J.B.S. Haldane (1957) introduced the cost of substitution concept, though its usage became hampered by various confusions, and it eventually fell into common disuse. It was criticized for requiring constant population size, and constant selective values, and for its reliance on “genetic death” and “genetic load,” whose physical interpretation is dubious. Such issues severely limited its deployment. Those difficulties were addressed and eliminated in ReMine (2005), which clarified cost theory and generalized the cost of substitution to have a concrete physical interpretation, without reliance on genetic death or genetic load, and while allowing fluctuations in any parameters. This paper applies that conceptual clarification to obtain more precise mathematical results.

For any given evolutionary scenario, cost theory calculates the required reproduction rate (referred to as the “cost of evolution”) and compares it with the species actual reproduction rate (referred to as the “payment”). If the species cannot “pay the cost,” then the scenario is not plausible. That concept is general, and can apply to any model of any evolutionary scenario, because they all require some level of reproduction rate. For ease of comprehension, calculation, and discussion, cost theory partitions the cost of evolution into a sum of various costs, with each cost named according to its specific role. Thus, the cost of substitution is one of many costs that each add extra reproduction rate to the amount required by the scenario (ReMine, 2005)

Evolutionary theory requires that some traits originate as rare beneficial mutations and then, through reproductive means, these increase in number of copies. This increase requires extra reproduction rate. Under the clarified definition, the cost of substitution (CS) is the extra reproduction rate required to increase a trait (or traits) at the rate given by an evolutionary scenario. This paper uses that clarified cost concept to study single substitutions (non-overlapping in time), under genetic circumstances of broad interest (the same cases studied by Haldane, 1957), and derives equations that are more general, more precise, and well-grounded in concrete physical principles.

Let Q be a specific genotype. At the start of generation i, let P be the “effective starting count”—the effective number of individuals who produce genotype-Q progeny. As the cycle of that generation comes to a close, let P´ be the “ending count” of genotype-Q individuals due solely to the reproduction of the former group. (Throughout this paper, a primed quantity, such as P´, denotes a quantity as the generation comes to a close.) The increase is ΔP = P´-P. Then the cost of substitution for genotype-Q, in generation i, is: …

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