Natural selection drives populations towards higher fitness, but crossing fitness valleys or plateaus may facilitate progress up a rugged fitness landscape involving epistasis. We investigate quantitatively the effect of subdividing an asexual population on the time it takes to cross a fitness valley or plateau. We focus on a generic and minimal model that includes only population subdivision into equivalent demes connected by global migration, and does not require significant size changes of the demes, environmental heterogeneity or specific geographic structure. We determine the optimal speedup of valley or plateau crossing that can be gained by subdivision, if the process is driven by the deme that crosses fastest. We show that isolated demes have to be in the sequential fixation regime for subdivision to significantly accelerate crossing. Using Markov chain theory, we obtain analytical expressions for the conditions under which optimal speedup is achieved: valley or plateau crossing by the subdivided population is then as fast as that of its fastest deme. We verify our analytical predictions through stochastic simulations. We demonstrate that subdivision can substantially accelerate the crossing of fitness valleys and plateaus in a wide range of parameters extending beyond the optimal window. We study the effect of varying the degree of subdivision of a population, and investigate the trade-off between the magnitude of the optimal speedup and the width of the parameter range over which it occurs. Our results, obtained for fitness valleys and plateaus, also hold for weakly beneficial intermediate mutations. Finally, we extend our work to the case of a population connected by migration to one or several smaller islands. Our results demonstrate that subdivision with migration alone can significantly accelerate the crossing of fitness valleys and plateaus, and shed light onto the quantitative conditions necessary for this to occur.
Natural selection drives populations towards higher fitness (i. e. reproductive success), but crossing fitness valleys or plateaus may facilitate progress up a rugged fitness landscape. Rugged fitness landscapes arise from epistasis, i. e. interactions between genetic variants. For instance, two mutations together can yield a benefit while each of them alone is detrimental: such reciprocal sign epistasis can give rise to a fitness valley [1], [2]. While the high dimensionality of genotype space makes it challenging to probe the structure of fitness landscapes [3], [4], evidence has been accumulating for frequent landscape ruggedness, especially in recent years [1], [2], [4]–[15]. Population structure can play an important role in evolution [16]–[24]. In particular, the time taken to cross a fitness valley or plateau depends on population size since stochastic effects such as genetic drift have an increased importance in small populations, allowing neutral and deleterious mutations to fix with increased probability [25]–[28]. Population subdivision into demes can allow the maintenance of larger genetic diversity due to increased genetic drift as well as to the quasi-independent explorations of the fitness landscape that are run in parallel by each deme. Subdivision may thereby facilitate valley or plateau crossing locally and subsequent migration can then spread beneficial mutations throughout the entire subdivided population (" metapopulation' ' ). This idea was first discussed by Wright in his shifting balance theory [29]–[32] and the importance of this effect has been the subject of a long debate [33]–[42]. In this work, we investigate the role of subdivision with global migration alone, without additional effects such as strong dependence of deme size on fitness, including extinction and refounding of demes, which played a crucial role in Wright' s theory. Our generic and minimal model enables us to quantatively determine the conditions under which population subdivision accelerates fitness valley or plateau crossing. Studying quantitatively the effect of subdivision on evolution may help in inferring fitness landscape structure from evolution experiments [43]. Work on structured populations has been used as qualitative proof of landscape ruggedness [16]. Current experiments investigating the evolution of subdivided populations at various migration rates have produced mixed results, some demonstrating faster adaptation of subdivided populations [44], and others not [45]. It is therefore important to determine under what conditions subdivision accelerates fitness valley or plateau crossing. Additionally, population subdivision is extremely common in natural systems. For instance, evidence has recently been found for compartmentalization of HIV in different organs of a single patient [46], [47]. Here we show that subdivision can significantly accelerate fitness valley or plateau crossing over a wide parameter range, both with respect to a non-subdivided population and with respect to a single deme. Intuitively, deleterious or neutral intermediate mutations may fix in individual demes, allowing for the maintenance of a larger proportion of these mutants in a metapopulation than in a well-mixed population. We first determine the optimal speedup of valley or plateau crossing by subdivision, in the best possible scenario where valley or plateau crossing by the metapopulation is driven by that of its fastest deme. This enables us to demonstrate that isolated demes must be in the sequential fixation regime for subdivision to significantly accelerate crossing. We then determine the conditions under which the best possible scenario can be realized. Using Markov chain theory, we obtain analytical expressions for the parameter range where valley or plateau crossing by a metapopulation is as fast as that of its fastest deme. Our analytical predictions are verified using stochastic simulations. Furthermore, we discuss the effect of varying the degree of subdivision of a population, and investigate the trade-off between the magnitude of the optimal speedup and the width of the parameter range over which it occurs. Finally, we extend our work to weakly beneficial mutations and to a population connected to smaller islands, and we discuss the magnitude and robustness of the effect for realistic parameter values.
We focus on asexual individuals, characterized by their genotype and associated fitness. Each individual has a division rate proportional to, and a death rate, which is the same for all. We consider an ensemble of identical demes, each with a constant number of individuals. The division rate averaged over the individuals of a deme is thus equal to the death rate. We treat migration as a random exchange of two individuals between two different demes, occurring at rate per individual. In our model, exchange between any two demes is equally likely, as in Wright' s" island model' ' [29]. This constitutes a generic and minimal model of subdivision with migration, without any dependence of migration rate on the average fitness of a deme (in contrast with models where demes containing beneficial mutants increase significantly in size and migrate more rapidly [30], [33]), or additional effects of extinction and re-founding of demes [30], [32], [33], specific geographic structure [16], [17], [19]–[21], or spatially heterogeneous environments [18], [22]–[24], on which previous studies focused. We consider the simplest fitness valley or plateau, involving three successive genotypes denoted by ‘0' , ‘1' and ‘2' (see Fig. 1A). The initial genotype is taken as reference for fitness: . We denote the fitnesses of the subsequent genotypes by and. The first mutation is assumed to be either neutral (), which yields a fitness plateau, or deleterious (), which corresponds to a fitness valley, while the second mutation is assumed to be beneficial (). We focus on first mutations that are not too strongly deleterious: . We only allow forward mutations, and note that including back mutations does not qualitatively affect crossing times [28]. Finally, we assume that all mutations have probability per division, but generalization to different mutation probabilities is straightforward. In this paper, we focus on the average time required for the whole metapopulation to cross the fitness valley or plateau, i. e. to fix mutation ‘2' in all demes, starting from an initial state where all individuals have genotype ‘0’. For small enough migration rates, each deme in the metapopulation performs a quasi-independent trial at crossing the valley or plateau. At best, the valley or plateau crossing time of the whole metapopulation is dominated by that, , of the “champion' ' deme in the metapopulation, i. e. the deme that crosses the fitness valley or plateau fastest. We now focus on this best possible scenario, which is illustrated schematically in Fig. 1B: first, the champion deme crosses the valley or plateau by sequential fixation, and then the beneficial mutation rapidly spreads by migration of through the whole metapopulation. Once this best possible scenario is characterized, the crucial question will be whether, and under what conditions, it can be attained: this point will be addressed in the following section. The previous section was dedicated to the study of the best possible scenario, where the valley or plateau crossing time of the whole metapopulation is dominated by that, , of the champion deme in the metapopulation (i. e. the one that crosses fastest). We now determine analytically the conditions under which this best possible scenario is attained. For this, we focus on migration rates much smaller than division/death rates, , such that fixation or extinction of a mutant lineage in a deme is not perturbed by migration. In addition, we assume that isolated demes are in the sequential fixation regime, since we showed above that it is a necessary condition for subdivision to significantly accelerate crossing, and that it is a sufficient condition for subdivision to accelerate crossing in the best scenario. In a nutshell, migration must be rare enough for demes to evolve quasi-independently, but frequent enough to spread the beneficial mutation rapidly. The analytical results below allow for predicting the range of migration rates such that subdivision maximally accelerates valley or plateau crossing. We now present numerical simulations of the evolutionary dynamics described above, which enable us to test our analytical predictions, and to gain additional insight in the process beyond the optimal scenario. Our simulations are based on a Gillespie algorithm [48], [49], and described in detail in Methods, Sec. 1. Let us first focus on the example presented in Fig. 1D, which shows an example plot of as a function of the ratio of migration to mutation rates, , obtained through our simulations when varying only the migration rate. With the parameter values used in this figure, the interval of Eq. 14 is. Note that here, and in the following examples, we use the general expressions of and given in Methods, Sec. 3, to compute the interval of Eq. 14. Fig. 1D features a minimum right at the center of this theoretically predicted optimal interval. Moreover, this minimum corresponds to, while: hence, the metapopulation crosses the valley on average 6. 54 times faster than an isolated deme. This is very close to the limit of the best possible scenario, where the metapopulation would cross 7 times faster than an isolated deme (since here). This example illustrates that speedups tend towards those predicted in the best scenario, when the interval in Eq. 14 is sufficiently wide (here the ratio between its upper and its lower bound is 359). Besides, here: comparing it to the above-mentioned value of yields a 3. 47-fold speedup of valley crossing by subdivision. The simulation results in Fig. 1D also show that significant (albeit smaller) speedups exist beyond the optimal parameter window. Fig. 2 shows heatmaps of the valley crossing time of a metapopulation as a function of the migration-to-mutation rate ratio, (varied by varying), and of the fitness valley depth, . Fig. 2A shows that the optimal interval of Eq. 14 (solid lines) describes well the region where the ratio of the crossing time of the metapopulation to that of an isolated deme is smallest and tends to the best-scenario limit. For migration rates lower than those in this interval, the ratio increases when decreases. This can be understood qualitatively by noting that if, is determined by the valley crossing time of the slowest among the independent demes. In the opposite case of migration rates larger than those in the optimal interval, increases with, and it tends to the non-subdivided case, , at high values of, as expected. Above a threshold value of (dashed line), becomes smaller than, in which case large values of, such that tends to, give a low (see Fig. 2A). Fig. 2B plots the ratio of the crossing time of the metapopulation to that of the non-subdivided population, which directly yields the speedup obtained by subdividing a population. It shows that, for the parameter values chosen, subdivision accelerates valley crossing over a large range of valley depths and migration rates, extending far beyond the optimal range given by Eq. 14, and that the metapopulation can cross valleys orders of magnitude faster than a single large population. In addition, above a second, larger threshold value of (dotted line in Fig. 2), isolated demes enter the tunneling regime [28]: Fig. 2B shows that sufficiently above this threshold, the metapopulation no longer crosses the valley faster than the non-subdivided population, as predicted above. While having isolated demes in the sequential fixation regime is a necessary condition to obtain significant speedups by subdivision, the non-subdivided population is not required to be in the sequential fixation regime (see above, and Fig. 1C–D). The value of above which the non-subdivided population enters the tunneling regime is indicated by a dash-dotted line in Fig. 2: significant speedups are obtained both below and above this line. The highest speedups are actually obtained above it, i. e. when the non-subdivided population is in the tunneling regime. With the parameter values used, Eq. 8 predicts a minimum of for (solid line in Fig. 2B), which agrees very well with the results of our numerical simulations. (Note that this value of satisfies, and is such that the non-subdivided population is in the tunneling regime. These conditions were used in our derivation of Eq. 8.) In the Results section, we have shown that having isolated demes in the sequential fixation regime is a necessary condition for subdivision to significantly accelerate crossing. This requirement limits the interval of the ratio over which the highest speedups by subdivision are obtained. The extent of this interval can be characterized by the ratio, , of the upper to lower bound in Eq. 14. Let us express the bound on imposed by the requirement of sequential fixation in isolated demes. If, the threshold value below which an isolated deme is in the sequential fixation regime satisfies [28]. Let us also assume that, and that while, to be in the domain of validity of Eqs. 15 and 16. Combining the condition with the expression of in Eq. 16 yields (19) For plateaus, isolated demes are in the sequential fixation regime if their size is smaller than [28]. In the regime of validity of Eqs. 17 and 18 (while, and,), this condition can be combined with Eq. 18, which yields (20) Both Eq. 19 and Eq. 20 show that increasing the number of demes decreases the range where the highest speedup by subdivision is reached. This is because having more subpopulations makes the spreading of the beneficial mutation slower. In addition, we find that the bound on is proportional to. Hence, despite this bound, the interval where subdivision most accelerates plateau crossing can span several orders of magnitude, given the small values of the actual mutation probabilities in nature. An interesting question raised by our results regards the optimal degree of subdivision. Given a certain total metapopulation size, into how many demes should it be subdivided in order to obtain the highest speedup possible? We first attack this question using our analytical results, and then we present simulation results, which allow for going beyond the best scenario and its associated parameter window. Let us consider a metapopulation of given total size. Our analytical results show that increasing subdivision, i. e. increasing the number of subpopulations at constant, leads to stronger speedups of valley crossing (see Eqs. 4 and 7, with). However, Eqs. 16 and 18, and the previous paragraph, show that when is increased, the parameter range where the speedup by subdivision tends to the best-scenario value becomes smaller and smaller. Eventually, this parameter range ceases to exist altogether: this occurs when becomes of order 1 and below. This sheds light on an interesting trade-off in the degree of subdivision, between the magnitude of the optimal speedup gained by subdivision and the width of the parameter range over which the actual speedup is close to this optimal value. This effect can be observed qualitatively in Fig. 3A, where the valley crossing time of a metapopulation with fixed total size is shown versus the migration-to-mutation rate ratio, , for different values of: when is increased, the minimum becomes deeper but less broad. In addition, Eqs. 15 and 17 show that when is increased, the lower bound of the interval where the speedup by subdivision tends to the best-scenario value decreases, as for plateaus (Eq. 17) and even more rapidly for deep valleys (Eq. 15). Qualitatively, this is because spreading of the beneficial mutation gets longer when increases. Conversely, the upper bound of this parameter range is independent of for deep valleys (Eq. 15), and grows only logarithmically with for plateaus (Eq. 17). Hence, when is increased, the center of the interval where the actual speedup is close to the optimal value shifts towards higher migration rates. This effect, which can be observed in Fig. 3A, is studied more precisely in Fig. 3B: at fixed migration rate, the crossing time of a metapopulation exhibits a minimum at an intermediate value of. Indeed, the crossing time of the metapopulation first decreases when is increased because the minimum crossing time then decreases. But beyond a certain value of, the migration rate that yields the highest speedup becomes larger than the fixed migration rate, so increases when is increased further. Next, we study the dependence on of the valley crossing time minimized over for each, again for a metapopulation with fixed total size. For values of small enough for the interval in Eq. 14 to be broad, we expect to be close to the optimal scenario value. But, as discussed above, as increases, this interval will become smaller and then vanish. In such a regime, our analytical results are no longer sufficient to predict the dependence of on, but our simulations can provide additional insight. Fig. 3C shows that, while (left of the dashed line), is close to the best-scenario value. When is increased beyond this point, decreases slower than the best-scenario value. Indeed, the interval in Eq. 14 is no longer wide enough for the best-scenario limit to be approached. Note also that when demes become small enough, verifying (right of the dotted line in Fig. 3C), mutation ‘1' becomes effectively neutral in individual demes, as tends to (see Eq. 1). For even higher values of, is observed to saturate rather than exhibiting a unique minimum. Interestingly, this occurs for such that the interval in Eq. 14 fully vanishes (i. e. when passes below 1, right of the solid line on Fig. 3C). While we do not have rigorous proof of the generic existence of this saturation, we have explored this point for other parameters, and found similar behavior (data not shown). Importantly, this indicates that there is a whole class of nearly optimal population structures. Our work has focused on fitness valleys (), such that mutation ‘1' is deleterious, and on fitness plateaus (), such that mutation ‘1' is neutral. For, mutation ‘1' is effectively neutral, as far as valley crossing is concerned, in a population with individuals [28]. (This condition holds both in the sequential fixation regime and in the tunneling regime.) This implies that our arguments and our results obtained in the case of the fitness plateau also hold for weakly beneficial intermediates. This point is illustrated in Fig. 4A. Thus far, we focused on demes of equal size for simplicity, but demes of different sizes are relevant in practice. As a step toward more general populations structures, we now consider a population connected by migration to smaller satellite populations of identical size, assumed to be in the sequential fixation regime. We only allow migration between the large population and each of the smaller islands, and the total migration rate is denoted by. The small island affected by migration is chosen randomly at each migration event. It is straightforward to adapt our work to this case (see Methods, Sec. 4). We obtain an interval of over which the crossing time for the large population is dominated by the crossing time of the champion island. This is corroborated by our simulations (see Fig. 4B). Let us consider the example of Escherichia coli, for which the mutation probability per base pair per division is [50]. In order to gain a speedup of crossing by subdivision, we require demes to be in the sequential fixation regime. For plateaus, this condition reads. Let us consider deme sizes such that this condition is satisfied. First, let us choose, which is within the smallest range of sizes used in current evolution experiments. For instance, it is the number of bacteria transferred at each dilution step for small populations in [27]. For this value of, all plateaus with are in the sequential fixation regime (from the condition). Let us also consider, since 96-well plates are often used in these experiments [27], [51]. This yields a total population size of individuals, which is in the tunneling regime for all plateaus with. For, isolated demes are in the sequential fixation regime for. (Subdivision cannot significantly accelerate crossing for deeper valleys since isolated demes are then in tunneling, but those valleys take longer to cross than shallow ones and are thus probably less often crossed in practice.) The ratio of the bounds of the interval in Eq. 14 satisfies throughout this range of valleys, with for the plateau and for the deepest valleys in the range. Thus, actual speedups will approach the best-scenario one, and significant speedups will exist in a wide parameter window. Eq. 8 predicts that the highest speedup is obtained for, and Eq. 9 then yields a speedup factor by subdivision of. (Using instead the full expression of obtained from Eq. 23 (see Methods, Sec. 2) yields, i. e. a correction of 7%.) Moreover, for all valleys with, the best-scenario speedup ranges from 18 to. Thus, subdivision significantly accelerates crossing for this entire class of valleys. It should be noted that the timescales obtained in this example are long compared to experimental ones. For instance, for the plateau, corresponds to divisions while is divisions. However, can become smaller if the number of subpopulations is increased, as discussed in our previous section. Besides, we have chosen to focus on standard Escherichia coli for simplicity. Organisms with a higher mutation rate, e. g. viruses such as HIV, or mutator strains, would have much shorter timescales, but smaller subpopulations would then be required for demes to be in the sequential fixation regime. Our example thus far focused on a small but realistic deme size, . Experimentally more frequent values of are in the range – [27], [51]. Increasing at fixed decreases the range of for which demes are in the sequential fixation regime. For a plateau, this condition reads. For, this yields, and for, this yields. Hence, the range of plateaus (and similarly, of valleys) for which subdivision accelerates crossing becomes more restricted when is increased. Nevertheless, if these increasingly stringent conditions on are satisfied, significant speedups by subdivision are still expected. Indeed, Eq. 9 shows that the smallest value of the ratio is proportional to, so if one increases while decreasing as, the maximal speedup by subdivision will remain unchanged. In this work, we have considered the crossing of one particular valley or plateau corresponding to a specific pair of two mutations. Given the complexity and high dimensionality of actual fitness landscapes, there may be a large number of parallel valleys or plateaus, so that one of these could be crossed quite frequently even though the crossing time for a single valley or plateau remains large. Our work shows that, under specific conditions, subdivision can significantly accelerate crossing for whole classes of valleys and plateaus. Furthermore, in a generic, high-dimensional fitness landscape that contains both valleys and/or plateaus and uphill paths, subdivision can provide an additional effect: it “shields' ' some demes in the metapopulation from adaptation via the uphill paths, leaving them time to explore valley-crossing paths that may be better in the longer term. While this effect is outside the scope of the present paper, it could lead to additional advantages of subdivision in evolution on rugged fitness landscapes. Our study of a generic and minimal model of population subdivision with migration demonstrates that subdividing a population into demes connected by migration can significantly accelerate the crossing of fitness plateaus and valleys, without the need for additional ingredients. We have derived quantitative conditions on the various parameters for subdivision to accelerate crossing, and for the resulting speedup to be maximal. In particular, isolated demes have to be in the sequential fixation regime for a significant speedup to occur. This condition is quite strong, but provided that it is met, significant speedups can be obtained in a wide range of migration rates, with the fastest deme driving the crossing of the whole metapopulation in the best scenario. We have derived the interval of migration rates for which this best scenario is reached. In addition, we have shown that increasing the degree of subdivision of a population enables higher speedups to be reached, but that this effect can saturate. Our quantitative assessment of the conditions under which subdivision significantly speeds up valley or plateau crossing can aid in optimally designing future experiments, enabling one to choose the sizes and the number of demes, as well as the migration rates, such that subdivision can accelerate valley and plateau crossing. Further directions include investigating the evolution of a metapopulation with a distribution of deme sizes on a more general rugged landscape, as well as assessing the impact of specific geographic structure. Our work could also be extended to sexual populations, where recombination plays an important role in valley or plateau crossing [52]. The interplay between recombination and subdivision, which respectively alleviate and exacerbate clonal interference, would be interesting to study.
Our simulations are based on a Gillespie algorithm [48], [49] that we coded in the C language. Here we will describe our algorithm for the case of a metapopulation of demes of identical size, which is the primary situation discussed in our work. In our simulations, each deme has a fixed carrying capacity –we discuss this choice further in this section. In this section, we give more details on the calculation of the average valley or plateau crossing time by the champion deme amongst independent ones. We show in the Results section that, in the best scenario, the crossing time of the whole metapopulation is determined by this time. is the average shortest crossing time of independent demes. This minimum crossing time, which we denote by, is also called the smallest (or first) order statistic of the deme crossing time amongst a sample of size [53]. Let us denote by the probability density function of valley or plateau crossing time for a single deme, and let us introduce (it satisfies where is the cumulative distribution function of valley or plateau crossing by a single deme). The probability that is larger than is equal to the probability that the crossing times of each of the independent demes are all larger than. By differentiating this expression, one obtains the probability density function of the crossing time by the champion deme (see e. g. [53]): (21) We now express explicitly. Since demes are assumed to be in the sequential fixation regime, valley or plateau crossing involves two successive steps. The first step, fixation of a ‘1' -mutant, occurs with rate, and the second step, fixation of a ‘2' -mutant, occurs with rate (see the Results section for expressions of these rates). The total crossing time is thus a sum of two independent exponential random variables, with probability density function given by a two-parameter hypoexponential distribution [53]: (22) Combining Eqs. 21 and 22, we obtain (23) with given by Eq. 22. can then be determined for any value of the parameters by computing the average value of over this distribution. Since mutation ‘1' is deleterious or neutral while mutation ‘2' is beneficial, the first step of valley crossing is much longer than the second one over a broad range of parameter values. In this case, we can approximate with a simple exponential distribution, (24) Eq. 21 then yields (25) i. e. is distributed exponentially with rate. In this case, we simply have, which can be written as, where is the average crossing time for an isolated deme. Hence, in this case, on which our analytical discussion focuses, the champion deme crosses the valley times faster on average than an isolated deme. For this approximation to be valid, the second step of valley crossing must be negligible even for the champion deme, i. e. , . For very large, the value of will not be as small as, since the second step will no longer be negligible (see [52] for a discussion of similar issues). The crossover to this regime can be determined by computing the average of the distribution in Eq. 23 and comparing it to. In our Results section, we have derived an interval of the ratio of migration rate to mutation rate over which subdivision most reduces valley or plateau crossing time (see Eq. 14). The upper bound involves, the average number of migration events required for the ‘1' -mutants to be wiped out by migration, starting from a state where one deme has fixed genotype ‘1' , while all other demes have genotype ‘0' . Similarly, the lower bound involves, the average number of migration events required for the ‘2' -mutants to spread by migration to the whole metapopulation, starting from a state where one deme has fixed genotype ‘2' , while all other demes have genotype ‘0' . In our Results section, we have provided intuitive derivations of the simple expressions of and, valid for and, and (see Eq. 15). However, it is important to derive more general expressions, especially since subdivision generically most accelerates valley crossing in the intermediate regime where (see Results, Eq. 8). Here, we derive general analytical expressions for and, both for fitness plateaus and for fitness valleys. These more general expressions are those used for numerical calculations of the bounds in our examples. Throughout this section, we consider a metapopulation of demes composed of individuals each, and we assume that individual demes are in the sequential fixation regime (see Results). Let us consider a population of individuals connected by migration to smaller population islands with individuals each. These islands of identical size are assumed to be in the sequential fixation regime. For the sake of simplicity, we consider that migration only occurs between the large population and the islands: a migration step is a random exchange of two individuals between the large population and one of the islands (chosen at random at each migration event), and the total migration rate is denoted by. Here, we focus on the valley or plateau crossing time of the large population. We demonstrate that the evolution of a large population can be driven by that of satellite islands. In the optimal case, the crossing time of the large population is determined by that of the champion island, i. e. , that which crosses the fitness valley or plateau fastest. We now determine the conditions under which this optimum is achieved, focusing on migration rates much smaller than division/death rates, , such that fixation or extinction of a mutant lineage in either the large population or an island is not significantly perturbed by migration. Again, migration should be rare enough for islands to remain effectively shielded from migration events while they have fixed the intermediate mutation, until the final beneficial mutation arises. Second, migration should also be frequent enough for the spreading time of the final beneficial mutation from the champion island to the large population to be negligible with respect to the crossing time of the champion island. These two criteria again provide upper and lower bounds on. The average time (with from Eq. 1) required for an island of ‘1' -mutants to fix the beneficial mutation ‘2' must be smaller than the average time, , for an island of ‘1' -mutants to be wiped out by migration from the large population, which still exhibits genotype ‘0' . The rate of migration events between the island of ‘1' -mutants and the large population is. Hence, , where is the probability of fixation of the lineage of a single migrant with genotype ‘0' in an island where all other individuals are ‘1' -mutants: for valleys, it is given by Eq. 1, while for plateaus, it is equal to. The first condition, , thus yields (47) The second condition is that the average spreading time, , for the final beneficial mutation to fix in a large population after it has fixed in the champion island, must be smaller than the average valley or plateau crossing time, , of the champion island. Similar to previously, we obtain, where is the probability of fixation of a migrant with genotype ‘2' in the large population, which is assumed to exhibit genotype ‘0' before migration (see Eq. 1). is the average of the minimum crossing time among independent islands. We again focus, for simplicity, on the limit where the first step of valley or plateau crossing, which occurs at rate, is much longer than the second. Then, we simply have (see Results). In this expression, (with obtained from Eq. 1) is the average crossing time for an isolated island. Hence, the champion island crosses the valley times faster on average than a single isolated island. The second condition, , finally yields (48) Together, Eqs. 47 and 48 yield the interval of over which we expect subdivision to maximally accelerate crossing: (49) In this range, we expect the valley or plateau crossing time of the large population to be dominated by the crossing time of the champion island, so that. This prediction is confirmed by our simulations (see Fig. 4B).
Experimental evidence has recently been accumulating to suggest that fitness landscape ruggedness is common in a variety of organisms. Rugged landscapes arise from interactions between genetic variants, called epistasis, which can lead to fitness valleys or plateaus. The time needed to cross such fitness valleys or plateaus exhibits a rich dependence on population size, since stochastic effects have higher importance in small populations, increasing the probability of fixation of neutral or deleterious mutants. This may lead to an advantage of population subdivision, a possibility which has been strongly debated for nearly one hundred years. In this work, we quantitatively determine when, and to what extent, population subdivision accelerates valley and plateau crossing. Using the simple model of an asexual population subdivided into identical demes connected by gobal migration, we derive the conditions under which crossing by a subdivided population is driven by its fastest deme, thus giving rise to the maximal speedup. Our analytical predictions are verified using stochastic simulations. We investigate the effect of varying the degree of subdivision of a population. We generalize our results to weakly beneficial intermediates and to different population structures. We discuss the magnitude and robustness of the effect for realistic parameter values.
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Quantifying the Role of Population Subdivision in Evolution on Rugged Fitness Landscapes