We develop mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we formulate a complete kinetic framework for age-structured interacting populations undergoing birth, death and fission processes in spatially dependent environments. We define the full probability density for the population-size age chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalization of the McKendrick-von Foerster equation, which describes mean-field deterministic behavior. Our approach utilizes mixed-type, multidimensional probability distributions similar to those employed in the study of gas kinetics and with terms that satisfy BBGKY-like equation hierarchies.

Cell division is a process that involves many biochemical steps and complex biophysical mechanisms. To simplify the understanding of what triggers cell division, three basic models that subsume more microscopic cellular processes associated with cell division have been proposed. Cells can divide based on the time elapsed since their birth, their size, and/or the volume added since their birth-the timer, sizer, and adder models, respectively. Here, we propose unified adder-sizer models and investigate some of the properties of different adder processes arising in cellular proliferation. Although the adder-sizer model provides a direct way to model cell population structure, we illustrate how it is mathematically related to the well-known model in which cell division depends on age and size. Existence and uniqueness of weak solutions to our 2+1-dimensional PDE model are proved, leading to the convergence of the discretized numerical solutions and allowing us to numerically compute the dynamics of cell population densities. We then generalize our PDE model to incorporate recent experimental findings of a system exhibiting mother-daughter correlations in cellular growth rates. Numerical experiments illustrating possible average cell volume blowup and the dynamical behavior of cell populations with mother-daughter correlated growth rates are carried out. Finally, motivated by new experimental findings, we extend our adder model cases where the controlling variable is the added size between DNA replication initiation points in the cell cycle.

Multiple somatic rearrangements are often found in cancer genomes; however, the underlying processes of rearrangement and their contribution to cancer development are poorly characterized. Here we use a paired-end sequencing strategy to identify somatic rearrangements in breast cancer genomes. There are more rearrangements in some breast cancers than previously appreciated. Rearrangements are more frequent over gene footprints and most are intrachromosomal. Multiple rearrangement architectures are present, but tandem duplications are particularly common in some cancers, perhaps reflecting a specific defect in DNA maintenance. Short overlapping sequences at most rearrangement junctions indicate that these have been mediated by non-homologous end-joining DNA repair, although varying sequence patterns indicate that multiple processes of this type are operative. Several expressed in-frame fusion genes were identified but none was recurrent. The study provides a new perspective on cancer genomes, highlighting the diversity of somatic rearrangements and their potential contribution to cancer development.