The Geometry of Population Genetics

211 Pages·1979·4.18 MB·English

by  Ethan Akin (auth.)

Lectu re Notes in
Biomathematics
Managing Editor: S. Levin
31
Ethan Akin
The Geometry of
Population Genetics
Springer-Verlag
Berlin Heidelberg New York 1979
Editorial Board
W. Bossert· H. J. Bremermann . J. D. Cowan . W. Hirsch
S. Karlin' J. B. Keller' M. Kimura' S. Levin (Managing Editor)
R. C. Lewontin . R. May' G. F. Oster' A S. Perelson
T. Poggio . L. A Segel
Author
Ethan Akin
Mathematics Department
The City College
137 Street and Convent Avenue
New York City, NY 10031
USA
Library of Congress Cataloging in Publication Data
Akin, Ethan, 1946-
The geometry of population genetics.
(Lecture notes in biomathematics; 31)
Bibliography: p.
Includes index.
1. Population genetics--Mathematical models. I. Title. II. Series.
QH455.A38 575.1 79-26084
AMS Subject Classifications (1970): 92AlO, 34C05, 34 [}20,
70G05,62JlO
ISBN-13: 978-3-540-09711-2 e-ISBN-13: 978-3-642-93128-4
DOT: 10.1007/978-3-642-93128-4
This work is subject to copyright. All rights are reserved, whether the whole
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a fee is payable to the publisher, the amount of the fee to be determined by
agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1979
Softcover reprint of the hardcover 1s t edition 1979
Table of contents
Introduction .1
I. The Vectorfield Model of Population Genetics 3
1. The Equations of Selection, Recombination and
Mutation. 5
2. Multivariate Analysis and Types of Epistasis. 11
3. Euclidean vector spaces and Riemannian Manifolds. 17
4. The Shahshahani Metric. 37
5. The Product Theorems and Epistasis. 43
6. The Selection Field. 51
7. The Recombination Field. 58
8. The Mutation Field. 64
9. The Combined Field. 67
II. The Geometry of Epistasis 80
1. Orthogonal Decompositions. 80
2. The Product Model. 91
III. Selection, Recombination and Mutation 119
1. Selection and Epistasis. 119
2. Recombination and Entropy. 134
3. Recombination and Epistasis. 145
4. position Effects. 153
5. Mutation. 160
IV. The Hopf Bifurcation 173
1. The Hessian. 173
2. The wright Conjecture. 181
IV
Appendix 191
1. Proper Mappings. 191
2. Partially Defined Distributions. 194
3. Game Dynamics 197
Bibliography 201
204
Introduction
The differential equations which model the action of selection
and recombination are nonlinear equations which are impossible to
solve explicitly. It is even difficult to describe in general the
qualitative behavior of solutions. Recently, Shahshahani began using
differential geometry to study these equations [28]. with this mono
graph I hope to show that his ideas illuminate many aspects of pop
ulation genetics. Among these are his proof and clarification of
Fisher's Fundamental Theorem of Natural Selection and Kimura's
Maximum Principle and also the effect of recombination on entropy.
We also discover the relationship between two classic measures of
genetic distance: the x2 measure and the arc-cosine measure.
There are two large applications. The first is a precise
definition of the biological concept of degree of epistasis which
applies to general (i.e. frequency dependent) forms of selection.
The second is the unexpected appearance of cycling. We show that
cycles can occur in the two-locus-two-allele model of selection plus
recombination even when the fitness numbers are constant (i.e. no
frequency dependence).
This work is addressed to two different kinds of readers which
accounts for its mode of organization.
For the biologist, Chapter I contains a description of the
entire work with brief indications of a proof for the harder results.
I imagine a reader with some familiarity with linear algebra and
systems of differential equations. Ideal background is Hirsch and
Smale's text [15]. In Section 3 we introduce what manifold theory is
necessary together with a review of the underlying linear algebra and
2
calculus.
The remaining Chapters are more demanding though the epistasis
examples and discussion of position effects in Chapter III are worth
a look.
For the mathematician, the technical Chapters II and IV are
the heart of the work with Chapter I serving as an introduction and
biological orientation. However, some acquaintance with the rudi-
ments of genetics is needed. I recommend "An Introduction to
Genetics" by sturtevant and Beadle (Dover--1962). This is a reprint
of a book published in 1939 and so is uncluttered by the fallout
of the recent explosive growth of the field.
Here I would like to thank Ms. Kate March for her typing of
the manuscript (twice) and the NSF for their support of this work.
I. The Vectorfield Model of Population Genetics
We consider a large population of diploid organisms among whose
gametes we distinguish n different types, indexed by a set I. So
we describe a member of the population by telling its genotype, a pair
ij (= ji) with i and j elements of I. We can describe the pop-
ulation by telling the frequencies of the different genotypes, x ..
~J
the number of organisms with genotype ij. The information in this
frequency table is equivalently described by the total population
number L: x .. and the distribution of diploid types (p .. } where p;J' is
~J ~J ~
the fraction of the total population having genotype ij. The diploid
zygotes which make up the population are obtained by the pairing of
haploid gametes. We will assume that this pairing is random in the
Hardy-weinberg sense. This means that the two gametes in the zygote
are independent of one another. It is then sufficient to know the
distribution of the haploid gamete types, (p.}, and their total number,
~
which we will denote by lxi, because Pij = 2PiPj (i I j), Pii = P~ and
L: x .. = Ixl/2. If we let RI denote the n-dimensional vector space of
~J
real valued functions on I then the gamete distribution is a vector
p in the simplex 6 = (p E RI : Pi 2 0 and L: Pi = l}
The genes of the gametes occur on the chromosomes. At each of
£ different positions, or loci, on the chromosomes are the genes
which in the zygote will determine its biological characteristics.
For the a position (a = 1, ... £) the na different possible genes
which can occur constitute a finite set Ia. Thus, Ia is the set of
alleles at the a locus. A haploid genotype i is a list of £
choices ia E Ia for a = 1, ... ,£. So the set I of genotypes is the
£
cartesian product I ITa=lIa. The number of genotypes n is the
4
I
Now let Ra denote the space R a and 6a be the corresponding
simplex. The gamete distribution p E 6, is a probability distribution
a
on the product I. It induces a distribution p E 6a, namely the
a
marginal distribution on the factor Ia. If k E Ia then Pk (also
written pd(k)) is the probability that a random gamete has gene k at
the a locus. The map Ea(p) = pa from 6 to 6a is the restriction
o f t h e l ~' near mapp~,nga E: RI ~ Ra def'~ ned by
L
(0.1) Ea(X) (k) = (x(i): for all i with ia k)
Note that we use Xi and x(i) interchangeably for notational conven-
ience.
This just means that the probability that k occurs at the a
locus is the sum of probabilities Pi where the sum is taken over all
genotypes with ia = k.
More generally, if S is any subset of the set of loci
=
L (l, ... ,L), let IS be the product of the factors Ia for a in S.
So IS = IT I is the collection of partial genotypes obtained by
aES a
ignoring all but the loci in S. For i E I let i denote the projec-
S I
tion of i to IS. So (isa = ia for all a E S. Define Rs = R Sand
6S to be the correspond~ng simplex. p induces a distribution
S S S
P - E (p) on the subp roduct IS· E : 6 ~ 6s is the restriction of
the linear map E S : RI ~ RS defined by:
(0.2) ES(X) (k) =~ (x(i): for all i with is k)
S
So P (k) is the probability that the allele ka occurs at locus a for
5
all of the loci a in S.
If T is another subset of L, disjoint from S, then for
is E IS and jT E IT we denote by isjT the element of ISUT whose value
at locus a agrees with (is)a for a E S and with (jT)a for a E T. In
particular, if we denote by S the complement of the set S in L
then, for i E I, i
These notations are just bookkeeping devices to keep from writ-
ing genotypes and partial genotypes as lists of genes. We turn now to
the substance of the model.
1. The EqUations of Selection, Recombination and Mutation.
In the vectorfield or differential equation model of population
genetics, evolution is regarded as due to the sum of the effects of
selection, recombination and mutation. Assuming the Hardy-Weinberg
condition, we represent each of these by a vectorfield on the space
of gametic genotype distributions, 6.
We have assumed that the diploid genotype of a member of the
population determines it biological characteristics, among these are
two rates: a reproductive ,rate and a death rate. Each zygote of type
ij is assumed to have an average of b .. dt offspring in a time interval
~J
of length dt and to have probability d .. dt of dying in the same time
~J
interval. By an offspring of a zygote we mean two gametes given to
newborns which are zygotes receiving complementary gametes from other
members of the population. On average the two gametes contributed
will be an i and a j. Since we are only counting gametes we can
think of an offspring of an ij zygote as a gain of an ij zygote.
The gain or loss of an ij zygote causes the gain or loss of one
6
(if i I j) or two (if i = j) i gametes. Thus, if we define fitness
m. . b .. - d.. the change in the number of i gametes in time dt is
~J ~J ~J
given by:
dx. (2m .. x .. + ~ m .. x .. )dt = x.m.dt.
~ ~~ ~~ ~J ~J ~ ~
j i
Here we define m. = E. m. . p. and get the last equation from the Hardy-
~ J ~J J
Weinberg assumption in the form x .. = 2p.p .. (lxl/2) = x.p. (i I j) and
J J
~J ~ ~
2x .. = 2p2. (lxl/2) xiPi' where xi = Pi· (Ixl) is the number of gametes
~~ ~
of type i.
So we get the first selection equation:
dx.
(1. 1) ~ x.m ..
dt
~ ~
Recall that I x I E xi is the total number of gametes. So:
.s.w
(1. 2) Ixlm.
dt
Since Pi x./lxl the quotient rule implies that:
~
(1. 3)
Applying this to (1.1) and (1.2) we get:
(1. 4)
Here m = E.p.m. = E .. p.p.m .. is mean fitness. Note that we write b .. ,
J
~ ~ ~ ~,J ~ ~J ~J
mij, etc. as functions of unordered pairs or, as in the latter equa-
tion, as symmetric functions of ordered pairs.
Recall that the offspring of an ij zygote consisted of i and