Kinetic theories of state- and generation-dependent cell populations

We formulate a general, high-dimensional, partial integrodifferential equation (PIDE) 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. Using our framework, we are able to marginalize the high-dimensional PIDEs in a number of different ways to derive equations which can be PIDEs themselves) 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.