Simulation of stem cell survival in small crypts

Monte Carlo simulations of the number of stem cells in human colon crypts allow for fluctuations which kill the population after sufficiently long times. Experiments and computer simulations [1, 2] have elucidated the fate of stem cells and of maturing cells within small crypts in the human colon. The present note restricts itself on the simulation of the number of cells, not on their genealogy. Yatabe et al. [1, 2] assume that at each iteration (" day ") each stem cell splits into two stem cells with probability q, into two maturing cells with the same probability q, and into one stem cell plus one maturing cell with probability p = 1 − 2q. The special case p = 1, q = 0 is deterministic and not dealt with by us. For simplicity we assume p = 0, which means q = 1/2 in the model studied by Yatabe et al. As soon as probabilities larger than zero and smaller than one are used, one introduces fluctuations. Sometimes the fluctuations can become very large and in the present model kill all stem cells [3]. From that moment on the crypt remains without stem cells, and after several iterations also the remaining maturing cells will have reached their life span, causing the whole crypt to die. For a functioning biological system this death of the whole crypt should not happen too often within reasonable time. For infinitely long times, every finite system living according to the above rules will finally die out; we are interested here in the survival for 365 iterations (one " year "). In addition to the above random splitting, we also allow for random deaths happening with a probability of one percent per iteration for each cell, independent of age and type. Moreover, maturing cells die when their life span of three iterations is exceeded. In this case, the number of cells in a system of initially 100 million cells decreases by one percent per iteration, and we balance this decrease by making the model slightly asymmetric: With 1