Abstract
We consider disease-causing mutations that are lethal when homozygous. Lethality involves the very strongest form of negative selection, with the selection coefficient against the disease-carrying homozygote having its maximum value of unity. We determine results for the behavior of the frequency of a lethal allele in an effectively infinite population. This includes an estimate of the time it takes to achieve equilibrium, and a description of transient behavior associated with a sudden change in the fitness of the heterozygote. We determine analogous results for a finite population, showing that a lethal disease-causing allele needs to be described by a modified Wright-Fisher model, which deviates from the standard model, where selection coefficients are assumed small compared with 1. We show that a by-product of the dynamics, resulting from the absence of the disease-allele carrying homozygote in adults, is the general constraint that the frequency of the disease-causing allele cannot exceed . The results presented in this work should prove useful to a number of areas including analysis of lethal/near lethal mutations in Mendelian disorders and, in particular, for exploring how mutation-selection-drift balance explains the current spectrum of mutation frequencies in humans. While the number of empirical examples of overdominance in lethal genetic disorders is not large, relatively high observed heterozygote frequencies may be a hint of transient heterozygous advantage in nature. For lethal disorders with anomalous frequencies, such as cystic fibrosis and Tay-Sachs, our analysis lends further support to the role that transitory episodes of weak overdominance may play in the evolution of lethal mutations.
Highlights
The population genetics of single-gene diseases, where a gene is typically considered to have two alleles and three genotypes, is generally an oversimplification
Before we give results based on numerical calculations, we note that analytical insight into the Wright-Fisher model, and the phenomena occurring in a finite population, can be gained using a diffusion approximation of this model (Kimura, 1955)
In this work we have provided an analysis of the implications of lethal mutations in both effectively infinite and finite populations
Summary
The population genetics of single-gene diseases, where a gene is typically considered to have two alleles and three genotypes, is generally an oversimplification. If we consider the textbook example of the F508 mutation of the CFTR gene, which is responsible for the majority of cystic fibrosis cases, we have a gene with a recessive cc genotype whose fitness is close to zero, relative to the CC and Cc genotypes. With the discovery of modifier genes that modulate CFTR (Guggino and Stanton, 2006), this gene can no longer be considered within a simple two-allele/threegenotype framework. As of October 2019, the Online Mendelian Inheritance in Man (OMIM) database, reported 6, 516 single-gene phenotypes with known molecular basis. The proportion of these that can be described using a simple two-allele/three-genotype model is likely to be small.
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