Stochastic differential equations used to model conjugation

Stochastic differential equations (SDEs) are used to model horizontal transfer of antibiotic resis- tance by conjugation. The model describes the concentration of donor, recipient, transconjugants and substrate. The strength of the SDE model over the traditional ODE models is that the noise can be split into measurement noise and system noise. The system noise is used to compensate for those biological processes not explicitly described by the model. Many authors model conjugation by a simple mass action model first proposed by Levin et al. (1979). Also Michaelis-Menten dependence on the recipient concentration has been used to mathematically describe conjugation (Andrup et al. (1998)). We find that it is important to include substrate depletion to model conjugation for a system with exhaustible media and implement the substrate dependence as a Michaelis-Menten expression. This is supported by an experiment conducted with E. faecium. In addition, we suggest that a 3rd order time-delay must be included in the model to account for the delay before a newly conjugated plasmid is expressed. A ML estimate of the parameters based on experimental data is found using the software CTSM. The conjugation rate is estimated to 1.4e−9 ± 0.38e−9 1/h.