On the Validity of the Stochastic Quasi-Steady-State Approximation in Open Enzyme Catalyzed Reactions: Timescale Separation or Singular Perturbation?

Abstract

The quasi-steady-state approximation is widely used to develop simplified deterministic or stochastic models of enzyme catalyzed reactions. In deterministic models, the quasi-steady-state approximation can be mathematically justified from singular perturbation theory. For several closed enzymatic reactions, the homologous extension of the quasi-steady-state approximation to the stochastic regime, known as the stochastic quasi-steady-state approximation, has been shown to be accurate under the analogous conditions that permit the quasi-steady-state reduction in the deterministic counterpart. However, it was recently demonstrated that the extension of the stochastic quasi-steady-state approximation to an open Michaelis–Menten reaction mechanism is only valid under a condition that is far more restrictive than the qualifier that ensures the validity of its corresponding deterministic quasi-steady-state approximation. In this paper, we suggest a possible explanation for this discrepancy from the lens of geometric singular perturbation theory. In so doing, we illustrate a misconception in the application of the quasi-steady-state approximation: timescale separation does not imply singular perturbation.

Appendix A. Fenichel Theory: Projecting Onto a Slow Manifold

In this brief appendix, we briefly describe how to construct $\Pi ^M$, the projection operator. For more technical details, we invite the reader to consult (Wechselberger 2020; Goeke et al. 2015, 2017). Following (Wechselberger 2020), we begin with the perturbation form

$$\begin{aligned} {\dot{z}}= w(z) + \varepsilon G(z,\varepsilon ),\quad z\in {\mathbb {R}}^n. \end{aligned}$$

and let M be a compact subset of $w(z)=0$ that forms a k-dimensional manifold ($k<n$) comprised of non-isolated stationary points such that:

• For all $z\in M$, the algebraic and geometric multiplicities of the zero eigenvalues of Dw(z) are equal with $T_zM = \ker Dw(z)$ and $\dim \ker Dw(z)= k$,

• If $\lambda _j$ is a nonzero eigenvalue of Dw(z), then $\mathfrak {Re}(\lambda _j) < 0\quad \forall z\in M$,

then, M is locally attracting and there exists a splitting

$$\begin{aligned} {\mathbb {R}}^n:=T_zM \oplus N_z \quad \forall z \in M, \end{aligned}$$

(A.1)

where $T_zM$ is the tangent space of M at z given by $\{ x \in {\mathbb {R}}^n: x \in \ker Dw(z)\}$, and $N_z$ is the complement to $T_zM$ and coincides with range of the Jacobian, ${\mathcal {R}}(Dw(z))$. The objective from the point forward will be to exploit the splitting (A.1) and construct an oblique¹⁰ projection operator:

$$\begin{aligned} \Pi ^M: {\mathbb {R}}^n \mapsto T_zM, \qquad I-\Pi ^M: {\mathbb {R}}^n \mapsto N_z. \end{aligned}$$

To construct $\Pi ^M$, we invoke the factorization

$$\begin{aligned} w(z) = P(z)f(z), \end{aligned}$$

(A.2)

where P(z) is a rectangular matrix function with full rank at every point in M, and the zero level set of f(z) coincides with the critical manifold, M. The derivative of f(z), Df(z), has full rank $\forall z \in M$ and:

• The columns of P(z) form a basis for the range of the Jacobian ${\mathcal {R}}(Dw(z))=N_z$, for all $z\in M$.

• The rows of Df(z) form a basis for the orthogonal complement of $\ker Dw(z)=(T_zM)^{\perp }$ for all $z \in M$.

It follows that

since $T_zM$ and $N_z$ are complementary subspaces, the matrix

$$\begin{aligned} \Pi ^M := I- P(Df P)^{-1}Df \end{aligned}$$

(A.3)

defines the oblique projection onto $T_zM$. To leading order in $\varepsilon$, the reduced flow on the slow manifold is

$$\begin{aligned} {\dot{z}} = \Pi ^M G(z,0), \end{aligned}$$

(A.4)

which is the QSS reduction obtained from Fenichel theory (Fenichel 1979).

Remark 4

The critical manifold is said to be normally hyperbolic if the matrix DfP (the eigenvalues of which are the non-trivial eigenvalues of Dw(z)) is hyperbolic $\forall z\in M$, meaning that $\mathfrak {Re}(\lambda _i) \ne 0.$ In most applications, we are interested in attracting critical manifolds, and thus, we often require $\mathfrak {Re}(\lambda _i) < 0.$ However, normally hyperbolic critical manifolds are repelling if $\mathfrak {Re}(\lambda _i) > 0$, or of saddle type if the real parts of the eigenvalues of DfP are both positive and negative.

As an example, we will explicitly compute the sQSSA for the open MM reaction mechanism. In standard form, we have

$$\begin{aligned} \begin{pmatrix} {\dot{s}} \\ {\dot{c}}\end{pmatrix} = \begin{pmatrix}k_{-1} +k_1s \\ -k_{-1}-k_2-k_1s \end{pmatrix}c + \varepsilon \begin{pmatrix} k_0^*-k_1e_T^*s\\ k_1e_T^*s \end{pmatrix} \end{aligned}$$

(A.5)

so that $f(s,c)=c$ and

$$\begin{aligned} P \equiv \begin{pmatrix}k_{-1} + k_1s \\ -k_{-1}-k_2-k_1s\end{pmatrix}. \end{aligned}$$

(A.6)

The derivative of f(s, c) is $[0 \;\;1]$, and thus $(DfP)^{-1}$ is

$$\begin{aligned} -\frac{1}{k_1(K_M+s)}, \end{aligned}$$

(A.7)

which is of course a scalar. The product, PDf, is given by

$$\begin{aligned} \begin{pmatrix} 0 &{} k_1(K_S+s)\\ 0 &{} -k_1(K_M+s)\end{pmatrix}, \quad K_S:=k_{-1}/k_1, \quad K_M:=(k_{-1}+k_2)/k_1. \end{aligned}$$

(A.8)

Computing $\Pi ^M$ from (A.3) yields

$$\begin{aligned} \Pi ^M:=\begin{pmatrix} 1 &{} \frac{(K_S+s)}{(K_M+s)}\\ 0 &{} 0\end{pmatrix}. \end{aligned}$$

(A.9)

To recover the sQSSA, we simply project the perturbation onto the tangent space of the critical manifold:

$$\begin{aligned} \begin{pmatrix} {\dot{s}} \\ {\dot{c}}\end{pmatrix} = \Bigg [\begin{pmatrix} 1 &{} \frac{(K_S+s)}{(K_M+s)}\\ 0 &{} 0\end{pmatrix}\begin{pmatrix} k_0^*-k_1e_T^*s\\ k_1e_T^*s \end{pmatrix}\Bigg ]\Bigg |_{c=0} =\begin{pmatrix} k_0^*-\frac{k_1k_2 e_T^*s}{k_{-1}+k_2+k_1s}\\ 0\end{pmatrix}.\nonumber \\ \end{aligned}$$

(A.10)

The QEA (26) that corresponds to small $k_0$ and small $k_2$ is computed in a similar manner. For more details see (Eilertsen et al. 2021).