## 2 Iterative Computation of Attracting Invariant Manifolds of ODEs
The concentration of S at the steady state satisfies an equation that could be dominated by different terms, depending on the parameter values. Since \(K_{R}\) controls the concentration at which RNA synthesis is half-maximal, i.e., the middle of the switching range from a low synthesis rate to a high synthesis rate, it defines a key concentration scale for the dynamics of this system. Accordingly, the scaling  
\[k\]3  
is at least a sensible choice. If we substitute these new variables into the rate equations, we get  
\[k\]4 \[k\]5 (8b) \[k\]6 (8c) \[k\]7  
Proteins typically have much longer lifetimes than their mRNAs [66; 67]. Moreover, both enzyme catalysis and transport are usually fast processes, so the time scales that could be extracted from Equation (8c) would be fast time scales. It follows that the slowest time scale will, in most cases, be \(k_{dE}^{-1}\). Accordingly, define the dimensionless time  
\[k\]8  
Also define the dimensionless parameters  
\[k\]9 \[S\]0 (10b) \[S\]1 (10c) \[S\]2 (10d) \[S\]3 (10e) \[S\]4  
where \(K_{E}=k_{-1}/k_{1}\) and \(K_{S}=(k_{-1}+k_{-2})/k_{1}\) are the classical Michaelis constants, the first in the equilibrium approximation, and the second in the steady-state approximation. The final set of dimensionless equations is  
\[S\]5 \[S\]6 (11b) \[S\]7 (11c) \[S\]8where  
\[S\]9  
and dots represent differentiation with respect to \(\theta\). Typical time scales for these processes would be such that \(\varepsilon_{1}\ll\varepsilon_{2}\ll 1\). The magnitudes of \(\alpha\), \(\beta\) and \(\kappa\) on the other hand would be highly problem-specific. In particular, if \(\beta\ll 1\), the fastest time scale would be associated with \(\beta\varepsilon_{1}\), i.e., \(c\) would be the distinguished \"fast\" variable. This is arguably the most common case in practice. An aside on \"fast\" variables: Singular perturbation theory has proven to be a rich framework for discussing the invariant and inertial manifolds of differential equations [2; 12; 68; 69; 70; 71; 72; 73]. The standard setting is a set of differential equations in which some derivatives are multiplied by small parameters, as in Equation (11) above. Informally, we say that the corresponding variables are \"fast\" [74]. This is a bit sloppy, for a number of reasons, not the least of which are, first, that the differential equation for a chemical species can contain both fast and slow components, and second, that the fast and slow components of a reactive system can change over time as the concentrations change. However, it is a useful shorthand for \"the concentration whose rate of change is multiplied by a small constant\", so we retain this terminology. Note that, from their definitions, we must have \(\gamma>\alpha\). Moreover, \(k_{\text{in}}\) must be smaller than the maximum clearance capacity of the system, \(k_{-2}E_{\text{max}}\), or else the concentration of substrate grows without bound. Transforming the latter relationship to the dimensionless parameters, we have  
\[\mathrm{R}\xrightarrow{k_{t}}\mathrm{R}+\mathrm{E}. \tag{1b}\]0  
[PERSON]'s theorem [6; 75; 76], one of the foundational theorems of singular perturbation theory, provides a key tool for determining the slow-manifold structure of this problem. If \(\varepsilon_{1}\ll\varepsilon_{2}\), \(\varepsilon_{1}\ll 1\) and \(\beta=O(1)\), system (11) is of the form  
\[\mathrm{R}\xrightarrow{k_{t}}\mathrm{R}+\mathrm{E}. \tag{1b}\]1 \[\mathrm{R}\xrightarrow{k_{t}}\mathrm{R}+\mathrm{E}. \tag{1b}\]2  
with \(x=(r,e)\) and \(z=(s,c)\). [PERSON]'s theorem then states that, for sufficiently small \(\varepsilon_{1}\) and after decay of fast transients, the solution of system (11) tends to the solution of the differential-algebraic system consisting of Equation (13a) with the algebraic equation \(g(x,z,0)=0\). In other words, the solutions approach a slow manifold which could be expressed in the form \(z=\mathcal{Z}(x)\) [for this problem: \((s,c)=(\mathcal{S}(r,e),\mathcal{C}(r,e))\)] such that \(\mathcal{Z}(x)\) solves \(g(x,\mathcal{Z}(x),0)=0\).