Ornstein-Uhlenbeck models of phenotypic selection and cellular evolution*

The century-old Ornstein-Uhlenbeck (OU) process describes stochastic diffusion processes with a linear restoring force. The OU process can model a Brownian particle on a spring, or electrical current fluctuations in an RL circuit. Here we generalize the OU process to model stochastic protein level fluctuations in single cells that can divide or die. If each cell's probabilities of division and death per unit time depend on the cell's current protein level, we obtain models of phenotypic selection and evolution. We investigate how such models can predict or interpret the results of cellular evolution experiments and phenotypic selection experiments.