## 3 Computation of Inertial Manifolds for DDEs
### Elimination of the Fastest Relaxation Mode  
We now return to the original four-variable model including RNA, enzyme, substrate and enzyme-substrate complex, Equation (21). Let us suppose that we want to develop a state-space model, i.e., carry out a minimal reduction in which we eliminate the delay and obtain ODEs for the dynamics in a three-dimensional state space by computing a three-dimensional inertial manifold in the four-dimensional state space. As discussed previously, for \"normal\" parameters, i.e., parameter sets with \(\varepsilon_{1}\ll\varepsilon_{2}\ll 1\), the system has three distinct time scales: fast enzyme kinetics, moderately fast mRNA dynamics, and slow dynamics of the total amount of enzyme. In this parameter range, the fastest variable, which is the one whose geometry will be most appropriate to this reduction, will be either \(s\) or \(c\), depending on the values of \(\beta\) and \(\gamma\). (Either \(\beta<1\) or \(\gamma\gg\alpha\) would tend to make \(c\) the fastest variable.) However, the delay appears in the equation for \(r\), so this is the variable in which we need to approximate the dynamics on the slow manifold in order to obtain the desired reduction. Let us say, for the sake of argument, that \(c\) is the fastest variable. We can compute an (approximate) inertial manifold in the form \(c=\mathcal{C}(e,r,s)\) as we did in the previous section. We could then, in principle, have a reduced model that consists of the following set of differential-algebraic equations:  
\[\varepsilon_{1} \dot{s}=\kappa-s(e-c)+\alpha c, \tag{29a}\] \[\beta\varepsilon_{1} \dot{c}=s(e-c)-(\gamma+\beta\varepsilon_{1})c,\] (29b) \[\dot{e}=r-e,\] (29c) \[c=\mathcal{C}(e,r,s). \tag{29d}\]  
As we did for the three-variable model, the last of these equations could in principle be numerically inverted to give us a value of \(r\) for any given \(c\), \(e\) and \(s\), thus closing the system. The problem is that \(c\) depends only indirectly on \(r\) through \(e\). This has the result of making \(\partial c/\partial r\) on the manifold small and difficult to compute accurately. The inversion process is therefore numerically ill-conditioned, so that a useful reduction is not obtained. Accordingly, in the usual parameter regime, a three-variable ODE model is not available. The assumption that \(\varepsilon_{2}\) is larger than \(\varepsilon_{1}\) will be true for the vast majority of enzyme synthesis/catalysis systems, but is not necessarily so. Values of \(\varepsilon_{2}\) in a mouse cell line can be calculated directly from the data of [PERSON] and coworkers [67], yielding values in the range of \(1.5\times 10^{-3}\) to 46, with a median of 0.20. The same data set gives a maximum value of \(k_{dE}\) of \(3.9\times 10^{-4}\,\mathrm{s}^{-1}\). For prokaryotes, values larger than this by an order of magnitude are possible [89]. Rate constants for enzyme-substrate association (\(k_{1}\)) can be as low as \(10^{3}\,\mathrm{M}^{-1}s^{-1}\)[90; 91]. Since enzyme concentrations in excess of \(1\,\mathrm{\SIUnitSymbolMicro m}\) are routinely seen in vivo [92], values of \(\varepsilon_{1}>\varepsilon_{2}\) are certainly within the realm of possibility. In the extreme, this parameter regime leads to the three-variable model. Since \(\varepsilon_{2}\) is never vanishingly small, it is worth reexamining this case using the manifold construction techniques we have been studying to compute an ODE state-space model valid when \(\varepsilon_{2}\) is small, but not vanishingly so. If \(\varepsilon_{2}\) is smaller than both \(\varepsilon_{1}\) and \(\beta\varepsilon_{1}\), then \(r\) is the fastest variable in the system, and we should be able to directly eliminate it by computing an inertial manifold of the form \(r=\mathcal{R}(e,s,c)\).