Model of differential equations describing the dynamics of plasmid-free bacteria and nonadapted and adapted plasmid bearersa

ProcessBacterial dynamicsReaction rate
010fψ(1 − α)P
001fψ[1 − α(1 − β)]A
Mortality−100(ω + υ)F
Segregation1−10fτψ(1 − α)P
10−1fτψ[1 − α(1 − β)]A
Mutation0−11fχψ(1 − α)P
  • a The model consists of three ordinary differential equations describing the dynamics of plasmid-free bacteria, F, nonadapted plasmid bearers, P, and adapted plasmid bearers, A, respectively. The derivatives of the compartments (F, P, or A) are determined by the reaction rates of the contributing processes, such as growth and mortality. Resource availability: f =1 – [(F + P + A)/k]. The matrix notation indicates these reaction rates as well as the directions of the particular effects (1, positive; 0, no effect; and −1, negative). Model versions for plasmid mutations and chromosomal mutations differ in two ways. (i) In the first way, conjugation initiated by adapted plasmid bearers, A, turns plasmid-free cells, F, into bacteria of type A (indicated as p [superscript]), when the compensatory mutation is located on the plasmid (“plasmid mutation”), or into nonadapted plasmid bearers, P, when the mutation is located on the chromosome (“chromosomal mutation,” indicated as c [superscript]). Note that c is only valid for chromosomal mutation (otherwise 0), and p is only valid for plasmid mutation (otherwise 0). (ii) In the second way, compensatory mutations are acquired proportional to replication events, which occur proportional to bacterial growth (given for “chromosomal mutation” and “plasmid mutation”) and conjugation (only given for “plasmid mutation,” p). For a parameter description, see Table 2. The equation form of each model version (for “no mutation,” “chromosomal mutation,” and “plasmid mutation”) is presented in equations E1, E2, and E3 in Text S1.