Cell population balance, ensemble and continuum modeling frameworks: Conditional equivalence and hybrid approaches

The cell population balance, ensemble and continuum modeling frameworks are widely used for the mathematical description of microbial populations. Each of these approaches focuses on different aspects of the processes of growth, division and intracellular reaction occurrence. Therefore, each framework can output different information at different computational expense. Continuum models assume lumped biomasses, capture the dynamics of bulk intracellular concentrations and are easy to simulate. Ensemble models account for heterogeneity due to different initial conditions or kinetic constants, and are more computationally expensive. Finally, cell population balances capture the partitioning of the intracellular contents in detail, but can quickly become intractable, as the number of biochemical species taken into consideration increases. It is thus natural to ask whether one can adequately simulate cell populations with a simpler approach.In this paper, starting from an (n+1) cell population balance, for n biochemical species and the cell volume, we investigate biologically plausible conditions, under which two exact hybrid models are derived. Provided that the conditions hold true, both hybrid models are exact but much simpler alternatives of the cell population balance; no approximation is involved. In the first, the evolution of the species concentrations is captured by an ensemble model, and in the second by a continuum model. Both hybrid models also contain a 1-dimensional population balance for the cell volume. This work should provide a guideline for choosing the modeling approach which is the most appropriate for the particular application.