An Analysis of the Cost-of-Selection Concept

Abstract

It is shown for a continuous haploid model that the common standard assumptions usedin calculating the cost of gene substitution, namely, large constant population size and small constant selective value, are unnecessary. Population size may fluctuate during the course of substitution without affecting the calculated total cost. The selective intensity does not need to be small and constant to give the standard result for substitution cost. Diploid models with multiple alleles are analyzed and contrasted with standard two-allele models in respect to calculation of substitution cost. The influence of population structure on the probability of occurrence of complete gene substitution is discussed on the basis of a numerical example. The robust nature of the cost-of-selection concept is examined in the light of a conservation principle.The original model of Haldane (1, 2) for the cost of selection and more recent standard models (3) assume constant large population size and constant small selective value. Using continuous deterministic models, both haploid and diploid, we have arrived at a more general formulation that does not require these assumptions. A number of other models have been analyzed in recent papers (see ref. 4 for references), but none of the deterministic models previously considered encompass the same generality as that presented here. In addition, we consider the cost-of-selection concept in terms of population structure and a general conservation principle. HAPLOID MODELThe cost of selection in the haploid or asexual case is first discussed for a discrete generation model, and then for a continuous model with random births and deaths. The calculation of cost for the discrete model follows the analysis of Crow and Kimura (3) closely, except that the presentation here begins explicitly with the difference equations describing the dynamic behavior of the number of individuals of each type in the population. The derivation for the continuous model is analogous to the discrete model argument. However, in the continuous case few restrictive assumptions or approximations are required. In particular, the latter approach for computing cost is valid for quite general population models, including those containing density-dependent fitness. In addition, the common assumptions of weak selection and stable population size during the process of the substitution are not required.Discrete Generation Model. We shall adopt the following customary notation for the fitnesses, frequencies, and numerical strength of the two competing types:The number of individuals of each type in successive generations is described by a pair of difference equations:

Fixation and Adaptation in the Lenski E. coli Long Term Evolution Experiment

“…The fixation process is a conservative phenomenon. Flake et al [11] showed that the cost of substitution is associated with a potential energy function and Kimura [12] describes diffusion models to model the changes in gene frequencies in populations over generations. It is well known that diffusion equations are obtained by applying the principles of conservation of energy or mass to control volumes.…”

Surprisingly Low Limits of Selection in Plant Domestication

“…Selection comes at a cost in that some organisms must die before reproducing each generation for their genes to be selected against the wild-type allele, therefore causing a reduction in the overall population size. [28][29][30][31] A consequence of this cost is that the amount of selection that a population can withstand, without dropping to unsustainably low demographic levels, is limited. Haldane noted that for this reason, there is a limit to the number of traits that plant breeders are able to select at a given time, and that the pace at which evolution can be driven by natural selection is limited because of the cost of selection.…”

Random Mutation and Natural Selection In Competitive and Non-Competitive Environments

“…The Lenski experiment demonstrates how his bacteria must compete with each other in order to evolve to the selection pressure, starvation. In fact, differential reproductive fitness is simply a restatement of the law of conservation of energy, as shown by Flake and Grant in their paper An Analysis of the Cost of Selection Concept [9]. However, RMNS is not a conservative phenomenon.…”