representation of nonepistatic selection models and analysis of multilocus hardy-weinberg...

J. Math. Biology 7, 353-374 (1979) Journal of

�9 by Springer-Verlag 1979

Representation of Nonepistatic Selection Models and Analysis of Muitilocns Hardy-Weinberg Equilibrium Configurations

Samuel Karlin 1 and Uri Liberman 2

Department of Mathematics, Stanford University, Stanford, CA 94305, USA 2 Department of Statistics, Tel-Aviv University, Ramat-Aviv, Israel

and Department of Mathematics, Stanford University, Stanford, CA 94305, USA

Summary. The paper develops conditions for the existence and the stability of central equilibria emanating from selection recombination interaction with generalized nonepistatie selection forms operating in multilocus multiallele systems. The selection structure admits a natural representation as simple sums of Kronecker products based on a common set of marginal selection components. A flexible parametrization of the recombination process is introduced leading to a canonical derivation of the transformation equations connecting gamete frequency states over successive generations. Conditions for the existence and stability of multilocus Hardy-Weinberg (H.W.) type equilibria are elaborated for the classical nonepistatic models (multiplicative and additive viability effects across loci) as well as for generalized nonepistatic selection expressions. It is established that the range of recombination distributions maintaining a stable H.W. polymorphic equilibrium is confined to loose linkage in the pure multiplicative case, but is not restricted in the additive model. In the bisexual case we ascertain for the generalized nonepistatic model the stability conditions of a common H.W. polymorphism.

Key words: Recombination-segregation distribution - - Nonepistasis selection - - Hardy-Weinberg equilibria - - Bisexual models.

O. Introduct ion

The early developments pertaining to quantitative genetics were loosely founded on the assumption of a single gene. The single locus results are often applied in the multilocus case by adding or multiplying over loci. Already the two-locus theory well establishes that the multilocus Hardy-Weinberg (H.W.) property (linkage

This paper was supported in part by NIH Grant GM 10452-14 and NSF Grant MCS 75-23608.

0303-6812/79/0007/0353/$04.40

354 S. Karlin and U. Liberman

equilibrium), meaning that gamete frequencies are products of their respective marginal gene frequencies, does not hold in general. The existence of a stable multilocus Hardy-Weinberg equilibrium form is mostly tied to nonepistatie selection systems or generalized symmetric viability patterns for special parameter specifications. Nonepistasis reflects the situation where the fitness value of a given genotype is attributable to ' independent' effects conferred by the separate loci.

Results on the two-locus two-alleles multiplicative and additive nonepistatic models have been set forth by a number of authors (e.g. Bodmer and Felsenstein (1967), Moran (1968), Karlin (1975)). When the contributions of the loci to the viabilities are additive, it is proved (Karlin and Feldman [1970], Karlin and Liberman [1978a]) that there is a single two-locus Hardy-Weinberg polymorphism which is globally stable for all sets of nonzero recombination values provided the separate loci are overdominant, i.e., when the viability effects at each locus are such that a stable polymorphism would be established at this locus if operating in isolation. When the viabilities are multiplicative over loci, it is established that only for significantly loose linkage, heterozygous advantage at the separate loci assures global stability of the Hardy-Weinberg equilibrium (Moran [1968]). Roux [1974] was the first to recognize the relevance of Kronecker product matrix repre- sentations in treating multiplicative nonepistasis especially in discerning the stability conditions of a Hardy-Weinberg polymorphism.

In this paper we investigate a hierarchy of generalized nonepistatic multilocus multiallele systems that admit a representation as sums of Kronecker products based on a common set of marginal (individual loci) selection components. In Sections 1 and 2, in the context of multiple alleles and loci, we introduce a flexible parameterization of the recombination process and selection regimes, leading to a succint expression of the transformation equations connecting gamete frequency arrays over successive generations. Section 3 elaborates necessary and sufficient conditions for existence of H.W. polymorphic equilibria for the classical nonepistatic models. Sections 4 to 6 report a series of results concerning the existence and stability properties of H.W. equilibria for mixed (additive and multiplicative) nonepistatic selection expression. The level of recombination maintaining a stable H.W. equilibrium is always confined to loose linkage in the pure multiplicative case, but is not restricted in the additive model. Complementary characterizations and delineations of the stable equilibrium realizations in the presence of tight linkage under a multiplicative nonepistatic regime is set forth in Karlin [1978]. A generalized nonepistatic model is presented in Section 6. A discussion of nonepistatic bisexual models involving autosomal characters occurs in Section 7.

For further biological orientations and interpretations of the results derived here the reader is referred to Karlin and Liberman [1979a] and [1979b].

1. The General Multilocus Multiallele Model

Consider in a large diploid monoecious population a trait determined at n loci with mk possible alleles A~ k), A(2 k), . �9 -~m~A<k> at locus k (k = 1, 2,. . . , n). In developing a

Representation of Nonepistatic Selection Models 355

f ramework that is tractable for delineating and interpreting recombinat ion and selection effects it is essential to operate with a natural coordinate system. We cannot just order all the gamete types 1, 2 , . . . , R, R = m~m2. �9 rn~, as is customary in the two-locus two-allele model. Without recognizing the inherent intra- and inter-locus symmetries in the n-locus context the t ransformat ion equations and their analyses become prohibitive. In the natural parameterization, the analysis is sub- stantially for thcoming and revealing.

Let i~ ~), i~ k) index the possible alleles at the k th locus. Associated gametes (haplo or chromosomal-t)Tes) are described by the n-tuples.

i0 = (i(01), i(o2),..., i(o ")) and i~ = (i~ ~), i~ ~) . . . . . i~")). (1)

A typical genotype composed of the two gametes above is displayed in the form

(io) [i(ol),i~o ~) . . . . . i(o~)~ = \i11,, ii~,,. ] . i ( ~ ] (2)

signifying that the allelic composi t ion at locus k consists of alleles i(0 ~) and i~ ~). We do not preclude that i(o ~) and i~ ~) refer to the same alleles at locus k.

The fitness value of the genotype (2) is denoted by

(,.'_o] ,;,1, w = w i ~ - ~'77-~t = w(io, i l ) . (3)

\ l U \ q - ' , . . , t ~ . ,

The array (3) where io = (i(01> . . . . . i(0 ")) and il = (il 1) . . . . . il ")) cover all gamete combinations, generates the fitness matrix F of order ( ~ = ~ mk) • ( I ~ = ~ mk).

The recombination-segregation distribution depicts the frequency rates o f the gamete output resulting from recombinat ion and segregation. The outcome of meiosis involving the 'ma le ' gamete (i(o 1), i(02) . . . . . i(o ")) and the ' female ' gamete (il 1), i~ 2>, . . . , il ")) can be any of the 2" gametes ,-~(;(1), -~2 i(z), - . . , i~, ">) where each ek = 0 or 1, k = 1, 2 . . . . , n. The recombination-segregation distribution ~ prescribes probabilities to these 2" mutually exclusive events, and, thus, it consists of the 2" nonnegat ive parameters {R(e)} where

(4)

stands for the probabil i ty of the recombination-segregation gamete product (i<1) i(2) i(~)~ and ~ = (~1, e2,. en). The representation (4) for the combined recombination-segregation distribution was formalized in this manner first by Geiringer (1944). As the two parental gametes contr ibute symmetrically to the gamete product and since the parameters {R(e)} correspond to mutually exclusive recombinat ion events we have the two intrinsic relations

R(~) = R ( 1 - e), ~ R ( e ) = 1. ( H e r e l = (1,1 . . . . ,1)) . (5)

Mostly, we refer to R = {R(6)} simply as the recombination distribution.


Example 1. In the case of two segregating loci i f r is the probability of recombination between the two loci, then

1 - - r r R(0,0) = R ( 1 , 1 ) = 2 ' R(0,1) = R ( 1 , 0 ) = ~. (6)

Example 2. In the case of three loci let r denote the frequency of the event of recombination only between loci 1 and 2, s only between loci 2 and 3 and let t be the frequency of simultaneous recombination between loci 1, 2 and loci 2, 3, then

1 - r - s - t R(0, 0, 0) -- R(I, 1, 1) = 2 '

8 R(0, 0, 1) = R(1, 1, 0) = -~

r R(0, 1, l) = R(1, 0, 0) = ~, t

R(0, l, 0) = R(1, 0, 1) = ~.

(7)

Let x(io) = x(i~ 1~ . . . . . ico "~) denote the frequency of the chromosomal type io = (i~o 1~ . . . . , ico "~) in the population of the current generation. Under random mating, subject to the effects of fitness selection, the relative proportion of the genotype (2) among mature individuals is w(io/i~)x(io)x(i,). The segregation process joined with the recombination contingencies leads to the frequency of the gamete type in the next generation x'(io) = x'(i~ ~, . . . , iCo ~) computed by the formula

g �9 �9 (8)

where w(x) is the mean fitness funetion for the population composition x, namely

w(x) = ~ w X(,o)X(,1). IO,Zl

(9)

It may be helpful to illustrate the reasoning underlying formula (8) and the formula itself for the two-locus two-allele case with alleles A, a at the first locus and B, b at the second. Let io = (i~o 1~, i~o 2~) refer to the gamete ab. There are four classes of genotypes which can produce by recombination-segregation the gamete ab. They are listed next with the corresponding frequencies�9

[ a, b ~, la, i~ 2~] (il 1), b] |,zi;<l', ,z;c2)\l Genotype class /;,t~ ;,2~/ \ ~ 1 , i ~ ) ] , \ a, b ] '

Frequency of gamete outcome ab

R(0,0) R ( 0 , 0 R ( l , 0 ) R(I, 0


where (ill), ii2)) runs over the four possible gametes AB, Ab, aB, ab. Accordingly, the resulting frequency of the ab gamete xl(a, b) in the next generation is

.(I) b + R(l, 0) w (%-) x ( i ~ ' , b)x (a, ii2))

(,~l,,,,,) a, P

which is exactly (8).

Remark I (Bisexual model). Using the same notations and conceptualization as above it is possible to derive the transformation equations of gamete frequencies in the corresponding bisexual model (entailing two separate sexes).

Let x and y denote the vectors of male and female gamete frequencies. Paralleling (3), the male and female fitness matrices I', and I', are prescribed by wm(io/il) and w,(io/il), respectively. With a two-sex autosoma1 character, the recursion formulas connecting gamete frequencies over two successive generations become

where Rm = {R,(E)) and R, = {R,(E)) comprise the recombination distributions and wrn(x, y) encompass the fitness functions associated with the male and female populations, respectively. Specifically,

2. The Transformation Equations in the Nonepistatic Cases

In this paper we concentrate on a number of manifestations of nonepistatic selection. Nonepistasis reflects the situation where the fitness value of a given genotype is attributable to 'independent' effects conferred by the separate loci. The two


classical prototypes of nonepistasis are the multiplicative and the additive fitness regimes.

For multiplicative nonepistatic selection we have

�9 o . . . . __ = Wtk)(i(oU), i~k)) w [i(~ ' i~2),. S i~")] k =1 (13)

where

(14)

can be construed as the marginal fitness matrix associated with locus k. In line with (13) the fitness of the genotype (io/i~) is determined by multiplication of the independent fitness effects from the separate loci.

For additive nonepistatic selection the evaluation on the right side of (13) replaces product by sum, so

/ i(1) " " i(~ w[ ~ . , . - ~ ) ] = ~ ~ w~)(i(o~, ii~)). 05)

Notation 1. Let A be an m x n matrix and B an l • k matrix. We denote by A Q B the Kronecker product of A and B, namely the partitioned matrix

( a l l B a12B. . . a1 ,B \

a21B a22B .a2 ,B} (16) A | ~

1 XamlB am2B" "am,B]

Employing this notation we recognize the multiplicative fitness matrix (13) in the Kronecker product form

M = W~ | W2 | . . . | W, (17)

where Wk, k = 1, 2 . . . . , n are the marginal fitness matrices (14).

In the additive case we denote by Ek for all k -- 1, 2 , . . . , n the m~ x mk matrix whose elements are all unity. Ek conforms to a (marginal) neutral selection (no selection differences) associated with locus k. In this notation, we may identify the summand w(k)(i(o ~), i!f )) with the n-locus fitness matrix Sk = E1 | E2 | | Ek-1 | Wk | Ek+l | - ' " | E,. Therefore, we can express (15) as the special sum of Kronecker products

s = ~ s~. 08) k = l


The Two-Locus Specialization. In the two-locus two-allele case, the marginal fitness matrices have the general form

, �9 Us 02 /38

Then, the multiplicative two-locus fitness matrix becomes

W~ | W~ =

UlVl UlV2 U2Vl U2~2

UlV2 UlV3 U2V2 u2v31

U2Vl U202 UsVl HsV2

U2V2 U2V3 UsV2 H3V3

while the additive fitness matrix reduces to

l'J'zl | E2 + E1 @ Wz =

where

ilu~+v~ u~+v2 u2+v~ u2+v2~[

iluJ_ + V2 Ul + V3 U2 + V2 U2 + V3 J/ U2 + Vx U2 + V2 u 8 + V 1 U 8 + v~

U2 + V2 U2 + V8 U3 + V2 U8 + V3

1 1 E1 = E2 = 1 1 "

Notation 2. Let a = (al, a2 . . . . , ap) and b = (bl, b2, . . . , bp). We denote by a o b the Schur product of a and b namely,

a o b = (albl, alb2 . . . . . avb~).

The next theorem provides a succinct symmetric representation of the transformation equations (8) for the special selection regime (13).

Theorem 1. For the multiplicative nonepistatic selection regime, the transformation equations (8) attain the form

w(x)x ' = ~ R(~) (w~I | w ~ | . . | w~..x) o (w~-~l| w d - ~ | . . | rv~.-~x)

where

w(x) : <x, rVl | | W,x>.

and (a, b> denotes the inner product of the given vectors.

(19)

(20)

Proof Inserting the multiplicative fitness matrix from (13) into (8) the transformation equations become

w(x)x'(io) = ~ R(e) ~ I-~ w(k'ti ~ , i ~ ,~)x(i,)x(i,_,). e il k=l


For fixed 8, we examine a typical sum

~-I w(ku i .~ , i[~ ,k)x( i.)x( it _ ~) (21) 11 k = l

which can be judiciously cast in the form

~, ~ [w,~,(i,g,, ~,'~))bx(~,) �9 Z I-~ Ew(~)(i(o~), i[k)]l-e~x(it-,) I k = l l l k = l

(22)

where the summation set I entails the different allelic compositions of i~ at those loci k corresponding to ek = 1, while the summation over H runs over the complementary set of loci for which e~ = 0. (Recall that W~ is a symmetric matrix.) I f we adopt the convention that W ~ is the identity matrix of the same order as We then (22) equals

Z ~-I [w(k)( i(o~', i[k')]"~x(i~) " ~. ~-I [w(~)( i(ok', i[~')l ~-"~x(il). i i k = l il k = l

The first sum is recognized as the components of the vector Wp | W~2 | �9 �9 | Wg,x while the second sum as the components of WI -~1 @ W~-'2 | �9 �9 | W~-~,x. Hence, summation of the right-hand side of (8) with reference to the properties of R(e) results in (19). []

Illustration of(19) in the Two and Three Loci Cases. In the two-locus multiplicative model the transformation equations in the representation (19) read as

w(x)x' = ( 1 - r)(I | Ix)o (WI | W2x) + r(W~ | Ix)o (I @ W2x)

w(x) = (x, rV~ | W~x). ( 2 3 )

In the corresponding three-locus case, we secure

w(x)x' = (1 - r - s - t)(I Q I Q Ix)o(W1 @ W2 | W3x) + r(W~ | o( I | W2 @ W3x) + s(W~ | W~ | o ( I | 1 7 4 V~x) + t(w1 | 1 7 4 W 3 x ) o ( l | w~ |

with

(24)

w(x) = (x, Wl | W2 @ W3x).

In both cases, as well as for the general representation (19), we emphasize the natural symmetric forms in which the recombination parameters appear.

As a direct consequence of Theorem 1, we derive for the additive case:


Corollary 1. The transformation equations (8) with additive nonepistatic selection can be expressed in the form

w(x)x' = ~ ~ R(~)(E~ | | ~ | W~ | | . . . | k=l

o (E~-~I | @ W~- ~ | 1 7 4

and

(25)

w(x) = ~ wk(x~) (26) k = l

where w~(xk) is the marginal mean fitness engendered at the kth locus.

Proof As the transformation equations (8) are linear in the fitness parameters, (25) ensues from Theorem I in conjunction with the representation (18). For the fitness function, we have

w x,: (x <x x>

But

{x, Skx) = (x, E1 | | EL-~ | W~ | E~+ 1 @ " " | E ,x ) = (x~, Wkx~)

where xk is the marginal allelic frequency vector determined by x at the kth locus So {xk, Wk xk) = w~(xk) is simply the marginal fitness function corresponding to the kth locus and accordingly (26) follows. []

3. Existence of Hardy-Weinberg Equilibria

Definition 1. A frequency vector x of order 1-I~-= 1 me is said to be a multilocus Hardy-Weinberg type (abbreviated H.W.) if x admits the Kronecker product representation

x = xl | x2 | | x~ (27)

where xk is a vector of order mk for k = 1, 2 . . . . . n. (We sometimes refer to (27) as a K-product population state.)

Remark 2. In the case of two loci involving two alleles at each locus it is readily checked that x = (xl, x2, x3, x4) is of H.W.-type if the disequilibrium function D(x) = xlx~ - x2x3 vanishes, i.e., x is in linkage equilibrium.

We next circumscribe conditions for the existence of H.W-type equilibria in the context of nonepistatic selection.


Let :~k be an equilibrium for the marginal fitness matrix Wk. Equivalently, .~ is an mk-tuple frequency vector satisfying

.~ o W k ~ = ~ k . ~ ( 2 8 )

where

~k = Wk(~k) = (~k, Wk~k).

Theorem 2. Let ~k, k = 1, 2 . . . . . n, satisfy (28). Consider the multilocus regime under multiplicative or additive nonepistatic selection with marginal viability matrices Wk, k = 1, 2 , . . . , n. Then the 1-1. W. frequency state (i.e. K-product frequency state)

is an equilibrium for any prescription o f the recombination parameters {R(e)} entailing mean f i t ne s s~ = ~1~'2 " . ff~, in the multiplicative case a n d # = ~1 + ~2 + " " + if', in the additive case.

Proo f Suppose first that selection is multiplicative. We implement the transformation equations (19) on ~ = ~1 | ~2 | | Invoking the properties of Kronecker products produces

(W~ | W~ | | W,~-)(~I | | ~,) = W ~ | W ~ | | W,~-~,

(w~-~l | w~-~2 | | w~-~.)(:~ | | ~,)

= w ~ - ~ l ~ | wd-~2:~ | | w,~- ' ,~ ,

and so

(Wfl | W~ | | Wg,x) o ( w ~ - ~ | | w~,-~,~)

= (e~ o w ~ 0 | ( ~ o w:e2) | | (~, o w ,e , )

= (ff~ff'2. �9 �9 ~',):~1 | ~2 | | ~,

the last identity resulting on account of (28).

As ~Y. R(e) = I, we secure that .~ = ~ | �9 . . | ~2, is an equilibrium with associated mean fitness f f~ ' : . . . if,,.

In the presence of additive nonepistatic selection, we note that

:~ o E~:~ = ~ , (:~, E ~ ) = I

Therefore, paraphrasing the argument of the multiplicative case leads to

(E~ | | W~:~ | | E,%~) o (E~-~, | | W~ - ~ | | E~-%$)

= ~ ( . ~ | | .~,).

Combining these in line with (25), we find that ~ | �9 �9 �9 | :~, is a H.W. equilibrium entailing mean fitness ~,~ + rT, z + �9 .. + rg,. []

A converse of Theorem 2 is available as attested to next.


Theorem 3. I f 2 = 2~ | . . . | 2,~ is an equilibrium for the muhiplieative or the additive nonepistatic selection regimes (17) or (18) and 2k is a frequency vector o f order m~, then 2k is an equilibrium of Wk, k = 1, 2 . . . . . n.

Proof Suppose first that selection is multiplicative, then retracing the analysis o f T h e o r e m 2, we achieve the identity

( ~ , ~ - . - ~,.)& | 2~ | | 2. = (& o w,&) | | (2. o w.2.)

which plainly implies

c~ff'k2g = 2k o Wk2~, k = 1, 2 . . . . . n (29)

for appropr ia te scalars ck. Since 2k is a frequency vector (with respect to the kth locus), it follows that c~ = 1, Compar ing (28) and (29) we see that 2k qualifies as an equil ibrium for W~. In the additive case we deduce in a parallel fashion the equat ion

• r | | 2.) = ~ 2 (~) (30) k=l k=l

where

2 (~) = 21 | --- | 2k- i | (2k o l'Vk2~) | 2~+i | | 2,, k = I, 2 , . . . , n.

Fix k and consider .Ok = 2k o Wk2k. In the presence of (30) we infer when the ith componen t of 2k equals zero, i.e. ,#k(i) = 0 then )Vi ) = 0. I f :?k(i) ~ 0 then dividing (30) by this factor we find that fk(i)/2k(i) is invariant with respect to all componen t s i for which 2k(i) r 0. Therefore .Oh = dk-f'k having d~ as a c o m m o n scalar. Since 2e is a f requency vector, it follows that

and so #k2~ = 2k o I4~2k establishing 2k as an equil ibrium of I4~. The p r o o f of T h e o r e m 3 is complete. [ ]

4. Stability of a H.W. Polymorphism with Multiplicative Selection Effects Across Loci

Let 2 = 21 | . - . | 2n be a H.W. po lymorphic equilibrium. Then by Theorem 3 each 2k is necessarily a po lymorph i sm for the one locus multiallele selection model with fitness matr ix Wk. It is a familiar proposi t ion (e.g., see K ingman [1961]) that the stability of 2~ requires the eigenvalues {hk,~}~'=~l and corresponding right eigenvectors {ak,~}?=~l of the matr ix

Wk = 2 k o W ~ wk ' (31)


(the nota t ion in (31) stands for the matr ix Wk multiplied on the left by the diagonal matr ix having the components of the vector :r on its diagonal) to satisfy

? 'k.1, ' ' ' ,~k,m~ are real and Ak,, < 0 , i = 2 . . . . . mk, (32a)

1 = a~ , l > la~, ,I , i = 2, 3 , . . . , m~;

ak.~ = ~k up to a scalar multiple and all ak,~ for i /> 2 are or thogonal to

e~ = (1, 1 . . . . . 1) o f m~ components . (32b)

Emana t ing straightforwardly, on the basis o f e lementary operat ions with Kronecker products and reliance on condit ions (32a) and (32b), we obtain

Lemma 1. (i) A complete set o f eigenvectors for fie~ | fie2 | "" �9 @ fie is generated by all Kronecker product vectors

~(i~, i2 . . . . . in) = az,,1 | a2,~2 | | an,c, (33a)

with associated eigenvalues 1-I~=~ Ak,r respectively, corresponding to all prescriptions o f n-tuples

(il, i2 . . . . . in); 1 ~ ik ~< ink, k = 1, 2 , . . . , n. (33b)

(ii) The subspace ~ comprised o f all vectors in E R, R = m~mz . �9 �9 mn, orthogonal to e = (1, 1 . . . . . 1) of R components has a basis consisting o f the R - 1 vectors ~(iz, i2 . . . . . in) corresponding to all prescriptions o f (iz, i 2 , . . . , in) as in (33b) ex- cluding il = i2 . . . . . in = 1.

We are now prepared to offer the main theorem o f this section.

Theorem 4. (c f Roux [1974]). Let {Ak,~}?$1 be the eigenvalues o f fie~ defined in (32). With multiplicative nonepistatic selection, the H. W. polymorphism ~ = 2~ @ �9 �9 �9 | .On is locally stable provided

2 ~ R(e)A~h~A~,,~..-a~-.,, < 1 (34)

holds for all choices o f (iz, iz . . . . . i,) as in (33b) except i~ = i2 . . . . . i, = 1.

Proof The local stability endowments of .~ are discerned by calculating the eigenvalues of the gradient matrix (local linear approximat ion) o f the t ransformat ion (19) at the po lymorphic equilibrium point .~. Consider a small per turbat ion ~ of the frequency vector 2 of the fo rm x = .~ + ~. As both x and .~ are frequency vectors, then (g, e) = 0 and so ~ e s defined in L e m m a 1. Discarding second order terms in the components of g, we secure

~ R(~)(w~I | | wg~ -~ | | w~.-'.~)

_~ ~ R(~)(w~ | | w.'-~)o (w~ -~ | | w.~-~-~)

+ 2 ~ R ( e ) ( W { ~ | @ Wg,k) o (W~ - ~ | | W~- '"g)


with help o f the relat ionship R(e) = R ( 1 - ~ ) . As ~ is a po lymorphism, following T h e o r e m 2, we have

R(~)(W~ | | Wg,~) o (W~-,I | | W~- ' , ] ) = (~ ,~ . . . ~ , ) ~ . r

Also, as -~k for any k = 1, 2 . . . . . n is a po lymorph i sm with respect to W~ we have W ~ i ~ = ~ , g ~ - ~ where x ~ = e~ and x~ = xk. Deleting second order terms in the componen t s o f g and using the fact that (~, e) = 0 leads to the approx imat ion

w ( x ) = ( x , w~ | | W , x ) ": ~ , ~ , ~ . . . r

Therefore , the linear approx imat ion of (19) is conver ted into

~" + 2 ~ R(6) ~ - "~ @ ~ - "= | | 2~- "" ,~,~-~,,~,1-~=... #~,-'" o (w~-"~ | | wk-"-g).

In terms of the matrices I~k defined in (31), the above equals

+ 2 ~ R(~)(W~-,~ | | W. ~-~-)g B

and again by virtue of R(e) = R(1 - e),

= aT" + 2 ~ R ( e ) ( f f ' ~ | | ff'~-lg.

As ~ + 1~ is mapped into 2 + ~' we can identify the gradient matr ix at 2 as

(35)

The po lymorph i sm 2 is locally stable provided all the eigenvalues of L confined to the subspace s are in magni tude less than 1. In view of L e m m a 1 the vectors {g(i~, iv . , . . . , i,)}, taking all possible prescriptions of (il, i2 . . . . . i,) except i~ = i2 = . . . . i, = 1, consti tute a basis o f ~ a. By their determinat ion (see (33))

( f f / ~ l @ . . . @ I ~ n , ~ ) ~ ( i , . . . . , i n ) = A]~.q )t~2,~2 . . . A~" , , ,~ ( ia . . . . . i n ) .

Therefore ~ is locally stable provided the (1-IL ~ m3 - 1 relevant eigenvalues of L, namely

2 ~ R(e)I~I.,1A~,,~.. - 1",.-,,

cor responding to the eigenvectors g(il, i 2 . . . i,), where (il, i2 . . . . . i,) traverse all feasible n-tuples precluding il = i2 . . . . . i, = I, have magnitudes smaller than unity. The p r o o f of Theo rem 4 is complete. [ ]


Example 3. In the case of two loci with two possible alleles at each locus, let the eigenvalues of l ~ be 1 and h and those of I~2, 1 and/~. As R(0, 0) = R(1, 1) = (1 - r)/2, R(O, 1) = R(1, O) = r/2, where r is the recombination rate between the two loci, the conditions (34) for the stability of .~ | ~2 reduce to the three in- equalities

I I + A [ < I , I1 +t~l < 1

I(1 - r) + (1 - r)A/~ + rA + r~l < 1. (36)

The first two express the stability of.~l and -~2 separately at loci 1 and 2 respectively, and these are synonymous with A, t~ < 0 - - t h e exact one locus conditions for stability, namely heterozygote advantage at each locus. The final condition of (36) agrees with the stability criterion derived by Bodmer and Felsenstein [1967].

5. Stabi l i ty o f a H.W. P o l y m o r p h i s m with Addit ive Se lec t ion Effects Across Loc i

It is established in Karlin and Liberman [1978a] where each of the separate locus polymorphisms s and s is stable, then the H.W. polymorphic equilibrium s | :~2 for the two-locus multiallele additive selection model is globally stable for positive recombination. In the presence of a general multilocus additive selection regime, with the aid of the special representation of the transformation equations (25), we can discern the local stability nature of the H.W. polymorphism. The problem of global stability for the additive model with 3 or more loci is unresolved.

Theorem 5. Suppose selection is additive and each of the single locus polymorphisms ~ (at locus k) is stable. Suppose positive recombination rates, then the H. I47. polymorphism ~ = ~ @ . . . | ~ is locally stable.

Proof As each summand in (25) is a Kronecker product, we can adapt the derivation of the local linear approximation at s of the multiplicative case in order to calculate the gradient matrix at s for the additive model. This procedure yields

2 E R(e) _~ ff'~(E~ 1 | | | I ~ | /~++~ | | E,9)g ~' = A ~ = ~ ~=~

k = l

where

and ~ represents an arbitrary vector in the subspace A ~ (defined in Lemma 1) orthogonal to e = (1, 1 . . . . . 1). We observed in Lemma 1 that s is spanned by the basis comprised of the vectors ~(il . . . . . i,) = al.il | a2,~2 | " '" | a~.~,, 1 ~< i~ ink, k -- 1, 2 , . . . , n precluding the specification il = i2 . . . . . & = 1. To prove


local stability we will show now that each of these ~(i~, . . . i,) is an eigenvector o f A with corresponding eigenvalue smaller than 1 in magnitude. Since ak,~ for 2 ~< i~ ~< m~ is or thogonal to e~ = (1 . . . . . 1), we see that

E ~ a ~ , ~ = O f o r 2 ~< i~ ~< m~ "~k E~a~,~ = 2~ o E ~ 2 ~ = 2~ = a ~ , ~ ) = 1, 2, . . . , n.

Thus the eigenvalues {F~,~}m~ o f ~ consist o f 0 occurring with multiplicity m~ - 1, and 1 having simple multiplicity. Using this fact, we find for g(i~ . . . . . i,)

(E~ | | E~_-~ ~ | r ~ | E~+? | �9 | E~, ~(i~ . . . . . ,,) = V~(~)~(i~ . . . . , i,)

I" ] l Z = 1 ,~e.,,: /

= l ~ ~ ' i ~ | ~ ( i l . . . . . in). L e,~,~ j

Concomitant ly , we have

r~(1 - ~)~(~ . . . . . ~,) = (s | 1 7 4 ~ : ~ - ~ | ff~-,~ |163

z m 1,iS ~' ,~1 - ~,, . ,

= - .--i.~-.--~-~ ~c.,k ~ ( i z , . . in) k Fk,~

(37)

Let

Ck(e,)~(i~ . . . . , i,) = [Fk(~ ) + F~(1 -- e)l~(iz . . . . . i,).

N o w if i~ = 1 then as (i~ . . . . . i,) -r (1, 1 . . . . . 1), f rom the evaluation o f the Fz,~,, we determine that at least one of the products 1-[?= 1 ~- ~' /zl,i z or 1--[?= 1 ~l /zz m is zero. More- over as m,~, = 0 or 1, we secure ICk(~)l ~< 1. I f ik :~ 1 then ICk(e)] equals 0, 1, ]1 + Ak,~] or [Ak,~l and as I1 + Ak,~d < 1 because o f the individual locus stability we have again ]Ck(~)l ~< 1. For all contingencies, with any given (i~ . . . . . i,), there is always an ~ for which ]Ck(e)] < 1. Appealing to the symmetry property R ( ~ ) = R ( 1 - F,), we deduce that

E R(~) ~ #~C~(~) A g ( i ~ . . . . . i,~) = " k= 1

~. ~'~ k = l

~(il . . . . , in), (38)

displaying ~(il . . . . . i , ) as an eigenvector o f A whose eigenvalue in magnitude, on account of (38) and the stipulations R(~) > 0 and ~ R(,) = 1, is strictly less than one. [ ]


6. Generalized Nonepistatic Selection

We formulate here a natural generalized nonepistatic model extending the multiplicative, additive and neutral structures.

Let Wu, as previously, be the me x m~ marginal fitness matrix associated with locus k, k = 1, 2 . . . . . n and let Eg be the m~ x mk matrix composed of all unit elements. For each vector TI = (~ , ~72 . . . . , ~7~) with ~ = 0 or ~k = 1 consider the Kronecker product matrix

WOI) = W(fl ~) | W(z "~) | | W~ "") (39)

subject to the convention

w2 ' = w~, w 2 ) = E~. (40)

The matrix W0q ) presents a pure multiplicative nonepistatic fitness matrix with marginal selection forces operating at those component loci where ~/~ = l, while no selection differentials are contributed by the other loci involved.

Definition 2. The fitness matrix I" or order 1-I~=~ me x [-I~=1 mk is called a generalized nonepistatic fitness matrix, based on the marginal fitness matrices W1, W2 . . . . . W., if it admits the representation

r - - ~ c(~)w(,1) = ~ c ( ~ ) ( w b , | w,~,~, | | w'.,.,) n n

(41)

where the sum extends over all n-tuples ~q = (71, ~Tz . . . . . n~), -% = 0 or 1. The coefficients {c01)} are free provided the resulting matrix I" exhibits nonnegative entries. Where c(1) = c(1, 1 . . . . . 1) = 1 and all other c(~)'s equal zero we have the pure multiplicative selection regime M of (17). In the case c ( 1 , 0 , . . . , 0 ) = c(0, 1, 0 . . . . . 0) . . . . . c(O, O, . . . , O, 1) = 1 and all other c(vl)'s equal zero then (41) reduces to the additive nonepistatic selection form S as in (18), while if c(O) = c (0 , . . . , 0) = 1 and all other c0q)'s are zero we obtain the neutral selection mode E = E1 | E2 | - - . | E~. Clearly, the family of generalized nonepistatic fitness matrices embraces the mixture

I' = a M + flS + ~E (42)

of the pure multiplicative, additive and neutral selection structures. Another relevant example with three loci of a generalized nonepistatic selection mode is the

case

r = W~ | E~ | E~ + E~ | W~ | W~ (43)

which reflects the phenomenon of multiplicative nonepistasis between the last two loci, but additive nonepistasis between the gene complex consisting of the last two loci and the first locus.

Adapting the methods of Theorems 1, 2 and 4 leads to the conclusion.


Theorem 6. Let F = ~ c(~) WO]) be a generalized nonepistatic fitness matrix based on the marginal fitness matrices W~, W2 . . . . . W~. Let x~ be a polymorphic equilibrium

for Wk with mean f imess ~'k, k = I, 2 . . . . . n. Then (i) .~ = .~1 | -~2 | �9 �9 " | x~ is a H . W . polymorphic equilibrium o f the multilocus system (8) under the generalized nonepistasic selection regime F with mean fitness ff = ~ c(YI)~'01) where

(ii) .~ = 2~ @ 22 | " " �9 @ -r. is locally stable provided the quantities

2 ~ c(~)ff(rl ) ~ R(~)t,~g.kk 1-,k~.~ (44) W

T] e

are less than one in magnitude for all possible specifications o f il, i2 . . . . . i, precluding

il = i2 . . . . . i, = 1.

The eigenvalues (Ak,~, tz~,~) were defined previously. In fact, for any k = l, 2 , . . . , n ()~k,1,/zk.1) = (1, 1) while for i = 2, 3 . . . . . mk, tzk.~ = 0 and (h~,~}~'~ consist of the eigenvalues distinct f rom 1 of the matr ix ff'~ defined in (31).

We highlight next two applications of Theorem 6.

Example 4. Felsenstein [1974] studied a special two-locus two-allele fitness model of the explicit s tructure

BB Bb bb

A A / I - s - u + ksu 1 - s 1 - s - v + ksv \ I

A a [ 1 - u 1 1 - v

aa \ 1 t - u + k tu I - t 1 t - v + ktv ]

(45)

and 0 < s, t, u, v, k < 1. I f we prescribe the marginal fitness matrices

1 - s I t , 1 - u 1 [ E = I 11 W I = 1 1 - - W 2 = 1 1 - v '

then the fitness matr ix associated with (45) can be expressed as

kW~ | W2 + (1 - k)[W1 | E + E | W2] - (1 - k ) E | E

manifestly of the form (41). In this special case the two-locus H.W. polymorphism is ~ | -~2 where

~ = + t' s + t u + v' u + v "


The calculations (44) determine that this H.W. polymorphism is locally stable provided the recombination frequency r satisfies

s t uv r > k - - - -

s + t u + v "

E x a m p l e 5. Consider the fitness matrix (43) W1 | Ez Q E3 + E~ | W2 @ W3 which corresponds to additive effects between locus 1 and the pair of loci 2 and 3 while multiplicative nonepistasis operates between loci 2 and 3. Let the recombination rates be determined by the crossover probabilities r, s, t as defined in Example 2. The conditions (44) assert that the seven eigenvalues displayed below are of magnitude less than one.

(i) [o + o~o.a~.,~1/o, [o + o#.a~, , j /o , [o + o~a~..1/o,

(ii) [(1 - r - s)O + (r + s)ffqA~, a + (r + s)O208~3.~a]/O

(iii) [(1 - r - 0 0 + (r + t)o~a~,,~ + (r + t)0~0~a2,j/o

(iv) [(1 - s - 0 0 + (1 - s - t)O20312,~1a.,~ + 0203(s + t)(a2,,~ + aa.,~)]/O

(v) [(1 - r - s - 0 0 + rO~A~,,~ + rOz0aA2,,2Aa.,a + 020a(sAa,,a + tA9.,,2)]/0.

Assuming single locus stability - 1 < A1,~1. A2,~2, Aa,~3 < 0 then the three eigenvalues in (i) lie in the interval (0, I). The eigenvalues in (ii) and (iii) are in magnitude less than one provided r + s > 0 and r + t > 0 which are the conditions analogous to those in the additive model. We are thus left with the two eigenvalues of (iv) and (v), which are the conditions determined by the across loci interaction of additive-multiplicative viability effects. We readily reduce these latter conditions to the more tractable expression

O < r ~ � 8 9 ~ < s + t < < . � 8 9

where

0 2 0 a A 2 , i 2 , ~ 3 , i z r = m a x

A2,,~,A3.,3 01 + Oz0a(1 - A z . , a ) ( 1 - A 3 . , a ) "

7. The Bisexual Case

In this section we extend the results of the previous sections to the bisexual model represented in (10). Here Pro, I~m and I'r, R r designate the fitness matrices and the recombination distribution associated with the male and female populations, respectively. We establish first a general theorem on bisexual systems, which applies independent of the nature of the selection forces. The special consequences accruing in the context of the generalized nonepistatic selection mode will be discussed afterwards.

The following assumptions apply throughout this section. Suppose ~ is a common polymorphic equilibrium of(8) with respect to the two separate


sexes, operated upon by the fitness matrices I'm and Pt, respectively, which persists independently of the recombination distributions. Let Lr~ and Lf be the local linear approximation transformations of(8) at s with fitness matrices Pm, F I and recombination distributions ~,,, ~f, respectively. We ascertain the following result which generalizes some special cases t reated by Strobeck [1975] and Roux [1978].

Theorem 7. (i) The population state (~ 20), where ~ = 20 = ~, is a polymorphie equilibrium of the bisexual system (10). independent of the male and female recombination distributions. (ii) (2, 20) is locally stable provided the eigenvalues of �89 + �89 r, corresponding to eigenvectors a satisfying (a, e) = O, are less than one in magnitude.

Proof. (i) ~ satisfies the two sets o f equat ions

~v,~(io) ~ ~. R.(~.)wr i~_~) = ~(i~)~(O" _~), a = m (males) or ~ = f (females)

(46)

where ff~ = ( i , F~i} for a = m, f a n d then(2,20) w i t h 2 = 20 = ~ is a po lymorphic equil ibrium o f the bisexual system (10). (ii) We examine first the local linear approx imat ion of (8) at ~ for the two separate sexes. Let g be a per turbat ion vector having small componen t s obeying the constra int <g, e} = 0 where e = (1, 1 . . . . ,1) , so tha t z = i + ~; persists as a frequency vector. Observe that the fitness function obeys

w,~(z) = w~(~ + ~) = <~ + ~, r~(~ + ~)>

= (~, P~s + 2(~, P ~ > + (l~, P~tj> a = m , f (47)

Since s is a po lymorph ic equil ibrium of (8) for the two separate sexes, independent of the recombina t ion frequencies, it is also a po lymorphic equil ibrium where no recombina t ion occurs. Therefore

~ ,~ = ~ o P ~ , ~ = (:~, P . ~ ) , a = m,f (48)

and so F~:~ = ~ e . As (~, e> = 0 we conclude tha t (~, F ~ > = 0. The t e rm (g, P ~ > is quadrat ic in the componen t s of g and therefore up to second order terms w~(z) = }~,. A direct calculat ion produces

�9 �9

Ra(6)w ~ Z(I~)Z(ll_r = ff'~Z(io) + 2 R~(6)w, z(ir162 Ii r I I e

a = m , f (49)

where we used the facts that F~ is a symmetr ic matr ix and R~(6) = R~(1 - 6) for all 6.

Combin ing (47) and (49) we find that the local linear approximat ions ~' = L,,~ and ~' = LIT' at ~ are, respectively,

2 / , . - - R o ( . ) , , o o = m , f (50)


Let x = 2 + g, y = .0 + ~1, where 2 = j3 = ~ and 1~, YI designate small perturbation vectors satisfying (g, e ) = (YI, e ) = 0. Paraphrasing the usual argument we determine that the local linear approximation of (10) at (2, .f) is

~' = (�89 + ~)

~' = (�89 + n)

( ~ , e ) = ( ~ , e ) = 0 (51)

where Lm and Le are defined by (50). The local stability of (2,)3) is determined by the eigenvalues of (51). Using the coordinates a = ~ + ~1, b = g - ~1 equations (51) equivalent to

a' = (�89 + �89

b' = (�89 - �89

( a , e ) = ( b , e ) = 0 (52a)

which possesses the partitioned matrix form

Lm + �89 I

[�89 �89 (52b)

whose eigenvalues consist of a multiplicity of zero values and the eigenvalues of �89 + �89 Thus, the local stability endowments of (2, ~) depend on the eigenvalues of �89 m -t- �89 I associated with eigenvectors a satisfying (a, e) = 0. []

We draw now some corollaries.

Corollary 1. Suppose the fi tness matrices in the two sexes coincide P m = F: = F, but the recombination distributions are different, then the stability conditions for (2, .~), where 2 = ~ = ~, coincide with the stability conditions for ~ in the one-sex model with fi tness matrix F and recombination distribution �89 3v �89

Proof In the case where Fm= F: = F we have ~'m = wl = ~' and in view of (50) the transformation ~' = (�89 + �89 is simply

g'(i0) = ~_ ~. ~ �89 + R:(~)]w z(l~)~(O-.). (53)

This is exactly the one-sex gradient linear transformation (50) at s with the fitness matrix F coupled to the 'effective' recombination distribution �89 + ~:)" []

Corollary 2. I f the recombination distributions are the same in the two sexes Rm = ~: = ~, but the fitness matrices differ, then the stability conditions for (x, y), where 2 = f~ = ~, reduce to the stability conditions for ~ in the one-sex model with recombination distribution ~ and fitness matrix I" = ~mI't + ~?F,~.


Proof. The local linear transformation ~' = (�89 -F �89162 is in this case

g'(io) = 2 ~ ~,. R(e) ~ w,~ ~ + ~ w~ ~ ~(t~)~(t,_ ~). (54)

It is manifest that (54) is the one-sex local linear approximation (50) at :~ corresponding to recombination distribution R and fitness matrix �89 + Pr/ff's]" Multiplying by a constant we can reduce considerations to the fitness matrix P = ~,.F I + ~rFr.. []

Consider next that Fm = ~nc~(vl)W(~l), r s = Y.~cfOi)w(71) are generalized nonepistatic selection regimes based on the same marginal fitness matrices I411, W2 . . . . , W,, allowing for sex dependent coefficients {Cm(YI)} and {cf(~l)}, cf. (41). As- sume that each locus is 'overdominant' , meaning that for each separate locus k, k = I, 2 , . . . , n a polymorphism :~k exists with respect to W~. We then know that the H.W. polymorphic equilibrium .~ | | | 2, exists for the selection models corresponding to I'~ and F,, independent of the recombination frequencies. Accordingly Theorem 7, Corollaries 1 and 2 apply to the bisexual population state (.~, p) where .~ = 33 = 21 | -~2 | | -f:~.

In the case where the coefficients {cm(Yl)} and {cr(vl)} are sex independent and I'~ = I' r = I', Corollary 1 implies that the bisexual H.W. polymorphic equilibrium is stable, provided the one-sex H.W. state .~ | ~2 | | ~, is stable for the selection matrix F and recombination distribution �89 + �89 Hence, the two-sex stability conditions reduce simply to the one-sex conditions (44) with ~ replaced by �89 + �89

Where I~m 5 & I~e but the recombination distributions coincide, i.e. ~m = ~S, then with regard to the stability nature of the H.W. polymorphism Corollary 2 provides the one-sex equivalent selection mode F = ~'~r'm + r This 'effective' selection mode is plainly a generalized nonepistatic selection structure F = _Sn c(rl)W(rl) based on the same marginal fitness matrices such that

col) = ff'ecm(~l) + ff' mCs(~) for all ~1. (55)

We can also write down explicitly the stability conditions of the bisexual H.W. polymorphism in the general setting entailing selection as well as recombination differences. Applying Theorems 6 and 7 and noting that L~ and Lr share their eigenvectors, we obtain

Corollary 3. With generalized nonepistatic selection regimes Fm and P r and recombination distributions ~m and ~I, the H. W. polymorphism (~, .0) where 2 = .~ = ~1 | " �9 �9 | x , is stable provided the quantities

374 s. Karlin and U. Liberman

are in magnitude less than one for all possible specifications of il, iz . . . . . i~ except i 1 = i 2 . . . . . i ~ = 1 .

Remark 3. Based on (56) we observe that if the H.W. polymorphism is stable for the separate male and female models, its bisexual version is also stable. The rate of

approach in the bisexual model is always as large as the min imal rate among the

two separate models.

Remark 4. The si tuat ion under which there exists a polymorphism independent of the recombinat ion frequencies c o m m o n to the sexes is no t representative only of

nonepistat ic selection regimes, but also works under generalized symmetric

selection; see Kar l in [1978, 1979].

References

Bodmer, W. F., Felsenstein, J.: Linkage and selection: Theoretical analysis of the deterministic two-locus random mating model. Genetics 57, 237-265 (1967)

Felsenstein, J.: Uncorrelated genetic drift of gene frequencies and linkage disequilibrium in some models of linked overdominant polymorphisms. Genetic Res. 24, 281-294 (1974)

Geiringer, H. : On the probability theory of linkage in Mendelian heredity. Ann. Math. Stat. 15, 25-57 (1944)

Karlin, S. : General two-locus selection models: some objectives, results and interpretations. Theor. Pop. Biol. 7, 364-398 (1975)

Karlin, S.: Theoretical aspects of multilocus selection balance I. In: Mathematical Studies in the Life Sciences (S. Levin, Ed.), Amer. Math. Soc. 1978

Karlin, S., Feldman, M. W.: Convergence to equilibrium of the two-locus additive viability model. J. Appl. Prob. 7, 262-271 (1970)

Karlin, S., Liberman, U.: The two locus multiallele additive viability model. J. Math. Biol. 5, 201-211 (1978)

Karlin, S., Liberman, U. : Central equilibrium in multilocus systems I. Generalized nonepistatic selection regimes. Genetics, 91 : (1979a)

Karlin, S., Liberman, U.: Central equilibrium in multilocus systems II. Bisexual models. Genetics, 91: (1979b)

Karlin, S., Campbell, R." Analysis of central equilibrium configurations for certain multilocus systems in subdivided populations. Genet. Res. 32, 151-169 (1978)

Karlin, S." Principles of polymorphism and epestasis for multilocus systems. PNAS, Jan. 76. 1979

Kingman, J. F. C.: A mathematical problem in population genetics. Proc. Camb. Phil. Soc. 57, 574-582 (1961)

Moran, P. A. P.: On the theory of selection dependent on two loci. Ann. Hum. Genet. 32, 183 (1968)

Strobeck, C.: The two-locus model with different recombination values in the two sexes. Adv. in Appl. Prob. 7, 23-26 (1975)

Roux, C. Z. : Hardy-Weinberg equilibria in random mating populations. Theor. Pop. Biol. 5, 393-416 (1974)

Roux, C. Z.: Sex differences in linkage between autosomal loci. Theor. Pop. Biol. 13, 295-303 (1978)

Received September 11, 1978

representation of nonepistatic selection models and analysis of multilocus hardy-weinberg...

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