Globally asymptotic properties of proliferating cell populations

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Abstract
This paper presents a general model for the cell division cycle in a population of cells. Three hypotheses are used: (1) There is a substance (mitogen) produced by cells which is necessary for mitosis; (2) The probability of mitosis is a function of mitogen levels; and (3) At mitosis each daughter cell receives exactly one-half of the mitogen present in the mother cell. With these hypotheses we derive expressions for the α and β curves, the distributions of mitogen and cell cycle times, and the correlation coefficients between mother-daughter (ρmd) and sister-sister (ρss) cell cycle times. The distribution of mitogen levels is shown to be given by the solution to an integral equation, and under very mild assumptions we prove that this distribution is globally asymptotically stable. We further show that the limiting logarithmic slopes of α(t) and β(t) are equal and constant, and that ρmd⩽0 while ρss⩾0. These results are in accord with the experimental results in many different cell lines. Further, the transition probability model of the cell cycle is shown to be a simple special case of the model presented here.