A mathematical analysis of an activator-inhibitor Rho GTPase model

Figure 1.  Schematic representation of an activator-inhibitor system. A: An illustrative activator-inhibitor network model. In this set-up $ R $ activates $ M $; which in turn inhibits both $ R $ and $ G $. $ R $ and $ G $ form a positive feedback loop [15,22]. B: Flow diagram representing the interactions between active and inactive forms of species. The active species are denoted respectively by the variables; $ R(t) $, $ M(t) $ and $ G(t) $. Their total concentrations are respectively denoted, $ R_T $, $ M_T $ and $ G_T $ and these are conserved. Dotted arrows represent the catalytic activity while full arrows represent chemical reactions

Figure 2.  Typical shapes for nullclines corresponding to System (8) using different values of $ G_T $. As the parameter $ G_T $ varies, we obtain up to three different nullcline intersections. The blue curve indicates the M-nullcline, while the brownish to redish curves indicate R-nullclines at different values of $ G_T $

Figure 3.  Qualitative forms of nullcline intersections corresponding to System (8) as the parameter $ G_T $ varies. (a) For $ G_T = 0.5 $, the steady state $ E_1 $ is globally asymptotic stable (G.A.S.). (b) With $ G_T = 1.2 $, there is possibility of $ \mathbb{H}_2 $ and hence periodic solutions might occur around $ E_2 $, or $ E_2 $ is asymptotically stable. (c) For $ G_T = 7 $, the steady state in the form of $ E_3 $ is G.A.S. (d) For $ G_T = 20 $, there exists a bistable behaviour ($ E_4 $ and $ E_6 $ are locally asymptotically stable (L.A.S.) and $ E_5 $ is a saddle point)

Figure 4.  Numerical simulations illustrating time-series dynamics of $ R(t) $ and $ M(t) $ for System (8) corresponding to different dynamic regimes. As the value of $ G_T $ increases, the system transitions from stable ((a), $ G_T = 0.5 $), oscillatory ((b), $ G_T = 2 $), excitable ((c), $ G_T = 7 $) and then to bistable ((d), $ G_T = 15 $). A minimum perturbation of $ 0.09 $ from the steady state has to be taken to exhibit excitable dynamics. In the bistable regime, we used initial conditions $ (0.4,0.7) $ and $ (0.6,0.3) $ to approach both L.A.S. equilibria. Base values for parameters are shown Table 1

Figure 5.  Bifurcation diagrams for System (8). (a) Equilibrium values of the $ R- $component are plotted as a function of the parameter $ G_T $. Colors of plain and dashed lines indicate the nature of the equilibria: a blue plain line represents a stable equilibrium, a red dashed line represents an unstable equilibrium and a yellow dashed line indicates an unstable saddle point. The green doted lines show the minimum and maximum amplitudes of limit cycles. Hopf bifurcations are labelled (HB) while fold bifurcations are labelled (LP). The values of $ G_T $ at Hopf bifurcations are $ 1.002672, $ $ 2.55130 $ and $ 7.576458 $, while at the fold bifurcation, $ G_T = 6.841074 $. (b) Two-parameters bifurcation diagram: the oscillatory region is represented in red, the bistable region in yellow and the stable region is in white

Figure 6.  Local sensitivity profiles for System (8) with $ G_T = 0.25 $ and $ G_T = 0.4 $ selected to represent stable and oscillatory regimes respectively. Other parameters are fixed as shown in Table 1. The sensitivity matrix is bounded in the stable regime, but unbounded in the oscillatory regime. The colour codes used in (d) and (e) are the same as those used in (a) and (b)

Figure 7.  Period sensitivity results corresponding to System (8). (a) shows convergent time series of period sensitivity calculated from Equation (22) while (b) shows a bar graph of normalised period sensitivities extracted from (a) after convergence

Figure 8.  Cleaned-out and amplitude sensitivity results for System (8). (a) The blue curve is the cleaned-out sensitivity obtained from Equation (23). The red curve is the corresponding unbounded state sensitivity before SVD was applied as shown in Figures 6 (d) and (e). (b) $ R(t) $ and $ M(t) $ amplitude sensitivities to parameters calculated with Equation (25)