Equilibrium frequency of endosymbionts in multiple infections based on the balance between vertical transmission and cytoplasmic incompatibility

Cytoplasmic incompatibility (CI)-inducing endosymbiotic bacteria, such as Wolbachia and Cardinium, have been well studied through field data and validations on the basis of numerical simulations. However, the analytically derived equilibrium frequency of multiple infections has not yet been determined, although the equilibrium for cases of single infection has been reported. In this study, we considered the difference equation for endosymbionts using three parameters: the probability of the failure of vertical transmission (μ), CI strength (z), and the level of host inbreeding (p). To analyze this model, we particularly focused on Q_N, i.e., the frequency of host individuals completely infected with all N -bacterial strains in the population. Q*_N, Q_N at the equilibrium state, was analytically calculated in the cases where N=1,2 and N is any arbitrary value. We found that Q*_N can be described using two parameters: N and α, which is identical to μ/pz. Q*_N has a larger value in a system with a smaller α. In addition, α determines the maximum number of strains that infect a single host. Our results revealed the following: i) three parameters can be reduced to a single parameter, i.e., α and ii) the threshold of the maximum number of infections is defined by α, which prevents additional invasions by endosymbionts.