Validity of the total quasi-steady-state approximation in stochastic biochemical reaction networks

TL;DR

This paper addresses the assumption that stochastic tQSSA universally preserves dynamics when reducing stochastic biochemical reaction networks. It analyzes deterministic tQSSA for fast reversible binding and derives stochastic reduced models, extending the analysis to spatially homogeneous and heterogeneous settings, including PDE and compartmental simulations. The key finding is that stochastic tQSSA can distort dynamics even when deterministic tQSSA is valid, with distortions arising under conditions like $n_{D_T} K_d \Omega < 10$ or in local compartments where counts are comparable; the stochastic low-state QSSA ($lQSSA$) and, in some cases, time-delay schemes (ETS) offer more robust alternatives. These results provide practical guidance for reliable stochastic model reductions in cellular systems, especially when spatial structure creates local violations of reduction validity, and they advocate compartment-wise assessment and adaptive use of alternative reductions to maintain fidelity in simulations.

Abstract

Stochastic models for biochemical reaction networks are widely used to explore their complex dynamics but face significant challenges, including difficulties in determining rate constants and high computational costs. To address these issues, model reduction approaches based on deterministic quasi-steady-state approximations (QSSA) have been employed, resulting in propensity functions in the form of deterministic non-elementary reaction functions, such as the Michaelis-Menten equation. In particular, the total QSSA (tQSSA), known for its accuracy in deterministic frameworks, has been perceived as universally valid for stochastic model reduction. However, recent studies have challenged this perception. In this review, we demonstrate that applying tQSSA in stochastic model reduction can distort dynamics, even in cases where the deterministic tQSSA is rigorously valid. This highlights the need for caution when using deterministic QSSA in stochastic model reduction to avoid erroneous conclusions from model simulations.

Figures (2)

• Figure 1: Stochastic tQSSA can distort dynamics even when the deterministic tQSSA is valid under spatial homogeneity. (a) When the binding ($k_f / \Omega = 1~\mathrm{s^{-1}}$) and unbinding ($k_b = 100~\mathrm{s^{-1}}$) rates are much faster than the other reactions ($\alpha_m = 0.1~\mathrm{s^{-1}}$ and $d_m = 0.001~\mathrm{s^{-1}}$), $M$ simulated with the deterministic full model (Eq. \ref{['eq:odefull']}, red solid line) and the reduced model (Eq. \ref{['eq:odereduced']}, blue dashed line) precisely match. (b) Under the same conditions, $n_M$ simulated with the stochastic full model (Table \ref{['table:1']}, red solid line) and the reduced model (Table \ref{['table:2']}, blue dashed line) also match closely, as various prior studies expected. Here, the lines with shaded regions represent the mean $\pm$ standard deviation, and the histograms depict the stationary distribution of $10^4$ trajectories. (c-d) However, when the amounts of the rapidly binding species are similar ($n_{D_T} = n_{P_T} = 10$) and the binding becomes tight ($k_f / \Omega = 100~\mathrm{s^{-1}}, k_b = 1~\mathrm{s^{-1}}$), the deterministic full and reduced models still precisely match (c), but the stochastic reduced model fails to replicate the dynamics of the full model (d). Here, $\Omega = 1$ (arbitrary unit), and the initial condition is $[D, P, D\text{:}P, M] = [10, 10, 0, 0]$.
• Figure 2: Stochastic tQSSA can distort dynamics even when deterministic tQSSA is valid under spatial heterogeneity. (a) When the binding ($k_f / \Omega_i = 1~\mathrm{s^{-1}}$) and unbinding ($k_b = 100~\mathrm{s^{-1}}$) rates are much faster than the other reactions ($\alpha_m = 0.5~\mathrm{s^{-1}}, d_m = 0.0005~\mathrm{s^{-1}}$) and diffusion ($\delta_M = \delta_P = 0.002~\mu\mathrm{m^2/s}$), $M$ at $t=5000$ and the spatial mean $M$ trajectory ($\overline{M}$; inset) simulated with the deterministic full model (Eq. \ref{['eq:pdefull']}, red solid line) and the reduced model (Eq. \ref{['eq:pdereduced']}, blue dashed line) are in precise alignment. (b) Under the same conditions, $n_{M,i}$ at $t=5000$ and the spatial total $n_{M,i}$ trajectory ($n_M$; inset) simulated with the stochastic full model (Table \ref{['table:3']}, red solid line) and the reduced model (Table \ref{['table:4']}, blue dashed line) also show close agreement. Here, $n_{M,i}$ were plotted at the center of the corresponding compartment on the $x$-axis and then interpolated. The lines with shaded regions represent the mean $\pm$ standard deviation, and the histograms illustrate the stationary distribution from $10^3$ trajectories. (c-d) However, when binding becomes tight ($k_f / \Omega_i = 500~\mathrm{s^{-1}}, k_b = 10~\mathrm{s^{-1}}$), the deterministic full and reduced models still match precisely (c), but the stochastic reduced model fails to replicate the dynamics of the full model (d). Notably, while the spatial total amounts of the reversibly binding species are not comparable ($n_{P_T} = 31$ and $n_{D_T} = 2$), their local amounts become comparable ($n_{P_T,16} \approx 1$ and $n_{D_T,16} = 2$), causing local violations of the stochastic tQSSA. Here, $L = 10~\mu\mathrm{m}$, $n = 31$, and $\Omega_i = 1$ (arbitrary unit). The initial conditions are $D(x,0) = 2I_{\{5-h/2 < x < 5+h/2\}}(x)$, $P(x,0) = I_{\{0 < x < 10\}}(x)$, and $D{:}P(x,0) = M(x,0) = 0$.