Mathematical Modelling Determines Post-Segregational Killing Efficacy of Toxin-Antitoxin Systems in Commensal Escherichia coli Nissle 1917

Plasmids are commonly used as vectors for introducing synthetic circuits into bacterial hosts. These circular pieces of extra-chromosomal DNA are easily editable and provide the modularity that is one of the central goals of synthetic biology. However, ensuring plasmids are stably inherited can be problematic. Uneven distribution of plasmids within a cell at the time of division leads to the creation of plasmid-free cells which often outgrow the plasmid-bearing population. Antibiotic selection, commonly used in the laboratory, cannot be used as a plasmid stability mechanism in environments such as the gastrointestinal tract, where the disruption of other bacterial communities is undesirable. Naturally occurring plasmids have evolved several mechanisms for enhancing plasmid stability, such as active partitioning and post-segregational killing. Here we implement deterministic and stochastic models describing the population dynamics of plasmid-bearing bacteria. Using flow cytometry and automated clustering with mixture-models, we quantify the change in the plasmid-bearing fraction of populations of the commensal bacteria Escherichia coli Nissle 1917 in the presence of different toxin-antitoxin systems. Bayesian methods are used to fit our models to the data, with a GPU accelerated tau-leaping implementation enabling simulation of the stochastic model. Our methods allow for the separation of growth rate differences, between plasmid-bearing and plasmid-free bacteria, from the plasmid stability parameters. We also demonstrate the difficulty of separating the rate of plasmid loss from the efficacy of an employed post-segregational killing mechanism. Further, our results show that the non-native Axe/Txe toxin-antitoxin provides stability beyond the period for which native, and more commonly used, Hok/Sok was effective. As synthetic biological applications progress to more complex environments consideration for the stability of synthetic circuits becomes essential. We anticipate that the use of mathematical and statistical modelling will become increasingly useful to rationalise the design of synthetic circuits as more components become available, and to predict their function as the complexity of the target environment increases.