The General Model: Multilocus Introgression at the Migration-selection Equilibrium

Our basic model is a generalization of Barton’s [17] model of low rate gene exchange between two demes. Consider a large population in which a small proportion m of residents is replaced by an equal fraction of immigrants each generation. All immigrants have the same genotype consisting of a finite number of discrete genes, that is, the genomic distance between any two genes far exceeds the size of the gene itself, and recombination is unlikely within the gene. Following migration, there is direct selection against the foreign genotypes in the resident population, which generally becomes weaker as the average number of the foreign genes per individual decreases due to recombination, but remains strong enough to counter-balance gene flow and maintain a migration-selection equilibrium. The total fraction of recombinant individuals is still assumed to be small even after the equilibrium is reached, so that we can safely ignore unions between gametes that both carry foreign genes. Selection is therefore only acting on heterozygotes: even if the immigrant individuals were homozygous, their progeny will be heterozygous after just one round of selection. The population at equilibrium is characterized by the mean population fitness , estimated at the haploid stage, where 1−m is the resident genotype frequency after migration, and is the frequency of the foreign genotype i weighted by its relative fitness . Due to migration and selection, is less than the fitness of the resident genotype , imposing the migration load on the population. Let be the unspecified (for now) probability of transition between the non-identical genotypes labeled i and j, carrying the numbers k and l of foreign genes, respectively, which we will call the lengths of the genotypes. Note that because immigrants only mate with the residents, j cannot carry more foreign genes than i (i.e. , for any j>i), and let be the initial immigrant type, which has the maximum length (Fig. 1). Over time, this transition probability will completely determine the distribution of genotypes within a set of individuals sharing a common ancestry and carrying the introgressed genetic material, which we will call a lineage [29]. Since the transition probability does not change through time, and there is no direct interaction between the foreign genotypes that belong to different lineages, the dynamics of the population can be classified as the multi-type branching process [29]–[31]. We will now demonstrate that the knowledge of the mechanisms of selection and recombination that take place within a single lineage is also sufficient to characterize the genetic structure of the population at migration-selection equilibrium. First, the mean population fitness can be expressed alternatively as:(1)

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 1. A subset of the lineage descending from an immigrant haplotype

carrying 10 genes. After selection, the genotypes carrying foreign genes (solid black dots) recombine with the resident genotypes (shown as the rows of open dots), so that the average number of foreign alleles per genotype is reduced every generation . Once recombination has broken the foreign genotype down to one gene, it is certain to eventually be eliminated by selection in the single foreign allele state.

https://doi.org/10.1371/journal.pone.0088651.g001

The term , which we will call the relative size of the lineage j, represents the contribution to the mean fitness made by all future descendants of the genotype . Correspondingly, is the size of the lineage descending from that survived the first round of selection and did not recombine. The dynamics of the lineage size over one population life cycle is described by:(2)

Eq (2) can be solved for and the solution substituted into (1). The resulting equation makes it possible to express the mean fitness , a population characteristic, in terms of parameters that govern the dynamics within a single lineage.

A Specific Model: Introgression of Linear Blocks

Fitness function with epistasis.

Approximating the multilocus genotypes as continuous gene blocks requires that the fitnesses of individual genotypes must be chosen such so they only depend on the number of foreign alleles and not on their positions. This is a reasonable assumption given a large degree of independence of gene function from the minor changes in its genomic localization. We will parameterize the fitness of a block of length k as follows (Nick Barton, pers. comm.):(8)

We use the ratio , where n is the arbitrarily chosen maximum number of loci that contribute to the local adaptation in the resident environment ( = the length on the foreign chromosome in Barton [17]), to establish a lower limit to the fitness of the worst possible genotype, which for is , where S is the selection acting on n genes. At this point, the fitness is independent of the parameter of epistasis θ. For any smaller block, , however, the fitness is an increasing function of θ, (θ <1), and at θ = 0, selection acts according to the additive scheme, (Fig. 2). When negative, the parameter θ also imposes relatively stronger selection on smaller blocks than on larger blocks (positive epistasis). Positive values of θ in the feasible range of 0< θ <1 impose relatively stronger selection on the smaller blocks than on the larger blocks (negative epistasis, Fig. 2).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 2. Genotype fitness as a function of the block length.

The fitness of the genotype is plotted against the proportion of the chromosome occupied by the foreign genes: corresponds to additive fitness (no epistasis), while and represent the positive and the negative epistasis, correspondingly. The points indicate the fitness of the initial block carrying 4 genes at . Point 1: n = 10, fitness is calculated according to eq. 12. Point 2: n = 10, fitness is rescaled by the length of the initial block. Point 3: initial block carries the maximum number of genes, .

https://doi.org/10.1371/journal.pone.0088651.g002

Allowing for the initial block to carry only a fraction of the maximum number of selectable genes, , makes it possible to compare the introgression of blocks of different lengths within the same parameter space. This is useful in the case where the populations involved in gene exchange have undergone incomplete divergence (i.e. the hypothetical diverging source population is still climbing the corresponding adaptive peak), as opposed to a scenario where further divergence is unlikely (the source population has settled on the peak). If n is constant and is varied, selection pressure on the initial block increases with its length, whereas if , and both are varied, the same selection pressure is distributed among different number of genes, that is, a polygenic trait can be compared to the phenotype controlled by only a few major loci. Unless specified otherwise, the following results were obtained holding n constant and varying , .

Numerical Results: Branching Process and Simulations

The Eqs (1) and (4–5) can be solved numerically to obtain the characteristics of the population at migration-selection equilibrium, termed the Branching Process Approximation (BPA) in the following. Note that the existence of the equilibrium itself depends on the numerical values of the corresponding parameters, and even when the equilibrium exists, the branching process can adequately describe the population dynamics only when the foreign genotypes are rare. While finding the conditions of existence for the equilibria in the multilocus system is outside the scope of our paper, we established the validity of the numerical results by comparing them with the deterministic simulations where some or all of the BPA assumptions were lifted. The times to compute the numerical solutions are of orders of magnitude shorter than those required to perform the corresponding simulations: this allowed for a faster and deeper exploration of the parameter space. We concentrated on low values of the migration (m<10⁻²) and recombination (r <10⁻²) rates, and medium to strong selection (S >0.4), since in this region of parameter space the introgression of the foreign genes is most likely to follow the branching process (Fig. 3A, Fig. 4B). Restricted recombination between loci that contribute to hybrid incompatibility has been inferred among many interbreeding taxa, with most obvious cases often resulting from the chromosomal rearrangement events [33].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 3. Migration load at the migration-selection balance.

A – Comparison of the migration load (L) imposed by the introgression of 2 to 6 genes in the initial block, calculated from the Branching Process Approximation (BPA, green dots), Panmictic Simulation (PS, red dots), Introgressive Simulation (IS, blue dots) and Single Crossover Simulation (SCS, black dots). Parameter values are: S = 0.7, θ = 0.5, m = 0.003, r = 0.015, n = 10. B – Migration load (L), calculated from the BPA, as a function of the length () of the initial block. The curves (from bottom to top) correspond to θ = −10, −3, −1, 0, 0.5, 0.8. Parameter values are: m = 0.003; r = 0.015; S = 0.7; C – the same as in B, but .

https://doi.org/10.1371/journal.pone.0088651.g003

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 4. Distribution of the foreign genotype frequencies at the migration-selection equilibrium.

A – Result of the introgression of the initial block composed of 15 genes, obtained through the Branching Process Approximation (BPA) by solving the Eqs (3) and (8–9) numerically. Note that the actual frequency of the single-gene blocks at θ = 0.8 is much higher than the corresponding bar height on the plot (indicated by the larger font on the Y-axis). B – Comparison of the frequency distributions, with the initial block composed of 6 genes, obtained through BPA (green line) against Panmictic Simulation (PS, red line), Introgressive Simulation (IS, blue line) and Single Crossover Simulation (SCS, black line) as the recombination rate (r) increases. For the simulation results to be comparable with the BPA, different genotypes were grouped together by the number of genes (j) they carry. Each plot represents variation in the frequency of the corresponding gene block (1–6). Parameter values are: S = 0.5, θ = 0.2, m = 0.001.

https://doi.org/10.1371/journal.pone.0088651.g004

When compared to the simulations, the branching process model was found to be a good approximation for the slow introgression at multiple loci [29]. Details are available in the Mathematica notebook (File S1). Note that although Eq (1) for the mean population fitness has k roots, corresponding to k linear block genotypes, only the largest root is both stable and globally attractive, and was therefore used in the analysis.

Distribution of Block Frequencies and the Effective Selection on Individual Genes.

Irrespective of the value of the epistasis parameter, the frequency distribution of the block lengths is always bimodal, with the peaks corresponding to the smallest (k = 1) and the largest blocks (Fig. 4A). Note that the frequency of the largest (initial) blocks is replenished every generation by migration, and we choose to calculate the block frequencies consistently after migration but before selection, i.e. at the same time when the migration load is measured. While estimating the frequencies after selection would result in a decreased frequency of the initial blocks, it otherwise does not affect our conclusions. There is a gradual transition from the domination of the largest blocks under the strong positive epistasis (θ <0) to the domination of the single-gene blocks under the negative epistasis (θ >0, see Fig. 4A). For most part of its feasible range, the largest and the smallest block frequencies are of comparable scale: however, as θ approaches 1, the frequency of the single-gene blocks increases very rapidly: for example, at (θ = 0.8, S = 0.7, r = 0.01 and m = 0.001) p₁ is an order of magnitude higher than p₁₅ (Fig. 4A). To demonstrate the extent of introgression by a single variable, we calculated the effective selection pressure on the individual genes, s* [17]. This is simply the ratio of the migration rate, m, to the average weighted frequency of the single alleles at equilibrium, :(9)where(10)

Note that here accounts for the non-additive selection on the individual alleles, whereas in Barton [17] only the additive fitness was considered. The effective selection pressure roughly shows how much of the foreign genetic material is present in the resident population, relative to the number of migrants that arrive every generation. It takes the maximum value of 1 if there is no introgression (i.e. with positive migration rate, m, the fitness of the initial block, , must be 0) and drops down to m if gene flow swamps the resident population. We found that the effect of epistasis on the effective selection pressure is different from that on the distribution of block frequencies above. For the most part of the feasible range of θ, s* stays almost constant, but then drops abruptly when epistasis becomes strongly negative (θ >>0): this corresponds to the rapid increase in the frequency of the single gene blocks in the same parameter range (Fig. 4A, Fig. 5A). The change in s*, however, is much more gradual when the size of the initial block is less than the maximum allowed block length , Fig. 5B).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 5. Effective selection (s*) on the individual genes.

A – s* as a proportion of the total selection on the initial block, with the number of genes in the initial block varying from 1 to 15, and epistasis θ ranging from −9 to 0.8. The maximum number of genes is fixed at n = 15. B – the same as in B, but the maximum number of genes, n, is set to equal in each case. Points along the curve for 15 loci correspond to the frequency distribution plots on A. Other parameters: S = 0.7, r = 0.01, m = 0.001.

https://doi.org/10.1371/journal.pone.0088651.g005

Invasion of the Modifiers

Since we are ignoring the gene positions on a linear block, it is not possible to estimate the strength of selection acting on the individual foreign genes. It is feasible, however, to follow the fate of the marker gene x located at a considerable distance from the foreign block, such as that the recombination rate between the marker and the edge of the linear block, , is much larger than the rate of recombination within the block itself . The barrier to the gene flow at the neutral marker linked to a block of genes under selection has been calculated by Barton and Bengtsson [18], by considering a matrix of all possible foreign backgrounds which the marker can be associated with. We use the same method here, but assume that the marker is a novel mutation equally likely to originate on both the foreign and the resident backgrounds. Instead of focusing on the barrier to gene flow, we investigate the fate of the mutation that reduces, to a certain degree, the deleterious effect of the foreign genes in a resident environment. In the fitness formulation used here, such mutation (modifier) can alter two key parameters in the Eq. (8): the strength of selection (S) and epistasis (θ), by the corresponding replacements and . We assume that the modifier’s effect is manifested at the diploid stage, and since the foreign genes are found only in heterozygotes, the mutation is analogous to the modifier of dominance [34]. Note that in case the dominance at several loci needs to be modified at once: in principle, this is possible in polygenic systems with a few genes responsible for most of the variation in a trait and the remaining variation being affected by a much larger number of loci. For example, the distinctive wing coloration patterns in a butterfly Heliconius erato are controlled by three loci of major effect, expressed early in the wing development, while multiple QTLs shape the minor details during the later stages of development [35]. A mutation targeting genes early in the pathway would therefore interact with multiple genes expressed later on. Alternatively, the modifier can affect the recombination rate on its background genotype by the replacement , without changing the fitness of the foreign genotype [36]. It follows from Eqs (5–6) that reduction of the recombination rate always limits the introgression of the foreign block into the resident population, thus improving the population mean fitness. The following recursions describe dynamics of the frequency of the modifier on the resident background:(11)and the frequency of the modifier associated with the foreign block of length l :

(12)Here, the «hat» superscript indicates the frequency of the corresponding type after migration took place, e.g. for all and for ; is the fitness of the foreign block associated with the modifier of selection or epistasis. For the first time, we consider the frequency of the resident gametes, calculated simply as . Since the modifier effect is manifested in diploids, the mean population fitness has to be estimated after the unions of gametes, and is calculated as:(13)

In the case where modifier only alters the recombination rate, the fitness of the block remains unchanged:, , but the background recombination rate r is replaced by ρ in the presence of the modifier. Differentiating the system (11–12) in respect to results in a matrix of linear coefficients: the leading eigenvalue of this matrix, λ, represents the strength of selection acting on the modifier [20] while the population is still in the migration-selection equilibrium. As follows from the Eqs (11–12), the dynamics of the invading modifier is determined both by its own effect on the background genotype and by the population structure/parameters at the migration-selection equilibrium.

Modifiers of selection, epistasis and recombination in a common parameter space.

We systematically explored the parameter space in the Eqs (5–6) and (11–12) and numerically estimated the leading eigenvalue of the invasion matrix λ. Note that in our parameterization, the selective advantage due to the modifier mutation cannot exceed the fitness of the resident genotype: the modifier invasion rate λ is therefore bounded by the difference between the (background) parameters of the migration-selection balance (i.e. S, θ, r) and their corresponding altered values in the presence of the modifier . That is, the favorable conditions for the invasion of selection, epistasis and recombination modifiers are , and , respectively. The effects of background parameters on the invasion rates of all three modifier types are complex: we investigate these to some extent in the SI. In the following, we concentrate on the effect of the modifier’s own properties, and arbitrary choose common set of background conditions that allow for the comparison between the different modifier types.

In general, the modifier of selection and epistasis are able to invade with the comparable rates under a wide range of conditions. The parametric plot on the Figure 6A shows that increasing the selection and epistasis differentials have similar effects on the invasion threshold λ = 1: the space where invasion is possible gets narrower as the linkage between the modifier and the foreign gene block (r_x) becomes loose (Fig. 6A). Even an unlinked modifier that simultaneously confers large selective advantage and switches the epistasis from positive to negative can invade the resident population. Reaching the invasion threshold for the unlinked modifier of recombination, however, requires much more stringent conditions, most of all a very high background recombination rate (r ≈ 0.2). At these values of r, our model based on the branching process is no longer a good approximation to the multilocus introgression (Fig. 3A, 4B): we therefore assumed a tighter linkage to the foreign block (r_x = 0.1) to ensure that invasion is possible in the following examples. Throughout the parameter space, the invasion rate λ is an exponentially increasing function of the altered epistasis parameter, , and a linear function of the modifier selection parameter, . In the example shown on Fig. 6B, the maximum strength of selection acting on the modifier approaches (but can never exceed) the migration load. The background parameter set on Fig. 6B intersects with that on Fig. 6C, where λ is plotted as a function of the altered recombination rate ρ. Although comparable, the invasion rate of the recombination modifier is much less than that of selection/epistasis, and the threshold for invasion is only reached if the recombination on a block is heavily suppressed or completely arrested (ρ ≈ 0). Note that due to our model being a good approximation only when the linkage between foreign loci is sufficiently tight, we could not compare the case where the modifier of recombination is expected to spread with the fastest rate, i.e. where the recombination between the unlinked immigrant genes is arrested by the modifier. To find out if the modifier arresting recombination (from the value of r = 0.05) on the larger blocks is favored over that on the smaller blocks, we set ρ = 0 and estimated λ against k_π under the varying background conditions. The invasion rate was indeed increasing with the initial block length, though still being of order of magnitude smaller than the corresponding migration load (Fig. 6D). Under the negative epistasis (θ >0) the modifier arresting recombination on very large blocks (k_π >10) invades marginally slower than that on the blocks of moderate length: this parallels reduction in migration load imposed by large initial blocks (Fig. 3, Fig. 6D).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 6. Selection acting on the invading modifier.

A – view of the parameter space with the modifier selection strength and epistasis plotted on the X and Y axes, the leading eigenvalue of the invasion matrix (λ) equals one along the lines shown for the following values of (top to bottom): 0.5, 0.3, 0.1, 0.05. The modifier is able to invade in the space above each line, and cannot invade below the line(s). Other parameters: S = 0.7, θ = 0.2, m = 0.003, r = 0.001, k_π = 5. B – λ as a function of the modifier epistasis, , plotted on the bottom axis; and selection strength , plotted on the top axis. The migration load (which is independent of ) is shown by the horizontal line on the top. Solid lines indicate k_π = 5, dotted lines indicate k_π = 7. Other parameters: S = 0.7, θ = –6, m = 0.001, r = 0.05, r_x = 0.1, n = 10. C – Invasion rate of the modifier of recombination, as a function of the imposed recombination rate ρ. Markers indicate different values of the background epistasis, solid lines: k_π = 5, dotted lines: k_π = 7. Other parameters are as in B. D – Modifier of recombination invades non-monotonically with the length of the initial block, k_π. Solid lines: k_π = 5, dotted lines: k_π = 7; markers indicate different values of the background epistasis. Migration load is shown at the top of the plot. Other parameters: S = 0.7, m = 0.003, r = 0.05, r_x = 0.1, n = 12.

https://doi.org/10.1371/journal.pone.0088651.g006

Comparison to Barton (1983)

Modeling introgression as a branching process where the linear gene blocks, explicitly characterized by the number of genes they carry, are being broken by recombination goes back to Barton [17]. He treated the resident population as (i) an unlimited pool of mating partners, which (ii) has essentially no effect on the dynamics of branching lineages. In the present paper, we kept the first assumption, but lifted the second one: that is, the foreign genotypes were still allowed to mate freely, and exclusively, with the residents, but selection against the migrants was normalized by the mean fitness of the resident population, . While Barton’s study mostly concerned the balance between selection and recombination rates, which ultimately determined whether introgression is limited or indefinite, we concentrated on the effect of the (limited) introgression on the resident population structure. Unless the immigrant lineages can be treated as independent, valid approximation of this effect by the branching process is not possible: this is why our numerical results are presented for such parameter ranges (low migration and recombination rates, and strong selection) that keep the total fraction of the introgressed genetic material low at equilibrium. Other distinctive feature of our model is that multiple crossovers per gene block are allowed, rather then just a single crossover [17], [29]. While our approach leads to correct (binomial-like) distribution of the daughter block lengths following recombination of the initial immigrant block of genes, it underestimates the recombination rate at the discontinuous daughter blocks that contain «gaps» between the foreign genes. The single crossover model, however, gives an incorrect (uniform) distribution of the daughter block lengths, and ignores the discontinuous blocks altogether. The position-independent model, and the single crossover model, respectively, under- and overestimate the frequencies of the small blocks, but the differences between the two models are small as long as the recombination rates are low (Fig. 3A, 4B).

In our model, selection on different genotypes within the immigrant lineage depended significantly on the epistatic interations within a gene block. Yet, the effect of introgression estimated on the population level (migration load, Fig. 3) and on the level of individual genes (the effective selection pressure s*, Fig. 5A,B), revealed that epistasis only has a strong overall effect if the deviation from additive fitness extends to the initial immigrant genotype. If, however, the initial genotype is insensitive to epistasis, as in the case of , the consequences of fitness non-additivity are much more subtle, and are visible primarily on the shape of the block frequency distribution (Fig. 4A). This is because the smaller blocks represented at higher frequency under the negative epistasis experience relatively weaker selection than they do under the posisitive epistasis, despite being less abundant in the latter case. Our results thus suggest that under a specific assumption that epistasis is manifested in the offspring of the initial block but has little of no effect on the initial block itself, the same amount of load and per-gene effective selection can be generated with very different observable patterns of introgression.

Alternative Microevolutionary Processes at the Migration-selection Balance

In the presence of the maladaptive gene flow, selection should favor those resident genotypes that have less chance of being associated with the immigrant genes. An allele that contributes to a stronger reproductive barrier between the residents and the migrants, through genetic, ecological, behavioral or other incompatibility, should therefore invade: current theory and empirical evidence suggest that that reinforcement or prezygotic isolation can occur in many systems [10], [37], though it is not particularly likely in a one-way migration scenario [9]. Alternatively, a rare variant, which improves fitness while being found with the immigrant allele, relative to the most widespread type in the population, can also be favored: whether this mechanism operates in nature remains to be demonstrated, but models of the evolution of dominance in spatially heterogeneous environment [21], [25] and invasion of gene duplication in response to gene flow [26] suggest that it is at least likely under a wide range of conditions. Moreover, an analysis of fitness gradients for mate choice versus the modification of dominance [38] suggested that the latter is more strongly selected when allele frequencies at the single locus with heterozygote disadvantage are unequal: the situation where rare migrants are introduced into a large resident population satisfies this criterion. Here, we showed that robustness to the deleterious effect of immigrant alleles in a multi-locus system can evolve at the migration-selection balance, but the question of whether this scenario poses a likely alternative to the reinforcement of the reproductive barriers remains open. A direct comparison of selection strengths acting on the modifier(s) of assortment [22] with the modifiers of selection/dominance/recombination is therefore needed: a study on this subject is currently under way.

Evolution of Recombination Rate

In a multilocus system, a strong barrier to gene flow exists even in the absence of any mechanisms of pre-mating isolation [18]. This is due to the fact that recombination is generally a much slower process than selective elimination of individuals carrying the maladaptive foreign alleles in linkage disequilibrium: physically reducing the rate of recombination between the foreign genes would therefore make the barrier even stronger. A number of models of migration-selection balance suggest that suppression of recombination/clustering of the locally adaptive loci in the genome is favored by selection [27], [28], [39], [40]. Here, we have demonstrated that tightening linkage between the immigrant genes will help to eliminate them from the population early in the lineage, thus decreasing the proportion of the population that carries any deleterious material. The modifier that reduces the recombination rate will have a relatively higher chance to be preserved on the resident background and therefore is favored by selection (Fig. 6C,D). However, under the premise that the linkage on the introgressing block is already tight, further reduction, or even complete arrest of recombination cannot be selected too strongly. This caveat could explain why a direct comparison between the invasion rates of the modifiers of selection/epistasis and the modifier of recombination revealed much weaker selection for the latter (Fig. 6B,C).

Effect of the Initial Block Length

Despite the fact that, after recombining into the resident gene pool, the foreign blocks of different length are selected independently at any single time point, selection on the larger blocks negatively affects the smaller blocks in a descending lineage. Increasing the number of genes in the initial block has therefore a dual effect on the lineage size: even though maladapted genes carried by large chunks of the genome are inherited by a larger number of descendants, elimination of the large blocks early in the lineage can be so effective that their impact on the equilibrium population fitness, measured as the migration load, becomes less relative to that caused by the introgression of the blocks carrying moderate number of genes (Fig. 3). This non-monotonic effect on the migration load is also manifested on the invasion rates of various modifier types, though with much weaker intensity (Fig. 6D). While a modifier mutation rescuing a (single) large block has a stronger selective advantage, it can still have less chance of being associated with the foreign alleles if the introgression is limited by the initial block size. Note that the standard matrix method of calculating the invasion rate employed here does not distinguish between the modifier originating on any specific genetic background. An alternative method, and the only feasible approach of examining the modifier that is tightly linked to the foreign block, would be to calculate the speed of the branching process starting from the modifier mutation associated with the particular genotype, and verify the results against simulations: we are currently working on this. So far, our results only seem to suggest that the gene flow dynamics in heterogeneous populations may depend non-linearly on the size of the genomic region containing the locally adapted loci.

The recent advance of population genomics, with large datasets becoming available for a number of hybridizing taxa, allows for an empirical application of the theoretical results such as [17], [29], [30] and ours. In particular, the introgression of genomic blocks on the flanks of the hybrid zone between the eastern and western subspecies of the house mouse (Mus musculus musculus and Mus m. domesticus) in Central Europe has now been traced in unprecedented detail [41]. Modeling this process based on the existing block size data is currently under way.

Derivation of Eq (1) for Mean Population Fitness

Let us examine a lineage descending from a single initial immigrant genotype introduced at the generation , and assume for simplicity that exactly one immigrant arrives every generation. A subset will represent all members of the lineage present after migration and before selection, at each of the successive generations following the introduction: , where is the generation when the last member of the lineage becomes deterministically extinct. The ultimate extinction of the lineage is guaranteed by our previous assumption that recombination cannot break up the foreign genotypes beyond a single gene; once this limit has been reached; the single gene is guaranteed to eventually be eliminated by selection [42]–[44]. Because the number of generations after which the lineage is sampled, t, is counted relative to the time of introduction , and the migration-selection process is at equilibrium, varying the time of introduction will have the same effect as varying the time of sampling. It is easy to see that in our model, where no interaction between the foreign genotypes is assumed, population at any time can be represented as a set of independent lineages with different times of origin, but otherwise identical (Fig. 7). Moreover, since the initial immigrant genotype is introduced at the same rate every generation, the set of all lineages sampled at time t will be equivalent to the set representing a single lineage:(14)

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 7. Migration load at the migration-selection balance.

The reduction in mean population fitness caused by the descendants of the migrants arrived in the generation 20 is measured every generation that follows (large black dots); this is equivalent to the load caused by the progeny of the migrants arrived in all preceding generations, calculated at point t = 20 (smaller black dots). The mean population fitness at any time can therefore be calculated from the branching process within a single lineage, according to Eq (1).

https://doi.org/10.1371/journal.pone.0088651.g007

Eq (14) will hold for any number of immigrants arriving each generation, as long as the migration rate m is constant. The left-hand side of (14) can be re-written in the frequency representation, as a sum , where is the frequency of the genotype i in the current generation after migration but before selection and is genotype fitness. Adding the fraction of the resident genotype after migration, , we obtain the expression for the mean population fitness, :(15)

We must now re-write the right-hand side of (15) in terms of the relative frequencies of all the genotypes within a lineage, sampled at different time points. It will include the frequency of the initial genotype that immigrated and have been selected against in the current generation, , minus the fraction of the population mean fitness, , occupied by the progeny of destined to become extinct in the future (Fig. 8A), resulting in the Eq (1).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 8. Schematic representation of the terms in Eqs (1) and (2).

A – representing the mean population fitness from the branching process within foreign lineages. The future load caused by the independent lineages (white contour) is subtracted from the fitness contribution of the migrants, measured at the generation of sampling (contour filled by dark grey). Note that the proportion of migrants culled before reproduction, , is not a part of the mean population fitness. B – Dynamics of the lineage size, , determined by the Eq. (2). is reduced by migration and selection, but increased as the foreign genotype recombines and gives rise to sub-lineages. Note that the term , i.e. the contribution of the individuals culled by selection, is included in the lineage size calculation.

https://doi.org/10.1371/journal.pone.0088651.g008

Derivation of Eq (2) for the Lineage Size Z

The concept of the lineage size Z is similar to Haldane’s [44] concept of the number of selective deaths necessary to eliminate a deleterious variant from the population. We calculate Z by adding the contributions made by the lineage members through time, and account for the effects of migration, selection, and recombination at each of the consequent generations. The factor in the right-hand side of the recursive Eq (15) accounts for migration, invariably reducing the lineage size throughout its lifespan. The denominator represents the fraction culled in the present generation, while the last term in the right-hand side of (15) is a product of the fraction that survives and enters the next generation (hence divided by ) and the sum of sizes of the daughter lineages descending from (Fig. 8B). While the explicit knowledge of the transition probability is required to obtain the numerical value of Z, in a degenerative case where there is no recombination within the immigrant genotype, , for all j, the lineage size can be found exactly:(16)and the only stable solution for the mean population fitness is , which is a standard result for one-locus migration-selection balance model [45].

Simulations Used to Verify the Branching Process Approximation

We verified the validity of Branching Process Approximation against the deterministic, frequency-based simulations of the migration-selection process as it was described in the previous sections. The initial population consisted of the resident genotypes only; the foreign genotypes, represented as identical linear blocks of equally spaced genes, were introduced every generation at the rate m, followed by selection and reproduction. The simulations were run using the Multilocus package [46] in Mathematica until the equilibrium was reached, at which point the stable genotype frequencies were compared with the corresponding results of BPA. Three models were used. In the Introgressive Simulation (IS), the foreign genotypes could only mate with the resident genotypes, assuming that there an unlimited supply of the residents. At the stage of selection, however, the frequency of the resident genotype was calculated as , and the mean population fitness as . The IS therefore exactly follows Eqs (1) and (3), which describe a general branching process, without the simplifying assumption of all foreign genotypes are represented as continuous gene blocks. If the initial genotype consists of only one (k_π = 1) or two (k_π = 2) genes, the results of the introgressive simulation match the BPA precisely (Fig. 3A, Fig. 4B), but starting from k_π = 3, the IS includes genotypes that carry the same number of genes located at different positions. The deviation of the IS from BPA increases as the number of genes, migration and recombination rates increase. As a modification of the IS, we included the case where the number of crossovers per block was restricted to one, referred to as the Single Crossover Simulation (SCS) in the following. The probability of recombination on a block of length k was calculated in SCS as (k –1)r, which corresponds exactly to the recombination model used by Barton [17].

In the Panmictic Simulation (PS), all mating types, including those between foreign genotypes, were allowed, and the positions of individual genes were recorded in all genotypes. The PS is the most realistic way to describe the population at the migration-selection balance, following the frequencies of genotypes composed of two (the resident and the foreign) alleles at k loci. Selection in PS was acting on haploids, consistent with the selection regime in BPA and IS. Since matings between the foreign genotypes were still very rare, a switch to diploid selection (acting differently on zygotes that carry foreign genes on both homologous sets) had very little effect on the results. As expected, the PS shows larger deviation from BPA than the Introgressive Simulation, with the differences between PS and IS typically being greater than between IS and BPA (Fig. 3A, 4B). The SCS, however, consistently deviated most from the PS in comparison to the IS and BPA. Nevertheless, the deviation remains small in the parameter range where our numerical results were obtained (Fig. 3A, 4B), suggesting that the branching process model is a good approximation for the introgression at multiple loci [29].

Derivation of the Eq for the Mean Population Fitness at the Diploid Stage

In order to estimate the leading eigenvalue of the modifier’s invasion matrix, we need to calculate the frequency of the resident type, as . This does not affect the frequencies of the foreign gene blocks, since we still assume that the foreign genotypes do not compete which each other for mating with the resident type. The mean population fitness is thus calculated as the frequency of heterozygotes formed by the foreign gametes after selection, , plus the residual fraction of the resident gamete pool that remains after all heterozygotes have been formed, . This simplifies to Eq (13).