Phenomenology of stochastic exponential growth

Stochastic exponential growth is observed in a variety of contexts, including molecular autocatalysis, nuclear fission, population growth, inflation of the universe, viral social media posts, and financial markets. Yet literature on modeling the phenomenology of these stochastic dynamics has predominantly focused on one model, geometric Brownian motion (GBM), which can be described as the solution of a Langevin equation with linear drift and linear multiplicative noise. Using recent experimental results on stochastic exponential growth of individual bacterial cell sizes, we motivate the need for a more general class of phenomenological models of stochastic exponential growth, which are consistent with the observation that the mean-rescaled distributions are approximately stationary at long times. We show that this behavior is not consistent with GBM, instead it is consistent with power-law multiplicative noise with positive fractional powers. Therefore, we consider this general class of phenomenological models for stochastic exponential growth, provide analytical solutions, and identify the important dimensionless combination of model parameters, which determines the shape of the mean-rescaled distribution. We also provide a prescription for robustly inferring model parameters from experimentally observed stochastic growth trajectories.

Figure 1

Log-linear plot of ensemble trajectories of stochastic exponential growth for different values of the two dimensionless parameters that affect the distributions, namely $\gamma$ and ${\mathrm{\Sigma }}_{\gamma }^2$. The values are chosen from $\gamma \in \{0,1/2,3/4,0.9\}$ and ${\mathrm{\Sigma }}_{\gamma }^2\in \{0.01,0.1,1\}$. The trajectories corresponding to ${\mathrm{\Sigma }}_{\gamma }^2=1$ are shown in cyan (light gray), ${\mathrm{\Sigma }}_{\gamma }^2=0.1$ in orange (medium gray), and ${\mathrm{\Sigma }}_{\gamma }^2=0.01$ in gray (dark gray). For small values of $\gamma$, the trajectories are more stochastic at the beginning and more likely to be absorbed by the boundary, but become more or less deterministic when they grow large. In contrast, as $\gamma$ approaches 1, the trajectories stay stochastic for longer time, as the mean rescale distribution takes longer and longer to settle to a stationary distribution. For a fixed $\gamma$, the over all variance grows with ${\mathrm{\Sigma }}_{\gamma }^2$. For a comparison of the effect of $\gamma$ on the distribution, when ${\mathrm{\Sigma }}_{\gamma }^2$ kept constant, see Fig. 2. Simulation variables: $x_0=1$ and $\kappa =1$.

Figure 2

Plots of the mean-rescaled distribution, $q(y,t)$; it depends only on the dimensionless exponent $\gamma$, and the time-dependent dimensionless factor, $v\left(t\right)={\mathrm{\Sigma }}_{\gamma }^2\tau \left(t\right)$. For small $v\left(t\right)$, as shown in (a), the dynamics are almost deterministic, and the distributions for all $\gamma$'s are well-approximated with a narrow Gaussian centered around the mean value, which is unity at all times. When the variance of the mean-rescaled distribution is kept constant (a, b, and c), $\gamma$ determines its skewness.

Figure 3

Plots of the ratio, $r_3(\gamma ,v)$, defined in Eq. (65), as a function of $\gamma$, for three choices of $v=\text{var}\left[y\left(t\right)\right]=0.01$ (dashed, orange), 0.1 (dotted-dashed, gray), and 0.2 (dotted, cyan). These show that $r_3$ is sensitive to the value of $\gamma$, and changes approximately linearly with $\gamma$, but it is extremely insensitive to ${\mathrm{\Sigma }}_{\gamma }^2$ (for small ${\mathrm{\Sigma }}_{\gamma }^2$). As a result, $r_3$ can be used to directly deduce the value of $\gamma$ from a distribution, independent of the method of inference used to find the value of ${\mathrm{\Sigma }}_{\gamma }^2$.

Figure 4

Plot of the density $q\left(y\right)$ given in Eq. (18) (gray curve) for the stochastic exponential growth process with constant noise Eq. (13). As shown in text, this can be represented as the difference of two Gaussian distributions, centered at $y=\pm 1$ (dashed curves); see Eq. (A5).

Figure 5

Sample path (orange curve) of the process Eq. (B1) with parameters $\sqrt{D}=0.01, \kappa =0.5$, and $x_0=0.01$. The dashed gray curve shows the corresponding deterministic exponential process, $x\left(t\right)=x_0{e}^{\kappa t}$.

Figure 6

The log-likelihood function, $L_n(D,{\kappa }_0;\left\{x_i\right\})$, for the dataset of Fig. 5. The $D$ parameter is determined by minimization of this function.