Generalization of Powell's results to population out of steady state

TL;DR

This paper extends Powell's classical results on bacterial population dynamics from steady to unsteady states, deriving a time-dependent Euler-Lotka equation and analyzing fitness landscapes for non-exponentially growing populations.

Contribution

It generalizes Powell's steady-state relationships to transient states, providing new formulas for population growth and fitness landscapes in unsteady conditions.

Findings

Derived a time-dependent Euler-Lotka equation.

Generalized the inequality between mean generation time and doubling time.

Calculated fitness landscapes for non-exponentially growing populations.

Abstract

Since the seminal work of Powell, the relationships between the population growth rate, the probability distributions of generation time, and the distribution of cell age have been known for the bacterial population in a steady state of exponential growth. Here, we generalize these relationships to include an unsteady (transient) state for both the batch culture and the mother machine experiment. In particular, we derive a time-dependent Euler-Lotka equation (relating the generation time distributions to the population growth rate) and a generalization of the inequality between the mean generation time and the population doubling time. To do this, we use a model proposed by Lebowitz and Rubinow, in which each cell is described by its age and generation time. We show that our results remain valid for a class of more complex models that use other state variables in addition to cell age and generation time, as long as the integration of these additional variables reduces the model to Lebowitz-Rubinow form. As an application of this formalism, we calculate the fitness landscapes for phenotypic traits (cell age, generation time) in a population that is not growing exponentially. We clarify that the known fitness landscape formula for the cell age as a phenotypic trait is an approximation to the exact time-dependent formula.