HITCHHIKING: A COMPARISON OF LINKAGE AND PARTIAL SELFINGI

PHILIP W. HEDRICK

Division of Biological Sciences, University of Kansas, Lawrence, Kansas 66045

Manuscript received May 2, 1979 Revised copy received September 17, 1979

ABSTRACT

Genetic hitchhiking occurs when alleles at unselected loci are changed in frequency because of an association with alleles at a selected locus. This asso- ciation may be mediated either by linkage or partial selfing (inbreeding) and can affect the gene frequency and gametic disequilibrium at the neutral loci. Hitchhiking from partial selfing (unlinked loci) occurs more quickly than linkage hitchhiking and generally has a greater effect. In addition, partial- selfing hitchhiking can cause increases or changes in sign in gametic disequi- librium between neutral loci. The effects of the two types of hitchhiking with different levels of dominance, zygotic frequencies and number of selected loci are also examined. The general conditions for linkage and partial-selfing hitchhiking are outlined and the implications of hitchhiking are discussed for marker or electrophoretic loci.

I T is often assumed that regular changes in gene frequency are an indication of selection between the alleles at these loci. However, selection at a locus

linked to a neutral locus can change the frequency of alleles at the neutral locus, a phenomenon known as hitchhiking (KOJIMA and SCHAFFER 1967). Various aspects of this phenomenon have recently been explored or discussed, such as the relationship to gene frequency change in selection experiments or natural populations (NEI 1975, p. 149; STAM 1975; KAUFFMAN and LEE 1976; THOMSON 1977), the level of heterozygosity in natural populations (MAYNARD SMITH and HAIGH 1974; OHTA and KIMURA 1975; HAIGH and MAYNARD SMITH 1976; OHTA and KIMURA 1976; THOMSON 1977), gametic disequilibrium between neutral loci (THOMSON 1977), estimation of selection coefficients ( OHTA and COCKER- HAM 1974; THOMSON 1977; CLEGG, KAHLER and ALLARD 1978) and the inci- dence of genetic diseases in humans (WAGENER and CAVALLI-SFORZA 1975).

Since, in some instances, linkage and partial selfing have analogous affects (JSARLIN 1969; WEIR and COCKERHAM 1973), I have examined a model with partial selfing to determine the importance of hitchhiking in a nonrandom mating system. Partial selfing is an important reproductive system in many plants and some animals (for a comprehensive review of the effect of plant mat- ing systems on genetic variation, see BROWN 1979). Furthermore, results in a partial-selfing model should be generally valid for other inbreeding systems.

Research partially supported by NSF Grant DEB73-01305 A02.

Genetics 94: 791-808 March, 1980.

792 P. W. HEDRICK

Therefore, the focus of this study will be on the effect of hitchhiking due to partial selfing on gene frequency and gametic disequilibrium change and a comparison to the effect of hitchhiking resulting from linkage.

DESCRIPTION OF T H E MODEL

Let us assume an infinite population of organisms with n loci, each locus having two alleles. The total possible number of gametes is then given by N = 2". This model can of course be extended to multiple alleles. Let the frequency of the diploid genotype composed of gamete types i and i be equal to gij and gii = gi i . Similarly, the relative viability of genotype ij will be denoted by W i i , where wij = wji. These frequencies and viabilities are given in Table 1 for the two-locus case, as is the notation for gametic frequencies in the two- and three- locus cases. Assume that generations are discrete and nonoverlapping, and via- bility selection operates on the pre-mating diploid organisms. The frequency of genotype ii after selection, but before mating, is then given by

where

is the mean relative fitness of the population. Inbreeding is imposed on the population by assuming that selfing occurs with

probability s and random outcrossing with probability 1 -s. Let Hu:dj be the probability that a genotype ii produces a gamete of type U. Obviously, this is a function of the probability of recombination among all the loci. Using this

TABLE 1

Genotypic frequencies, relatiue fitnesses and gametic frequencies in two- and three-locus cases

* Notation used for two-locus genotypic frequencies (gif) and relative fitnesses (wij). ** Notation used for gametic frequencies in the two- and three-locus cases.

HITCHHIKING 793

recombination information, the frequency of genotype uu after mating, gwv, is N N

guv =s.z 0.=13=1 .E ( H u : i r ) ( H v : i i ) gir + (1-3) z'uz 'v , (2) where

is the frequency of gamete type U in the post-recombination parent population. The equations above can be used to follow changes in genotypic, gametic and

allelic frequencies over time for any number of loci. I will, however, consider only the cases of two and three loci with two alleles at each locus. The equations for two loci are given below where we denote the recombination fraction between loci A and B by c, where 0 5 c 5 %.

where,

For n = 3, let CAB and cBc denote the recombination probabilities between loci A and B and between B and C, respectively. By assuming noninterference of the loci, the recombination probability between A and C is then equal to

CAC = CAB + CBO - 2 c A B c B C . Since the three-locus equations are a straightforward, but cumbersome, exten- sion of (3), I will not give them here. Because even the two-locus, partial-selfing model is algebraically extremely complex, the results will be given as numerical iterations of particularly pertinent situations of the above equations.

Allelic frequencies at the ith locus will be denoted by pi. Since I am assuming only two alleles per locus, the frequency of the second allele, denoted by 44, is

(4)

4' = 1 - p . 1.

The allele frequencies, when there are two loci, are

pl = x1 + x2 and p z = x1 + x3 , (5 1

794 P. W. HEDRICK

whereas for n = 3,

p1= x1 + 5 2 + XB + x4; pz = XI + x., + 2.; + 51; and

p3 = XI+ 2 3 + x5 + 5 7 . (6) To aid in the description of multilocus changes over time, we will use the

following measures of interlocus association or gametic disequilibrium (see also HEDRICK, JAIN and HOLDEN 1978). A useful and the most common alge- braic measure of gametic disequilibrium is the parameter D, introduced by LEWONTIN and KOJIMA (1 960) and defined for two loci as

D = xi - pipz

D = - xzxs .

(7)

or equivalently as

Since this measure is a function of the allelic frequencies at each locus, we will also use the standardized disequilibrium measure, D', introduced by LEWONTIN (1964) as

D' = D/Dmax (8) where

min(p1q27 qIpZ), > Dmax =

min(plp2, q1q2), D < 0 This measure, unlike D, has the same range (-1 to 3.1) for all gene frequen-

cies, making it particularly useful for situations in which there are large changes in allelic frequencies.

For the two-locus case, the notation above is sufficient; however, when we consider three loci, we have three pair-wise disequilibrium measures

D A B = D,, = (Xl + x*) - p1pz De0 = D23 (xl f x5) - pZp3 D A C = 0 1 3 = (21 $- 23) - p1p3

Similarly, the pairwise standardized disequilibrium measures are

where

(9)

HITCHHIKING 795

In addition to pairwise association, a three-way gametic disequilibrium

(11)

parameter (BENNETT 1954) can be defined as

DABO = D1m = ~ 1 - pl& - p J A , - p& - ~ 1 ~ 2 ~ 3 - One can assume that DaBc measures that gametic disequilibrium not “accounted for” by the pairwise measures.

RESULTS

The magnitude of the hitchhiking effect is strongly dependent upon the initial gametic array in the population and particularly upon the type and amount of gametic disequilibrium. Specifically, when considering only two loci, if there is no gametic disequilibrium between the selected locus and the neutral locus there is no hitchhiking. However, if an allele at a neutral locus, say B,, is asso- ciated with an allele favored by selection at another locus, say A,, then gene frequency change of B1 can result from this association.

The intent of the first part of this section is to examine the relative effects of other factors on hitchhiking. Therefore, I will use a gametic array that has the maximum initial gametic disequilibrium, D’ = 1, and the favored allele at the selected locus and the hitchhiking allele have low initial frequenc:es. Such an initial array could be generated by migration From another population con- sisting primarily of A,B, gametes. At this point, I will consider only the changes at the two loci, A and B. Assume that the fitness of the genotypes, A I A I , A,A, and A,A,, are 1 4- t, 1 -I- ht and 1, respectively. The initial genotypic frequencies are always in multilocus Hardy-Weinberg proportions (random association of gametes into zygotes) for the linkage examples. When examining partial self- ing, the initial zygotic array is 100% homogametic zygotes, i.e., only A,B,/A,B, and A,B,/A,B, genotypes, except where it is indicated that Hardy-Weinberg proportions are used; furthermore, it is assumed in the partial-selfing examples that the selected locus is unlinked to the other loci.

Gene frequency As mentioned before, partial selfing and linkage have some analogous effects

when there is no selection; with selection on but one of the loci, it will be obvious from the outset that they can have very different effects. For example, Figure 1 gives a comparison of linkage and selfing, each resulting in a similar change in gene frequency of approximately 0.6 in both examples. In this case, c = 0.01 for the linkage example and s = 0.95 for the partial-selfing example. In this example, and all others except where noted, the selection coefficient, t , is 0.2, there is additive gene action, h = 0.5, and the initial gametic array will have frequencies of 0.01 and 0.99 for A,B, and A,&, respectively.

The major difference between linkage and partial selfing is that hitchhiking uia partial selfing changes the frequency at the neutral locus much more quickly. The frequency of B , nears its asymptote in approximately half the number of generations for partial selfing as for linkage. This can be related back to the

796 P. W. HEDRICK

Generations

FIGURE 1.-The change in gene frequency for a selectively favored allele (A , ) and a neutral allele (B , ) associated with it (the initial gametic frequencies for A,B, and A,B, are 0.01 and 0.99, respectively). The broken lines indicate the frequencies when A and B are one map unit apart (c = 0.01) with random mating, and the solid lines represent the changes when there is 95% selfing (s = 0.95) with no linkage.

faster change at the selected locus A for partial selfing that results from the decreased frequency of heterozygotes in highly selfed populations.

In examining the asymptotic rate of decay of gametic disequilibrium in a system with partial selfing and linkage, KARLIN (1969) and WEIR and COCKER- HAM (1973) showed that partial selfing and linkage contribute equally, i.e., if the two factors are scaled similarly (partial selfing, s, ranging from 0.0 to 1.0 and linkage, 1-2c, also ranging from 0.0 to 1.0), their effects are equivalent. However, for hitchhiking, the two factors are quite different. In Figure 2, the final gene frequency for B , after hitchhiking is given for different levels of partial selfing or linkage.

First of all, the final gene frequency after hitchhiking for partial selfing is quite dependent on the initial zygotic constitution. When there are initially only homogametic types, then partial selfing always has a greater hitchhiking effect than linkage. But when there are initially Hardy-Weinberg proportions of the zygotes, linkage has, at high values, a greater effect. However, an argu- ment can be made that when there is high partial selfing, it would be reasonable that the initial zygotic proportions would be nearly homogametic proportions.

As pointed out by MAYNARD SMITH and HAIGH (1974) and discussed by THOMSON (1977), the level of dominance is critical when hitchhiking OCCUTS because of linkage to an allele undergoing directional selection. When the favored allele is recessive ( h = O.O), hitchhiking will occur only with very tight linkage,

HITCHHIKING

r 0.8 1

2,

c 0

0

t 6 - 0 E

ii

0.7 0.0 0.9

s or I-2c

1.0 -

0.8 -

2,

c

0

LL

% 0.6 - ?!

6 E 0.4 ii

- -

0.2 -

0.7 0.0 0.9

s or I-2c

797

3

FIGURE 2.-The final gene frequency after hitchhiking for different levels of selfing (solid lines) or recombination (broken line). The homogametic example assumes that initially all individuals were A,B,/A,B, or A,B,/A,B,, while the Hardy-Weinberg example assumes that homogametic and heterogametic (A,B,/A,B,) types were in binomial proportions.

1.0 -

0.0 - h 0

5 0.6 - g t m 0 04 .5 LI

- -

- I

Recessive,

I

0.2 -

/ 0.0 - - - - -* - - - - -1 - - - -1 - - - - -1 - - - - - 1 - d 0.7 0.0 0.9 1.0

s or I-2c

FIGUEE 3.-The final gene frequency after hitchhiking for dominance and recessivity when there is selfing (solid lines) or linkage (broken lines).

798 P. W. HEDRICK

while with additivity ( h = 0.5) or dominance ( h = l.O), hitchhiking occurs fo r a range of values. The broken lines in Figure 3 give the amount of linkage hitch- hiking for recessivity and dominance (additivity was given in Figure 2 and has only a slightly less effect than dominance). Obviously, with recessive gene action, linkage must be very tight (c < 0.01, in this case) for there to be any effect. This is partially a result of the low initial frequency. Since selection is so inef- fective in changing the gene frequencies, a long time is required for the initial gametic disequilibrium to break down. When the gene frequency of A , does begin to change significantly (after about 400 generations in this case), gametic disequilibrium is essentially zero for all but the tightest linkage.

On the other hand. when there is partial-selfing hitchhiking, the effect of different levels of dominance is quite small. Hitchhiking occurs for all levels of dominance with directional selection to a fairly similar extent as shown by the solid lines in Figure 3, which give the amount of hitchhiking for dominance and recessivity. Additivity, which was given in Figure 2, is almost exactly inter- mediate between dominance and recessivity. The similarity is the result of a low frequency of heterozygotes with high selfing, making dominance obviously a secondary factor for partial-selfing hitchhiking.

A related phenomenon is that there is little hitchhiking with partial selfing when there is symmetrical overdominance, as used by THOMSON (1977) to inves- tigate linkage hitchhiking. Because high partial selfing reduces the frequency of heterozygotes to a low level, there is little differential selection (the homozy- gotes have the same relative fitness). As a result, there is only slow gene frequency change and, therefore, minimal hitchhiking.

The overall selective advantage could be spread over several loci instead of only one. Assume that the selective advantage is spread over two loci making

TABLE 2

The initial gamete frequencies used as examples

Gamete a

AIBIC, 0.01 AIBIC, 0.0 A A C , 0.0 A A C , 0.0 A,B,C, 0.0 A * B A 0.0 A,B*Cl 0.0 A P , G 0.99

D*, 0.0099 D*c 0.0099 DBC 0.0099

D1*13 1.0 D’AG 1.0 DIBG 1 .o

DAB0 O.GO97

Initial frequency b c

0.25 0.125 0.0 0.0 0.0 0.0 0.25 0.125 0.0 0.0 0.25 0.375 0.25 0.375 0.0 0.0

0.0 0.0 0.0 0.0 0.0 -0.125 0.125 0.0938 0.0 0.0 0.0 0.0 0.0 -0.5

HITCHHIKING 799

a 10% selection differential at each ( t = 0.1) and there is a third neutral locus. The three-locus gametic array (a) of Table 2 will be used. In order to avoid confounding hitchhiking with two-locus selection problems, only nonepistatic models were examined. The curves for additive and multiplicative independence are virtually identical, so that ensuing discussion applies to both situations.

Again, linkage hitchhiking and partial-selfing hitchhiking are very different in some respects. Obviously, the map position of the selected loci relative to the neutral locus will be of major consequence in linkage hitchhiking. Since the partial selfing case assumes the selected loci are unlinked to the neutral locus, map position is of no consequence. Figures 4 and 5 illustrate the results of spread- ing the selective advantage over two loci for linkage hitchhiking and partial- selfing hitchhiking, respectively. These graphs are drawn on the same scale, i.e., s = 1 - 2c, for easy comparison, where c is the recombination between the selected locus ( A ) or loci ( A and C) and the neutral locus ( B ) .

Figure 4 gives the final gene frequency for linkage hitchhiking when there is one locus (solid line) or two loci five map units apart (broken line). Of course, the two-locus case will become the one-locus case if the map distance is zero between the selected loci. First, it is obvious that the region where there is sig- nificant hitchhiking is greater when the selective advantage is spread over two loci. For example, with two loci, there is a region of more than seven map units

\ ' \ ' \

1

\ L' \

0.0 I I I I I I I One locus ( 1 - 2 ~ ~ ~ ) 0.9 0.95 Lo a95 0.9

(fq2cAm) 095 1.0 0.9s 0.90 0.85 0.9 0.95 1.0 0.95 Two loci (I-ZCncI 0.85

FIGURE 4.-The final gene frequency after linkage hitchhiking when the neutral locus ( B ) is linked to one selected locus (solid line) or two selected loci (broken line) where each have half the advantage of the single locus. The horizontal axis gives the location of the neutral locus relative to the position of the selected locus (loci).

800

I .o

0.8

0.2

0.0

P. W. I-IEDRICK

S

FIGURE 5.-The final gene frequency after partial-selfing hitchhiking when selection is equally divided between two loci (broken lines) as compared to the case where selection i s at one locus (solid lines). Homogametic and Hardy-Weinberg use are as defined before.

where the frequency of B, is increased to more than 0.5, while, when selection is at only one locus, this region is less than half this size (about three map units). Second, there is a synergistic effect such that hitchhiking is much greater when the neutral locus is between the selected loci, making the amount of hitchhiking much greater. For example, when the neutral locus is 2.5 map units from one selected locus (1 - 2cAB = 0.95), the final gene frequency is 0.34. With the two selected loci and the neutral locus exactly intermediate (1 - 2cAB = 1 - 2cBc = 0.95) or 2.5 map units to the side (1 - 2cAB = 0.95 and 1 - 2cBc = 0.85 or vice versa), the final gene frequencies are 0.52 and 0.22, respectively. This syn- ergism for “embedded” loci is different from that seen because of finite popu- lation size (SVED 1968; OHTA and KIMURA 1969) or in equilibrium populations (LEWONTIN 1964), since this is an infinite population and applies for both addi- tive and multiplicative independence and, presumably, for epistatic directional selection arrays as well. Spreading the selective advantage over two loci also increases partial-selfing hitchhiking (Figure 5). The effect is not so interesting or so complicated as with linkage, but is fairly substantial for high selfing.

Therefore, for both linkage and partial selfing, hitchhiking is increased by spreading a given selective advantage over several loci. With linkage, the map distance between loci and their relative map positions are important. If a number of loci, each with small effect, are spread over a region with linkage hitchhiking or throughout the genome with partial-selfing hitchhiking, then substantial hitchhiking should occur at neutral loci in the region or genome, respectively.

HITCHHIKING 801

Gametic disequilibrium THOMSON (1977) demonstrated that gametic disequilibrium can be gener-

ated between two neutral loci as a result of linkage hitchhiking. The basis for this effect is the simultaneous hitchhiking of two loci that exhibit gametic dis- equilibrium with a selected locus. This may be because the initial pairwise disequilibria are nonzero, DABc is nonzero, or both. As for gene frequency, a comparison of the effects of linkage and partial-selfing hitchhiking on gametic disequilibrium will be made.

Figures 6 and 7 illustrate the changes in gametic disequilibrium parameters for linkage and partial-selfing, respectively. The gametic array (a) in Table 2 is used so that the initial gametic disequilibrium is maximized and the favored allele at the selected locus and the hitchhiking alleles at the two neutral loci have low initial gene frequencies. All other parameters are identical to those used for Figure 1, so that the gene frequency hitchhiking is comparable for link- age and partial selfing. The selected locus for the linkage example is between, and one map unit from each of, the two neutral loci.

As indicated by THOMSON (1 977) , pairwise D values may increase over time for linkage; this occurs in Figure 6, and also occurs in Figure 7 for partial selfing. However, D’ values always appear to approach zero between selected loci and neutral loci (DAB’, D A ~ ’ ) , as illustrated in Figures 6 and 7. D,,’, the gametic

Generations

FIGURE 6.-The change in gametic disequilibrium is measured by D and D’ when the selected locus is in between and one map unit from the two neutral loci. The change in gene frequency for these loci are given in FIGURE 1. The change at locus C is identical to that at locus B .

802 P. W. HEDRICK

1.0

0.8

0.6

9 0.4

0.2

0.0

1 I I I I

Generations

FIGURE 7.-The change in gametic disequilibrium is measured by D and D when there is 95% selfing. The changes in gene frequency for these loci are given in Figure 1. The change at locus C is identical to that at locus B.

disequilibrium between the neutral loci, always declines in this example, but it may, as pointed out by THOMSON (1977), and as I will demonstrate in other examples (see, for instance, Figure a), increase over time. The behavior of DABC is quite complicated in this example. Unlike THOMSON’S example, DABC, in this case, changes sign before beginning to approach zero.

As for changes in gene frequency due to hitchhiking, the hitchhiking effect on gametic disequilibrium is much faster for partial selfing than for linkage. This is again based on the faster gene frequency change at the selected locus. Another interesting point is that the conclusions based on a comparison using D and D’ give somewhat different results. That is, the painvise gametic disequilibrium using D’ is always higher between the selected and neutral loci than between neutral loci; using D, the pairwise gametic disequilibrium is higher between the selected and neutral loci for linkage and during approximately the first 30 generations for partially selfing, but after this it is much higher between the neutral loci. Since D is gene-frequency dependent and there are large changes in gene frequencies, D’ would seem a more appropriate parameter to use.

Figure 8 shows some interesting properties, using another gametic array, (b), in Table 2. This array has equal proportions of four of the initial gametes, such that there is no painvise gametic disequilibrium, but DAB, is nonzero. Gene fre- quency and gametic disequilibrium values over time are given in Figure 8 for 99 % selfing and initial homogametic zygotic proportions. The gene frequencies

HITCHHIKING 803

Generations '

FIGURE 8.-The changes in gene frequency (broken lines) and gametic disequilibrium (solid lines) for the initial gametic frequencies (b) given in Table 1 with 99% selfing and initially only homogametic types. Only D is given, since D' = 4D in this case.

at the two neutral loci do not change and remain at 0.5. Because of this, DBC' = 4 D B C ; thus, only DBc is given.

The gametic disequilibria between the selected and neutral loci, DAB and D A C , are initially zero and remain at zero. However, the disequilibrium between the two neutral loci, DBc, rises rapidly, so that by generation 25 it is over 85% of its maximum value. Simultaneously, DAsc quickly approaches zero. DBc remains high for an extended period of time because of high selfing and eventually decays slowly at the rate expected for two neutral loci with high selfing.

Since there is no gene-frequency change, this example was used to determine the relative effects of linkage and partial-selfing hitchhiking on gametic dis- equilibrium. By plotting the maximum gametic disequilibrium values obtained for different: amounts of linkage and partial selfing, a graph strikingly similar to that given for gene frequency hitchhiking in Figure 2 was obtained. This, of course, is reasonable because changes in gametic disequilibrium are really the result of hitchhiking at two separate loci. For this gametic array, the level of dominance again makes little difference for partial-selfing hitchhiking. How- ever, since the change in gene frequency for a recessive with random mating is faster than for dominance at these higher gene frequencies, linkage hitch- hiking is more effective for recessivity than for dominance, a reversal from the low gene-frequency example shown in Figure 3.

Another interesting initial gametic array is given by (c) in Table 2. For either tight linkage or high partial selfing, the gametic disequilibrium between the

804 P. W. HEDRICK

neutral loci starts initially as a negative value and becomes positive before declining to zero. For example, with 99% selfing, the initial DBc’ of -0.5 increases to 0.8 before starting to decline to zero. THOMSON (1977) mentioned the possi- bility of changing sign of gametic disequilibrium over time for linkage hitch- hiking; obviously, it can also occur with partial selfing.

Surprisingly, for linkage hitchhiking, the position of the selected locus between the two neutral loci generally makes little difference in the amount of gametic disequilibrium (THOMSON 1977). This appears generally true for D, but in some situations, e.g., gametic array (a), the level of D’ depends upon the relative position of the selected locus between the neutral loci, Using this example, D’ was found to become somewhat lower as the selected locus departs from being exactly intermediate between the neutral loci. Of course, if the selected locus is not between the two neutral loci, then gametic disequilibrium decreases as linkage between the selected locus and the neutral loci becomes less.

DISCUSSION A N D CONCLUSIONS

From these results, it is apparent that both linkage and partial selfing can contribute significantly to changing the gene frequencies at neutral loci and altering the amount and type of gametic disequilibrium between neutral loci. Presumably, such hitchhiking could be important for near-neutral loci as well. As a result, it seems necessary to recognize hitchhiking as a factor in addition to selection, migration and genetic drift that can significantly alter gene frequency and gametic disequilibrium.

As listed below, linkage and partial selfing have some similarities in their hitchhiking effects, as well as a number of differences from the results of these numerical iterations. These are listed below, although many of the characteristics for linkage were given by THOMSON (1977).

(1) Hitchhiking from partial selfing occurs more quickly and has a greater effect than that from linkage. It is important to realize that the selected loci in a partial-selfing system can be anywhere in the genome, whereas the selected loci must be fairly tightly linked to have an effect from linkage.

(2) The initial zygotic proportions are quite important for partial selfing and relatively unimportant for linkage. In particular, if only homogametic types are initially present, the amount of hitchhiking from partial selfing is greatly enhanced.

(3) The level of dominance for directional selection is very important for linkage and unimportant for partial selfing. For example, at low initial gene frequencies, linkage hitchhiking with a recessive is very ineffective, but it is effective for partial-selfing. Also, partial-selfing hitchhiking is ineffective when there is symmetrical or nearly symmetrical overdominance, while linkage hitchhiking is effective for such selection models. (4) When the selective advantage is split between two (and presumably

more) loci, hitchhiking is greater than when selection is concentrated in one locus for both partial selfing and linkage. The map position is important for link-

HITCHHIKING 805

age, having a synergistic effect when the neutral locus is between two selected loci.

(5) Gametic disequilibrium between two neutral loci can change from zero if the three-way gametic disequilibrium is nonzero, with a third locus under- going selective change with either linkage or partial selfing. Furthermore, the gametic disequilibrium between neutral loci can increase even though there is no selection and no gene-frequency change at the loci.

(6) Gametic disequilibrium between two neutral loci and the three-way gametic disequilibrium with a third selected locus can change signs from negative to positive, or uice uersa, €or both linkage and partial-selfing hitchhiking.

The extent of hitchhiking in either gene or gametic frequency change due to linkage or the mating system depends on several factors. First, there must be an initial gametic disequilibrium between alleles at the selected neutral or near- neutral loci. Second, the change in gene frequency at the selected locus needs to occur before the gametic disequilibrium between the selected and neutral or near-neutral loci decays substantially by either recombination or outcrossing. This general relationship for linkage-induced hitchhiking can be expressed for one selected and one hitchhiking locus as

2 A pz = f ( Z @ I , D’, , c> (12) where Z Ap, and 2 Ap, are the total gene frequency changes over time at the hitchhiking and selected alleles, B, and A,, before gametic disequilibrium has substantially decayed. Do’ is a measure of the initial gametic disequilibrium and c is the amount of recombination between the loci. A similar expression can be written for hitchhiking due to partial selfing

Apz f(x Qi, D*,, s> (13) where s is the proportion of self-fertilization and Dz is a measure of the amount of initial disequilibrium that encompasses both gametic disequilibrium and non-Hardy Weinberg genotypic frequencies.

Although these expressions are not specific, they include the basic elements necessary for hitchhiking. For example, c must generally be small in (12) and s large in (13) for substantial hitchhiking. Furthermore, if Z A p l is near zero, then there is minimal hitchhiking potential. That the change at the selected locus must occur before disequilibrium has decayed is exemplified by the situation of a recessive favorable allele at low initial frequency that is linked to and in gametic disequilibrium with a neutral allele (Figure 3). In this case, the initial change in gene frequency at the selected locus is so slow that the gametic disequilibrium is broken up before hitchhiking can occur. In addition, Do in (12) and Do* in (13) generally need to be quite different from zero for substantial hitchhiking. As illustrated previously, a less frequency-dependent measure of gametic disequilibrium, such as D’ or a correlation coefficient, may be preferable to D to indicate potential hitchhiking. These generalizations are for two loci, and it should be recalled that, when three or more loci are involved, there may be situations where there is no initial pairwise gametic disequilibrium

806 P. W. HEDRICK

between the selected and neutral locus, but still substantial hitchhiking (e.g., Figure 8).

From this discussion, it is obvious that either tight linkage o r a high proportion of selfing is necessary to retard the decay of disequilibrium in order for hitch- hiking to occur. In addition, however, it is important to keep in mind that the initial gametic disequilibrium must have been generated by some other factor, since neither linkage or inbreeding can generate gametic disequilibrium. The factors that can are the same as those that can cause gene-frequency change, i.e., selection, genetic drift, mutation and migration. HEDRICK, JAIN and HOLDEN (1978) have reviewed the potential for generating disequilibrium from each of these factors but, obviously, their relative importance in a particular organism is dependent on such parameters as population size, migration, etc.

It is also important to realize that hitchhiking is an important alternative to selection at monitored loci when examining changes in gene frequency or gametic disequilibrium for closely linked genes or in partial-selfing species. In particular, results documenting gametic disequilibrium for closely linked loci, such as those of CANNON (1963) in Drosophila [see HEDRICK, JAIN and HOLDEN (1978) fo r other examples], may be due to hitchhiking. For example, in CAN- NON’S study, D’ is 1.0 initially and declines over time. while D and the gene frequency increase, much like the results from gametic array (a).

ALLARD and his co-workers have observed changes in gene frequency and disequilibrium between alleles at different electrophoretic loci in cultivated bar- ley (ALLARD, KAHLER and WEIR 1972; WEIR, ALLARD and KAHLER 1972) and high levels of gametic disequilibrium in the slender wild oat (ALLARD et al. 1972). As illustrated elsewhere (HEDRICK and HOLDEN 1979), these observa- tions may be explained by hitchhiking due to linkage and/or partial selfing. In both of these examples, it appears that hitchhiking is a realistic alternative to epistatic selection o r co-adaptation when examining changes in gene frequency or gametic disequilibrium for closely linked genes or in a partial-selfing species. In addition, it may be the most parsimonious and simplest alternative since it does not necessitate any complicated form of selection.

One can imagine situations where maladaptive alleles are increased in fre- quency because of hitchhiking with selectively favored alleles. This could be a mechanism by which disease alleles could be increased, as suggested by WAGENER and CAVALLI-SFORZA (1975). It also could result in the reduction of the rate of gene substitution of favorable alleles by slowing down their passage. In fact, in a highly-selfed organism, this could act as a drag, creating inertia against evo- lutionary change. Additionally, hitchhiking with selection in opposite direc- tions at two loci could explain reversals in gene frequencies as, for example, in coat-color alleles in several experimental mouse populations ( HEDRICK and COMSTOCK 1968).

One of the frustrating aspects of population genetics is the difficulty in obtain- ing estimates of genetic parameters to use in explaining or predicting popula- tion genetic phenomena. By illustrating the importance of hitchhiking in affect- ing genetic change, a nearly impossible dimension is added to the genetic system.

HITCHHIKING 807

Obviously, there are no one-locus or two-locus organisms, so that the effects of other loci must be considered. In most cases, however, the genes that cause hitch- hiking are unidentified, making the scope of their effect unknown. This is cer- tainly an unsettling effect in our efforts to achieve a coherent theory to explain evolutionary change. In fact, hitchhiking may become a reasonable, but catchall, hypothesis for unpredictable or unexplainable population genetic phenomena.

I am grateful to LARRY HOLDEN for writing the computer program and for many helpful discussions. MICHAEL CLEGG and GLENYS THOMSON made a number of useful suggestions on the manuscript.

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