On the reduction of stochastic chemical reaction networks

TL;DR

It is demonstrated that eigenvalue disparity does not guarantee the accuracy of the reduced LNA, known as the slow scale LNA (ssLNA), however, the inaccuracy of the ssLNA can often be eliminated with a proper coordinate transformation.

Abstract

The linear noise approximation (LNA) describes the random fluctuations from the mean-field concentrations of a chemical reaction network due to intrinsic noise. It is also used as a test probe to determine the accuracy of reduced formulations of the chemical master equation and to understand the relationship between timescale disparity and model reduction in stochastic environments. Although several reduced LNAs have been proposed, they have not been placed into a general theory concerning the accuracy of reduced LNAs derived from center manifold and singular perturbation theory. This has made it difficult to understand why certain reductions of the master or Langevin equations fail or succeed. In this work, we develop a deeper understanding of slow manifold projection in the linear noise regime by answering a straightforward but open question: In the presence of eigenvalue disparity, does the appropriate oblique projection of the LNA onto the slow eigenspace accurately approximate the first and second moments of complete LNA, and if not, why? Although most studies concentrate on the role of eigenvalue disparity arising from the drift matrix, we go further and examine the interplay between disparate ``drift" eigenvalues and the eigenvalues of the diffusion matrix, the latter of which may or may not be disparate. Furthermore, we place the previously established reductions of the LNA into a more general framework and formulate the necessary and sufficient conditions for the projected LNA to accurately approximate the first and second moments of the complete LNA.

Figures (4)

• Figure 1: The composite expansion \ref{['composite']} approximates the exact solution \ref{['exact']} over fast and slow timescales. In this example, $A_0=-2\;\;\;1\;\;\;2-1$, $A_1=0\;\;\;00-\varepsilon$, and $Y(0)=11$. A simple calculation reveals $\pi_0=1/31/32/32/3$, which yields $y_1(t,\tau) \approx (1/3)\exp(-3t) + (2/3)\exp(-(2/3)\tau)$ and $y_2(t,\tau)\approx -(1/3)\exp(-3t) + (4/3)\exp(-(2/3)\tau)$. The solid curves are the numerical solutions to the components of full system \ref{['exact']}, and the dotted curves are the solutions to the components of the composite expansion \ref{['composite']} with $\varepsilon = 0.1$. It is straightforward to see that the composite expansion approximates the exact solution over the fast and slow phases of the timecourse, and improves as $\varepsilon \to 0$.
• Figure 2: When diffusion occurs on the slow timescale, the approach to the slow eigenspace can be approximated with the expectation, $\mathbb{E}\{Y\}$, which obeys a deterministic ordinary differential equation. In both panels the thick, solid curve is the expected value, $\mathbb{E}\{Y\}$, and the dashed/dotted curves are the mean $\pm$ the standard deviation obtained from the full LNA \ref{['LNA0']}. The thick green line is the approximate mean for $y_1$, obtained from the composite solution \ref{['compY1']}, and the dotted green curves are the composite solution for $y_1$, $\pm$ the standard deviation obtained from projected LNA \ref{['redcov']}. The red, blue, orange and magenta curves are numerically-integrated realizations of the LNA \ref{['LNA0']}. Parameter values used in the simulations (obtained via numerical integration of \ref{['LNA0']}) are (in arbitrary units): $k_0=k_1=k_2=k_3=1.0$, $\varepsilon = 0.001$, and $y_1(0)=y_2(0)=5.0$. Panel a: Note that the LNA realizations do not deviate significantly from $\mathbb{E}\{Y\}$ during transient decay; only after the decay of transients do we start to see significant deviations from the expectation due to influence of diffusion. Moreover, the variance obtained from the projected LNA is highly accurate. Panel b: A close-up of the panel a.
• Figure 3: When diffusion occurs on the fast timescale, realizations of the LNA can depart from the mean immediately. The thick, solid back curve is the expected value, $\mathbb{E}\{Y\}$, and the thin, dashed/dotted black curves are the mean $\pm$ the standard deviation. The red, blue, orange, and magenta curves are numerically integrated realizations of the LNA \ref{['LNA0']}. The parameter values used in the simulations (obtained through the numerical integration of \ref{['LNA2']}) are (in arbitrary units): $k_0=k_1=k_2=k_3=1.0$, $\varepsilon = 0.001$, and $y_1(0)=y_2(0)=5.0$. Note that the LNA realizations deviate significantly from $\mathbb{E}\{Y\}$ during transient decay due to diffusion that occurs on the fast timescale. This behavior is markedly different than the behavior presented in figure\ref{['FIG2']}.
• Figure 4: When diffusion occurs the fast and slow timescales, the projected LNA \ref{['SDE0P']} will underestimate the variance of the LNA by a factor of $(1+\delta_0)$. Panel a: The thick black curve is the limiting variance of substrate, $s$, as the eigenvalue ratio $\lambda_+/\lambda_-\to 0$ computed from the LNA of the open MM reaction mechanism discussed in Example 3. The parameters used in each simulation are (in arbitrary units): $s_0=10.0,k_1=10.0,k_{-1}=5.0,\alpha=0.5$, and $e_0=5.0$. The catalytic rate constant, $k_2$, is varied from $0.0001$ to $10.0$ As the eigenvalue ratio vanishes, the limiting steady-state variance of substrate converges to $Z_0^{\infty}$ (dashed/dotted line). However, the projected LNA converges to $Z_p^{\infty}\neq Z_0^{\infty}$. The difference (bold arrows) is the variance of substrate resulting from fast timescale diffusion as $\varepsilon \to 0$, which is $\frac{\alpha e_0k_{-1}(1-\alpha)}{k_1e_0(1-\alpha)^2+k_{-1}}$. Panel b: The solid black curve is the steady-state covariance $Z^{\infty}$ of the total substrate, which converges to $Z_0^{\infty} = Z_p^{\infty}$ as the eigenvalue ratio vanishes. The coordinate transformation results in a new concentration (the total substrate) $s_T=s+c$, which diffuses on the slow timescale only. Consequently, the total substrate variance obtained from the projected LNA will converge to the total substrate variance obtained from the LNA as the eigenvalue ratio vanishes and $(\varepsilon,\delta)\to (0,0)$.

Theorems & Definitions (13)

• Remark 1
• Proposition 1
• proof
• Proposition 2
• proof
• Example 1
• Example 2
• Example 3
• Example 4
• Proposition 3