Switching, Multiple Time-Scales and Geometric Blow-Up in a Low-Dimensional Gene Regulatory Network

TL;DR

This work analyzes how switching and time-scale separation interact in a low-dimensional GRN by recasting the model as a smooth singular perturbation problem and applying geometric blow-up to resolve non-smooth switching. It proves the existence of a 2D slow manifold $\mathcal{M}_{\mu}(\sigma)$ for $0<\mu=\varepsilon/\sigma\ll1$, showing the reduced dynamics on this manifold can differ qualitatively from the quasi-steady-state reduction, including Hopf bifurcations that require $\mu>\alpha\sigma$. The results reveal that the limit $\varepsilon\to0$ and $\sigma\to0$ is path-dependent, with global protein-only reductions failing in some regions of parameter space, while local reductions away from switching sets remain valid. The analysis provides a rigorous framework for selecting appropriate reduced models in GRNs and suggests that the relative magnitudes of the small parameters crucially shape the qualitative dynamics, with potential extensions to higher-dimensional networks.

Abstract

ODE-based models for gene regulatory networks (GRNs) can often be formulated as smooth singular perturbation problems with multiple small parameters, some of which are related to time-scale separation, whereas others are related to 'switching' (proximity to a non-smooth singular limit). This motivates the study of reduced models obtained after (i) quasi-steady state reduction (QSSR), which utilises the time-scale separation, and (ii) piecewise-smooth approximations, which reduce the nonlinearity of the model by viewing highly nonlinear sigmoidal terms as singular perturbations of step functions. We investigate the interplay between the reduction methods (i)-(ii), in the context of a 4-dimensional GRN which has been used as a low-dimensional representative of an important class of (generally high-dimensional) GRN models in the literature. We begin by identifying a region in the small parameter plane for which this problem can be formulated as a smooth singularly perturbed system on a blown-up space, uniformly in the switching parameter. This allows us to apply Fenichel's coordinate-free theorems and obtain a rigorous reduction to a 2-dimensional system, that is a perturbation of the QSSR. Finally, we show that the reduced system features a Hopf bifurcation which does not appear in the QSSR system, due to the influence of higher order terms. Taken together, our findings suggest that the relative size of the small parameters is important for the validity of QSS reductions and the determination of qualitative dynamics in GRN models more generally. Although the focus is on the 4-dimensional GRN, our approach is applicable to higher dimensions.

Figures (9)

• Figure 1: The Hill functions $h^-(p_i; \theta_i, n_i)$ and $h^+(p_i; \theta_i, n_i)$ are shown in in the left and right panels respectively, for $\theta_i = 1$ and three different values of $n_i$. Both functions converge do a discontinuous step function when $n_i \to \infty$, as in \ref{['eq:Hill_fns']}.
• Figure 2: Conceptual schema for the combined approach to model reduction in the (generally high dimensional and highly nonlinear) GRN model \ref{['eq:grn_network']}. Applying partial reductions in different orders (or equivalently, taking a different path from the original model \ref{['eq:grn_network']} to the final reduced model) can lead to different reduced models, and therefore to different predictions.
• Figure 3: Two different particular solutions of system \ref{['eq:main0']}. In (a) and (c): Projections onto the $(p_a,p_b)$-plane. In (b) and (d): Time series of solutions. The parameter values are given by \ref{['eq:para']} and $\varepsilon=5\times 10^{-5}$ in (a) and (b) and $\varepsilon=5\times 10^{-3}$ in (c) and (d). The presence of oscillations appears to depend upon $\varepsilon$.
• Figure 4: The positive quadrant of the $(\sigma,\varepsilon)$-plane is shown. The line $\varepsilon = \mu_0 \sigma$, i.e. $\mu = \mu_0$, defines a boundary below which our first main result (Theorem \ref{['thm:thm0']} assertion \ref{['existence']}) guarantees the existence of $\mathcal{M}_{\mu}(\sigma)$. The quadratic curve $\varepsilon = \alpha \sigma^2$ in dark blue divides the region defined by $0<\varepsilon < \mu_0 \sigma$ into two distinct (asymptotic) subregions. Hopf bifurcations are possible (impossible) in the red (blue) subregion; see assertion \ref{['kappa']} of Theorem \ref{['thm:thm0']} and the discussion in the text for details.
• Figure 5: Schematic representation of the sequence of blow-up transformations which allow us to resolve the loss of smoothness along the singular set $\Sigma_a\cup \Sigma_b$ in system \ref{['eq:main0']} when $\sigma = 0$. In (a), $\Sigma_a$ and $\Sigma_b$ are illustrated (as projections) within the $(p_a,p_b)$-plane. To resolve the lack of smoothness, we first resolve the intersection $\Sigma_a\cap \Sigma_b$ (red) with a spherical blow-up transformation. This gives the intermediate blown-up space $\overline{\mathbb P}$ in (b). Subsequently, we blow-up the remaining singularities $\overline \Sigma_b^\pm$ and $\Sigma_a^\pm$ along the axes, using cylindrical blow-ups. This gives the final blown up space $\overline{\overline{\mathbb P}}$ in (c).

Theorems & Definitions (21)

• Lemma 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Proposition 2.4