Fixation times in differentiation and evolution in the presence of bottlenecks, deserts, and oases

TL;DR

This paper models the stochastic process of cellular differentiation and evolution using multi-type branching processes, analyzing how bottlenecks and oases affect the time it takes for cells to reach specific states, with implications for understanding cancer and resistance.

Contribution

It provides analytical and numerical insights into first passage times in complex evolutionary landscapes, accounting for non-linear effects of proliferation and population dynamics.

Findings

First passage times are highly sensitive to bottlenecks and oases.

Proliferation introduces nonlinearity affecting fixation times.

Mean-field approximations may be insufficient in certain scenarios.

Abstract

Cellular differentiation and evolution are stochastic processes that can involve multiple types (or states) of particles moving on a complex, high-dimensional state-space or "fitness" landscape. Cells of each specific type can thus be quantified by their population at a corresponding node within a network of states. Their dynamics across the state-space network involve genotypic or phenotypic transitions that can occur upon cell division, such as during symmetric or asymmetric cell differentiation, or upon spontaneous mutation. Waiting times between transitions can be nonexponentially distributed and reflect e.g., the cell cycle. Here, we use a multi-type branching processes to study first passage time statistics for a single cell to appear in a specific state. We present results for a sequential evolutionary process in which $L$ successive transitions propel a population from a "wild-type" state to a given "terminally differentiated," "resistant," or "cancerous" state. Analytic and numeric results are also found for first passage times across an evolutionary chain containing a node with increased death or proliferation rate, representing a desert/bottleneck or an oasis. Processes involving cell proliferation are shown to be "nonlinear" (even though mean-field equations for the expected particle numbers are linear) resulting in first passage time statistics that depend on the position of the bottleneck or oasis. Our results highlight the sensitivity of stochastic measures to cell division fate and quantify the limitations of using certain approximations and assumptions (such as fixed-population and mean-field assumptions) in evaluating fixation times.