LINKAGE DISEQUILIBRIUM, SELECTION AND RECOMBINATION AT THREE LOCI

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1 Copyright by the Genetics Society of America LINKAGE DISEQUILIBRIUM, SELECTION AND RECOMBINATION AT THREE LOCI ALAN HASTINGS Defartinent of Matheinntics, University of California, Davis, Calijornia Manuscript received May 20, 1983 Revised copy accepted September 26, 1983 ABSTRACT Limits to the relationship among linkage disequilibrium, selection and recombination at equilibrium in three-locus, two-allele, deterministic, discrete generation models are determined using linear programming techniques. These results show that the commonly used measures of linkage disequilibrium are not appropriate for a multilocus setting. Additionally, interactions among three loci are important in reducing the strength of selection necessary to maintain a given level of disequilibrium, relative to a two-locus model. NE of the most important problems in theoretical population genetics 0 during the past decades has been determining the extent to which results for one or two loci carry over to larger numbers of loci (LEWONTIN 1974; EWENS 1979). A particular example of such a question is the extent to which the relationship between selection and disequilibrium in two-locus models describes truly multilocus situations. Is the amount of selection necessary to maintain disequilibrium at a large number of loci given (approximately) by the product of selection coefficients at pairs of loci, or is it determined additively in some sense or is some other process occurring? It is just this question that is the central one in this paper. In an earlier paper, I discussed the relationship among seiection, recombination and linkage disequilibrium in the context of two-locus, two-allele models (HASTINGS 198 la). The goal of this paper is to consider similar questions within the context of three-locus models. Often, attempts are made to measure disequilibrium for pairs of loci. How do unseen loci affect inferences made from observations of linkage disequilibrium? The possibility that two-locus models do not necessarily give a good indication of the behavior of models with many more than two loci has been a theme of many papers in theoretical population genetics, notably FRANKLIN and LEWONTIN (1970). The questions I will examine-are there new features found in three-locus models not found in two- locus models-are similar to those raised by FRANKLIN and LEWONTIN. The results of my earlier paper, which considered only the existence of equilibria, can be summarized as indicating a very strong restriction on the level of disequilibrium present in a two-locus model at equilibrium. More precisely, let all fitnesses in a deterministic, two-locus, two-allele model lie between 1 + s and 1 - s times that of the double heterozygote. Then, if r is the Genetics January, 1984

2 154 A. HASTINGS recombination and D is the disequilibrium, I showed that at all equilibria of this model, rldl < s/10. If, to maintain large amounts of disequilibrium at many loci, such strong selection would be required (in a multiplicative sense), then linkage disequilibrium (caused by selection) would likely be unimportant in the population genetics of natural populations. Before even considering the relationship between disequilibrium and selection in a three-locus context, one must examine the measures of disequilibrium. These measures, which are unambiguous in a two-locus model, are not so in a three-locus context. Defining pairwise equilibria in a three-locus context is simply an application of the two-locus definitions. The three-locus disequilibrium term also has a standard definition due to BENNETT (1954). What is unclear is whether an increase in (the absolute value of) one of these four numbers represents an increase in the total disequilibrium. In the two-locus case, the relationship between the disequilibrium D and deviations from linkage equilibrium is clear. That such a clear-cut relationship is not to be expected for three loci i's evident from the work of HILL (1974) on the expected value of the covariance among the linkage disequilibrium at three loci under a neutral model. METHODS AND MODEL The model I will employ here is the standard three-locus, discrete generation deterministic population genetic model. I will employ the same mathematical techniques to analyze equilibria of this model as used in the paper by HASTINGS (198 la). I will first specify equilibria of the model. Such equilibria are specified either by eight chromosomal frequencies (constrained to add to one) or by three gene frequencies, three pairwise disequilibria and one three-way disequilibrium. For each such equilibrium I then determine the minimum strength of selection necessary to maintain such an equilibrium using linear programming techniques. The results reported are for this minimum strength of selection which is defined as follows. All fitnesses (viabilities) are constrained to lie between 1 - s and 1 + s. Then, the minimum value of s for which the specified equilibrium exists is found (for details see HASTINGS 1981a). Thus, the strength of selection can be thought of as 2s. The notation employed is the following. In all cases studied there are three loci, A, B, C, located in that order on the chromosome. The loci have alleles A,n, B,b, C,c respectively. Let PA be the frequency of the allele A at locus A; LAC be the frequency of chromosomes with allele A at locus A and C at locus C. Other frequencies are defined similarly. The pairwise linkage disequilibrium between A and B is denoted DAB. It is defined as DAB = PABpob - PAbpnB. The two other pairwise disequilibria are denoted and defined similarly. The third order disequilibrium among A, B and C is denoted as DABC and is defined as: DABC = PABC - PADBC - PBDAC PAPBPc. The recombination probability between loci A and B is denoted as?-ab; that between B and C is denoted as re(;. I have assumed no interference, so TAC = TAB + rbc - 2TABTBC. A number of equilibria were then examined to determine the stability of

3 THREE LOCI AND DISEQUILIBRIUM 155 the equilibria thus obtained. This was done by numerically calculating the modulus of the largest eigenvalue of the appropriate Jacobian matrix, evaluated at the equilibrium in question. RESULTS With the curse of dimensionality upon us, a fairly complete listing of equilibria (disequilibrium values) with minimum strengths of selection is quite out of the question. Instead, I will report observations based on a fairly coarse survey of the equilibrium space, along with a more detailed analysis of some special cases. Most of the special cases reported have gene frequencies of 0.5 FIGURE l.-minimum D BETWEEN LOCI A RND C strength of selection required for an equilibrium as a function of DAC. Here, PA = PB = PC = 0.6, TAB = rk = DAB = DE = 0.05; A. DAB = DBC = 0.10; 0. DAB = DBC = 0.15; +. The lines merely connect points at which the calculations were made.

4 A. HASTINGS 0 0 dbi FIGURE 2.-Minimum D BETWEEN LOCI A AND C strength of selection required for an equilibrium as a function of DAc. Here, PA = PB = PC = 0.5, TAB = rbc = DAB = Dm = 0.05; A. DAB = DBc = 0.10; 0. DAB = DBC = 0.15; +. The lines merely connect points at which the calculations were made. at all loci and TAB = rbc = Unless specifically mentioned, similar results are obtained for different gene frequencies and recombination rates. Since I am particularly interested in two-locus marginal properties, the cases examined closely reflect this view. In most cases either one or two pairwise disequilibria are specified, and then either one or two disequilibria are varied and the minimum strength of selection is calculated. A preliminary case has pairwise disequilibrium between only two of three loci and all three other disequilibria (pairwise and three-way) zero. Here, the minimum strength of selection required is the same as if only the two loci with disequilibria were present. Adding the third locus here has no effect. I will next consider cases in which two pairwise disequilibria, gene frequencies and the three-way disequilibrium are specified, while the final pairwise

5 THREE LOCI AND DISEQUILIBRIUM 157 %LOO OLl D BETWEEN LOCI R AND B FIGURE 3.-Minimum strength of selection required for an equilibrium as a function of DAB. Here, pa = pb = pc = 0.5, TAB = rsc = Dsc = DAc = 0.05; A. DK = DAc = 0.10; 0. The lines merely connect points at which the calculations were made. disequilibrium is varied. The two pairwise disequilibria held fixed can either both be large or one can be large and the other small. The value DUC, likewise, can be small or large. The final variable is the position of the loci on the chromosome. The following observations include all of these possibilities. Observation I: Let there be high pairwise linkage disequilibrium between adjacent loci, with DAB = DBC > 0. Then, the amount of selection required actually decreases when DAC increases away from zero. The minimum minimum strength of selection occurs for a value of DAG close to (but not equal to) DAB. See Figures 1 and 2. Making DAC negative does require additional selection. Observation ZI: Change the order of the loci from observation I. Let DBC = DAC > 0. Again, as DAB varies, the minimum minimum strength of selection occurs for a positive value (see Fighre 3).

6 158 r- 0 A. HASTINGS \ ""1 \ W $a :05 0!06 D BETWEEN LOCI A AND C FIGURE 4.-Minimum strength of selection required for an equilibrium as a function of DAC. Here, = pb = p, = 0.5, DAB = LIB(: I= 0.10, TAB = TBC = DMC = 0.0; A. DMc = 0.01 OR ; 0. The lines merely connect points at which the calculations were made. Observation III: The minimum "minimum strength of selection" for the cases considered in I and I1 depends not only on the other pairwise disequilibria, but also on the three-way disequilibria (Figure 4) and, more interestingly, on both the order of the loci and the recombination rates between them. See Figure 5 and compare it with Figure 2. Observation N: Let DAC be fixed, and let DABC = 0. Add a third locus inbetween, but keep DBC = 0. Then, the minimum strength of selection required increases as DAB increases (Figure 6). Also, a similar result obtains if DAC is decreased from zero. If, however, DABC < 0, a different result obtains. Let PA = 0.9, PB = Pc; = 0.5, DAB = -0.03, DBC = 0, DABC = -0.01, TAB = 0.10, rbc = Then, with DAC = 0, the minimum value of s is close to 1, whereas with DAC = 0.03, the minimum value of s is 0.25.

7 THREE LOCI AND DISEQUILIBRIUM 159 (D m D BETWEEN LOCI B AND C FIGURE 5.-Minimum strength of selection required for an equilibrium as a function of DE. Here, PA = fa = Pc = 0.5, DAB = DAc = 0.10, rm = rsc; = 0.001; A. rx = 0.01; 0. The lines merely connect points at which the calculations were made. Obseruation V: By combining the information available in the previous observations, one can examine the case in which DAB is positive and a third locus is added outside these two. Then, varying DBC or DAC away from zero while holding the other fixed at zero increases the minimum strength of selection required. Next I will consider the effect of varying DABC with all other values held constant. Observation VI There are cases in which varying DABc away from zero may decrease the minimum strength of selection required, as the following example shows. Let PA = 0.9, PB = pc = 0.5, DAB = -0.03, DBC = 0.15, DAc = -0.03,?+AB = 0.10, YBC = Then, with DABc = 0, the minimum value of s is 0.23, whereas with DABC = 0.01, the minimum value of s is The following 24

8 A. HASTINGS FIGURE 6.-Minimum D BETWEEN LOCI A AND B strength of selection required for an equilibrium as a function of DAB. Here, PA = PB = PC = 0.5, DBC = 0, rm = rk = DAC = 0.05; A, DAC = 0.10; 0. DAC = 0.2; +. The lines merely connect points at which the calculations were made. example is even more surprising. Let PA = p~ = pc = 0.5, DAB = 0.15, DBC = DAC = 0.0, TAB = rbc = Then, with DABC = 0, the minimum value of s is , whereas with DMc = -0.01, the minimum value of s is Here, the marginal observation of the quantity rd/s at loci A and B is increased from to , by the interactions with the third locus. Note that the value is the same as the one in the true two-locus model. Compare these situations to Figure 4, in which increasing DABC does increase the minimum value of s. Finally, I will consider the effect of adding a third locus and varying two pairwise disequilibria simultaneously. Observation VIZ: Let DAB > 0 be fixed. Add a third locus, with DBC = DAC > 0. If p = 0.5 at all loci, adding such a locus requires increasing the strength 24

9 THREE LOCI AND DISEQUILIBRIUM 161 P D BETWEEN LOCI A AND C, B AND C FIGURE 7.-Minimum strength of selection required for an equilibrium as a function of DAc = Dw;. Here, p A = p~ = pc = 0.5, rab = rx = DM = 0.05; A. DM = 0.10; 0. D,Q = 0.15; +. The lines merely connect points at which the calculations were made. 24 of selection (Figure 7). However, if p = 0.6, there is a range of values over which little additional selection is required (Figure 8). Observation VIII Let there be two loci with significant disequilibrium between them. Then, maintaining a third locus between these two with large pairwise disequilibria between the interior locus and the outer two requires essentially no additional selection (Figure 9). The results on the stability of the equilibria obtained by the linear programming techniques are somewhat disappointing. All of the cases examined were unstable. However, in all cases the spectral radius is extremely close to 1, typically about 1.O 1. Hence, small changes in the parameters and, consequently, the equilibrium should make the equilibrium stable. Thus, the results

10 A. HASTINGS - "b:oo 0:04 OL08 0:12 0:lS 0:20 0'. 24 B ET W E E N LOCI - FIGURE 8.-Minimum strength of selection required for an equilibrium as a function of DAC = DEC. Here, PA = PB = PC = 0.6, TAB = rbc = DAB = 0.05; A. DAB = 0.10; 0. DAB = 0.15; +. The lines merely connect points at which the calculations were made. reported here, although not exact limits for stable equilibria, likely provide a good guide to the limitations on stable equilibria. DISCUSSION There are two classes of conclusions possible from the conclusions presented. One deals with the measurement of linkage disequilibrium in a multilocus context. The second concerns the interaction among three loci, selection and recombination in the maintenance of linkage disequilibrium and evolutionary consequences. The use of the usual disequilibrium measures, the ones employed here, is flawed in a multilocus context in which selection is present. This occurs because observations of a single pairwise disequilibrium in a three-locus context can be

11 THREE LOCI AND DISEQUILIBRIUM 163 *O I 4 2 %a D BETWEEN LOCI A AND B e B AND C FIGURE 9.-Minimum strength of selection required for an equilibrium as a function of DAB = DEC. Here, PA = = PC = 0.5, = re(: = DAC = 0.05; A. DAC = 0.10; 0. DAC = 0.20; +. The lines merely connect points at which the calculations were made. misleading. It is not the case that more disequilibrium between a pair of loci necessarily implies more selection (observations I, 11, IV and VI). Also, any attempt to modify the measure of linkage disequilibrium must include the recombination rates between the loci in question to be valid, as the minimum minimum strength of selection can depend on recombination, as in observation 111. Thus, attempts to use log-linear-type models to study the interactions among loci, as in SMOUSE (1974), have only suggestive value when selection is operating. The major finding of this paper on the interaction among selection, recombination and linkage disequilibrium is the following. The minimum amount of selection necessary to maintain three loci in high pairwise disequilibria is primarily determined by the disequilibrium between the outer two, as in observation VI11 (see also I and 11). If this result does extend to more loci, then

12 164 A. HASTINGS linkage disequilibrium may be an important feature of the population genetics of natural populations. This is a major difference between the results reported here and those for two loci reported by HASTINGS (198 la), which appeared to place strong limits on disequilibrium. It is tempting to say that the three-locus results reported here do provide a good guide to situations with more loci. However, two-locus systems can exhibit surprising behavior not found in onelocus systems (HASTINGS 1981b,c, 1982), and the results here on three-locus systems differ from those on two-locus systems. Even marginal two-locus observations of the quantity rd/s are affected by a third locus, as in observation VI. Here, this figure is increased beyond that possible in a truly two-locus model. Thus, the limits to the relationship among r, D and s presented for two loci by HASTINGS (1981a) may not carry over simply to more loci. If, in fact, linkage disequilibrium may be important as suggested here, why have attempts to measure it been so unsuccessful? Perhaps the reason lies only with the statistical problems pointed out by BROWN (1975). Additionally, the results here are in qualitative agreement with the observation that linkage disequilibrium between an inversion and an allozyme locus is more common than between two allozyme loci (LANGLEY, TOBARI and KOJIMA 1974). Many questions concerning the role of linkage disequilibrium remain unanswered. I thank JOHN GILLFSPIE and JIM QUINN for helpful discussions. Supported by Public Health Service grant 1 R01 GM LITERATURE CITED BENNETT, J. H., 1954 On the theory of random mating. Ann. Eugen. 184: BROWN, A. H. D., 1975 Sample sizes required to detect linkage disequilibrium between two or three loci. Theor. Pop. Biol EWENS, W. J., 1979 Mntheinnticnl Populntio11 Genetics. Springer-Verlag, Berlin. FRANKLIN, I. and R. LEWONTIN, 1970 Is the gene the unit of selection. Genetics 65: HASTINGS, A., 1981a Disequilibrium, selection, and recombination: limits in two-locus, two allele models. Genetics 98: HASTINGS, A., 1981b Marginal underdominance at a stable equilibrium. Proc. Natl. Acad. Sci. USA 78: HASTINGS, A., 1981c Stable cycling in discrete-time genetic models. Proc. Natl. Acad. Sci. USA HASTINGS, A., 1982 Unexpected behavior in two locus genetic systems: an analysis of marginal underdominance at a stable equilibrium. Genetics 102: HILL, W. G., 1974 Disequilibrium among several linked neutral genes in finite population. 11. Variances and covariances of disequilibria. Theor. Pop. Biol LANGLEY, C. H., Y. N. TOBARI and K. I. KOJIMA, 1974 Linkage disequilibrium in natural pop ulations of Drosophila?nelanognsler. Genetics 78: LEWONTIN, R., 1974 York. The Geiietir Bnsis of Evolutioitary Change. Columbia University Press, New SMOUSE, P., 1974 Likelihood analysis of recombinational disequilibrium in Inultiple-locus gametic frequencies. Genetics 76: Corresponding editor: W. J. EWENS