Download Polymorphism due to selection of varying direction

237
POLYMORPHISiV[ DUE TO SELECTION OF VARYING
DIRECTION
BY J. 13. S. H A L D A N E AZ~I~S. D. J A Y A K A R
Genetias and BiomeOy Laboratory, Gover~me~ztof Orissa, Bhuba~eswar
INTRODUCTION
Geneticists are showing increasing interest in geneti~ally determined polymorphism.
This is especially true of h u m a n geneticists, since polymorphism is often associated
with a loss of fitness, sometimes described as genetic load. In recent discussions the
following causes of more orless stable polymorphism have been distinguished.
(1) T h e conflict between selection and mutation. An equiIibrium is reached when
in each generation in a population of given number as fhany " m u t a n t " genes are destroyed b y selection as appear b y mutation. T h e conflict between selection and the
abnormal Mendelian segregation now called ~meiotic drive" is analogo.us.
(2) T h e conflict between selection and segregation. T h e best known exampte
occurs When a heterozygote is fitter than either of the t~,o corresponding homozygotes.
But where several alIelomorphs are concerned matters are more complicated, and,
as we hope to show later in soN4 detail, the heterozyg6te need not be fitter in order
to stabilize polymorphism when seIective intensities differ in the two sexes.
(3) The conflict between fitness and abundance. A genotype may become tess
fit as it becomes more abundant2 ~ The most striking case is perhaps genetically determined self-sterility in plants, but Batesian mimicry probably acts in a similar way,
mimics being better protected when greatly outnumbered by d~stastefnl modeIs than
when mimics are common. So does resistance to infectious dhease if the pathogen
ca~ a d a p t itself fairly rapidly so as to attack the commonest genotypes preferentially(4.) T h e conflict between selection and migration. I f each of t xvo genotypes %
fitter in difl~rent areas, migration between them m a y lead to a cIine which m a y b e
polymorph.ic.
(5) T h e conflict between selections in different directions in th.e diploid generation
and the haploid generatioi~ (us~aIIy pollen tubes) or in the two sexes.
Finally a polymorplg,~m m a y be transient A genotype which was ibrmerly common
m a y be dhappearing as the result of changed conditions.
These are not, howeve:r, the only conditions which m a y lead to long-lasting genetic
polymorphism, nor is the account of the five situations described above very satisfactory.
We hope, i~1 a series of papers, of which this is the first, to add some precision to the
accounts of polymorphism so far given by iLIaldane and others. The prescnL paper
deals with polymorphism caused by selection which sometimes favours one phenotype
and sometimes another, or in the gel~eral terminology of this introduction by conflict
between selection in two directions.
Selection and Polymorphism
238
TK~ v G~NERAL TI-mOI~EM
Suppose that tllepopuladon consists of two phenotypes P and Q, and that the
of Q. relative to P is F~ in the ~th generatim~, but that F~ sometimes exceeds
a n d sometimes falls short ofiK I f P and Q a r e clones or p u r e lines which do n o t
breed; or genotypes o f a haploid differing in respect o f a single gone pair, and
ratio of P to Q in generation n is u,, then lg,~+l--~-2~,~-lz/,~,
So
~,~ =
rfoFrJ
uo
fitness
unity,
interif the
(1.1)
that is to say u~ is equal to u o divided by the product of the vahtes of F~. I t is most
i m p r o b a b l e that this will be e~xactiy unity. As time goes on it is almost certain that
the p r o d u c t of the values of F,,~ will be m a c k greater or m u c h less than unity, and P
or Q, as the case m a y be, will disappear. We can also think in terms of the geometric
mean. I f this differ s even slightly from unity over sufficiently m a n y generations,
one or other of the phenotypes will disappear. I n practice of course it is possible
that tile rarer phenotype will be restricted to a niche where it is at an advantage, leading
to an equilibrium of type (4) considered above.
Now consider a large* diploid r a n d o m mating popuIation, so 1argo that the probability of the extinction of a temporariiy rare gene by r a n d o m fluctuation is negligible.
Let a pair of allelomorphs A a n d - a at an autosomal locus occur in the gametes which
form the nth generation in the ratio u~ A: 1 a. Let a be fuliy recessive as regards
5tness, a n d let the fitness of a a in year n relative to ~ a n d A a be F~. T h a t is to say
let P be A A or An, and Q be aa. T h e n (Haldane, 1924)
u~+l-
~,, (~.+l)
..+F.
0.2)
First consider the case yvhen aIi vaIues of u~ are large.
--1
F r @ F'' ( F ' ' - I)
Hence, by summation
n--I
n--I F~(F.,.-- 1)
u~--uo:n-- 2
F,,+
X
~
r:0
r=0
u~+F~
(1.3)
I f all values of u~ are su~ciently large, then only the first two terms need be considered.
For a n y finite value of n this can be assured b y choosing a large enough value of u~,
Hence if 2 F ~ > n , or, what.is' ecluivalent , ift!~e mean v a h e of F~ is greater than unity,
u,~ wilt be less than u o provided % is sufficiently large.
*Mutation gives a slmple crlteri'on for "largeness" 9f a population. If a popu[atiqn is, say, ten t.:imes
the reciprocal of the frequency with which a gene appears by mutation, the probability that no mutaat~
will appear it~ it in any given generatlog is e-~~ or 5 • I0-~. tn a population ,f~fthis size we may base
our calculations on selection and mntatior~ rates, neglecting ihe possibility of random extinction. The
size will boor the order of a million for many human mutation rates.
J. B. S. H A L D A N Z & S. I).JAVAKA <
Now
let ur be small.
Then
From
239
(1.2)
u,,.
log u~+~ -- Iog u, = -- log F,. - - l o g .
F,,q-u,,
F1 q- F~.E (1F,>sI
d-u,)J
So b y s u m m a t i o n
- *Iog
log' u,, -- log zgo = -- Z iog F,.-- " Z
r=0
r=0
[hl
k]
+ F~. (1 q-u,.)J
(1
As before, if all values of u,. are suflqclently small, the second t e r m is negligible unless
some value of F~ is zero. I f however a value of F~ i s z e r o the corresponding term
in the s u m is log (u,.-lq ~ 1). This is not infinite, but can be m a d e as large as we
choose b y a suitable choice of uo. W e conclude that if go issufficiently small, u,~ Will
exceed uo provided Z iog F , < 0 , or /7 F r < l , or the geometric mean of F,. is less
than unity,
r=0
it follows that, provided the arithmetic mean of the values of F.~ is greater than unity,
a n d the geometric mean less, z,.,~is bounded, that is to say, it must be between certain
values. Neither of the genes A a n d a can disappear. I f ' t h e values of F~ form a cycle,
so t h a t F~.+~ = F=, then the various values of u,~ in a cycle converge towards fixed
values. An example wiE be given later.
I t m a y be asked whether .~e condition that the arithmetic mean should exceed
unity, and the.geometric m e a n be less, is not highly artificial. It is easy to give an
el<ample w h e r e it would be fulfilled. Suppose that the recessive a a is normally 5 %
to I 0 % fitter t h a n the dominants, b u t that e-v-cry twen W generations, on an average,
an. epidemic disease kills off all the recessives; then the arithmetic m e a n exceeds unity,
b u t the geometric mean is zero. After each epidemic the frequency of recessive genes
is h a t f the frequency of hetero zygotes in the previous generation, that is to say somewhat
less than a half. It then increases until the next epidemic. I t is not of.course necessary to suppose that all the reeessives are killed off. A n d the killil~ agent nlJght be
a d r o u g h t or fi-ost. It is sufficient that the recessive should be fitter than the dominant
on the whole, but liable to be killed or sterilized in some a b n o r m a l but rare circumstances. I t is h a r d to suppose t h a t this.condition :is never fulfilled in nature.
I f b o t h the arithmeticaI and geometrical means of th.eir fitness are less than unity,
recessives will disappear in the course of time (except insoPar as they are replaced by
m u t a t i o n ) . And if both these means are greater than. unity, d o m i n a n t s wiI/. disappear.
Since the arithmetic ~mean. necessarily exceeds the geometric, unless th.e fitnesses are
constant, there is no situation w h e r e either gene m a y disalSpear according to its initiaI
frequency, as when heterozygotes are less :fit t h a a either hon~ozygote.
tqnally we m a y remark that the arithmetic m e a n of the relative fituesses of recessives
is the reciprocal of the h a r m o n i c m e a n of the relative fitnesses ofdon~inants. For
stability this harmonic mean m u s t be less than unity. As the h a r m o n i c m e a n is a
less familiar idea than 'the arithmetic, we have pret~r/~ed to use the latter.
Sdection and PoLymo~phisrn
260
~IVOLTINE O R OAtgI S.~IS
Let us suppose that, like some insects, an organism has just two generations per year.
In the spring generation the fitness of a a relative to AA and A a is F, in the a u t u m n
generation f . Suppose the spring generation of year n to be formed f r o m gametes
with u,~ A : t a, the a u t u m n generation f~'om gametes with z,'~ A : i a. T h e n
u%(,*',~+ 1)
(2.1)
(,< +F) [.g- + (~ -r/),,,, §
--u,~ [ ( F + f - - 2 ) u,,~--2 (1 --Ff)u,,--F ( 1 At equilibrium
Ff)]
(2.2)
1
(~.,,.+v) [~,,,g ~-(1
+/),,~,+U]
*"~+*-'<
u~ = U, where
(F+f-.2)
whence U = I - - F f •
U=0,
U=co,
or
U~--2 ( 1 - - F f ) U - - F ( i - - F f ) = 0
(1--Ff)~
F+f--2
(2.s)
(2.4)
Since the roots are real ~, Ff< I. Since one must ~]Sepositive, F@f>2. T h e fignre
shows the area within which the point (F, f ) must lie to ensure an equilibrium w i t h
both genes present. H e n c e the right hand side of (~.2) is negative w h e n u~ is large,
positive when it is small. H e n c e the equilibrium is stable. (2-3) has only one positive
r o o t . . I f F < 1 the ambiguity in (2'4) must have the negative sign. T h e equiIibrium
recessive gene frequencies in spring a n d a u t u m n are:
0=(o-+i)-~
= (i - 7 ) - (1 - F f ) ~
f (F--l)
02=
Thus if F = ~, f
spring and a u t u m n
T h e deviations
9Similar calculations
(2.5)
F - 1 - ( t --Ff) ~ .
F (l-f)
= ~,1 Q = - 2 7 5 3 , . Q ' - - -3101, so the frequencies of recessives in
are 7-580/0 add 9-82~
ti'om equilil)rium fail off roughly in a geometric progression.
could be m a d e for a cycle of three or more generations.
OvER-nOMr>'A~ce
It is well known (Fisher, 1922) that i f t h e reiative fitnesses ofA~A : A a : a a izl each
generation are F : 1 : f , being the same in each generation, there is a stable
j . B. S. HALDAWZ & S. D. J_~-zAKar<.
2~51
f
I
'~
2
5
F
Fig. 1. A bivQIthle population reaches a stable annual oscillation if the- poln~ (F, t') is in the shaded are~<.
e q u i l i b r i u m if and. only if F alad f a r e b o t h less tl~an unity.
case w h e n F a n d f a r e
So
u,~+~
L e t us now consider tlTe
variable.
Z[~+I =
l[,rz @.~1~
= F~, q_
~l--F"&
(3.2,
=Z_ ~ +
(Eft, - 1)~,,
S, (f,, +~,,)
(a.5}
W e see t h a t we can choose zgU so large t h a t u,=+, za,,-~ is as close as we wish to F~,. t~{ence
z~o>z~,, if the g e o m e t r i c m e a n o f F~. is less t h a n unity. S i m i l a r l y we can choose z~o so
small t h a t
u,,+: is as cIose as we wish t o f ~ S .
gTr
Hence u~,2Cz, o if the g e o m e t r i c mea~
off~, is Iess t h a n unity. 9T h u s p o l y m o r p h i s m will persist if the geometric m e a n timesses
of each h o m o z y g o t e relative to the heterozygotes are each leas tllm~ unity. O n c e
again this m a y oecu:r, even i f the m e a n fitness of h o m o z y g o t e s relative to heterozygotes
exceeds unity, p r o v i d e d the relative fitness o f laomozygotes occasionally fh.lls v e r y low..
242
Sdecaio,z and P@raor]ohism
I t is not suMcient or necessary that the arithmetic mean fitness of the heterozygotes
relative to tmmozygotes (that i~ to say the harmonic mean ofF,, orf~) should exceed
unity.
For example an occasional severe epidemic offalciparum malaria might suffice to keep
the gene tbr haemogIobi~l S fi'om disappearance, even if for generations on end heterozygotes tbr sickling were at a disadvantage compared with persons homozygous for
haemoglobin A.
Discussion
There is no great difficulty in extending tke argument to a pair of sex-linked genes
wkert dominance {s complete in the homogametic sex. Here however a condition
for equilibrium is that the relative fitness of hecerogametie " m u t a n t s " should exceed
mlity, that of ho~mogametie mutants being Iess than unity. It will be best discussed
along with other cases where selection is in opposite directi~ in the two sexes.
We think that we have proved two novel points, first that a mere series of changes
in the direction of selection m a y be enough to secure polymorphism, secondly that
one result of Mendelian inheritance is that polymorphism m a y be permanent, if no~
exactly stable, when it is not so for a difference inherited elonally, cytoplasmically,
or in a haploid. Finally we have shown that a statistical treatment of the intensity
of selection can iead to definite results.
T h e results of this paper can, and doubtless will, be extended. Thus it is ctear that
inbreeding makes the conditions for polymorphism more stringent, and that ~he sums
in (I.3) and (i-4) can be replaced by integrals when generations overlap.
SUMMARY
In a targe r a n d o m mating population segregating for a pair of allelomorphs with
full dominance, the condition that neither allelomorph should disappear is that the
arithmetic mean of the fitnesses of recessives in different generations shouId exceed
unity~ their geometric mean fall below it.
REFERENCES
HAI.OAI~Z,J. B. S. (1924). A mathematical theo~." of namrm and artificial selcction~ Part I.
CaTnb. Phil Soc., 23, t9-41.
FIshER, R. A. (1922). On the dominance ratio. Proc. Roy. Soc. Edin., 42, 321-341.
Trans.