Abstract
Phenotypic memory can predispose cells to physiological outcomes, contribute to heterogeneity in cellular populations, and allow computation of environmental features, such as nutrient gradients. In bacteria and synthetic circuits in general, memory can often be set by protein concentrations: because of the relative stability of proteins, the degradation rate is often dominated by the growth rate, and inheritance is a significant factor. Cells can then be primed to respond to events that recur with frequencies faster than the time to eliminate proteins. Protein memory can be extended if cells reach extremely low growth rates or no growth. Here we characterize the necessary time scales for different quantities of protein memory, measured as relative entropy (Kullback-Leibler divergence), for a variety of cellular growth arrest transition dynamics. We identify a critical manifold in relative protein degradation/growth arrest time scales where information is, in principle, preserved indefinitely because proteins are trapped at a concentration determined by the competing time scales as long as nongrowth-mediated protein degradation is negligible. We next asked what characteristics of growth arrest dynamics and initial protein distributions best preserve or eliminate information about previous environments. We identified that sharp growth arrest transitions with skewed initial protein distributions optimize flexibility, with information preservation and minimal cost of residual protein. As a result, a nearly memoryless regime, corresponding to a form of bet-hedging, may be an optimal strategy for storage of information by protein concentrations in growth-arrested cells.