Selection in a Cyclical Environment: Possible Impact of Phenotypic Lag on Darwinian Fitness

Abstract

We investigated the effect of generation time (as controlled by chemostat flow rate) and temporal variability in nutrient (arginine) availability on selection at a regulatory locus in Escherichia coli. We first determined the fitness conferred by argR^K12 (which regulates the arginine regulon) relative to argR^B (a weak constitutive) in constant environments at several generation times across a range of concentrations of arginine. The relative fitness of argR^K12 with respect to argR^B declines with longer generation times in the absence of arginine yet becomes independent of generation time in the presence of excess arginine. Control experiments show this differential response in selection is entirely attributable to transcriptional regulation by argR^K12. Temporal variability in the supply of arginine generates fluctuations in selection. A simple model, based on the assumption that relative fitness tracks changes in arginine availability instantaneously, captures many of the features of the oscillating allele frequencies and accurately predicts the direction and intensity of selection in environments where arginine concentrations fluctuate frequently or infrequently. However, the model fails to predict the direction and intensity of selection in environments that fluctuate at moderate frequencies. This suggests that phenotypic lag, wherein cellular physiology changes more slowly than the environment, may be influencing the outcome of competition in this experimental system.

Appendix

The Basic Chemostat Model

Let the growth of the two competing populations in a chemostat have a simple birth death process:

$$ dN_{K12} /dt=(\mu _{K12} - D)N_{K12} $$

(A1)

$$ dN_B /dt=(\mu _B - D)N_B $$

(A2)

where D is the chemostat dilution rate. N _K12 and N _B are the densities of the competing strains, and μ _K12 and μ_B are their growth rates. The latter are given by

$$ \eqalign{\mu _{K12}=&\left( {{{\mu _{K12.{\rm{max}} } G} \over {\gamma_{K12} + G}}} \right)\left( {1 - {{\beta _{K12} } \over {\kappa _{K12} + R}}} \right) \cr &- \left( {{{\chi _{K12} } \over {\kappa _{K12} + R}}} \right)\,\,} $$

(A3)

$$ \mu _B=\left( {{{\mu _{B.{\rm{max}} } G} \over {\gamma_B + G}}} \right)\left( {1 - \beta _B } \right) - \chi _B \, $$

(A4)

where G and R are the concentrations of glucose and arginine. The model is a modification of the classic Monod model where μ_i.max is the maximum growth rate of strain i (assuming expression of the arginine regulon imposes no cost to fitness) and γ_i is the concentration of glucose sufficient to produce μ_i.max/2. Two costs associated with expressing the arg regulon have been introduced: β _i is proportional to growth rate and χ _i proportional to absolute time. These costs are fixed in strains carrying argR^B. In strains carrying argR^K12, κ _K12 is the concentration of arginine necessary to halve the cost of regulon expression.

Let the rates of glucose and arginine consumption be proportional to growth rates:

$$ dG/dt=D(G_0 - G) - \left( {{{\mu _{K12} } \over {Y_G }}N_{K12} + {{\mu _B } \over {Y_G }}N_B } \right)\, $$

(A5)

$$ dR/dt=D(R_0 - R) - \left( {{{\mu _{K12} } \over {Y_R }}N_{K12} + {{\mu _B } \over {Y_R }}N_B } \right)\, $$

(A6)

where G₀ and R₀ are the concentrations glucose and arginine in the feed medium and Y _G and Y _R are the yield coefficients (number of cells produced per amount of resource consumed).

Selection at Quasi-Steady-State

After inoculation, the conditions in the chemostat growth chamber rapidly approach a quasi-steady-state where competition is intense and the environment is characterized by very slowly shifting states. The growth rate of the total population. N _T is zero ($dN_T/dt=dN_{K12}/dt + dN_B/dt=0$ ) and (A1) and (A2) sum to

$$ D\hat N_T=\hat \mu _{K12} N_{K12} + \hat \mu _B N_B $$

(A7)

where hats (^) denote quasi-steady-state values. The ambient concentrations of glucose and arginine are found by setting dG/dt = 0 and dR/dt = 0 and solving

$$ \hat G=G_0 - \hat N_T /Y_G $$

(A8)

$$ \hat R=R_0 - \hat N_T /Y_R $$

(A9)

With the environment essentially unchanging, (A1) and (A2) can be integrated to yield

$$ N_{K12} (t) =N_{K12} (0)e^{(\hat \mu _{K12} - D)_t } $$

(A10)

$$ N_B (t)=N_B (0)e^{(\hat \mu _B - D)t } $$

(A11)

Taking the log _e ratio produces

$$ \log _e \left[ {{{N_{K12} (t)} \over {N_B (t)}}} \right]=\log _e \left[ {{{N_{K12} (0)} \over {N_B (0)}}} \right] + \hat s(Dt)\, $$

(A12)

Thus the slope of a plot of the log_e ratio of strains against time (measured in Dt population generations) provides a direct estimate of the selection coefficient per generation, $\hat s=(\hat \mu _{K12} - \hat \mu _B )/D$. Relative fitness is simply

$$ \hat w_B^{K12}=1 + \hat s\, $$

(A13)

In the quasi-steady-state virtually all glucose is consumed and G << γ _B (Dykhuizen and Dean 1994). Let argR^B be by far the most frequent competitor. Then equations (A3) and (A4) can be rewritten

$$ \hat \mu _{K12}={{\mu _{K12.{\rm{max}} } \hat G} \over {\gamma_{K12} }}\left( {1 - {{\beta _{K12} } \over {\kappa _{K12} + \hat R}}} \right)\, -{\chi_{k12}\over\kappa_{k12}+\hat{R}} $$

(A14)

$$ \hat \mu _B={{\mu _{B.{\rm{max}} } \hat G} \over {\gamma_B }}\left( {1 - \beta _B } \right)\, - \chi _B=D\, $$

(A15)

Fitness is given by

$$\eqalign{ \hat w_B^{K12}=&{{\hat \mu _{K12} } \over {\hat \mu _B }}=w_{B.{\rm{max}} }^{K12} \left( {1 - {{\beta _{K12} } \over {\kappa _{K12} + \hat R}}} \right) \cr &+ w_{B.{\rm{max}} }^{K12} \left( {\chi _B /D} \right)\left( {1 - {{\beta _{K12} } \over {\kappa _{K12} + \hat R}}} \right) - {{\chi _{K12} /D} \over {\kappa _{K12} + \hat R}}\,} $$

(A16)

where $w_{B.{\rm{max}} }^{K12}=\left( {\mu _{K12.{\rm{max}} } / \gamma_{K12} } \right)/\left( {\left( {\mu _{B.{\rm{max}} } / \gamma_B } \right)\left( {1 - \beta _B } \right)} \right)$ is a constant that could be interpreted as the (fictional) fitness when arginine is present in excess, the dilution rate is infinite and the glucose concentration is zero. Fitting the model to the fitness data reveals that χ _B is tiny and not significantly different from zero. Hence, expression of the arg regulon in strains carrying argRB does not impose a cost proportional to absolute time. The model simplifies to

$$ \hat w_B^{K12}={{\hat \mu _{K12} } \over {\hat \mu _B }}=w_{B.{\rm{max}} }^{K12} \left( {1 - {{\beta _{K12} } \over {\kappa _{K12} + \hat R}}} \right) - {{\chi _{K12} /D} \over {\kappa _{K12} + \hat R}}\, $$

(A17)

Transient Arginine Kinetics

Whereas virtually all glucose is consumed (i.e.,$\hat G \to 0$), arginine concentrations will range from 0 to 1 mM. The transient behavior of arginine during periods between alternating steady-states needs to be described. Substituting equation (A7) into (A6) and integrating produces

$$ \eqalign{R(t)=&\left( {R_0 - \hat N_T /Y_R } \right) + \big( R(0)\cr &- \left( {R_0 - \hat N_T /Y_R } \right) \big) e^{ - Dt} \quad {\rm{for}}\;R_{\rm{0}} \, > \,\hat N_T /Y_R} $$

(A18)

where R(t) and R(0) are the concentrations of arginine at times t and 0.

When R₀ = 0 the time (t _crit ) needed to consume the remaining arginine, R(0) > 0 is

$$ t_{crit}={1 \over D}\log _e \left[ {1 + {{R(0)} \over {\hat N_T /Y_R }}} \right]\quad {\rm{for}}\;R_{\rm{0}}=0\, $$

(A19)

Substituting R ₀  = 0 into (A18) provides one solution when t < t _crit , otherwise

$$ R(t)=0\quad {\rm{for}}\;{{t > t}}_{crit} $$

(A20)

Transient Fitness

Fitness in a variable environment can be predicted from the relationship between fitness and arginine, (A17), and the kinetics of arginine in the chemostat growth chamber, (A18)–(A20). Assume argR^K12 is very rare so that μ _B  = D. Then the growth rate of argR^K12 is simply μ _k12 (t) = Dw ^K12 _B (t), and equations (A1) and (A2) become

$$ \eqalign{&dN_{K12} /dt \cr &\quad=\left( {\mu _{K12} (t) - D} \right)N_{K12} \cr &\quad=\left( {w_{B.{\rm{max}} }^{K12} \left( {1 - {{\beta _{K12} } \over {\kappa _{K12} + \hat R}}} \right) - {{\chi _{K12} /D} \over {\kappa _{K12} + \hat R}} - 1} \right)DN_{K12} } $$

(A21)

$$ dN_B /dt =0 $$

(A22)

Following integration the log _e ratio of strain densities is found to be

$$\eqalign{& \log _e \left[ {{{N_{K12} (t)} \over {N_B (t)}}} \right] \cr &\quad=\log _e \left[ {{{N_{K12} (0)} \over {N_B (0)}}} \right] \cr &\qquad\quad+ \left( {w_{B.{\rm{max}} }^{K12} - {{\left( {w_{B.{\rm{max}} }^{K12} \beta _{K12} + \chi _{K12} /D} \right)} \over {\kappa _{K12} + R_0 - \hat N_T /Y_R }} - 1} \right)Dt \cr &\qquad\quad - {{\left( {w_{B.{\rm{max}} }^{K12} \beta _{K12} + \chi _{K12} /D} \right)} \over {\kappa _{K12} + R_0 - \hat N_T /Y_R }}\cr &\qquad\times\log _e \left\lfloor {\left( {1 + {{R_0 - \hat N_T /Y_R - R(0)} \over {\kappa _{K12} + R(0)}}\left( {1 - e^{ - Dt} } \right)} \right)} \right\rfloor \quad\cr &\qquad\qquad\quad\quad\quad \rm{for \ }t<t_{{\rm{crit}}}} $$

(A23)

$$ \eqalign{ & \log _e \left[ {{{N_{K12} (t)} \over {N_B (t)}}} \right]\cr &\quad=\log _e \left[ {{{N_{K12} (0)} \over {N_B (0)}}} \right] \cr &\qquad\quad+ \left( {w_{B.{\rm{max}} }^{K12} - 1 - \left( {w_{B.{\rm{max}} }^{K12} \beta _{K12} + \chi _{K12} /D} \right) / \kappa _{K12} } \right)D(t - t_{crit} ) \cr &\qquad\quad+ \left( {w_{B.{\rm{max}} }^{K12} - {{\left( {w_{B.{\rm{max}} }^{K12} \beta _{K12} + \chi _{K12} /D} \right)} \over {\kappa _{K12} - \hat N_T /Y_R }} - 1} \right)Dt_{crit} \cr &\qquad\quad - {{\left( {w_{B.{\rm{max}} }^{K12} \beta _{K12} + \chi _{K12} /D} \right)} \over {\kappa _{K12} - \hat N_T /Y_R }}\cr &\qquad\quad\times\log _e \left[ {1 - {{\hat N_T /Y_R - R(0)} \over {\kappa _{K12} + R(0)}}\left( {1 - e^{ - Dt_{crit} } } \right)} \right]\quad \cr &\qquad\qquad\quad\quad\quad{ \rm{for}}>t_{{\rm{crit}}} \cr}$$

(A24)

Fitness in a Cyclical Environment

Let the delivery of two arginine concentrations, R_0.1 and R_0.2, in the fresh medium alternate between two half-cycles, lengths t₁ and t₂. Then the ambient arginine concentration at time t₁, R(t₁), is the R(0) at the beginning half-cycle 2, while the ambient arginine concentration at time t ₂ (R(t2), is the R(0) at the beginning of half-cycle 1. Thus,

$$ \eqalign{R(t_1)=&\left( {R_{0.1} - \hat N_T /Y_R } \right) \cr &+ \left( {R(t_2 ) - \left( {R_{0.1} - \hat N_T /Y_R } \right)} \right)e^{- Dt_1} \cr &{\rm{for}}R_{\rm{0}} > \,\hat N_T /Y_R} $$

(A25)

$$ \eqalign{ R(t_2 ) =&\left( {R_{0.2} - \hat N_T /Y_R } \right) \cr &+ \left( {R(t_1 ) - \left( {R_{0.2} - \hat N_T /Y_R } \right)} \right)e^{ - Dt_2 } \cr &\quad {\rm{for}}\;R_{\rm{0}} > \,\hat N_T /Y_R} $$

(A26)

which yield

$$ \eqalign{ R(t_1 ) =&\left( {R_{0.2} - \hat N_T /Y_R } \right) \cr & - {{\left( {R_{0.2} - R_{0.1} } \right)\left( {1 - e^{ - Dt_1 } } \right)} \over {\left( {1 - e^{ - Dt(t_1 + t_2 )} } \right)}}\quad\cr & {\rm{for}} \, t < t_{crit}} $$

(A27)

$$ \eqalign{R(t_2 ) =\ &\left( {R_{0.1} - \hat N_T /Y_R } \right) + {{\left( {R_{0.2} - R_{0.1} } \right)\left( {1 - e^{ - Dt_2 } } \right)} \over {\left( {1 - e^{ - Dt(t_1 + t_2 )} } \right)}}\quad\cr &{\rm{for}}\; t < t_{crit} } $$

(A28)

$$ R(t_i ) =0 \, {\rm{for}}\; R_{0.i}=0\;{ \rm{and}} t > t_{crit} $$

(A29)

With very long cycles most of the time is spent at one or the other quasi-steady-state, characterized by R(t ₁ ) = $\hat R_1$ = R_0.1− $\hat N_T /Y_R$ and R(t ₂ ) = $\hat R$₂ = R_0.2 − $\hat N_T /Y_R$. With very short cycles there is hardly time to displace arginine far from its arithmetic mean, and both (A27) and (A28) converge on $\overline R=\left( {R_{0.1} t_1 + R_{0.2} t_2 } \right)/\left( {t_1 + t_2 } \right) - \hat N_T /Y_R.$

Expected fitness in cyclical environments is calculated as the sum of the differences in log _e ratios of the strains over a full cycle divided by the total number of generations (D(t₁ + t₂)),

$$ {w_B^{K12}}={{\log _e \left[{N_{K12}(t_2)/N_B (t_2)}\right]} - \log _e {\left[ {N_{K12} (t_0)/N_B(t_0)}\right]} \over {{D(t_1 + t_2)}}} $$

(A30)

using the relationships in (A23), (A24), (A27), (A28), and (A29).

Our experiments are confined to half-cycles of equal length, t = t ₁ = t ₂ , with one feed arginine concentration set at zero, R_0.2 = 0. The time (t _crit ) needed to consume the remaining arginine is given by

$$ t_{crit}={1 \over D}\log _e \left\lfloor {{{R_{0.1} } \over {\hat N_T /Y_R }} - 1} \right\rfloor $$

(A31)

and fitness by

$$\eqalign{w_B^{K12}&=w_{B.\max}^{K12} - \left({w_{B.\max }^{K12} \beta_{K12} + \chi_{K12} /D}\right) {{\left( {\kappa_{K12} + R_{0.1} /2 - {\hat{N}}_T /Y_R } \right)} \over {\left( {\kappa_{K12} + R_{0.1} - {\hat{N}}_T /Y_R } \right)\left( {\kappa_{K12} - {\hat{N}}_T /Y_R } \right)}} \cr &\qquad- {{\left({w_{B.\max }^{K12} \beta_{K12} + \chi_{K12} /D} \right)} \over {2Dt}} \cr &\quad\times\left( {{1 \over {\kappa _{K12} + R_{0.1} - {\hat{N}}_T /Y_R }}log_e \left( {1 + {{R_{0.1} \left( {1 - e^{-Dt}} \right)^2 } \over {\left( {\kappa_{K12} + R_{0.1} - {\hat{N}}_T /Y_R } \right)\left( {1 - e^{-2Dt} } \right) + R_{0.1} \left( {1-e^{-Dt} }\right)}}} \right) + {1 \over {\kappa_{K12} - {\hat{N}}_T /Y_R }}log_e \left( {1 - {{R_{0.1} \left( {1 - e^{-Dt} } \right)^2 } \over {\left( {\kappa_{K12} - {\hat{N}}_T /Y_R } \right)\left( {1-e^{-2Dt}} \right) + R_{0.1} \left( {1-e^{ - Dt} } \right)}}} \right)} \right)\cr &\qquad\qquad{\rm{for \ }} t\leq t_{crit}} $$

(A32)

and

$$\eqalign{ w_B^{K12}&=w_{B.{\rm{max}} }^{K12} - {{\left( {w_{B.{\rm{max}} }^{K12} \beta_{K12} + \chi_{K12} /D} \right)} \over {2\kappa_{K12} }}\left( {{{2\kappa_{K12} + R_{0.1} - \hat{N}_T /Y_R } \over {\kappa_{K12} + R_{0.1} - \hat{N}_T /Y_R }}} \right) \cr &\qquad - {{\left( {w_{B.{\rm{max}} }^{K12} \beta_{K12} + \chi_{K12} /D} \right)} \over {2D\kappa_{K12} t}} {{\hat{N}_T /Y_R } \over {\kappa_{K12} - \hat{N}_T /Y_R }}\times log_e \left( {1 + {{R_{0.1} - \hat{N}_T /Y_R } \over {\hat{N}_T /Y_R }}\left( {1 - e^{ - Dt} } \right)} \right)\cr &\qquad + {{\left( {w_{B.{\rm{max}} }^{K12} \beta_{K12} + \chi_{K12} /D} \right)} \over {2Dt}}{{R_{0.1} } \over {\left( {\kappa_{K12} + R_{0.1} - \hat{N}_T /Y_R } \right)\left( {\kappa_{K12} - \hat{N}_T /Y_R } \right)}}\cr &\quad\times log_e \left( {1 + {{R_{0.1} - \hat{N}_T /Y_R } \over {\kappa_{K12} }}\left( {1 - e^{ - Dt} } \right)} \right)\quad {\rm{for}}\ t\,>\,t_{crit}}$$

(A33)

For t → 0 (A32) fitness converges on ${w_B^{K12} \left( {\overline R } \right)}=\left( {w_{B.{\rm{max}} }^{K12} -\left( {w_{B.{\rm{max}} }^{K12} \beta _{K12} + \chi _{K12} /D} \right)/\left( {\kappa _{K12} + \overline R } \right)} \right)$, which is the function of an expected ambient arginine concentration, while for t → ∞ (A33) fitness converges on the expected mean fitness $\ \overline {w_B^{K12} }=(w_B^{K12} (R_{0.1} ) + w_B^{K12} (0))/2$, which comprises the first row of (A33), selection in the transitions (second and third rows) being insignificant.

Assumptions

We assume that the rate of consumption of arginine is dependent on culture growth, $D\hat N_T$, and independent of the genotypes present and the ambient arginine concentration, R(t). The approximation is justified because: (1) $\hat N_T$ is almost constant (the maximum mass of arginine consumed is only 3.5% of the glucose consumed), (2) the quantity of arginine consumed is independent of genotype (Fig. 4), (3) the quantity of arginine consumed is constant for $R_0 > \hat N_T /Y_R=0.02\ {\rm{ m}}M$ (Fig. 3), and (4) very low concentrations of arginine saturate the arginine transporters (K _m ≈ 20 nM [Celis 1977]). Only at the last moment as the last trace of arginine disappears is this model unrealistic. The second assumption is that the growth rates of argR^B strains are independent of arginine availability. The third assumption is that argR^B strains are at sufficiently high frequency and that they determine the steady-state concentration of glucose in the chemostat growth chamber, $\hat G$.

The model should not be used to describe long periods of selection at very low ambient concentrations of arginine, in the general vicinity of 0 < R₀ < $\hat N_T /Y_R$. Fluctuations between absence and presence of high ambient arginine concentrations help minimize the impact on selection of low ambient arginine concentrations in our experiments.

Our experiments are designed to test whether selection in transient states can be predicted from a knowledge of selection at quasi-steady-states. The key assumption is that fitness changes instantaneously with changes in ambient arginine concentrations—there are no delays and fitness is confined to the surface depicted in Fig. 6.