©
DOI: 10.1534/genetics.104.036053
Protecting Haploid Polymorphisms in Temporally Variable Environments
Antony M. Dean
1BioTechnology Institute and Department of Ecology Evolution and Behavior, University of Minnesota, St. Paul, Minnesota 55108
Manuscript received September 7, 2004 Accepted for publication November 11, 2004
ABSTRACT
Analysis of a continuous-time model shows that a protected polymorphism can arise in a haploid pop-ulation subject to temporal fluctuations in selection. The requirements are that poppop-ulation size is regulated in a density-dependent manner and that an allele’s arithmetic mean relative growth rate is greater than one when rare and that its harmonic mean relative growth rate is less than one when common. There is no requirement that relative growth rate be frequency dependent. Comparisons with discrete-time models show that the standard formalism used by population genetics ignores forced changes in generation time as rare advantageous alleles sweep into a population. In temporally variable environments, frequency-dependent changes in generation times tend to counteract these invasions. Such changes can prevent fixation and protect polymorphisms.
T
HE means by which temporal fluctuations in selec- 2002), and even nontransitive fitness relations that allow clones to endlessly pursue one another across uniform tion might maintain polymorphisms were firstex-plored by Dempster (1955). He showed that a rare surfaces (Kerret al.2002). None invoke temporal fluc-tuations in selection for haploids.
allele in an infinite randomly mating diploid population
will increase in frequency if the geometric mean fitness Competition between haploid clones is conceptually no different from competition between species. Ecolo-of the heterozygote exceeds that Ecolo-of the common
homo-zygote. Neither allele can fix if the geometric mean fit- gists long ago showed that temporal variability in the en-vironment can promote coexistence (StewartandLevin ness of the heterozygote is greater than those of both
homozygotes, and so the polymorphism persists indefi- 1973; Levins 1979; Armstrong and McGehee 1980; Chesson1985, 2000), as when a resource fluctuates either nitely. In a haploid population there are only two
geno-types and the allele with the larger geometric mean side of the intersection of two nonlinear growth curves (Figure 1). Realistic models tend to be particular, mak-fitness inevitably sweeps to fixation—there is no third
genotype to counteract this invasion. Dempster con- ing generalizations difficult. Other models, perhaps un-realistic, are concocted to illustrate principles. Some cluded that temporal fluctuations in selection can
pro-tect polymorphisms in diploids but not in haploids. general criteria leading to coexistence have been exam-ined, but lack intuitive appeal.
Later,HaldaneandJayakar(1963), and then
Gilles-pie(1972, 1973a,b), analyzed similar models and came Using a continuous-time model of chemostat compe-tition, I derive the conditions under which temporal to exactly the same conclusion.
Dempster’s (1955) model is widely cited as one of fluctuations in selection can maintain a genetic poly-morphism in a haploid species subject to density-depen-several mechanisms to maintain polymorphisms in
dip-loids. Haploid monomorphism barely gets a mention. dent population regulation. Nevertheless, the notion that temporal fluctuations in
selection cannot protect polymorphisms in haploids is
deeply ingrained (Felsenstein1976;Hedricket al.1976; BACKGROUND
Hedrick1986;Gillespie1991;Maynard Smith1998).
Chemostat competition: The model of competition All recent investigations invoke alternative selection
between two haploid clonal populations for a single schemes: fitness fluctuations in space (either implicitly,
growth-limiting resource inhabiting a chemostat is
e.g.,Dean 1995, or explicitly, e.g., Rainey and
Travi-sano1998), cooperativity (Rosenzweiget al.1994;Rainey dN
1
dt ⫽ (1(S)⫺D)N1 (1)
andTravisano1998), stabilizing frequency-dependent selection (Bohannanand Lenski2000; Lunzeret al.
dN2
dt ⫽ (2(S)⫺D)N2 (2)
1Address for correspondence: BioTechnology Institute, University of
Minnesota, 1479 Gortner Ave., St. Paul, MN 55108-6106. dS
dt ⫽ D(S0⫺S)⫺
1(S)
Y1
N1⫺ 2(S)
Y2
N2, (3)
E-mail: [email protected]
N1⫽0). After inoculation, the density of clone 2 grows
as it consumes resources. Eventually, the resource con-centration becomes sufficiently reduced that growth rate slows and a stable steady state is approached where
dN2/dt⫽ 0,dS/dt⫽ 0, and
2(Sss.2)⫽D (7)
N2⫽N2.ss⫽Y2(S0⫺Sss.2). (8)
Here,S⫽Sss.2is the resource concentration in the
chem-ostat growth chamber when clone 2 is at its steady-state carrying capacityN2 ⫽N2.ss.
This population is subject to density-dependent popu-lation regupopu-lation. The mechanism of regupopu-lation is re-source depletion. In the absence of mutation or ex-ternally imposed change, the population will remain at
N2⫽N2.ssand will continue to grow at2(Sss.2)⫽Dwith
generation timeg2(Sss.2)⫽Loge2/D, indefinitely. A simple clonal sweep:Now imagine that a second, fitter clone is introduced at very low density to this Figure1.—Coexistence of both types is possible if the re- steady-state chemostat population. At the beginning of source concentration moves either side of the intersection of
the sweep, whent⫽0, the growth rate of the invading
the growth curves (growth rate curves) so that each type is
type 1 must exceed that of the resident clone 2 and
favored alternately. This phenomenon was first explored by
StewartandLevin(1973).
1(Sss.2)⬎ 2(Sss.2)⫽D forN1⬇0,N2⫽ N2.ss; (9)
otherwise invasion is not feasible. Toward the end of where N1 and N2 are the densities of the competing the sweep, whent→∞, the growth rate of clone 1 must
clones,1(S) and2(S) are their rates of growth, andD slow to the chemostat dilution rate and
is the chemostat dilution rate (the fractional rate of
re-placement of medium in the growth chamber).S0and 2(S)⬍ 1(S)⫽ D asN1 →N1.ss,N2 →0; (10)
S are, respectively, the concentrations of the
growth-otherwise the population grows without bound. Loss of limiting resource enteringthe growth chamber and in
clone 2 brings the population to a new steady state the growth chamber. Y1 and Y2 are yield coefficients
characterized by that determine the biomass produced per amount of
resource consumed. 1(Sss.1)⫽ D (11)
To connect growth rates to resource levels assume
N1⫽ N1.ss⫽ Y1(S0⫺Sss.1), (12)
that the former are concave monotonic functions of the
latter. For example, let where S ⫽ S
ss.1 is the resource concentration in the
chemostat growth chamber with clone 1 alone at steady
i(S)⫽
maxiS Ki⫹S
, i⫽1, 2, (4)
state. As with clone 2, clone 1 is subject to density-dependent population regulation generated by re-where maxi is the maximum rate of growth of clonei source depletion.
when the resource is in excess, andKiis a half-saturation In simple models of resource competition such as constant, the concentration of resource sufficient to this, the superior competitor reduces the resource con-allow growth at exactly half the maximum rate. centration to a point where the inferior competitor can Cell generation times are defined as the average time no longer sustain itself. With Sss.1 ⬍ Sss.2 the inferior
for one cell to become two (Kubitschek1970), competitor is washed from the chemostat growth cham-ber. This is true for any simple concave monotonic
gi(S)⫽ Loge2
i(S)
, i⫽ 1, 2, (5) growth function such as Equation 4 (Tilman1982). Relative fitness:The fitness of clone 1 relative to clone 2 is defined as a ratio of growth rates (Lunzer et al.
with the number of cell generations elapsed during time
2002),
tgiven simply as
w1
2(S)⫽ 1
(S)
2(S)
. (13)
Gi(S)⫽ i(S)
Loge2
t, i⫽ 1, 2. (6)
the appearance of frequency dependence in relative The population generation timeis proportional to the reciprocal of the average growth rate
fitness. This does nothing to affect the outcome of com-petition, however, and the favored clone sweeps
inexor-ably to fixation. Loge2
(S) ⫽ Loge2
D . (21)
Changes in growth rates do not necessarily imply
changes in relative fitness. Following inoculation of a Being a function of the dilution rate, the population chemostat the concentration of the limiting resource is generation time is under the direct control of the exper-commonly reduced far below the half-saturation con- imenter. Whereas the population generation time is a stant (SⰆ K1,K2;Lunzeret al.2002) and reciprocal of the average growth rate, theaverage cell
gen-eration time,g1(S)p⫹g2(S)q⬆Loge2/Dand is an aver-w1
2(S)⬇
max 1/K1
max 2/K2
. (14) age of the reciprocals of growth rates.
This case illustrates why, during clonal sweeps, relative
CHEMOSTAT THEORY FOR
fitness will often remain independent of the slowdown
VARIABLE ENVIRONMENTS
in growth rates produced by the decline in resource
Clonal competition in a variable environment: Now abundance.
imagine the same two clones competing in a chemostat, Estimating relative fitness:A small change in the
den-but allow the environment to vary from time to time. sity of an exceedingly rare clone has a negligible impact
As before, imagine that clone 1 is exceedingly rare and on resource abundance. With a second clone at its
car-attempting to invade a numerically dominant clone 2 rying capacity the system enters a quasi-steady state,
at steady state. Let the growth rate of clone 1 in environ-characterized bydS/dt⬇0. Over short periods of time
mentjbe1.j(Sss.2.j), the dilution rate in environmentj
the growth rates remain virtually constant. Equations 1
beDj, and the time spent in environment jbe⌬tj. As and 2 can be integrated:
before, assume1.j(Sss.2.j) is constant over each interval N1(t)⬇N1(0)e(1(S)⫺D)t (15) ⌬tj. Afternenvironmental changes the density of clone
1, initiallyN1(0) and very rare, is
N2(t)⬇N2(0)e(2(S)⫺D)t. (16)
N1(T)⫽N1(0) ·e兺
n
j⫽1(1.j(Sss.2.j)⫺Dj)·⌬tj forN
1.j⬇0,N2.j⫽N2.ss.j
Taking Logeratios yields
(22)
Loge
冢
N1(t)N2(t)
冣
⬇Loge
冢
N1(0)N2(0)
冣
⫹s(S)t. (17) at timeT⫽ 兺jn⌬tjand withN1(T) remaining very small.
Similarly, when clone 2 is very rare and clone 1 is at steady state
The slope of a plot of the Logeratio of clone densities
against time is commonly used to estimate the selection N
2(T)⫽N2(0) ·e兺
n
j⫽1(2.j(Sss.1.j)⫺Dj)·⌬tj forN
1.j⫽N1.ss.j,N2.j⬇0.
coefficients(S)⫽(1(S)⫺ 2(S)) per hour. The fitness
(23) of clone 1 relative to clone 2 is simply
This model is approximate. It assumes that tiny changes in the densities of rare clones have negligible impacts
w1
2(S)⫽1⫹
s(S)
2(S)
. (18)
on growth rates. It also assumes that selection during the transitions between environments is insignificant Relative fitness and generation times:Relative fitness
compared to the selection at quasi-steady state within can also be described as a ratio of cell generation times
environments where the dominant clone, growing at or as a ratio of the number of generations per unit time
rateDj, remains close to its carrying capacity,N2.ss.j.
Fi-nally, Equations 22 and 23 are concerned only with the
w1
2(S)⫽
g2(S)
g1(S)
⫽G1(S)
G2(S)
. (19)
fates of very rare clones. The interior dynamics when both clones have comparable frequencies and dynamics The average growth rate at quasi-steady state is approxi- far away from quasi-steady state shall not concern us.
mately Coexistence in a variable environment: Coexistence
is assured if the densities of each clone increase when
(S)⫽ 1(S)p⫹ 2(S)q⬇ D, (20)
rare. This requires where p⫽ 1 ⫺ q⫽ N1/(N1 ⫹ N2) is the frequency of
兺
nj⫽1
(1.j(Sss.2.j)⫺ Dj) ·⌬tj⬎0 forN1.j⬇0,N2.j⫽ N2.ss.j
clone 1. In practice this approximation is excellent
be-cause the difference in steady-state resource levels is (24) but a tiny fraction of the total resource entering the
chemostat growth chamber [i.e., (Sss.2⫺ Sss.1)/S0⬇ 0].
兺
n
j⫽1
(2.j(Sss.1.j)⫺Dj) ·⌬tj⬎0 forN1.j⫽N1.ss.j,N2.j⬇0.
Thus, the population neither increases nor decreases
A further simplification uses 1⫽兺
n
j⫽11.j(Sss.2.j)⌬tj/ 兺n
j⫽1⌬tj,2⫽兺
n
j⫽12.j(Sss.1.j)⌬tj/兺 n
j⫽1⌬tj, and D⫽兺
n j⫽1Dj·
⌬tj/兺nj⫽1⌬tjto produce
1⬎D forN1.j⬇0,N2.j⫽ N2.ss.j (26)
2⬎D forN1.j⫽N1.ss.j,N2.j⬇0. (27)
Although they turn out to be less useful than (24) and (25), (26) and (27) state the obvious: that the average growth rate of a rare type must exceed the average dilution rate if it is to persist.
Define ␥k⫽Dk⌬tk/兺 n
j⫽1Dj⌬tj as the proportion of
population generations spent in environmentkand sim-ply rewrite (24) and (25) as
兺
nj⫽1
␥jw12.j(Sss.2.j)⬎1 forN1.j⬇0,N2.j⫽ N2.ss.j (28)
兺
nj⫽1
␥jw21.j(Sss.1.j)⬎1 forN1.j⫽N1.ss.j,N2.j⬇0. (29)
Figure 2.—A protected polymorphism (shaded region)
We see immediately that two clones can coexist when- arises in a haploid population subject to density-dependent ever their weighted arithmetic mean relative fitnesses, regulation when the arithmetic mean fitness (relative growth rate) of an allele is greater than one when rare and its
har-when rare, are⬎1. Since fitness is defined as a ratio of
monic mean fitness is less than one when common. The
popu-growth rates, sow2
1.j(Sss.1.j)⫽1/w12.j(Sss.1.j) and Equation
lation is assumed to spend half its time in each environment.
29 can be rewritten as
1
兺
nj⫽1␥j/w12.j(Sss.1.j)
⬍1 forN1.j ⫽N1.ss.j,N2.j⬇0. (30) The generation time effect is explained as follows.
Without loss of generality, assume relative fitness remains constant withw1
2.j⫽w12.j(Sss.1.j)⫽ w12.j(Sss.2.j). The growth
This makes plain that the reciprocal of the arithmetic
rate of clone 2 when common is 2.j(Sss.2.j) ⫽ Dj and
mean relative fitness of the invading type is the
har-when rare is 2.j(Sss.1.j)⫽ Dj/w12.j. The number of cell
monic mean relative fitness of the resident clone.
generations that clone 2 spends in an environment In the special case where relative fitness is constant,
when common is (2.j(Sss.2.j)/Loge2)⌬tj⫽(Dj/Loge2)⌬tj
andw1
2.j(Sss.2.j)⫽w12.j(Sss.1.j) regardless of clone frequency,
and when rare is (2.j(Sss.1.j)/Loge2)⌬tj⫽(Dj/w12.jLoge2)·
Equations 28 and 30 provide the conditions necessary
⌬tj. As clone 1 invades so clone 2 spends fewer and to protect both clones from loss (Figure 2). Temporal
fewer generations in environments where clone 1 is fluctuations in selection can protect genetic
polymor-favored (since Dj⌬tj/w1
2.j⬍Dj⌬tj whenw12.j⬎1), yet ever
phisms in haploid clones in the absence of
frequency-more generations in environments where clone 1 is dependent selection. All that is needed is for each allele,
disfavored (whereDj⌬tj/w1
2.j⬎ Dj⌬tjwhenw12.j⬍ 1).
Fre-when rare, to have a weighted arithmetic mean relative
quency-dependent changes in cell generation times, fitness⬎1.
driven by changes in resource levels, counteract the The generation time effect: Using␥k⫽Dk⌬tk/兺
n j⫽1Dj·
invasion. If sufficient they can prevent clone 1 reaching
⌬tjand␥2.k⫽ 2.k(Sss.1.k)⌬tk/兺 n
j⫽12.j(Sss.1.j)⌬tjas the
pro-fixation. portion of cell generations clone 2 spends in each
envi-A haploid polymorphism can be protected in a tempo-ronment when common and when rare, rewrite (24)
rally variable environment when relative fitness and and (25) as
selection coefficients remain independent of allele fre-quencies so long as the population is subject to
density-兺
nj⫽1
␥jw12.j(Sss.2.j)⬎1 forN1.j⬇0,N2.j⫽ N2.ss.j, (31) dependent regulation.
兺
nj⫽1
␥2.jw12.j(Sss.1.j)⬍1 forN1.j⫽N1.ss.j,N2.j⬇0. (32)
DISCRETE-TIME MODELS
We see that when relative fitness is constant (w1
2.j(Sss.1.j)⫽ The conclusion that a haploid polymorphism can be
w1
2.j(Sss.2.j)) frequency-dependent changes in growth protected in a temporally variable environment when
rates, reflected as changes in the proportion of cell fitness is independent of allele frequency is seemingly generations clone 2 spends in each environment when at variance with the conclusions drawn from analyses of common and when rare (␥j ⬆ ␥2.j), promote coexis- fluctuating selection using discrete-time models. Here,
A simple clonal sweep revisited:The continuous-time appears to be frequency dependent when time is mea-sured on a per population generation (i.e., an absolute) model of a simple clonal sweep can be recast in discrete
time with scale. This frequency dependence is entirely attribut-able to an unacknowledged inversion of relative fitness from the continuous-time model.
p(t)⫽ p(0)W(S,t)
p(0)W(S,t)⫹ q(0), (33) When dealing with fitnessrelativeto clone 2 it is natu-ral to measure time inclone 2generations, even though wherep(t)⫽1⫺ q(t)⫽ N1(t)/(N1(t)⫹N2(t)) is the
these change with clone frequency. To do this set time frequency of clone 1 and
in Equation 35 tot⫽ g2(Sss.2)⫽ Loge2/Dwhen clone W(S,t)⬇e(1(S)⫺2(S))·t (34)
2 is common andt⫽g2(Sss.1)⫽w12(Sss.1)Loge2/Dwhen
clone 2 is rare. The result is is fitness at resource concentrationS over the intervalt
(Hartl and Clark 1997). Just as with Equations 15 Loge(W(Sss.2,g2(Sss.2)))⫽ (w12(Sss.2)⫺ 1)Loge2
and 16, this approximation is very accurate whenevert
forN1.j⬇ 0,N2.j⫽N2.ss.j (38)
is sufficiently small that changes inS, and hence changes
in the growth ratesi(S), are negligible. Nevertheless, Loge(W(Sss.1,g2(Sss.1)))⫽ (w12(Sss.1)⫺ 1)Loge2
as the clonal sweep proceedsW(S,t) must vary withS
forN1.j⫽ N1.ss.j,N2.j ⬇0. (39)
as it declines from Sss.2 to Sss.1. If variation in W(S, t)
with clone frequency is interpreted as frequency depen- These equations show that the frequency-dependent dence, then the chemostat model of a clonal sweep is, by selection in the discrete-time model disappears when definition, a model of frequency-dependent selection. w1
2(Sss.2)⫽w12(Sss.1) and time is measured units of clone
Measuring time: The fitness of a clone is commonly 2 generations.
expressed relative to another, with time measured on a Returning to time measured per population gener-per generation basis (HartlandClark1997). Sett⫽ ation, letG(Sss.2)⫽ (D/Loge2)t ⫽ 1 be one clone 2 cell
Loge2/D so that time in Equation 34 is measured in generation when clone 2 is common. Then clone 2 spends
terms of one population generation. Then Logefitness G(Sss.1)⫽(D/Loge2w12(Sss.1))t⫽1/w12(Sss.1) cell
genera-over one population generation is simply tions over period t when rare. Now rewrite Equations 36 and 37 as
Loge(W(S, 1))⫽ (1(S)⫺ 2(S))
Loge2
D . (35) Loge(W(Sss.2, 1))⫽ G(Sss.2) · (w12(Sss.2)⫺ 1)
forN1.j⬇0,N2.j⫽ N2.ss.j (40)
At the beginning of the sweep, whenN2 ⫽ N2.ss,N1⬇
0,2(Sss.2)⫽D, andw21(Sss.2)⫽ 1(Sss.2)/D, Equation 35 Log
e(W(Sss.1, 1))⫽ G(Sss.1) · (w12(Sss.1)⫺ 1)
becomes
forN1.j⫽N1.ss.j,N2.j⬇ 0. (41)
Loge(W(Sss.2, 1))⫽(w12(Sss.2)⫺ 1)Loge2
This formulation presents the two causes of frequency
forN1.j⬇0,N2.j⫽ N2.ss.j. (36) dependence in the discrete-time model. First is the passive
decrease in the number of clone 2 cell generations per Toward the end of the sweep, whenN1⫽ N1.ss,N2⬇0,
unit time, fromG(Sss.2)⫽1 toG(Sss.1)⫽1/w12(Sss.1),
attrib-1(Sss.1) ⫽ D, and w12(Sss.1)⫽D/2(Sss.1), Equation 35
utable to density-dependent population regulation. Sec-becomes
ond are biological mechanisms that cause changes in rela-Loge(W(Sss.1, 1))⫽(1⫺ w21(Sss.1))Loge2 tive fitness such thatw12(Sss.2)⬆w12(Sss.1).
The 1955 Dempster classic:Comparing the geomet-for N1.j ⫽N1.ss.j,N2.j⬇0. (37)
ric mean fitnesses of the haploid discrete-time model Inspecting Equation 37 reveals that the relative fitness (W(Sss.i.j,⌬tj)⫽ n
√
⌸nj⫽1(W(Sss.i.j,⌬tj))) to the haploid
con-of the continuous-time model [w2
1(Sss.1) on the right] is tinuous-time model (after being Logetransformed)
re-not that of clone 1 with respect to clone 2 but rather veals that that of clone 2 with respect to clone 1. This quirk arises
because we measured time in units of one population Loge(W(Sss.2.j,⌬tj))⫽ 1
n
兺
n
j
(s(Sss.2.j) ·⌬tj)
generation (t⫽Loge2/D) and this corresponds to one
cell generation of the numerically dominant clone. In forN
1.j⬇0,N2.j⫽N2.ss.j (42)
the course of the selective sweep the numerically
domi-nant clone changes from clone 2 in Equation 36 to Log
e(W(Sss.1.j,⌬tj))⫽
1
n
兺
n
j
(s(Sss.1.j) ·⌬tj)
clone 1 in Equation 37. Factoring1(Sss.1)⫽D[instead
of2(Sss.2)⫽ D] to produce Equation 37 forces fitness for
N1.j⫽N1.ss.j,N2.j ⬇0, (43)
of the continuous-time model to be inverted. When w1
2(Sss.2)⫽w12(Sss.1) there is no frequency de- wheres(Sss.i.j)⫽ 1.j(Sss.i.j)⫺ 2.j(Sss.i.j) is the selection
co-efficient per hour. pendence in relative fitness in the continuous-time
allele frequency by settingW(Sss.2.j,⌬tj)⫽W(Sss.1.j,⌬tj). rate of a rare heterozygote exceeds those of the common
homozygotes. As a consequence the clone with the largest geometric
By convention (e.g., see Felsenstein1976), the fit-mean fitness wins the competition.
nesses of the homozygotes are usually presented relative Applying the same definition to the continuous-time
to that of the heterozygote: chemostat model forces the selection coefficient per
hour to be independent of allele frequency,s(Sss.2.j)⫽
s(Sss.1.j). During a clonal sweep the growth rates slow
1
兺
nj⫽1␥j/wAAAa.j
⬍1 forpa⬇0 (46)
with the decline in S. As a consequence, the selection
coefficients per clone 2 cell generation become frequency 1
兺
nj⫽1␥j/waaAa.j
⬍1 forpA⬇0. (47)
dependent:s(Sss.2.j)/2.j(Sss.2.j)⬆s(Sss.1.j)/2.j(Sss.1.j).
De-fining fitness per unit time as independent of allele
Hence, a protected polymorphism can be established frequency in the discrete-time model forces the
selec-in a temporally variable environment when the weighted tion per cell generation to become frequency
depen-harmonic mean fitnesses of both homozygotes are each dent in the continuous-time chemostat model.
less than that of the heterozygote. The discrete-time model can be modified to
accom-modate the slowdown in growth rates during a clonal
sweep by definingW(Sss.i.j,⌬tj)⫽(V(Sss.i.j))G2.j(Sss.i.j), where EXPERIMENTAL EVIDENCE V(Sss.i.j)⫽e(w
1
2.j(Sss.i.j)⫺1) is relative fitness and G
2.j(Sss.i.j) ⫽
The above theory makes two predictions: (1) the num-(2.j(Sss.i.j)/Loge2)⌬tjis the number of cell generations
ber of cell generations varies with time and (2) both experienced by clone 2 in environment j. The
condi-clones can increase in frequency when rare in a tempo-tions necessary to protect a polymorphism can now be
rally variable environment. Neither is predicted by the rewritten as
classical population genetics theory.
Frequency-dependent changes in generation time
兺
nj⫽1
Loge(W(Sss.2.j,⌬tj))⫽
兺
nj⫽1
(G2.j(Sss.2.j) · Loge(V(Sss.2.j)))
have been observed directly. Equations 16 and 17 state that when relative growth rates are independent of
fre-⫽
兺
nj⫽1
冢
Dj⌬tj Loge2· (w1
2.j(Sss.2.j)⫺1)
冣
⬎0quency [i.e., whenw1
2(Sss.2)⫽w12(Sss.1)] and proportional
to resource abundance, the selection coefficient per
forN1.j⬇0,N2.j⫽N2.ss.j (42⬘) population generation will decline during a selective
sweep, froms(Sss.2)⫽ ((max 1/K1)Sss.2⫺(max 2/K2)Sss.2)
兺
nj⫽1
Loge(W(Sss.1.j,⌬tj))⫽
兺
nj⫽1
(G2.j(Sss.1.j) · Loge(V(Sss.1.j)))
to s(Sss.1)⫽ ((max1/K1)Sss.1 ⫺ (max 2/K2)Sss.1).
Chemo-stat competition experiments between Escherichia coli
⫽
兺
nj⫽1
冢
Dj⌬tj
w1
2(Sss.1.j)Loge2 · (w1
2.j(Sss.1.j)⫺1)
冣
⬍0 strains TD10 and TD2 for growth-limitingconcentra-tions of the sugar methylgalactoside produce sufficiently strong selection that the change in the intensity of
selec-forN1.j⫽N1.ss.j,N2.j⬇0. (43⬘)
tion can be discerned (Figure 3). Whenever selection The right-hand sides of these two inequalities can be re- is very intense, frequency-dependent changes in cell arranged using␥k⫽Dk⌬tk/兺
n
j⫽1Dj⌬tj to produce Equa- generation time need to be taken into account when
tions 28 and 29 that define the conditions for protecting estimating relative fitness (Lunzeret al.2002). the polymorphism. The discrete-time model has been Studying selection in a temporally variable environ-reconciled with the continuous-time chemostat model ment directly is immensely difficult, even in near ideal-by simply accounting for the change in number of cell ized laboratory populations ofE. coli(Suiteret al.2003). generations per unit time generated by changes in clone Instead, our evidence is indirect, the argument taking an interesting, if circuitous, route. First, a model of frequencies.
authentic stabilizing frequency-dependent selection in The diploid model: The diploid model is a simple
a uniform environment is described and the evidence extension of the haploid model—we need consider only
for it presented. Second, the model is extended to ac-invasion of an equilibrium population of homozygotes
commodate temporal variability. Third, it is shown that by those few heterozygotes carrying an exceedingly rare
the stabilizing frequency dependence seen in a uniform allele. The conditions for invasion are
environment, and the balancing selection generated in a temporally variable environment, are synonymous.
兺
nj⫽1
␥jwAaAA.j⬎ 1 forpa⬇0 (44)
Fourth, since the one implies the other, demonstrating stabilizing frequency dependence in a uniform
environ-兺
nj⫽1
␥jwAaaa.j⬎ 1 forpA⬇0. (45) ment implies that the selection generated in a
Figure3.—The intensity of selection diminishes during a
selective sweep as growth rates slow down (data fromLunzer
Figure 4.—Frequency-dependent selection generated by
et al.2002). Pooled results from four replicate chemostat
ex-differential consumption of substitutable resources. TD10 is
periments in whichE. colistrain TD10 competes with strain TD2
fitter on methylgalactoside and TD2 is fitter on lactulose. A for limiting concentrations of methylgalactoside are shown;
zone of coexistence arises between 24 and 30.5%
methylgalac-w1
2.MG⫽1.312⫾0.006. The solid line is a second-order
poly-toside where strains are fitter than their competitor when rare nomial fitted to the data. The dashed line is drawn for
com-(arithmetic mean fitness), but less fit when common (harmonic parison.
mean fitness).
lection in constant environments. Theory predicts that, ronments is insignificant compared to selection at steady when growth rates are proportional to resource concen- state within environments. Define, as before, the num-trations (e.g.,i(SLU,SMG)⫽ ␣i.LUSLU⫹ ␣i.MGSMG), coexis- ber of population generations spent in each
environ-tence is possible whenever the arithmetic mean fitness ment,␥k⫽Dk⌬tk/兺n
j⫽1Dj⌬tj. Then it follows that
coexis-exceeds one and the harmonic mean fitness is less than tence is possible if one (Lunzeret al.2002):
兺
nj⫽1
␥j· (luj·w102.LU⫹mgj·w102.MG)⫽lu ·w102.LU⫹mg ·w102.MG⬎1
lu ·w10
2.LU⫹ mg ·w102.MG⬎ 1 forpTD10⬇0 (48)
forpTD10⬇0 (50)
1 lu/w10
2.LU⫹ mg/w102.MG
⬍ 1 forpTD2⬇0. (49)
1
兺
nj⫽1␥j· (luj/w102.LU⫹mgj/w102.MG)
⫽ 1
lu/w10
2.LU⫹mg/w102.MG ⬍1
Here, the super- and subscripts refer to E. coli strains TD10 and TD2,w10
2.LUandw102.MGare the relative growth forpTD2⬇0, (51) rates obtained during competition in chemostats for
where lu⫽兺nj⫽1␥jluj and mg⫽兺 n
j⫽1␥jmgj are average
100% lactulose (w10
2.LU⫽ ␣10.LU/␣2.LU⫽0.91) or 100%
resource abundances. Hence, coexistence is assured methylgalactoside (w10
2.MG⫽ ␣10.MG/␣2.MG⫽1.32), and lu
when the expected (i.e., mean) resource supply lies and mg⫽1⫺lu are the proportions of each resource
within the zone of coexistence. in the fresh medium supplied to the chemostats.
Experi-Third, there is no requirement for a realized resource ments confirm that a narrow zone of coexistence exists
supply to ever reside within the zone of coexistence. between 23 and 30.5% methylgalactoside (Figure 4). As
Suppose only one resource is ever present in the envi-long as the resource supply lies within this zone neither
ronment at one time (i.e., lu ⫽ 1, mg⫽ 0 or lu ⫽ 0, strain can fix.
mg⫽1), and let␥LUand␥MGrepresent the time spent
Second, temporal environmental variability plays no
consuming each. Then rewrite (50) and (51) as role in producing this balanced polymorphism.
Never-theless, it is instructive to introduce it to the model. ␥
LU ·w102.LU⫹ ␥MG·w102.MG⬎1 forpTD10⬇ 0 (52)
Assume, as before, that tiny changes in the frequencies
of rare types have negligible impacts on growth rates, 1
␥LU/w102.LU⫹ ␥MG/w102.MG
⬍1 forpTD2⬇0. (53)
envi-Figure5.—A protected polymorphism of two types competing for a single variable resource. (A) Relationships between growth
rates resource abundance. As clone 1 sweeps through a population with dilution rateDA⫽1.5 the resource abundance drops
from Sss.2.A ⫽ 45 m to Sss.1.A ⫽ 30 m, forcing growth rates to slow and relative fitness to decline from w12(Sss.2.A)⫽1.5 to
w1
2(Sss.1.A)⫽1.125. Despite evident frequency dependence, coexistence is not possible at a fixed dilution rate. (B) Regions of
coexistence (shaded) calculated assuming the population spends equal time atDA⫽1.5 andDB⫽0.5. The regions were calculated
assuming changes in death rates are infrequent. Individuals cannot respond to very rapid changes in dilution rates, which will
be perceived to converge on a single death rate,D⫽(DA⫹DB)/2, where coexistence is no longer possible.
We have just transited from a balanced polymorphism of selection can protect a polymorphism in a haploid species whose population size is subject to density-attributable to stabilizing frequency-dependent
selec-tion in a uniform environment [(50) and (51)] to a pro- dependent regulation. The growth rate of an invading tected polymorphism in a variable environment where type must exceed the dilution rate when rare, else inva-relative growth rates,w10
2.LU⫽0.91 andw102.MG⫽1.32, are sion fails, just as it must eventually slow to match the
independent of frequency [(52) and (53)]. dilution rate, else the population grows without bound. Fourth, the only difference between these equations Hence, density-dependent population regulation forces is that the former use lu and mg⫽1⫺lu as the propor- growth rates to slow as a selective sweep proceeds. Decel-tions of each resource available for consumption in a erating growth rates, which cause changes in generation uniform environment, whereas the latter use ␥LU and times, can be attributed to a variety of ecological
mecha-␥MG⫽1⫺ ␥LUto represent the proportion of time spent nisms. One of the most familiar, and the one modeled
consuming each resource in a variable environment. here, is resource depletion: an invading type consumes In both cases the same proportions of lactulose and a limiting resource more efficiently, reducing its abun-methylgalactoside are delivered into the environment, dance, which in turn lowers the growth rates of all com-and in both cases the same quantities of lactulose com-and petitors. This causes a decline in the intensity of selection methylgalactoside are consumed. Hence, the same mech- per population generation (but not per cell genera-anism, the differential consumption of resources, pro- tion) as a selective sweep proceeds (Figure 3).
motes coexistence both in constant and in variable en- A direct demonstration that temporal fluctuations in vironments. The only difference is whether competitors selection can protect a polymorphism is most improba-consume resources concurrently or consecutively. These ble. Measuring selection coefficients accurately in natu-models are synonymous, despite selection being fre- ral populations is extraordinarily difficult, and the gen-quency dependent when resources are consumed con- eration times of most species are too long for sufficient currently and frequency independent when they are data to be gathered. Laboratory populations of micro-consumed consecutively. organisms inhabiting defined environments provide better opportunities but here again time is limited. With reproduction clonal, periodic selection of new mutants
DISCUSSION
is likely to confound all long-term experiments. Nevertheless, indirect evidence has been obtained. Analyses of both continuous- and discrete-time
substitut-able resources promotes coexistence through stabilizing Dempster’s model is that temporal fluctuations in selec-tion cannot protect polymorphisms in haploid species frequency-dependent selection generated by
differen-unless selection is frequency dependent (Dempster tial resource depletion (Lunzer et al. 2002). Readily
1955; Haldane and Jayakar 1963; Gillespie 1972, transmogrified into a model of competition for two
al-1973a,b, 1991;Felsenstein1976;Hedricket al.1976; ternating substitutable resources, the frequency
depen-Hedrick1986; Maynard Smith1998). If variation in dence dissipates (fitness on a single resource is not a
W with clone frequency is interpreted as frequency de-weighted average) even as the ability to promote
coexis-pendence, then all clonal sweeps in all populations sub-tence is retained. Hence, stabilizing
frequency-depen-ject to density-dependent regulation are, by definition, dent selection produced by concurrent consumption
frequency dependent. We conclude that Dempster’s of resources is evidence that temporal fluctuations in
1955 discrete-time model is irrelevant to the vast major-selection, generated by consecutive consumption of
re-ity of haploid species subject to densre-ity-dependent pop-sources, can promote coexistence.
ulation regulation. What is true for one mechanism of selection is not
In definingW as relative fitness the standard formal-necessarily true for all. Consider the case where the
ism used by population genetics amalgamates two phe-growth rate of type 2 is a concave function of resource
nomena: frequency-dependent changes in generation abundance [e.g., k.j ⫽ maxk.jSss.i.j/(Kk.j ⫹ Sss.i.j),
Fig-time and frequency-dependent changes in relative fit-ure 5A]. Fitness now changes with clone frequency
be-ness. In ascribing all changes inW to frequency depen-cause the equilibrium resource levels (referred to as
dence in relative fitness, this formalism ignores the role
R* byTilman1982) differ according to which clone is
played by changing generation times in populations dominant [e.g., whenDA⫽1.5 hr⫺1,Sss.1.A⫽1.5/0.05⫽
regulated by density-dependent mechanisms. In
con-30mandSss.2.A⫽ 15 · 1.5/(2⫺1.5) ⫽45m]. This
founding relative fitness and cell generation time an frequency dependence changes the intensity of
selec-unnecessarily restrictive condition has been inadver-tion, but withDAheld constant it cannot change the
direc-tently imposed on the ability of temporal variations in tion of selection and so is unable to promote coexistence.
selection to protect polymorphisms. Thus was born the When the dilution rate moves across the intersection of
myth that temporal fluctuations in selection cannot pro-the growth curves a change in pro-the direction of selection
tect a polymorphism in a haploid species. is produced. Seasonal switches in the dilution rate between
DAandDBare necessary, although not sufficient, for coexis- I am deeply indebted to Ben Kerr for many hours of useful
discus-tence (Figure 5B). This example nicely illustrates two facts: sion that greatly clarified my thinking. I also thank Dick Hudson for criticizing an admittedly confusing first draft. This work was supported
(1) alone, frequency-dependent selection need not
pro-by National Institutes of Health grants to A.M.D.
mote coexistence, and (2) fluctuations in the availability of a single limiting resource are necessary, although not
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