## 3 Computation of Inertial Manifolds for DDEs
### The \"Fast RNA\" Case  
Before analyzing system (21), we consider a special limit from which we will learn some valuable lessons: Suppose that the RNA dynamics is fast, i.e., that \(\varepsilon_{2}\) is vanishingly small. Then, using a standard result of singular perturbation theory for delay-differential equations [87; 88], we can write the simplified system  
\[\varepsilon_{1}\dot{s} =\kappa-s(e-c)+\alpha c, \tag{22a}\] \[\beta\varepsilon_{1}\dot{c} =s(e-c)-(\gamma+\beta\varepsilon_{1})c,\] (22b) \[\dot{e} =\rho\left(s(\theta-\delta)\right)-e. \tag{22c}\]  
This system is almost identical to one studied previously [39], except that the older model assumed that only the free enzyme decayed. This allowed the enzyme-substrate complex (C) to act as a protected reservoir of the enzyme, so that there was no upper limit on the allowed values of \(k_{\text{in}}\). This is not the case for the family of models studied in this contribution, which are subject to the constraint (12). Like the previously studied model [39], the system (22) displays an Andronov-Hopf bifurcation. Thus, it should have a two-dimensional (slow/center/unstable) inertial manifold in the vicinity of this bifurcation. Let us suppose, for the sake of argument, that \(\beta\varepsilon_{1}\ll\min(\varepsilon_{1},1)\). Then we would expect to have an inertial manifold parameterizable in the form \(c=\mathcal{C}(e,s)\). Our objective is to obtain an ODE model for the motion on the inertial manifold. Thus, we must eliminate Equation (22c) which contains the delayed term. Suppose that we have obtained the equation of the manifold, \(c=\mathcal{C}(e,s)\). We can then obtain an ODE system representing the motion on this manifold by using Equation (22a,b), and solving the equation \(c=\mathcal{C}(e,s)\) for \(e\) to close the system. Our first task then is to obtain an approximation for the equation of the inertial manifold. We start, as usual, from the invariance equation  
\[\dot{c}=\frac{\partial\mathcal{C}}{\partial e}\dot{e}+\frac{\partial\mathcal{C }}{\partial s}\dot{s}. \tag{23}\]  
In this system, \(\dot{e}\) depends on delayed values of \(s\). This is a problem because Equation (23) is, in principle, a PDE for a manifold in the state space. Consider however expanding the nonlinearity in small powers of the delay:  
\[\rho\left(s(\theta-\delta)\right)\approx\rho(s)-\delta\left[\frac{d\rho}{ds}s \right]_{\delta=0}+\frac{\delta^{2}}{2}\left[\frac{d^{2}\rho}{ds^{2}}s^{2}+ \frac{d\rho}{ds}s\right]_{\delta=0}-\frac{\delta^{3}}{6}\left[\frac{d^{3}\rho }{ds^{3}}s^{3}+3\frac{d^{2}\rho}{ds^{2}}ss+\frac{d\rho}{ds}\ddot{s}\right]_{ \delta=0}+\ldots \tag{24}\]  
As previously mentioned, we cannot substitute this expansion beyond the \(O(\delta)\) term into Equation (22c) and numerically integrate the equations, as shown by [PERSON] [36] (pp. 284-285). For this reason, previous studies had truncated the small-delay expansion after the \(O(\delta)\) term [49; 32]. However, we _can_ use this expansion to approximate the vector field implied by the delay equations [38], and in particular to compute inertial manifolds of the DDEs [33; 40]. The quantities \(\ddot{s}\) and \(\dddot{s}\) (both evaluated at \(\delta=0\)) appearing in Equation (24) can be obtained by repeated differentiation of Equation (22a). For this particular model, we get  
\[\ddot{s}|_{\delta=0}=\varepsilon_{1}^{-1}\left\{(c-e)\dot{s}-s\left[\rho(s)-e \right]+(s+\alpha)\dot{c}\right\} \tag{25}\]and  
\[\left.\vec{s}\right|_{\delta=0}=(\beta\varepsilon_{1}^{2})^{-1}\Big{\{}2 \beta\varepsilon_{1}\hat{s}\left[\dot{c}-\rho(s)+e\right]+\left[\beta(e-c)+s+ \alpha\right]\left[\hat{s}(e-c)+s(\rho(s)-e)\right]\\ -\beta\varepsilon_{1}s\left(\frac{d\rho}{ds}\hat{s}-\rho(s)+e \right)+\dot{c}\left[\beta(c-e)(s+\alpha)-(s+\gamma+\beta\varepsilon_{1})( \alpha+s)\right]\Big{\}}. \tag{26}\]  
The approximate manifolds \(\mathcal{C}_{k}^{(m)}\) will now have two indices: \(k\) to label the iterate, and \(m\) to indicate the last power of \(\delta\) included in the expansion (24).