Diffusive waves, dynamic instability, and chromosome missegregation: dimensionality, discreteness, stochasticity

Abstract

Modeling biological systems is challenging because many processes act at the same scale, and because biological systems span many scales. Because it is difficult or impos- sible to know everything about a given cell, tissue, or organism, enterprising theorists must construct models that are either insensitive to details or that capture some (but not all) vital details. Moreover, the heterogeneity and noise inherent to biological sys- tems require the modeler to construct stochastic models. Here, I discuss three projects in which I have used this approach. First, I study the dynamics of diffusive cell signaling waves. Many organisms employ such waves in order to communicate faster than the diffusion limit. I detail the scaling laws and dimension-dependent dynamics that emerge within a broad class of diffusive wave models, showing in the process that dimensionality, discreteness, and disorder play crucial roles in such systems. Next, I study the stochastic growth-and-collapse dynamics of dynamically unstable polymers. Dynamically unstable polymers grow until they reach a point of “catastro- phe”, at which point they shrink before regrowing. I show that within a broad class of models, the distribution of times between catastrophe events is exponential. This universal distribution shape can be altered if the polymer grows from a small pool of monomers, and I compute analytic formulae for the altered distribution shape. Finally, I study chromosome missegregation, which is the unhappy failure of a mother cell to endow both of its daughter cells with one copy of a given chromosome. Using stochastic models, I derive closed-form solutions for the temporal dynamics of chro- mosome missegregations, then heavily constrain the possible model space by using a small dataset. In doing so, I show that missegregating mother cells have, on average, a “preferred” daughter cell that inherits more of the chromosomes. I also show that there must be either correlated chromosome missegregations or cell-to-cell variability in the initial state of the mitotic spindle’s connections to the chromosomes.