markovian
Bulmer M 1991 The selection-mutation-drift theory of synonymous codon usage. Genetics 129:897-907.
- Kimura (1981, 1983) considered a model of stabilizing selection on codon usage
- there is an optimal state of highest fitness when the relative frequencies of synonymous codons exactly match those of the cognate tRNAs
- the rationale of this model is unclear
- this model also fails to take into account differences between codons recognized by the same tRNA
- there is some weakness in the link between its verbal formulation and its mathematical development (Li 1987)
- we must consider not the equilibrium gene frequency P but the equilibrium distribution of gene frequencies ƒ(p) with Expected value P
- a classical result in population genetics
- ƒ(p) ∝ eSppV − 1(1 − p)U − 1
- S = 2Nes
- V = 2Nev
- U = 2Neu
- if U + V is large, the distribution will be clustered about the deterministic equilibrium
- if U + V is small, the population is likely to be at or near one of the boundaries so that the expected gene frequency is the probability of being near 1 rather than 0
- given by P = eSV / (eSV + U)
- the number of new B2 mutants arising per generation in B1 populations which are ultimately fixed is NuPφ(− S)
- the number of new B1 mutants arising per generation in B2 populations which are ultimately fixed is Nv(1 − P)φ(S)
- at equilibrium these flux rates are equal
- the problem seems to arise from the Hill-Robertson effect
- this factor will be implicitly allowed for in the present context by estimating the effective population size empirically
