Multi-type branching process models of cell proliferation
Abstract(Axelrod et al., 1993) carried out experiments in which colonies of mouse fibroblast cells were dispersed and seeded to form secondary colonies. They found that there existed highly significant correlations between sizes for the primary and secondary colonies, which were accompanied by high variances of colony sizes. Simulation experiments by these same authors showed that the data are fitted only by a computer simulation model in which the lifetimes of cells in a secondary clone are, to a large extent, determined by the lifetimes of the founder cell of the clones, selected randomly from a primary colony. To mathematically analyze this unexpected result, we derive, for previously uninvestigated multi-type Bellman-Harris and Galton-Watson branching process models, "sampling formulas" which make it possible to find the mixed second moment of the cell counts in the primary and secondary colonies.
The derived expressions are difficult to evaluate explicitly for all but fairly simple cases; hence, it was necessary to write computer code to do so numerically. Despite an inherent trade-off between correlation and variance, it was possible to match the Galton-Watson model to the observations. Numerical results for the Bellman-Harris process are still pending; however, some simplified asymptotic equivalents were derived and analyzed. It was found analytically that, in these models, the correlation goes to zero asymptotically. We note that we have added to the number of explicit expressions for branching processes. We present these results and some of their consequences