Grow now, pay later: When should a bacterium go into debt?

Microbes grow in a wide variety of environments and must balance growth and stress resistance. Despite the prevalence of such trade-offs, understanding of their role in nonsteady environments is limited. In this study, we introduce a mathematical model of "growth debt," where microbes grow rapidly initially, paying later with slower growth or heightened mortality. We first compare our model to a classical chemostat experiment, validating our proposed dynamics and quantifying Escherichia coli's stress resistance dynamics. Extending the chemostat theory to include serial-dilution cultures, we derive phase diagrams for the persistence of "debtor" microbes. We find that debtors cannot coexist with nondebtors if "payment" is increased mortality but can coexist if it lowers enzyme affinity. Surprisingly, weak noise considerably extends the persistence of resistance elements, pertinent for antibiotic resistance management. Our microbial debt theory, broadly applicable across many environments, bridges the gap between chemostat and serial dilution systems.