Branching and fragmenting particle systems in biology

We introduce and study two new classes of branching and fragmenting particle systems in biology. The first class of branching particle systems with interactions model spatially-structured populations with density-dependent birth and death mechanisms. Our spatial model describes spatially heterogeneous populations using point measures, considering birth and death rates based on spatial position and local population density. Notably, our model includes a juvenile phase, where offspring mature with a density-dependent probability. This results in novel scaling limits: an interacting superprocess, a nonlocal partial differential equation (PDE), and a classical PDE (in which the diffusion term is nonlinear). Furthermore, using a lookdown representation, we retain information about genealogies and, in the case of deterministic limiting models, use this to deduce the backwards in time motion of the ancestral lineage of a sampled individual. We observe that knowing the history of the population density is not enough to determine the motion of ancestral lineages in our model. In our second class of models, motivated by using cell-free DNA (cfDNA) as a biomarker, we capture three mechanisms governing cfDNA fragmentation in the bloodstream. Our Markovian model, FRIME (Fragmentation, Immigration, and Exit), allows the simulation of cfDNA fragment profiles by sampling from its stationary distribution. We validate the model by comparing simulated profiles with mitochondrial and genomic cfDNA fragments, finding consistency with experimental observations. We employ probabilistic tools and the theory of fragmentation processes to establish the first moments of FRIME processes mathematically. This simulation framework improves our computational understanding of DNA fragmentation dynamics in the bloodstream and has the potential to underpin tools for the analysis of liquid biopsy data.