of sexual reproduction between two parents. All loci are unlinked, i.e., offspring produced sexually sample each locus independently with equal probability from their parents (again, with possible mutation). We use this recombination model primarily for simplicity, to cut down on the proliferation of model parameters and possible dynamical regimes. It is also a realistic description for recombination in most eukaryotes and segmented viruses, as most sets of loci will be on different chromosomes or segments. However, it is not a good model of homologous recombination in bacteria. Even in eukaryotes or segmented viruses, while random loci are likely to be unlinked, epistatic interactions may be enriched among loci within a single gene, where linkage will be important. Fortunately, none of our conclusions will depend on the details of the recombination model beyond numerical prefactors that do not affect scaling behavior. One may be concerned that this model elevates the rate of double crossovers compared to one in which loci recombine separately. But for K = 3 even banning double crossovers entirely in our model has only a minor effect, as only one of the three possible pairings between double-mutant and single-mutant genotypes requires a double crossover to produce a triple-mutant, i.e., most paths to the triple-mutant involve only single crossovers. Although f is strictly speaking the frequency of sexual reproduction, we will generally refer to it as the rate of recombination, since recombination is the key process and we expect all our results to hold in more general models, including those where recombination is independent of reproduction. We will focus on finding the expected time T until the all1 genotype first makes up the majority of the population. For simplicity, we will also refer to the ‘‘rate’’ of crossing the plateau, defined as T −1 , even though it is not a true rate, as the distribution of time to cross the plateau is not exponential in general. The definitions of the most important symbols are collected in Table 1. Exact simulations were done in Python (Figs. 3, 4, and 5) and Mathematica (Figs. 7 and 8). 3. Results There are two fundamentally different plateau-crossing dynamics, depending on the relative rates of selection and recombination (Eshel and Feldman, 1970; Feldman, 1971; Suzuki, 1997; Jain, 2009). If recombination is weak relative to selection (f ≪ s), the adaptive genotype is rarely broken up by recombination and can spread rapidly once formed even if the individual mutant alleles are very rare in the population. If, on the other hand,

Fig. 1. A visualization of the fitness landscape in the case of K = 3. The nodes represent the fitness of wild type, single-, double-, and triple-mutant genotypes. Wild-type alleles are denoted by 0 and mutants by 1. The {1, 1, 1} genotype has a fitness 1 + s > 1 and the rest have fitness 1.

Fig. 2. Schematic diagram of the asymptotic dynamics by which a population crosses a three-mutation (K = 3) fitness plateau to acquire a complex adaptation providing advantage s, as a function of frequency of sexual reproduction f and population size N. Color qualitatively represents the expected time T for the population to cross; for quantitative expressions, see Table 2. For f ≪ s, selection can drive the triple-mutant genotype to fixation while the other mutant genotypes remain rare, while for f ≫ s the population always remains close to linkage equilibrium; plateau crossing is fastest for intermediate frequencies of sex. The time to cross the plateau decreases as population size increases from the ‘‘sequential fixation’’ regime to the ‘‘deterministic’’ regimes. The ‘‘stochastic tunneling (sexual)’’ regime is a combination of several different regimes that can be practically indistinguishable, with boundaries that depend on the value of µ/s — see Appendix A.3 .. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

recombination is strong (f ≫ s), the population is kept in quasilinkage equilibrium, with the dynamics determined by the allele frequencies. Because the dynamics are so different, we consider these two regimes separately. In this section, we will provide a summary of our main mathematical results, reserving the details and derivations for the Analysis section below, and the biological implications for the Discussion. Fig. 2 and Table 2 summarize the different possible scaling behaviors of T over all of parameter space. Many of the possible scaling behaviors are quite close to each other and only practically distinguishable for extreme parameter values. 3.1. Rare recombination, f ≪ s For f ≪ s, we must track genotype frequencies rather than just allele frequencies, so the complexity of the dynamics increases

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M. Ghafari and D.B. Weissman / Theoretical Population Biology 129 (2019) 54–67 Table 1 Symbol definitions. Symbol Definition N

µ f s T

Ri (t) ni (t) pi (t)

Haploid population size Mutation rate per locus per generation Frequency of sexual reproduction Selective advantage of the all-mutant genotype Expected time until the all-mutant genotype makes up the majority of the population The rate at which i-mutant individuals arise in the population at time t Total number of i-mutant individuals The probability that an i-mutant lineage arising at time t will be successful

Table 2 Rate of plateau-crossing (inverse of expected crossing time T ) for asymptotic regimes shown in Fig. 2, organized by main divisions of parameter space. Final column lists section in text where scaling is derived and figures where it is compared with simulation results. Rows labeled with ‘‘2009’’ are shown in Fig. 6A of Weissman et al. (2009). Divisions f
Nµ ≫ 1

Name

T −1 scaling

Section & Figures

Triple-mutants deterministic

s/ln(s/µ)

4.1.1, Fig. 3

Single-mutants deterministic (sexual)

( 3 )1/3 Nµ s ( 3 )1/4 N µ fs ( 2 5 )1/4 N µ s ( 2 5 3 )1/7 N µ f s

Stochastic tunneling (asexual) Stochastic tunneling (sexual)

N µ (s/µ) ∼ N 1.4 µ1.9 f 0.4 s0.1

Double-mutants deterministic (asexual) Double-mutants deterministic (sexual) Single-mutants deterministic (asexual) Nµ ≪ 1

2

Semi-linkage-equilibrium tunneling Tunneling after 1 mutation fixes (asexual) Tunneling after 1 mutation fixes (sexual) f ≫s

√

N ≪ 1/ µs

1/4

( )1/4 N µ2 N 3 µfs µ µ ( 2 )1/3 µ s

4.1.2, Fig. 3, 2009 4.1.2, Figs. 3 and 4 4.1.3, Fig. 4, 2009 4.1.3, Figs. 3 and 4 4.1.4, 2009 A.3, Fig. 3 4.1.4, Fig. 3 4.1.5, 2009 4.1.5

Nµ ≫ 1

Alleles deterministic

Nµ ≪ 1

Linkage equilibrium (LE) tunneling LE tunneling after 1 mutation fixes

N µ2 (Ns)1/3

µ

4.2.2, Fig. 5 4.2.2, Fig. 5

Sequential fixation

µ

4.2.2, Fig. 5

Fig. 3. Expected time T for a rarely recombining population, i.e. f ≪ s, to cross a three-mutation fitness plateau, as a function of the mutation rate µ. Points show simulation results (averaged over 100 simulations per data point), curves show analytical predictions. Note that in some regions, several analytical expressions give almost the same exact expected time. Typical plateau-crossing dynamics for two of the asymptotic regimes are illustrated in Figs. 7 and 8. In all regimes, T has a weaker dependence on µ than the rate ∝ µ3 for the three mutations to happen simultaneously (triangle). Parameter values: N = 1010 , f = 10−4 , s = 0.1. Error bars are smaller than the size of the points.

rapidly with the width of the plateau; we therefore focus on the simplest case that has not yet been fully characterized, K = 3. Even for this simplest case, there are many possible dynamical regimes (left side of Figs 2, 3), depending on how difficult it is for the population to generate the adaptive genotype. For all but the largest population sizes, plateau-crossing becomes faster with increasing recombination rates, so the optimal rate is f ≲ s. The equations in this section all apply to this regime as well, with s replaced by the average rate of increase of triple-mutants

4.2.1, Figs. 4 and 5

Fig. 4. Expected time T to cross a three-mutation fitness plateau as a function of the frequency of sexual reproduction f . Points show simulation results, curves show analytical predictions. Parameter values: N = 1012 , µ = 10−10 , s = 0.1. Plateau-crossing is fastest for f ≲ s. Error bars are smaller than the size of the points.

when rare, s˜ = s − 87 f . (The factor of 7/8 is the probability that an individual produced by sexual reproduction will not inherit the same parent’s alleles at all three loci.) For more frequent recombination, f ≳ s, s˜ is negative, meaning that the frequency of the triple-mutant genotype is held in check until the other mutant genotypes also increase in frequency. This slows crossing, leading to the optimal value of f mentioned above. If mutation and recombination are so frequent and the population is big enough that the triple-mutant genotype is generated effectively instantaneously, then the expected plateau-crossing time T is just the time for a selective sweep, and depends primarily on s. For smaller µ, f , and N, most of T is waiting for the successful triple-mutant to be produced and the strongest

M. Ghafari and D.B. Weissman / Theoretical Population Biology 129 (2019) 54–67

Fig. 5. Expected time T for a frequently recombining population, i.e. f ≫ s, to cross a three-mutation fitness plateau, as a function of the population size N. Points show simulation results, curves show the analytical prediction, Eq. (17). Parameter values: f = 0.5, µ = 10−7 , s = 0.05. The time to cross the plateau depends strongly on N for N ≲ 1/µ, and levels off for large or very small populations. Error bars are smaller than the size of the points.

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triple- and double-mutants are important, but single-mutants can still be treated deterministically (the ‘‘single deterministic (sexual)’’ regime). Both of these regimes could plausibly apply to many microbial populations – the requisite recombination rates are not too high – but see the Discussion for the difficulties in assessing the relevant size of natural populations. If the recombination rate is lower than the thresholds in the second and third lines of Eq. (1), the population is effectively asexual and T follows the scaling behavior described in Weissman et al. (2009) and Eqs. (6) and (9). If the mutation supply is low (N µ ≪ 1), then T is approximately the expected waiting time for the first successful mutation. Since exploration of genotype space is more of a challenge for populations when mutations are rare, recombination has the potential to make more of a difference. When the recombination is very rare, the population is effectively asexual, with plateau−1 crossing rate Tasex ≈ N µ2 (s/µ)1/4 (Eq. (10), see also Weissman et al., 2009). As the recombination rate increases, it becomes easier for mutations to be successful, and plateau-crossing speeds up. There are eight different asymptotic scaling regimes for rare recombination as N → ∞, depending on exactly how µ, f , s → 0, but for reasonable parameter values they are generally fairly similar (see Appendix A.3), with the expected rate of plateaucrossing roughly given by T −1 ≈ N 1.4 µ1.9 f 0.4 s0.1 . As recombination becomes more frequent (but still f ≪ s), pairs of large single-mutant lineages are able to succeed by reaching linkage equilibrium with each other and then recombining with a smaller third lineage (‘‘semi-linkage-equilibrium tunneling’’), and the rate of crossing increases further to T −1 ≈ (N µ)2 (µfs/N)1/4 (Eq. (11)). This is the regime where recombination speeds plateau-crossing the most; comparing Eqs. (11), we see that it increases ( (10))and 1/4 , which could exceed an order the rate by a factor ∼ N 3 µ2 f of magnitude if Nf > 106 and N µ ∼ 0.1. 3.2. Frequent recombination, f ≫ s

Fig. 6. The number of mutations that must be fixed sequentially by drift for a population to find an adaptive combination requiring K = 3 mutations, as a function of the size of the population. The last mutation will always be fixed by selection. Larger populations can tunnel through the earlier mutations, fixing them only together with the last one. The points show the results of the simulations shown in Fig. 5 and the curve shows the analytical prediction, Eq. (2). We show only population sizes N ≤ 1/µ = 107 ; larger populations tunnel through all the mutations.

dependence is on µ. N and f are most important at intermediate levels of diversity, where producing triple-mutants is difficult but there are opportunities for simultaneous polymorphisms at multiple loci to recombine. Quantitatively, when the mutation supply is large (N µ ≫ 1), then the expected plateau-crossing time is approximately:

⎧ ⎪ if N ≫ s2 /µ3 ⎨(ln(s/µ))/s − 1 / 4 T ≈ N µ3 rs if f ≫ µ(Ns)1/3 and N ≫ µs3 /f 5 ⎪ ⎩( 2 5 3 )−1/7 N µ f s if (N 2 µ5 s)1/4 ≪ f ≪ (µs3 /N)1/5 .

(1)

The first line of Eq. (1) corresponds to the approximately deterministic dynamics of very large populations, which are insensitive to rare recombination (because the only substantial linkage disequilibrium is that generated by selection on the triple mutants after they are already on their way to fixation). This regime requires high mutation rates, and so is likely only applicable to viruses. In the second line, fluctuations in the number of triplemutants are important, but single- and double-mutants can be treated deterministically (the ‘‘doubles deterministic (sexual)’’ regime). In the third line, fluctuations in the numbers of both

For frequently recombining populations (f ≫ s), we find the expected time for plateau crossing across the full spectrum of possible plateau widths K , mutation rates µ, population sizes N (Fig. 5), and selective coefficients s. These population will be in quasi-linkage equilibrium and selection will therefore act on alleles rather than genotypes. In this regime, the plateau-crossing time depends primarily on the mutation rate and is typically ∼ O(1/µ). When mutations are frequent (N µ ≫ 1), the population crosses the plateau nearly deterministically and solving the deterministic mutation-selection dynamics gives plateau-crossing time: T ≈ (s/µ)1/K /µ. When mutations are rare (N µ ≪ 1), stochasticity is important and the dynamics typically proceed in two stages: first, K − m of the necessary mutations sequentially drift to fixation by chance; then, once the population is sufficiently close to the adaptive genotype, it relatively rapidly acquires the last m mutations together via stochastic tunneling (see Fig. 6). The typical value of m is the largest integer such that the probability of a new mutation triggering a tunneling event of m mutations is higher than the probability 1/N of fixation by drift:

⌊ m≈

1 2

√ +

1 4

+

2 ln(Ns)

|ln(N µ)|

⌋ ,

(2)

where ⌊.⌋ represents the floor function. Therefore, the plateaucrossing time is typically dominated by the time for K − m mutations to drift to fixation, unless m ≥ K , in which case the population tunnels directly. Summarizing these regimes, the

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M. Ghafari and D.B. Weissman / Theoretical Population Biology 129 (2019) 54–67

expected time for a frequently recombining population to cross a fitness plateau is:

T ≈

⎧ ( ) 1 µ 1/K ⎪ ⎪ ⎪ ⎪ µ s ⎨ N(Ns)

− K1

⎪ ⎪ ln(K /m) ⎪ ⎪ ⎩ µ

if N µ ≫ 1 1 − K+ 2

(N µ)

CK

if N µ ≪ 1 and m ≥ K

(3)

if N µ ≪ 1 and m < K ,

where CK in the second line (pure tunneling) is a combinatorial factor that depends only on K (see Eq. (15)). The first line of Eq. (3) shows that even the large populations that cross plateaus the quickest can only ‘‘see’’ a limited distance in the fitness landscape: for, e.g. K = 10, T will be close to the neutral time 1/µ for any reasonable values of the other parameters. Equivalently, the fact that Eq. (2) depends only logarithmically on the parameters means m < 10 for real populations. However, it may be that standing genetic variation significantly changes this picture — see the Discussion and Trotter et al. (2014). Comparing the Eq. (3) to the expected time for an asexual 1−K population to cross the plateau (T ≈ N 1µ2 (s/µ)−2 for N ≪

1/µ, T ≈ Ks ln(s/µ) for N ≫ sK −1 /µK , with additional asymptotic regimes for intermediate population sizes (Weissman et al., 2009)), we see that frequent recombination tends to speed up adaptation in small populations (relative to asexuality), where the primary challenge is producing the beneficial genotype, while slowing it down in large populations, where most of the time is spent fixing the genotype after it has been produced. 4. Analysis 4.1. Rare recombination (f ≪ s)

In this section we will consider the plateau-crossing process in populations with rare recombination, starting with very large populations and progressively decreasing in size. As N decreases, the population’s ability to efficiently explore genotype space (measured by N, µ, and f ) becomes more important, and its ability to exploit its discoveries (s) less so. At the largest population sizes, T is essentially determined by s. For all the lower population size regimes, there will be at least some genotypes that are only rarely produced, and T will be approximately the waiting time for the production of the first successful lineage of a rare genotype. 4.1.1. Very large populations: deterministic dynamics For extremely large population sizes, the number of single-, double-, and triple-mutant individuals are well approximated by their expected values after the first few generations. Triple mutants are produced almost instantaneously, and the plateau-crossing time is dominated by the time it takes them to sweep to fixation. This can be found by solving the deterministic equations for the dynamics of the genotype frequencies under mutation, recombination, and selection (Appendix A.1): T ≈

3 s

( ) ln

s

µ

.

(4)

(Recall that s should be replaced by s˜ when f ≲ s; this slight slowdown is the only effect of recombination in this regime.) It is straightforward to generalize Eq. (4) to arbitrary plateau widths K: T ≈

K s

( ) ln

s

µ

.

For this deterministic approximation to be accurate, the production rate of triple mutants, R3 (t), must be large at the time

t ∼ 1/s when the first triple-mutant lineage reaches number 1/s and becomes established. Triple mutants are produced by double-mutant individuals that either acquire another mutation or recombine with single mutants, so R3 is: R3 (t) = µn2 (t) +

f 24N

n1 (t)n2 (t),

(5)

where ni is the total number of i-mutant individuals, i.e., n1 (t) ≈ 3N µt and n2 (t) ≈ 3N µ2 t 2 . (The factor of 1/24 in Eq. (5) is because to make a triple mutant, each double mutant can only mate with 1/3 of the single mutants, and only 1/8 of the offspring will inherit the correct alleles.) At t ∼ 1/s, Eq. (5) gives R3 ≈ 3N µ3 /s2 . (The recombination term is smaller by a factor ∼ O (f /s).) So to be in this regime, the population must have size N ≫ s2 /µ3 . Note that recombination is almost irrelevant in this regime: mutants are produced so frequently that there is no need for recombination to generate new combinations. (It does slightly slow down adaptation, as technically s should be replaced in the equations by s˜ = s − 78 f .) 4.1.2. Large populations: single- and double-mutants common, triple-mutants rare Slightly smaller populations (with N ≪ s2 /µ3 ) will only occasionally be producing triple-mutants at the time they cross the plateau, so while the single- and double-mutant populations will have nearly deterministic dynamics, fluctuations in the number of triple-mutants will be important. Because triple-mutant lineages are rare, we can consider them in isolation, and T will be the waiting time for the first successful triple-mutant (Fig. 7). Since the probability that a triple-mutant lineage is successful once it has been produced is ∼ s (Ewens, 2004) (ignoring constants of O(1), as we will throughout the paper), the probability that a successful triple-mutant [ lineage will have ] been produced by time t is P3 (t) = 1 − exp −

∫t 0

dt ′ sR3 (t ′ ) . Therefore, the expected

waiting time T is given by:

∫ T =

∞

[

∫

dt exp −s

t

dt ′ R3 (t ′ )

]

0 {0( )−1/3 N µ3 s , f ≪ (N µ3 s)1/3 (asexual path) ≈ ( 3 )−1/4 N µ fs , f ≫ (N µ3 s)1/3 (sexual path).

(6)

In the first line of Eq. (6), the population is effectively asexual, i.e., the successful triple-mutant is likely to arise via mutation from a double-mutant. In the second line, it is more likely to arise via recombination between a double-mutant and a single-mutant. The boundary between the two regimes has a natural interpretation as the point at which recombination becomes likely within the expected time to cross, f T ∼ 1. The two expressions in Eq. (6)

)1

are generally close, differing by a factor of only ∼ f 3 /(N µ3 s) 12 : recombination can provide a mild increase in speed, but as in the previous section, the population is so large that the triple-mutant genotype will rapidly be produced by mutation alone anyway. In deriving Eq. (6), we ignored fluctuations in the number of double-mutants (as well as of single-mutants), so these must be negligible on time scales similar to T for the expressions to be valid. There are two ways that this approximation can hold. First, if the number of double-mutants is in fact close to its expectation with high probability. This will be true if the production rate R2 (t) of double-mutants is high, i.e., R2 (T ) ≫ 1, so that the doublemutant population is composed of many lineages and fluctuations in the individual lineage sizes average out. R2 is given by:

(

R2 (t) = 2µn1 (t) +

f 12N

n1 (t)2

(7)

M. Ghafari and D.B. Weissman / Theoretical Population Biology 129 (2019) 54–67

59

Fig. 7. Typical simulated plateau-crossing dynamics for large populations where T is dominated by the waiting time for the production of the first successful triple-mutant. The inset is a magnified view of the last 50 generations before the adaptive genotype fixes in the population which demonstrates the establishment time and sweep time of the triple-mutants. The parameters are N = 1011 , µ = 10−6 , f = 10−3 , and s = 1.

Fig. 8. Typical simulated plateau-crossing dynamics for moderately large populations where T is dominated by the waiting time for the arrival of the first successful double-mutant lineage. The frequency of sex is high enough that this lineage is typically produced by recombination between two single-mutant lineages. The inset is a magnified view of the last thousands of generations before the adaptive genotype fixes in the population showing that the population of single-mutants remains roughly constant while the double-mutant lineage is drifting. The parameters are N = 1011 , µ = 5 × 10−10 , f = 10−3 , and s = 1.

Plugging in( the first )1/3 line of Eq. (6) for t gives the requirement R2 (T ) ∼ µ N 2 /s ≫ 1 in the asexual regime. In the recombination-dominated regime in the second line of Eq. (6), there is an additional way for the fluctuations to be negligible: recombination can cap their size by preventing them from greatly exceeding linkage equilibrium with the much larger and approximately deterministic wild-type and single-mutant populations. In this case, if R2 ≪ 1 the number of double-mutants will be fluctuating as lineages are sporadically produced and die out, but no one lineage will drift for a time much exceeding ∼ 1/f before being broken up by recombination. We will see what condition this puts on the parameters in the following section, but for now, note that this mechanism requires the single-mutants to be approximately deterministic, so at a minimum we require R1 ∼ N µ ≫ 1. Finally, we must also check the conditions for our assumption that the triple-mutant lineages are rare enough to be considered in isolation. This is equivalent to R3 (T ) ≪ 1 — the flip side to the parameter condition in the preceding section requiring that triple-mutants be approximately deterministic. In the asexual regime, plugging in our expressions for R3 and T , we get µ(N /s2 )1/3 ≪ 1, which is indeed the reverse of the previous condition.

4.1.3. Moderately large populations: single-mutants common, double-mutants rare For populations slightly smaller than those in the previous section, the mutation supply will still be high (N µ ≫ 1), so singlemutants will still be approximately deterministic, but double-mutant lineages will be rare (R2 (T ) ≪ 1) and we must consider their fluctuations. Since they are rare, we can consider each lineage in isolation, and T will be the waiting time for the first successful double-mutant to arise (Fig. 8). A double-mutant lineage can be successful by either mutating or recombining with single-mutants to produce a successful triple-mutant. Since the single-mutants are deterministic (n1 (t) ≈ 3N µt), we can lump these two processes into a single f time-dependent effective mutation rate, µ ˜ ≡ µ + 24N n1 (t) ≈ ( ) 1 µ 1 + 8 ft . Since in this regime we expect the waiting time for the first successful double-mutant lineage to be long compared to the time for which that lineage must drift before producing the successful triple-mutant, we can further treat this effective mutation rate as being approximately constant over each lineage’s lifetime. (If t2 is the waiting time and t ′ ≪ t2 is the drift time, then the expected change in n1 , N µt ′ , will be small compared to the√starting value n1 (t2 ) ≈ N µt2 , as will the standard deviation, ∼ N µt2 t ′ .)

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With this approximation, the problem is reduced to that considered in Weissman et al. (2009): a lineage mutates at rate µ ˜ to a genotype with advantage s. Weissman et al. (2009) also allowed the lineage to have a selective disadvantage; in the present case, the double-mutant lineages have an effective selective disadvantage ∼ f due to being broken up by recombination with the wild type. The probability p2 (t) that a double-mutant lineage arising at time t will be successful is therefore (Weissman et al., 2009):

⎧√ ( ) ⎪ ( ) 1 ⎪ ⎪ ⎨ µs 1 + ft if µs 1 + 81 ft ≫ f 2 8 p2 (t) ≈ ( ) ⎪ ( ) µs 1 ⎪ ⎪ 1 + ft if µs 1 + 81 ft ≪ f 2 . ⎩ f

(8)

8

See Appendix A.2 for the intuition behind Eq. (8). In the first line of Eq. (8), a lineage is most likely to succeed by drifting for long enough to produce many (∼1/s) triplemutants. In the second line, recombination is too frequent and lineages are broken up before they can drift for that long. We therefore see that the condition for being able to ignore fluctuations( in the double-mutant numbers as in the previous section ) is µs 1 + 18 f T ≪ f 2 . Since this case is covered by that section’s

√

analysis, we will now focus on the case p2 (t) ≈ µs 1 + 81 ft where fluctuations are key. Combining the success probability ) ( p2 (t) with the production rate R2 (t) ≈ N µ2 t 2 + 43 ft , we can find the expected waiting time T for the first successful doublemutant (ignoring O(1) constants):

∫

∞

T ≈

(

)

] [ ∫ t ′ ′ ′ dt R2 (t )p2 (t ) exp − 0

{0(

)−1/4 N 2 µ5 s f ≪ (N 2 µ5 s)1/4 (asexual path) ≈ ( 2 5 3 )−1/7 N µ f s f ≫ (N 2 µ5 s)1/4 (sexual path).

(9)

Both R2 and p2 switch from being mutation-dominated to recombination-dominated at time t ∼ 1/f . In the first line of Eq. (9), T ≪ 1/f so the population is effectively asexual. In the second line, T ≫ 1/f so the successful double-mutant is likely both to be produced by recombination and to produce the successful triple via recombination. 4.1.4. Moderately small populations: occasional triple polymorphisms If the mutation supply is low (N µ ≪ 1), then the population will typically be monomorphic. The plateau-crossing time is dominated by the waiting time for a lucky single-mutant lineage that drifts long enough to either fix or encounter additional mutations that allow it to tunnel across the plateau. We will consider the latter process in this section. Call this mutation A, and let TA be the time for which this mutation’s lineage must drift to be likely to be successful; over this time, the lineage will typically reach a size nA ∼ TA (see Appendix A.2). The mutation will manage to drift this long with probability p1 ∼ 1/TA , so the expected T plateau-crossing time is T = 3N 1µp = 3NAµ . Note that if TA > N 1 (or, equivalently, T > 1/µ), the lineage is more likely to fix than tunnel. We now find expressions for the necessary drift time TA . First, we will review the ( asexual )−1/4 process. Weissman et al. (2009) showed that TA ∼ µ3 s (ignoring combinatoric factors) is long enough for the lineage to be likely to acquire two additional mutations (which we will call B and C ) and be successful. The expected time to cross the plateau is thus: T ∼

1 N µ2

( µ )1/4 s

.

(10)

Comparing ( )1/4 TA to N, we see that the population will only tunnel if N µ3 s > 1. This result therefore applies only to populations

within a fairly narrow band of sizes, with the lower limit of validity only a factor of (µs)1/4 smaller than the upper limit (N < 1/µ) — less than three orders of magnitude for realistic parameters. Recombination can speed up tunneling (i.e., reduce the necessary TA ) by allowing the original A lineage to acquire B and C from the ∼ N µTA independent mutant lineages that arise while it is drifting. Let B be the mutation carried by the largest such lineage; it will typically drift for TB ∼ N µTA generations, reaching a size nB ∼ N µTA . If TB ≫ 1/f , recombination will effectively reduce the linkage disequilibrium between A and B, i.e., there will be an average of nAB ∼ nA nB /N ≈ µTA2 AB individuals over most of the TB generations for which both mutations are drifting. During this time, there will be ∼ N µTB C lineages produced by mutation. Since the probability of drifting for at least tC generations falls off like 1/tC (Weissman et al., 2010), the largest of these ∼ N µTB C lineages will typically drift for TC ∼ N µTB ≈ (N µ)2 TA generations, reaching size nC ∼ TC . AB and C individuals will therefore coexist for ∼ TC generations, during which they will f generate ∼ N nAB nC TC ≈ N 3 µ5 fTA4 triple-mutant recombinants. Each of these has a probability ≈ s of being successful, so we see that for our original A lineage to be likely to be successful, its drift time TA must satisfy N 3 µ5 fTA4 ∼ 1/s. Solving for TA gives TA ∼ N 3 µ5 fs of:

(

T ∼

)−1/4

1 (N µ)2

, corresponding to an expected plateau-crossing time

(

N

µfs

)1/4

.

(11)

We refer to this as ‘‘semi-linkage-equilibrium tunneling’’, since the two most frequent mutations are in linkage equilibrium with each other while drifting, but the third mutation may not be, and the triple-mutant will produce large linkage disequilibria once it starts to sweep. The derivation of Eq. (11) assumed that A and B were close to being in linkage equilibrium with each other, i.e., fTB ≫ 1. Substituting in TB ∼ N µTA , this is equivalent to a condition that TA ≫ 1/(N µf ). However, it may be the case that the A and B lineages can produce enough recombinants to be successful before they approach linkage equilibrium. This is true for small values of f , where the time to approach linkage equilibrium becomes very long. In this situation, the analysis here overestimates how large TA must be, and therefore overestimates the time required to cross the plateau. The correct analysis of this ‘‘sexual stochastic tunneling’’ regime is even more involved, and we leave it for Appendix A.3. We also ignored the possibility that the AB individuals might produce a triple mutant directly via mutation. The ∼ µTA2 AB individuals existing for ∼ TB ∼ N µTA generations will typically produce ∼ N µ3 TA3 triple-mutants via mutation. Comparing this to the N 3 µ5 fTA4 produced by recombination, we see that recombination will dominate as)long as (N µ)2 fTA ≫ 1. Plugging in the ( −1/4 expression TA ∼ N 3 µ5 fs derived above shows that this is

]1/5

equivalent to the condition N ≫ s/(µf )3 . When f > (µs3 )1/4 , this is a weaker condition than the condition N ≫ TA required for the population to tunnel rather than fixing the A mutation by drift, i.e., populations that are small enough that the mutational path is more likely than recombination are so small that the A mutation is likely to drift to fixation before either happens. When f < (µs3 )1/4 , it is weaker than the set of conditions for semi-linkage-equilibrium tunneling to be more likely than the sexual stochastic tunneling described in Appendix A.3. Thus, for all parameter combinations where semi-linkage-equilibrium tunneling is likely, it is likely to involve recombination between AB and C individuals rather than mutation of AB individuals.

[

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4.1.5. Small populations: single-mutants drift to fixation For smaller populations, the most likely way for a singlemutant to be successful is for it to drift to fixation, which occurs with probability 1/N. The expected waiting time is therefore T = 1/(3µ). Once the single-mutant has fixed, the population only needs two additional mutations, so Weissman et al. (2010)’s twolocus analysis applies. The average time to tunnel will necessarily be small compared to the time for the first mutant to drift to fixation, so it can be neglected in T . The exception is for very √ small populations, N ≪ 1/ µs, where the second mutation is also more likely to drift to fixation than to tunnel (Weissman et al., 2010). In this case, the total waiting time is T ≈ 31µ + 21µ = 5 . 6µ

(The final fixation of the third mutation is relatively rapid as long as Ns ≫ 1.)

the longest-drifting mutation at a third locus will typically persist for T3 ∼ (m − 2)N µT2 = (m − 1)(m − 2)(N µ)2 T1 , and so on, with the mth mutation persisting for Tm ∼ (K − 1)!(N µ)m−1 T1 . The frequency xm of the m-mutant genotype will peak at: xm ∼

m ∏

(Tk /N)

k=1

( )m (m − 1)!m−1 m−1 T1 ∼ (N µ) 2 , N G(m + 1) where G is the double gamma function. For the mutations to establish, this peak frequency must exceed ∼ 1/Ns. Solving this condition for T1 gives the time scale over which the first mutation must drift to be likely to be successful: 1

T1 ∼ N(Ns)− m (N µ)−

4.2. Frequent recombination (f ≫ s)

4.2.1. Large populations (N µ ≫ 1): deterministic dynamics When the mutation supply is large, N µ ≫ 1, the mutant allele frequency trajectories are nearly deterministic, and therefore almost the same as each other, i.e., a single variable x(t) can describe the frequency of all the mutant alleles. When the mutations are rare (x ≪ 1), their frequencies increase according to: (12)

where xK is the frequency of the beneficial genotype. Solving Eq. (12) for t such that x(t) ≈ 1 gives the time to cross the plateau: T ≈

1 ( µ )1/K

µ

s

,

m−1 2

m

e− 4 (m − 1)

m−4 2

,

−m 4

If recombination is frequent (f ≫ s), selection will be too weak to generate linkage disequilibrium, and the population will stay close to linkage equilibrium (LE). We can therefore simply track allele frequencies, rather than genotype frequencies. For this much easier problem, we can consider plateaus of arbitrary width K . Note that this regime still exists in more general recombination models that include linkage as long as the recombination rates among all pairs of loci are large compared to s.

x˙ ≈ µ + sxK ,

61

(13)

where we have assumed µ ≪ s. We can understand this as the time it takes for mutation to drive the mutations to the frequency x ∼ (µ/s)1/K at which point selection takes over, after which fixation is rapid. 4.2.2. Small populations (N µ ≪ 1): sequential fixation + stochastic tunneling of mutant alleles When the mutational supply of the population is small (N µ ≪ 1), most loci will usually be monomorphic, with occasional drifting mutant lineages. To cross the plateau, the population needs some combination of mutations drifting to fixation, and others producing the beneficial genotype and tunneling together. We can think of the tunneling dynamics as allowing the population to ‘‘see’’ the adaptive genotype once the dominant genotype is within m mutations of it, for some m. We must first find how the maximum tunneling range m depends on N, µ, and s. As in Section 4.1.4, a population can cross the plateau via a rare mutant lineage that grows to a large size over an extend period of time. Suppose that such a lineage persists for ∼ T1 generations, typically growing to size ∼ T1 . There will be ∼ (m − 1)N µT1 mutations at other loci during this time, the largest of which will typically persist for T2 ∼ (m − 1)N µT1 while the original allele is still drifting. During T2 ,

(14) m−4 2

are approxwhere the final combinatorial factors e (m − 1) imations valid for large m, and negligible for small m. For the initial mutation to be more likely to tunnel than to fix, T1 must be small compared to N. Solving T1 ∼ N for m therefore gives the maximum tunneling range, Eq. (2)):

√

⌊ ( m≈

1

1+

1+

2

8 ln(Ns)

|ln(N µ)|

)⌋ ,

where ⌊.⌋ is the floor function. If m ≥ K , then a wild-type population can tunnel directly to the beneficial genotype. The probability that a mutation successfully tunnels is ∼ 1/T1 , so the expected waiting time is: T ≈

T1 KN µ 1

≈ N(Ns)− K (N µ)−

K +1 2

K

e− 4 (K − 1)

K −6 2

,

(15)

where we have substituted Eq. (14) with m = K for T1 . If 1 < m < K , the total plateau-crossing time is dominated by the time it takes for the population to fix K − m mutations via drift so that it can get close enough to the adaptive genotype to tunnel the rest of the way. (If m = 1, then the population cannot tunnel and must fix K − 1 mutations by drift, at which point the K th mutation becomes beneficial.) The kth mutation fixes after an expected waiting time of 1/(K − k + 1)µ, so the total expected waiting time for K − m mutations to fix is K −m

T ≈

≈

∑

1

k=1

(K − k + 1)µ

ln(K /m)

µ

for K − m ≫ 1.

(16)

4.2.3. Three-mutation plateaus Plugging in K = 3 to the above analysis, the expected time to cross the plateau is (with O(1) combinatorial factors included for clarity):

T

≈

⎧ 1 ⎪ ⎪ (µ/s)1/3 ⎪ ⎪ µ ⎪ ⎪ ( ) −1 ⎪ ⎪ ⎪ 3N µ2 (2Ns)−1/3 ⎪ ⎪ ⎪ ⎪ ⎨ 1

−1/4

1/(3µ) + (µ s) ⎪ ⎪ N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 5/(6µ) ⎪ ⎪ ⎪ ⎩ 3

for N µ ≫ 1 (deterministic) for (µ3 s)−1/4 ≪ N ≪ 1/µ (pure tunneling) for (µs)−1/2 ≪ N ≪ (µ3 s)−1/4 (tunneling after one mutation fixes) for N ≪ (µs)−1/2 (sequential fixation).

(17)

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The second term in the third line is the K = 2 tunneling time (Weissman et al., 2010). The final line is from Section 4.1.5; in the sequential fixation regime, since there are no simultaneous polymorphisms, recombination is irrelevant. 5. Discussion We have shown that even moderately large populations can acquire complex adaptations requiring three individually neutral mutations substantially faster than would be expected if mutations had to fix sequentially by drift. In other words, natural selection can at least somewhat effectively promote mutations that not only provide no direct selective benefit, but also do not directly increase evolvability, i.e., do not change the distribution of mutational effects. Considered together with the combinatorial argument for the potentially huge number of available complex adaptations, this raises the possibility that complex adaptations may be widespread in nature, particularly in large viral populations with high mutation rates. Recombination helps most at intermediate population sizes, where there can be simultaneous polymorphisms at multiple loci but triple mutants are rare. In this range, the rate of plateau-crossing is maximized when the rate of recombination is comparable to but smaller than the selective advantage conferred by the adaptation Across regimes, the rate of crossing the three-mutant fitness plateau scales sub-cubically in the mutation rate, i.e., complexity is not strongly suppressing the rate of adaptation, suggesting that even more complex adaptations could also potentially be acquired, particularly with moderate levels of sexual reproduction. However, analyzing even the three-mutation case for f ≲ s involved a proliferation of different dynamical regimes, so simply extending our analysis to wider plateaus is likely to be impractical. The asexual case (Weissman et al., 2009) and the case f ≫ s analyzed above are simpler but plateau-crossing is fastest for f ≲ s, meaning that these easier limiting cases may be missing essential dynamics. 5.1. Multiple mutations The sub-cubic dependence on the mutation rate indicates that the tunneling dynamics described here are a far more probable mode of adaptation than one individual simultaneously acquiring all the necessary mutations independently. But in fact mutations need not be independent, particularly for loci within a single gene. Suppose that a fraction α of mutational events affect multiple nucleotides within a range of l bases of each other on the genome, and that a fraction β of these events affect more than two nucleotides. Then a potential complex adaptation involving three mutations within l bases will arise directly due to multinucleotide mutation at a rate of ∼ µαβ/l2 per individual per generation. (The factor of 1/l2 is because the mutations must be precisely placed on the necessary nucleotides.) It is estimated that α is on the order of a percent in eukaryotes; while β and l are harder to measure, values of β ∼ 10% and l ∼ 100 would be consistent with observations (Schrider et al., 2011, 2013). This gives a rate of ∼ 10−7 µ, far larger than µ3 for cellular organisms, and giving rates of adaptation comparable to the slower rates of tunneling shown on the left-hand sides of Figs. 3 and 4. This suggests that multinucleotide mutations could play an important role in crossing fitness plateaus where the necessary mutations are, for example, all to the same protein domain or local RNA structure. While tunneling rates in the optimal regions of parameter space are substantially faster than pure multinucleotide mutation, hybrid paths involving lineages that acquire two mutations in a

multinucleotide event before drifting long enough to acquire the third via recombination or additional mutation may be fastest of all. It is unclear what fraction of potential complex adaptations involve loci close enough to each other to be affected by multinucleotide mutations. On the one hand, it seems reasonable that epistatic interactions should be more likely among loci within the same gene. But on the other hand, our original combinatorial argument motivating the consideration of complex adaptation applies much more strongly to widely-space loci: there are ∼ LK possible combinations of these, as compared to only ∼LlK −1 possible combinations of nearby loci. 5.2. Implications for natural and experimental populations How practically important could adaptive paths across threemutation plateaus be? Could we hope to observe experimental populations following them? Viruses often have large populations and high mutation rates; if we consider an RNA virus with a mutation rate of ∼ 10−4 per base per replication (Acevedo et al., 2014) and a potential adaptation providing a ∼ 10% fitness advantage (higher than typical rates of sex (Neher and Leitner, 2010)), a population size of N > 109 – fewer than might be present in a single infected host (Haase et al., 1996) – would be enough for the population to deterministically acquire the triple-mutant genotype within tens of generations. On the other hand, if we consider a yeast population in which the relevant mutations have target sizes of ∼ 300 bases, for an effective mutation rate of ∼ 10−7 (assuming a per-base mutation rate of ≈ 3.3 × 10−10 (Lynch et al., 2008)), it would be difficult to maintain a large enough experimental population for long enough to reliably acquire a triple-mutant complex adaptation via any of the paths we have described here. To apply our results to natural populations, it is important to be clear about what is meant by the population size N. Often when discussing natural populations, it is used interchangeably with the ‘‘effective population size’’ Ne inferred from pairwise diversity, i.e., the mean pairwise coalescence time. If this number is similar to the simple census size, there is no ambiguity, but it is often much smaller. But when there is a mismatch, the Ne that is ‘‘effective’’ for one process (pairwise coalescence) is generally not ‘‘effective’’ for other processes (Weissman and Barton, 2012), e.g., the number of Drosophila melanogaster individuals who can contribute to the population’s adaptation is far larger than the usual Ne (Karasov et al., 2010). Roughly speaking, under many reasonable models of population dynamics, the faster a process occurs, the larger its corresponding ‘‘effective population size’’ is, ranging from the standard Ne for processes taking ∼Ne generations to the census size for processes occurring in a single generation (Cvijović et al., 2018). Note that the relevant time scales in our case are the drift times of the mutant alleles, not the waiting time for the first successful one. In the tunneling regimes, alleles are rare and therefore the drift times are ≪ Ne , but they could still be much larger than 1 (Figs. 7 and 8). This would suggest an intermediate effective size. But since the different lineages typically have very different drift times, there may be no one single effective size that can accurately describe the dynamics, and a new approach may be required that explicitly models the additional sources of noise in allele trajectories — see below. As a starting point, we can consider both extreme limits for N (the heterozygosity Ne and the census size). For humans, this gives us a range of possible values for N between ∼104 and ∼1010 . If the relevant mutations are standard point mutations, we have µ ∼ 10−8 . The probability of fixation of a beneficial

M. Ghafari and D.B. Weissman / Theoretical Population Biology 129 (2019) 54–67

combination is unlikely to be more than s ∼ 0.1. We are therefore somewhere between the sequential fixation regime and the asexual double-mutants deterministic regime, potentially including the sequential fixation regime for weaker adaptations. We can be on either side of the f ∼ s divide, depending on how close the mutations are to each other on the genome. The most optimistic parameter values therefore put us in the sexual double-mutants deterministic regime, with a time to cross the plateau of T ∼ 3 × 103 generations. This is not much smaller than Ne , which means that it would be inconsistent to use N ≫ Ne to describe past evolution, but it might still be valid for future evolution. The potentially many more combinations that involve mutations on different chromosomes and are in the f > s deterministic alleles regime take much longer, T ∼ 5 × 105 generations. Very tightly linked mutations, i.e., those in a single gene, are in the asexual single-mutants deterministic regime and are similarly slow, with T ∼ 2 × 105 . Together, these results suggest that evolution is best able to find the unusual complex adaptations that rely on mutations that happen to be near each other on a chromosome without being in the same gene. The most pessimistic parameter values put us in the sequential fixation regime, where the population has no ability to ‘‘see’’ complex adaptations. 5.3. Standing genetic variation Throughout our analysis, we have assumed that the population is initially monomorphic. This is a reasonable approximation for populations with low mutation supplies (N µ ≪ 1), but we have found that plateau-crossing is most rapid in populations with high mutation supplies (N µ ≫ 1). In some such populations, the approximation may still be reasonable, such as the experimental microbial populations and infections discussed above, which are often recently descended from a single individual. To the extent that the gap discussed above between the standard Ne and the census size N is caused by fluctuating demography, our analysis may be widely applicable, when taken to apply to the period immediately following a bottleneck. (This would be the case for the future human evolution example discussed above.) But in populations with high, stable mutation supplies, every locus should almost always be polymorphic. Our calculated rates of plateau-crossing are therefore underestimates. For frequent recombination, f ≫ s, the problem of complex adaptation from standing variation is addressed in Trotter et al. (2014), who found that it may be common, depending on the number of potential adaptations. For rare recombination, f ≪ s, our calculation is still basically accurate for extremely large (deterministic) populations, where the timescale of adaptation is essentially set by the strength of selection, s. For moderately large populations however, the true rates of plateau-crossing may be much higher than those found here. Accurate calculations would require considering the steady-state spectrum of genotype frequencies and how it shifts over time — roughly the approach of Trotter et al. (2014), but made much more complicated by the importance of linkage disequilibrium. 5.4. Other potential adaptations and environmental shifts The major limitation in seriously applying any of our analysis to real populations is that we have considered the necessary loci in isolation. As mentioned in the Introduction, a major part of our motivation in considering the possibility of complex adaptation is that a combinatorial argument suggests that there are potentially very many of them available. Thus, even if the rate of acquiring any particular complex adaptation may be very low, the rate of

63

acquiring any complex adaptation may be high. But if there are in fact very many possible complex adaptations, then the first one that actually fixes in the population is likely to be one that happened unusually quickly, potentially by different dynamics than those considered here (Weissman et al., 2010). Thus at a minimum, we would need to consider the distribution of plateaucrossing times rather than just the mean. More precisely, we would need to describe the left tail of the distribution. This may in fact simplify the analysis – there may be only a few ways for a lineage to get a lot of mutations quickly, regardless of the population parameters (Weissman et al., 2010) – and thus provide a way forward to analyzing wider plateaus. The fact that the population is likely to be adapting at more than just K loci does not only mean that we need to think about the left tail of crossing-time distribution; it also means that we need to think about how adaptation elsewhere in the genome may affect evolution at the focal loci. If there is substantial fitness variance due to genetic variation in the rest of the genome and limited recombination, the dynamics of the mutant lineages can be completely different due to hitchhiking (Neher and Shraiman, 2011; Kessinger and Van Cleve, 2018). In addition, the complex adaptation may be lost due to clonal interference once it is produced, reducing its fixation probability. The fixation probability in this case is likely to require a careful calculation in its own right, as the background fitness is likely to systematically differ from the mean because of the required conditioning on long-lasting lineages carrying the intermediate mutations. Perhaps even more importantly than clonal interference is the potential epistatic interference. When we consider just the K focal loci, substitutions at other loci should turn the fitness landscape into a constantly shifting ‘‘seascape’’ (Mustonen and Lässig, 2009). To a first approximation, this should not make any difference to the overall rate of plateau-crossing, as each substitution will push the population away from some complex adaptations and towards others. But to put the combinatorial argument on a firmer footing, we have to think about how different complex adaptations may interact with each other, or, to put it another way, how they are arranged in genotype space. In one extreme, we could imagine a set of K -adaptations evenly spaced around the initial wild-type, so that any mutation would take the population towards some peak. In the other extreme, we can imagine the same number of adaptations all clustered on one side, sharing many of the same mutations, so that if the population starts moving in the wrong direction it will never find any of them. Implicit in this second extreme is the idea that other mutations outside the focal K may disrupt the potential complex adaptation, so that adaptation may not simply be a matter of getting the right mutations but also avoiding the wrong ones. We have no understanding of how this should affect the probability of complex adaptation in anything beyond the simplest possible case of a single beneficial mutation blocking a two-mutation complex adaptation in an asexual population (Ochs and Desai, 2015). We can already see that these considerations are likely to substantially change the interpretation of our results by looking at Table 2 and our results for generic K with f ≫ s and N µ ≪ 1. The regimes in which the population only tunnels through the last mutations while initially fixing the others via drift appear to have roughly the same rate of plateau-crossing as the sequential fixation regime in which all mutations but the very last must drift to fixation. But this is because our model assumes that all populations reach the adaptive genotype eventually. In a more realistic model in which populations can get diverted and miss potential adaptations entirely, being able to tunnel through m mutations greatly increases the zones of attraction of adaptive genotypes in the fitness landscape, and could make a large difference in the probability of finding them.

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In addition to epistatic interactions with other loci, the plateau could shift because of environmental changes (Masel, 2006; Kim, 2007). It is difficult to say which process is likely to be more important. We currently do not even know whether changes in the selective coefficient of single mutations are driven more often by environmental changes or changes in the rest of the genome, let alone what drives changes in selection on the rest of the genome. More generally, the basic difficulty in analyzing more complex, realistic fitness landscapes is that we have no idea what they should look like. Even mapping out the local fitness landscape of a single gene requires a heroic experimental effort (e.g., Bank et al., 2016) — and then we only know it in a limited number of artificial environments. Our best hope may be to try to develop a theory that can reduce the full complexity of landscapes to a reasonable number of parameters describing their features that are most relevant for adaptation, but it is an open question whether such a theory exists. Acknowledgments DBW was supported in part by a Mathematical Modeling of Living Systems Investigator award from the Simons Foundation, USA. MG would like to thank Dr. Christopher Illingworth for hosting during his stay in Cambridge. Appendix A.1. Deterministic dynamics with rare recombination When recombination is rare (f ≪ s) and the population is so large that there are many K -mutants produced in time ≪ 1/s, i.e., before selection has a chance to boost their frequency, the genotype frequencies are close to the solutions of the deterministic evolution equations. When all mutations are rare, such that the wild-type frequency is close to one, and we neglect recombination, these are: x˙ 1 ≈ K µ x˙ 2 ≈ (K − 1)µx1

.. . x˙ K ≈ µxK −1 + sxK .

(18)

Here xk is the total frequency of all genotypes with k mutations. For k < K , the solution is:

( ) xk (t) ≈

K

k

(µt)k .

Equivalently, the frequency of each specific k-mutant genotype is ≈ (µt)k . Notice that there is no linkage disequilibrium generated (except for xK ), so this is also the solution to the more complicated equations including recombination. For the frequency xK of the beneficial combination, there is an exact solution to Eq. (18) in terms of the incomplete gamma function, but it is more useful to consider an approximate solution describing the two phases of its evolution. Initially, the first term in Eq. (18) dominates, and xK approximately follows the neutral trajectory xK (t) ≈ (µt)K . After time t ≈ K /s, the second term becomes larger, and the frequency begins growing exponentially:

( xK (t) ≈

Kµ es

)K

est .

Solving for xK (t) ≈ 1 and dropping numerical corrections inside the log gives Eq. (4) for K = 3 and the equation below it for generic K .

A.2. Intuition for multitype branching process dynamics Here we briefly describe some of the basic intuitions that help understand the dynamics of near-critical branching processes, as in Section 3 of Weissman et al. (2009) and Appendix A of Weissman et al. (2010). Consider a lineage of individuals that reproduce via binary fission at rate 1 and disappear at a rate 1 + δ , where 0 ≤ δ ≪ 1. Note that disappearance need not be due only to death — in particular, in the present context it can also be due to recombination. Since the net growth rate of the lineage is ≤ 1, it must eventually go extinct. But suppose that at rate µ ≪ 1 individuals in the lineage also produce type-2 individuals whose lineages can avoid extinction with probability s. Again, µ is not necessarily mutation — it could be migration to a new patch (Hermsen and Hwa, 2010), or, in the present context, recombination with a third type that is at roughly constant frequency. Now there is some probability that the original lineage succeeds in leaving (type-2) descendants at infinity. This probability can easily be found exactly using a first-step analysis (see Appendix C of Weissman et al., 2009), but because this branching process model is usually an approximation to some more complicated system of interest, it is helpful to also have an intuitive understanding of the dynamics to make sure that the approximation is valid. Assume that the focal lineage starts from a single individual. Until time t ∼ 1/δ , the dynamics are approximately neutral. The lineage avoids extinction up to time t with probability ∼ 1/t, and conditional on avoiding extinction typically grows to size n(t) ∼ t. This gives ∼ t 2 individual-generations, and therefore an expected number ∼ µst 2 of successful type-2 lineages produced. √ A lineage that manages to drift for t ≳ 1/ µs generations is thus likely to be successful, and our focal lineage has a probability √ ∼ √1/t ∼ µs of lasting this long. Note that we must have 1/ µs ≪ 1/δ for our assumption of approximately neutral dynamics to be consistent. For t ≳ 1/δ , the growth defect makes it exponentially unlikely √ that the lineage avoids extinction. So if 1/ µs ≫ 1/δ , i.e., δ 2 ≫ µs, the lineage’s best chance of success is to last for ∼ 1/δ generations, producing ∼ µ/δ 2 type-2 lineages. The probability that at least one of these avoids extinction is ∼ µs/δ 2 ≪ 1. Multiplying this by the probability δ that the original lineage persists for ∼ 1/δ generations gives the total probability of success, µs/δ . A.3. Moderately Small populations with rare recombination Here we focus on populations with low mutation supply, N µ ≪ 1, and rare recombination, f ≪ s. In particular, we focus on those that fall in between the asexual and semi-linkage equilibrium cases discussed above, for which recombination is frequent enough to speed plateau-crossing but too rare to bring even the largest mutant lineages into linkage equilibrium with each other. As above, the expected plateau-crossing time is dominated by the waiting time for the production of the first successful single-mutant lineage A which drifts for time TA , with the other possible mutations labeled B and C . All genotypes that drift for a time TX reach a typical size nX ∼ TX , so we will not need to distinguish between drift times and lineage sizes in the following. We will exploit our freedom in labeling the B and C mutations to always label the second-largest single-mutant lineage as carrying the B allele. Throughout, we will ignore O(1) numerical factors arising from combinatorics and integration, none of which change the results significantly. We can identify eight possible asymptotic scenarios, depending on the relative sizes of the different relevant lineages (Fig. 9):

M. Ghafari and D.B. Weissman / Theoretical Population Biology 129 (2019) 54–67

65

Fig. 9. A schematic plot of 8 different ways in which the plateau-crossing can occur given that all the mutant alleles are not frequently produced in the population. In case (i), the two bigger lineages, A and B, recombine with each other to produce the AB lineage which in turns recombines with the C lineage to produce the first successful ABC lineage after TA generations. Cases (ii) and (iii) also correspond to the same dynamics as case (i) except for that the second and third biggest mutant lineages could be different. Case (iv) occurs when lineages A and B recombine to produce an AB lineage which later mutates to produce the beneficial mutant. In both cases (v) and (vi), the A lineage mutates to produce the AC lineage which then recombines with the B lineage to produce the successful ABC lineage. In case (vii), the B lineage mutates to produce a BC lineage which later recombines with A. Case (viii) occurs when the B and C lineages recombine to produce a BC lineage which then recombines with the A lineage.

(i) TA ≫ TB ≫ TC ≫ TAB : While three independent mutant lineages are drifting, the larger two recombine, and then that recombinant recombines with the third lineage to produce the successful triple-mutant. Typical sizes are TB ∼ N µTA , TC ∼ N µTB , and TAB ∼ Nr TA TB TC (because we only consider the largest AB lineage that arises while C is drifting). The number of ABC individuals produced by recombination between C and AB during f 2 the ∼ TAB generations that they coexist is ≈ N TC TAB ; we need this quantity to be ∼ 1/s for success to be likely: fs 2 TC TAB N ( )2 ) f fs ( 2 3 ∼ (N µ) TA (N µTA ) N N

1∼

∼

fs N fs N

2 TB TAC

(N µTA )

(

f N

(N µ)

4

TA3

)2

≈ N 6 µ9 f 3 sTA7 . Solving for TA and the other drift times gives:

⎧ ( )−1/7 TA ∼ N 6 µ9 f 3 s ⎪ ⎪ ⎪ ( )−1/7 ⎨ TB ∼ µ2 f 3 s/N ( 3 )−1/7 ⎪ TC ∼ f s/(N 8 µ5 ) ⎪ ⎪ ( )−1/7 ⎩ TAB ∼ N 2 µ4 f 2 s3 .

1∼

≈ N 5 µ8 f 3 sTA7 . Solving for TA and the other drift times gives:

⎧ ( )−1/7 TA ∼ N 5 µ 8 f 3 s ⎪ ⎪ ⎪ ( )−1/7 ⎨ TB ∼ µf 3 s/N 2 ( )−1/7 ⎪ TC ∼ f 3 s/(N 9 µ6 ) ⎪ ⎪ ( ) ⎩ −1/7 TAB ∼ N µ3 f 2 s3 .

we need:

(19)

(ii) TA ≫ TB ≫ TC ≫ TAC : While three independent mutant lineages are drifting, the largest recombines with the smallest, and then that recombinant recombines with the middle lineage to produce the successful triple-mutant. Typical sizes are TB ∼ f N µTA , TC ∼ N µTB , and TAC ∼ N TA TC2 . To get ∼ 1/s triple-mutants,

(20)

(iii) TA ≫ TB ≫ TAB ≫ TC : Two single-mutant lineages recombine. While that recombinant double-mutant is drifting, a third single-mutant lineage arises and recombines with it to produce a successful triple-mutant. Typical sizes are TB ∼ N µTA , f TAB ∼ N TA TB2 , and TC ∼ N µTAB . To get ∼ 1/s triple-mutants, we need: 1∼

∼

fs N fs N

TAB TC2 (N µ)2

(

f N

∼ N 4 µ8 f 4 sTA9 .

(N µ)2 TA3

)3

66

M. Ghafari and D.B. Weissman / Theoretical Population Biology 129 (2019) 54–67

need:

Solving for TA and the other drift times gives:

⎧ ( )−1/9 TA ∼ N 4 µ8 f 4 s ⎪ ⎪ ⎪ ( )−1/9 ⎨ TB ∼ f 4 s/(N 5 µ) ( 2 )−1/3 ⎪ ⎪ ⎪TAB ∼( N µ fs )−1/3 ⎩ TC ∼ fs/(N 2 µ) .

1∼ (21)

(iv) TA ≫ TB ≫ TAB : Two single-mutant lineages recombine, and that recombinant lineage then mutates and succeeds. Typical f sizes are TB ∼ N µTA and TAB ∼ N TA TB2 . The AB lineage will 2 produce ∼ µTAB mutants while it is drifting; setting this equal to ∼ 1/s gives: 1∼µ

2 sTAB

∼ µs

(

f N

(N µ)

2

TA3

)2

∼ N 2 µ5 f 2 sTA6 . Solving for TA and the other drift times gives:

fs N fs

(

(viii) TA ≫ TB ≫ TC ≫ TBC : While three single-mutant lineages are drifting, the smaller two recombine. The recombinant then successfully recombines with the largest single-mutant linf eage. Typical sizes are TB ∼ N µTA , TC ∼ N µTB , and TBC ∼ N TB TC2 . To get ∼ 1/s triple-mutants, we need:

)2

1∼

∼

N fs N

TAC TB2

( )2 µTA2 N µ2 TA2

∼ N µ5 fsTA6 . Solving for TA and the other drift times gives:

⎧ ( 5 )−1/6 ⎪ ⎨TA ∼ (N µ fs ) −1/3 TAB ∼ N µ2 fs ⎪ ( 2 )1/3 ⎩ TC ∼ N µ/(fs) .

(

)2

⎧ 1 ( 3 3 )−1/7 ⎪ ⎪ µ f s TA ∼ ⎪ ⎪ N ⎪ ⎨ ( µ3 3 )−1/7 TB ∼ µ f s ( )−1/7 ⎪ ⎪ TC ∼ N µ µ 3 f 3 s ⎪ ⎪ ⎪ ( )1/7 ⎩ TBC ∼ N µ5 /(f 2 s3 ) .

(23)

(vi) TA ≫ TAC ≫ TB : A single-mutant lineage mutates. While the resulting double-mutant lineage drifts, a new lineage with a mutation at the third locus arises and successfully recombines with it. Typical sizes are TAC ∼ µTA2 and TB ∼ N µTAC . To get ∼ 1/s triple-mutants, we need: fs

2 TA TBC

TA N 4 µ5 fTA3 N ∼ N 7 µ10 f 3 sTA7 .

Solving for TA and the other drift times gives:

⎧ ( 2 )−1/5 1 ⎪ ⎨TA ∼ (µ N fs ) 1/5 TB ∼ N 3 /(fs) ⎪ ( )1/5 ⎩ . TAC ∼ N /(fs)2

fs N fs

(26)

For all of these cases, the expected plateau-crossing time is T ∼ TA /(N µ). All require that the double-mutant drift times TAB or TBC be small compared to 1/f , so that the lineage is not broken up by recombination. We collect the predicted rates and conditions here:

2 TB TAC

N µTA N µ2 TA2 N ∼ N 2 µ5 fsTA5 .

∼

(25)

Solving for TA and the other drift times gives:

(v) TA ≫ TB ≫ TAC : While two single-mutant lineages are drifting, the larger one mutates at the third locus. This doublemutant then recombines with the other single-mutant lineage. Typical sizes are TB ∼ N µTA and TAC ∼ µTA TB (because we only consider the largest-double mutant lineage that arises while C is drifting). To get ∼ 1/s triple-mutants, we need: 1∼

⎧ ( 3 6 )−1/5 ⎪ ⎨TA ∼ (N µ fs ) 1/5 TB ∼ N 2 /(µfs) ⎪ ( 4 3 )1/5 ⎩ TBC ∼ N µ /(fs)2 .

∼ (22)

2 TA TBC

N 4 µ6 TA5 N Solving for TA and the other drift times gives:

∼

1∼

⎧ ( 2 5 2 )−1/6 ⎪ ⎨TA ∼ (N µ f s ) −1/6 TB ∼ f 2 s/(N 4 µ) ⎪ ⎩ −1/2 . TAB ∼ (µs)

fs N fs

(24)

(vii) TA ≫ TB ≫ TBC : While two single-mutant lineages are drifting, the smaller one acquires an additional mutation at the third locus. This double-mutant lineage then successfully recombines with the larger single-mutant lineage. Typical sizes are TB ∼ N µTA and TBC ∼ µTB2 . To get ∼ 1/s triple-mutants, we

T −1

⎧ ( )1/7 ⎪ ⎪ N µ N 5 µ8 f 3 s , ⎪ ⎪ ⎪ ⎪ ( 6 9 3 )1/7 ⎪ ⎪ Nµ N µ f s , ⎪ ⎪ ⎪ ⎪ ( 4 8 4 )1/9 ⎪ ⎪ ⎪ Nµ N µ f s , ⎪ ⎪ ⎪ ⎪ ( ) 1/6 ⎨ , N µ N 2 µ5 f 2 s ∼ ( ) ⎪ 1 / 5 ⎪N µ2 N 2 fs ⎪ , ⎪ ⎪ ⎪ ( 5 )1/6 ⎪ ⎪ ⎪ N µ N µ fs , ⎪ ⎪ ⎪ ⎪ ( ) ⎪ 1 / 5 ⎪ ⎪ N µ N 3 µ6 fs , ⎪ ⎪ ⎪ ( ) ⎩ 1 / 7 (N µ)2 µ3 f 3 s ,

N ≫ f 5 /(µs)3 (i) N ≫

√

f 5 /(µ4 s3 ) (ii)

N ≫ f 2 /(µ2 s) (iii) f ≪

√

µs (iv)

N ≪ s /f 2

3

(27)

(v)

N ≫ f /(µ s) (vi) 2

2

N ≪ s2 /(µf )3

(

) 5 1/7

N ≪ s3 /(µf )

(

)1/4

(vii) (viii).

For parameter values where multiple cases apply, the predicted T value is the one corresponding to the case with the smallest TA — the rates for the different cases do not add. This is because they are all are dependent on the same initial dynamic of an unusually long-lived single-mutant, and the number of triplemutants produced by any path scales with a large power of TA , so roughly speaking any single-mutant lineage that is big enough to have a chance of succeeding via a sub-dominant path will be big enough to definitely succeed by the dominant one. If even the smallest TA is greater than N, single-mutants are more likely to fix than tunnel. For most reasonable parameter values, multiple different cases give similar values in Eq. (27), i.e., populations are not in the true asymptotic regimes corresponding to one case or another. In this situation, any of the roughly equivalent values can be used to predict T .

M. Ghafari and D.B. Weissman / Theoretical Population Biology 129 (2019) 54–67

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