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FEDER ET AL. ONLINE SUPPLEMENT
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ADAPTIVE CHROMOSOMAL DIVERGENCE DRIVEN BY MIXED GEOGRAPHIC
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MODE OF EVOLUTION
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ANALYTICAL APPROXIMATION
In addition to computer simulations, we also examined an analytical approach to estimate
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the probabilities of establishment for new inversions suggested to us by S. Yeaman (pers.
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comm.). The approach involved splicing the results of Kirkpatrick and Barton (2006; Eq. 3) for
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approximating the rate of increase in frequency of an inversion,
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λ = 1+[2r/((2r + n-1)ms)](n-1)m
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into Kimura’s diffusion equation for the probability of fixation of a new mutation (see Crow and
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Kimura 1970; Eq. 8.8.3.13) to estimate the probability of establishment of an inversion (Pr[fix]),
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as done with single locus models by Yeaman and Otto (2011). Substituting Kirkpatrick and
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Barton’s (λ – 1) for Kimura’s s yields:
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Pr[ fix] 
1  e 4 N(  1)p
1  e 4 N (  1)
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20  where, m is the migration rate between populations, s is the selection coefficient for each
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individual locus, n is the number of loci, N is the population size, r the recombination rate
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between adjacent loci, and p the starting frequency of the inversion (k/2N).
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Yeaman and Otto (2011) found that this splicing approach provided an accurate estimate
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of the establishment probability of a single mutation. We found, however, that this approach did
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not perform especially well when applied to predict the probability of establishment of an
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inversion. In general, the estimated probability of establishment for the analytical formula was
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on the scale of an order of magnitude (or more) higher than that estimated for the mixed mode
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simulations (see Figs. S1 and S2 for results for an inversion containing four loci under divergent
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selection). This was true whether or not we included a deleterious meiotic effect for single
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recombination events in inversion heterokaryotypes (data not shown). In addition, regardless of
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the rate of gene flow, the analytical approximation was not sensitive to differences in the level of
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divergent selection affecting loci (see dashed lines in Figs. S1 A-C). In contrast, the interplay
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between migration and selection strongly influenced establishment probabilities in the mixed
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mode simulations, as would be intuitively expected (Figs. S1 A-C). When selection is high
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relative to migration rate, reduced recombination is not as strongly favored (locally favored
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alleles are at high frequency for all loci), and the probability of establishment for an inversion
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drops (see right hand side of solid line curves for mixed mode simulations in Figs. S1 A-C). The
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same is true when selection is weak relative to migration, but in this case the cause is that gene
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flow tends to swamp local adaptation (see left hand side of solid line curves for mixed mode
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simulations in Figs. S1 B, C). The analytical and mixed mode simulations results did appear to
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converge, however, with weak selection acting on loci (s = 0.001) and low migration rate (m =
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0.001; Fig. S1 A). Moreover, the probabilities of inversion establishment derived from the
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analytical approximation equation and mixed mode simulations were similarly affected by
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recombination rate (Fig. S2) under conditions of relatively high migrations rate (m = 0.1) and
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moderate selection (s = 0.1). However, despite the curves for the analytical approximations and
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the simulations being similar in shape, the magnitudes of the difference in the probability of
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establishment for an inversion were still over an order of magnitude higher for the analytical
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approximation (Fig. S2).
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We suspect that two factors compromise the analytical approximations. First, the
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conditions most relevant to the mixed mode model, where migration rates (m) and selection
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coefficients (s) are high and s is not >> m, negate a number of simplifying analytical assumptions
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of Kirkpatrick and Barton’s estimate of λ and of this application of Kimura’s diffusion
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approximation. Second, the selective advantage of reduced recombination afforded by an
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inversion changes through time in relation to the genetic composition of populations 1 and 2 for
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locally favored alleles. This is particularly true for the mixed model, where populations are not at
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selection-migration equilibrium for locally favored alleles at the time of secondary contact and
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the rate and degree to which introgressed genes accumulate between populations are prime
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factors influencing the changing selective advantage of the inversion. As the analytical approach
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assumes a fixed s value, this could have compromised its effectiveness. In contrast, our
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simulations allowed for fluctuating selection favoring the inversion. Future analytical attempts to
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estimate the establishment probabilities for new inversions might concentrate on branching
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approximations in which probabilities of transitions are dependent on the states of populations.
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PRESTANDING INVERSIONS IN BOTH POPULATIONS
In the simulations reported in the main text, we considered standing inversion
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polymorphism to be present at low frequency (k = 1 to 200) in only one of the two populations.
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Prior to secondary contact, however, it is possible that both populations 1 and 2 contain standing
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inversion variation for a given genomic region. This will generally increase the probability of
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establishment of an inversion polymorphism (usually by a factor for k approximately equal to the
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sum of the number of inversion copies in the two populations combined prior to contact) (Fig.
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S3). But standing variation in the two populations can complicate the dynamics of the process, as
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it will usually require one or the other of the inversions (usually the one at lower initial
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frequency) to be essentially lost while the other is retained. The issues of standing and partially
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overlapping inversions in both populations prior to secondary contact are topics warranting
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further investigation.
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DELETERIOUS MEIOTIC EFFECTS
The simulation runs described in the main text considered heterokaryotypes to have a
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selective disadvantage of 10-5 due to meiotic irregularities associated with single exchange
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events. Varying the level of this selective disadvantage from 10-3 to 10-7 did not greatly affect our
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results, especially given that divergent selection pressures (s) between populations were over
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several orders of magnitude higher. However, this does not mean that meiotic problems in
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heterokaryotypes are unimportant for the dynamics of chromosomal evolution, even when they
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contribute only slight underdominance to fitness. This is because when migration rates are low
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relative to divergent selection (m < s), negative frequency dependent selection resulting from
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meiotic irregularities can still impede the establishment of an inversion polymorphism. Selection
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favoring reduced recombination is not as strong under this condition, so slight deleterious effects
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in heterokaryotypes can gain in significance for impeding the establishment of an inversion. In
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addition, negative frequency dependent selection against the rarer arrangement can also
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contribute to the fixation of chromosomal arrangements if populations experience a period of
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allopatry following secondary contact and introgression.
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SELECTION ACTING AFTER MATING
Divergent selection acting after mating is generally less effective than selection occurring
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immediately after migration in maintaining genetic differentiation between populations (Fry
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2003; Nosil et al. 2005). This is because when selection occurs after mating, migrant alleles are
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not selected against until after they have a chance to introgress into the gene pool of the alternate
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population in the form of F1 hybrids. But this is not the case when selection occurs prior to
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mating. Here, migrant genes are selected against before they occur in F1 hybrids.
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In the simulations conducted in the main text, we considered divergent selection to occur
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after migration and before mating. However, we also modeled the consequences of selection
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acting after mating for its effects on inversion establishment. Selection acting after mating tended
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to increase the probability of inversion establishment under the mixed mode and sympatric
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origins models for low (m = 0.001) and modest (m = 0.01) levels of migration (Fig. S4). This
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was true because the increased rate of effective introgression for low and modest migration rates
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resulting from selection following mating increased the selective advantage of reduced
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recombination associated with the inversion. For high migration rate (m = 0.1), there was little
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effect of when selection occurred on the probabilities of inversion establishment (Fig. S4), as
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effective introgression was similar between populations whether selection occurred prior to or
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after mating.
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GENE FLUX
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In the stochastic simulations reported in the main text, we did not allow for gene flux
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between inverted and standard arrangements (i.e., there was no double recombination or gene
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conversion in heterokaryotypes). In nature, gene flux does occur between inverted and collinear
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genomic regions. It is not uncommon to observe genetic exchange on the order of 10-6 to 10-9,
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and sometimes much higher, in heterokaryotypes (Navarro et al. 1997; Schaeffer and Anderson
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2005). Allowing for gene flux did not greatly affect the probabilities of retention of an inversion
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in our stochastic simulations (data not shown, spreadsheets of full results available upon
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request). This is because when an inversion is lost, it generally is lost in the first few generations
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after it occurs as a new mutation, especially for the sympatric origins model. At this time, the
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inversion is present at extremely low frequency in rare heterokaryotypes. Thus, under the
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sympatric origins model, if a new inversion failed to capture all locally favored alleles across
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loci, it is unlikely to obtain them through gene flux with the standard arrangement before being
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selectively lost. In contrast, inversions containing all favorable alleles will experience only a
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very slight drain of positively selected alleles and influx of deleterious alleles due to gene flux
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during the critical stages of establishment under the sympatric origins and mixed modes models.
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Gene flux can be an important factor under certain circumstances, however, in facilitating the
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eventual fixation of alternative arrangements between populations when they have become
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established. Once an inversion polymorphism is established, gene flux can help sort new, locally
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favored mutations differentially into inverted vs. standard arrangements when they arise in the
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wrong genetic background. But gene flux can also impede the evolution of intrinsic postmating
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isolation caused by negative incompatibilities between universally favored alleles (Navarro and
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Barton 2003). Future work could examine these possibilities more thoroughly.
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POSITIVE EPISTASIS
In the stochastic simulations in the main text, we considered loci to independently and
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multiplicatively affect fitness. We also examined how positive epistasis fitness interactions
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between a pair of alleles at two different loci 1 and 2 captured within a rearrangement containing
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a total of four loci influenced its establishment. We present results in Figures 5 A and B for the
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sympatric and mixed mode models, respectively, with a recombination rate of r = 0.1 between
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the four loci and a baseline level of divergent selection of s = 0.1 per locus. Positive epistasis
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was introduced by adding an epistasis term of e = 0.1 to the net fitness of locus 1 and 2
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genotypes in population 1 that contained the locally favored alleles a at both loci and subtracted
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a term e = 0.1 when the locally unfavored allele A was present at both loci. The reverse was true
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for populations 2, where the epistasis term e = 0.1 was subtracted or added to allele a and allele
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A containing genotypes at both locus 1 and 2. We then multiplied the fitnesses of locus 1 and 2
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genotypes by that for loci 3 and 4 to get the overall four locus fitness for each genotype.
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When positive epistatic fitness interactions between loci were considered, the
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probabilities of inversion establishment were affected in both models (Figs. S5A, B). For the
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sympatric origin, positive epistasis can help relax some of the constraint that a new inversion
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must capture all of the locally adapted alleles across all loci to establish as a polymorphism.
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However, even so, the inversion must still capture all of the favorable alleles having large effects
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on fitness in order to establish. Strong positive epitasis can also influence the dynamics of
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inversion establishment by changing gene frequencies at selection-migration equilibrium under
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the sympatric origins model. In this case, positive epistasis generally enhances genetic
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divergence between populations prior to the inversion arising, elevating the probabilities that a
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new rearrangement will capture locally favored alleles. However, it lessens the subsequent
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strength of selection favoring reduced recombination, thereby inhibiting the establishment of the
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new inversion. The latter argument also holds for the mixed mode model. The results showed
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that positive fitness interactions (e = 0.1) between two of the four loci reduced the probability of
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establishment of an inversion for low (m= 0.001) to modest (m = 0.01) migration rates. For high
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migration rate (m = 0.1), positive epistasis had minimal effects under the mixed mode model, but
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increased the probability of establishment under the sympatric origins model.
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NEGATIVE EPISTASIS
To examine the consequences of intrinsic postmating isolation, we performed an analysis
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of the mixed mode and sympatric models in which one locally adapted a allele at locus 1 that
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was favored in population 1 negatively interacted with an A allele at a second locus 2 that was
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favored in population 2. The two negatively interacting loci were considered to reside within a
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rearrangement that contained a total of four loci, with a recombination rate of r = 0.1 between the
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four loci, a migration rate m = 0.1 between populations, and a baseline level of divergent
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selection of s = 0.1 per locus. The fitnesses of all genotypes that contained an a allele at locus 1
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and an A allele at locus 2 were set equal to 1-e (where e was varied from 0.001 to 0.95)
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regardless of the other alleles present at locus 1 and locus 2. The two locus fitnesses for locus 1
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and locus 2 genotypes were then multiplied by that for loci 3 and 4 to get the overall four locus
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fitness values.
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Computer simulations of the mixed mode and sympatric origins models indicated that
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negative epistasis of this type generally made it harder for an inversion polymorphism to become
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established (Fig. S5C). Indeed, under the sympatric model incompatible alleles were effectively
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selectively eliminated, making it extremely improbable that a new inversion captured them. As a
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consequence, there was little or no chance for a new inversion to establish under the sympatric
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origins model that contained negative epistatically interacting loci. These results suggest that
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when intrinsic postmating isolation is associated with an inversion it may often evolve after the
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establishment of the rearrangement in sympatry (Navarro and Barton 2003) or later during a
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period of secondary allopatry after inversion fixation (Kirkpatrick and Barton 2006). The effects
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of epistasis warrant further analysis.
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Supplementary Figure Legends
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Figure S1. Comparisons of the probabilities of establishment between analytical approximation
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(dashed lines) and computer simulations for the mixed mode model (solid lines) for varying
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levels of divergent selection (s) acting on loci within the inversion under conditions of relatively
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(A) low migration rate (m = 0.001), (B) modest migration rate (m = 0.01), (C) and high migration
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rate (m = 0.1) between equal-sized populations.. Shown on a log scale are the probabilities of
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establishment for an inversion in population 1 starting from a single copy (k1 = 1) when the
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rearrangement contained four loci, with a recombination rate of r = 0.1 between loci.
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Figure S2. Comparisons of the probabilities of establishment between analytical approximation
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(dashed lines) and computer simulations for the mixed mode model (solid lines) for varying
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levels of recombination (r) between loci within the inversion. Shown on a log scale are the
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probabilities of establishment for an inversion in population 1 starting from a single copy (k1 =
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1) when the rearrangement contained four loci, divergent selection of s = 0.1 per locus, and
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migration rate m = 0.1 between equal-sized populations.
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Figure S3. The effects of prestanding rearrangements present in both populations 1 and 2 on the
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establishment of an inversion following secondary contact under the mixed mode model. Shown
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on a log scale are the probabilities of establishment for an inversion in either population 1 or 2
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estimated from 100,000 stochastic simulation runs with an initial copy number of k1 in
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population 1 and k2 in population 2. The rearrangement contained four loci, with a
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recombination rate of r = 0.1 between loci, divergent selection of s = 0.1 per locus, and migration
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rate m = 0.1 between equal-sized populations.
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Figure S4. Comparison of the effects of selection acting before vs. after mating (sel/mate vs.
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mate/sel) on the establishment of an inversion under the mixed mode (solid lines) and sympatric
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origins (stippled lines) models. Shown on a log scale are the probabilities of establishment for
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an inversion in population 1 derived from 100,000 stochastic simulation runs between equal-
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sized populations. The rearrangement contained four loci, with a recombination rate of r = 0.1
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between loci and divergent selection of s = 0.1 per locus.
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Figure S5. The effect of fitness interactions between a pair of loci on the establishment of an
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inversion under (A) the sympatric origins model with positive epistasis, (B) the mixed mode
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model with positive epistasis, and (C) the mixed mode model with negative epistasis. Shown on
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a log scale are the probabilities of establishment for an inversion in population 1 derived from
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100,000 stochastic simulations with m = 0.001, 0.01 and 0.1 for A) and B) and m = 0.1 for C)
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between equal-sized populations . The rearrangement contained four loci, with a recombination
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rate of r = 0.1 between loci, divergent selection of s = 0.1 per locus. For (A) and (B), there was
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positive epistasis of e = 0.1 between two of the four loci. For C), varying levels of negative
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epistasis of (e) were considered to act between two of the four loci to reduce fitness to 1-e for the
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two loci, regardless of habitat. For the sympatric origins model, selection acting against negative
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epistatic alleles quickly eliminates them from populations, making it extremely improbable that a
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new inversion will capture them and become established.