A GENERALIZED FUNDAMENTAL THEOREM
O F NATURAL SELECTION
W. J. EWENS
La Trobe University, Melbourne, Australia
Received December 30, 1968
classical result (FISHER
1930) of population genetics is that if, in a random-
A mating population, the fitness of any individual depends on his genetic con-
stitution at a single locus, then the mean fitness of that population increases (or
at worst remains stable) with time. This is true for an arbitrary number of
alleles at the locus and for arbitrary fitness values; (for the best proof of this
theorem, see KINGMAN
1961) .
Unfortunately, if it is supposed that fitness depends on the genetic constitution
at two loci, FISHER’S
result no longer holds (MORAN
19164). This disturbing result
opens up several lines of enquiry, one of which is LLAre
there any restrictions
which can be placed on the fitnesses which will secure an extension of FISHER’S
theorem to two (or more) loci?” The purpose of this paper is to state, prove, and
discuss in more generality than has been done before (EWENS1969), a theorem
which shows that, in a certain range of cases, an extension of FISHER’S
theorem
to many loci does exist. The discussion of this theorem opens up a connection with
KIMURA’S( 1965) concept of quasi-linkage equilibrium, and this connection is
discussed in some detail.
THE GENERALIZED FUNDAMENTAL THEOREM
Consider a population in which the fitness of any individual depends on his
genetic constitution at two loci, “A” and “B”, at which occur alleles A,, A, and
B,, B,, respectively. We suppose that the fitness of the nine possible genotypes
(coupling and repulsion double heterozygotes are assumed identical in all
respects) can be written in the following form:
AsAs
BiBs
BIB,
BA32
U1
U2
U3
+ v1
+ v1
+ v1
AiA,
vz
U2
v2
U1
U3
+
+
+
v2
AA,
v3
U2
v3
U3
v3
U1
+
+
+
(1)
If fitnesses can be expressed in the form (1), we shall say that fitnesses are
additive over loci; the fitness of the genotype A,A,BkBIcan be expressed as the
sum of two terms, one characteristic of the genotype A,A, and the other of the
genotype BB,.
If we suppose that the frequencies of the gametes A,B,, A,B,, AlB2 and A&
Genetics 63: 531-537. October, 1969.
532
W. J. EWENS
going to form the zygotes of generation t are c1,c2,c3and c4respectively, then the
mean fitness W of the population in this generation is
W
= U1 (Cl+C2)
2
+ 2uz
(c,+c4) + v1
+ 2vz(Cl+c3) (c2+c4) + v3(c2+c4)2.
(Cl+CP)
( C 3 f C 4 ) +U,
2
(Cl+C3)
2
(2)
Note that W depends on the gamete frequencies only through the gene frequencies
c1 cz and c1 c,; it is this observation which is fundamental to the theorem
below.
If we assume a recombination fraction R between A and B loci, the frequencies
dl,dZ,c ’ ~and d4 of the gametes going to make up the zygotes of generation
t 1 are found from the recurrence relations
+
+
+
+ + u2c4) + c1 {VICl + vzcz + v1c3 +
$- R
+
d2w= (UlClfUlC2 + +
f (v2c1 +
+ vZc3 + v3c4)
R
+
+ + + {vlcl + vZc2 + v1c3 f v2c4)
<w
R(uz + vz)
d4w c4
f
+ + (v2c1 + + + v3c4)
d1W = c1 {u1c1+u1c2
U#,
v2c4)
(U,
c2
uZc3
uZc4)
c3 {u?cl+U2c2
u3c3
U3c.4)
u3c3
u3c4)
~
CZ
-
{u2cl+U2c2
(3)
vSc2
(
~
2 v2) (
~
2 ~ ~
3
1
~
4
)
(4)
c3
(czcd
-
ZI
2 ( )~ 2 ~ 3~ 1 ~ 4 )
(34
+ R(uz +
v3cZ
cIc~),
(5)
v2c3
cic4),
(6)
v2) ( ~ 2 ~ 3
The above considerations are sufficient to enable us to state the following:
Theorem: If fitnesses are assumed additive over loci, then the mean fitness of the
population increases monotonically in time (or at worst remains stable), irrespective of the number of loci on which fitness depends, of the number of alleles at
each locus, and of the linkage arrangements between the loci.
Proof: The proof of this theorem is surprisingly simple. We consider firstly the
proof for the two locus case (1) given above. The mean fitness W’ of the population in generation t 1 is found from equation (2) if we replace W by W’, c1 by
c‘~, . . . , c4 by c’~. Clearly W’ depends on the gamete frequencies only through
c ’ ~and C ’ ~ c’,, and equations (3)-(6) show that,
the gene frequencies c’]
once the frequencies c1, cz,c3 and c4 are given, the gene frequencies c ’ ~ c ’ ~and
C ’ ~ 4- c ’ ~do not depend on R. These gene frequencies, and hence also W’, are
therefore equal to the values obtaining in the particular case R = 0. But it is a
classical result that when R = 0 the popullation behaves as a single-locus population with four alleles Gl(for AIBl), Gz(for A,B,), G,(for AlB2) and G4(for
theorem shows that for this case,W’ 2 W. Then since W’
AZB2),and FISHER’S
is independent of R, we have W’2 W for all R.
The key point in the above proof is that W’depends on gamete frequencies
only through gene frequencies, and we now use this observation to extend the
proof to an arbitrary number of loci with arbitrary linkage arrangement among
loci. The first point to note here is that, given the frequencies of the gametes
forming the zygotes of generation t , the frequencies of the genes in generation
t 1 do not depend on the linkage arrangement between loci: this follows
essentially from the fact that genes relate to single loci rather than combinations
+
+
+
+
533
THEOREM ON NATURAL SELECTION
of loci (as is the case for gametes), The second point to note is that W’ depends,
as above, only on gene frequencies. To exemplify this, we consider a case where
fitness depends additively on three loci, A, B and D. Suppose that the eight possible gametes are A,BIDl, AIB,D,, A,B,D,, A,B,D,, A,B,D,, A,B,D,, A,B,D, and
A,BzD,, and that in the gametes forming the zygotes of generation t these have
frequencies c1, . . . . , cs, respectively. If in addition to the fitnesses displayed in
( 1 ) we associate the additive fitness components y, (for DIDl), yz(for DID,) and
y3(for D,D,) , (so that the fitness of AIA,B,BID,D,, for example, is u1 vz y3>
the mean fitness W of the population in genera tion t is
+ +
v1 (clfcZfc3fc4)
fU1 ( C 1 f C Z f C 5 f C G )
+ yl
(Cl+C3fCBfC7)
+ 2v2
+ 2u2
+ 2yZ
f v3 (C5+CG+C7fC8)
( C l f C p f C F i f C ~ )( C 3 f C 4 f C 7 f C 8 ) f U3 (C3+C4+C7fC8)
( C l f C 3 f C 5 + C 7 ) (c2fc4+ctj+cR) f yj (C2+C4+Cs+C8)
(ClfCZfC3fCB) (C5fCsfC?fC$j)
Clearly this W depends only on the frequencies c1+c2+c3+c4 (of the gene AI),
c1fc2Sc5+cs (of the gene B,) and c1+c3+c5fc7 (of the gene D,). Thus W’
depends on gamete frequencies only through gene frequencies and is therefore,
from the above argument, independent of the linkage arrangement among A, B
and D loci. In particular, W’ is equal to the value obtaining for the case where
all recombination frequencies are zero. But again, in this latter case, once c1, . . . . ,
cs are given, the value of W’ is identical to that for a certain case where fitness
depends on (eight) alleles at a single locus, so that using FISHEX’S
theorem again,
W’ 2 W . Since R’ does not depend on recombination frequencies once c,, . . . ,cS
are given, we have W! 2 W generally, and the theorem is proved.
It should be noted that the theorem does not mean that the evolution of the
population is identical to that of some population where fitness depends only on
a single locus. This evolution will depend on knowledge of the linkage arrangement and the gamete frequencies, and this accounts for the use of the phrase “once
gamete frequencies are given” at several points in the above proof. We have used
the analogue with a certain single-locus population merely to prove the result
W’ 2 W. As time goes on, the successive values of W will depend on the linkage
arrangement and the gamete frequencies.
The above theorem generalizes FISHER’S
Fundamental Theorem of Natural
Selection to the case where fitnesses depend additively (in the above sense) on
many loci. It appears quite possible that this is the only obtainable generalization
of this theorem, and might thus provide the only optimizing principle in population genetics for the case where fitness depends on more than one locus.
CONNECTION WITH QUASI-LINKAGE EQUILIBRIUM
Apart from the result AW 2 0, FISHER’S
theorem for the single-locus case
provides the further information that to a very high degree of accuracy,
AW = v,
(7)
where V is the additive component of the genetic variance at the locus under consideration. Since, when fitness depends additively on many loci, the change in
.+
c
wd
3
d
THEOREM O N NATURAL SELECTION
535
mean fitness is identical to that for a certain single-locus population, it follows
that in the many-locus, additive case an equation identical to (7) can be arrived
at, provided that V is interpreted correctly. Specifically, V must be interpreted
as the additive part of the genetic variance for that single-locus population whose
behaviour parallels that of the actual population under consideration. When
fitness depends additively on two loci, rows 4 and 5 in Table 1 show that this is
that part of the total genetic variance which is explained by a least-squares fit of
the parameters O, a,j? and y shown in the “Tc-fit’’ line of Table 1.
Since we can identify AW in this single-locus case with the AW for the two-locus,
additive case, we can state that when fitnesses are additive and depend on two
loci, the increase in mean fitness is equal to that part of the total genetic variance
removed by tlie “Tc-fit’’ parameters in Table 1. The extension of this result to
the case where fitness depends additively on many loci is immediate.
Now this component of the total genetic variance is identical to what KIMURA
( 1 965) has called the “total chromosomal variance” (denoted VTC), so that we
can restate the above result thus: “when fitnesses are additive over loci, the
increase in mean fitness is equal to the total chromosomal variance in fitness,”
and express this symbolically by the equation
AW=VTC
.
(8)
Equation (8) no longer holds exactly, of course, when fitnesses are not in the
additive form (1 ) , although there are cases when fitnesses depend on two loci
and are “almost” additive and for which equation (8) is “almost” true. An
example is supplied by the population whose behaviour is given in Table 7 of
KIMURA(1965): here AW and V, are reasonably close throughout the entire
evolution of the population being considered. It is not known to what extent
equation (8) is “almost” true when fitnesses are “almost” additive and depend
on many loci; the above analysis gives no answer to this difficult and important
problem.
The total chromosomal variance VTo can itself be partitioned into two components, a n additive chromosomal variance (VAC)and an epistatic chromosomal
(1965) paper is a discussion
variance (VEpc),and the main point of KIMURA’S
of the conditions under which the approximate relation
AW = VAC
(9)
will hold. The component VACis that part of the total genetic variance removed
by a least-squares fit of the parameters in the “AC-fit”row in Table 1. Now when
fitnesses are additive over loci, the value of y in the Tc-fit is identically zero, SO
that VAC= VTC.It follows that in this case, the approximation (9) becomes,
effectively, an equality.
KIMURA(1965) derived the approximation (9) by a completely different
approach than that outlined above. He showed, both empirically and theoretically,
that in a wide range of cases the quantity c1c4/c2c3should soon reach a fairly
stable value, so that one can assume that
A (c1c4/c2c3)= 0.
(10)
536
W. J. EWENS
KIMURAthen demonstrated that when equation (10) holds, (9) should normally
hold: thus the attainment of ( l o ) , called by KIMURAa state of quasi-linkage
equilibrium, implies the truth of (9) in a wider range of cases than those where
fitnesses are in the additive form (1). For a further discussion of the concept of
(in preparation).
quasi-linkage equilibrium, see FELDMAN
The above arguments suggest that a slightly different approach to the concept
of quasi-linkage equilibrium may be useful in some cases, since there are cases
where c1c4/c2c3+ CO and yet (9) holds. (See, for example, the population treated
in Table 7 in KIMURA(1965)). This suggests that (9) may be true in a wider
range of circumstances than those for which (10) is true, so that (9) is not
necessarily dependent on (10) for its validity and must sometimes be reached
by an independent argument. Such an independent argument may sometimes be
provided by the results of this paper: that (9) holds exactly when fitnesses are
additive over loci, and should therefore hold as a reasonable approximation when
fitnesses are almost additive over loci, at least when fitnesses depend on twr, loci.
We exemplify this argument by considering KIMURA’S
population, referred to
above, in generation 120. Here there are two genotypes whose frequencies are
rather larger than those of the remaining genotype, namely AIRl/AzBz and
A,A,B,B,. The fitnesses of these two genotypes can trivially be fitted exactly by
additive parameters of the form uz v2 and u3 v3, respectively. Since these
two genotypes account almost entirely for VACand AW, the above theory asserts
that we should have
+
VAC= VTc
+
AW.
(11)
Further, it suggests that as time goes on, and the total frequency of these two
genotypes becomes even higher, that (11) should be more and more exact as
time goes on. This behaviour is actually observed. This latter observation leads
to a further prediction, since very tight linkage between A and B loci implies
an even larger total frequency of AIBl/AzBz and A,A,B,B, at this stage than is
the case for loose linkage, that (9) holds best for very tight linkage (in the
example under consideration). But it is precisely for tight linkage that c1c4/c2c3
approaches infinity most rapidly. I n other words, while (9) was originally
derived by KIMURAunder arguments which relied entirely on the validity of
(1 0) ,there are cases when (9) holds best when ( 10) is least true.
This observation is intended to extend somewhat the range of cases in which
we can expect mean fitness to increase beyond those cases accounted for by the
concept of quasi-linkage equilibrium.
SUMMARY
It has been shown recently that, when the fitness of the individuals in a population is supposed to depend on the genetic constitution at two loci, the mean
fitness of the population can decrease monotonically in time. This had led to
various discussions concerning the nature of those fitness values for which mean
fitness increases. In this paper it is shown that when fitnesses are additiue ouer
THEOREM O N NATURAL SELECTION
537
loci (a condition sometimes assumed for independence of the effects of the loci)
the mean fitness must increase with time. This result holds true irrespective of
the number of loci involved, of the number of alleles at each locus, and of the
linkage arrangement among loci. The analysis has points of contact with the
concept of quasi-linkage equilibrium, and several of these points are discussed.
LITERATURE CITED
EWENS,W. J., 1969 Mean fitness increases when fitnesses are additive. Nature 221 : 1076.
FISHER,
R. A., 1930 The Genetical Theory of Natural Selection. Clarendon Press, Oxford.
KLMURA,M., 1965 Attainment of quasi-linkage equilibrium when gene frequencies are changing
by natural selection. Genetics 52 : 875-890.
KINGMAN,J. F. C., 1961 A matrix inequality. Quart. J. Math. 12: 78-80.
MORAN,
P. A. P., 1 9 M On the nonexistence of adaptive topographies. Ann. Human Genet. 27:
383-393.