Nearly neutral theory of molecular evolution

Theory

The rate of substitution, ${\displaystyle \rho }$ is

${\displaystyle \rho =ug{N}_e{\overline{P}}_{fix}}$,

where ${\displaystyle u}$ is the mutation rate, ${\displaystyle g}$ is the generation time, and ${\displaystyle N_e}$ is the effective population size. The last term is the probability that a new mutation will become fixed. Early models assumed that ${\displaystyle u}$ is constant between species, and that ${\displaystyle g}$ increases with ${\displaystyle N_e}$. Kimura’s equation for the probability of fixation in a haploid population gives:

${\displaystyle P_{fix}=\frac{1-e^{-s}}{1-e^{-s{N}_e}}}$,

where ${\displaystyle s}$ is the selection coefficient of a mutation. When ${\displaystyle |s|\ll \frac{1}{N_e}}$(completely neutral), ${\displaystyle P_{fix}=\frac{1}{N_e}}$, and when ${\displaystyle -s\gg \frac{1}{N_e}}$ (extremely deleterious), ${\displaystyle P_{fix}}$ decreases almost exponentially with ${\displaystyle N_e}$. Mutations with ${\displaystyle -s\simeq \frac{1}{N_e}}$ are called nearly neutral mutations. These mutations can fix in small-${\displaystyle N_e}$ populations through genetic drift. In large-${\displaystyle N_e}$ populations, these mutations are purged by selection. If nearly neutral mutations are common, then the proportion for which ${\displaystyle P_{fix}\ll \frac{1}{N_e}}$ is dependent on ${\displaystyle N_e}$

The effect of nearly neutral mutations can depend on fluctuations in ${\displaystyle s}$. Early work used a “shift model” in which ${\displaystyle s}$ can vary between generations but the mean fitness of the population is reset to zero after fixation. This basically assumes the distribution of ${\displaystyle s}$ is constant (in this sense, the argument in the previous paragraphs can be regarded as based on the “shift model”). This assumption can lead to indefinite improvement or deterioration of protein function. Alternatively, the later “fixed model” fixes the distribution of mutations’ effect on protein function, but allows the mean fitness of population to evolve. This allows the distribution of ${\displaystyle s}$ to change with the mean fitness of population.

The “fixed model” provides a slightly different explanation for the rate of protein evolution. In large ${\displaystyle N_e}$ populations, advantageous mutations are quickly picked up by selection, increasing the mean fitness of the population. In response, the mutation rate of nearly neutral mutations is reduced because these mutations are restricted to the tail of the distribution of selection coefficients.

The “fixed model” expands the nearly neutral theory. Tachida classified evolution under the “fixed model” based on the product of ${\displaystyle N_e}$ and the variance in the distribution of ${\displaystyle s}$: a large product corresponds to adaptive evolution, an intermediate product corresponds to nearly neutral evolution, and a small product corresponds to almost neutral evolution. According to this classification, slightly advantageous mutations can contribute to nearly neutral evolution.

The "drift barrier" theory

Michael Lynch has proposed that variation in the ability to purge slightly deleterious mutations (i.e. variation in ${\displaystyle N_e}$) can explain variation in genomic architecture among species, e.g. the size of the genome, or the mutation rate. Specifically, larger populations will have lower mutation rates, more streamlined genomic architectures, and generally more finely tuned adaptations. However, if robustness to the consequences of each possible error in processes such as transcription and translation substantially reduces the cost of making such errors, larger populations might evolve lower rates of global proofreading, and hence have higher rates of error. This may explain why Escherichia coli has higher rates of transcription error than Saccharomyces cerevisiae. This is supported by the fact that transcriptional error rates in E. coli depend on protein abundance (which is responsible for modulating the locus-specific strength of selection), but do so only for high-error-rate C to U deamination errors in S. cerevisiae.