### 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 (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.

Original language | English |
---|---|

Article number | e94900 |

Journal | PloS one |

Volume | 9 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 18 2014 |

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### All Science Journal Classification (ASJC) codes

- Biochemistry, Genetics and Molecular Biology(all)
- Agricultural and Biological Sciences(all)

### Cite this

*PloS one*,

*9*(4), [e94900]. https://doi.org/10.1371/journal.pone.0094900

**Equilibrium frequency of endosymbionts in multiple infections based on the balance between vertical transmission and cytoplasmic incompatibility.** / Kawasaki, Yuuki; Ito, Hiroshi; Kajimura, Hisashi.

Research output: Contribution to journal › Article

*PloS one*, vol. 9, no. 4, e94900. https://doi.org/10.1371/journal.pone.0094900

}

TY - JOUR

T1 - Equilibrium frequency of endosymbionts in multiple infections based on the balance between vertical transmission and cytoplasmic incompatibility

AU - Kawasaki, Yuuki

AU - Ito, Hiroshi

AU - Kajimura, Hisashi

PY - 2014/4/18

Y1 - 2014/4/18

N2 - 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 QN, i.e., the frequency of host individuals completely infected with all N -bacterial strains in the population. Q*N, QN 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.

AB - 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 QN, i.e., the frequency of host individuals completely infected with all N -bacterial strains in the population. Q*N, QN 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.

UR - http://www.scopus.com/inward/record.url?scp=84899679569&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84899679569&partnerID=8YFLogxK

U2 - 10.1371/journal.pone.0094900

DO - 10.1371/journal.pone.0094900

M3 - Article

C2 - 24747216

AN - SCOPUS:84899679569

VL - 9

JO - PLoS One

JF - PLoS One

SN - 1932-6203

IS - 4

M1 - e94900

ER -