Individual histories and selection in heterogeneous populations

Abstract

The strength of selection in populations has traditionally been inferred by measuring changes in bulk population parameters, such as mean reproductive rates. Untangling the effect of selection from other factors, such as specific responses to environmental fluctuations, poses a significant problem both in microbiology and in other fields, including cancer biology and immunology, where selection occurs within phenotypically heterogeneous populations of cells. Using "individual histories"--temporal sequences of all reproduction events and phenotypic changes of individuals and their ancestors--we present an alternative approach to quantifying selection in diverse experimental settings. Selection is viewed as a process that acts on histories, and a measure of selection that employs the distribution of histories is introduced. We apply this measure to phenotypically structured populations in fluctuating environments across different evolutionary regimes. Additionally, we show that reproduction events alone, recorded in the population's tree of cell divisions, may be sufficient to accurately measure selection. The measure is thus applicable in a wide range of biological systems, from microorganisms--including species for which genetic tools do not yet exist--to cellular populations, such as tumors and stem cells, where detailed temporal data are becoming available.

Fig. 1.

Overview of the history formulation. An example of a growing population is shown in which individuals can be in two different phenotypic states. The population’s lineage tree is colored to indicate individuals’ phenotypes (green or red). The environment is indicated by the background color. In each type of environment (green or red), the adapted phenotype, i.e., the fastest reproducing one, matches the environment’s color.

Fig. 2.

Individual histories in two-phenotype models. (A) Individual histories are shown for responsive and stochastic switching models, for normal and 10% improved historical conditions. For each model and historical condition, three independent simulations were run, and a randomly chosen individual history is shown from each simulation (colored segments), with the same color scheme as in Fig. 1. Tick marks display cell divisions. Temporal intervals during which an individual was in the adapted state are shown (gray segments), and the percent of time spent in the adapted state is indicated. Parameters used were s = 0, s_r = 1, f_a = 0.88 (responsive), and s = 0.01, s_r = 0, f_a = 1 (stochastic); τ = 10 and f_na = 0 in both cases. The parameters were chosen such that for both models, the analytically computed value of is the same (); simulation results reflect this, with slight deviations due to the small number and short length of histories shown. Direct estimates of are in good agreement with analytically computed values (responsive) and (stochastic). (B) Histories in normal historical conditions from A are shown with historical divisions only (phenotypic states and environments are not displayed). Selection here is stronger for slow stochastic switching, seen via its high value of and non-Poissonian (patchy) distribution of historical divisions.

Fig. 3.

Dependence of on the environmental duration τ for the two-state models of stochastic and responsive switching described in the text. Fitness values for all models shown are f_a = 1 and f_na = 0.1. The dashed line in all panels is the curve 1/τ, shown for reference. All rates, including f_a, f_na, s, s_r, and , are given in arbitrary units of 1/time, with τ shown in the same units of time. (A and B) Analytically computed curves are shown for pure stochastic switching (A), for given values of s, and for responsive switching (B), for given values of s_r with s = 0.1. (C) Measurements of (filled circles) using statistics of cell divisions from stochastic population simulations are compared with analytically computed values (solid curve). Blue: pure stochastic switching with s = 0.01; magenta: responsive switching, with s_r = 0.95 and s = 0.05. Each point is an average over 100 estimates of that were generated (see Methods); the bars on three representative points show the standard deviation of the estimates. The blue curves in A and C are identical.