Department of Entomology, College of Agriculture, University of Kentucky, Lexington, KY 40546-0091, USA

Department of Biology, Center for Ecology, Evolution and Behavior, University of Kentucky, Lexington, KY 40502-0225, USA

Abstract

Background

The evolutionary success of

Results

The results predict the

Conclusions

The model described here takes a novel approach to understanding the spread of

Background

The success of obligate endosymbiotic organisms depends on their ability to invade, establish and persist in their host.

Unidirectional cytoplasmic incompatibility crossing pattern

**Unidirectional cytoplasmic incompatibility crossing pattern**. White circles represent uninfected individuals and black circles represent

Prior models highlight three

Previous studies predict that the successful invasion of

The relative importance of

To better understand population replacement by CI-inducing

Methods

The model simulates a panmictic population that is closed to immigrants and emigrants. Consistent with previous studies, the model assumes mating is random and that

Brief Description of Equations

The following is a brief overview of all equations and parameters implemented in the model presented here. Additional development details, initial parameter values, and sensitivity analysis are provided in Additional File

**Model parameters, detailed equation appendix and sensitivity analysis**. Portable Document File (pdf) containing all parameters, initial parameter values, and equations utilized by the model. Model development is discussed, and includes references from which each equation was developed/parameterized. Also includes the sensitivity analysis of all population dynamic parameters and discussion about the robustness of model predictions

Click here for file

Larval development rate ^{-1}),

Larval survival, _{L}: μ is the baseline mortality rate of mosquito larvae in the absence of competition (units of (time)^{-1}). ^{-1}). ^{-1}),

Mosquito body mass, _{x }(units of mass) is the theoretical maximum mass of a given mosquito at time _{x }is linked to _{x }that is attainable. _{0 }(dimensionless) is the development time at which mass at pupation is _{x}/2 days, and

Female survivorship, _{s}: ^{-1}) and

Egg production, ^{-1}); _{f }is the body mass of the ovipositing female (units of mass);

Immature Life Stages

To simulate variation in egg hatch, the model assumes that some eggs (proportion equal to _{3}, Table S1 in Additional file _{3}) hatch on day four (Figure

Immature population structure

**Immature population structure**. a) Eggs develop through four discrete stages and each stage is one day. There are two cohorts of eggs, _{E}, daily egg survivorship (Table S1). All eggs hatch after four days except a proportion of eggs hatch at day three (_{3}, Table S1). b) Larvae develop through _{2}, to their next developmental stage, _{2+R}, is equal to the product of the number of larvae in a developmental stage and larval survival (Equation 2). Larval survival and development are density dependent. If larvae are _{P}, daily pupal survivorship (Table S1).

Larvae develop through discrete developmental stages, where the development rate is affected by density dependence, and larval survival is subject to both stage-dependent mortality and density-dependence (Figure

Uninfected larval cohorts progress through development subject to stage-dependent mortality and density dependent effects only. Infected larval cohorts are subject also to a reduction in viability associated with

Glossary of notation, including the initial values for each key parameter

**symbol**

**definition**

**initial value**

proportion of embryos not hatching in incompatible CI crosses

0.999

proportion of offspring receiving infection (maternal inheritance)

0.999

relative fecundity of infected females to uninfected females

0.999

relative larval viability of infected larvae to uninfected larvae

0.999

initial frequency of gravid infected females to the total adult population

0.500

In all subsequent model runs, each value remains constant while one key parameter is varied. (For a list of all population dynamic parameters, see Table S1 in Additional file

Following the completion of larval development stages, individuals become non-feeding pupae, which have a daily survival that is independent of population density (_{p}, Table S1; Figure

Adult Life Stages

Six variables are tracked over time and determine the state of individual females: the blood meal state (time since last feeding), age (days since emerging),

The probability that a female obtains a blood meal is determined by the frequency of potential blood meals per unit area, and each blood meal is associated with an additional mortality risk, regardless of mosquito age (Table S1). In the panmictic population simulated here, the availability of potential blood meals is assumed to be constant, but the model will allow downstream population structuring and geographic variation of bloodmeal availability.

Adult female daily survivorship _{s }is age-dependent and probabilistic (Equation 4)

Adult males, which are dead end hosts for _{M}, Table S1). The proportion of

Simulations

The model was written in MATLAB 7 (The MathWorks Inc., Natick, MA). A single simulation of the model produced population dynamics that are tracked over time (Figure

Example of typical population dynamics produced by a simulation of the model

**Example of typical population dynamics produced by a simulation of the model**. a) Populations begin with an uninfected cohort of eggs. The population is allowed to persist and self-regulate for 800 days, at which time

Results

Figure

Due to the stochastic nature of the model, the number of individuals within each lifestage fluctuates considerably over time (Figure

Five parameters associated with

Maternal inheritance (

The probability of population replacement for five

**The probability of population replacement for five Wolbachia specific parameters**.

A different functional relationship is observed with the level of incompatibility (

The results obtained from the model here were compared to a previously published stochastic model

The probability of population replacement for given parameter values

**
MI
**

**
RF
**

**1.0**

**0.9**

**0.8**

1.0

0.1023/0.0359

0.0224/0.0089

0.0004/0.0007

0.9

0.0158/0.0060

0.0004/0.0000

/

0.8

0.0001/0.0004

/

/

Jansen

The probability of population replacement for given parameter values, assuming perfect CI and the release of a single infected adult female into an uninfected population with a size of 100. The fixation probabilities from our model are generally lower than the values predicted by Jansen

The probability of population replacement by

**The probability of population replacement by Wolbachia given different initial infection frequencies**. This figure assumes that the relative fecundity of infected females is 0.95, with perfect CI and maternal inheritance. The dashed line indicates the probability of population replacement as calculated by Jansen

Discussion

The model presented here examines the probabilities of

The relative larval viability between

Recent work has highlighted the prevalence of

The level of CI in insects varies widely

High maternal inheritance rates have been observed consistently in natural populations

The effect of

For all parameters, the probability of population replacement approached an absolute maximum of 90% given the conditions defined in Table

The model presented here predicted lower population replacement probabilities than those predicted by previous stochastic models (Table

The model here addresses a single, panmictic, isolated population but could be expanded to include metapopulation structure. If introduction events can be assumed to occur randomly, then the surrounding subpopulations should generally tend to inhibit population replacement, because migration between subpopulations would dilute the proportion of infected individuals. However, as demonstrated here, genetic drift may influence the invasion of

The majority of models that address the invasion of

Conclusions

The rapid decline in the probability of population replacement associated with reduced larval viability indicates that empirical studies directed toward quantifying the effects of endosymbionts on immature insects are important for understanding and predicting

Authors' contributions

The original model was conceptualized by PRC, JWM and ES in a class taught by PHC. The model was revised by YH, PHC, and SLD. JWM and ES parameterized equations, which were derived by PHC. PRC coded the model and simulations in MATLAB, which was later optimized by PRC, PHC, and YH. The manuscript was written by PRC, PHC, and SLD. All authors have read and approved the final manuscript.

Acknowledgements

The authors would like to thank Michael Turelli and Peter Hammerstein for comments and suggestions on this project. This research was supported by grants from the National Institutes of Health [AI-067434] and the Bill and Melinda Gates Foundation [#44190]. This is publication 11-08-042 of the University of Kentucky Agricultural Experiment Station.