ON THE NUMBER O F STABLE EQUILIBRIA AND THE SIMULTANEOUS STABILITY OF FIXATION AND POLYMORPHISM
I N TWO-LOCUS -MODELS1
MARCUS W. FELDMANZ AND URL LIBERMANN3
Manuscript received February 13, 1979 Revised copy received May 31, 1979
ABSTRACT
It is shown that in simple symmetric two-locus, two-allele constant fitness models the bound of four simultaneously stable equilibria previously accepted for general two-locus, two-allele models is exceeded. Situations with five and six stable equilibria are exhibited. These involve four chromosomal fixations and either one or two polymorphic stable equilibria.
H E location, qualitative properties and number of stable equilibria in multi- locus genetic models under the influence of selection and recombination has been a focus of interest in population genetics since the original two-locus studies
of KIMURA (1956) and LEWONTIN and KOJIMA (1960). Comparisons between one-locus selection and more complex schemes have been made in terms of
FISHER'S (1958) fundamental theorem of natural selection (MORAN 1964), the role of the number of alleles at each locus (FELDMAN et al. 1975; CHRISTIANSEN and FELDMAN 1975; KARLIN 1979), and the validity of an induced overdomi- nance principle (KARLIN 1975).
I n assessing two-locus, two-allele results, it is appropriate, for tight linkage, to consider the four-allele situation in the one-locus case (KARLIN and
FELDMAN
1970). For larger values of the recombination, the correct standard becomes more difficult to discern and in fact may not exist. Thus, for example, there is only one completely polymorphic equilibrium in the one-locus case. But with two diallelic loci, there can simultaneously be two stable polymorphisms, each with linkage disequilibrium (for example, in the symmetric and multiplicative models) or one in linkage equilibrium, with the other in linkage disequilibrium
(FRANKLIN and FELDMAN 1977; KARLIN and FELDMAN 1978).
Of more interest to us in the present note is the fact that, with a single diallelic locus and constant fitnesses, it is impossible for a state of fixation to be simul- taneously stable with a polymorphism. Of course, HALDANE and JAYAKAR (1963) showed that with fluctuating viabilities this state of affairs can exist, and it is well known that the simultaneous action of more than one evolutionary force can produce this outcome. There has been little discussion in the literature of
'Research supported In part by Public Health Service grant 10452 and Nauonal Science Foundation grant DEB
77-05742
Department of Biological Sciences, Stanford University, Stanford, California 94305. a Department of Statistics, Tel Aviv Umversity, Tel Aviv, Israel.
1356 M. W. FELDMAN AND U. LIBERMANN
the extent to which, with two loci for the same recombination fraction and con- stant selection, both boundary and polymorphic equilibria can be stable. KARLIN
and CARMELLI (1975) found that six of their 100 randomly chosen two-locus
viability matrices allowed the simultaneous stability of a single boundary equilibrium and a single polymorphic equilibrium for sufficiently tight linkage. In all of these cases, the stable polymorphism became unstable or disappeared as the recombination increased. Of their 50 randomly chosen symmetric viability matrices, four exhibited a single polymorphism stable simultaneously with at least two chromosomal fixation states. Finally, KARLIN (1975) demonstrated
that, for very tight linkage, gene fixation and polymorphism can simultaneously be stable; the polymorphism in this case moves in from a boundary equilibrium as recombination increases from zero.
The basic two-locus, two-allele model involves the four chromosomes AB, Ab, aB and ab in frequencies xl, x2, x 3 and x 4 , respectively. The recombination fraction is
R .
The genotypic fitnesses are specified by the viability matrixI
[wij[I
.:{ = 1, with wij=
wji and w14 = wZ3.The standard recursions for the chromosomal frequencies are
I?
X: = xlW1. - Rw14Dw
xz'
= x2Wz.4-
Rw14D Wx4
= x3W3.+
Rw14DW
xi
= x4W4.-
RwlJl,
( l a )
(1b)
(IC)
( I d )
where
D
=Of special interest to us here will be the symmetric viability matrix ( LEWONTIN and
KOJIMA 1960;
BODMER andFELSENSTEIN
1967; KARLIN andFELDMAN
1970), which has
-
22x3,Wi.
= Z w i j x i ,W
=I:
XjWj..
3 3
w,, = w44 = 1
-
6, WZZ = w33 === 1 - a!w1z=w34=1-p, w13=w24=1-y (2)
w,4 = w23 = 1
.
It is necessary to distinguish three categories of equilibrium states of the sys- tem ( 1 ) . First, there are the four states of chromosomal fixation: xi = 1
(i
= 1, 2, 3, 4). Second, there are four states of gene fixation: x1 fx2
= 1, x1 -I- x 3 = 1,xz
4-
x4 = 1 , x 34-
x4 = 1. Finally, there are polymorphic equilibria with xi>
0 for alli.
Based on the known equilibrium properties of the corresponding multi- allele models (for zero recombination), it was conjectured (FELDMAN,FRANK-
LIN and THOMSON 1974) that there are at most seven of these for any recombina-
FIXATION A N D POLYMORPHISM I N TWO-LOCUS MODELS 1357
of ( I ) , including all three equilibrium classes, is four. We show that in the symmetric viability system (2) the bound is in fact six and that five can also be achieved. In particular, for loose linkage, simultaneous chromosome fixation and stable polymorphism is possible.
RESULTS A N D DISCUSSION
Simultaneous stability of gene fixation and polymorphism: I n the simplest version of the symmetric viability model (2), usually called the Lewontin- Kojima model, we set 6 = a. Then referring to (1) and ( 2 ) , it is known that
there may be three polymorphic equilibria for tight linkage; two of these have non-zero values of the linkage disequilibrium
B,
and the third has 23 = 0. Carets will be used to denote equilibrium values. Explicitly the three equilibria are:f , = f 4 = l / 2 - f , = 1 / 2 - f 3 = 1 / 4 { 1 - 8R
}
,
(3b)2, = 2, = f 3 = 24 = 1/4
,
(3c)where Z=2(J3++-cr)
.
(4)
8 1 - y
>
cr (Le.,I >
0) ( 4 4and R < 1 / 8
.
(4b)Now consider the edge equilibrium involving, say, fixation of the A/a locus in allele A . At
R
= 0, the conditions for stability of the point(5a)
are
f f > 8 ,
y > c r + J 3 . (6)Exactly the same conditions ensure the stability of the “a”-allele fixation equilibrium
2 3 = & = 1 / 2 . (5b)
Note that (6) automatically implies (4a). Thus, even in the Lewontin-Kojima model, for tight linkage, with y
>
a+
$3,>
8,
it is possible for two allele fix- ation states to be simultaneously stable with the two fully polymorphic equi- libria (3a) and (3b).It might also be remarked that the simultaneous stability of the two edge equilibria, (5a) and (5b), continues for all recombination values. I n particular, throughout the range of stability of the “high complementarity” polymorphic equilibria, (3a) and (3b), these gene fixation equilibria are also stable. For R
small enough, this type of coexistence of stable chromosomal polymorphism and stable gene fixation had been shown by KARLIN (1975, p. 386) for an eight- parameter viability matrix. The classical symmetric viabilities model under con- ditions (6) is a special case of KARLIN’S for which the precise range of co-stability can be given. Qualitatively, the conditions y
>
1-8,
a>
$3 entail that at leastone single heterozygote should be more fit than the double homozygotes, i.e.,
1 -/3
>
1 -CY, while the other single heterozygote must be relatively unfitClearly the existence of (3a) and (3b) requires
1358 M. W. F E L D M A N -4ISD U. LIBERMANN
( y
>
a:f
8 ) .
These conditions are not extreme; the following fitness matrix for example satisfies the conditions0.98 0.99 0.96 1
0.99 0.98 1 0.96
0.96 1 0.98 0.99
1 0.96 0.99 0.98
It is obvious from simple four-allele theory that, f o r absolute linkage, gene fixa- tion cannot be stable simultancously with a bona fide polymorphism. From this and the result of the following section, we might conjecture that should a poly- morphic two-locus equilibrium become stable only for R larger than R,
>
0, then it cannot be stable simultaneously with gene fixation equilibria.Simultaneous stability of chromosome fixation and polymorphic equilibria:
T o illustrate the conclusions, we again use the symmetric viability model (2). The point can be made with a: = 6 as before and we can also assume 8 = y . In this case it is easily seen that the gene fixation equilibria cannot be stable. How- ever, the conditions for the stability of (3a). (3b) and (3c) are particularly simple. (3a) and (3b)
,
the “high complementarity” equilibria, are stable if(i) 2/3
>
a: and (ii) R<
8/2 - a/4 (7) (KARLIN and FELDMAN 1970). On the other hand (3c), the central equilibrium, is stable ifR
>
/3/a - a/4.
(8)Now consider the four chromosome fixation equilibria 9, = 1, 9, = 1, 9, = 1,
z4
= 1 . In view of the extreme symmetry of the assumptions, the local stability conditions of each is the same. The relevant eigenvalues at each corner are(BODMER and FELSENSTEIN 1967)
(1 -
8)/(1
- a ) and (1 - R)/(1 -a:).
(9)(i) @ > a and (ii) R
>
a:,
(10)O < a < R < < / 2 - a / 4 , (11) Thus, if
then all four corners are locally stable. From (8) and ( l o ) , it is clear that if
F I X A T I O N A N D P O L Y M O R P H I S M I N TWO-LOCUS MODELS 1359
I n fact (12) is sufficient for the existence of [he region allowing six stable equilibria. It should be noted that, as first shown by EWENS (1968), if
/3
>
(2 -I-dza.
a gap in the stability range of (3a) and (3b) will exist.If R is increased beyond the upper bound in (1 1 ), the central point, (3c), becomes stable, while both (3a) and (3b) disappear. Thus, if
8
>
a, then in the rangeR
>
max ( p / 2 - a/4, a)(3c) and the four chromosome fixations are stable simultaneously, for a total of five stable equilibria. It might be noted that the condition ( I O ) (i) for the stability of the chromosome fixations ensures that, as R increases, both the six- equilibrium and five-equilibrium phases are encountered. Once again the condi- tions allowing six (or five) equilibria are not extreme. Qualitatively, the single heterozygotes should be deleterious relative to the double homozygotes, with the
double heterozygotes being most fit.
KARLIN and J. ROUGHGARDEN for their criticism of an earlier draft.
We are grateful to R. C LEWONTIN for a particularly stimulating discussion, and to S.
L I T E R A T U R E C I T E D
RODMER, W. F. and J. FBLSENSTEIN, 1967 Linkage and selection: Theoretical analysis of the deterministic two locus random mating model. Genetics 57: 237-265.
Multiple alleles and interactions between two loci. J. Math. Biol. 2: 179-204. CHRISTIANSEN, F. B. and M. W. FELDMAN, 1975
EWENS, W. J., 1968
FELDMAN, M. W., I. R. FRANKLIN and G. THOMPSON, 1974
Selection in complex genetic systems, IV:
-4 genetic model having complex linkage behavior. Theoret. Appl. Genet.
Selection in complex genetic systems, I: The symmetric equilibrium of the three locus symmetric viability model. Genetics 76: 135-162.
Selection in complex genetic systems, 111: An effect of allele multiplicity with two loci. Genetics
79: 333-347.
38: 140-143.
FELDMAN, M. W., R. C. LEWONTIN, I. R. FRANKLIN and F. B. CHHISTIANSEN, 1975
FISHER, R. A , 1958
FRANKLIN, I. R. and M. W. FELDMAN, 1977
The Genetical Theory of Natural Selection, 2nd ed. Dover, New York.
Two loci with two alleles: Linkage equilibrium and linkage disequilibrium can be simultaneously stable. Theoret. Popul. Biol. 12: 95-113. Polymorphism due to selection of varying direction. HALDANE, J. B. S. and S. D. JAYAKAR, 1963
J. Genet. 58: 237-242.
KARLIN, S., 1975 General two locus selection models: Some objectives, results and interpreta- tions. Theoret. Popul. Biol. 7 : 364-398. -- , 1979 Principles of polymorphism and epistasis for multilocus systems. Proc. Natl. Acad. Sci. U.S. 75: 541-546.
KARLIN, S. and D. CARMELLI, 1975 Numerical studies on two-loci selection models with gen- eral viabilities. Theor. Popul. Biol. 7 : 399-421.
Linkage ancl selection: Two locus symmetric viability model. Theoret. Popul. Biol. 1: 39-71. ---, 1978 Simultaneous stability of D=O
1360 M. W. FELDMAN AND U. LIBERMANN
KIMURA, M. 1956
tion. Evolution 10: 278-281. LEWONTIN, R. C. and K. KOJIMA, 1960
Evolution 14: 458472. MORAN, P. A. P., 1964
A model of a genetic system which leads to closer linkage by natural selec- The evolutionary dynamics of complex polymorphism.
