Ann. Hum. Uenet., Lond. (1977), 41, 219

219

Printed in Great Britain

Selection at a multiallelic locus : Feller’s transformation BY A. W. F. EDWARDS

Department of Community Medicine, University of Cambridge INTRODUCTION

A common representation of the gene-frequencies a t a triallelic locus is by trilinear coordinates, in which the three coordinates of a point in the plane are taken to be the perpendicular distances from the point to each of the three sides of an equilateral triangle, known as the reference triangle. If the reference triangle is chosen to have unit height, the sum of the three coordinates is unity, and we have a perfect representation of gene-frequencies, which also sum to unity. Alternatively we may use the related concept of areal coordinates, in which the three coordinates of a point are given by the three areas into which the lines from the point to the three vertices divide the reference triangle. If the reference triangle is again equilateral, it is easy to show that trilinear coordinates and areal coordinates are the same, except for a constant factor depending on the area of the triangle. The two systems are examples of homogeneous coordinates, and further details may be found in Edwards (1977). An advantage of areal coordinates which we shall exploit is that they are invariant under linear transformation, except for a constant factor, since under linear transformation all areas are transformed by the same factor. Since the mean viability w of a population of diploid organisms in which each genotype aiai has an associated viability wii ( = wji) is a quadratic function of the gene frequencies, lines of equal mean viability at a triallelic locus will be conic sections in the plane in a homogeneouscoordinate representation. Under certain conditions on the matrix of viabilities W = {wij}, given in Edwards (1977), these conic sections will be ellipses. Feller (1969) proved that if areal coordinates are used and the reference triangle is chosen to have sides related in a certain way to the wij, the ellipses for varying w are all transformed into concentric circles, resulting in a much-simplified diagram with many interesting properties. Feller’s proof, however, involves Cartesian rather than homogeneous coordinates, and is thus inelegant through its lack of symmetry. Edwards (1977) supplied the corresponding proof using areal coordinates. Neither of these proofs indicates the precise linear transformation involved; Feller’s is not well suited to an extension to the case of an arbitrary number of alleles, and mine, though it may readily be extended, relies on a theorem in areal coordinates which is no longer widely known; moreover neither proof provides a clear intuitive understanding of the reason why the particular choice of reference triangle renders otherwise elliptical lines circular. The object of the present paper is t o explain why the ellipses become circles and t o provide a proof in the more familiar language of linear algebra rather than in the language of homogeneous coordinates. NOTATION AND EARLIER RESULTS

We use the notation of Edwards (1977), and here repeat only the most central definitions and results. References to Edwards (1977) are given as follows: (E, p. 1). Let 15

aij= 2wij-wil.-wj,

( i +j), H G B

41

A. W. F. EDWARDS

220

the c6,thus being measures of pairwise dominance in an additive sense. The mean viability may be written w = pTWp, (1) where p T is the vector of gene frequencies (p, p, p3).It may then be shown (E, p. 52) that, for varying w,( 1 ) is a concentric family of homothetic conic sections (i.e. of the same shape and orientation), being ellipses, parabolae or hyperbolae according to whether 4 3

+ + 4 3

- 2 ~ 1 g31 2 - 2 a 2 3 ala - 2a23~

1

3

is less than, equal to, or greater than zero. It follows from this (E, p. 54) that the conics are ellipses if and only if the uij are of the same sign and there exists a triangle with sides Li such that Lf : La : Li = : CTl3 : U12. Feller’s result is that these ellipses may be rendered circular if, using areal coordinates, the reference triangle is chosen with sides L,, L, and L3 (E, p. 56). We may think of this in terms of a linear transformation: the linear transformation which maps an equilateral triangle into a triangle with these sides will simultaneously map the ellipses into circles. Our first task is to explain this result. EXPLANATION OF FELLER’S RESULT

We will refer to the ellipses of equal mean viability as ‘contours’, thus evoking the picture of a hill or depression. The expanded form of (1)is = wll Pf

+ W 2 2 Pi + w33 P: + 2w23

P2 P3

+ 2w13 P1 P3 + 2w12 P1 P29

which may be written = a23p2

P3 + u13

Pl P3

+ g12 PI P2 + wllP1 -k w 2 2 P2 + w33p3-

(2)

With reference to an equilateral triangle, consider the values of w along one side of the triangle, sayp3 = 0. On that side w = ~l2PlP2+wllPl+w22P2 = g 1 2 P1 ( 1 -PA +w11 P l + w22(1 -PJ since then p1+p, = 1. Thus along the side p3 = 0 the surface is a parabola whose quadratic term has coefficient - g12. Similarly along p, = 1 we find a parabola with coefficient -g13,and along p1 = 0 a parabola with coefficient - g 2 3 . The measure with respect to which these curves are parabolae is straightforward linear measure along the respective side of the triangle, from zero a t one vertex to unity at the other. If a linear transformation is applied to the triangle such that the length of a particular side is multiplied by a factor c, then with respect to the original measure the coefficient of the corresponding parabola will be divided by a factor c2, as may easily be seen from the form of its equation. Consequently, if the length of the side p3 = 0 is multiplied by Jlalzl,that of the side p2 = 0 by JIrl3(, and that of the side p, = 0 by ,/la,\,the transformed parabolae will each have coefficient + 1 or each have coefficient - 1. I n other words, the parabolae will be congruent. Now one of the characteristics of a quadratic surface which has circular contours is that along any line its curve is a parabola, and all such parabolae are congruent. Conversely, if a quadratic

Selection at a multiallelic locus

221

surface exhibits congruent parabolae along three distinct lines no two of which are parallel, it has circular contours. But that is precisely what we have achieved by the linear transformation of the triangle introduced above. With respect to the new triangle, therefore, which has sides L,, L, and L, where Lq:Li:L: = v2,:c13: cr,,, the original equation (2) will, for any allowable w, represent a circle, since areal coordinates are unaffected by linear transformation. The argument can be extended to any number of alleles if the analogous homogeneous coordinate system to that of areal coordinates in two dimensions is used (thus for four alleles, and three dimensions, the four coordinates are given by the volumes of the four tetrahedra formed by the subdivision of the reference tetrahedron by the planes joining the point to the six edges). This completes our demonstration of Feller’s result: we see that it is achieved by adjusting the length of each side so that all the corresponding parabolae are congruent, from which the rest follows. For a formal proof, however, we prefer to use the methods of linear algebra which will reveal the nature of the linear transformation involved.

PROOF OF FELLER’S RESULT

If the contours are elliptical the surface of mean viability is either concave (stable equilibrium, d s positive) or convex (unstable equilibrium, d s negative) (E, ch. 5). I n the unrestricted space with orthogonal axes p , , p 2 ,p 3 , pTWp=w

(w>O)

is a central quadric (since changing the signs of all the pimakes no difference) and, according to standard theory, will be an ellipsoid if W has three positive latent roots, a hyperboloid of one sheet if W has just two positive latent roots and a hyperboloid of two sheets if W has just one positive latent root. I n the case of an ellipsoid cut by the plane Xpi = 1, as w increases the ellipse grows larger and hence the surface of mean viability is convex; in the case of a hyperboloid of one sheet the intersection is a hyperbola, a situation we do not consider; and in the case of a hyperboloid of two sheets cut by the plane, as w increases the ellipse grows smaller and hence the surface of mean viability is concave. We take the convex case first, where the equilibrium is unstable, the d s negative, and the quadric an ellipsoid. Accordingly, W must have positive latent roots and may be written W = BA2BT,

where B is an orthogonal 3 x 3 matrix and A2 is a 3 x 3 diagonal matrix. Thus we may write W = BARBT = BABTBRBT= A2,

where Ais the symmetrical matrix B ABT,the elements of A being taken positive for convenience. Then the linear transformation p’ = Ap maps the quadric pTW p = w into ( ~ ‘ ) ~ A - l w A - l p=’W,

which is ( ~ ’ ) ~=pw,’ a sphere centred on the origin. The homogeneous coordinate system’s reference triangle has vertices (1, 0, 0 ) , (0, 1, 0 ) , (0, 0 , l ) which A maps into (a,,, u12,a13), (al2,a22,US,), (al3,a23, a,,), where ai3 is the (i,j ) t h element of A. Since any plane which cuts a sphere does so in a circle, A has transformed the elliptical contours 15-2

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A. W. F. EDWARDS

in the plane of the original triangle into circular contours in the plane of the new triangle, and since a linear transformation leaves areal coordinates unchanged (except for a multiplier), all that remains t o be shown is that the new triangle has sides L,, L2,L3, where

Thus

Similarly (r13 = - L; and u12 = - L;and the proof is complete for negative d s . I n the other case (stable equilibrium, u ' s positive) the function

pTWp

=

w

(1 bis)

represents a central hyperboloid of two sheets, and W has one positive and two negative latent roots. Consequently it cannot be linearly transformed into a sphere. However, as we show below, we can construct a central ellipsoid which intersects the plane p1+p2+p3= 1 in the same ellipse as does ( l ) ,and similarly a family of ellipsoids corresponding to the family of hypcrboloids for varying w. The contours in the plane will be the same, though relabelled, and mill correspond to an unstable case, to which the above treatment may then be applied. For consider the central quadric

PT(kU-W)p

=

k-w,

(3)

where U is the 3 x 3 matrix of ones and k is a positive number larger than the largest element of W. From (3) k p T U p - p T W p = k-w, and in the plane in question pz'l = 1, p T U p = 1, whence p T W p = w. Thus in this plane every elliptical contour k - w of (3) coincides with the contour w of ( l ) , and as the contours of ( 1 ) are decreasing outwards in w from the centre, the contours of (3) are increasing outwards in k - w, and (3) is a family of ellipsoids for varying w.

Selection at a multiallelic locus

223

The transformation A that maps (3) into the sphere pTp = k - w is now such that

Now v23

=

2w23-

=

-2(k-w,,)+(k-w,,)+(k-w33)

= -2

w22-

w33

-

-

~ 1 2 ~ 1 23 ~ 2 2 ~ 2 32a2,a3,

+a!z +ai2+ ui3+ a%+ a;, +ai3 -

(a12-a13)2

+ (a22-a23)2+

(a23-a33)2

= L!.

Similarly gI3= Li and v12= Li and the proof is complete for positive d s . By similar methods any number of alleles may be treated. Hyperellipsoidal contours in the gene-frequency space will occur only when all the latent roots of W are positive (unstable equilibrium, d s negative) or when just one is positive and all the others negative (stable equilibrium, d s positive: see Mandel’s theorem, E, p. 43), and these are the extensions of the two cases we have treated. The transformation A that we have used does not itself constitute a linear transformation of homogeneous coordinates, because it does not map the plane p 1 + p 2 + p 3= 1 into itself; for that it would be necessary to use a further, orthogonal, transformation with matrix C such that the product CA was a matrix whose columns each summed to unity. However, C is not then uniquely defined, and in any particular case it is simpler to proceed from first principles using Feller’s result, as follows. The transformation p‘=f S t

: ).

0

0

1-s-t

maps the plane p1 + p 2 + p 3 = 1 into itself and maps the vertices of the reference triangle into (1,0,0), (0, l , O ) , ( s , t , 1 -s-t).Thusonlythevertex (0,0, 1)ismoved.Ifthenewreferencetriangle is to have sides with the required lengths then r g 2 3 = 8 2 + ( 1 - t ) 2 + (1 - s - t ) 2 ,

r v I 3= ( l - ~ ) ~ + t ~ + ( l - s - t ) ~ , r v 1 2= 2 ,

where r is a constant ( = 2,/u12). These equations may readily be solved for s and t to obtain the transformation.

224

A. W. F. EDWARDS SUMMARY

An explanation of Feller’s result enabling the contours of mean viability at a triallelic locus to be rendered circular is offered, and a proof given which does not involve the direct use of homogeneous coordinates. In a subsequent paper we propose to use the fact that the same transformation also maps the contours of equal average effect for each gene into circles to establish further results. REFERENCES

EDWARDS, A. W. F. (1977). Foundations of Mathematical Qenetim. Cambridge University Press. FELLER, W. ( 1 969). A geometrical analysis of fitness in triply allelic s y s t e k . Mathematid Bioscielzces 5, 19-38.

Selection at a multiallelic locus: Feller's transformation.

Ann. Hum. Uenet., Lond. (1977), 41, 219 219 Printed in Great Britain Selection at a multiallelic locus : Feller’s transformation BY A. W. F. EDWARD...
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