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.