THEORETICAL
POPULATION
3, 324-346 (1972)
BIOLOGY
Selection
for Linkage
I. Random
Mating
Modification: Populations*
MARCUSW. FELDMAN Department
of Biological
Sciences, Stanford
Received
University,
October
Stanford,
California,
94305
4, 1971
The diploid system with chromosomes MAB, MAb, MaB and Mab is at equilibrium under selection and recombination between A/a and B/b loci. The M locus controls this recombination. Using local analysis near the original equilibrium, the fate of a selectively neutral recombination modifying allele ‘m’ introduced at the ‘M’ locus is investigated in the cases where the selective regimes are additive, multiplicative and symmetric. It is found that if initially there is linkage disequilibrium, ‘m’ will increase if it produces tighter linkage between the other two loci. The existence of epistasis, either additive or multiplicative, is shown not to be sufficient for recombination to be modified at a geometric rate.
1. INTRODUCTION
AND HISTORY
Fisher (1930; 1958, p. 117) presented a verbal argument to demonstrate that “the presence of pairs of factors in the same chromosome, the selective advantage of each of which reverses that of the other will always tend to diminish recombination and therefore to increase the intensity of linkage in the chromosomes of that species.” Haldane (1931) analysed a model in which there was a reversal of the selective values of alleles present at one locus depending on the allele present at a second, and concluded that no stable polymorphism is possib1e.l This difficulty was pointed out by Kimura (1956) who constructed a two locus selection model which allowed a stable polymorphic equilibrium for sufficiently tight linkage. He then showed that if an inversion arose from one of the existing chromosomes which eliminated recombination completely it would increase in frequency. The selection model used by Kimura was a * Research supported in part under Grant NIH 10452 at Stanford University. 1 Haldane’s analysis is not strictly valid since it was in terms only of gene frequencies.
324 Copyright All rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
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MODIFICATION:
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325
particular case of one used by Wright (1952) and comes under the general class of symmetric viability models (see, e.g., Karlin and Feldman (1970)). Bodmer and Parsons (1962, p. 81) included a brief discussion of linkage modification. In the symmetric viability model one of the four chromosomes is modified while the population is at equilibrium, so that the frequency with which it recombines is reduced. A series of approximations produced the result that the chromosome with reduced recombination will increase in frequency. Turner (1967a) made a qualitative analysis of the problem of linkage modification between two loci. From his analysis he conjectured that “pairs of loci on are always subjected to which selection acts epistatically (i.e., non-additively) selection for gradual or sudden decreases in recombination when the population is at a stable equilibrium.” No analytical proof was given. Perhaps the most exhaustive analysis of the linkage modification problem so far is that made by Nei (1967, 1969). Nei (1967) constructed two models for the way in which linkage might be modified. These were applied to both haploid and diploid populations. His first model is similar to that of Kimura (1956) and Bodmer and Parsons (1962) in that one of the chromosomes is modified so that it does not recombine. Unfortunately in the haploid case there is no stable polymorphic equilibrium of the original chromosomes where this modification can be made (Feldman, 1971). In the diploid case the result should depend on the original equilibrium at which the modification is made. Despite these difficulties in the formulation and analysis the conclusion was again reached that reduced recombination is favoured by natural selection. The second model introduced by Nei (1967) is the one with which this paper is concerned. Here there is a third locus which controls the recombination fraction between the other two while not altering the selection scheme operating on the controlled loci. Nei’s conclusion that the frequency of a recombination reducer will always increase depends on his use of the result that the sign of D, the linkage disequilibrium, and E, the additive epistasis, eventually become the same. The correct result for the model used, however, is that of Felsenstein (1965), namely, that D and E - 1 eventually have the same sign where E is the multiplicative epistasis. It is possible that c and E - 1 have different signs. In his second paper Nei (1969) allowed for an arbitrary degree of linkage between the modifier and the two modified genes where no such linkage was assumed in the earlier paper. In all cases the conclusion was that at the modifying locus a recombination reducer would be favoured so long as there is additive epistasis in the selection acting on the modified loci. A recent note by Lewontin (1971) examines a particular case of the linkage modification problem in an indirect way, through the use of mean fitness. It is shown that when the equilibrium gamete frequencies are continuous functions of the recombination parameter for very tight linkage (i.e., in the neighbourhood
326
FELDMAN
of zero recombination) the mean fitness is greatest when there is no recombination. In this analysis no mechanism for the reduction is specified. As Lewontin points out, what happens when the original degree of linkage is moderate or is loose cannot be answered in this way. In this paper I shall take a different approach to Nei’s three-locus model of recombination modification. The question posed is the following. Suppose two loci with alleles A and a at the first and B and b at the second are in a state of stable polymorphic equilibrium while at the recombination modifying locus the genotype is MM. A small amount of the gene ‘m’ appears at the modifying locus and ‘m’ alters the recombination fraction between A and B although it has no effect on the selection scheme. Under what conditions does m increase in frequency ? As I shall show, the answer depends on whether the A and B loci are in linkage equilibrium or not. The basic result is that, under the above hypothesis, in all known caseswhere the two loci are not in linkage equilibrium before ‘m’ is introduced, ‘m’ will increase in frequency if and only if it reduces the recombination fraction.
2. THE THREE-LOCUS MODEL
The model is basically the three locus model of Nei (1967, 1969). However, instead of arguing in terms of the frequency of the recombination modifiers at an arbitrary stage, as Nei did, we consider the evolution of the modifier gene shortly after its arrival in the population. The diploid population originally contains only the four chromosomes MAB, MAb, MaB and Mab in frequencies xi , xa , x3 and x4 , respectively. These chromosomes evolve under the influence of the usual selection scheme for the two loci A/a and B/b. The selection matrix is given in (1) below. MAB
MAb
MaB
Mab
MAB
Wll
w12
w13
w14
MAb
w21
w22
w23
w24
MaB
w31
w32
w33
w34
Mab
w41
w42
W43
w44
(1)
Here, for example, wr2 is the relative fitness of the genotype MAB/MAb, etc. The recombination fraction between the A/a and B/b loci is rr at the initial stage when MM is the genotype at the modifying locus.
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MODIFICATION:
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321
Under the influence of (1) and recombination we assume the population to have reached a stable polymorphic equilibrium with MAB, MAb, MaB and Mab in frequencies 32’,, $a ,4, and 4, , respectively. Now at the M locus a small amount of an alternative gene ‘m’ is introduced, possibly through mutation. Denote the frequencies of the new chromosomes mAB, mAb, maB and mab There are now three loci segregating and by x5 , x6 , x7 and x3, respectively. x4 have moved very slightly away from the original equilibrium Xl , x2 9 x3 9 values. It is assumed that the presence of m instead of M in a chromosome does not alter the selection matrix (1). However, we postulate that when the genotype at the modifying locus is Mm, the recombination fraction between the ‘A’ and ‘B’ loci is r2 . When the genotype is mm the recombination fraction between ‘A’ and ‘B’ is r3 . The degree of linkage between M and A must also be specified, and in this paper we shall assume the ordering of loci M, A, B and the recombination fraction between the M/m and A/a loci to be Y. The final specification necessary to complete the model is the probability of a double recombination event. With the ordering MAB in this paper we shall assume that the probability of the double recombination is the product of r and the recombination fraction between A/a and B/b. In a later paper it will be seen how this assumption can be relaxed. In terms of the model, whether linkage will be modified depends on the fate of ‘m’ in the population. Thus if ‘m’ increases when r2 < ri , the average recombination fraction will be reduced. If ‘m’ increases when r2 > r1 it will be increased. If ‘m’ does not increase, but is lost from the population there will be no change from rl , the original recombination fraction.
3.
RESULTS
The method by which we shall ascertain whether ‘m’ will increase or not from its originally very small frequency has been used by many workers for many different problems. The procedure is to use local linear analysis in the neighbourhood of the original equilibrium. The recursion system for (xi’, x2’,..., x8’) in one generation in terms of (xi , x2 ,..., x8) in the previous, is linearized about the equilibrium % = (91 ,5, , f, ,4, ) 0, 0, 0,O). The eigenvalues of the resulting matrix are determined. The stability of % determines the fate of ‘m’. If f is unstable then ‘m’ increases away from f. If it is stable then ‘m’ is lost from the population and there is no change in the recombination fraction from the initial value rl . The eigenvalues of the local matrix determine the stability of P. In the present situation it turns out that the result may be obtained in terms of the largest eigenvalue. If the largest eigenvalue is greater than unity in absolute value,
328
FELDMAN
the equilibrium is unstable and ‘m’ will increase initially at the geometric rate specified by this eigenvalue. On the other hand, if all the eigenvalues are less than unity in absolute value j; is stable and ‘m’ will disappear at the asymptotic rate specified by the largest eigenvalue. If the largest eigenvalue is unity we can say that there can be no change at a geometric rate. These eigenvalues will be functions of the equilibrium values ii , 9, , 4, and $ and of all the recombination fractions. But there are only certain cases of the selection matrix (1) for which the equilibria are known. These are the additive viability model, the multiplicative viability model and the symmetric viabilities model. It is worthwhile at this stage to recall briefly what the relevant equilibria are. In the case of additive viabilities there is a single polymorphic equilibrium which is globally stable (Karlin and Feldman, 1970a) and which has a = a,$ - Ra4, = 0. In the multiplicative viabilities model, for loose linkage between the A/a and B/b loci there is a globally stable equilibrium with B = 0 (Moran, 1968) but for tighter linkage the equilibria are, in general, not known. In the particular case where the multiplicative viabilities are also symmetric there are two locally stable equilibria for tighter linkage with B # 0 (Lewontin and Kojima, 1960). In the symmetric viabilities model the most general results are those of Karlin and Feldman (1970). For the fitnesses chosen by Lewontin and Kojima (1960) there is a stable D = 0 equilibrium for loose linkage and two stable equilibria with D # 0 for tighter linkage. For the Wright (1952) model, for relatively tight linkage, there is a single stable equilibrium with D # 0. These equilibria all have ii = $ , 2, = 4, . Karlin and Feldman demonstrated further that stable equilibria not of this form and with D # 0 may exist. These are called unsymmetric equilibria. The above include all of the general classes of models about whose equilibrium nature is known. In behaviour anything of a theoretical (i.e., nonnumerical) this paper all results will be with reference to these models.
Result 1 If the initial stable equilibrium ir = (a, , f, , ia , S4 , 0, 0, 0, 0) is such that D = S& - a # 0 then the newly arisen modifying gene ‘m’ increases in frequency if and only if the recombination fraction ra associated with Mm is less than the original value ri . Tighter linkage is therefore favoured.
Result 2 If the initial equilibrium 2 is such that fi = 0, the largest eigenvalue of the stability matrix is unity. In this case, a neutral curve of equilibria exists and it appears that there will be convergence to a point on this curve, depending on the initial frequency of ‘m’.
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MODIFICATION:
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I
ANALYSIS
Consider the general three locus selection scheme given by the matrix MAB MAB MAb MaB Mab
maB
mab
MA\,
MaB
Mab
mAB
mAb
w12
w13
w14
w15
2%
w17
%8 W28
Wll
(2)
w12
w22
w23
w24
w25
W26
w27
w13
w23
w33
w34
w35
w36
w37
w38
w14
w24
w34
w44
w45
w46
W47
w48
w15
w25
w35
w45
%5
w56
w57
w58
W16
W26
w36
w46
w56
2066
w67
W68
w17
w27
W37
w47
W57
w67
w77
w78
w18
w28
W38
w48
w58
W68
w78
W88
(2) mAB mAb maB mab
Here x1 , x2 ,..., x8 are the frequencies of the chromosomes in the order given across the top of (2) and wij is to be interpreted in a similar manner to (1). Write x1’, x2’,..., x8’ as the same frequencies in the next generation. Then the relationship between the frequency vector x’ and x is given by Eq. (3) below. xlwl.
+
r[x2x5w25
+
yl[x2x3w23
-
$
y2[x2x5w25
+
+
yy2[2w16xlx6
+
x3x5w35
+
x4x5w45
-
w17x1x7
-
w18x1x8
-
w16x1x61
w14xlx41 x2x7w27 -
-
x1x8w18
2%5%%
+
-
xlx6w161
x3x6w36
+
x1x8w18
-
'4'gw45
-
w27x2x71?
3(i) x2w2.
+
r[x1x6w16
+
y1[x1x4w14
-
+
T2[xlx6w16
+
+
%[2%5%%
+
x3x6w36
+
x4x6w46
-
x2x5w25
-
x2x7w27
-
x2x8w281
x2x3w231 x1x8w18 -
-
x2x5w25
2w16xlx6
+
-
x2x7w271
x2x7w27
+
x4x5w45
-
w18x1x8
-
w36x3x61y
3(ii) x3w3.
+
T[xlx7w17
+
yl[xlx4w14
-
+
y2[x4x7w47
+
+
rr2[x1x8w16
+
x2x7w27
+
x4x7w47
-
x3x6w36
-
'3'Sw38
-
x3x5w351
x2x3w231 x4x5w45 +
-
x3x6w36
x3x8w38 +
-
x3x6w361
2w38x3x8
-
x4x5w45
-
x2x7w27
-
2x4x7w471,
3(iii) x4w4.
+
r[x,x8w,,
+
yl[x2x3w23
-
+
y2[x3x6w36
+
+
rr2[x2x7w27
+
x2x8w26
+
x3x3w38
-
x4x5w45
-
X4X6W46 -
x4x7w471
xlx4w141 x3xSw36 +
xqxgw45
-
x4x5w45 -
x1x8w16
-
x4x7w471 -
x3x6w36
-
2x3x3w36
+
2x4x7w471,
3(iv)
330 ax,
FELDMAN
:
x5w5.
+ +lx6wlG
+ ~1~7W17 +
+ T2[xlx6w16
+ x3xSw36
+ ~3h?X7%7
-
+
-
XlX8Wl8
x2x5w25
-
-
x 2x 5w 25 -
-
x3xsw35
xxw 4 5 45]
x4x5w451
X5%W531
rr2[2x,x,w,,
-
2xlx6w16
+
x2x7w27
+
x4x5w45
-
xIx8wlS
-
x3x6w361,
3(v) WX6’:
‘gw6.
+
y[x2x5w25
+
T2[x2x5w25
+
+
r3[x5xL7w53
-
+
rr2[2xlx6w16
+
x2x7w27
x4x5w45
-
+
x2%w28
-
xlx6w16
-
‘3’Bw361
xlx6wlS
-
x3x6w36
-
x4x6w461
x6x7w671 -
2x2x5w25
+
xlx8w18
+
x3x6w36
-
x2x7w27
-
x4x5w451~
3(vi)
fax,’ :
x7w7.
+
r[x3x5w35
+
x3x6w36
+
T2[xlx8w18
+
x3x8w38
+
y3[x5x8w58
-
x6x7w671
+
rr2[~~4w7w47
-
-
+
x3x8w38
x2x7w27
2x,%w3,
+
-
-
xlx7w17
-
x2x7w27
-
x4x7w4,1
x4x7w471
x4x5w45
+
x2x7w27
-
xlx8w16
-
x3x6w361,
3(vii)
iiTX8’=
xSw8.
+
T[x4x5w45
+
X4X6W46 +
+
r2[x2x7w27
+
x4x7w47
+
r3[x6x7w67
-
x5x8w581
+
~~2[+%3wl6
+
x3x6w36
-
x4x7w47
xlxEw18 -
x2x7w27
-
-
xlxSw18
-
x2x8w28
-
x3x8w331
x3x8w381 -
x4x5w45
+
2x3%w38
-
2x4x7w471.
3(viii) Here
8 w, = c wijxj ) j=l
and
Y is the recombination fraction between M/m and A/a; ri is the recombination fraction between A/a and B/b when MM is the genotype at the modifying locus. r2 is that when the genotype is Mm and r3 is that when the genotype is mm. It is assumed that there is no interference. The reason for the horizontal and vertical lines through (3) is to draw attention to the fact that in what follows we shall assume that the four matrices so partitioned are identical. This means simply that the M/m locus has no selective effect. According to the model, the population is originally assumed to be in the stable equilibrium with MAB, MAb, MaB and Mab present in frequencies II II and g5 = 2s = 9, = $, = 0. A very small Xl 9 x2 , 9, and g4 , respectively, amount of ‘m’ is introduced in the neighbourhood of this equilibrium. Since
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MODIFICATION:
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I
‘m’ does not affect the relative fitnesses of its carriers the only selective effect on ‘m’ is through its effect on recombination. To determine the fate of m just after its introduction (3) is linearized in the neighbourhood of ji = (x1 , x2 , xa , x4 , 0, 0, 0,O). The linearized system breaks up into two separate linear systems. One is in terms only of xj , xg , x, and xs , the originally very small variables and the other is in terms of the high frequency xi = I the system is really only seven variables x1 , xa , xa and xq (since & dimensional). This type of partitioning was also noticed in two loci by Bodmer and Felsenstein (1967) in their analysis of the fate of a new gene linked to a stable one-locus polymorphism. The linear system in x1 , xa , xs and xq , however, is precisely that which would determine the stability of the equilibrium 3i’, , f, , 1 xa , GG~ as a two-locus equilibrium in MAB, MAb, MaB and Mab. Since as part of the model it is assumed that this equilibrium be locally stable all its eigenvalues are less in absolute value than unity. The conditions on rl and the selection parameters which ensure this stability are important to the fate of ‘m’ as we shall see. The analysis therefore boils down to the analysis of the linearized system involving xs , xs , x7 and xs . This is written out as (4) below; it comes from 3(v)-(viii) neglecting quadratic terms. &x5’ = x5[ws* + xsp
i2w2&
+ r2 - 2rra) -
+ y2 - 2rr,)
~PlS
+
+ Y2 -
&Was(Y
r,(l
-
rY2)
- Y&W,,]
r> QJ3a]
+ e%w,, + ~~,%,%I+ %L+,,~(l - r,)l, WXs’
= %[“25~2(Y + r2 - 2rr2) + Wa&,(l
- r)l
f,W,,(Y + r2 + x&G* + X,P2W2,Y(1- r2)1 + %Jr~2w2, + ~Y2%%31, s&W,,@
$x7’ = x&%w,,
+
r2 -
2rr,)
+ Y~2%4w451 + x&f&&
-
-
rr2)
-
r3~w*6]
(4)
- r2)1
+ x,[w,” - (r + r2 - rr2) g2w2, - &w,,(r + r2 - 2rr,) - r&w,,] + %[~2W,,(~ + y2 - 2rr2) + r2u - r) eJ,,l, zix,’ = %[4%(1 - y2)l + %m%l + ~~,%%,I + X,[32’,W,,(r+ r2 - 2rr2) + 3i’,w,,r,(l - r)] + “&s” - (r + Y2 - YY2) ~lwls - (r + r2 - 2rr,) 22w2s In (4), due to the identity
of the four submatrices
Y3Z’,W,,].
in (2) we have
i = 5, 6, 7, 8, (5)
332
FELDMAN
The local stability of the equilibrium, that is, whether the frequency of ‘m’ will increase or not is determined by the eigenvalues of the matrix on the right side of (4). These are the solutions of d(h) = 0 where d(h) is the determinant / dii(A)I specified by (6) below.
A,,(X) = w5* - i2w2Jr + Y2- 2rr,) - 3i’,w,,(r + r2 - rrJ - rS&w,, - M, A,,@) = (r + r2 -
2rr2) flw16 + r2(l - r) i3ws6 ,
A,,@) = r&q,
+ rr2w27~2 ,
A,,@) = r(l -
r2) 4wls ,
A,,(h) = w2$,(r
+ r2 - 2rr2) + w&r2(l
A,,(h) = w6* -
f,w,,(r
A,,(4
+ r2 -
- I>,
2rr,) - 5&w&r + r2 - rrJ - r&w,,
- hti,
= r(l - r2) R2w2, ,
A,,(A) = ri2w2s + rr24w13 ,
(6)
A,,(h) = w3523r+ rr23i.4w45 , A,,(4 = r(l - r2>~3w36, A,,(h) = w,* - (r + r$ - rr2) 4,w,, - &w4,(r + rg - 2rr,) - r&w,, A,,(h) = ~3w33(r + r2 - 2rr2) + r2(l - r) Qs A,,@) = 4w4,(l A,,@) = r&s
-
,
r2),
+ rr2%w3, ,
A,,@) = 3Z.,w,,(r + r2 A,,(h) = w8* -
- heir,
2rr2) + i2w2,r2(l
(r + r2 - rr2) 3z’,w,, -
- r),
(r + r2 -
2rr.J ZSwS8- r3z’,w,, - M.
Even though we have restricted the 8 x 8 fitness matrix (2) to four identical 4 x 4 matrices, it is obvious from (6) that we cannot continue with a completely general 4 x 4 matrix. To obtain the eigenvalues we must find the roots of a fourth degree polynomial. It is at this stage that we specialize to the fitness schemes mentioned in the previous section. We treat the additive fitnesses model first, then multiplicative viabilities and finally the symmetric viabilities model. Case 1: Additive viabilities. of the matrix (7) below
In this case (2) reduces to four identical
%+A
%
-tB2
011 +I%
011 +
012+ Bl
a2
012 +
012 +ps
83
012 +
Pl
012 +
I32
%
B2
a8
83
A
013 +P2
+
+
copies
(7) P2
-+B2
OIS + 05
+P2
05
+
P3
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FOR LINKAGE
MODIFICATION:
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I
It is assumed that 01s> o[i , 01s; /3s > & , /3s . In this case, along the hypersurface x5 = xs = x, = xs = 0, there is global convergence to the equilibrium with B = 3i’,& - &$, = 0 specified by
with $A = (a3 -
~2)/(9
+ 013-
4B = (83 -
2a2),
B2MPl
+ P3 -
282).
Substituting (7) and (8) into (5) it is seen that ws* = we* = w,* = ws* = 6. Make this simplification in (6), then add the second, third and fourth rows to the first. The elements of the first row are now all equal to ti(l - h). Add the fourth row to the second. Then subtract the third column from the first and the fourth column from the second, Finally subtract the fourth column from the third. We are now left with the determinant (9) where elements marked X do not affect the result. 0 K
0 L
M s
N T
0 - X) + (r + r2 - 2rr,)
-ti(l
ti(1 - h) X
x [(a2+ B2)BA+ (013 + P2>(1 - $A)1 X X
*
(9)
X x
Here,
K = (r + rz - 2~~2)[4,(~1- 4 + &(a2 - 41, L
=
(r +
M
=
-
f.2 -
-
s = &(a, T =
2rr,)[i2@2
+ B2) r(l %j(“l
al) + &3
-
a2)1,
4 + ~(a2 + ,&)$B
+ (r + r2 N = f&2
-
2rr,p&,
~(1 -
-
+ &)(r
+ B,) r(l
r2> + r2 -
-
r2> -
-
3i.&3 + a2>(r + r2 -
-
z;( 1 -
+ Bz) + ~‘l(o”3 + P2)l + ry2@2 + B2)(1 3i’,b2 + B2) r,(l
Now substituting are therefore
2rr2)M~2
r)
-
r)
2rr,), ~2@2 + B2) yz(l 2rr,),
4 + ~(012 + A)(1
+ (r + r2 -
-
i&J>
- j&J
+ B2) + %b3
+ P2)l + rr2b2
+ B2)A3 .
from (8) we have K = L = 0. The first two eigenvalues
A, = 1 and
(10) A2 =
1 -
@(r
+ r2 -
2rr2>G2
+ P,>$A
+ (a3 + Is2)U -
4.4)).
334
FELDMAN
To obtain the other two eigenvalues we must consider the determinant MN s T
i
I
=a
Adding the two rows and substituting again from (8) we have M + S = N + T. The remaining eigenvalues are then simple to extract. They are A, = 1 -
@Y{PB(ar,
A, = 1 -
&l{(Y
+ a> + (1 - $l3)(~2 + 82),
and + f-2- 2TT2)[$A(OL2 + 82) + (1 - $A)(+ + B2)l
(11)
- 41 - y2)@2+ B2)U - Ai) + rr2b2 + P,>bB + +2 + Ml - m. Assume Y < l/2 and r2 < l/2. Then it is not difficult original stability conditions 01~> (or , as ; /32 > /3r , & -1
< 4) ha, h, < 1.
to see that under the
(12)
The conditions on the recombination fractions can be relaxed slightly in this model but biologically these conditions are not really conditions at all. The result for the additive case is, therefore, that the largest eigenvalue is exactly one while all of the others are less in absolute value than one. This means that there can be no increase of ‘m’ at a geometric rate in the usual sense. This is discussed in more detail in Section 5.
Case 2 : Multiplicative identical
viabilities.
The fitness matrix
(2) now becomes four
copies of (13) below.
aA alb2
alb2 4,
a2bl a2b2
a2b2 a2b2
a2bl
a2b2
4
4,
a2b2
a2b3
a&,
4,
(13)
With these viabilities the full equilibrium behaviour of the original “two-locus” population MAB, MAb, MaB and Mab is not known. As in the additive case we shall assume overdominance at the two single loci; a2 > a, , as and b, > bl , 6, . Moran (1968) has shown that the equilibrium of the form (8) with fiA = (a, - a&al + a3 - 2a,), jB = (6, - b,)/(b, + 6, - 2b,) is globally stable for rl near l/2. The condition given by Bodmer and Felsenstein (1967) for local stability includes Moran’s condition.
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For tighter linkage and general multiplicative viabilities little is known except that for very tight linkage two stable equilibria exist (S. Karlin, private communication). When the viabilities are multiplicative and symmetric the full equilibrium structure is known. This is deferred to the section on general symmetric viabilities. Insofar as the explicit form of the equilibria for tight linkage in this model is not known it is possible, according to our program, only to treat the B = 0 equilibrium (8). As with additive viabilities, substituting from (13) and (8) into (5) we find *,w+=w*z on the WE* = zi, and exactly the same manipulations WEI determinint (6) broduce the eigenvalues
As before, when u2 > a, , us ; 6, > b, , b, ; r < l/2, r < l/2 we have - 1 < X, , ha , /\a < 1. The result for the B = 0 equilibrium of multiplicative viabilities is the same as for the additive case. More details are presented in Section 5. Case 3: Symmetric viubilities. (15) with 01,6, y, S all positive. received a great 1-S 1-p l--y 1
The selection matrix in this case is given by There are two special cases of (15) which have
l-/3 1-a 1 l--y
l--y 1 1-a 1-p
1 l--y l-/3 1-S
(15)
deal of attention. The first is the Lewontin and Kojima (1960) model which takes cx = 6. The second is of the form considered by Wright (1952) and has I = 2(/3 + y) - (a + S) = 0. We shall treat these separately. Case 3A: Lewontin and Kojimu model, 01= 6. As usual consider the population MAB, MAb, MaB and Mab with r1 the recombination fraction between A/a and B/b. Lewontin and Kojima showed that in this case the equilibrium with b = 0 and 3i’, = 4, = 2s = 4, = l/4 is stable if S > 1/I - y 1 and ri > l/S where I = 2(fi + y) - (CY+ 6). If rl < Z/8 two equilibria of the form 32’,= 4, and 9, = ffa (equilibria of this form are called symmetric) and having D # 0 exist. If S > 1/I - y 1 these are stable near rl = 0 and rl = Z/8 but, if there is strong underdominance at the single loci, Ewens (1968) showed that for an 6531313-7
336
FELDMAN
interval of yi values (1970) showed that are unstable in this which have stability
in between they might be unstable. Karlin and Feldman four unsymmetric equilibria might also exist but these model. The two cases ij = 0 and b = +J(l - (SY~/Z))~/~ domains will be treated separately.
Case 3A(i): B = have in (5) z+,* = same operations on models produce the
Obviously in this case we 0; +?r= 3ja = $a = 2, = l/4. ws* = wg* = ws* = w = (4 - 6 - /3 - y)/4. Exactly the the determinant (6) as in the additive and multiplicative eigenvalues
A, = 1, & = 1-
(Y + r2 - 2yr,)(4 -
2/3)/(4 - S - /3 - y),
(16)
AI = 1_ 2 i (1- m + Y2- 2rr,) + y(l- Y)+ Y&I 4-s-p-y \’ l Again - 1 < ha , ha , h, < 1 and the largest eigenvalue is one. As before, the reader is referred to Section 5 for a discussion of this case. Case 3A(ii): Lewontin and Kojima where
model B # 0.
It suffices to treat the case
(17) Return to the stability determinant reduces to the simpler form
B
IA-M d(X) =
(6). Using
; D
(15) we find the determinant
C G F-M B
F-X G c
D I H E ’ A - hzi,
where A = w5* B =
(Y
+
- p) 32’,-
(Y
+ y2 -
- 2yra) 3i’,(l - p) + yz(l -
Y)
4, ,
(Y r2
+
Y2
- 2yy,)(l
c = Y(1 - y) 3i’, + rr&,
YY‘J
3i’, - Y(l
)
D = y(1 - ~a) 32’,, E = (y + F = w,* G = ~(1 -
r2
(Y YJ
~YY,) $(I + rz -
- /3) + y,(l - Y) 3i.r,
2ry,)(l
A$ ,
H = ~(1 - y) $a + YY& ,
- /?) 3i’, -
(y + r2 - ryz) $a - y(l
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with * = w8* = 1 - 6/2 -
7J5
l/8(1 T 2/( 1 -
W6* = w7* = 1 - S/2 - Z/8(1 + d(1 22 = 1 - s/2 - r1 .
-
8rJZ); 8rJZ);
(20)
It is important to note from (19) that the stability matrix is strictly positive except when Y :=- 0, when it breaks down into identical strictly positive 2 x 2 matrices. Elementary manipulations on the rows and columns of d(X) produce the very useful finding that the fourth degree polynomial A(/\) is the product of two quadratics Ql(X) and Q2(X) with Q#)
= h2d2 - hd(A + D + F + G) + [(F + G)(A + D) - (B + C)(E + H)],
(21) and
Q2(h) = h2# - h&(A - D + F - G) + [(A - D)(F - G) - (C - B)(H - E)]. (22) It remains to examine the roots of Qt(X) = 0 and Q2(X) = 0. Q2,(h) obviously has real roots and the larger of these, A1 , is positive. Substituting from (17), (19) and (20) into (21) we find that if and only if
A, > 1
rr > ra ,
(23)
provided only that rl < l/8 which we have assumed as part of the stability conditions on 2. It is not difficult to show that the other root of Q1(X), h, , is always less than one. Further, since A + D + F + G > 0, A, > 1A2 /. Q2(A) is considerably harder to analyse primarily because its roots may not be real. The discriminant is
[(A - D) - (F - G)12 + 4(B - C)(E - H),
(24)
which is easily seen to be positive if p = y. But if /3 is considerably larger than y, it appears that B - C might be negative and E - H positive. Thus (24) could conceivably be negative. After some algebra we find that when r < l/2, r2 < l/2
(A - D)(F - G) - (C - B)(H - E) < ti2, so that from the form of Q,(h) the absolute value of the complex roots (should they exist) is less than unity. When Q2(1\) has real roots h, and X, , say, with X, > X, then h, is positive, and h, < 1 if and only if
V/8
-
r,)[2r(l +
r(r
+
r2) r2
+ -
r2 2rr2)[l
-
rl] -
r/2
-
/3/Z +
@#~,/l)l
>
0.
(25)
338
FELDMAN
Obviously if Y = 0, (25) becomes ra > y1 . (This can also be seen from the form of A(/\) when r = 0). When l/4 < Y 6 l/2 the condition is obviously true. In fact for h, > 1 it is necessary that Y < (ri - r,)/2(1 - r2) < rl - r2 when ri > r2 and ra < l/2. But under this condition on r it is possible to show that h, > ha . Thus hi is the dominant eigenvalue as ra becomes less than ri . Since A - D + F - G > 0, Qa(h) achieves its minimum when X > 0. Thus when the roots are real h, > 1h, I. For the Lewontin-Kojima D # 0 case we have shown that a necessary and sufficient condition for the largest local stability eigenvalue to be larger than one is ~a < r1 .
Remark. Since it has been assumed that % is stable to begin with, if the selection coefficients are such that the gap of recombination value might exist in which f is not stable, then we have in fact assumed ri not to be in this gap. Clearly r2 may fall in this interval without invalidating the result. Case 3B: Wright model 1 = 2(fi + y) - (a + 6) = 0. With this restriction on the symmetric viabilities there is exactly one symmetric equilibrium possible for the original population. It is given by 2l = i4 = l/4 + ri/m -
l/4( 1 + 16r,a/~a)ii~;
i2=93=1/2-21,
(26)
where without loss of generality ti = 6 - OL> 0. This equilibrium is stable for an interval of ri values around the origin with the bound given by formula (4.5) of Karlin and Feldman (1970). When ‘m’ is introduced, the stability determinant (6) again takes the form (18) with (19), where
(9 (ii)
$?=Iwj*
a+6 4
rl + T (1 + 16~r~/i~i~)l’~
= ti + %/8[4r,/?ii -
1-
(1 + 16r12/%2)i12] = wg* -
m/4.
(27)
Again d(h) = Q,(h) Q2(h) where Qi and Q2 are as in (21) and (22). Again Q,(h) has real roots. The larger of these, Xi , is positive, and substituting from (19) (26) and (27) we find, as before, A, > 1
if and only if
ri >
r2
.
(28)
It is not difficult to show that the other root X2 of Q,(h) is always less than one and greater than minus one when r < l/2, r2 < l/2. The discriminant of Q2(‘\) could conceivably be negative but once again, analysis of the constant term of Q2(/\) sh ows that any complex roots will be less than unity in absolute value if r < l/2, r2 < l/2. When the roots of Qz(X) are
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real a condition analogous to (25) arises. Thus, the larger root, h, , of Qa(X) = 0 is less than one if and only if +ii/8 (1 + 16r12/iii2)1’2 - +] I +
T(T +
r2 -
2w,)[(l
- r2) +
(24
-
r/2)(1
-
r2
- r1)
- rP(4 - a2Yl > 0.
P/2)
(2%
The same remarks apply here as in the Lewontin-Kojima case. Thus h, > 1 implies Xi > 1 but not conversely. Indeed, using the fact that for (29) to be false it is necessary that rl > r2 and Y < ri - r2 , we can show that h, > X, . Since A - D + F - G > 0 the other root h, of Q,(h) is smaller in absolute value than X, . Therefore the condition necessary and sufficient for at least one eigenvalue to be greater than one is r2 < ri . Case 3C: Unsymmetric equilibria;
g=
/3 = y.
2p B - - 4242- +6 W2(r1 4 6r,/2(r,-
8) - S) ’
Define g
=
,$2
-
‘II’ 1 1 f”
)
(30)
where ri > 6 and it is assumed that cl/2 Then Karlin
Sr,/2(r, - S) > s > p
and
8 > 0.
(31)
and Feldman (1970) showed that the equilibria
are stable. To complete our study of linkage modification assume that MAB, MAb, MaB and Mab are at this equilibrium under the above conditions, when ‘m’ appears. In light of (32) we have we* = w,*, but wg* # ws*. Return to the basic stability determinant (6). Since 32’,# 3i’, there is no simplification analogous to are possible but (18). Making use of w6* = w,* some small simplifications these do not allow for a partition of d(X) into factors like (21) and (22). What I have done in this case is to examine two special cases analytically, namely, when r = 0 and 7 = l/2. I have also taken for (32) the values given in the numerical example used by Karlin and Feldman, and solved the equation d(X) = 0 numerically for a range of r and r2 values. When r = 0 in this unsymmetric case A(X) factors into two quadratics, A(x) = P,(h), P2(X) which have real roots. Using (30) and (32) the larger roots of Pi(X) and P2(X), namely h, and X, , respectively, are greater than one if and only if r2 < y1 .
340
FELDMAN
When r = l/2 so that the modifier segregates independently of the genes being modified, d(h) does not break down into the product of quadratics. Instead after some manipulation d(h) d ecomposes into the product of a linear term and a cubic factor. The linear term gives us the eigenvalue A” = [w7* - 9, -
- P)/2]/~ > 0.
(6 + $)(I
Here ti = 1 - St - /3(1 - 5) and application of (30) produces the conclusion A* < 1. Only a partial treatment of the remaining cubic, A *(A) has been made and perhaps more can be done with it. It is possible to show that O*(X) is negative for h large and positive and positive for h large and negative. In addition, a20 ‘*cl)
=
[
- 6) %(l 8+-, - a)2
0”
+
a20
-
P) ‘31
f3@1-
-
1
(rl
0”
6)
_
y2)
which is obviously positive for rl > r2 . This proves that r2 < ri is sufficient for a root of d(h) = 0 to be greater than unity. It seems likely that on closer examination of A *(A) this would also prove to be necessary. For general r, i.e., 0 < Y < l/2, d(h) d oes not simplify. For a numerical example, I chose the parameter set in Example I on p. 49 of Karlin and Feldman (1970), that is (Y = 0.03, ,8 = 0.004, 6 = 0.005 and rr , the original recombination fraction, was set at 0.05. Because these numbers are small double precision was used in all computations. In Table I below are recorded the results for three of the many values of r that were run for the specified range of r2 values. In all runs the same phenomenon was observed; as r2 increased TABLE Leading
Eigenvalues
of Stability
Matrix
I at an Unsymmetric
Equilibrium
Y
Y2
0.1
0.3
0.5
0.0 0.01 0.025
1.OOOOO0864 1.000000632 1.000000350
1.000000280 1.000000219 1.000000132
1.000000167 1 .000000132 1.000000081
0.05 0.075
1.oOOOoOOOO 0.999999746
1.OOoOOOOOO 0.999999880
1.000000000 0.999999922
0.200
0.999999097
0.999999428
0.999999581
0.500
0.99999863
0.999998842
0.999998997
cd = 0.03, f9 = 0.004, 6 = 0.005, (12 places) and corrected.
Yl
= 0.05. c omputations
done
in
double
precision
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I
the leading eigenvalue changed from being greater than one for r2 < 0.05 to be equal to one at r2 = 0.05 and then to be less than one for r2 > 0.05.
5. REMARK ON STABILITY In the analysis of Section 4 it was shown that when D # 0 in the Lewontin and Kojima model and the Wright model, r2 < y1 is necessary and sufficient for the largest eigenvalue of the local stability matrix to be greater than one. At the unsymmetric equilibria the condition r2 < y1 has only been shown to be sufficient, although the numerical examples indicate that it is also necessary. It was mentioned in Section 4 that the matrix of the linearized recursion system (4) is strictly positive unless Y = 0. (When r = 0 it decomposes into two independent strictly positive matrices). Under these circumstances the Perron-Frobenius theorem (see, e.g., Gantmacher, 1959, p. 53) ensures that associated with the largest eigenvalue h, are strictly positive left and right eigenvectors. Using the spectral decomposition of the stability matrix it is then obvious that if h, > 1 the frequency of some chromosome containing ‘m’ must eventually increase over its original value. This proves that r2 < rl is sufficient for the increase of ‘m’ in all cases analysed since r2 < rl implies h, < 1. The above argument applies to the stability analysis in the case of a new gene linked to a stable one locus polymorphism for which Bodmer and Felsenstein determined conditions on selection and degree of linkage under which one eigenvalue was greater than one. In all models examined it was shown that when the original chromosomes MAB, MAb, MaB and Mab are in linkage equilibrium prior to the arrival of ‘m’, the largest eigenvalue is exactly h, = 1. When ij = 0 in a standard two-locus system the frequencies of chromosomes AB, Ab, ab and ab may be written j&B , bA$b , ha&, and $a$~ , respectively. Now in the three-locus system the frequencies of AB, Ab, aB and ab are x1 + X, , These pairs remain associated in x2 + X6 , x3 + x7 and x4 + x8 , respectively. the expressions for w and wi because of the form of (2). Therefore, if at equilibrium ij = 0, we have 4, + R, = $&, R, + 4, = $&, , 2s + 2, = $&& and 4, + 3i’s= $&, . A family of equilibria then exists2 with 3i’, = p&A& ,
4,
=f’d’tt$b
,
$3
=$‘,$a$,
,
9.4 =p&s$b
,
3i’, =
j’ra$,$‘,
,
3i’,
=
j’m$&b
,
where p, is the frequency of M. If pm is very small initially, it seems (although it has not been proved in all cases) that there will be convergence to the curve above. Since the gene frequency of m does not change to linear order the change is presumably of second order in the perturbation analysis.
4,
=
prnja$~
> g’s =
pm$a$b
2 The author thanks Professor
,
S. Karlin
for pointing
out the existence
of this curve.
342
FELDMAN
DISCUSSION
Properties of B and & For all models analysed it has been shown that a selectively neutral recombination reducer, introduced at an equilibrium with D # 0 will increase in frequency at a geometric rate. If the equilibrium has D = 0 then the analysis indicates that it is highly unlikely for a recombination modifier to be favoured. The latter follows from the fact that the largest eigenvalue is one when a = 0 in the additive, multiplicative and symmetric viabilities models. It is worthwhile noting from (6) that no matter what the selection model assumed, if B = 0 there will be an eigenvalue of one although it might be difficult to prove it to be the largest. The tj = 0 case is of further interest because, in the symmetric viabilities model it occurs when both additive and multiplicative epistasis exist. Thus the assumption of the existence of epistasis is not enough to guarantee that the recombination reducer will increase geometrically, It is the condition on f, which is critical. When D = 0 equilibrium chromosome frequencies are not functions of the recombination fraction so that ti is independent of the degree of linkage. In terms of Lewontin’s (1971) formulation this provides something of an anomaly which may be resolved by the remarks in the preceding paragraph. However, when b # 0, in the three cases examined here, & is indeed a decreasing function of rl in the whole range of stability of the equilibrium. This is particularly interesting for the unsymmetric equilibria which exist only when ri is sufficiently large. It is tempting to conjecture that (&?/a~~) < 0 is equivalent to the largest eigenvalue of our analysis being greater than unity; this has not been shown. A point that should be emphasised is that in the preceding analysis the recombination reducer was introduced at an equilibrium of the original system and the analysis and results are not valid for any other starting conditions. It would be desirable to have a theory which allowed for the modifier to be introduced at any transient stage in the evolution of the MAB, MAb, MaB, Mab system. At present, since we cannot iterate the two locus recursion system analytically it seems that this problem might best be attacked by numerical computation. Limits to linkage reduction. There are many well documented examples of genes with related functions, in bacteria and higher organisms, which are closely linked. These include such diverse cases as the genes controlling flagella in Esckerickia Coli and those controlling the T alleles in mice. It is also well known that metabolically related genes (i.e., those coding for different steps of a biosynthetic pathway) are commonly closely linked. This is not, however, universally true as in E. Coli, for example, the genes controlling the arginine pathway are not adjacent (Maas, 1961; Gorini et al., 1961). It is also true that the genes coding for sequential steps in a biochemical pathway which are
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adjacent in some species of bacteria need not be adjacent in other bacteria or higher organisms such as fungi. As pointed out by Bodmer (1970) the latter indicates that there must at some time have existed selective forces favoring such reorganisation of the chromosome structure as the development of linkage groups. There are two obvious ways in which linkage of genes may have come about. In the first a gene may have been duplicated with subsequent gradual differentiation of the originally similar segments. Later a recombination mechanism could develop, but after linkage was established. In the second way the original genes are not linked but some interaction develops between them in such a way that linkage of the genes becomes advantageous. Aspects of both are incorporated into the general model proposed by Horowitz (1965) for the development of gene clusters, but it is the second which is pertinent to the model of Nei which we have analysed here (see Nei, 1967, for more details). Obviously if recombination occurred nowhere in a chromosome, no further selection for tight linkage would be possible; a limit state would have been reached. In previous theoretical treatments the conclusion has always been reached that when (additive) epistasis exists tight linkage will be favoured. In practice, of course, whole chromosomes are rarely observed to be completely without recombination. This has led Turner (1967b) and Lewontin (1971) to ask why such chromosomes have not reached the limit that theory indicates is necessary. That is “why has the genotype not congealed ?” With a little indulgence from the reader our theory might be able to shed some light on this paradox. In the first place, the evidence from the genetics of Schizophyllum Commune (Simchen, 1967; Stamberg, 1969) and Neurospora Crassa (Catcheside, 1968) is that the ret- genes which reduce recombination are usually dominant. Admittedly these are haploid species and the recombination may be intragenic. But if in our model the recombination reducer ‘m’ were dominant over its allele M so that mm also produced a recombination fraction ra , then since r2 < rr , ‘m’ would increase initially at a geometric rate but would fix at a rate slower than geometric and presumably algebraic. Thus the final stages of reduction would be very slow indeed so that changes in the selective regime might occur which could prevent or delay further reduction. A second escape from the paradox was mentioned by Turner and Lewontin. That is, on biological grounds, there is likely to be a stage at which further reduction of recombination must have a selective effect on the organism, for example by causing a breakdown of the meiotic mechanisms. If this is sufficiently deleterious the modifier might be expected not to succeed at all. Another way of overcoming the paradox has been to invoke some presumed advantage, imparted by recombination, in enabling adaptation to a changing environment. This could balance or overcome the selective forces favoring
344
FELDMAN
tighter linkage. A difficulty with this line of reasoning has been pointed out by Eshel and Feldman (1970), Karlin and McGregor (1971) and Eshel (1971). These authors have shown that in some deterministic models favorable combinations of genes do not necessarily occur more frequently with recombination than without although in small populations recombination may be of short term advantage. Bodmer (1970) h as made an analysis pertinent to the last remark. It is important to note that the paradox does not arise when the original population is at one of the stable unsymmetric equilibria, so that rr > 6. Here gradual reduction of recombination will take the population into a state where the new recombination fraction is too small for the equilibrium to be stable and the whole theory breaks down. If ra were zero, of course, then in one drastic step the limit would have been reached. Thus in discussing the linkage reduction paradox it appears that reference must be made to the equilibrium at which the modifier is introduced. Neutral genes. That genes may increase to polymorphism by virtue of their linkage to other genes which are selected has been pointed out in the deterministic context by many authors (see, e.g., Bodmer and Parsons, 1962, and Bodmer and Felsenstein, 1967, for discussions). Although the linkage modifier in the above treatment is selectively neutral it evolves by virtue of its effect on the linkage disequilibrium at the other two loci. This point is amplified elsewhere (Feldman and Balkau, unpublished data). Recombination-recombination polymorphism. It is of theoretical interest to consider the case where mm has a higher recombination frequency than Mm so that ra > ra < rr and the stability conditions are satisfied by ra as well as rr . Then in our cases 3A(ii) and 3B, for example, both boundary states (& , xa , xa , Y!?,,, 0, 0, 0, 0) and (0, 0, 0, 0, z5 , J!?a, 2, , 2s) are locally unstable. This would normally imply the existence of an interval stable polymorphism whose existence and stability would be due to the balance between the recombination fractions. This is particularly pertinent to the evolution of inversion polymorphisms and is the subject of a manuscript in preparation. Recent work by Feldman and Karlin A case where loose linkage is favored. (1971) has shown that the initial and final rates of evolution of dominance in the classical Fisher (1928)-Wright (1929) model are faster with increased recombination. However, differences caused by the modification are very small, of the order of the mutation rate in the initial stages and of the square root of the mutation rate in the final stages. Further problems. Certain aspects of the theory remain unresolved. One of the most important was mentioned earlier and concerns the fate of the modifier introduced at a transient state in the evolution of the original population.
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A second problem is to treat the model when the modifier lies between the modified genes. Another problem concerns the fate of recombination modifiers in inbreeding systems. The last two are the subjects of a manuscript in preparation.
ACKNOWLEDGMENT The author is grateful to Professor S. Karlin for a number of stimulating discussions during the preparation of the manuscript and to the Weitzman Institute of Science, Israel, for its hospitality during part of the preparation. The author is also indebted to Mrs. B. Balkau for her critical reading of the manuscript.
REFERENCES BODILIER, W. F. 1970. The evolutionary significance of recombination in prokaryotes, in Symposia of the Society for General Microbiology, XX. BODMER, W. F. AND FELSEZNSTEIN,J. 1967. Linkage and selection: Theoretical analysis of the deterministic two-locus random mating model, Genetics 57, 237-265. BODMER, W. F. AND PARSONS, P. A. 1962. Linkage and recombination in evolution, Advan. Genet. 11, I-100. CATCHESIDE, D. G. 1968. The control of genetic recombination in Neurospora Crassa, in “Replication and Recombination of Genetics Material” (W. J. Peacock and R. D. Brock, Eds.) Australian Academy of Science, ESHEL, ILAN. 1971. On evolution in a population with an infinite number of types, Theor. Population Biol. 2, 209-236. ESHEL, ILAN AND FELDMAN MARCUS W. 1970. On the evolutionary effect of recombination, Theor. Population Biol. 1, 88-100. EWENS, W. J. 1968. A genetic model having complex linkage behaviour, Theor. Appl. Genet. 1, 140-143. FELDMAN, MARCUS W. 1971. Equilibrium studies of two-locus haploid populations with recombination, Theor. Population Biol., to appear. FELDMAN, MARCUS W. AND KARLIN SAMUEL. 1971. The evolution of dominance: a direct approach through the theory of linkage and selection, Theor. Population Biol. 2, 482-492. FISHER, R. A. 1928. The possible modification of the response of the wild type to recurrent mutations, Amer. Natur. 62, 115-126. FISHER, R. A. 1958. “The genetical theory of Natural Selection,” Dover, New York. GANTMACHER, F. R. 1959. “Matrix Theory,” Vol. II, Chelsea, New York. GORINI, L., GUNDERSUN, W. AND BURGER, M. (1961). Genetics of regulation of enzyme synthesis in the arginine pathway of Escherichia coli, Cold Spring Harb. Symp. Quant. Biol. 26, 173. HALDANE, J. B. S. 1931. A mathematical theory of natural selection: VIII. Metastable populations, Proc. Camb. Phil. Sot. 27, 137-142. HOROWITZ, N. 1965. The evolution of biochemical synthesis-retrospect and prospect, In “Evolving Genes and Proteins,” Academic Press, New York. KARLIN, S. AND FELDMAN, MARCUS W. 1970a. Convergence to equilibrium of the twolocus additive viability model, J. Appl. Prob. 7, 262-271.
346 KARLIN, metric KARLIN, locus
FELDMAN
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