Large Differences Between Stochastic and Deterministic Kinetics in a Simple Autocatalytic Reaction Network

TL;DR

The paper addresses when stochastic descriptions based on the Chemical Master Equation (CME) diverge from deterministic kinetics in small, well‑mixed systems, focusing on the Finke–Watzky autocatalytic scheme $\mathrm{A} \rightarrow \mathrm{B}$ and $\mathrm{A}+\mathrm{B} \rightarrow 2\mathrm{B}$ and its reversible generalization. It derives a concise, explicit analytical solution of the CME for the irreversible FWM, including the time‑dependent probability distribution $P_n(t)$ and its first two moments, and extends the analysis to reversible reactions and two related networks via parameter mappings. The main contributions are closed‑form expressions for $P_n(t)$ and $\mu_{1A}(t)$, $\mu_{2A}(t)$, a detailed characterization of stochastic delay and its dependence on initial conditions, and a steady‑state solution for the reversible CME expressed through hypergeometric functions. The findings show that, for certain parameter regimes (notably $k_1 \ll k_2$ and $N_B(0)=0$), stochastic delays can be exceptionally large and persist even at relatively large system sizes, highlighting limitations of deterministic descriptions for a wide class of autocatalytic networks with competing steps. These results have implications for modeling epidemics and intracellular chemical processes, and more broadly for population dynamics where discreteness and stochasticity play a crucial role.

Abstract

In small systems, quantitative discrepancies between stochastic and deterministic descriptions of chemical kinetics can be significant, with their magnitude depending on the specific reaction network. Here, we study the Finke-Watzky model-an irreversible autocatalysis, A + B -- > 2B, supplemented by an irreversible first-order process, A -- > B. This model has been used to describe the formation of transition metal nanoparticles and protein misfolding and aggregation, but it may also serve as a minimal model for the spread of a non-fatal but incurable disease. We show that, for certain parameter values, exceptionally large deviations can arise between stochastic and deterministic kinetics of the Finke-Watzky model. Moreover, its stochastic time evolution may be highly sensitive to initial conditions. These properties are retained in the generalization of the model to reversible reactions. To quantify the differences between the predictions of deterministic and stochastic kinetics, we derive the explicit analytical solution of the Chemical Master Equation for the Finke-Watzky model. This solution also allows us to derive analogous solutions for two related reaction networks: A + A -- > A + B, A -- > B, and A + A -- > A + B, A + B -- > 2B. Our findings may have implications for modeling epidemics and intracellular chemical processes, and more broadly for models of population dynamics.

Figures (10)

• Figure 1: Stochastic versus deterministic kinetics for the reactions (\ref{['1st_reaction_WF']})--(\ref{['2nd_reaction_WF']}), with initial condition $N_B(0) = 0$, and rate constants $k_1 = 0.001$, $k_2 = 0.1$. The concentration of A given by $a(t)$ (\ref{['Det_Chem_Kin_WF_only_a_solution_of']}) is shown as a red dashed line, and decays significantly faster than its stochastic counterpart $a_s(t) = \mu_{1A}(t)/V$ (\ref{['stochastic_concentration']}), plotted as a thick blue solid line. The light blue dotted lines indicate the single standard deviation envelope around the mean, $\mu_{1A}(t) \pm \sigma(t)$, providing a quantitative measure of stochastic fluctuations. The system volume is set to $V = 1$, so that $k_2 = \mathcal{K}_2$ and $a_s(t) = \mu_{1A}(t)$, with $\mu_{1A}(t)$ given by Eq. (\ref{['first_moment_non_deg_WF']}). All quantities are plotted as functions of the dimensionless time variable $\tau = k_2 t$.
• Figure 2: Stochastic versus deterministic kinetics for reactions (\ref{['1st_reaction_WF']})–(\ref{['2nd_reaction_WF']}) with $N_B(0)=1$, $k_1 = 0.001$, and $k_2 = 0.1$. When a single B molecule is initially present, the difference between the deterministic concentration $a(t)$ (\ref{['Det_Chem_Kin_WF_only_a_solution_of']}) (red dashed line) and its stochastic counterpart $a_s(t) = \mu_{1A}(t)/V$ (\ref{['stochastic_concentration']}) (thick blue solid line) is much smaller than in Fig. \ref{['sd_WF_nu_0p01_M_100_N_100_up_label']}. Fluctuations around the mean, $\mu_{1A}(t) \pm \sigma(t)$ (light blue dotted lines), are also significantly reduced compared to that figure. Here, $V=1$ so that $k_2 = \mathcal{K}_{2}$ and $a_s(t) = \mu_{1A}(t)$, with $\mu_{1A}(t)$ from (\ref{['first_moment_non_deg_WF']}). All quantities are plotted as functions of the dimensionless time $\tau = k_2 t$. Note the different $\tau$ range compared to Fig. \ref{['sd_WF_nu_0p01_M_100_N_100_up_label']}.
• Figure 3: Stochastic versus deterministic kinetics for reactions (\ref{['1st_reaction_WF']})–(\ref{['2nd_reaction_WF']}) with $N_B(0)=5$, $k_1 = 0.001$, and $k_2 = 0.1$. The difference between the deterministic concentration $a(t)$ (\ref{['Det_Chem_Kin_WF_only_a_solution_of']}) (red dashed line) and its stochastic counterpart $a_s(t) = \mu_{1A}(t)/V$ (\ref{['stochastic_concentration']}) (thick blue solid line) is smaller than in Fig. \ref{['sd_WF_nu_0p01_M_100_N_99_up_label']}. All quantities are plotted as functions of the dimensionless time $\tau = k_2 t$.
• Figure 4: Stochastic versus deterministic kinetics for reactions (\ref{['1st_reaction_WF']})–(\ref{['2nd_reaction_WF']}) with $N_B(0)=10$, $k_1 = 0.001$, and $k_2 = 0.1$. The difference between the deterministic concentration $a(t)$ (\ref{['Det_Chem_Kin_WF_only_a_solution_of']}) (red dashed line) and its stochastic counterpart $a_s(t) = \mu_{1A}(t)/V$ (\ref{['stochastic_concentration']}) (thick blue solid line) is now negligible. Fluctuations around the mean (light blue dotted lines) are comparable to those in Fig. \ref{['sd_WF_nu_0p01_M_100_N_90_up_label']}. All quantities are plotted as functions of the dimensionless time $\tau = k_2 t$.
• Figure 5: Stochastic versus deterministic kinetics for the reactions (\ref{['1st_reaction_WF']})--(\ref{['2nd_reaction_WF']}) for several values of the total number of molecules $M$, with system volume scaling as $V \propto M$, initial condition $N_B(0) = 0$ ($b_0 = 0$), and macroscopic rate constants $\mathcal{K}_{1} = 0.001$ and $\mathcal{K}_{2} = 0.1$. Consequently, $k_1 = 0.001$ in all cases, while $k_2 = \mathcal{K}_{2}/V$ varies with $M$. The deterministic concentration of A, $a(t)$ (\ref{['Det_Chem_Kin_WF_only_a_solution_of']}), is shown as a red dashed line, and its stochastic counterpart, $a_s(t) = \mu_{1A}(t)/V$ (\ref{['stochastic_concentration']}), is plotted as thick solid lines for $M = 100$ (blue, 1), $200$ (light blue, 2), $500$ (dark green, 3), $1000$ (light green, 4), and $2000$ (orange, 5), ordered from left to right. The corresponding system volumes are $V = 1$, $2$, $5$, $10$, and $20$, respectively. For clarity, the standard-deviation envelope around the mean is omitted. All quantities are plotted as functions of the rescaled time variable $\tau \equiv \mathcal{K}_{2}t$.