Theory for transitions between log and stationary phases: universal laws for lag time

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. Quantitative characterization of a cellular state, in terms of cellular growth rate, concentration of external resources, as well as abundances of specific components, has long been one of the major topics in cell biology, ever since the pioneering study by Monod [1]. Quantitative growth laws have been uncovered mainly by focusing on the microbial ∗[email protected] †[email protected] 1 ar X iv :1 60 7. 03 24 6v 1 [ ph ys ic s. bi oph ] 1 2 Ju l 2 01 6 log phase in which the number of cells grows exponentially, including Pirt’s equation for yield and growth [2] and the relationship between the fraction of ribosomal abundance and growth rate (experimentally demonstrated by Schaechter et al.[3], and theoretically rationalized by Scott et al. [4]), among others [5, 6, 7, 8], in which the constraint to maintain steady growth leads to general relationships[9, 10, 11]. In spite of the importance of the discovery of these universal laws, cells under poor conditions exhibit different growth phases in which such relationships are violated. Indeed, in addition to the death phase, cells undergo a stationary phase under conditions of resource limitation, in which growth is drastically suppressed. Once cells enter the stationary phase, a certain time span is generally required to recover growth after resources are supplied, which is known as the lag phase. Although several quantities have been measured to characterize these phases, such as the length of lag time for resurrection, and the tolerance time for starvation or antibiotics [12, 13, 14], there has been no theory put forward to characterize the phase changes, and no corresponding quantitative laws have been established. To develop a theory for bacterial physiology beyond the log phase, we first constructed a simple mathematical model that exhibits the changes among the lag, log, stationary, and death phases. We then uncovered the quantitative characteristics of each of these phases in line with experimental observations. Including bacterial growth curve, quantitative relationships of lag-time with starvation time and the maximal growth rate, exponentiallytailed distribution of lag-time, and trade-off between the growth rate and tolerance for the starvation. These are formulated by the changes in inhibitor (or mistranslated proteins) chemicals in addition to changes in ribosomal proteins (ribosomes). The proposed model also allowed us to reach several experimentally testable predictions, including the dependence of lag time on the speed of the starvation process.