Rigorous estimates for the quasi-steady state approximation of the Michaelis-Menten reaction mechanism at low enzyme concentrations

Abstract

There is a vast amount of literature concerning the appropriateness of various perturbation parameters for the standard quasi-steady state approximation in the Michaelis-Menten reaction mechanism, and also concerning the relevance of these parameters for the accuracy of the approximation by the familiar Michaelis-Menten equation. Typically, the arguments in the literature are based on (heuristic) timescale estimates, from which one cannot obtain reliable quantitative estimates for the error of the quasi-steady state approximation. We take a different approach. By combining phase plane analysis with differential inequalities, we derive sharp explicit upper and lower estimates for the duration of the initial transient and substrate depletion during this transitory phase. In addition, we obtain rigorous bounds on the accuracy of the standard quasi-steady state approximation in the slow dynamics regime. Notably, under the assumption that the quasi-steady state approximation is valid over the entire time course of the reaction, our error estimate is of order one in the Segel-Slemrod parameter.

Figures (7)

• Figure 1: The $(s,c)$ phase plane geometry of the Michaelis--Menten reaction mechanism. The thick red curve, the graph of $g(s)$, is the QSS variety (i.e., the $c$-nullcline) and the thick blue curve, the graph of $g_1(s)$, is the $s$-nullcline. The thick black curve is the invariant slow manifold, $\mathcal{M}$. In the shaded violet region between the graphs is the slow invariant manifold, $\mathcal{M}$, that connects the stable equilibrium at the origin with a saddle equilibrium at infinity. The vector field in the red shaded region below Graph($g(s)$) satisfies $\dot{c}>0$ and $\dot{s} <0$. On Graph($g(s)$), $\dot{c}=0$ and $\dot{s}<0.$ In the magenta region that lies above Graph($g(s)$) and below Graph($g_1(s)$), $\dot{s}<0$ and $\dot{c}<0.$ On $g_1(s)$, $\dot{s}=0$ and $\dot{c}<0$. In the blue shaded region above Graph($g_1(s)$) and below $c=e_0$, $0<\dot{s}$ and $\dot{c}<0.$ The dotted black curve in the top panel is a single trajectory obtained via numerical integration of the mass action equations \ref{['eqmmirrev']} with parameters (in arbitrary units): $s_0=100,e_0=5.0, k_1=1.0,k_2=k_{-1}=10.0$. The trajectory approaches and intercepts the graph of $g(s)$ at $t=t_{\rm cross}.$ For $t>t_{\rm cross}$, the trajectory lies above Graph($g(s))$, but below $\mathcal{M}$. Top: The trajectory enters the the magenta region and approaches $\mathcal{M}$ as time evolves forward. Bottom: Closeup of the top panel near the QSS variety. The trajectory still lies below $\mathcal{M}$, but becomes effectively indistinguishable from $\mathcal{M}$ as $t\to\infty$.
• Figure 2: Numerical simulations indicate that $t_u^\dagger(1)$, defined in \ref{['EST']}, is a reasonable estimation of $t_{\rm cross}$ when $e_0\ll K_M$. In all panels, $e_0 \in [0.025, 0.05, 0.075, 0.1, 0.25, 0.5, 0.75, 1.0, 2.5, 5.0, 7.5, 10]$, $k_1=1.0$, $k_2=k_{-1}=100$, thus $K_M=200$, and $\sigma=s_0/K_M$. The parameter $\varepsilon$ corresponds to the Reich-Selkov parameter $\varepsilon_{RS}=e_0/K_M$. The solid black diamonds are the numerically estimated crossing times. The densely dashed line is obtained from (\ref{['EST']}). The dotted line is obtained from (\ref{['tlowdagas']}). Top Left:$s_0 =2$. Top Right:$s_0=20$. Bottom Left:$s_0=200$. Bottom Right:$s_0=2000$. Observe the noticeable difference between (\ref{['ESTasex']}) and (\ref{['tlowdagas']}) when $s_0$ is much larger than $K_M$. This is due to the difference in the constant terms of the expansions. One also sees that the lower estimate $t_\ell^\dagger$ from \ref{['tlowdagas']} is worse than the upper estimate; compare Remark \ref{['badsestrem']}.
• Figure 3: Numerical simulations suggest that (\ref{['optimalguess']}) provides an upper bound on the normalized error between the $s$-component of the mass action equations and the sQSSA for the complete time course when initial conditions lie on the QSS variety, $c=g(s)$. In both panels, the black curve is the numerically-estimated normalized absolute error, $|\xi-s|/s_0$. The dash-dotted and dotted lines correspond to $\varepsilon_L$ and $\varepsilon_{LW}$, respectively, and the red line is $\varepsilon_{\rm opt.}$ On the $x$-axis, $t$ has been mapped to $t_{\infty}=1-1/\log(t+e)$, and initial conditions for the mass action equations and the sQSSA satisfy $(s,c)(0)=(s,c)(t_{\rm cross})$ and $\xi(0)=s(t_{\rm cross})$, respectively ($t_{\rm cross}$ is estimated numerically). Top: The parameters used in the simulation are (in arbitrary units): $s_0=10.0$, $e_0=10.0$, $k_1=2.0$, $k_2=100.0$ and $k_{-1}=100.0$. Bottom: The parameters used in the simulation are (in arbitrary units): $s_0=$$10.0$, $e_0=1.0$, $k_1=2.0$, $k_2=100.0$ and $k_{-1}=100.0$. The estimate $\varepsilon_{\rm opt}$ is not a sharp error estimate due to the choice of initial conditions.
• Figure 4: Numerical simulations confirm that (\ref{['normER']}) provides a sharp bound on the normalized error between the $s$-component of the mass action equations and the sQSSA. In both panels, the black curve is the numerical solution the mass action equations. The dashed/dotted curve is the numerical solution to the sQSSA. The admissible region given by the error bound (\ref{['normER']}) is shaded in blue. The blue line is the (normalized) numerical solution to the right-hand side of (\ref{['normER']}) with numerically-estimated (a priori) $q$ that corresponds to the upper boundary of (\ref{['normER']}). Time has been rescaled by $\tau = t/t_{\rm cross}$, where $t_{\rm cross}$ has been numerically-estimated. Top: The parameters used in the simulation are (in arbitrary units): $s_0=10.0$, $e_0=10.0$, $k_1=2.0$, $k_2=100.0$ and $k_{-1}=100.0$. Bottom: The parameters used in the simulation are (in arbitrary units): $s_0=100.0$, $e_0=1.0$, $k_1=2.0$, $k_2=100.0$ and $k_{-1}=100.0$.
• Figure 5: Numerical simulations confirm that (\ref{['optimalguess']}) is a reasonable estimation of the normalized error when $t=t_{\rm cross}$ for small $\varepsilon_{SSl}$. In all panels, $e_0 \in [0.025, 0.05, 0.075, 0.1, 0.25, 0.5, 0.75, 1.0, 2.5, 5.0, 7.5, 10]$, $k_1=1.0,$$k_2=k_{-1}=100.0$, thus $K_M=200$, and $\sigma=s_0/K_M$. The solid black crosses are the numerically--computed normalized error $|z-s|/s_0$ at $t=t_{\rm cross}$. The orange diamonds correspond to $\varepsilon_{opt}$. Top Left:$s_0 =2.0$. Top Right:$s_0=20.0$. Bottom Left:$s_0=200.0$. Bottom Right:$s_0=2000.0$.

Theorems & Definitions (50)

• Lemma 1
• Lemma 2
• Remark 1
• Lemma 3
• proof
• Remark 2
• Lemma 4
• proof
• Remark 3
• Remark 4