The relation between reproductive value and genetic contribution

Abstract

What determines the genetic contribution that an individual makes to future generations? With biparental reproduction, each individual leaves a "pedigree" of descendants, determined by the biparental relationships in the population. The pedigree of an individual constrains the lines of descent of each of its genes. An individual's reproductive value is the expected number of copies of each of its genes that is passed on to distant generations conditional on its pedigree. For the simplest model of biparental reproduction (analogous to the Wright-Fisher model), an individual's reproductive value is determined within ∼10 generations, independent of population size. Partial selfing and subdivision do not greatly slow this convergence. Our central result is that the probability that a gene will survive is proportional to the reproductive value of the individual that carries it and that, conditional on survival, after a few tens of generations, the distribution of the number of surviving copies is the same for all individuals, whatever their reproductive value. These results can be generalized to the joint distribution of surviving blocks of the ancestral genome. Selection on unlinked loci in the genetic background may greatly increase the variance in reproductive value, but the above results nevertheless still hold. The almost linear relationship between survival probability and reproductive value also holds for weakly favored alleles. Thus, the influence of the complex pedigree of descendants on an individual's genetic contribution to the population can be summarized through a single number: its reproductive value.

Figure 1

The relative contribution of four ancestors from a population of N = 1000, plotted against time. This is defined as the number of pedigree descendants, divided by 2^t. The top and bottom sets of points show the ancestors with highest and lowest reproductive value, respectively; the two intermediate sets are randomly chosen individuals. Note that the distribution of v has mean and variance equal to one.

Figure 2

The bars show the distribution of reproductive values for a single population of N = 1000, compared with the theoretical expectation (curve). The latter is calculated by expanding the generating function for the distribution of numbers, E[yⁿ] as a Taylor series in y and then rescaling using n = v2^t. This calculation was done at t = 10 generations; however, the correlation with the ultimate reproductive value is extremely close (0.99957).

Figure 3

The relation between probability of survival, P, and reproductive value, v, at times t = 10 (top), t = 30 (middle), and t = 50 (bottom). For each time, there are 1000 dots, each representing a single ancestor. Each dot gives the probability that a single copy of a gene in the ancestor will survive to time t, plotted against the ancestor’s relative contribution to the pedigree up to time t. The straight lines show the linear relation vP_N,t where P_N,t is the probability of survival of a neutral allele in a branching process with growth rate λ = 1. The curves show the approximation is ${\text{P}}_{\text{t}}=1-e^{-\upsilon {\tilde{P}}_t}$, where ${\tilde{P}}_t$ is an effective value determined by $P_{N,t}=1-\mathbb{E}\left[e^{-\upsilon {\tilde{P}}_t}\right]$. This calculation is simplified by using the fact that $\mathbb{E}\left[e^{-\upsilon \tilde{P}}\right]$ is the generating function for v (Equation 5) evaluated at $e^{-\tilde{P}}.$

Figure 4

(A) The distribution of numbers of copies left by a single copy in an ancestor, after 50 generations in a population of 1000. The usual outcome—loss of the allele—is not shown. Ancestors are classified by their reproductive value (<0.5, 0.5−1, 1−1.5, 1.5−2, >2, bottom to top). (B) The distribution, conditional on survival, is almost independent of reproductive value. These distributions are for a single pedigree; they are estimated by simulating the flow of genes through that pedigree, using 1000 replicates. At the start of each replicate, every allele is labeled by a unique integer that denotes the individual that carries it. Thus, 1000 allele frequencies are estimated in each of the 1000 replicates.

Figure 5

The coefficient of variation of the distribution of numbers of copies, conditional on survival, plotted against relative contribution, v. This is calculated for a single pedigree, with N = 1000, at t = 10, 30, and 50 generations (bottom to top). Each dot represents one individual.

Figure 6

The probability of fixation of an allele with advantage s = 0.05 plotted against reproductive value, v. The curve gives the expected relationship, $1-{\left(1-\tilde{P}\right)}^{\upsilon },$ where $\tilde{P}$ is chosen such that the average fixation probability equals the standard value $\left(\mathbb{E}\left[1-{\left(1-\tilde{P}\right)}^{\upsilon }\right]=0.0937\right).$ The fixation probability is estimated from 400 replicates, each starting with 10 favorable alleles in a population of N = 1000 and iterated for 30 generations. Each of the 4000 alleles is classified by the reproductive value of the individual that first carried it; the points show the mean reproductive value and the mean fixation probability for each class (±1 SE).

Figure 7

The joint distribution of log relative fitness, $z-\overline{z},$ and reproductive value, v, under the infinitesimal model. This distribution is taken from generations 60−100 of a simulated population of N = 1000 individuals, with genetic variance var(z) = 1. The variance in reproductive value is 73.45. Seventy-six percent of individuals have zero reproductive value; these are shown by the distribution at the right.

Figure 8

The mean reproductive value, as a function of log fitness, $z-\overline{z}.$ The line is the theoretical prediction exp$\exp \left(2\left(z-\overline{z}\right)-2V\right).$ Note that for very low z, the average reproductive value falls below the prediction. This is because almost all such individuals have zero reproductive value, so that the mean is determined by very rare individuals with high value and is correspondingly poorly estimated. These data are taken from the same simulation as in Figure 7 with var(z) = 1.

Figure 9

The probability of survival of a neutral allele, plotted against reproductive value, v, under the infinitesimal model (V = 1, N = 1000), for times t = 10, 20, 30, 40, and 50 (top to bottom).

Figure 10

With intermediate migration rates, population structure slows the convergence of survival probability to strict dependence on reproductive value. The vertical axis shows the correlation between probability of survival to time t and the expected genetic contribution at time t; this correlation measures the tightness of the scatter in plots such as that in Figure 4. The correlation is plotted against time and against migration rate, m. There are 100 demes of 10 haploid individuals; parents are chosen from within the deme with probability 1 − m and randomly from the whole population with probability m.

Figure 11

The relation between an individual’s contribution to inbreeding, $\overline{F},$ and its immediate fitness (W, left) or its reproductive value (v, right). $\overline{F}$ is the average probability that two randomly chosen lineages will coalesce in a particular ancestor, 50 generations before; population size is N = 100 diploid individuals. The curve on the left is the best fit, $\overline{F}=0.00090W\left(W-1\right).$ The curves on the right are the best quadratic fits, for individuals with immediate fitness W = 2, 3, 4: 0.00205v², 0.00271v², and 0.00323v², respectively.