## 3 Computation of Inertial Manifolds for DDEs
### Computing a Two-Dimensional Inertial Manifold  
Given the Andronov-Hopf bifurcation in this system, the lowest-dimensional description of the dynamics that captures the behavior to either side of the bifurcation is motion on a two-dimensional inertial manifold. As argued previously, the most likely parameter regime is one in which \(c\) and \(s\) are the fast variables. However, as in the three-variable model, this leads to an implicit manifold representation problem which will be left for future study. Rather, we consider the regime where \(\beta\ll 1\) and \(\epsilon_{2}\ll\{\epsilon_{1},1\}\). This is the easiest biochemically plausible case since \(r\) is then one of the fast variables, which means that the delay can be directly eliminated. The other fast variable in this case is of course \(c\). These considerations imply a parameterization of the manifold \(c=\mathcal{C}(e,s)\), \(r=\mathcal{R}(e,s)\). The corresponding invariance equations are  
\[\dot{c} =\frac{\partial\mathcal{C}}{\partial e}\dot{e}+\frac{\partial \mathcal{C}}{\partial s}\dot{s}, \tag{36a}\] \[\dot{r} =\frac{\partial\mathcal{R}}{\partial e}\dot{e}+\frac{\partial \mathcal{R}}{\partial s}\dot{s}. \tag{36b}\]  
Figure 8: Trajectories of (**a**) the DDE inducible enzyme model, (**b**) the reduced model for \(m=1\), (**c**) the reduced model for \(m=2\), and (**d**) the reduced model for \(m=3\). For the reduced models, the \(k=3\) iterate was used. The parameters are the same as in Figure 7.  
The delayed term does not appear in Equation (36a), so the functional iteration for \(\mathcal{C}(e,s)\) is the same regardless of the delay expansion order:  
\[\mathcal{C}^{(m)}_{k+1}=\frac{es-\beta\epsilon_{1}\frac{\partial\mathcal{C}^{(m )}_{k}}{\partial e}\left(\mathcal{R}^{(m)}_{k}-e\right)-\beta\frac{\partial \mathcal{C}^{(m)}_{k}}{\partial s}(\kappa-es)}{s+\gamma+\beta\epsilon_{1}+ \beta\frac{\partial\mathcal{C}^{(m)}_{k}}{\partial s}(s+\alpha)}. \tag{37}\]  
(\(\mathcal{C}^{(m)}_{k}\) will eventually depend on \(m\) because of its dependence on \(\mathcal{R}^{(m)}_{k}\).) The functional equations for \(\mathcal{R}(e,s)\) can also be derived analytically, although for \(m>1\), the derivation is best left to a symbolic algebra system. For \(m=1\), we have  
\[\mathcal{R}^{(1)}_{k+1}=\frac{\rho(s)-\left(\frac{\delta}{ \epsilon_{1}}\frac{d\rho}{ds}+\frac{\epsilon_{2}}{\epsilon_{1}}\frac{\partial \mathcal{R}^{(1)}_{k}}{\partial s}\right)\left[\kappa-es+\mathcal{C}^{(1)}_{ k+1}(s+\alpha)\right]+\epsilon_{2}e\frac{\partial\mathcal{R}^{(1)}_{k}}{ \partial e}}{1+\epsilon_{2}\frac{\partial\mathcal{R}^{(1)}_{k}}{\partial e}}, \tag{38}\]  
assuming chained iteration, \(\mathcal{C}\) first. Even starting with our usual naive initial function of \(\mathcal{R}^{(m)}_{0}=0\), we see that \(\mathcal{R}^{(1)}_{1}\) will depend on \(\delta\). Corrections to the slow manifold due to the delay therefore enter at a level equivalent to the application of the steady-state approximation. From the functional Equations (37) and (38), we can see that at this level of approximation of the vector field, corrections to the manifold due to the delay depend on \(\delta/\epsilon_{1}\). Thus, the delay will become less significant to the slow dynamics at small values of \(\delta\) or large values of \(\epsilon_{1}\). Here we see an example where the analysis tells us something that might have been difficult to intuit: In terms of the original dimensional parameters, the condition \(\delta/\epsilon_{1}\ll 1\) becomes \(k_{1}E_{\max}\ll\tau^{-1}\). Thus, if the association of the enzyme and substrate is sufficiently slow compared to the gene expression cycle, the delay associated with this cycle should have a negligible effect on the geometry of the slow manifold and, at least from the perspective of the motion on the two-dimensional slow manifold, an ODE model neglecting the transcription and translation delays would be appropriate. Note that this says nothing about the transients, which could still be very different between the models with and without delays.