  We formulate a general, high-dimensional kinetic theory describing the
internal state (such as gene expression or protein levels) of cells in a
stochastically evolving population. The resolution of our kinetic theory also
allows one to track subpopulations associated with each generation. Both
intrinsic noise of the cell's internal attribute and randomness in a cell's
division times (demographic stochasticity) are fundamental to the development
of our model. Based on this general framework, we are able to marginalize the
high-dimensional kinetic PDEs in a number of different ways to derive equations
that describe the dynamics of marginalized or "macroscopic" quantities such as
structured population densities, moments of generation-dependent cellular
states, and moments of the total population. We also show how nonlinear
"interaction" terms in lower-dimensional integrodifferential equations can
arise from high-dimensional linear kinetic models that contain rate parameters
of a cell (birth and death rates) that depend on variables associated with
other cells, generating couplings in the dynamics. Our analysis provides a
general, more complete mathematical framework that resolves the coevolution of
cell populations and cell states. The approach may be tailored for studying,
e.g., gene expression in developing tissues, or other more general particle
systems which exhibit Brownian noise in individual attributes and
population-level demographic noise.
