  Quantitative characterization of bacterial growth has gathered substantial
attention since Monod's pioneering study. Theoretical and experimental work has
uncovered several laws for describing the log growth phase, in which the number
of cells grows exponentially. However, microorganism growth also exhibits lag,
stationary, and death phases under starvation conditions, in which cell growth
is highly suppressed, while quantitative laws or theories for such phases are
underdeveloped. In fact, models commonly adopted for the log phase that consist
of autocatalytic chemical components, including ribosomes, can only show
exponential growth or decay in a population, and phases that halt growth are
not realized. Here, we propose a simple, coarse-grained cell model that
includes inhibitor molecule species in addition to the autocatalytic active
protein. The inhibitor forms a complex with active proteins to suppress the
catalytic process. Depending on the nutrient condition, the model exhibits the
typical transition among the lag, log, stationary, and death phases.
Furthermore, the lag time needed for growth recovery after starvation follows
the square root of the starvation time and is inverse to the maximal growth
rate, in agreement with experimental observations. Moreover, the distribution
of lag time among cells shows an exponential tail, also consistent with
experiments. Our theory further predicts strong dependence of lag time upon the
speed of substrate depletion, which should be examined experimentally. The
present model and theoretical analysis provide universal growth laws beyond the
log phase, offering insight into how cells halt growth without entering the
death phase.
