Differential Equation Model for the Population-Level Dynamics of a Toggle Switch with Growth-Feedback

Bistable genetic circuits that can be toggled between two states have been engineered in bacterial cells for a variety of applications. These circuits often impose state-dependent resource loads on the cell, creating growth feedback. In the context of a population of cells, each with a copy of the genetic circuit, cells in either circuit state grow at different rates, thereby affecting the emergent population-level dynamics. It is generally difficult to predict how this growth heterogeneity will affect the composition of the population over time. In this work, we consider an ODE population model and evaluate its ability to predict the transient dynamics of the fraction of cells in either state. These dynamics are driven by two processes. The first is due to the difference in growth rate between the cells in the two states, while the second process arises from the probability that the circuit switches state. For the latter, we compute switching rates for the toggle switch using a Markov chain two-dimensional model and exploit the system's structure for efficient computation. We demonstrate via simulations that the ODE model well approximates the dynamics of the system obtained by a published population simulation algorithm for sufficiently large molecular counts and population sizes. The ability to approximate via ODEs the population-level dynamics of cells engineered with multi-stable circuits will be especially relevant to forward engineer such circuits for desired population dynamics.