Kinetic theory for structured populations: application to stochastic sizer-timer models of cell proliferation*

Structured population models have been widely used to model cell
proliferation that depends on cell age, size, and/or added size since
birth. However, few models have take into account the effects of
stochasticity in both cell growth and randon cell division and death
times (demographic stochasticity). We derive the full kinetic
equations describing the evolution of the probability density for a
structured population such as cells distributed according to their
ages and sizes. The kinetic equations for such a "sizer-timer" model
incorporates both demographic and individual cell growth rate
stochasticities. Marginalizing over the densities yields corrections
to existing structured population models via hierarchies of equations,
some of which can be closed. For example, we derive a second order PDE
that describes mean-field cell population density evolution which
involves stochastic growth rates. Our kinetic framework is thus a more
complete model that subsumes both the deterministic PDE and
birth-death master equation representations for structured
populations.