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On the rate of genetic adaptation under natural selection
J. theor. Biol. (1979) 77, 307-316
On the Rate of Genetic Adaptation Under Natural Selection L. R. GINZBURG~ Department of EcologJJ and Evolution, State University of New York at Stony Brook, Stan.v Brook, N. Y. 11794, U.S.A. AND
R. F. COSTANTINO Department of Zoology, University of Rhode Island, Kingston, R.I. 02881, U.S.A. (Received 27 March 1978, and in revisedjbrm
16 August 1978)
For a one locus n-allele genetic model of natural selection two theoretical predictions are obtained: first, that the non-equilibrium average fitness value, W(t) is always between the equilibrium value, W*, and W*-H(po, p*)/t until finally W(t) = W* and, secondly, that the area bounded by W* and the curve W(t) is equal to H(p,,p*), the entropy distance between initial (p,,) and equilibrium (p*) gene frequency distributions. The experimental observations corresponding to these predictions are discussed.
1. Introduction Consider the process of genetic adaptation of a diploid population to a new stationary environment. We will describe the genetic structure of the population by the vector of gene frequencies p = {pl, . . ., p,} and use the notation W for the average fitness of the population. If the population is placed into a new environment E, it will usually change its structure p and the average fitness W under the new form of natural selection so that PO, 6 PO) p*w 1-3 Wt,
E, PO) f’?
W*(E,
PO) po)
(1) I
where t is time and p,, is the initial genetic structure. The presence of p,, as the argument for p* and W* means that we have included in this consideration t This work was supported in part by NSF Grant Service Biomedical Research Grant 5 S)7 RR 07067-I?.
MCS77--03255
and U.S.
Public
Health
Press Inc. (London)
Ltd.
307 0022-5193/79/070307
+ 10 $02.00/O
i” 1979 Academic
308
L.
R.
GINZBURG
AND
R.
F.
COSTANTINO
the possibility of multiple stable steady states so that the new equilibrium can be different depending on the initial point. We will consider the rate of adaptation in the same sense as Maynard Smith (1976) i.e. as the change of the average fitness of the population. One of the properties of the adaptation curve IV(t, E, pO) is known as the Fisher Theorem of Natural Selection (Fisher, 1930). It has been proved for a number of cases (Kimura, 1965; Nagylaki, 1976) and basically it says that
i.e. the average fitness tends to the equilibrium clear from this theorem that WO, E, ~0) 6 W,
value monotonically.
E, po) G W*(E, po),
It is (3)
for all t 2 0. The question is whether there is a non-trivial time-dependent estimate for W(t, E, p,,)? The adaptation rate theoretically can assume any value, so over all possible environments we cannot state anything but the trivial estimate (3). Let us reformulate the problem fixing one of the properties of the environment, namely, p*. The question now is whether the population can “move” from the initial structure p0 to the equilibrium structure p* with any preassigned rate? The answer is “no”. We can obtain for the one locus n-allele genetical model a result in the form of the inequality : W* -f(p,,
P*, t) < Wf, E, P,,) 6 W*,
(4)
where “fu, PO?P*) - 0. t+cc
So, the relative position of the points p. and p* restricts the adaptive rate under natural selection to be higher than the rate defined by the function fk
PO?P*).
Suppose that we observe the average fitness curve W(t, E, po). This may be accomplished by genetic analysis or perhaps by measuring the population growth curve N(t) W(t, E, po) = $ g.
The problem is what can we say about the distance between the initial and equilibrium genetic structures from W(t, E, po)? We will answer this question and we will attempt to support the theoretical predictions with experimental data.
RATE
OF
GENETIC
ADAPTATION
UNDER
NATURAL
309
SELECTION
2. The Basic Inequality We will base our discussion on the continuous-time selection (Crow & Kimura, 1970) which has the form fii = pi(M(-W),
i = I,.
mode1 of natural
.. II.
(6)
where pi is the frequency of ith allele, w = i
w is the allelic fitness (
wijpj , i
j=l
W = i ejpipj
W is the average fitness of the population i
, 1
PQ is the symmetric fitness-matrix, which is the complete description of the environment in this mode1 [~j = IQ(E)]. Under natural selection a population starting from the arbitrary point PO = IPOl,. . ., pan} will tend to one of the stable equilibria. Formally, it can also tend to an unstable equilibria along some special directions. We will not consider this case. Suppose p* = {pf, . . ., p,*) is the stable equilibrium point our trajectory tends to. We allow some p* to be zero, i.e. the point p* can belong to the boundary of the simplex i?,=
p>/o,
&4
i
i=l
i
Without loss of generality suppose p* > 0, i = 1, . . ., k and p* = 0, i = k + 1, . . ., n. The necessary condition of stability for the boundary point is that q(p*)
(Crow & Kimura,
< W*,
i = k+ 1,. . ., n
(7)
1970). Let us show now that k
c pTw,(p)
< W*.
i=l
We will refer to (8) as the basic inequality. explicitly :
Because of the symmetry of the matrix changing the order of summation: i j=l
Pj&(P*)+j=$+l
To prove (8) let us rewrite it
Wj we can continue
PjWj(P*).
as follows
310
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R.
GINZBURG
AND
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F. COSTANTINO
I\‘ow at the r:quilibrium point all wj(p*) = W* forj = 1. . . . k (see equation (t ;). Taking into account (7) we get finally h W* C pj+ ~ pj)j(P*) 6 W*. (9) j=k-+ 1 j= I The equality in (8) is true if and only if all pr are positive and we do not have the second term in (9). In the form of an equality for non-zero polymorphisms it was noticed by Ginzburg (1972). 3. The Entropy Distance
We introduce the formula for the distance between the arbitrary and the polymorphic frequency distributions as
(10)
H(p, p*) = - i p* In +. i= 1
I
H is zero when p = p* and positive for any p # p*, H tends to cc when p tends to the boundary of the simplex (one of pi’s tends to zero). Note that if k < n only the part of the vector p participates in the formula. This distance is known in information theory (Kullback, 1960). It was introduced to population genetics by Ginzburg (1972) and used in connection with the concept of selective delay in Ginzburg (1977) and Costantino, Ginzburg and Moffa (1977). Calculating the time-derivative of H taking into account the system (6), we have I9 = - i; p*; = - ; pj+(W.;.- W) = wi=l I i= I
It follows then from the basic inequality
i
p*w,.
(8) that
fi> w-w*. Integrating
(11)
i=l
(12)
(12) with respect to time from 0 to CYJwe have [ (W*-
W)dt
< H(a)-H(0).
(13)
0
H(cc) = H(p*, p*) = 0 and H(0) = H(p,, p*). Changing the sign we obtain i (W* - WI dt 2 H(p,, p*).
(14)
This is our first result. It gives the estimate of the distance between p. and
RATE
OF
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ADAPTATION
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311
p* from the measurement of the growth-rate curve (Fig. 1). It will actually give the exact value of the distance if the equilibrium is a complete polymorphism (k = n). In this case all the inequalities starting from (8) become equalities. The exact result has the form ‘i (I$‘*-W)dt
= H(JJ~,P*)+
0
[W*-WJ/J*)] 7 Pjdt,
E j=k+
1
where the second term of the right side is positive conditions (7)). Note that for pO sufficiently close to p*
pjtr) SZPjOexp {[T(P*)-
(‘5)
0
(see the stability
w*lr}
so that (15) can be rewritten in the form
which shows how strong the inequality (14) is in the neighborhood of the equilibrium. We do not know a simple global estimate for the second term in (15).
FIG. 1. Shaded H(p,. p*) = j (W*-
area W(t))
between dt.
the
graph
of
W(t)
and
W*
is the
genetic
distance,
312
L.
K.
GINZBURG
AND
R.
F. COSTANTINO
4. The Adaptation Rate Consider the case of complete polymorphism k = n. In this case we will obtain a simple estimate of the adaptive process in the form (4). We have f (W* - W) dt = H(p,, p*). b
(16)
Let us show that w*
_
H(Po,
P*) 6
w(t).
(17)
t
To prove (17) consider s (W*-W)dt=
; (W*-W)dt+
0
0
j (W*-W)dt. f If at the moment to PV(t,) < W* -H(p,, p*)/t, it should be true also for all 0 < t < t, by the Fisher theorem (2). Therefore i(W*-W)dt>
y(W*-W)>H(po,p*),
0
0
which is in contradiction with (16). So, the inequality (17) is proved. On Fig. 1 we have shown the hyperbola which bounds all possible adaptation curves W(t, E, po). The shaded area is H(p,, p*). 5. Experimental
Observations
The design of experiment centers on cultures of the flour beetle Tribolium castaneum Herbst initiated with identical age structures but with different frequencies of the unsaturated fatty acid sensitive allele. Demographic and genetic data of continuously growing populations were collected for 68 weeks. A complete discussion of these data is presented by Moffa & Costantino (1977). At this time, we shall briefly review the experimental procedure and then focus attention on those data relevant to measuring the rate of adaptation. Cultures were established in which the initial frequency of the unsaturated fatty acid sensitive allele, symbolized cos, range from 0 to 1 in increments of 0.1. The initial genotypic arrays were constructed from combinations of the +/+ and cos/cos homozygotes ; the initial demographic array consisted of IO newly emerged adult females and 10 newly emerged adult males. Each population was grown on 20 grams of corn oil medium which was changed every two weeks. The beetles were cultured in chambers maintained at
RATE
OF
GENETIC
ADAPTATION
UNDER
NATURAL
SELECTION
313
33 f 1°C and 42+60, relative humidity. The number of small and large larvae, pupae and adults were counted every two weeks; in addition, the frequency of the sensitive allele and the genotypic array at this locus were estimated. The measurement of the growth rate curves, l%(t), was obtained by combining two data items : estimates of the total numbers of offspring (S) of 3.97, 54.32 and 46.53 for the cos/cos. + /cos and +/igenotypes, respectively, and the genotypic array data (see Table 3 in Moffa & Costantino, 1977). The predicted stable equilibrium frequency of the sensitive allele, based on the above estimates, was p* = 0.13. Experimentally we observed p* = 0.25 +003. This discrepancy was discussed by Moffa & Costantino (1977) and it appears that the predicted value represents a lower boundary of p*, consequently, we used p* = 0.25 in our computations. The preceding theory is done for the continuous model. If Si are the absolute fitnesses for the discrete case and Wi are the fitnesses in the continuous model the relationship is exp (wit) = Si > 0, where t is generation time. We can take t = 1, so w = log, Si. Consequently, we have used w as the logarithm of the absolute fitnesses and calculated average fitness with them. We direct our attention now to the inequality (17). The hyperbola which bounds the possible adaptation curves was computed with W* = 3.85 and H(p,. p* = 025). This lower bound together with the experimental data are sketched in Fig. 2. It is clear from the picture that if the initial value of the average fitness, W(O), is much less than W* the adaptation should go fast. The closer p. is to p* the closer the adaptation curve should be to a step
Time (t)
FIG. 2. A comparison of the interval between W* and W* -H(p,, p* = 0.25)/t, solid lines, with the experimental values of W(t), circles, for cultures with initial cos allele frequencies ofO1. @3. 0.5, 0.7 and 0.9.
314
L.
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GINZBURG
AND
R.
F.
COSTANTINO
function. The data points were fitted to a function of the form V%‘(t)= u-/?/f (see Fig. 2) by the method of least squares. The overall fit of the data was significant at the 0.05 level of probability in each case. For the cultures with po = 0.1.05 10.7 and 0.9, t@(t) is above the lower bound of W* -H(p,, p*)/t. For the culture with p0 = 0.3. the data are below this bound. All populations definitely show the trend of converging to W*. The shaded area in Fig. 1 is H(p,, p*). The corresponding experimental observation is given by the area bounded by W* and the locus of W(t) as it approaches W*. The latter area was computed by substituting the estimated functions, I%(t), into the left-hand side of equation (16) and then performing the integration. The lower limit of integration was set at 0.1 and the upper limit was set at the time such that t@(t) was within 0.02 of W*. This same measure of entropy was used by Costantino, Ginzburg & Moffa (1977) in the experimental evaluation of the concept of selective delay (Ginzburg, 1977). namely, H(p,, p*) = j [W* - W(r)] dt = W*t.
For both data sets (Fig. 3) the hypothesis that the data fit the theoretical H(p,, p*) was accepted using a chi-square test at the 0.05 level of probability, although for the area data the computed chi-square value was just less than the critical value.
PO-
. I.5 I.0 -
I
. 0.5 . 0.0
-
.
I
0
/
.c*
I
0.1 0.2
I
0.3
.
l
/
l
I
I
I
I,
0.4
0.5
0.6
0.7
Inltlal
.
allele
11
0.0
0.9
I
frequency
FIG. 3. An experimental check of H(p,, p* = 0.25), solid line, with W*s(p,, p*), circles, and J (W* - W(t)) dt, squares.
RATE
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GENETIC
ADAPTATION
UNDER
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SELECTION
315
6. Discussion The first theoretical prediction given in this paper is that the average fitness, I@(f), is bounded by the function W*-H(po, p*)/t until finally @(r/(t)= W* [equation (17)]. For the cultures with p. = 0.1, 0.5, 0.7 and 0.9. this inequality was satisfied. The second theoretical forecast is that the area bounded by W* and the locus of I@(r) as it approaches W* is equal to the entropy distance [equation (16)]. For the experimental cultures with p0 = 0.1 and 0.9, this criterion was met: for p0 = 0.3, 05 and 07, @t(t) did not approach W* as rapidly as theoretically predicted. Fisher’s theorem states that W(t) is a monotonically increasing function of time. The observed average fitness values display some oscillations and fail to “stick” to W*. This latter behavior of I@(t) can explain why { [W* - r;i/(t)] dt is greater than the expected H(p,, p*). Why does W(t) oscillate? Our interpretation is that the distribution of genotypes is changing dramatically and although each culture is converging to the equilibrium a&/t array of 05 +/+ and 0.5 +/cos, genetic segregation alters the equilibrium genotypic array in the immat~ve stages, when we experimentally measured the populations, resulting in our estimates of U’(t) wavering somewhat in the vicinity of IV*. Another explanation of why the areas were greater than H(p,, p*) is stated in equation (15). Our genetic analyses have revealed two alleles at the unsaturated fatty acid sensitive locus; however, the presence of additional “hidden” alleles not yet identified would result in the observed areas being greater than H(p,, p*). Different information from the same overall experiment (Moffa & Costantino, 1977) has yielded two estimates of the genetic distance. In the first case we examined W*T = H(p,, p*)? where T is the asymptotic timedelay of the real growth curve in comparison to the equilibrium growth curve. To estimate this parameter we compared the time required of each initial non-equilibrium population to reach the number of adults predicted by the equilibrium culture at week 4. I@(t) was computed directly from the observed number of adults. In the present paper we have equation (16). It was suggested that W(f) can be measured from the population growth curve [equation (5)]. The theory requires the identification of small changes in this curve, dN/dt, to be reflected in W(f). These changes may be difficult to extract experimentally. In T. castaneum, the initial growth rate reflects the initial genotypic array when populations are started with young adults only. However, as p. -+ p* the demographic array (eggs, larvae, pupae, adults) controls population growth
316
L.
and changes in I?(t) these data we did estimates of the total observed genotypic
R.
GINZBURG
AND
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COSTANTINO
are difficult to observe. Consequently, to obtain i@(t) in not use population growth information but rather number of offspring for each genotype together with the arrays. Both data sets agree reasonably well.
This is a contribution No. 290 from the Program in Ecology and Evolution at the State University of New York at Stony Brook. REFERENCES COSTANYINO, R. F., GINZBURG, L. R. & MOFFA, A. M. (1977). J. rheor. Biol. 68, 317. CROW. J. F. & KIMURA. M. (1970). An Introduction to Population Generics Theory. New York: Harper and Row. FISHER, R. A. (1930). The Genetic Theory qf Natural Selection. Oxford: Clarendon Press. GINZBURG, L. R. (1972). J. gen. Biol. 33, 77 (in Russian). GINZBURG, L. R. (1977). J. theor. Biol. 67,671. KIMURA, M. (1965). Genetics 52, 875. KULLBACK, S. (1960). Information Theory and Sfatbtics. New York: John Wiley and Sons, Inc. MAYNARD SMITH, J. (1976). Am. Nar. 110, 331. MOFFA, A. M. & COSTANTINO, R. F. (1977). Generics 87, 785. NAGYLAKI, T. (1976). Genetics 83, 583.
On the Rate of Genetic Adaptation Under Natural Selection L. R. GINZBURG~ Department of EcologJJ and Evolution, State University of New York at Stony Brook, Stan.v Brook, N. Y. 11794, U.S.A. AND
R. F. COSTANTINO Department of Zoology, University of Rhode Island, Kingston, R.I. 02881, U.S.A. (Received 27 March 1978, and in revisedjbrm
16 August 1978)
For a one locus n-allele genetic model of natural selection two theoretical predictions are obtained: first, that the non-equilibrium average fitness value, W(t) is always between the equilibrium value, W*, and W*-H(po, p*)/t until finally W(t) = W* and, secondly, that the area bounded by W* and the curve W(t) is equal to H(p,,p*), the entropy distance between initial (p,,) and equilibrium (p*) gene frequency distributions. The experimental observations corresponding to these predictions are discussed.
1. Introduction Consider the process of genetic adaptation of a diploid population to a new stationary environment. We will describe the genetic structure of the population by the vector of gene frequencies p = {pl, . . ., p,} and use the notation W for the average fitness of the population. If the population is placed into a new environment E, it will usually change its structure p and the average fitness W under the new form of natural selection so that PO, 6 PO) p*w 1-3 Wt,
E, PO) f’?
W*(E,
PO) po)
(1) I
where t is time and p,, is the initial genetic structure. The presence of p,, as the argument for p* and W* means that we have included in this consideration t This work was supported in part by NSF Grant Service Biomedical Research Grant 5 S)7 RR 07067-I?.
MCS77--03255
and U.S.
Public
Health
Press Inc. (London)
Ltd.
307 0022-5193/79/070307
+ 10 $02.00/O
i” 1979 Academic
308
L.
R.
GINZBURG
AND
R.
F.
COSTANTINO
the possibility of multiple stable steady states so that the new equilibrium can be different depending on the initial point. We will consider the rate of adaptation in the same sense as Maynard Smith (1976) i.e. as the change of the average fitness of the population. One of the properties of the adaptation curve IV(t, E, pO) is known as the Fisher Theorem of Natural Selection (Fisher, 1930). It has been proved for a number of cases (Kimura, 1965; Nagylaki, 1976) and basically it says that
i.e. the average fitness tends to the equilibrium clear from this theorem that WO, E, ~0) 6 W,
value monotonically.
E, po) G W*(E, po),
It is (3)
for all t 2 0. The question is whether there is a non-trivial time-dependent estimate for W(t, E, p,,)? The adaptation rate theoretically can assume any value, so over all possible environments we cannot state anything but the trivial estimate (3). Let us reformulate the problem fixing one of the properties of the environment, namely, p*. The question now is whether the population can “move” from the initial structure p0 to the equilibrium structure p* with any preassigned rate? The answer is “no”. We can obtain for the one locus n-allele genetical model a result in the form of the inequality : W* -f(p,,
P*, t) < Wf, E, P,,) 6 W*,
(4)
where “fu, PO?P*) - 0. t+cc
So, the relative position of the points p. and p* restricts the adaptive rate under natural selection to be higher than the rate defined by the function fk
PO?P*).
Suppose that we observe the average fitness curve W(t, E, po). This may be accomplished by genetic analysis or perhaps by measuring the population growth curve N(t) W(t, E, po) = $ g.
The problem is what can we say about the distance between the initial and equilibrium genetic structures from W(t, E, po)? We will answer this question and we will attempt to support the theoretical predictions with experimental data.
RATE
OF
GENETIC
ADAPTATION
UNDER
NATURAL
309
SELECTION
2. The Basic Inequality We will base our discussion on the continuous-time selection (Crow & Kimura, 1970) which has the form fii = pi(M(-W),
i = I,.
mode1 of natural
.. II.
(6)
where pi is the frequency of ith allele, w = i
w is the allelic fitness (
wijpj , i
j=l
W = i ejpipj
W is the average fitness of the population i
, 1
PQ is the symmetric fitness-matrix, which is the complete description of the environment in this mode1 [~j = IQ(E)]. Under natural selection a population starting from the arbitrary point PO = IPOl,. . ., pan} will tend to one of the stable equilibria. Formally, it can also tend to an unstable equilibria along some special directions. We will not consider this case. Suppose p* = {pf, . . ., p,*) is the stable equilibrium point our trajectory tends to. We allow some p* to be zero, i.e. the point p* can belong to the boundary of the simplex i?,=
p>/o,
&4
i
i=l
i
Without loss of generality suppose p* > 0, i = 1, . . ., k and p* = 0, i = k + 1, . . ., n. The necessary condition of stability for the boundary point is that q(p*)
(Crow & Kimura,
< W*,
i = k+ 1,. . ., n
(7)
1970). Let us show now that k
c pTw,(p)
< W*.
i=l
We will refer to (8) as the basic inequality. explicitly :
Because of the symmetry of the matrix changing the order of summation: i j=l
Pj&(P*)+j=$+l
To prove (8) let us rewrite it
Wj we can continue
PjWj(P*).
as follows
310
L.
R.
GINZBURG
AND
R.
F. COSTANTINO
I\‘ow at the r:quilibrium point all wj(p*) = W* forj = 1. . . . k (see equation (t ;). Taking into account (7) we get finally h W* C pj+ ~ pj)j(P*) 6 W*. (9) j=k-+ 1 j= I The equality in (8) is true if and only if all pr are positive and we do not have the second term in (9). In the form of an equality for non-zero polymorphisms it was noticed by Ginzburg (1972). 3. The Entropy Distance
We introduce the formula for the distance between the arbitrary and the polymorphic frequency distributions as
(10)
H(p, p*) = - i p* In +. i= 1
I
H is zero when p = p* and positive for any p # p*, H tends to cc when p tends to the boundary of the simplex (one of pi’s tends to zero). Note that if k < n only the part of the vector p participates in the formula. This distance is known in information theory (Kullback, 1960). It was introduced to population genetics by Ginzburg (1972) and used in connection with the concept of selective delay in Ginzburg (1977) and Costantino, Ginzburg and Moffa (1977). Calculating the time-derivative of H taking into account the system (6), we have I9 = - i; p*; = - ; pj+(W.;.- W) = wi=l I i= I
It follows then from the basic inequality
i
p*w,.
(8) that
fi> w-w*. Integrating
(11)
i=l
(12)
(12) with respect to time from 0 to CYJwe have [ (W*-
W)dt
< H(a)-H(0).
(13)
0
H(cc) = H(p*, p*) = 0 and H(0) = H(p,, p*). Changing the sign we obtain i (W* - WI dt 2 H(p,, p*).
(14)
This is our first result. It gives the estimate of the distance between p. and
RATE
OF
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ADAPTATION
UNDER
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SELECTION
311
p* from the measurement of the growth-rate curve (Fig. 1). It will actually give the exact value of the distance if the equilibrium is a complete polymorphism (k = n). In this case all the inequalities starting from (8) become equalities. The exact result has the form ‘i (I$‘*-W)dt
= H(JJ~,P*)+
0
[W*-WJ/J*)] 7 Pjdt,
E j=k+
1
where the second term of the right side is positive conditions (7)). Note that for pO sufficiently close to p*
pjtr) SZPjOexp {[T(P*)-
(‘5)
0
(see the stability
w*lr}
so that (15) can be rewritten in the form
which shows how strong the inequality (14) is in the neighborhood of the equilibrium. We do not know a simple global estimate for the second term in (15).
FIG. 1. Shaded H(p,. p*) = j (W*-
area W(t))
between dt.
the
graph
of
W(t)
and
W*
is the
genetic
distance,
312
L.
K.
GINZBURG
AND
R.
F. COSTANTINO
4. The Adaptation Rate Consider the case of complete polymorphism k = n. In this case we will obtain a simple estimate of the adaptive process in the form (4). We have f (W* - W) dt = H(p,, p*). b
(16)
Let us show that w*
_
H(Po,
P*) 6
w(t).
(17)
t
To prove (17) consider s (W*-W)dt=
; (W*-W)dt+
0
0
j (W*-W)dt. f If at the moment to PV(t,) < W* -H(p,, p*)/t, it should be true also for all 0 < t < t, by the Fisher theorem (2). Therefore i(W*-W)dt>
y(W*-W)>H(po,p*),
0
0
which is in contradiction with (16). So, the inequality (17) is proved. On Fig. 1 we have shown the hyperbola which bounds all possible adaptation curves W(t, E, po). The shaded area is H(p,, p*). 5. Experimental
Observations
The design of experiment centers on cultures of the flour beetle Tribolium castaneum Herbst initiated with identical age structures but with different frequencies of the unsaturated fatty acid sensitive allele. Demographic and genetic data of continuously growing populations were collected for 68 weeks. A complete discussion of these data is presented by Moffa & Costantino (1977). At this time, we shall briefly review the experimental procedure and then focus attention on those data relevant to measuring the rate of adaptation. Cultures were established in which the initial frequency of the unsaturated fatty acid sensitive allele, symbolized cos, range from 0 to 1 in increments of 0.1. The initial genotypic arrays were constructed from combinations of the +/+ and cos/cos homozygotes ; the initial demographic array consisted of IO newly emerged adult females and 10 newly emerged adult males. Each population was grown on 20 grams of corn oil medium which was changed every two weeks. The beetles were cultured in chambers maintained at
RATE
OF
GENETIC
ADAPTATION
UNDER
NATURAL
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33 f 1°C and 42+60, relative humidity. The number of small and large larvae, pupae and adults were counted every two weeks; in addition, the frequency of the sensitive allele and the genotypic array at this locus were estimated. The measurement of the growth rate curves, l%(t), was obtained by combining two data items : estimates of the total numbers of offspring (S) of 3.97, 54.32 and 46.53 for the cos/cos. + /cos and +/igenotypes, respectively, and the genotypic array data (see Table 3 in Moffa & Costantino, 1977). The predicted stable equilibrium frequency of the sensitive allele, based on the above estimates, was p* = 0.13. Experimentally we observed p* = 0.25 +003. This discrepancy was discussed by Moffa & Costantino (1977) and it appears that the predicted value represents a lower boundary of p*, consequently, we used p* = 0.25 in our computations. The preceding theory is done for the continuous model. If Si are the absolute fitnesses for the discrete case and Wi are the fitnesses in the continuous model the relationship is exp (wit) = Si > 0, where t is generation time. We can take t = 1, so w = log, Si. Consequently, we have used w as the logarithm of the absolute fitnesses and calculated average fitness with them. We direct our attention now to the inequality (17). The hyperbola which bounds the possible adaptation curves was computed with W* = 3.85 and H(p,. p* = 025). This lower bound together with the experimental data are sketched in Fig. 2. It is clear from the picture that if the initial value of the average fitness, W(O), is much less than W* the adaptation should go fast. The closer p. is to p* the closer the adaptation curve should be to a step
Time (t)
FIG. 2. A comparison of the interval between W* and W* -H(p,, p* = 0.25)/t, solid lines, with the experimental values of W(t), circles, for cultures with initial cos allele frequencies ofO1. @3. 0.5, 0.7 and 0.9.
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function. The data points were fitted to a function of the form V%‘(t)= u-/?/f (see Fig. 2) by the method of least squares. The overall fit of the data was significant at the 0.05 level of probability in each case. For the cultures with po = 0.1.05 10.7 and 0.9, t@(t) is above the lower bound of W* -H(p,, p*)/t. For the culture with p0 = 0.3. the data are below this bound. All populations definitely show the trend of converging to W*. The shaded area in Fig. 1 is H(p,, p*). The corresponding experimental observation is given by the area bounded by W* and the locus of W(t) as it approaches W*. The latter area was computed by substituting the estimated functions, I%(t), into the left-hand side of equation (16) and then performing the integration. The lower limit of integration was set at 0.1 and the upper limit was set at the time such that t@(t) was within 0.02 of W*. This same measure of entropy was used by Costantino, Ginzburg & Moffa (1977) in the experimental evaluation of the concept of selective delay (Ginzburg, 1977). namely, H(p,, p*) = j [W* - W(r)] dt = W*t.
For both data sets (Fig. 3) the hypothesis that the data fit the theoretical H(p,, p*) was accepted using a chi-square test at the 0.05 level of probability, although for the area data the computed chi-square value was just less than the critical value.
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FIG. 3. An experimental check of H(p,, p* = 0.25), solid line, with W*s(p,, p*), circles, and J (W* - W(t)) dt, squares.
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6. Discussion The first theoretical prediction given in this paper is that the average fitness, I@(f), is bounded by the function W*-H(po, p*)/t until finally @(r/(t)= W* [equation (17)]. For the cultures with p. = 0.1, 0.5, 0.7 and 0.9. this inequality was satisfied. The second theoretical forecast is that the area bounded by W* and the locus of I@(r) as it approaches W* is equal to the entropy distance [equation (16)]. For the experimental cultures with p0 = 0.1 and 0.9, this criterion was met: for p0 = 0.3, 05 and 07, @t(t) did not approach W* as rapidly as theoretically predicted. Fisher’s theorem states that W(t) is a monotonically increasing function of time. The observed average fitness values display some oscillations and fail to “stick” to W*. This latter behavior of I@(t) can explain why { [W* - r;i/(t)] dt is greater than the expected H(p,, p*). Why does W(t) oscillate? Our interpretation is that the distribution of genotypes is changing dramatically and although each culture is converging to the equilibrium a&/t array of 05 +/+ and 0.5 +/cos, genetic segregation alters the equilibrium genotypic array in the immat~ve stages, when we experimentally measured the populations, resulting in our estimates of U’(t) wavering somewhat in the vicinity of IV*. Another explanation of why the areas were greater than H(p,, p*) is stated in equation (15). Our genetic analyses have revealed two alleles at the unsaturated fatty acid sensitive locus; however, the presence of additional “hidden” alleles not yet identified would result in the observed areas being greater than H(p,, p*). Different information from the same overall experiment (Moffa & Costantino, 1977) has yielded two estimates of the genetic distance. In the first case we examined W*T = H(p,, p*)? where T is the asymptotic timedelay of the real growth curve in comparison to the equilibrium growth curve. To estimate this parameter we compared the time required of each initial non-equilibrium population to reach the number of adults predicted by the equilibrium culture at week 4. I@(t) was computed directly from the observed number of adults. In the present paper we have equation (16). It was suggested that W(f) can be measured from the population growth curve [equation (5)]. The theory requires the identification of small changes in this curve, dN/dt, to be reflected in W(f). These changes may be difficult to extract experimentally. In T. castaneum, the initial growth rate reflects the initial genotypic array when populations are started with young adults only. However, as p. -+ p* the demographic array (eggs, larvae, pupae, adults) controls population growth
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and changes in I?(t) these data we did estimates of the total observed genotypic
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are difficult to observe. Consequently, to obtain i@(t) in not use population growth information but rather number of offspring for each genotype together with the arrays. Both data sets agree reasonably well.
This is a contribution No. 290 from the Program in Ecology and Evolution at the State University of New York at Stony Brook. REFERENCES COSTANYINO, R. F., GINZBURG, L. R. & MOFFA, A. M. (1977). J. rheor. Biol. 68, 317. CROW. J. F. & KIMURA. M. (1970). An Introduction to Population Generics Theory. New York: Harper and Row. FISHER, R. A. (1930). The Genetic Theory qf Natural Selection. Oxford: Clarendon Press. GINZBURG, L. R. (1972). J. gen. Biol. 33, 77 (in Russian). GINZBURG, L. R. (1977). J. theor. Biol. 67,671. KIMURA, M. (1965). Genetics 52, 875. KULLBACK, S. (1960). Information Theory and Sfatbtics. New York: John Wiley and Sons, Inc. MAYNARD SMITH, J. (1976). Am. Nar. 110, 331. MOFFA, A. M. & COSTANTINO, R. F. (1977). Generics 87, 785. NAGYLAKI, T. (1976). Genetics 83, 583.