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Genetic Drift in a Cline Felsenstein GENETIC DRIFT IN A CLINE MONTGOMERY SLATKIN Department of Biophysics ana! Theoretical Biology, The University of Chicago, 920 East 58th Street, Chicago: Illinois 60637 and TAKE0 MARUYAMA National Institute of Genetics, Yata 1,111, Misima, Sizuoka-ken 411, Japan Manuscript received November 26, 1974 Revised copy received May 12, 1975 ABSTRACT A model is developed of genetic drift in a cline maintained by spatially varying natural selection and local dispersal of individuals. The model is analyzed by an approximation scheme which is valid for weak selection and small migration rates. The results, which are based on numerical iterations of the approximate equations, are that the cline is less steep than predicted on the basis of the deterministic theory but that for weak selection the correlation between random fluctuations in neighboring colonies is approximately the same as in models of migration and drift in the absence of selection. PATIAL patterns of gene frequencies in natural populations are determined by S the combination of genetic drift, spatially varying natural selection, and gene flow between local populations. A quantitative theory of the interaction of these mechanisms is necessary for interpreting observed spatial patterns and for understanding the possibilities for local genetic differentiation and speciation in the presence of a small amount of gene flow. Most of the previous theoretical work on gene flow has ignored either genetic drift or natural selection. We present here the results of a simple model of genetic drift in a cline main- tained by selection and gene flow and reach come conclusions which do not seem to depend on the details of the model and which have some implications for real populations. We find that genetic drift can significantly alter the structure of a cline maintained by selection and gene flow, but that spatially varying selection makes little difference in the expected correlation in gene frequencies between local populations based on the theory of gene flow and drift alone. FELSENSTEIN(1975) has considered the same problem, but from a different point of view. To analyze the model, he uses the approximation that the ex- pected gene frequencies at each location are the same as in the absence of random drift. This is quite accurate if the local population size is large, but with this technique it is not possible to determine the effect of drift on expected gene frequencies in the cline. We use a different method, which is accurate only if * SupDorted in part b4 AEC grant AT (11-1)-2472 Genetics 81: 209-222 September, 1975. 21 0 M. SLATKIN AND T. MARUYAMA the local population size is small; so with our model, we can consider a different range of parameters than does FELSENSTEIN. HASTINGSand ROHLF (1974) consider the effect of genetic drift when there is a linear change in the selection gradient. They consider a different case than ours in which drift cannot change the expected gene frequencies and determine the relationship between the expected and actual clines as a function of migra- tion. They- show that for the case they analyze migration must be relativelg- large before the actual cline observed would be a good predictor of the under- lying selection. THE MODEL For a diploid species with discrete non-overlapping generations we consider a single locus with two alleles. A and a. We assume that there are IZ local popula- tions, each of effective size N, arranged in a one-dimensional array. For def- initeness in notation. we will aways assume that n is odd. Selection acts inde- pendently in each population and there is no dominance; we will consider later the effect of different degrees of dominance. The selection coefficient in the ith population is s1. We assume that there is migration only between adjacent populations in the array and that a fraction m/2 of the alleles is exchanged each generation. This is KIMURA’S(1953) “stepping-stone” model. The assumption of gene flow between adjacent populations only is for computational simplicity and is not restrictive. It is known both for the cases of gene flow and selection and for gene flow and genetic drift that only the variance in the dispersal distance is significant, at least to the first approximation (SLATKIN1973; MARUYAMA1972). Therefore, in the present model, more general types of local dispersal would be expected to produce approximately the same results. We ignore long-distance migrants which could have significant effects on the spatial pattern of gene frequencies because we are interested here only in the effects of local gene flow. By ignoring long-distance migrants, we are underestimating the importance of gene flow. For a finite collection of populations, the only equilibrium state is complete fixation of one of the alleles. unless there are mutations. However. differential selection in different locations could retard fixation greatly in much the same wav as heterosis (ROBERTSON1962). Therefore, the structure of the population before fixation occurs is of interest. One way to analyze that structure is to con- sider a “quasi-equilibrium” state, before fixation occurs. in which there is a relatively permanent cline with some fluctuation due to genetic drift. FELSEN- STEIN (1975) uses this method. Another possibility is to introduce some mecha- nism which prevents fixation and analyze the true equilibrium of the modified model. This is reasonable if the results are not sensitive to changes in the stabiliz- ing mechanism. We use this approach here and assume that the end populations in the array exchange a fraction m/2 of their genes with reservoirs kept at fixed gene frequencies po and p,. This is an appropriate model for a chain of islands between two mainland areas, but, as we will show, our results apply to other situations as well. GENETIC DRIFT IN CLINES 21 1 We denote the frequency of the A allele in generation t by x1 (t). If m and the si are small, the basic recursion equation for the model is m zi(t+l)=(l-m)si(t)+ - (~i-~(t)~i+~(t)) 2 + +SiZi(t) (l-zi(t>) + . (1) With weak selection and low migration rates, this equation applies whether selection occurs before or after dispersal. The first two terms represent the effects of gene flow; the third, natural selection; and the last, genetic drift, modeled by a random variable, ti, which ,has mean 0 and variance xi (t)( 1-xi (t)) /2N (CROWand KIMURA1970, Chapter 7). In (1) , terms of the order of magnitude of (ml), (ms), and (9)are omitted. Throughout, we assume that m,lsil<<l and that, notationally, zo=poand zntl=pl, according to the boundary conditions we have specified. To consider the expected equilibrium shape of the cline, we used the first and second moments of (1 ) . There is no solution to (1) with xi (t+ 1) =xi (t)because & is assumed not to be constant in time, but we can expect the ensemble averages of xi to approach constant values. If we take the expected value of (1) and assume Fi (t+ 1) =Zi (t),we get where the bar indicates the expected value. We will assume that the expected value or ensemble average is the same as the time average, which could, in principle, be measured. The last term of the right-hand side of (2) can be rewritten as Sifi (1-fi) - siui2 (3) where uiz = so we have Equation (4), except for the last term, is the same as found in the absence of genetic drift. Since ui2 is always positive, the effective selection at each location is weakened and the cline will be less steep than in the absence of genetic drift. PELSENSTEIN(1975) reached the same conclusion using an intuitive argument.- To take the second moment of (1) we define a new variable yi (t) =xi (t)-xi, the deviation from the expected value of the gene frequencies. The equation for yi (t)in matrix form is where - - l-m m/2 0 0 - - - m/2 1-m m/2 0 - Y (1) M= 0 m/2 . 21 2 M. SLATKIN AND T. MARUYAMA and Note that the boundary conditions do not enter directly in (5) but only through - xi. We multiply (5)- by its transpose and take expected values to get the equation for the matrix A = yyT where 1 D,=- 2N and un2 . - The diagonal elements of A are the variances, uL2. In general, the system of equations derived in this way is not closed; (7) involves the higher moments of yL. However, the terms with those higher moments are all mutiplied by s, or st2. If selection is weak (Ns,<<l), then, to a first approximation, these terms can be ignored. This is equivalent to assuming that, while genetic drift affects the expected gene frequencies, there is no direct interaction of selection and genetic drift in determining the variance and correla- tions in gene frequencies. GENETIC DRIFT IN CLINES 21 3 We can estimate the magnitude of the terms in (7) that depend on the selec- tion coefficients to estimate the error made with this approximation scheme. The elements of =and MysT are the expectations of sisj [yi (1-2%;) + (ai2-yi2) ] [yj ( 1-2&) + (ajz-yjz) 1 (8) and We can place an upper bound on (8) and (9) by using the fact that the moments of yi are a maximum when one or the other allele is fixed at each location and when adjacent locations are completely correlated. For example aiz<zi (1 -YO. Thus, (9) can be shown to be bounded by the maximum of sj(l-2fi + &(l-Zj)) , (10) which is always less than Ssj/4.
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