The Spread of an Advantageous Allele Across a Barrier: the Effects of Random Drift and Selection Against Heterozygotes

The Spread of an Advantageous Allele Across a Barrier: The Effects of Random Drift and Selection Against Heterozygotes

Jaroslav Piiilek*>+and Nick H. Barton* *Institute of Cell, Animal, and Population Biology, University of Edinburgh, Edinburgh EH9 3p,United Kingdom and iAcademy of Sciences of the Czech Republic, Institute of Animal Physiology and Genetics, CZ-675 02 Studenec 122, Czech Republic Manuscript received December 19, 1995 Accepted for publication October 16, 1996

ABSTRACT A local barrier to gene flow will delay the spread of an advantageous allele. Exact calculations for the deterministic case show that an allele that is favorable when rare is delayed very little even by a strong barrier: its spread is slowed by a time proportional to log( (B/o)&S)/S, where B is the barrier strength, o the dispersal range, and fitnesses are1:l + X1 + 2s. However,when there is selection against heterozygotes, such that the allele cannot increase from low frequency, a barrier can cause a much greater delay. If gene flow is reduced below a critical value, spread is entirely prevented. Stochastic simulations show that with additive selection, random drift slows down the spread of the allele, below the deterministic speed of 06s.The delay to the advance of an advantageous allele caused by a strong barrier can be substantially increased by random drift and increases with B/ (2Spa2)in a onedimensional habitat of density p. However, with selection against heterozygotes, drift can facilitate the spread and can free an allele that would otherwise be trapped indefinitely by a strong barrier. We discuss the implications of these results for the evolution of chromosome rearrangements.

N a spatially structured population, adaptation de- by a barrier. These results are relevant to the key ques- I pends on the spreadof genes throughout theregion tion of how far adaptation and divergence are affected in which they are favored. If the species is sufficiently by population subdivision. abundant that any particular mutation occurs fre- Gene flowmay be impeded by a physical barrier, quently, or if selection acts on abundantpolygenic vari- caused by a reduction in density or dispersal, or by a ation, then adaptation will not be muchslowed by spa- genetic barrier, caused by incompatibilities at otherloci tial subdivision. However, in less numerous species, or that cause a reduction in the fitness of hybrid popula- where alleles are favored only in scattered habitats, lack tions. Provided that selection on the locus in question of gene flow may limit the availability of the necessary is weak, both physical and genetic barriers can be seen variants. Even if the population does adapt rapidly, this as causing a sharpchange in gene frequency, Ap, whose adaptation may be based on different genes in different sizeis proportional to the gradient in frequency on places, which may lead to reproductive isolation if these either side, +/ax; the strength of the barrier is given genes turn out to be incompatible with each other. by a single quantity with the dimensions of a distance: There is a particular difficulty if adaptation depends B = Ap/(ap/ax) (NAGYWU 1976; BARTON 1979a on gene combinations that cannot increase from low 1986).We use this approximation throughout, and thus frequency and that therefore tend to be contained be- do not consider the detailed effects of barriers that are hind narrow “tension zones” (KEY 1968). The spread of not localized, and that may be due to selection on many such gene combinations requires a “shifting balance” loci. We only consider short-range dispersal; our simula- between different evolutionary forces (WRIGHT1932). tions involve exchange between nearest neighbours, Here we consider the effect of a local barrier to gene and ouranalytic results approximate gene flow by diffu- flow on the spread of a favorable allele through a con- sion. Thus, we avoid the complexities of long-range dis- tinuous habitat. We consider thetwo qualitatively differ- persal (see MOLLISON1977; SHAW1995). The final re- ent cases, first, where the favorable allele can increase striction is that we deal with the spread of an allele at from low frequency, and second, where it must increase a single locus; this choice is made partly for simplicity, above some threshold before it can be established by but also because our study is motivated by observations selection. We begin by setting out exact calculations for on the distribution of underdominant chromosomal infinite populations, and then use simulations to show rearrangements. how random genetic drift can increase the delay caused In the next section, we review analytic results based on the diffusion approximation, and use deterministic calculations for a onedimensional habitat to test their Cmmpmdzngauthor: Nick H. Barton, Institute of Cell, Animal, and Population Biology, University of Edinburgh, Edinburgh EH9 3JT, accuracy.Since the effects of spatial subdivision are United Kingdom. E-mail: [email protected] likely to be most important in sparse populations, we

Genetics 145 493-504 (February, 1997) 494 J. Piaek and N. H. Barton then use stochastic simulations in one and two dimen- ap - a2a'p - - " (1) sions to find how random drift affects the spread of at 2 82) favorable alleles. In the DISCUSSION,we consider the implications of our results for the movement of tension where x is distance, and covers an infinite range (FISHER zones, and in particular, for the spreadof chromosome 1937; HAL.DANE 1948). This equation has a solution rearrangements. given by the integral of the normal distribution, with variance a". The collapse of the gradient can be mea- sured by the cline width w,which is defined as the THE DETERMINISTIC CASE inverse of the maximum slope of the gene frequency cline p(x) (MAY et al. 1975). In this case, w = M. Diffusion through a continuous habitat: Provided Equation 1 can also be solved explicitly if the mixing that allele frequencies change slowly in space and time, of populations is impeded by a barrier. The solution is and provided that long-range dispersal is sufficiently a sum ofGaussian integrals; it shows that the delay rare, the effects of gene flow can be approximated as caused by thebarrier is proportional to (BAR- a diffusion througha continuous habitat (NAG- TON 1979a). 1975). With discrete generations, the rate of diffusion Spread of an advantageous allele: Suppose now that is given by a', the variance of the distance moved by an one allele has an advantage S over the other and there individual from its birthplace to a new place of breeding is no dominance; thediffusion approximation requires along a particular axis. (With overlapping generations, that S be small. Equation 1 now includes an extra term a' is the variance of distance moved per unit time). In representing selection: one-dimensional stepping stone models, we define the migration rate m as the probability that an individual dp=" a2a'p + SPq. (2) will move to either of the two adjacent demes. In two at 2 a2 dimensions, we define m as the chance that an individ- ualmoves to any of the four adjacent demes. [This FISHER(1937) showed that Equation 2 has a family of definition was used by SLATKINand BARTON(1989) ; in solutions, each consisting of a wave with constant shape, two dimensions, it differs by a factor of two from that which advances at a characteristic speed. He argued used in BARTONand ROUHANI(1991) and other pa- that the relevant solution was that with the minimum pers]. If the deme spacing is E, then the relationship speed, c = a&S. Subsequent numerical and analytic between the two parameters is 0' = m' in one dimen- work has confirmed that thoughsolutions exist in which sion, and a' = m2/2in two dimensions. a broad wave advances rapidly, random fluctuations at Under the diffusion approximation, a barrier that is the leading edge,or boundary effects, tend to slow this restricted to a narrow region causes a sharp step in down to Fisher's solution. (This kind of wave is referred gene frequency A$. The gradients in gene frequency to as a "pulled" wave, since its speed is determined by on either side ((@/ax) ~,(ap/ax)+) are proportional the leading edge;STOKES 1976). Thediffusion approxi- to the flux of genes flowing in either direction (J+ = mation (Equation 2) is only valid if long-distance dis- (02/2)(8p/8x)+;NAGW 1976; BARTON 1986) and persal is sufficiently rare: thedispersal distribution must are also proportional to the size of the step. The ratio fall away at least exponentially (MOLLISON1977). Oth- between step size and the gradient in allele frequency erwise, long-range migrants can establish new centres (B+= Ap/ (@/ax) + ) gives a measure of the barrier to of increase, so that advance isby sporadic outbreak gene flow in either direction. This has the dimensions rather than by a smooth wave (MOLLISON1977; SHAW of a distance. If the barrier is asymmetrical (B, f B-), 1995). then the gradients of allele frequency will differ on the Since Fisher's solution to Equation 2 has not been two sides. In a stepping stone model, a local barrier derived analytically, it is unlikely that the effect of a corresponds to a reduction in the migration rate be- barrier can be found explicitly. On dimensional tween adjacent demes by a factor k [ie., from m/2 to grounds,the delay T should depend on the dimen- m/(2k)in one dimension, or from m/4 to m/(412)in sionless parameter = (B/a)&, having the form ST two]. This causes an increase in the gradient of allele = f(B).BARTON (1979a) gave the approximation ST = frequency by a factor k, and hence produces an extra log (&8/2) for a very strong barrier. This is the time step in allele frequency ofA$ = (k- l)~(ap/dx),where taken to increase to p = 0.5 from a threshold frequency t is the deme spacing. Hence, the barrier strength is B N. This threshold is chosen to be the time at which = (k - 1)E. the allele begins to increase faster by selection than by Spread of a neutral allele: Iftwo populations that the flux across the barrier. The argument ignores the are fixed for different alleles meet in a sharp step, the effect of diffusion. A slightly more accurate argument, discontinuity between them will be smoothed out by which takes into account diffusion and avoids using an gene flow. The decay of this initial step with time can arbitrav threshold frequency, is given in APPENDIX A. be approximated by a diffusion equation: This also scales with log(B)/S: the crucial point is that of an Advantageous Allele 495 AlleleAdvantageous an Spread of

A ministic calculations for S = 0.025 and S = 0.1. Results 5- for these two selection pressures fall on the same line (compare circles and squares), showing that the diffu- a sion scaling is accurate. The delay does indeed increase logarithmically with barrier strength, but is much less than predicted by either approximation: thebest fit for S = 0.1 is ST = 0.53 log (0.53(B/a)J(2S)), and 0.52 log(0.71(B/a)&) for S = 0.025. The delay is less than s expected because when the barrier is strong, the allele has time to spread over a large distance before it increases to appreciable frequency. Hence,the initial wave of advance is broad, andso advances more quickly, reducing the delay; it settles to a solution to Equation a 2, but one with higher speed than the minimum.Over time, it converges on the solution to Equation 2 that 0 has minimum speed, a6S (STOKES1976). This can be *100 1000 1 10 1o4 seen in Figure lB, which shows the difference in total (B/o)d 2s allele frequency between a freely advancing wave, and a wave that is impeded by a barrierof strength B = 1000. B The difference in positions increases to a maximum as 15 r the wave meets the barrier, but thendecreases, because the wave moves faster for some time after it escapes from thebarrier. Because therate at whichthis “pulled” wave slows down depends onrandom fluctuations at the leading edge, the delay may be sensitive to details of the modelsuch as deme spacing and dispersal distribution. Selection against heterozygotes: If heterozygotes are less fit than the average of the two homozygotes (ie., if there is “underdominance,” with fitnesses 1:l + S - s:l + 29, then the allele will spread more slowly than the rate a&S predicted by Equation 2. There are two alternative predictions for the wave speed. First, when 0 100 200 300 400 500 the allele can increase from low frequency (S > s) , time consideration of the leading edge (p 4 1) shows that FIGURE1.-{A) The delay to the advance of an advanta- there is a family of solutions, with minimum speed c = geous allele, T, as a Re- plotted function of barrier strength. ad=; these arepulled waves (HADELER1976). sults are scaled so that if the diffusion approximation of Equa- tion 2 is valid, the same relation should be obtained fo? all Second, there is always a solution that follows a logistic 0, 3 thus, ST is plottedagainst log(@, where B = (or tanh) curve, with p = 1/(1 + exp(-4(x - ct)/w)), (B/(T)&. The heavy line gives the approximation derived and width w = Gs;this has speed c = &/a;a from Equation 4, and the light line the approximation ST = perturbation analysis shows that this is the solution for log(&B/2) of BARTON (1979a). Circles give results from a S s (BARTON1979b). This solution is referred to as one-dimensional stepping-stone model with m = 0.5 and S = e 0.1; squares give results for S = 0.025. One hundred forty a “pushed” wave, since its speed depends on the way demes were used, with a barrier to the right of deme 40; selection acts over the whole cline. Numerical results initially, the 10 leftmost demeswere fixed. The delay was from stepping stone models show that the wave speed calculated from the difference in total number of alleles be- is givenis by for S and by when tween the solutions with and withouta barrier, after the wave Sa/& < 2s, had passed the barrier. (B) The difference in position be- S > 2s (Figure 2). These results alsoshow that the tween the waves with and without the barrier. Note that the diffusion approximation is accurate for fairly strong se- delay decreases from a maximum after the wave passes the lection (S = 0.1). barrier. This is because the wave is broadened by the barrier If heterozygotes are fitter than the original homozy- and hence moves faster. gote (S> s) , then the invading allele can increase expe nentially from an indefinitely low frequency. The delay the delay only increases logarithmically with barrier therefore increases logarithmically withbarrier strength, strength, and hence is small even for a very strong bar- as inthe caseof additive selection (Figure 1A). The rier. effect of a barrier is qualitatively different if the allele Figure 1A compares these approximationswith deter- cannot increase from low frequency (ie., if S < s). In 496 J. Pialek and N. H. Barton

4 Sls = 0.2 Sjs = 0.5 Sjs = 0.8 \ J i 40 3

mI 20

0 2 4 6 a Trapped SIS

FIGURE2.-The speed of a wave of advance, as a function 0 u of the degree of underdominance (S/s). The upper straight 0 10 20 B+ 30 40 line shows the speed of the tanh solution, c = Sa/&, while the lower curve shows FIGURE3,"These graphs show the critical combinationsof dicted by considering barrier strengths that prevent an allele from invading;barrier speed c is scaled relative to strengths are scaled relative to the distance o/&. From left and squares are for s = 0.005. These results are from a step to right, the curves are for S/s = 0.2, 0.5, 0.8. The allele is ping-stone model with m = 0.5, starting with the leftmost 10 spreading from right to left; the horizontal axisshows the demes fixed, as for Figure 1. The speed was estimated by critical B,, while the vertical axis shows the critical B-. The regressing total allele frequencyZp against time. allele will be trapped if the barrier to flow to the right is too strong (B, right of the curve) or if the barrier to flow to the left is too weak (B- below the curve). If the barrier to flow this case, a sufficiently strong barrier can prevent spread to the right is too strong, the allele cannot advance, even if altogether: an equilibrium will be reached in which the there is free flow in the opposite direction (B- = 0; threshold flux across the barrier is balanced by selection against B+ is 2.92, 7.69, 35.69 for S/s = 0.2, 0.5, 0.8, respectively). the invading alleles. We first find this equilibrium, and hence the critical barrier strength needed to prevent invasion, and then gwe numerical results for the delay caused when the barrier is below the critical value. Including selection against heterozygotes inEqua- tion 2 gives

(P- - P+) (2)+= - B+

where = (s - S)/2s is the threshold frequency below which the allele is selected against; as before, the range These equations can be solved numerically for given of x is from --OO to CQ.At equilibrium, Equation 3 can barrier strengths (B-, B+) and threshold frequency be integrated (Po).To find the critical barrier strengths which- prevent ap 1 2s invasion, rewrite Equation 5a as an expression for B-, - = JC + p2(6p0 - 4p(l + p,,) + 3p2). (4) as a function of q-. This has a minimum, which gives ax -4.u the critical B- for given p,. Equation 5b then gives the Applying the boundary conditions thatp tends to1 on value of B, as a function of p+ at the critical point. the left,and 0 on the right, determines the constantof Figure 3 shows the critical combinations of barrier integration, C, on either side of the barrier. At the strengths; the three curves correspond to S/s = 0.2,0.5, barrier (taken to be at x = 0), the gradient is equal 0.8. If B+ is sufficiently strong (right of Figure 3), or if to the ratio between the step in allele frequency and B- is sufficiently weak (below curveson Figure 3), then thebarrier strength. This givestwo simultaneous the allele cannot invade. Figure 4 shows the delay as a equations for the allele frequencies on either side of function of barrier strength,calculated from a stepping- the barrier, p-, p+, stone model. This increases to infinity at the critical of an Advantageous AlleleAdvantageous an Spread of 497

100 demes and position of the barrier were chosen to avoid edge effects; most runs used 41 demes, with a barrier between demes 6 and 7. ST To increase speed, simulations in two dimensions were run using a program written in Pascal; 20 X 10 10 demes were simulated, with the allele advancing along the axis of length 20 demes. Runs were terminated at 10,000 generations, or when the advantageous allele was completely fixed; the time at which the latter oc- 1 curred was taken to be the time to fixation. The two- dimensional simulation differed slightly, in that migra- (l3P)KS tion is by a small number of adults, rather than by a large number of juveniles. Random fluctuations there- FIGURE4.-The delay to the spread of an allele as a func- fore occur both through the random sampling of mi- tion of barrier strength,for a onedimensional stepping stone model with m = 0.5. Squares show results for S = 0.025, s = grants between demes and through random variation 0.05 giving fitnesses 1:0.975:1.05;circles show results for S = in reproductive success within demes. Such stochastic 0.1, s = 0.2 giving fitnesses 1:0.9:1.2. Results are scaled relative migration can alter spatial correlations, and increases to the time l/s and distance o/& the graph should thus be the variance of gene frequency above that expected independent of due to sampling drift alone (see EPPERSON1994). For strength is at B, example, in systems where the stochastic variance occurs from random sampling of a migrant group of size barrierstrength and can be large belowthis value. 2Nm, the variance of gene frequencies can be increased (Note that the delay is plotted as ST, on a log scale). up to double that for the corresponding case of deter- The effect of a barrieris much greater than for allelean ministic migration; this is because there are two rounds that is favored in the heterozygote, because the allele of sampling rather than one (EPPERSON1994). must build up above a threshold frequency before its In each generation, the number of migrants from a spread is assisted by selection. deme was taken from the binomial distribution with parameters m, N, where m is migration rate and N is the THEEFFECTS OF RANDOM DRIFT number of adult individuals in the deme before migration. Genotypes of migrants wereassigned randomly We now consider the effects of random drift on the from the genotypes within the deme of origin after ran- spread of an advantageous allele, treating themodels in dom drift. Sampling was without replacement. Parents the same order as inthe previous section. The stochastic were expected to produce an infinite number of gametes problem is harder to treat, because simulations are and therefore selection on gametes was a deterministic slower and must be replicated. Moreover, the effects of process, as in the onedimensional simulations. random drift are qualitatively different in one and two All procedures used to simulate migration, random dimensions (~ECOT1948), and so both cases must drift and selection were checked against theoretical ex- be treated. pectations. The probability and rate of fixation of a Thestochastic simulation: The effect of barriers, neutral allele without migration agreed with a Marko- deme size and selection on the time needed forfixation vian transition probability matrix for a single deme; the of a favored allele was studied using a stepping-stone fixation probability of an advantageous allele (with and model. Throughout, we use the convention that the without migration) was 2Sfor small S. Stochastic migra- deme spacing is E = 1, so that the deme size is N = p. tion was checked against the decay of cline width of a Initially, different alleles were fixed on either side of a neutral allele (from Equation 1) and also agreed closely barrier at x = 0, representing secondary contact be- with theory. tween diverged populations. There was no migration Spread of anadvantageous allele: Mathematical beyond the bounds of the array. The time delay, T, due analysis of the interaction between selection, gene flow to a barrierwas estimated by subtracting themean time and drift is in general intractable. However, we can use to fixation without a barrier from that with a barrier. some heuristic arguments to illuminate the simulation In one dimension, simulations were run using Mathe- results. First, consider the spread of an advantageous matica (WOLFRAM1991). Selection acted on a large allele in the absence of a barrier. Taking expectations number of diploid juveniles; the recursion for change of Equation 2, in allele frequency is p' = p(1 + 2S - qs))/ (1 + 2pS - 2pqs). A fraction m of these then migrated, half to the left and half to the right. N diploid parents were then -2 a2r sampled; the next generation was formed by random U - "uy+m(l-F). (6) union of a large number of gametes. The number of 2 a2 498 J. Pialek and N. H. Barton

Since the variance in allele frequency produced by ran- A dom drift reduces the proportion of heterozygotes by 0.3 , - a fraction F, it will reduce the response to selection and 2 e, slow the advance of the wave; the effective selection 9 coefficient is reduced to S( 1 - F). In a spatially uniform 0.2 polymorphism with stability S and population density p = N/E', the standardized variance F= var(p)/&would be constant, and equal to 1/(1 + 4paJ2S) in one di- 0.1 mension, and I/( 1 + 4npa2/ln (1/&)) in two dimensions (WECOT1948). NAGYLAKI(1978) showed that in a one-dimensional cline, Fdepends on the dimensionless quantity pad%, being inversely proportional to 0' 1 I I I I 0 10 20 30 40 50 it for large values, and tending to 1 for small values. If P we assume F has the same form for a wave of advance, B then we expect a speed 0.15 r (onedimension), (7a) c=J"3&s

(two dimensions),(7b) c=)% (4npu') 0.05 1 where ~0 = a&% is the deterministic speed corrected for high value of selective advantage used in simulations 0 (S = 0.1). Figure 5 compares simulated values of the 0 20 40 60 80 100 wave speed with these predictions; A is calculated by P least-squares regression of l/z on l/p. There is a good FIGURE5.-(A) The speed of a wave of advance in one fit, with A estimated as3.87 in one dimension, and dimension, plotted against deme size. 0 show mean speeds from 10 of the with con- 12.96 in two. This is much greater than the value A = replicates stochastic simulation, 95% fidence limits. (S = 0.1, m = 0.5; 60 demes, startingwith the describing the effect of drift on a uniform polymor- leftmost six fixed). The curve shows the predicted relation- phism in one dimension, or A = In (1/&') = 0.80 in ship from Equation 9, with A = 3.87 calculated by least-squares two. It seems that random drift reduces heterozygosity regression of l/c' against l/p. The horizontal line shows the in the advancing wave, and hence its rate of advance, deterministicspeed, 06,while thesteep line to the left much more than would be expected from its effect on shows the approximation for small p, c = 2pa'S. (B) The corresponding graph,for two dimensions. (S = 0.1, m = 0.2; neutral polymorphisms. 60 X 10 demes, starting with the leftmost 6 X 10 fixed; A In the alternative limit ofvery low density, most 12.96). demes are likely to be fixed for one or otherallele. The probability of fixation of a favorable allele that enters dent, thevariance in position around theexpected loca- a demefixed for its homologue is 2S/ (1 - exp (-4") ) ; tion increases linearly with time (Figure 6). The vari- the converse probability is 2S/ (exp(4NS) - 1). There are, on average, Nm such migrants in each direction, ance in position increases atratea 1.19a/p& and so the expected rate of advance of the boundary (APPENDIX B). This prediction agrees with one-dimen- between demes fixed forthe different alleles is sional simulations for high densities (p > lo), but over- estimates the increase in variance for low density (Fig- (Nm)(2S/(1 - exp(-4NS)) - 2S/(exp(4NS) - I)) = 2NmS (see SLATKINand CHARLESWORTH1978). This ap- ure 7). This may be because at low density, drift proximation is shown by the steep line on the left of substantially reduces heterozygosity, andhence wave Figure 5A. speed (Figure 5a). In two dimensions, fluctuations at As well as slowing the advance of the favored allele, different positions along the wave tend to cancel, so drift introduces random variation in its position. Al- that the variance in overall position should increase though the meanlocation of a set of clines will increase with A, rather than linearly with time (BARTON1979b). linearly with time, at the rate shown in Figure 5, the However, we were unable to simulate large enough position of any one cline will vary around this expected areas to confirm this argument. position. In each generation, drift introducesa random Spread of an advantageous allele past a barrier: In perturbation that advances or retards the cline by some the deterministic case, we showed that even a strong amount. Since successive perturbationsare indepen- barrier has little effect on an advantageous allele: be- Spread of an Advantageous Allele 499

I I I I 0' I 0 10 20 40 50 30 P FIGURE7.-The rate of increase in variance around the 0 100 time 200 expected position in a one-dimensional habitat,plotted against density, p. The curve shows the expectation from FIGURE6.-The variance around the expected position, plotted aainst time. The expectation from Equation B1, Equation 10, 1.19/po&. Circles show mean speeds from 10 replicates of the stochastic simulation, with 95% confidence l.l9/po?- 2s is shown by the straight line. Based on 10 repli- limits. Parameters are as in Figure 6. cates of the stochastic simulation in one dimension. (p = 40; S = 0.1, m = 0.5; 60 demes, starting with the leftmost six fixed). fixation, is given by following Equation 2, with initial inoculum 1/4pS; this converges to a wave of advance, cause such an allele can increase exponentially from with some delay. Scalingarguments show that this delay low frequency, the delay only increases logarithmically is, on average, a functionflpafi/S, where the dimen- with barrier strength. SLATKIN(1976) considered the sionless quantity is a measure of the relative stochastic case and demonstrated a relation between strengths of drift and selection (see NAGVWU1978). the delay in time to fixation with decreasing migration Deterministic simulations for S = 0.1, m = 0.5 show that rate and the numberof individuals within a population. flx] 3.50 + 0.55 In(%)+ 0.01461n(x)' (estimated by His argument was based on the approximate time to least squares fit of the delay to ln(p)).Overall, therefore, fixation of an advantageous allele within a single popu- the expected delay is B/(2Spa2) + flpafi/S. This lation, (l/S)In [2Np(O)],where p(0)is the initial allele increases linearly withbarrier strength, rather thanloga- frequency. However, SLATKIN'Sformula applies to flow rithmically, showing that drift can substantially amplify into asingle deme, rather than the rateof spread across the effect of a barrier. an array of demes. Here, we combine a heuristic argu- This prediction is compared with simulation results ment with simulation results to show that random drift in Figure 8. The lower curve in Figure 8A shows the can substantially enhance the effect of a strong barrier, delay in an infinite population (from Figure 1). This above that expected in a dense population. only increases logarithmicallywith barrier strength,and When the barrier is very strong, migrants cross it only so is much less than in finite populations. The simula- rarely. The flux across is J = (a2/2)/B,the density of tions show that the delay increases linearly with barrier genes is 2p, and so the expected number of genes cross- strength, and is greater when density is low (circles: p ing per generation is (pa'/B). Each of these genes has = 20, squares: p = 10). There is reasonable agreement probability 2S of fixing,and so the expected time before with the prediction described above, which is shown by an allele begins tofix is B/ (2Spa'); we assume B S the straight lines for each of the two densities. However, 2Spa2,so that only one allele fixes. [Note that the proba- the delay is somewhat lower than predicted for one bility of fixation is 2S, regardless of the presence of the dimension. The two-dimensional simulations were nec- barrier (assumed impermeable) onone side (MA- essarily run with a short barrier,and so the above argu- RUYAMA 1972)].There is then a furtherdelay before the ment should apply, with p replaced by pL, where L is allele rises to high frequency and begins to spread as a the length of the barrier. If the barrier were very long, steady wave. This second delay can readily be calculated. alleles would crossand begin to fix at many places simul- When the allele is rare, its expected frequency (includ- taneously. One should then take into account the time ing cases of loss and fixation) is given by Equation 2, taken for alleles advancing from different points to and depends on the initial total allele frequency, 1/2p. spread along the barrier from their separate origins; After a time long enoughfor the fate of the allele to be this might increase the delayabove the one-dimen- decided, but short enough thatit is still rare, its expected sional prediction, as is seen in Figure 8B. The relation- frequency giuen that it will bejixed is equal to the overall ship between the time delay and the length of the bar- expectation, divided by the probability of fixation, 2s. rier, L, is given in Figure 8C. Thegraph shows a Hence, theeventual expected frequency, conditional on situation with a strong barrier (B = 500). Here, the 500 and J. Pialek N. H. Barton

delay is primarily a function of probability that a mi- T/ grant will cross the barrier. It can be seen the delay decreases inversely with pL, as argued above. 20 Escape of a tension zone from abarrier: Suppose now that selection acts against heterozygotes, such that b v3 the invading allele cannot increase from low frequency. In the deterministic case, a sufficiently strong barrier 10 will prevent the invasion altogether; instead, theincom- ing allele will reach an equilibrium between migration across the barrier and selection against it, given by Eq. 5. With finite population size, there is some chance that the allele will drift to high enough frequency to escape the barrier. While it would be hard to calculate the rate -10 1 of escape, we expect it to decrease exponentially with 0 deme size (see BARTONand ROUHANI1987, 1991). O0 (B/o)d2s 200 Figure 9 shows the average delayas a function of barrier strength. For (S/s) = 0.5, the critical barrier 30r B strength is (B/o)& = 6.82, above which the delay be- b comes infinite in an infinite population (heavy curve ?A in Figure 9). All the simulations were for barriers stronger than this critical value. Random drift can free 20 ' the favorable allele and so in (almost) allcases the delay decreases as deme size becomes smaller. A similar pattern is seen in two dimensions; there, the delay is closer to the deterministic expectation for p = 96, and 10 . considerably reduced for p = 12 (Figure 9b). Figure 1OA shows the same data,plotted against deme size instead of barrier strength. As expected, the delay increases exponentially with deme size when the barrier is strong (as shown by the approximately linear increase 0 on this log scale for B = 20, 30, 40). However, there is 0 2oo (B/o)d% 400 little increase in the delaywith deme sizewhen the 100 barrier is only just stronger than the critical value [B= 10 (circles) us. the critical value of B = 6.82a/& = 7.621. Figure 10B shows corresponding results for two 80 dimensions; again, these roughly linear graphs suggest that the delay increases exponentially with deme size. 60 Here, however, the lines for different barrier strengths b v3 are morenearly parallel than in one dimension, indicat- 40 ing that an increase in barrier strength increases the delay by the same factor, independent of deme size. . 20 DISCUSSION

I I I I The effect of population structure on adaptation de- 0' I I I I I 0 20 40 60 80 pends very much on whether alleles tend to increase barrier length, L even when rare, ormust instead rise abovesome threshold frequency before being favored. These twocases FICLJRE8.- (A) The average delay, T, caused by a barrier reflect the contrast between Fisher's view of evolution, in one dimension, plotted against barrier strength. Results in which adaptation is most efficient in large popula- arescaled to the dimensionless quantities ST and B = tions, with Wright's, in which random drift and re- B&%/o. Simulations were run for S = 0.1, at a density p = 20 (0) and p = 10 (W). Confidenceintervals (95%) are shown, derived from 20 replicates. The straight lines show = 12 (@, 50 replicates) and p = 96 (W, 25 replicates), m = the theoretical predictions fora strong barrier,B/(2Spu2) + 0.2, S = 0.02. (C) The relationship between the delay caused f[po&/S, for p = 10 and 20. The deterministic results (p by the local barrier (B= 500) and the length of a barrier L. = m) are shown by the lower heavy curve, which is interpo- Squares indicatea scaled delay, ST, for S = 0.02, while circles latedfrom the results in Figure 1 for S = 0.025. (B) The for S = 0.08 coming from 100 replicates for L 5 20 and 50 corresponding graph fortwo dimensions. Density herewas p replicates otherwise. m = 0.2, p = 12. Spread of an Advantageous Allele 50 1

A r B=40 500 p=25 B=30

100 : B=20

2: B=10 t I 2 // p=20

10 :

n" 20 30 0 10 20 30 40 lo P (B/O).l2s B=40 B=30

B=20

100 : 2: B=10

0 20 30 (B/o)G lo P FIGURE9.-(A) The average delay as a function of barrier FIGURE10.-The delay, plotted on a log scale against deme strength with heterozygote disadvantage s = 0.2, and selective size, for B = 10, 20, 30, 40; the data are thesame as in Figure advantages S = 0.1, in one dimension; m = 0.5. For these 9. (A) One dimension. (B) Two dimensions. parameters, the critical barrierstrength is B+ = B- = 6.82a/& (see Figure 3). Above this value, the delay becomes rier decreases substantially as the length of contact be- infinite in an infinite population. Values are based on 20 repli- tween two races increases. Thus, genetic differentiation cates, for densities p = 2, 5, 10, 20, 25. (B) Results for two dimensions, using an array of 20 X 10 demes. (s = 0.04, S = will be greater where gene flow is confined to a narrow 0.02, m = 0.2; for these arameters, the critical barrier strength corridor such as a mountain valley or shoreline. is B+ = B- = 6.82a/ ,re2s, as before). Values are based on 50 Our results could be used to describe the spread of and 25 replicates, for densities p = 12 and 96, respectively. any beneficial allele. However, our study was motivated originally by the evolution of chromosome rearrange- stricted gene flow are essential for continued progress. ments, and in particular, by the geographic distribution In the first case, the spread of an allele through a con- of centromeric(Robertsonian) fusions in mice and tinuous habitat is somewhat slowed by random drift shrews (SEARLE1993). Here,populations are usually (Figure 5). In a dense population, a local barrier causes fixed for a particular combinationof metacentrics over a slight delay (elog (B)) , which can be substantially substantial areas. This suggests that new metacentric increased when deme size islow (=B Figure 8). In fusions areeither lost or spread rapidly to fixation. contrast, when selection against the heterozygote pre- Moreover, the large number of metacentrics that have vents increase from low frequency, a local barrier to been derived from the ancestral acrocentric karyotype gene flow can prevent an advantageous allele from suggests that they increase the fitness of their carriers. spreading through a dense population. Drift can then This selective advantage could arise due to a novel re- facilitate the spread and can free an allele that would shuffling of genes (WILSONet al. 1974), meiotic drive otherwise be trapped indefinitely. Two-dimensional re- (NACHMANand SmE1995), reduction in recombina- sults are qualitatively similar; the delay caused by a bar- tion between epistatically interacting genes (CHARLES 502 and J. Pialek N. H. Barton

WORTH and CHARLESWORTH 1980;for a mechanism of increase the effective interdeme distance; we chose 20 recombination suppression see DAVISSONand AKESON here as a conservative estimate. Hence, physical barriers 1993) or competitive superiority (CUANNAet al. 1984; could decrease migration rate by “20 times. Taking SCRIVEN1992). No direct estimates of selective advan- selection against hybrids as s = 0.3 and m as 0.2, it leads tage are available.However, analysis of overlapping to a value of the scaled barrier (B/o)& = 48. The clines in the hybrid zone between the Oxford and Her- critical values for barriers with S/s = 0.5 and 0.8 are mitage races of shrewssuggested an advantage of meta- 7.69 and 35.69, respectively. Boththe theoretical values centrics over acrocentrics of S = 2.6% (HATFIELD et al. are below the estimates from the field data, which 1992). The hybrids originating from crosses between means that the barrier could be overcome by a combi- two chromosome races are expected to suffer partial nation of drift and a rather high selective advantage to sterility due to problems at meiosis. Two types of hybrids the homozygote (S > 0.2). While the field data have are found. “Simple” heterozygotes are detected, if one supported the presence of random drift, there is no parental race is fixed for one ormore fusions while the evidence for the high advantage that would be neces- corresponding arms in the other are acrocentric. These sary to break these barriers. Therefore we conclude that hybrids usually suffer only weak selection (VIROUXand the distribution of both POS and W races will be stable BAUCHAU1992; WALLACE1994). “Complex” heterozy- until the population is disrupted by some extrinsic ca- gotes could arise if their parental races share one arm in tastrophe. two different fusions; this could be due to independent Many other animal species have their range divided fusions of the same chromosome arm, or whole-arm into subpopulations differing in chromosome arrange- reciprocal translocation. Where one arm is shared in ment (WHITE1978; BARTON and HEWITT1985; KING two fusions (monobrachial homology), hybrids are then 1993). These chromosomal races are often associated expected to suffer strong selection because they form with physical barriers such as streams, rivers, roads, gul- chain or ring configurations at meiosis (CAPANNAet al. lies or inhospitable areas [see SAGE et al. (1993) for a 1976), and hence asubstantial decrease of heterozygote review on mice; MERCERand SEARLE(1991): shrews in fitness (RED1 and CAPANNA 1988; SEARLE1988). Scotland; BRUNNERet al. (1994): shrews in Switzerland; A particularly well-studied example is found in the PA~ON(1993): pocket gophers in northern America; Upper Valtellina valley in northern Italy, where four NEVOand BAR-EL (1976): mole rats in Israel; BARTON Robertsonian races meet with the all-acrocentric race of and GALE(1993) : grasshoppers in the Alps Maritimesl. the house mouse (Mus domesticus) (HAUFFE and SEARLE Such barriers increase the chance that a chromosomal 1993).The distribution is patchy, one race being usually mutation will rise to high frequency despite selection restricted to one or two villages. In samples collected against the heterozygote (MDE1979), butas our anal- between 1989 and 1991, most individuals were homozy- ysis shows, impedes its subsequent spread. The fate of gous, yet hybrids were found in almost every village chromosome races, or more generally, of novel “adap- (HAUFFE andSEARLE 1993). Most villages inUpper Val- tive peaks,” may therefore depend more on the chance tellina are divided by natural barriers such as streams, factors that influence the distribution of the species, gullies or pasture, which may prevent movement of than on the adaptive competition between gene combi- chromosome races. What can one say on the future nations envisaged by WRIGHT(1932) in his theory of evolution of this system? If two races occupy different the “shifting balance.” villages, their stability will depend on the strength of We are specially grateful to H. C. HAUFFE for allowing us to present both physical and genetic barriers and population den- her unpublished data. B. NURNBERGER,J. B. SEARLE,H. C. HAUFFE, S. sity (ie., on B, s, and p). For example, the Poschiavo BAIRD,L. KRUUK and two anonymous referees gave constructive com- (POS, 2n = 26) and Upper Valtellina (W,2n = 24) ments on the manuscript. The work was supported by the European races are fixed in four neighboring villages (POS in Union (Human Capital and Mobility ContractNo. ERB4050PL922765). Migiondo, W in Sontiolo, Tiolo and Lago). What is LITERATURE CITED the chance that the UV race will invade the POS race? Due to monobrachial homology the fitness of the hy- BARTON,N. H., 1979a Gene flow past a cline. Heredity 43: 333- 339. brids between the two races is estimated to be reduced BARTON,N. H., 197913 The dynamics of hybrid zones. Heredity 43: by 30-40% due to chromosome nondisjunction at mei- 341-359. osis (HAUFFE 1993).Based on capture-recapture study BARTON,N. H., 1986 The effects of linkage and densitydependent regulation on gene flow. Heredity 57: 415-426. in the same area, the deme size varies between 1 and BARTON,N. H., and K, S. GALE,1993 Genetic analysis of hybrid 23 mice, and there are -50 potential breeding sites zones, pp. 13-45 in Hybrid Zones and the Evolutionaly Process, ed- (demes) in each village with the average distance be- ited by R. G. HARRISON. Oxford University Press, Oxford. BARTON,N. H., and G. M. HEWITT, 1985 Analysis of hybrid zones. tween demes E = 33 m (H. HAUFFE,unpublished data). Ann. Rev. Ecol. Syst. 16: 113-148. Tiolo and Sontiolo are isolated by a distance of 500 m BARTON,N. H., and S. ROUHANI,1987 The frequency of shifts be- from Migiondo, corresponding to 15interdeme dis- tween alternative equilibria.J. Theor. Biol. 125 397-418. BARTON,N. H., and S. ROUHANI,1991 The probability of fixation tances. In addition, these villages are separated by a of a new karyotype in a continuous population. Evolution 45: fast-flowing river and a motorway. This should greatly 499-517. Spread of an Advantageous Allele 503

BRUNNER,H., L. FUMAGALIand J. HAUSSER,1994 A comparison of the case of the common shrew, pp. 507-531 in The Cytogmtics two contact zones between chromosomal races of Swex aranm of Mammalian Autosomal Rearrangements, edited by A. DANIEL.Alan inthe Western Alps: karyology and genetics. Folia Zool. R. Liss, New York. 43(Suppl. 1): 113. SEARLE,J. B., 1993 Chromosomal hybrid zones in Eutherian mam- CAPANNA,E., M. CORTI,D. MAINARDI,S. PARMIGIANI and P. F. BRAIN, mals, pp. 309-353 in Hybrid Zones and the Evolutionaly Process, 1984 Karyotype and intermale aggression in wild house mice: edited by R. G. HARRISON.Oxford University Press, Oxford. ecology and speciation. Behav. Genet. 14 195-208. SHAW, M.W., 1995 Simulation of population expansion and spatial WANNA,E., A. GROPP,H. WINKING,G. NOACKand M.-V. CMTELLI, pattern when individual dispersal distributions do not decline 1976 Robertsonianmetacentrics inthe mouse. Chromosoma exponentially with distance. Proc. R. SOC.London B 259: 243- 58: 341-353. 248. CHARLESWORTH,D., and B. CHARLESWORTH,1980 Sex differences SLATKIN, M.,1976 The rate of spread of an advantageous allele in in fitness and selection for centric fusions between sex chromo- a subdivided population, pp. 767-780 in Population Genetics and somes and autosomes. Genet. Res. 35: 205-214. Ecology, edited by S. KARLIN and E. NEVO.Academic Press, New DAVISSON,M. T., and E. C. AKESON,1993 Recombination suppres- York. sion by heterozygous Robertsonian chromosomes in the mouse. SLATKIN,M., and N. H. BARTON,1989 A comparison of three meth- Genetics 133: 649-667. ods for estimating average levels of gene flow. Evolution 43: EPPERSON,B. K., 1994 Spatial and space-time correlations in systems 1349-1368. of subpopulations with stochastic migration. Theor. Pop. Biol. SLATKIN,M., and D. CHARI~ESWORTH,1978 The spatial distribution 46: 160-197. of transient alleles in a subdividedpopulation: a simulationstudy. FISHER,R. A,, 1937 The wave of advance of advantageousgenes. Genetics 89 793-810. Ann. Eugenics 7: 355-369. STOKES,A. N., 1976 On two types of moving front in quasilinear HADELER,K. P., 1976 Travelling population fronts, pp. 585-592 in diffusion. Math. Biosci. 31: 307-315. Population Genetics and Ecology, edited by S. KARLIN and E. NEVO. VIROUX,M.-C., and V. BAUCHAU,1992 Segregation and fertility in Academic Press, New York. Mus musculus domesticus (wild mice) heterozygous forthe HAI.DANE,J. B. S., 1948 The theory of a cline. J. Genet. 48: 277- Rb(4.12) translocation. Heredity 68: 131-134. 284. WALLACE, B. M.N., 1994 A comparison of meiosis and gametogene- HATFIEID, T., N. BARTONand J. B. SEARLE,1992 A model of a hybrid sis in homozygotes and simple Robertsonian heterozygotes of the zone between two chromosomal races of the common shrew common shrew (Swex araneus) and house mouse (Mus musculus (Sorex aranm). Evolution 46: 1129-1145. domesticus). Folia Zool. 43(Suppl. 1): 89-96. HAUFFE,H. C., 1993 Robertsonian Fusions and Speciation in a House WHITE,M. J. D., 1978 Modes ofSpeciation. W. H. Freeman, San Fran- Mouse Hybrid Zme. PhD. Thesis, University of Oxford. cisco. HAUFFE,H. C., and J. B. SEARLE,1993 Extreme karyotypic variation WILSON,A. c.,V. M. SARICH and L. MAXSON, 1974 The importance found in a mottled hybrid zone between five chromosomal races of gene rearrangement in evolution: evidence from studies on of Mus musculus domesticus in Upper Valtellina, Northern Italy. rates of chromosomal, protein, and anatomical evolution. Proc. The tobacco mouse story revisited. Evolution 47: 1374-1395. Natl. Acad. Sci. USA 71: 3028-3030. KEY, K. H. L., 1968 The concept of stasipatric speciation. Syst. 2001. WOLFRAM,S., 1991 Mathematica. Addison Wesley, New York. 17: 14-22. WRIGHT,S., 1932 The roles of mutation, inbreeding, crossbreeding KING, M., 1993 Species Evolution: The Role of Chromosome Change. Cam- and selection in evolution. Proc. 6th Int. Cong. Genet. 1: 356- bridge University Press, Cambridge. 366. LANDE,R., 1979 Effective deme sizes during long-term evolution estimated from rates of chromosomal rearrangement. Evolution Communicating editor: M. SLATKIN 33: 234-251. MALECOT,G., 1948 Les Mathhatiques del’H6rbditi. Masson, Paris. MARLIYAMA, T., 1972 On the fixation probability of mutant genes APPENDIX A SPREADPAST A STRONGBARRIER in a subdivided population. Genet. Res. 15: 221-225. MAY, R. M., J. A. ENDLERand R. E. MCMURTRIE,1975 Gene fre- If the barrieris strong, anadvantageous allele spread- quency clines in the presence of selection opposed by gene flow. Am. Nat. 109 659-676. ing from the left will quickly be fixed on one side of MERCER,S., and J. B. SEARLE,1991 Preliminary analysis of a contact the barrier (x < 0). Integrating Equation 2 over x > 0, zone between karyotypic races of the common shrew (Swex ara- assuming q = 1, and applying the boundaq condition new) in Scotland. MCm. SOC.Vaud. Sci. Nat. 19: 73-78. MOLIJSON,D., 1977 Spatial contact models for ecological and epi- that @/ax = l/Bjust to the right of a barrier at x = demic spread. J. R. Statist. SOC.B 39 283-326. 0 shows that the total number of alleles on the other NACHMAN,M. W., and J. B. SEARLE,1995 Why is the house mouse side (x > 0) will increase as: karyotype so variable? Trends Ecol. Evol. 10: 397-402. NAGYIAKI,T., 1975 Conditions for the existence of clines. Genetics 80: 595-615. NAGYLAKI,T., 1976 Clines with variable migration. Genetics 83: 867- 886. NAGYIMI,T., 1978 Random geneticdrift in acline. Proc. Natl. Acad. Sci. USA 75 423-426. - --IT2 + S pdx. (Al) NEVO,E., and H. BAR-EL,1976 Hybridization and speciation in fos- 2B 1 sorial mole rats. Evolution 30: 831-840. PATTON,J. L., 1993 Hybridization and hybrid zones in pocket gophers (Rodentia, Geomyidae), pp. 290-308 in Hybrid Zones and Integrating over time, the Evolutionaly Process, edited by R. G. HARRISON. Oxford Univer- sity Press, Oxford. = - - REDI, C.A., and E. WANNA, 1988 Robertsonian heterozygotes in pdx IT2 (exp(St) 1) for p e 1. (A2) the house mouse and the fate of their germ cells, pp. 315-359 [ 2BS in The Cytoptics of Mammalian Autosomal Rearrangements, edited by A. DANIEL.Alan R. Liss, New York. Equation A2 accounts for theinflux of alleles, and their SAGE,R. D., W.R. ATCHLEYand E. CAPANNA,1993 House mice as subsequent increase. When St * 1, the influx becomes models in systematic biology. Syst. Biol. 42: 523-561. negligible, and the allele frequency approachesthe SCMN, P. N., 1992 Robertsoniantranslocations introducedinto an island population of house mice. J. 2001. 227: 493-502. Gaussian solution to Equation 2 with q = 1 and no SEARI.E,J. B., 1988 Selection and Robertsonian variation in nature: influx, so that (@I/&) = 0 at x = 0. Thus 504 J. Pihlek and N. H. Barton

this solutionby w (x - ct, t). These follow a linear diffren- tial equation that has one eigenfunction (Ad&/&), which corresponds to a shift in position by Ax, and has eigenvalue zero. Since thisis orthogonal to all the other eigenvalues, a perturbation E(X) causes an eventual shift This is only valid for small p, and St 4 1, but suggests that p(0, t) will approach 1 after a time given by setting Ax = S~(x)(d&/ax)dx/S(d&/dx~dx. The variance of Equation A3 to 1 at x = 0; this time is an approximation perturbation E is proportional to &q,,/2p in each genera- to the delay caused by the barrier and is greater than tion, and so the rate of increase of var(Ax) is the value of ST = log (&&'2) given by BARTON (1979a) (see Figure 1A).

APPENDIX B The rate of increase in variance of position can be calculatedusing the same method as for underdomi- Numerical solution of Equation 2 using the Runge- nance (BARTON1979b). Let Fisher's solution to Equation Kutta algorithm, followed by numerical integration of 2 be &(x - ct), where c = afi.Denote deviations from Equation B1, shows that this rate is 1.19a/p&.