Muller’s ratchet and gene duplication

Copy number of genes in gene families can be highly variable among individuals and may continue to change across generations. Here, we study a model of duplication–selection interaction, which is related to Haigh’s mutation–selection model of Muller’s ratchet. New gene copies are generated by duplication but fitness of individuals decreases as copy number increases. Our model comes in two flavors: duplicates are copied either from a single template or from any existing copy. A duplication–selection equilibrium exists in both cases for infinite size populations and is given by a shifted Poisson or a negative binomial distribution. Unless counteracted by synergistic epistasis, finite populations suffer from loss of low copy-number haplotypes by drift, forcing them into a regime called ‘run-away evolution’ in which new copies accumulate without bound nor equilibrium. We discuss a few empirical examples and interpret them in the light of our models. Generally, large gene families appear too over-dispersed to fit the single template model suggesting a dynamic, and potentially accelerating, duplication process.