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Accuracy versus Predominance: Reassessing the validity of the quasi-steady-state approximation

Kashvi Srivastava, Justin Eilertsen, Victoria Booth, Santiago Schnell

Abstract

The application of the standard quasi-steady-state approximation to the Michaelis--Menten reaction mechanism is a textbook example of biochemical model reduction, derived using singular perturbation theory. However, determining the specific biochemical conditions that dictate the validity of the standard quasi-steady-state approximation remains a challenging endeavor. Emerging research suggests that the accuracy of the standard quasi-steady-state approximation improves as the ratio of the initial enzyme concentration, $e_0$, to the Michaelis constant, $K_M$, decreases. In this work, we examine this ratio and its implications for the accuracy and validity of the standard quasi-steady-state approximation as compared to other quasi-steady-state reductions in its proximity. Using standard tools from the analysis of ordinary differential equations, we show that while $e_0/K_M$ provides an indication of the standard quasi-steady-state approximation's asymptotic accuracy, the standard quasi-steady-state approximation's predominance relies on a small ratio of $e_0$ to the Van Slyke-Cullen constant, $K$. Here, we define the predominance of a quasi-steady-state reduction when it offers the highest approximation accuracy among other well-known reductions with overlapping validity conditions. We conclude that the magnitude of $e_0/K$ offers the most accurate measure of the validity of the standard quasi-steady-state approximation.

Accuracy versus Predominance: Reassessing the validity of the quasi-steady-state approximation

Abstract

The application of the standard quasi-steady-state approximation to the Michaelis--Menten reaction mechanism is a textbook example of biochemical model reduction, derived using singular perturbation theory. However, determining the specific biochemical conditions that dictate the validity of the standard quasi-steady-state approximation remains a challenging endeavor. Emerging research suggests that the accuracy of the standard quasi-steady-state approximation improves as the ratio of the initial enzyme concentration, , to the Michaelis constant, , decreases. In this work, we examine this ratio and its implications for the accuracy and validity of the standard quasi-steady-state approximation as compared to other quasi-steady-state reductions in its proximity. Using standard tools from the analysis of ordinary differential equations, we show that while provides an indication of the standard quasi-steady-state approximation's asymptotic accuracy, the standard quasi-steady-state approximation's predominance relies on a small ratio of to the Van Slyke-Cullen constant, . Here, we define the predominance of a quasi-steady-state reduction when it offers the highest approximation accuracy among other well-known reductions with overlapping validity conditions. We conclude that the magnitude of offers the most accurate measure of the validity of the standard quasi-steady-state approximation.
Paper Structure (22 sections, 6 theorems, 108 equations, 6 figures)

This paper contains 22 sections, 6 theorems, 108 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Fenichel Theory, TFPV Theory, and Literature Review
  3. A brief introduction of Fenichel theory
  4. Persistence and Flow
  5. Critical Manifolds and Stability
  6. Approximating the Slow Flow
  7. Convergence and Initial Conditions
  8. From Theory to Practice: Tikhonov-Fenichel Parameter Values
  9. The Open Problem: Identifying the Predominant Reduction
  10. Anti-funnels and the Slow Invariant Manifold
  11. Trapping the Slow Manifold: The Standard QSSA
  12. Accuracy of the Standard QSSA
  13. Validity of the sQSSA for High Enzyme Concentrations
  14. Trapping the Slow Manifold: The Slow Product QSSA
  15. Revisiting the Reich-Sel'kov Condition
  16. ...and 7 more sections

Key Result

Theorem 1

Anti-funnel Theoremhubbard1997differential. Consider the first-order differential equation $y' = f(x,y)$ over the interval $I = [a,b)$ where $a<b \leq \infty$. If $\Gamma$ is an anti-funnel with a strong lower fence, $\alpha$, and a strong upper fence, $\beta$, then there exists a solution $y(x)$ to Furthermore, if $\Gamma$ is narrowing and $\frac{\partial f}{\partial y} (x,y) \geq 0$ in $\Gamma$,

Figures (6)

  • Figure 1: The sQSSA \ref{['sQSSA2']} and the slow product QSSA \ref{['slowp']} are indistinguishable for large $k_{-1}$.Left: $s_0 = 10$, $c_0 = 0$, $e_0 = 10$, $k_1 = 1$, $k_{-1} = 100$, $k_2 = 1.$Right: $s_0 = 10$, $c_0 = 0$, $e_0 = 10$, $k_1 = 1$, $k_{-1} = 500$, $k_2 = 1.$ Substrate depletion over time is shown in both panels. $s_c(t)$ is the numerical solution to \ref{['sQSSA2']} and $s_{SP}(t)$ is the numerical solution to \ref{['slowp']}.
  • Figure 2: The region, $\Gamma$, between the horizontal nullcline and the vertical nullcline curves is a trapping region on the phase plane. The numerical solutions to the mass action equations \ref{['MM']} for several initial conditions are denoted by black dashed lines. $s_0 \in \{1,2,3,4,5,6\}$, $c_0 = 0$, $e_0 = 6$, $k_1 = 1$, $k_{-1} = 1$, $k_2 = 1.$ The numerical solution of trajectories start on the $s$-axis and eventually enter $\Gamma$.
  • Figure 3: The region between the horizontal nullcline curve $\gamma_c(s)$ and the curve $\gamma(s)$ is a trapping region on the phase plane. The numerical solutions to the mass action equations \ref{['MM']} for several initial conditions are denoted by black dashed lines. $s_0 \in \{1,2,3,4,5,6\}$, $c_0 = 0$, $e_0 = 6$, $k_1 = 1$, $k_{-1} = 1$, $k_2 = 1.$
  • Figure 4: The slow product QSS curve $\gamma_{SP}$ lies above the horizontal nullcline $\gamma_c$ for $s < s^*$\ref{['eq:sp_intersection']} in the phase plane.Top Left ($k_2 > e_0k_1$): $k_1 = 0.1$, $k_2 = 1$, $k_{-1} = 1$, $e_0 = 6$, $s^* = -4$. Top Right ($k_2 = e_0k_1$): $k_1 = 0.1$, $k_2 = 0.6$, $k_{-1} = 1$, $e_0 = 6$, $s^* = 0$. Bottom Left ($k_2 < e_0k_1$): $k_1 = 1$, $k_2 = 1$, $k_{-1} = 1$, $e_0 = 6$, $s^* = 5$. Bottom Right ($k_2 < e_0k_1$): $k_1 = 1$, $k_2 = 0.4$, $k_{-1} = 1$, $e_0 = 6$, $s^* = 14$.
  • Figure 5: The region between the slow product QSS curve $\gamma_{SP}(s)$ and the curve $\gamma(s)$ is a trapping region for solutions in the phase plane when $e_0 < 8K_S$. The intersection point of $\gamma_{SP}$ and $\gamma_c$, $s^*$, is denoted by the blue star. The numerical solutions to the mass action equations \ref{['MM']} for several initial conditions are denoted by black dashed lines. $s_0 \in \{2,6,12\}$, $c_0 = 0$, $e_0 = 6$, $k_1 = 1$, $k_{-1} = 1$, $k_2 = 1.$
  • ...and 1 more figures

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • ...and 2 more