Abstract
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 (), CI strength (), and the level of host inbreeding (). To analyze this model, we particularly focused on , i.e., the frequency of host individuals completely infected with all -bacterial strains in the population. , at the equilibrium state, was analytically calculated in the cases where and is any arbitrary value. We found that can be described using two parameters: and , which is identical to . has a larger value in a system with a smaller . In addition, 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., and ii) the threshold of the maximum number of infections is defined by , which prevents additional invasions by endosymbionts.
Highlights
Endosymbiotic bacteria such as Wolbachia and Cardinium are well known to be reproductive manipulators infceting insect cells [1,2]
We demonstrated the analytic equilibrium solutions of the frequencies of individuals infected with all the bacterial strains in a population with single, double and arbitrary N infections
Our results provide of a qualitative insight into the symbiotant bacterial dynamics, in constrast to recently developed models that quantitatively simulate specific experimental results [4,32]
Summary
Endosymbiotic bacteria such as Wolbachia and Cardinium are well known to be reproductive manipulators infceting insect cells [1,2]. Cytoplasmic incompatibility (CI) is considered to be the most common and efficient form of manipulation that can spread an infection throughout the host population. CI causes males infected with a bacterial strain reproductively incompatible with uninfected females [3]. Because of CI, infected females exhibit a relatively higher fitness than uninfected females; the number of infected individuals gradually increases and they become dominant in the host population. Approximately only 10% of Drosophila simulans individuals were infected with Wolbachia in the middle of the 1980s in California; the infection rate increased to 95% in 1993 [4,5]. In a rearing system of Encarsia pergandiella infected with Cardinium, the infection gradually spread to almost fixation within the population, irrespective of the initial rate of infection [6]
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