Universality in Stochastic Exponential Growth

Recent imaging data for single bacterial cells reveal that their mean sizes grow exponentially in time and that their size distributions collapse to a single curve when rescaled by their means. An analogous result holds for the division-time distributions. A model is needed to delineate the minimal requirements for these scaling behaviors. We formulate a microscopic theory of stochastic exponential growth as a Master Equation that accounts for these observations, in contrast to existing quantitative models of stochastic exponential growth (e.g., the Black-Scholes equation or geometric Brownian motion). Our model, the stochastic Hinshelwood cycle (SHC), is an autocatalytic reaction cycle in which each molecular species catalyzes the production of the next. By finding exact analytical solutions to the SHC and the corresponding first passage time problem, we uncover universal signatures of fluctuations in exponential growth and division. The model makes minimal assumptions, and we describe how more complex reaction networks can reduce to such a cycle. We thus expect similar scalings to be discovered in stochastic processes resulting in exponential growth that appear in diverse contexts such as cosmology, finance, technology, and population growth.

Figure 1

Stochastic Hinshelwood cycle. (a) Schematic of the cycle. An example with three chemical species ($N=3$) is shown. The curved arrows indicate that the production of each species $X_i$ is catalyzed by the previous one $X_{i-1}$ with rate $k_i{x}_{i-1}$, where $x_{i-1}$ is the copy number of $X_{i-1}$. The box shows the corresponding reactions explicitly [see (1) for the general specification with $N$ species). (b) Stochastic exponential growth trajectories for the model shown in (a). A single composite time scale emerges asymptotically: $x_1$, $x_2$, and $x_3$ all grow with the same exponential growth rate $\kappa =(k_1{k}_2{k}_3{)}^{1/3}$, which is the mean slope for each curve in the log-linear plot, for $t\gg 1/\kappa$. We show the evolution of the three species for 100 stochastic trajectories. They are obtained from Gillespie simulations [24] of (2) for rate constants $\mathit{k}=(0.1,1.5,3.2)$ and initial copy numbers $\mathit{x}(0)=(20,20,30)$.

Figure 2

Copy number fluctuations are perfectly correlated in the asymptotic limit. (a) Ratios of the copy numbers of the components of the $N=3$ SHC from the trajectories shown in Fig. 1. As predicted by (6), for $t\gg 1/\kappa$, the ratios of the different $x_i$ tend to constant values in each ensemble member, a signature of the perfect correlations between component copy numbers in the asymptotic state. (b) Emergence of a single composite variable. The variable $s_i$ is the projection of the state vector $\mathit{x}$ onto the $i\text{th}$ eigenvector (${\mathit{\xi }}_i$) of the rate constant matrix, $\mathbb{K}$. We see that $s_3$ here (or more generally, $s_N$) tends to a constant nonzero level, while the remainder of the projections vanish.

Figure 3

Universality of growth fluctuations. The symbols mark the numerically obtained distributions of copy numbers, rescaled by their means, for the trajectories shown in Figs. 1 and 2, at the time indicated by the brown dotted line in the inset ($\kappa t=3$). The analogous distribution for the composite stochastic variable, $s_3$ (the projection of the state vector onto the dominant eigenvector), is also shown. The black dashed curves are the gamma distribution in (8). The inset superimposes the composite stochastic variable (gray curves) on the trajectories in Fig. 1. The fact that the trajectories do not converge or diverge with time in this representation also indicates that the distributions of all the $x_i$ are time invariant asymptotically when these variables are rescaled by their exponentially growing means.