Origin of Sex for Error Repair I. Sex, Diploidy, and Haploidy

Genetic damage is a fundamental problem for living systems. Recombination can repair a damaged gene, so long as there is an undamaged copy of the gene available in the cell. This requires that the cell be diploid for the damaged locus. During sex, outcrossing generates the diploid state by temporarily fusing two haploid cells (as in the case of meiosis) or by bringing DNA into the cell from outside (as in the case of bacterial transformation). But why should cells alternate between the haploid and diploid states in the first place? Why not just remain diploid, if damage repair is the only problem for a cell? The goal of our work is to understand if the problem of genetic damage would select for diploidy or for the alternation between diploid and haploid states-that is, sex-early in the history of life. Using mathematical models we study competition between asexual haploids (termed "haploids"), sexuals (termed "sexuals"), and asexual diploids (termed "diploids"). Haploid cells are efficient replicators, while diploid cells are resistant to damage. A sexual may combine the advantages of both: spending much of its life cycle in the haploid state, then temporarily fusing to become diploid, followed by splitting to the haploid state. During the diploid state DNA damage can be repaired, since there are two copies of the gene in the cell and one copy is presumed to be undamaged. We describe the competition in terms of mathematical models, employing five rate parameters which represent the life processes of cells most probably active at the time that sexuality arose: birth and death; genomic damage (for the haploids alone); and, for the sexual cell, fusion and splitting. Parameter space bifurcation diagrams for the equilibria are drawn in the three-dimensional space of damage, splitting, and fusion, and solutions of the equations (i.e., the outcomes of the competition) are described in terms of them. It turns out that those three parameters suffice to give an essentially complete description of the qualitative behavior possible, since one parameter can be scaled out of the equations we ultimately consider, and the other permits generic analysis, for the range of parameter values of interest, at a fixed value of that parameter. Each type of cell has a region of the parameter space that it occupies exclusively (given its initial presence in the competition). The haploid can win only in environments characterized by low damage (relative to mortality), while the diploid can win only in environments characterized by high damage (relative to mortality). However, the sexual may outcompete either of the asexuals in those domains assuming that the parameters of the sexual cycle are adjusted appropriately. In general, only a single type of cell occupies a given portion of the space. We find, however, that the competitive coexistence of a diploid and a sexual is possible in spite of the fact that they are competing for a single resource (nucleotide building blocks). This coexistence is the result of an overactive sexual cycle and so would presumably be selected against.