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A mathematical analysis of an activator-inhibitor Rho GTPase model

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  • Recent experimental observations reveal that local cellular contraction pulses emerge via a combination of fast positive and slow negative feedbacks based on a signal network composed of Rho, GEF and Myosin interactions [22]. As an examplary, we propose to study a plausible, hypothetical temporal model that mirrors general principles of fast positive and slow negative feedback, a hallmark for activator-inhibitor models. The methodology involves (ⅰ) a qualitative analysis to unravel system switching between different states (stable, excitable, oscillatory and bistable) through model parameter variations; (ⅱ) a numerical bifurcation analysis using the positive feedback mediator concentration as a bifurcation parameter, (ⅲ) a sensitivity analysis to quantify the effect of parameter uncertainty on the model output for different dynamic regimes of the model system; and (ⅳ) numerical simulations of the model system for model predictions. Our methodological approach supports the role of mathematical and computational models in unravelling mechanisms for molecular and developmental processes and provides tools for analysis of temporal models of this nature.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • 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)

    Table 1.  Parameters, their descriptions and values used for simulations and bifurcation analysis. Base line parameter values are taken from [43]. Some of them were adjusted to illustrate the qualitative dynamics hypothesised. These values could be estimated when considering experimental data through a parameter inference approach, this forms part of our current work (see for example [4]). The parameters $ k_i $, $ i = 0,\cdots, 7 $, represent the reaction constants, while $ K_{i} $, $ i = r0 $, $ r2 $, $ m5 $, $ g3 $, $ g4 $, represent Michaelis-Menten constants

    Parameter Description Base value
    $ k_0 $ R activation by G per unit time 4
    $ k_1 $ Rate of R baseline activation 0.6
    $ k_2 $ R baseline inhibition per unit time 1
    $ k_2' $ R inhibition by M per unit time 1
    $ k_3 $ Rate of G activation by R 1
    $ k_4 $ Rate of G inhibition by M 1
    $ k_5 $ M activation rate 0.035
    $ k_6 $ M decay rate 0.01
    $ k_7 $ M baseline recruitment rate 0.001
    $ K_{r0} $ Michaelis-Menten constant for R activation 1
    $ K_{r2} $ Michaelis-Menten constant for R self inhibition 1
    $ K_{m5} $ Michaelis-Menten constant for M activation 1
    $ K_{g3} $ Michaelis-Menten constant for G activation 0.12
    $ K_{g4} $ Michaelis-Menten constant for G inhibition 0.075
    $ R_T $ R total concentration 1
    $ M_T $ M total concentration 1
    $ G_T $ G total concentration variable
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