Regulation of stem cell dynamics through volume exclusion

Abstract: Maintenance and regeneration of adult tissues rely on the self-renewal of stem cells. Regeneration without over-proliferation requires precise regulation of the stem cell proliferation and differentiation rates. The nature of such regulatory mechanisms in different tissues, and how to incorporate them in models of stem cell population dynamics, is incompletely understood. The critical birth-death (CBD) process is widely used to model stem cell populations, capturing key phenomena, such as population asymmetry and scaling laws in clone size distributions. However, the CBD process neglects regulatory mechanisms. Here, we propose the birth-death process with volume exclusion (vBD), a variation of the birth-death process that takes into account crowding effects, such as may arise due to limited space in a stem cell niche. While the deterministic rate equations predict a single non-trivial attracting steady state, the master equation predicts extinction and a transient distribution of stem cell numbers that can be bimodal. We investigate the accuracy of the system-size expansion (including finite size corrections to the linear-noise approximation), the quasi-steady state approximation, and the WKB method to approximate the probability distribution solution of the vBD master equation, as well as the mean extinction time. Our study suggests that the size distribution of a stem cell population and its extinction dynamics bear signatures that may be useful to detect negative feedback mediated via volume exclusion.