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The authors have declared that no competing interests exist.

Analyzed the data: DM JE. Wrote the paper: DM JE. Designed the mathematical and computational framework: DM JE Implemented the model: DM.

There is ample empirical evidence revealing that fitness landscapes are often complex: the fitness effect of a newly arisen mutation can depend strongly on the allelic state at other loci. However, little is known about the effects of recombination on adaptation on such fitness landscapes. Here, we investigate how recombination influences the rate of adaptation on a special type of complex fitness landscapes. On these landscapes, the mutational trajectories from the least to the most fit genotype are interrupted by genotypes with low relative fitness. We study the dynamics of adapting populations on landscapes with different compositions and numbers of low fitness genotypes, with and without recombination. Our results of the deterministic model (assuming an infinite population size) show that recombination generally decelerates adaptation on these landscapes. However, in finite populations, this deceleration is outweighed by the accelerating Fisher-Muller effect under certain conditions. We conclude that recombination has complex effects on adaptation that are highly dependent on the particular fitness landscape, population size and recombination rate.

The emergence and persistence of recombination is a long-standing open question in evolutionary biology. Most previous theoretical studies assumed relatively simple fitness landscapes, i.e., simple relationships between allelic states at different loci and fitness. By contrast, empirically determined bacterial and viral fitness landscapes reveal pervasive complex interactions between alleles at different loci. In this study, we explore the effect of recombination on adaptation on fitness landscapes where some trajectories leading to a global fitness peak are interrupted by genotypes of very low fitness. We find that in infinitely large populations, recombination generally reduces the rate of adaptation. However, in finite populations and under certain conditions, recombination can substantially speed up adaptation. Our study provides insights into the effect of recombination on more realistic fitness landscapes. Moreover, it helps gain a better understanding of the dynamics of the spread of adaptive genes in recombining bacterial populations during niche expansion and colonization of new habitats.

Sex and recombination are widespread phenomena in nature

Among other factors, LD can be generated by epistasis and random genetic drift. Epistasis (in fitness) is a deviation of independent fitness effects of alleles at different loci. Magnitude epistasis refers to the case where the direction of selection is independent of the genetic background. Magnitude epistasis can either be positive (intermediate genotypes have a lower fitness than expected from the average of the extreme genotypes) or negative (higher fitness of intermediates). By contrast, with sign epistasis an allele can be selected for or against, depending on the allelic state at another locus

In addition to epistasis, LD can also be generated through stochastic effects in finite populations

In most theoretical studies on the evolutionary consequences of recombination, either no epistasis or only a simple type of magnitude epistasis is considered under which deviations from independence of fitness effects are the same for all genotypes with the same number of deleterious mutations. The topology of these fitness landscapes is smooth. However, empirically determined fitness landscapes are often complex in that some landscapes exhibit pervasive sign epistasis

In this study, we develop a mathematical/computational framework that allows us to examine the recombination effect on a special type of complex fitness landscapes that are characterized by sign epistasis. In these landscapes, we assume a single global fitness peak towards which the population can evolve, but we introduce a number of low fitness genotypes (‘LFG's) that make some mutational pathways inaccessible (or less accessible). Such limited peak accessibility has indeed been reported in some empirically obtained fitness landscapes (e.g.,

Motivated by the wealth of recent studies focusing on fitness landscapes and adaptation in bacteria

We consider a continuous time model of a population of infinite size. Each individual is characterized by a genotype comprising

Darker colors correspond to lower relative finesses. Arrows show point mutation steps directed toward fitter genotypes.

Mutations occur at a constant rate

The fitness

Recombination is assumed to occur through transformation. Cells release free DNA into the environment, and we assume that that (1) all DNA fragments are of length 1 (a single allele), and (2) that the allele frequencies within the pool of free DNA are the same as in the bacterial population. These DNA fragments may be taken up by the bacteria at a rate

Integrating all of the above assumptions, we arrive at the following set of differential equations:_{fix}, of the fixation time in the population with recombination to that in the population without recombination. Thus, _{fix} is a measure for the effect of recombination on the rate of adaptation. In the Supplementary Online Material (Figure S9 in

In order to investigate the dynamics of adaptation in finite populations, we employed a modified version of a previously developed ‘hybrid algorithm’

The size of compartment

The stochastic model converges to the deterministic one when the population size is very large and selection is moderate (see Figure S4 in _{fix}, defined as the ratio of the mean fixation time in the population with recombination to that in the population without recombination.

We start by considering the deterministic evolutionary dynamics in our model, first for the simplest case of only two or three loci, and then for all four-locus fitness topographies. We then investigate the evolutionary dynamics for a subset of the four-locus fitness landscapes in the stochastic model.

In the two-locus case and for given parameters

Panel A shows four two-locus fitness landscapes with no LFG, one LFG (strong sign epistasis) and two LFGs (strong reciprocal sign epistasis). In B, the frequency of the fittest genotype is shown for the three types of fitness landscapes (green: no LFG, blue: one LFG, red: two LFGs), without recombination (solid lines) and with recombination (dashed lines). Plot C shows the corresponding LD dynamics of the three fitness landscapes without recombination. Parameters take the values

In the first landscape, it is well known that with negative epistasis, recombination accelerates fixation of the fittest genotype and with positive epistasis, recombination slows down the adaptive process

When there is a single LFG in the fitness landscape, this implies strong positive sign epistasis and therefore, we would expect that recombination decelerates fixation of the fittest genotype, which is in accord with simulations of this case (e.g.,

Finally, the case of two LFGs has been widely studied, for example in the context of compensatory mutations

For three loci, there are already

We now consider the dynamics in the four-locus case. We define the following standard parameter set: _{fix}. Note that as in the two-locus case, high recombination rates may also prevent fixation of the fittest genotype on some fitness landscapes that are characterized by reciprocal sign epistasis. However, with the relatively low recombination rates that we assume here, the fittest genotype will always become fixed eventually and we therefore only focus on the time to fixation of that genotype rather than whether or not it becomes fixed.

We first investigate how the number of LFGs affects _{fix} (_{fix} across fitness landscapes with the same number of LFGs, indicating that the position of LFGs is crucial for the effect of recombination. There are even some fitness landscapes with a high number of LFGs where recombination has an accelerating effect. This clearly demonstrates that the above heuristic that positive epistasis produces a decelerating effect of recombination is not strictly valid (see also below and the

A) no baseline epistasis (_{fix}_{fix}_{fix}

When there is positive (negative) baseline epistasis, recombination decelerates (accelerates) adaptation in the landscape without LFGs (red dashed lines in _{fix}_{fix} becomes largely independent of the baseline epistasis.

We next explored how the different parameters affect _{fix}. To this end, we again used our standard parameter set and varied one parameter while keeping the others constant. In most of our fitness landscapes, recombination decelerates adaptation and this effect becomes more pronounced with increasing recombination rate (

In all plots, the standard parameter set was used and one parameter was varied. Solid lines shows independently ranked _{fix} values for all fitness topographies. For comparison, the dashed lines show _{fix} in the corresponding fitness landscape with no LFG. A) Effect of recombination rate. Red, green, orange and brown curves correspond to ^{−6}, 10^{−5} and 10^{−4}, respectively. D) Effect of selection coefficient. Orange, red and green curves correspond to

We can also ask to what extent the effect of recombination is a property of a specific fitness topography or an effect of other parameter values. In _{fix} values for two different parameters against each other. These plots indicate that the effect of recombination on the rate of adaptation is fairly robust with respect to the baseline selection coefficient, the mutation rate and the baseline epistasis parameter. However, we see that the effect of recombination rate can vary substantially for individual fitness topographies, and this variation is even more substantial in comparisons between more different recombination rates (e.g., we measured R-Squared

Each point in the above plots represents one fitness topography and its position is given by _{fix}

Our results indicate that LFGs in the fitness landscape have an effect similar to positive magnitude epistasis in that recombination slows down adaptation. We therefore ascertained whether measured epistasis on our fitness landscapes is a predictor for the effect of recombination. To this end, we regressed fitness against the number of deleterious mutations _{fix}. As anticipated, all landscapes are characterized by positive epistasis. However, there is no correlation between this measure of epistasis and the effect of recombination on adaptation, _{fix}. As an example, Figure S3 in

Each point corresponds to one landscape. Parameters take values

Due to computational limitations, an exhaustive study on all possible landscapes analogous to the deterministic part was not possible. Therefore, we randomly sampled 50 fitness topographies with 3, 5 and 7 LFGs and determined the fixation time for all of these topographies. We used the same standard parameter set as in the deterministic model. We focused on the region of the parameter space where

We screened a total of 150 randomly sampled fitness topographies with 3, 5 and 7 LFGs. _{fix}_{fix}

It is also evident that at least for low numbers of LFGs in the fitness topographies (3 or 5), the ranking of _{fix}_{fix}_{fix}

Comparing the different panels in _{fix}

We studied the effect of recombination on the tempo of adaptation. We focused on adaptation on adaptive fitness landscapes with limited peak accessibility, i.e., fitness landscapes with an underlying monotonic gradient of fitness values towards a single global fitness peak but where some genotypes have a very low fitness (see

In the absence of random effects, recombination slows down adaptation on most fitness landscapes. This finding is consistent with analytical results for two-locus fitness landscapes exhibiting a fitness valley

Unfortunately, it is very difficult to predict the impact of recombination on our as well as on other complex fitness landscapes from simple statistics derived from the landscape

The situation becomes more complicated when finite populations are considered. With stochastic mutation and random genetic drift, clonal interference between beneficial mutations at different loci can ensue, so that recombination can accelerate adaptation (the Fisher-Muller effect, which can be considered a special case of the Hill-Robertson effect

We have focused on a particular regime of the parameter space where selection is relatively strong and the number of mutations that arise in the population (

Our model was motivated by recent evolution experiments in bacteria (e.g.,

Only few studies are devoted to investigating the evolutionary effect of recombination on complex fitness landscapes. Here, we observed that including more features besides steepness and curvature in the structure of fitness landscapes results in rich dynamics and complex effects of recombination on the evolutionary process. More work is necessary to elucidate what properties of fitness landscapes are decisive for the impact of recombination and to quantify those properties in empirical fitness landscapes.

Supplemental figures mentioned in the text.

(PDF)

We would like to thank two anonymous referees for helpful comments on the manuscript.