Stem cell differentiation as a renewal-reward process: predictions and validation in the colonic crypt

Stem cells serve as persistent reservoirs for replenishment of rapidly renewing tissues, frequently also ensuring that the correct tissue morphology is maintained. This process is inherently stochastic due to the small number and stochastic division patterns within the stem cell compartments, as well as the essentially stochastic differentiation events that follow the initial stem cell expansion. Here we propose a new formalism to describe this process, by employing the approach known in statistics as the renewal-reward process. Using this approximation allows application of the mathematical apparatus developed for renewal-reward processes to the stochastic stem cell biology. We show in the context of colonic crypts that the resulting predictions match the experimental results, while also providing a convenient tool for analysis of normal and abnormal differentiation processes.