Fitness flux and ubiquity of adaptive evolution

Abstract

Natural selection favors fitter variants in a population, but actual evolutionary processes may decrease fitness by mutations and genetic drift. How is the stochastic evolution of molecular biological systems shaped by natural selection? Here, we derive a theorem on the fitness flux in a population, defined as the selective effect of its genotype frequency changes. The fitness-flux theorem generalizes Fisher's fundamental theorem of natural selection to evolutionary processes including mutations, genetic drift, and time-dependent selection. It shows that a generic state of populations is adaptive evolution: there is a positive fitness flux resulting from a surplus of beneficial over deleterious changes. In particular, stationary nonequilibrium evolution processes are predicted to be adaptive. Under specific nonstationary conditions, notably during a decrease in population size, the average fitness flux can become negative. We show that these predictions are in accordance with experiments in bacteria and bacteriophages and with genomic data in Drosophila. Our analysis establishes fitness flux as a universal measure of adaptation in molecular evolution.

Fig. 1.

Evolution in fitness landscapes and seascapes. The evolutionary history of a population is described by a series of genotype or allele frequency states x = (x₀, …, x_n) at times (t₀, …, t_n) (here, n = 3). Evolutionary time increases between the initial state (◇) and the final state (□). The cumulative fitness flux in each time interval (gray-filled vertical arrows) is the product of the frequency change Δx_i = x_i+1 – x_i between successive states (horizontal arrows) and the selection coefficient s(x_i, t_i) of this change; the cumulative flux Φ(x) of the entire history is the sum of these terms. The reverse history x^T = (x_n, …, x₀) evolves through the same states in reverse order from the initial state (□) to the final state (◇). Each transition has the opposite fitness effect as the corresponding transition of the original history, resulting in a cumulative fitness flux Φ(x^T) = – Φ(x) (the direction of all arrows is reversed). (A) Evolution in a fitness landscape F(x). The gradient of this function defines time-independent selection coefficients s(x) = ∇F(x). A linear landscape corresponds to a frequency-independent selection, and a nonlinear landscape as shown here corresponds to frequency-dependent selection. The cumulative fitness flux Φ(x) of a population history measures the fitness difference ΔF = F(x_n) – F(x₀) between initial and final population. In general, the function F(x) is not equal to the mean population fitness, as discussed in SI Text. (B) Evolution in a fitness seascape F(x, t). The gradient of this function defines time-dependent selection coefficients s(x, t) = ∇F(x, t). The cumulative fitness flux of a population history is defined in terms of selection coefficients and frequency changes as before. However, it no longer equals the fitness difference between initial and final population, because its definition does not include the explicit time dependence of fitness during the history that is unrelated to adaptation (unfilled vertical arrows). (C) Evolution in a fitness seascape with selection coefficients s(x, t) not of gradient form. The example shows cyclic selective advantage as in the rock-paper-scissors game (i.e., each of the transitions from x₀ to x₁, from x₁ to x₂, etc., and from x₂ to x_n = x₀ involves a positive selection coefficient). The fitness flux is defined as before and is again unrelated to a fitness difference between final and initial population state.

Fig. 2.

Fitness evolution of genomic population histories under three different scenarios of selection and demography. (A) Evolutionary equilibrium in a time-independent fitness landscape. (Upper Left) Fitness evolution of a single two-allele genomic locus (schematic). The two alleles a, b have time-independent fitness values f_a, f_b (black lines). The mean population fitness of this locus (red line) evolves by a series of beneficial or deleterious substitutions (red arrows), which have selection coefficients s = f_b – f_a and – s, respectively. This process obeys detailed balance (i.e., beneficial substitutions occur at the same rate as deleterious ones). (Lower Left) Fitness evolution of sequences with L = 12 independent two-allele loci, additive fitness and a uniform mutation rate of μ per locus. Fitness flux NΦ (red lines) and the negative of the log-likelihood change, (green lines), between initial and final population state in the interval (0, t) are shown as time series of an individual history (solid lines) and as ensemble averages over 10⁵ independently evolving populations (dashed lines). Each population history obeys the detailed balance relation (in the equilibrium ensemble, log likelihood equals scaled fitness NF). Evolutionary time is measured in units of the inverse neutral genomic mutation rate 1/μL. Polymorphism lifetimes are short, and substitution processes (vertical line segments and arrows) appear instantaneous on this time scale. For simulation details, see SI Text. (Lower Right) Histograms of NΦ (red), (green), and (blue) at a given time t = 28.8/μL for an ensemble of 10⁵ populations, with averages marked by dashed vertical lines. (B) Nonequilibrium stationary state in a stochastic fitness seascape. Diagrams are the same as in A. Selection coefficients s(t) = f_b(t) – f_a(t) at individual genomic loci fluctuate between two values following a Poisson process, which generates independent selection histories at each locus (for details, see SI Text). Because the rate of selection fluctuations is much smaller than the inverse polymorphism lifetime, a switch of selection generates a persistent window of positive selection. The average cumulative fitness flux N〈Φ〉 increases with time at a constant positive rate, signaling adaptive substitutions. Most individual population histories have a flux NΦ close to this average, but there are rare drift-dominated histories with . (C) Transitions between equilibria under demographic changes. Diagrams are the same as in A. Population size first decreases from an initial value N₀ to a bottleneck value N_b = N₀/2, remains constant during the bottleneck, and later increases to the original value N₀. This process results in time-dependent scaled-allele fitness values N(t)f_a, N(t)f_b and selection coefficients N(t)(f_b – f_a). The population decline generates a loss in scaled fitness, Δ₁H = Δ₁〈NF〉 < 0 and a negative scaled fitness flux N₀〈Φ₁〉 < 0 in the time interval (0, t₁ = 26.6/μL). The recovery in the time interval (t₁, t₂ = 57.6/μL) restores the initial fitness, Δ₂H = Δ₂〈NF〉 = – Δ₁H > 0, and generates a positive scaled fitness flux N₀〈Φ₂〉 that exceeds the flux N₀〈Φ₁〉 of the decline in magnitude (for details, see Methods and SI Text).