the two-locus multi-allele additive viability model

J. Math. Biology 5, 201-211 (1978) Journal of

C by Springer-Verlag 1978

The Two-Locus Multi-Allele Additive Viability Model*

Samuel Karlin 1 and Uri Liberman z

1 Mathematics Department, Stanford University, Stanford, California, USA-and The Department of Pure Mathematics, The Weizmann Institute of Science, Rehovot, Israel Mathematics Department, Stanford University, Stanford, California, USA-and Statistics Department, Tel-Aviv University, Ramat-Aviv, Israel

Introduction

Exploiting the concept of sums of Kroneeker product of matrices of simple form, analyses of a number of classes of multilocus multiallele regimes have been re- ported (Karlin, 1977a, 1977b). In this paper we use this approach in the develop- ment of the complete dynamic behavior of the two-locus multi-allele selection balance model involving non-epistatic additive viabilities. It is established that with positive recombination where the viability array at each locus in isolation would maintain a stable polymorphism, then the two locus system possesses a unique globally stable Hardy Weinberg equilibrium expressing the gamete frequencies as products of the corresponding gene frequencies. This extends earlier work of Karlin and Feldman (1970) who dealt with the two-locus two-allele model.

1. Kronccker Products of Matrices

We review first for ready reference the concept and some of the elementary properties of Kronecker product of matrices.

Definition 1. Let A be an m • n matrix and let B be an l • k matrix. The Kronecker product of A and B written A | B is the partitioned ml • nk matrix

allB, a l ~ B , . . . , a j , B \

a21B, a22B, , a2 ,B | A | 8 = ~ ]. (1)

\amlB, amzB, , amnB/

The operation A | B enjoys the following properties: (.4 + B) | C = A | C + B | (A | 1 7 4 @ ( B | A | C ) = A | |

* Research Supported in Part by N.I.H. Grant USPHS 10452-13, N.S.F. Grant, MPS71- 02905-A03, and The Volkswagen Foundation Grant No. 573.

Reprinted Issue without ~ Advertisements

0303-6812/78/0005/0201 / $02.20

202 S.K.~lin and U. Liberman

(L4) | B = A | (AB) = L4 | B. Specializing the definition, the Kronecker product of the vectors a = (al . . . . . am) and b = ( b l , . . . , b,) written a | b is the mn vector

a | b = (albl, alb2 . . . . . a l b , , . . . , amba . . . . . amb,). (2)

The main applications of Kronecker products are based on the identity (A | B). (C | D ) = AC @ BD, provided the matrix multiplications are feasible. Let (x, y) denote the scalar product of the two vectors x and y. We have

(.4 | B)(a | b) = Aa | Bb

(a | b, a' | b') = (a, a')(b, b').

Let ~ be a polynomial in two variables ~ and ~7 with real coefficients, namely ~b(~, ~/) = Y.~s=0 c~j~ j and for any two matrices A, B we form

ao

~(A, B) = ~ c~jA ~ | BJ. p =

Theorem 1. Let A~, . . . , Am be the eigenvalues of the m x m matrix A and let Iza, l~2 . . . . , t*, be the eigenvalues of the n x n matrix B, then the eigenvalues of the mn x mn matrix ~ (A ,B) are the mn numbers ~(A,,/zs), r = 1,2 . . . . . m; s = 1, 2 , . . . , n .

Corollary 1. i) The eigenvalues of A | B are the mn numbers {)q/~s}.

ii) I f Aa = Aa and Bb = tzb, then (tl | B)(a | b) = AtL(a | b). Thus, i f a and b are eigenveetors of A and B for the eigenvalues A and tz, respectively, then a | b is an eigenvector of A | B with eigenvalue At*.

Definition 2. Let a and a' be two m-vectors. The Sehur product of a and a' written a o a' is the m real vector

a o a' = (a~a~, a' a2 9. . . . . . area').

The following relation joins the Kronecker and Schur product operations

(3)

(a | b) o (a' | = (aoa') @(bob').

2. The Model

We consider the standard diploid population evolving under the influence of random mating, viability selection, recombination and Mendelian segregation. We assume that the trait in question is determined by two loci such that the viability effects across the two loci are additive. Accordingly, let A1 . . . . , Am designate the alternative alleles at the first locus and B1 . . . . . B, those at the second locus. Let U = (u~k) and V = (vjz) be the viability matrices associated with the first and second locus, respectively. Assuming additive selection the viability value of the genotype A~B/AkB~ is the sum of the fitness attributes of A~A~ and BIB,, namely

Two-Locus Multi-Allele Additive Viability Model 203

u,~ + vjz. Let E be a matrix whose elements are all unity, then it is easily checked that the viability matrix M of the two-locus system is

M = U | 1 7 4 (4)

(We should write M = U | + E2 | V where E~ is n • n and/?2 is m x m but we omit the indices as no confusion is likely.)

Example. Suppose there are two possible alleles at each locus and let

A2 B1 B2

Uig" 1 V_.= Bi ( vii vi21 u22/ B2 \V2i V22]

ml

A1 (uii U = A2 \u2i

then

A1BI

A~B2 M =

AzBI

A2B2

and in fact

M = (Un \U~i

A zBI A1Bz A2B1 A2B2

( uzl + vn ull + vl2 ul2 + v= u12 + v12\

Uil + Vzi Ull + V22 g12 + Vzi Ui2 + V22]

U21 JF 011 /'/21 + V12 U22 + Vli Z/22 ~- V12 I

U21 Jr V21 U22 "~- V22 1/22 "~- V21 /"/22 "q- V22]

1 1 (V l zV l2 t . U22/ \V21 v22/

Let x,j denote the frequency of the A~Bj gamete with x summarizing the vector of gamete frequencies, i.e.

x = (xii, x12,. �9 xi , , �9 �9 xml, xm2 . . . . . x,,,)

where

x~ ~> 0 and ~ x , j = 1.

Let A denote the simplex of all frequency vectors x.

Let r, 0 ~< r ~< 1, denote the recombination frequency between the two loci and let Tr (displaying the dependence on the recombination parameter) be the transformation law x ' = T#c evaluating the frequency vector x' in the next generation in terms of the frequency vector x in the present generation, that appropriately accounts for the action of random mating, additive selection, recombination and Mendelian segregation.

Employing the notions and devices of Kronecker and Schur products, it is readily verified that we can represent Tr succinctly as described in the following theorem.

Theorem 2. The transformation T~ is given by

Wx' = (I - r )xo(U | E + E | V)x (5)

+ r(U | Ix) o (I | Ex) + r(E | Ix) o (I | Vx)

204 S. Karlin and U. Liberman

where I stands for the Identity matrix and

W = W(x) = (x, Mx) = (x, U | Ex + E | Vx)

is the mean fitness function for the population state x.

3. The Fitness Function Increases

For any given frequency vector x let

n

p ~ = f r e q u e n c y o f A ~ = ~ x u i = 1 , 2 , . . . , m

q j = f r e q u e n c y o f B j = ~ x u j = 1,2 . . . . , n

denote the allele frequencies involved at the two loci and designate by P = (Pl . . . . . pr~) and q = (ql . . . . . q~) the vectors of allele frequencies associated with each of the two loci. The following lemmas are straightforward.

Lemma 1. The mean fitness expression is a function only of" the allele frequencies. Moreover,

W(x) = WI(p) + W2(q) (6)

where W1 and W2 are the marginal fitness funetions associated with each of the separate loci.

Proof. The fitness function W(x) is given by

W(x) = (x~ U @ Ex + E @ Vx).

But

U | Ex = U p | E | Vx = e | Vq

where the e vectors consist only of unit components. Therefore

W(x) = (x, Up | + (x, e | Vq)

= (v, Vp) + (q, Vq)

" = w ~ ( r ) + W2(q ) .

Lemma 2. The transformation equations of the allelic frequencies induced by T, coincides with that induced by To. Namely

T~(p, q) = To(p, q)

where To is the transformation involving no recombination events.

Proof

U | Ex = Up | e E | Vx = e | Vq

I | 1 7 4 E | 1 7 4


The t rans format ion TT given in (5) reduces to

Wx' = ( 1 - r )x o ( Up | e + e @ Vq)

+ r (p | e) o (U | lx) + r (e @ q) o (1 | Vx)

and the new allele frequencies at the first locus are given by

Wp~ = (1 - r)p,(Up), + (1 -- r) ~ (Vq)~x u + rp,(Up), + r ~ (Vq)sx u .*'=1 1=1

o r

Wp'~ = p,(Up), + ~ (Vq)jx u i = 1, 2 . . . . , m (7) J = l

and similarly for the second locus m

Wq'y = qs(Vq) s + ~ (fp)~x u j = 1, 2 . . . . . n. (8) |=1=

As W is independent o f r we have tha t T~(p, q) = To(p, q).

When r = 0 no recombina t ion occurs and the model is equivalent to a one locus m �9 n allele model. Hence To is basically a multiallelic one locus t ransformat ion and the mean fitness funct ion serves as a Lyapounov function for the t ransforma- t ion (e.g. Kingman , 1961). This means that the mean fitness of the popula t ion increases over successive generat ions with constancy occurring only at equilibrium points. Thus when r = 0,

W(Tox) >1 W(x) with equali ty iff x = Tox.

We prove (following Ewens (1969))

Theorem 3. For any 0 ~ r <~ I and any x, W(TTx) >1 W(x). Equality holds iff ToX~- X.

Proof. By L e m m a 1 the fitness funct ion is a funct ion of the allele frequencies. Hence for any 0 ~< r ~< 1

W(Trx) = W(T,(p, q)).

By L e m m a 2, T~(p, q) = To(p, q), so

W(T,.x) = W(To(p, q ) )= W(Tox).

But W(Tox) >I W(x) with equali ty iff x = Tox. Therefore

W(T,x) >I W(x)

with equali ty iff x = Tox.

4, Polymorphie Equilibria of Tr We look for po lymorph ic equilibria (internal fixed points) of the t ransformat ion Tr. Th roughou t the discussion we assume

Assumption 1. With respect to each of the two loci there exists a unique stable polymorphic equilibrium.


Let/~ represent the stable polymorphism at the first locus, and ~ that at the second locus. Then

o I: 0 = Wl ) 0, o = (9)

and each is globally stable in the corresponding simplex, respectively.

Let L be the set of frequency vectors for which the allele frequencies at the two loci coincide with the corresponding equilibrium allele frequencies.

L = { x z : e ( x ) = q ( x ) = #} . (10)

Theorem 4. i) The polymorphic equilibria o f To coincide with the interior o f L.

ii) L is invariant under T~ for any 0 <<. r <<. 1.

Proof. i) The transformation To is

Wx' = x o (Up | e + e | Vq) where p = p(x), q = q(x).

Let x be a polymorphic equilibrium of To so

W e | Up | e + e | Vq.

As W(x) = WI(p) + W2(q) by Lemma 1 then

Up = Wl(p)e, Vq = Wz(q)e.

But these are exactly the equations defining the polymorphic (allelic) equilibria at the Separate loci. It follows, by virtue of Assumption 1 that

p =/~, q = # implying x e L.

ii) According to (7) for any 0 <~ r <~ 1

Wp~ = p,(Up)t + ~ (Vq)sx~j i = 1, 2 . . . . , m. j = l

If x e L then r = / ~ , q = q , U/S= l~le, V ~ = I~2e and W = l ~ = I~1 + 1~2 where I~ 1 = W1 (/$), 1~2 = W2 (~). Thus

t = 1

andp~ = /~ , i = 1, 2 . . . . , m. Similarly, we deduce q~ = ~j for all j = 1, 2 , . . . , n. Therefore if x e L also Trx e L.

As L includes all polymorphic equilibria of To, it follows by Theorem 3:

Corollary 2. The polymorphic equilibria o f TT, for any 0 <<. r <~ 1, are included in L.

Let ~ =/~ | ~. Manifestly ~ e L and it is readily verified that .~ is a polymorphic equilibrium of T, for any 0 ~< r ~< 1. ~ is the Hardy-Weinberg type polymorphism as it represents each gamete frequency as the product of its allele constituent frequencies.

Two-Locus Multi-Allele Additive Viability Model

Theorem 5. The Hardy-Weinberg polymorphism ~ = ~ | morphic equilibrium of T, for any 0 < r <~ 1.

Proof. Let x be a polymorphic equilibrium of Tr, then by (5)

Wx = (l - r ) x o ( U | Ex + E | Vx)

+ r(U | Ix) o (1 | Ex) + r(E | Ix) o (I | Vx).

As x ~ L, p(x) = ~, q(x) = ~.and

U | Ex = U~ | e = ff'le | e, E | Vx = e | V~ = ff'2e | e

[ | Ex = ~ | e, E | l x = e | ~, W = Iff = I~l + IfV2.

So

207

is the unique poly-

5. Local Stability of

Theorem 6. I f ~ and ~ are stable, then ~ = ~ | ~ is locally stable with respect to Tr for any r > O.

Proof. The local stability properties of :~ are determined by the eigenvalues of the gradient matrix (local linear approximation) of the transformation TT at :~. Let x = ~ + e be a frequency vector close to k, in the manner that the components of e are 'small ' and their sum, (e, e @ e), equals zero.

The fitness function W(x) satisfies

IV(x) = (x, Mx) = (~, M$) + 2(e, M~) + (e, Me).

As ~ is a polymorphism of To maintaining thereby M~ = ff'e | e which implies (e, M~) = 0. Accordingly,

W(x) = if" + (e, Me)

~ x = (1 - r)ff'x + r([, | | Ix) + r(e | 1 7 4 Vx). ( l l )

As r > 0, we obtain

r = (~ | e) o (U | Ix) + (e @ # ) o ( I @ Vx).

Let 0 and 12 be the matrices

0 = ~ o u = (p,u,~) ~ = 4 o v = (4;vjz),

then

H/x = ( O | I | ~)x

and x is a strictly positive right eigenvector of the nonnegative irreducible matrix 1.7 | I + I | ~ (in fact all the elements of its square are strictly positive) with a positive eigenvalue if'. As ~ is a polymorphism of T, for any 0 ~< r ~< 1, :~ is also such an eigenvector.

By the Perron-Frobenius theory of positive matrices if" is necessarily the largest eigenvalue and its unique (up to scalar multipliers) right eigenvector is :~.


and as all the components of 8 are small the proper approximation to W(x) apart from quadratic terms in 8 is if/. Hence the linear approximation of TT in the neighborhood of .r has the form

1~8' = (1 - - r ) l ~ 8 + (1 - r ) ~ o ( U | E | V)8

+ r(U | 18) o (I | E~) + r(U | I~) o (I | Es) (12)

+ r ( E | 1 7 4 V~) + r ( E | 1 7 4 Vs).

Let • = ((.~o W)l f&) =do, IIP, u,dr~lll, 17 = ( ( i o V)l f /~) =do~ Ilq,v,,/~/~ll. Founded on one locus theory and the stability nature of/9 and q, we secure

a ~ = ~ I70=0 and for other eigenvalues,

then - 1 < ~, < 0, - 1 < t~ < 0 and (~, e) = (71, e) = 0.

Also the sets of eigenvectors {/~, ~ . . . . . ~,~- 1} and {0, 711 . . . . . 7h_ 1} constitute bases for the Euclidean spaces E m and E ~, respectively. Hence the collection of the mn - 1 vectors of the form/~ | ~, ~ @ 0 and ~ | ~ constitute a basis for the subspace of vectors 8 in E ~ satisfying (8, e | e) = 0. Moreover, by virtue of Corollary 1, we find that each of these vectors is an eigenvector of the linear mapping (12). The corresponding eigenvalues are

Eigenvector Eigenvalue

~ | 1 +tz~

g | 1 + ~ 2 W

(1 - r )~ + r ~ + r ~

Note that - 1 < A < O , - 1 < ~ < 0 implies O < 1 + A ~ . / W < 1 and o < 1 + ~ 1 / ~ < 1.

Also if r > 0 then [(1 - r)ff" + r)tl~l + r/~l~2] < l~. Therefore where r > 0 then all the eigenvalues of the gradient map (local linear approximation) are in magni- tude less than one and accordingly .r =/~ | ~ is locally stable.

6. Global Convergence for r > 0

We make use of the following theorem which is a standard result of a Lyapounov type (e.g. see Karlin and Feldman (1970)).

Theorem 7. Let T be a continuous transformation acting on a compact set ~ of some Euclidean space into itself. Let C1, C2 . . . . . Cz be closed disjoint subsets of J such that

i) T(CJ c G for any i = 1, 2 . . . . . I.


ii) W(Tp) >! W(p) for any p ~ J" with equality iff p ~ Ul= l Cifor some continuous real valued function W. Then for each p ~ 3- the sequence {T~p} converges to one of the sets C~, i = 1, 2 . . . . . L

Assumption 2. Corresponding to any subset of alleles, at each of the two loci, there exists a unique equilibrium point which is stable, relative to its set of alleles.

The transformation Tr maps A, the simplex of frequency vectors, continuously into itself. The fitness function W increases with T , W(Trx) > W(x), unless x is a fixed point of To. All the relevant fixed points of To are included in closed sets of the form of L. Each of these sets is invariant under T , and due to Assumption 2 there is a finite number of such invariant closed sets. Therefore

T h e o r e m 8. Let r > O. Then for any frequency vector x ~ A, the sequence {T~x}~~ 1 converges. It converges either to L or to some invariant closed set of equilibrium points of To on the boundary of A.

When convergence to L occurs we show

Theorem 9. Let r > 0 and assume that ~ and ~ are stable. I f

T~x ~ L then in fact T~x ~ ~ = ~ | ~.

Proof. We introduce the notation

x ~ = T~x, p~ = p(x~), q~ = q(x~), k = O, 1, 2 . . . . .

Then

T k r x ~ L iff p ~ W p and q ~ O .

Represent x ~ as x k = p~ | qk + g~ where

m

<~u = 0 1=1

e = 0

for a l l j = 1, 2 , . . . , n

for all i = 1, 2 , . . . , m.

As x ~ ~ L then for k large enough p~ ~/~, q~ ~ ~ and x k ~/~ | ~ + gk. Also

W k = W(x ~) = W(p ~, q~) ,~ I~. Therefore if k is sufficiently large

if/g,,+1 = ff/8~ + (1

+ r(U |

+ r(E |

wh~ere

~ ~--=-~ O.

So

-- r)W~ k + (1 - - r ) : ~ o ( U | E | V)~j k

I ~ ) o (Z | E~) + r(U | I~) o (S | E~ ~)

Ig ~) o (I | 112) + r (E | I.~) o (I | Vg u) (13)


where .4 is a linear transformation. Moreover, A is exactly the linear approximation (12) of T~ in the neighborhood of :~. Assuming ~0 and to be stable we know from Theorem 6 that for r > 0 all the eigenvalues of A are in absolute value less than one. Therefore there exists a norm I1" I1 in E m"-I for which

11.4 11 < pll~ll for any [ # 0

with 0 < p < 1. In this norm

ll § II ll § pll ll.

Hence as 0 < p < 1, limk., o0 11 ~kl[ = 0, [~ ~ 0 and x ~ ~_. co > ~-

7. Convergence to the Unique Polymorphic Equilibrium

Theorem 10. Let r > 0 and assume that ~ and ~ are stable. Then for each internal frequency vector x, T~rx k-. | > ~"

Proof. We show that starting with any internal frequency vector x the sequence {T~x} cannot converge to any boundary set of equilibrium points of To. Hence T~x ~ | > L and by Theorem 9, T~x ~ ~.

Let/~ and ~ be equilibrium points of U and V, respectively. Let • be the boundary set of equilibrium points determined by ~ and ~, namely

L = (x A:p(x) = q ( x ) =

Let I = { i :~ = 0} and J = {j: ~j = 0}. As L is a boundary set, we can assume without loss of generality that I # ~ (the empty set).

By (7)

I V ' ~ ., p~ = p~(Up)~ + (Vq)jx,~ i = 1, 2 , . . m. J=:t

I f x is close to L then for any i ~ I , p~ and x,j, for aU j = 1, 2 . . . . , n, are small. Therefore up to quadratic or smaller terms

#p~ = p,(U/~), + ~ (V~)~x,j i ~ I J = l

where t~ = W1(/3) + Wa(#).

But ~ is an equilibrium point of V hence (Vq)j = W2(q) for all j r J while by Assumption 2, (V#)j > W2(#) for j s J. So

l~p~ >1 p,(U/~), + W2(q)p, i e I

or up to quadratic or smaller terms

Pi >1 (U/~), + W2(q')p, i ~ I . (14) #

Using again Assumption 2 we know that for all i ~ I, (U/~)~ > W~ (~) and from (14) we conclude that p~ for all i e I increases when small and non-zero. Now if x is


internal, T~x is internal for all k, p~ # 0 for all k and i and thus {Tk, x} cannot converge to Z.

As the proof of Theorem 10 works also for the case r = 0 we have

Corollary 3. For each internal frequency vector x, T~x ~_. | > L.

8. Some Concluding Remarks

The nice behavior of the two-locus model with additive viabilities is, unfortunately, not characteristic of other viability models. In general it is rare that a unique stable polymorphism is maintained. Although this paper concentrates on the two- locus multiple allele situation, it appears that the analogous structure is shared by additive viability models with any number of loci and alleles. Actually, the proofs of Theorems 2, 3, 4, 8 carry over for the general case, mutatis mutandis. The key point in the proof of Theorems 5 and 9 is that the transformation Tr is linear on L. Unfortunately this property is no longer available with more than two loci. In Karlin and Liberman (1977) we generalize Theorem 6 and show that under analogous conditions, the Hardy-Weinberg type polymorphism is locally stable.

References

Ewens, W. J.: With additive fitnesses the mean fitness increases. Nature 221, 1076 (1969) Karlin, S. : Theoretical aspects of multilocus selection balance, I., in Mathematical Studies in the

Life Sciences (ed. S. Levin), Am. Math. Soc. (1977a) Karlin, S. : The studies of selection in multilocus systems III. Selection depending on aggregate

genotype heterozygosity. Proc. of Iowa Conf. on Quantitative Genetics, Iowa Univ. Press (1977b)

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. : Representation of some non-epistatic selection models and analysis of Hardy-Weinberg equilibrium configurations, to appear

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

Received May 26, 1977]Revised July 25, 1977

the two-locus multi-allele additive viability model

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