Theory for Transitions Between Exponential and Stationary Phases: Universal Laws for Lag Time

The quantitative characterization of bacterial growth has attracted substantial attention since Monod’s pioneering study. Theoretical and experimental works have uncovered several laws for describing the exponential 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, for which quantitative laws or theories are markedly underdeveloped. In fact, the models commonly adopted for the exponential phase that consist of autocatalytic chemical components, including ribosomes, can only show exponential growth or decay in a population; thus, phases that halt growth are not realized. Here, we propose a simple, coarse-grained cell model that includes an extra class of macromolecular components in addition to the autocatalytic active components that facilitate cellular growth. These extra components form a complex with the active components to inhibit the catalytic process. Depending on the nutrient condition, the model exhibits typical transitions among the lag, exponential, stationary, and death phases. Furthermore, the lag time needed for growth recovery after starvation follows the square root of the starvation time and is inversely related to the maximal growth rate. This is in agreement with experimental observations, in which the length of time of cell starvation is memorized in the slow accumulation of molecules. Moreover, the lag time distributed among cells is skewed with a long time tail. If the starvation time is longer, an exponential tail appears, which is also consistent with experimental data. Our theory further predicts a strong dependence of lag time on the speed of substrate depletion, which can be tested experimentally. The present model and theoretical analysis provide universal growth laws beyond the exponential phase, offering insight into how cells halt growth without entering the death phase.

Popular Summary

Cells originate from other cells. In order to grow, they must continuously synthesize their components by themselves; otherwise the components decompose, resulting in cell death. With this simple view, a population of cells either grows exponentially (if provided with sufficient nutrients) or dies out (if starved). In reality, however, cells that are starved often stop growing without dying. It is not clear how a self-reproducing cell can “sleep,” nor is it understood why a transition to such a sleeping state (known as a stationary phase) is ubiquitous in microorganisms. To address these mysteries, we introduce a simple model of a cell that includes extra macromolecular components used to inhibit catalytic processes, and we use it to provide new insight into how cells cease growth while avoiding death.

Our cell model consists of substrates, active proteins such as ribosomal proteins, and “waste” proteins (for example, inevitably produced mistranslated proteins). This model generally exhibits transitions among exponential growth, stationary, and death phases with a decrease in nutrient concentrations, and it reproduces well the cell growth curves seen in experiments. Furthermore, upon recovery of nutrients after starvation, the cell needs a certain time to grow. This lag time increases with the square root of the starvation time, in agreement with experiments, where cells “remember” the starvation time by the slow accumulation of waste. We also obtain a tailed distribution of lag time that agrees with experimental results, and we predict that lag time depends strongly on the speed of nutrient depletion.

Our theory demonstrates the universality of the transitions among the phases, as well as laws on lag time. It also explains why microorganisms generally enter the stationary phase under starvation, and it provides implications for cellular response upon antibiotics treatment and food sanitation.

Figure 1

(a) Schematic representation of the components and reactions in the present model. The concentration of each chemical changes according to the listed reactions. In addition, chemicals are spontaneously degraded at a low rate, and they become diluted because of the volume expansion of the cell. (b) Steady growth rate and the concentration of component A are plotted as functions of the external concentration of the substrate. (c,d) Growth curve of the model. Parameters are set as follows: $v=0.1$, $k_p=1.0$, $k_m={10}^{-6}$, $K=1.0$, $K_t=10.0$, $d_R=d_B={10}^{-5}$, $d_C={10}^{-12}$. The detailed numerical method for panels (c) and (d) is given in Appendix pp3.

Figure 2

(a,b) Lag time as a function of (a) starvation time or (b) preincubation time. The lag time is scaled by the maximum growth rate (inversely proportional to the shortest doubling time in the substrate-rich condition). Purple pentagons, cyan dots, and orange squares are adopted from Figs. 3, 6(a), and 6(b) of Augustin et al. [12], respectively, and the red triangles are extracted from the data in Table 1 of Pin et al. [40]. (c,d) Relationship between the lag time and maximum specific growth rate ${\mu }_{\max }$. Data are adopted from Table 1 of Oscar [39]. Parameters were set as follows: $S_{\mathrm{ext}}^{\mathrm{rich}}=10^4$, $S_{\mathrm{ext}}^{\mathrm{poor}}={10}^{-2}$, $v=0.1$, $k_p=1.0$, $k_m={10}^{-6}$, $K=1.0$, $K_t=10.0$, and $d_A=d_B=d_C=0$ (the same parameter values as in Fig. 1 except $d_i\mathrm{s}$). The lag time is computed as the time needed to reach the steady state under the $S_{\mathrm{ext}}=S_{\mathrm{ext}}^{\mathrm{rich}}$ condition from an initial condition in the inactive phase. In panel (c), it is obtained by varying $v(={\mu }_{\max })$.

Figure 3

(a) Dependence of lag time $\lambda$ on the time required to decrease the substrate $T_{\mathrm{dec}}$ and starvation time $T_{\mathrm{stv}}$. (b–d) Time series of starvation for different $T_{\mathrm{dec}}$ [$T_{\mathrm{dec}}=10^6$ (green line) and $T_{\mathrm{dec}}=1.0$ (purple line)] values—the internal concentrations of substrate $S$ (b), component A (c), and component B (d). (e) Time series of biomass during resurrection. The same parameter values as indicated in Fig. 2 were adopted. The batch culture model (which is used to compute a bacterial growth curve) was adopted to compute the time series of biomass accumulation (e). The time series of $\mu$ is shown in Fig. S3 of Ref. [34].

Figure 4

Movement of nullclines and time evolution of state variables [circles within the state space $(A,B)$]. (a) The case of a fast substrate decrease (the orange line indicates the orbit, and numbers in white boxes indicate the time points). The orbit of a slow substrate decrease is also plotted (black dashed line). (b–d) The case of a slow substrate decrease. Each point is the value of the state variable at the indicated time and substrate concentration. The vector field $v=(dA/dt,dB/dt)$ is also depicted. Parameters are identical to those described in Fig. 2.

Figure 5

Distribution of lag time obtained by model simulation (solid line) with experimental data (lag-time distribution of cultures starved for the indicated days) overlaid. The horizontal axis of each distribution was normalized by using its peak point $(\text{Peak})$ and the full width at half maximum (FWHM) as $\lambda \to (\lambda -\text{Peak})/\mathrm{FWHM}$. Experimental data were extracted from those presented in Fig. 1(e) of Reismann et al. [18]. Methods of stochastic simulations, the procedure used to compute $\text{Peak}$ and FWHM, and parameter values are given in Appendix C.

Figure 6

Comparison of the model results using estimated values (Table 2) with experimental values. (a) The specific growth rate is plotted as a function of the external substrate (glucose) concentration. Experimental data are adopted from Monod [1]. (b) Ratio of ribosomal proteins (component A) to total proteins as a function of the specific growth rate: Orange triangles are from Scott et al. [4], red circles are from Bremer and Dennis [48], and green squares are from Forchhammer et al. [49]. In panel (b), the theoretical curve from the model is plotted up to about 1.0 because we obtained the parameter values by fitting the $\mu -\varphi$ relation and the Monod equation with the maximum growth rate of ${\mu }_{\max }\sim 1.0$.

Figure 7

Schematic representation of a polymer elongation system with kinetic proofreading. The reactions, other than the synthesis part [$F_A(S)A$ and $F_B(S)A$], are identical to those of the original model (2).

Figure 8

$J_{\mathrm{A}}^{L=10}$ and $J_{\mathrm{B}}^{L=10}$ are plotted against the substrate concentration [$S$]. The ratio of $J_{\mathrm{A}}^{L=10}$ to $J_{\mathrm{B}}^{L=10}$ is also plotted in the inset of the figure. Parameters for the polymer elongation part are set to $\widehat{v}=0.1$, ${\rho }_C=1.0$, ${\rho }_D=10.0$, ${\widehat{k}}_0=10^5$, ${\widehat{k}}_1=10^2$, ${\widehat{k}}_2=10.0$, ${\widehat{l}}_0={\widehat{l}}_1=\exp (-1)$, ${\widehat{l}}_2=\exp (1)$, $K_a=10.0$, $K_S=1.0$, $[M{]}_{\max }=1.0$, and $[A_0]=1.0$.

Figure 9

Steady growth rate of the model with polymerization and kinetic proofreading. Here, $J_{\mathrm{A}}^{L=10}$ and $J_{\mathrm{B}}^{L=10}$ are adopted for the synthetic reaction rate of components A and B. Parameters for $J_{\mathrm{A}}^{L=10}$ and $J_{\mathrm{B}}^{L=10}$ are identical to those in Fig. 8, and others are set to be the same.