Evolution of multigene families by gene duplication. A haploid model

Evolution of multigene families by gene duplication and subsequent diversification is analyzed assuming a haploid model without interchromosomal crossing over. Chromosomes with more different genes are assumed to have higher fitness. Advantageous and deleterious mutations and duplication/deletion also affect the evolution, as in previous studies. In addition, negative selection on the total number of genes (copy number selection) is incorporated in the model. First, a Markov chain approximation is used to obtain formulas for the average numbers of different alleles, genes without pseudogene mutations, and pseudogenes assuming that mutation rates and duplication/deletion rates are all very small. Computer simulation shows that the approximation works well if the products of population size with mutation and duplication/deletion rates are all small compared to 1. However, as they become large, the approximation underestimates gene numbers, especially the number of pseudogenes. Based on the approximation, the following was found: (1) Gene redundancy measured by the average number of redundant genes decreases as advantageous selection becomes stronger. (2) The number of different genes can be approximately described by a linear pure-birth process and thus has a coefficient of variation around 1. (3) The birth rate is an increasing function of population size without copy number selection, but not necessarily so otherwise. (4) Copy number selection drastically decreases the number of pseudogenes. Available data of mutation rates and duplication/deletion rates suggest much faster increases of gene numbers than those observed in the evolution of currently existing multigene families. Various explanations for this discrepancy are discussed based on our approximate analysis.