## 2 Iterative Computation of Attracting Invariant Manifolds of ODEs  
Before we proceed to the delay-differential equation problem, it is useful to review the iterative method of solution for attracting invariant manifolds of ODEs. A simple model for an inducible enzyme will be used as an example. Suppose that the cell produces an enzyme, E, the purpose of which is to metabolize a substrate S, but E is only made when S is present. Examples of such enzymes include \(\beta\)-galactosidase, which is required to metabolize lactose and is only produced when lactose is available [54; 55], and Hmp, an enzyme involved in NO detoxification [56], which is only produced when NO starts to accumulate in a cell [57]. A simple model might be the following: The messenger RNA (R) that codes for E is synthesized at a rate that depends on the concentration of the substrate. The RNA has some natural decay rate, which may include the effect of dilution by growth. \[\xrightarrow{v_{R}(S)}\mathrm{R}\xrightarrow{k_{dR}}.\] (1a) In this and subsequent reactions, standard mass-action kinetics will be assumed, except for the empirical rate of transcription, \[v_{R}(S)\]. A \[k\] with an identifying subscript over an arrow denotes a mass-action rate constant. Concentrations of chemical species will be denoted by the corresponding italic letter, for example, \[S\] is the concentration of the substrate S.  
The RNA is translated to the enzyme E that will be responsible for metabolizing S.  
\[\mathrm{R}\xrightarrow{k_{t}}\mathrm{R}+\mathrm{E}. \tag{1b}\]  
We consider a constant source of substrate, either as a byproduct of metabolism, or entering the cell from the surrounding medium. \[\xrightarrow{k_{\mathrm{in}}}\mathrm{S}. \tag{1c}\]  
The enzyme converts S to a metabolite P by the usual Michaelis-Menten mechanism:  
\[\mathrm{E}+\mathrm{S}\xrightleftharpoons[k_{1}]{k_{-1}}\mathrm{C}\xrightarrow{ k_{-2}}\mathrm{E}+\mathrm{P}. \tag{1d}\]  
The enzyme is also subject to degradation and dilution by growth. We assume here that both the enzyme E and enzyme-substrate complex C have identical decay kinetics:  
\[\{\mathrm{E},\mathrm{C}\}\xrightarrow{k_{dE}}. \tag{1e}\]  
The corresponding mass-action rate equations are \[\frac{dR}{dt} =v_{R}(S)-k_{dR}R, \tag{2a}\] \[\frac{dE}{dt} =k_{t}R-k_{dE}E-k_{1}ES+(k_{-1}+k_{-2})C,\] (2b) \[v_{R}(S)\]0 (2c) \[v_{R}(S)\]1  
The model is completed by specifying the dependence of the RNA synthesis rate on the substrate concentration. We assume Hill-function kinetics, which can arise in a variety of biochemical scenarios [58; 59; 60], and which commonly appears in gene expression models:  
\[v_{R}(S)\]2  
Similar models have a long history in theoretical biology [61; 62]. Although not strictly necessary, it is convenient to consider the total amount of enzyme  
\[v_{R}(S)\]3  
instead of the free enzyme concentration, for reasons that will shortly become clear. In terms of this variable, the governing equations become  
\[v_{R}(S)\]4 \[v_{R}(S)\]5 (5b) \[v_{R}(S)\]6 (5c) \[v_{R}(S)\]7  
We now transform these equations to dimensionless form following typical principles of scaling [6; 63; 64; 65] to obtain equations on the system's slow time scale. From the point of view of computing invariant manifolds, this is not strictly necessary, but it will facilitate some later arguments about appropriate representations of manifolds. The maximum values of \(R\) and \(E_{T}\), assuming that the system is not \"spiked\" with one of these species from an external source, are as follows, obtained from the rate Equation (5), and considering that the maximum value of \(v_{R}(S)\) is \(V_{R}\):  
\[v_{R}(S)\]8 \[v_{R}(S)\]9  
Thus, we define \(O(1)\) dimensionless variables  
\[k\]0 \[k\]1 (7b) \[k\]2  
Note that Equation (4) implies that \(c\leq e\). There is not a single natural scale for \(S\). For some parameters at low initial concentrations of E, \(S\) might rise to very large values before induction of enzyme synthesis returned its value to the vicinity of the steady state.