Stability Analysis of a Signaling Circuit with Dual Species of GTPase Switches

Abstract

GTPases are molecular switches that regulate a wide range of cellular processes, such as organelle biogenesis, position, shape, function, vesicular transport between organelles, and signal transduction. These hydrolase enzymes operate by toggling between an active (“ON”) guanosine triphosphate (GTP)-bound state and an inactive (“OFF”) guanosine diphosphate (GDP)-bound state; such a toggle is regulated by GEFs (guanine nucleotide exchange factors) and GAPs (GTPase activating proteins). Here we propose a model for a network motif between monomeric (m) and trimeric (t) GTPases assembled exclusively in eukaryotic cells of multicellular organisms. We develop a system of ordinary differential equations in which these two classes of GTPases are interlinked conditional to their ON/OFF states within a motif through coupling and feedback loops. We provide explicit formulae for the steady states of the system and perform classical local stability analysis to systematically investigate the role of the different connections between the GTPase switches. Interestingly, a coupling of the active mGTPase to the GEF of the tGTPase was sufficient to provide two locally stable states: one where both active/inactive forms of the mGTPase can be interpreted as having low concentrations and the other where both m- and tGTPase have high concentrations. Moreover, when a feedback loop from the GEF of the tGTPase to the GAP of the mGTPase was added to the coupled system, two other locally stable states emerged. In both states the tGTPase is inactivated and active tGTPase concentrations are low. Finally, the addition of a second feedback loop, from the active tGTPase to the GAP of the mGTPase, gives rise to a family of steady states that can be parametrized by a range of inactive tGTPase concentrations. Our findings reveal that the coupling of these two different GTPase motifs can dramatically change their steady-state behaviors and shed light on how such coupling may impact signaling mechanisms in eukaryotic cells.

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Acknowledgements

This work was supported by Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) Grant FA9550-18-1-0051 (to P. Rangamani) and the National Institute of Health (CA100768, CA238042 and AI141630 to P. Ghosh). Lucas M. Stolerman acknowledges support from the National Institute of Health (CA209891). The authors would like to acknowledge Prof. Ali Behzadan (Sacramento State University) for his careful reading and insightful comments for this work.

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Appendices

Appendix A. Proof of Proposition 3.1

We must find nonnegative \(\widehat{[mG]}\), \(\widehat{[mG^*]}\), \(\widehat{{\mathcal {G}}^*}\), and \(\widehat{{\mathcal {T}}^*}\) satisfying the following system:

$$\begin{aligned}&\widehat{[mG]} - \rho ^{mG}_{off}[mGAP^*]\widehat{[mG^*]} - \rho ^{I}_{on} [tGEF_{tot}](1 - \widehat{{\mathcal {G}}^*})\widehat{[mG^*]} = 0 \end{aligned}$$
(A.1)
$$\begin{aligned}&\rho ^{tG}_{on}[tGEF_{tot}] \widehat{{\mathcal {G}}^*} (1 - \widehat{{\mathcal {T}}^*}) - \rho ^{tG}_{off}\widehat{{\mathcal {T}}^*} = 0 \end{aligned}$$
(A.2)
$$\begin{aligned}&(1 -\widehat{{\mathcal {G}}^*})\widehat{[mG^*]} = 0 \end{aligned}$$
(A.3)
$$\begin{aligned}&\widehat{[mG]} + \widehat{[mG^*]} + [tGEF_{tot}] \widehat{\mathcal {G^*}} = C \end{aligned}$$
(A.4)

From Eq. A.3, we must have \(\widehat{[mG^*]}=0\) or \(\widehat{{\mathcal {G}}^*} = 1 \). Thus, we divide the steady-state analysis in two cases.

\(\underline{\hbox {Case 1: }\widehat{[mG^*]}=0.}\)

From Eq. A.1, we must have \(\widehat{[mG]}=0\), and from Eq. A.4, we obtain \( \widehat{\mathcal {G^*}} = \frac{C}{[tGEF_{tot}]} \). Since \(\widehat{\mathcal {G^*}} \le 1 \) by definition, we conclude that

$$\begin{aligned} C \le [tGEF_{tot}]. \end{aligned}$$
(A.5)

Equation A.5 is also sufficient for \(\widehat{[mG^*]}=0\). Otherwise, if \(C \le [tGEF_{tot}]\) and \(\widehat{[mG^*]}>0\), then \(\widehat{\mathcal {G^*}} = 1\) (Eq. A.3) and from Eq. A.4, we would conclude that \(\widehat{[mG]}+\widehat{[mG^*]} \le 0\), which is impossible.

Finally, by substituting \( \widehat{\mathcal {G^*}}\) in Eq. A.2, we obtain \(\widehat{{\mathcal {T}}^*} = \frac{1}{1+\frac{\rho ^{tG}_{off}}{\rho ^{tG}_{on} C}}\), and therefore, the steady state is given by

$$\begin{aligned} \left( \widehat{[mG]},\widehat{[mG^*]},\widehat{{\mathcal {T}}^*},\widehat{{\mathcal {G}}^*} \right) = \left( 0, 0, \frac{1}{1+\frac{\rho ^{tG}_{off}}{\rho ^{tG}_{on} C}}, \frac{C}{[tGEF_{tot}]} \right) \end{aligned}$$

\(\underline{\hbox {Case 2: }\widehat{\mathcal {G^*}}=1}\)

In this case, \( \widehat{[mG^*]} \ge 0\) and from Eqs. A.1 and A.4, we obtain

$$\begin{aligned} \widehat{[mG^*]} = \frac{C- [tGEF_{tot}]}{1+\rho ^{mG}_{off}[mGAP^*]} \end{aligned}$$

and

$$\begin{aligned} \widehat{[mG]} = \frac{\rho ^{mG}_{off} [mGAP^*]}{1+\rho ^{mG}_{off}[mGAP^*]} \left( C- [tGEF_{tot}]\right) . \end{aligned}$$

In this case, since the steady state has to be nonnegative, we must have

$$\begin{aligned} C \ge [tGEF_{tot}]. \end{aligned}$$
(A.6)

which is also sufficient for \(\widehat{\mathcal {G^*}}=1\). Otherwise if \(C \ge [tGEF_{tot}]\) and \(\widehat{\mathcal {G^*}}<1\), then \(\widehat{[mG^*]}=\widehat{[mG]}=0\) (Eqs. A.1 and A.3), and from Eq. A.4, we would have

$$\begin{aligned} C = \widehat{[mG]} + \widehat{[mG^*]} + [tGEF_{tot}] \widehat{\mathcal {G^*}} < [tGEF_{tot}], \end{aligned}$$

which is impossible.

Finally, by substituting \(\widehat{\mathcal {G^*}}=1\) in Eq. A.2, we obtain

$$\begin{aligned} \rho ^{tG}_{on} [tGEF_{tot}] (1 - \widehat{{\mathcal {T}}^*}) - \rho ^{tG}_{off}\widehat{{\mathcal {T}}^*}= 0 \end{aligned}$$

which gives \(\widehat{{\mathcal {T}}^*} = \frac{\rho ^{tG}_{on} [tGEF_{tot}]}{\rho ^{tG}_{on}[tGEF_{tot}] + \rho ^{tG}_{off}}\) and therefore

$$\begin{aligned} \left( \widehat{[mG]},\widehat{[mG^*]},\widehat{{\mathcal {T}}^*},\widehat{{\mathcal {G}}^*} \right)= & {} \left( \frac{\rho ^{mG}_{off} [mGAP^*]}{1+\rho ^{mG}_{off}[mGAP^*]} \left( C- [tGEF_{tot}]\right) , \right. \nonumber \\&\left. \frac{C- [tGEF_{tot}]}{1+\rho ^{mG}_{off}[mGAP^*]}, \right. \nonumber \\&\left. \frac{\rho ^{tG}_{on} [tGEF_{tot}]}{\rho ^{tG}_{on}[tGEF_{tot}] + \rho ^{tG}_{off}} ,1 \right) . \end{aligned}$$
(A.7)

Appendix B. Proof of Theorem 3.2

We begin our proof by computing the steady states of the system, which are solutions of the algebraic system given by Eqs. 3.273.32. We also establish necessary and sufficient conditions involving the parameters \(C_1\), \(C_2\), and \([mGAP_{tot}]\) for the existence of each steady state. We then compute the Jacobian matrix of the system and determine the local stability of the steady state based on the classical linearization procedure (Strogatz 1994).

Steady States

We divide our analysis into four different cases that emerge from the preliminary inspection of the system given by Eqs. 3.273.32.

\(\underline{\hbox {Case 1: }\widehat{[mG^*]}=0\hbox { and }\widehat{[mGAP^*]}=[mGAP_{tot}].}\)

From Eq. 3.27, we have \(\widehat{[mG]} = 0\) and from Eq. 3.31, \(\widehat{[tGEF^*]} = C_1 - [mGAP_{tot}]\). Thus, \(C_1 \ge [mGAP_{tot}]\) since the steady state must be nonnegative. Now Eq. 3.32 gives \(\widehat{[tGEF]} = C_2 - C_1\) and that implies \(C_2 \ge C_1\).

Finally, Eq. 3.28 yields

$$\begin{aligned} \rho ^{tG}_{on}(C_1 - [mGAP_{tot}])(1- \widehat{{\mathcal {T}}^*}) - \rho ^{tG}_{off} \widehat{{\mathcal {T}}^*} = 0, \end{aligned}$$

and hence,

$$\begin{aligned} \widehat{{\mathcal {T}}^*} = \frac{\rho ^{tG}_{on}\left( C_1 - [mGAP_{tot}]\right) }{\rho ^{tG}_{on}(C_1 - [ mGAP_{tot}]) + \rho ^{tG}_{off}} \end{aligned}$$

The steady state is therefore given by

$$\begin{aligned} \widehat{\mathbf{x}}= & {} \bigg (0,0,\frac{\rho ^{tG}_{on}\left( C_1 - [mGAP_{tot}]\right) }{\rho ^{tG}_{on}(C_1 - [ mGAP_{tot}]) + \rho ^{tG}_{off}}, C_2 - C_1,C_1 - [mGAP_{tot}],[mGAP_{tot}] \bigg ). \end{aligned}$$

We now observe that the two parameter relations

$$\begin{aligned} C_1 \ge [mGAP_{tot}] \quad \text {and} \quad C_2 \ge C_1 \end{aligned}$$
(B.1)

are sufficient for \(\widehat{[mG^*]}=0\) and \(\widehat{[mGAP^*]}=[mGAP_{tot}]\). First, we observe that if \(C_2 \ge C_1\) then \(\widehat{[mG^*]}=0\). In fact, by subtracting Eq. 3.31 from Eq. 3.32, we obtain

$$\begin{aligned} \widehat{[tGEF]} - \widehat{[mG]} + \widehat{[mG^*]} = C_2 - C_1 \ge 0, \end{aligned}$$

and hence, \( \widehat{[tGEF]} \ge \widehat{[mG]} + \widehat{[mG^*]} \). On the other hand, from Eq. 3.29, we must have \( \widehat{[tGEF]}=0\) or \(\widehat{[mG^*]}=0\). Thus, if \( \widehat{[tGEF]}=0\) then \(\widehat{[mG]} + \widehat{[mG^*]} \le 0\), and hence, the nonnegativeness of the steady state implies \(\widehat{[mG]}=\widehat{[mG^*]}=0\). Now, Eq. 3.31 gives \(\widehat{[tGEF^*]} = C_1 - \widehat{[mGAP^*]}\) and from Eq. 3.30, we must have \(\widehat{[mGAP^*]}= [mGAP_{tot}]\) or \(\widehat{[tGEF^*]}=0\). If \(\widehat{[tGEF^*]}=0\), then \(\widehat{[mGAP^*]} = C_1 \ge [mGAP_{tot}]\), and hence, \(\widehat{[mGAP^*]} = [mGAP_{tot}]\). Therefore, we have shown that Eq. B.1 imply \(\widehat{[mG^*]}=0\) and \(\widehat{[mGAP^*]}=[mGAP_{tot}]\). Consequently, the steady state in this case must be given by Eq. 3.33.

\(\underline{\hbox {Case 2: } \widehat{[tGEF]}=0\hbox { and }\widehat{[mGAP^*]}=[mGAP_{tot}]}\)

From Eq. 3.32, \(\widehat{[tGEF^*]} = C_2 - [mGAP_{tot}] \), and hence, \([mGAP_{tot}] \le C_2\). From Eq. 3.31, we must have \( \widehat{[mG]} + \widehat{[mG^*]} = C_1 - C_2\) and that implies \(C_1 \ge C_2\). Now, Eq. 3.27 gives

$$\begin{aligned} \left( C_1 - C_2 - \widehat{[mG^*]} \right) = \rho ^{mG}_{off} [mGAP_{tot}] \widehat{[mG^*]} \end{aligned}$$

and therefore

From Eq. 3.28, we must have

$$\begin{aligned} \rho ^{tG}_{on}\left( C_2 - [mGAP_{tot}] \right) (1- \widehat{{\mathcal {T}}^*}) - \rho ^{tG}_{off} \widehat{{\mathcal {T}}^*} = 0 \end{aligned}$$

from which we obtain

$$\begin{aligned} \widehat{{\mathcal {T}}^*} = \frac{\rho ^{tG}_{on}(C_2 - [mGAP_{tot}])}{ \rho ^{tG}_{on}(C_2 - [mGAP_{tot}]) + \rho ^{tG}_{off}} \end{aligned}$$

and therefore, the steady state is given by

$$\begin{aligned} \widehat{\mathbf{x}}= & {} \left( \frac{\rho ^{mG}_{off} [mGAP_{tot}]}{1+\rho ^{mG}_{off} [mGAP_{tot}]} (C_1-C_2), \frac{(C_1-C_2)}{1+\rho ^{mG}_{off} [mGAP_{tot}]}, \right. \nonumber \\&\left. \frac{\rho ^{tG}_{on} (C_2 - [mGAP_{tot}])}{(C_2 - [mGAP_{tot}])+\rho ^{tG}_{off}} ,0,C_2 - [mGAP_{tot}], [mGAP_{tot}] \right) \end{aligned}$$

We now observe that the two parameter relations

$$\begin{aligned} C_2 \ge [mGAP_{tot}] \quad \text {and} \quad C_1 \ge C_2 \end{aligned}$$
(B.2)

are sufficient for \( \widehat{[tGEF]}=0\) and \(\widehat{[mGAP^*]}=[mGAP_{tot}]\).

In fact, if \(C_1 \ge C_2\) then \(\widehat{[tGEF]}=0\) from the same argument as in Case 1. Now, Eq. 3.32 gives \(\widehat{[tGEF^*]} = C_2 - \widehat{[mGAP^*]}\) and from Eq. 3.30, we must have \( \widehat{[mGAP^*]} = [mGAP_{tot}]\) or \(\widehat{[tGEF^*]}=0\). If \(\widehat{[tGEF^*]}=0\), then \( \widehat{[mGAP^*]} = C_2 \ge [mGAP_{tot}]\) (from Eq. B.2), and thus, \(\widehat{[mGAP^*]} = [mGAP_{tot}]\). Therefore, we have shown that Eq. B.2 imply \( \widehat{[tGEF]}=0\) and \(\widehat{[mGAP^*]}=[mGAP_{tot}]\). Consequently, the steady state in this case must be given by Eq. 3.34.

\(\underline{\hbox {Case 3: }\widehat{[mG^*]}=0 and \widehat{[tGEF^*]}=0.}\)

From Eq. 3.27, we have \(\widehat{[mG]} = 0\) and from Eq. 3.28, we also get \(\widehat{{\mathcal {T}}^*} =0\) since \(\rho ^{tG}_{off}>0\). Now, Eq. 3.31 gives \(\widehat{[mGAP^*]} = C_1\), and thus, we must have \(C_1 \le [mGAP_{tot}]\). Moreover, Eq. 3.32 results in \(\widehat{[tGEF]} = C_2 - C_1\) and since all steady states must be nonnegative, we obtain \(C_2 \ge C_1\). In this case, the steady state is given by

$$\begin{aligned} \widehat{\mathbf{x}}= & {} \left( 0,0,0,C_2-C_1,0,C_1\right) \end{aligned}$$
(B.3)

We now observe that the two parameter relations

$$\begin{aligned} C_1 \le [mGAP_{tot}] \quad \text {and} \quad C_2 \ge C_1 \end{aligned}$$
(B.4)

are sufficient for \(\widehat{[mG^*]}=0\) and \(\widehat{[tGEF^*]}=0\). In fact, \(C_2 \ge C_1\) implies \(\widehat{[mG^*]}=0\) from the same argument as in Case 1.

Now, Eq. 3.31 gives \(\widehat{[tGEF^*]} = C_1 - \widehat{[mGAP^*]}\) and from Eq. 3.30, we must have \(\widehat{[tGEF^*]}=0\) or \(\widehat{[mGAP^*]}= [mGAP_{tot}]\). If \(\widehat{[mGAP^*]}=[mGAP_{tot}]\), then \(\widehat{[tGEF^*]} = C_1 - [mGAP_{tot}] \le 0\) (from Eq. B.4), and thus, \(\widehat{[tGEF^*]} = 0\). Therefore, we have shown that Eq. B.4 imply \(\widehat{[mG^*]}=0\) and \(\widehat{[tGEF^*]} = 0\). Consequently, the steady state in this case must be given by Eq. 3.35.

\(\underline{\hbox {Case 4: } \widehat{[tGEF]}=0\hbox { and } \widehat{[tGEF^*]} = 0 }\)

From Eq. 3.32, we obtain \(\widehat{[mGAP^*]}=C_2\), and hence, \(C_2 \le [mGAP_{tot}]\). From Eq. 3.31, we have \(\widehat{[mG]} + \widehat{[mG^*]} = C_1 - C_2\) and that implies \(C_1 \ge C_2\) since the concentrations at steady state must be nonnegative. Eq. 3.27 then gives

from which we obtain

$$\begin{aligned} \widehat{[mG^*]} = \frac{1}{1+\rho ^{mG}_{off}C_2} (C_1 - C_2) \quad \text {and} \quad \widehat{[mG]} = \frac{\rho ^{mG}_{off}C_2}{1+\rho ^{mG}_{off}C_2} (C_1 - C_2). \end{aligned}$$

From Eq. 3.28, we have \(\widehat{{\mathcal {T}}^*} = 0\), and therefore, the steady state is given by

$$\begin{aligned} \widehat{\mathbf{x}} = \left( \frac{1}{1+\rho ^{mG}_{off}C_2} (C_1 - C_2), 0,0,0, C_2 \right) . \end{aligned}$$

We now observe that the two parameter relations

$$\begin{aligned} C_2 \le [mGAP_{tot}] \quad \text {and} \quad C_1 \ge C_2 \end{aligned}$$
(B.5)

are sufficient for \( \widehat{[tGEF]}=0\) and \(\widehat{[tGEF^*]} = 0\). In fact, if \(C_1 \ge C_2\) then by subtracting Eq. 3.32 from Eq. 3.31, we have

$$\begin{aligned} \widehat{[mG]} + \widehat{[mG^*]} - \widehat{[tGEF]} = C_1 - C_2 \ge 0, \end{aligned}$$

and hence, \( \widehat{[mG]} + \widehat{[mG^*]} \ge \widehat{[tGEF]} \). On the other hand, from Eq. 3.29, we must have \( \widehat{[tGEF]}=0\) or \(\widehat{[mG^*]}=0\). Thus, if \(\widehat{[mG^*]} = 0\) then \(\widehat{[mG]} = 0\) (from Eq. 3.27), and hence, the nonnegativeness implies \(\widehat{[tGEF]}=0\). Hence, we conclude that Eq. B.5 guarantee \(\widehat{[tGEF]}=0\).

Now, Eq. 3.32 gives \(\widehat{[tGEF^*]} = C_2 - \widehat{[mGAP^*]}\) and from Eq. 3.30, we must have \(\left( [mGAP_{tot}] - \widehat{[mGAP^*]} \right) =0\) or \(\widehat{[tGEF^*]}=0\). If \(\widehat{[mGAP^*]}=[mGAP_{tot}]\) then \(\widehat{[tGEF^*]} = C_2 - [mGAP_{tot}] \le 0\) (from Eq. B.5), and thus, \(\widehat{[tGEF^*]} = 0\). Therefore, we have shown that Eq. B.5 implies \(\widehat{[tGEF]}=0\) and \(\widehat{[tGEF^*]} = 0\). Consequently, the steady state in this case must be given by Eq. 3.36.

Local Stability Analysis

We begin reducing the ODE system with the conservation laws given by Eqs. 3.19 and 3.20. In fact, if we write

$$\begin{aligned}&[mG] = C_1 - [mG^*] - [tGEF^*] - [mGAP^*] \quad \text {and} \quad \\&[tGEF]= C_2 - [tGEF^*] - [mGAP^*] \end{aligned}$$

then Eqs. 3.213.26 can be written in the form

$$\begin{aligned} \frac{d[mG^*]}{dt}= & {} f_1([mG^*],{\mathcal {T}}^*, [tGEF^*],[mGAP^*]) \end{aligned}$$
(B.6)
$$\begin{aligned} \frac{d{\mathcal {T}}^*}{dt}= & {} f_2([mG^*],{\mathcal {T}}^*, [tGEF^*],[mGAP^*]) \end{aligned}$$
(B.7)
$$\begin{aligned} \frac{d[tGEF^*]}{dt}= & {} f_3([mG^*],{\mathcal {T}}^*, [tGEF^*],[mGAP^*]) \end{aligned}$$
(B.8)
$$\begin{aligned} \frac{d[mGAP^*]}{dt}= & {} f_4([mG^*],{\mathcal {T}}^*, [tGEF^*],[mGAP^*]) \end{aligned}$$
(B.9)

where

$$\begin{aligned} f_1([mG^*],{\mathcal {T}}^*, [tGEF^*],[mGAP^*])= & {} \left( C_1 - [mG^*] - [tGEF^*] - [mGAP^*] \right) \\&- \rho ^{tG}_{off}[mGAP^*][mG^*] \\&- \rho ^{I}_{on}\left( C_2 - [tGEF^*] - [mGAP^*]\right) [mG^*],\\ f_2([mG^*],{\mathcal {T}}^*, [tGEF^*],[mGAP^*])= & {} \rho ^{tG}_{on} [tGEF^*] (1 - {\mathcal {T}}^*) - \rho ^{tG}_{off}{\mathcal {T}}^*,\\ f_3([mG^*],{\mathcal {T}}^*, [tGEF^*],[mGAP^*])= & {} \rho ^{I}_{on}\left( C_2 - [tGEF^*] - [mGAP^*]\right) [mG^*] \\&- \rho ^{II}_{on} [tGEF^*] \left( [mGAP_{tot}] - [mGAP^*] \right) , \end{aligned}$$

and

$$\begin{aligned} f_4([mG^*],{\mathcal {T}}^*, [tGEF^*],[mGAP^*]) = \rho ^{II}_{on} [tGEF^*] \left( [mGAP_{tot}] - [mGAP^*] \right) . \end{aligned}$$

The eigenvalues of the Jacobian matrix can be thus calculated for each one of the four steady states given by Eqs. 3.333.36. We prove that all steady states are LAS by showing that the eigenvalues of the Jacobian matrix are all negative real numbers. We perform the calculations with MATLAB’s R2019b symbolic toolbox and analyze each case separately (see supplementary file with MATLAB codes). We analyze each case separately.

  1. (1)

    If \(C_1 > [mGAP_{tot}]\) and \(C_2>C_1\), the Jacobian matrix evaluated at the steady state given by Eq. 3.33 gives the eigenvalues

    $$\begin{aligned} \lambda _1 = -\rho ^{mG}_{on}(C_1 - [mGAP_{tot}]) \quad \text {and} \quad \lambda _2 = - \rho ^{tG}_{on} ( C_1 - [mGAP_{tot}] ) - \rho ^{tG}_{off} \end{aligned}$$

    which are negative. Moreover, the other eigenvalues \(\lambda _3\) and \(\lambda _4\) are such that

    $$\begin{aligned} \lambda _3 + \lambda _4 = \rho ^{I}_{on} (C_1 - C_2) - 1 - \rho ^{mG}_{off} [mGAP_{tot}] < 0 \end{aligned}$$

    and

    $$\begin{aligned} \lambda _3 \lambda _4 = - \rho ^{I}_{on}(C_1 - C_2) >0, \end{aligned}$$

    and thus, \(\lambda _3\) and \(\lambda _4\) are negative, and hence, the steady state is LAS.

  2. (2)

    If \(C_2 > [mGAP_{tot}]\) and \(C_1>C_2\), the Jacobian matrix evaluated at the steady state given by Eq. 3.34 gives the eigenvalues

    $$\begin{aligned} \lambda _1= & {} - \rho ^{tG}_{off} - \rho ^{tG}_{on} (C_2 - [mGAP_{tot}]), \quad \lambda _2 = - 1 - \rho ^{mG}_{off}[mGAP_{tot}],\\ \lambda _3= & {} -\rho ^{II}_{on} (C_2 - [mGAP_{tot}]) \quad \text {and} \quad \lambda _4 = -\frac{\rho ^{I}_{on} (C_1 - C_2)}{\rho ^{mG}_{off} [mGAP_{tot}] + 1} \end{aligned}$$

    which are all negative, and hence, the steady state is LAS.

  3. (3)

    If \(C_1 < [mGAP_{tot}]\) and \(C_2>C_1\),the Jacobian matrix evaluated at the steady state given by Eq. 3.35 gives the eigenvalues

    $$\begin{aligned} \lambda _1 = \rho ^{II}_{on} (C_1 - [mGAP_{tot}]) \quad \text {and} \quad \lambda _2 = -\rho ^{tG}_{off} \end{aligned}$$

    which are negative. Moreover, the other eigenvalues \(\lambda _3\) and \(\lambda _4\) are such that

    $$\begin{aligned} \lambda _3 + \lambda _4 = \rho ^{I}_{on}(C_1 -C_2) - C_1 \rho ^{mG}_{off} - 1 < 0 \end{aligned}$$

    and

    $$\begin{aligned} \lambda _3 \lambda _4 = -\rho ^{I}_{on}(C_1 - C_2)>0, \end{aligned}$$

    and thus, \(\lambda _3\) and \(\lambda _4\) are negative, and hence, the steady state is LAS.

  4. (4)

    If \(C_2 < [mGAP_{tot}]\) and \(C_1 > C_2\), the Jacobian matrix evaluated at the steady state given by Eq. 3.36 gives the eigenvalues

    $$\begin{aligned} \lambda _1 = - 1 - C_2 \rho ^{mG}_{off} , \quad \lambda _2 = \rho ^{II}_{on} (C_2 - [mGAP_{tot}]), \quad \lambda _3 = -\rho ^{tG}_{off} \end{aligned}$$

    and

    $$\begin{aligned} \lambda _4 = -\frac{\rho ^{I}_{on}(C_1 - C_2)}{C_2 \rho ^{mG}_{off} + 1} \end{aligned}$$

    which are all negative, and hence, the steady state is LAS.

Appendix C. Proof of Theorem 3.3

We proceed with the steady-state analysis in the same way of Theorem 3.2. We consider the same four different cases and calculate the \(\xi \)-dependent families of steady states, where \(\xi \ge 0\) represents the tG concentration. We also obtain necessary relationships for the conserved quantities \(\tilde{C_2}\), \(\tilde{C_1}\), and \([mGAP_{tot}]\), as well as admissible intervals for \(\xi \) that guarantee the existence of nonnegative steady states.

\(\underline{\hbox {Case 1: }\widehat{[mG^*]}=0\hbox { and }\widehat{[mGAP^*]}=[mGAP_{tot}]}\).

From Eq. 3.39, we have \(\widehat{[mG]}=0\) and subtracting Eq. 3.38 from Eq. 3.37, we get \(\widehat{[tGEF]} = \tilde{C_2} - \tilde{C_1} \ge 0\) only if \(\tilde{C_2} \ge \tilde{C_1} \). Substituting \(\widehat{[tGEF]}\) on the conservation law given by Eq. 3.38 and using Eq. 3.40 to write    \(\widehat{[tG^*]} = \frac{\rho ^{tG}_{on}\widehat{[tGEF^*]} \xi }{\rho ^{tG}_{off}},\) we obtain

$$\begin{aligned} \xi + \frac{\rho ^{tG}_{on}\widehat{[tGEF^*]} \xi }{\rho ^{tG}_{off}} + (\tilde{C_2} - \tilde{C_1}) + \widehat{[tGEF^*]}+[mGAP_{tot}] = \tilde{C_2}, \end{aligned}$$

and hence,

$$\begin{aligned} \widehat{[tGEF^*]} = \left( \tilde{C_1} - [mGAP_{tot}] -\xi ] \right) \frac{\rho ^{tG}_{off}}{\rho ^{tG}_{off} + \rho ^{tG}_{on}\xi } \end{aligned}$$

only if \(\tilde{C_1} - [mGAP_{tot}] \ge \xi . \) Therefore, in this case the \(\xi \)-dependent family of steady states is given by

$$\begin{aligned} \widehat{\mathbf{x }}_{\xi }= & {} \left( 0, 0, \xi , \frac{\left( \tilde{C_1} - [mGAP_{tot}] -\xi \right) \rho ^{tG}_{on}\xi }{\rho ^{tG}_{off} + \rho ^{tG}_{on}\xi }, \tilde{C_2} - \tilde{C_1}, \right. \\&\left. \left( \tilde{C_1} - [mGAP_{tot}] -\xi \right) \frac{\rho ^{tG}_{off}}{\rho ^{tG}_{off} + \rho ^{tG}_{on}\xi }, [mGAP_{tot}] \right. \Bigg ) \end{aligned}$$

\(\underline{\hbox {Case 2: } \widehat{[tGEF]}=0\hbox { and }\widehat{[mGAP^*]}=[mGAP_{tot}]}\)

Using Eq. 3.39 to write \(\widehat{[mG]} = \rho ^{mG}_{off} [mGAP_{tot}] \widehat{[mG^*]}\) and subtracting Eq. 3.38 from Eq. 3.37, we obtain the expressions for \([mG^*]\) and [mG]

$$\begin{aligned} \widehat{[mG^*]} = \frac{(\tilde{C_1} - \tilde{C_2})}{ \rho ^{tG}_{off} [mGAP_{tot}] + 1} \quad \text {and} \quad \widehat{[mG]} = \frac{(\tilde{C_1} - \tilde{C_2}) \rho ^{tG}_{off}[mGAP_{tot}] }{ \rho ^{tG}_{off}[mGAP_{tot}] + 1}, \end{aligned}$$

and thus, we must have \(\tilde{C_1} \ge \tilde{C_2}\). Now looking at Eq. 3.38 and substituting    \(\widehat{[tG^*]} = \frac{\rho ^{tG}_{on}\widehat{[tGEF^*]} \xi }{\rho ^{tG}_{off}}\) from Eq. 3.40, we obtain

$$\begin{aligned} \widehat{[tGEF^*]} = \left( \tilde{C_2} - [mGAP_{tot}] -\xi ] \right) \frac{\rho ^{tG}_{off}}{\rho ^{tG}_{off} + \rho ^{tG}_{on}\xi } \end{aligned}$$

only if \(\tilde{C_2} - [mGAP_{tot}] \ge \xi \). Therefore, in this case the \(\xi \)-dependent family of steady states is given by

$$\begin{aligned} \widehat{\mathbf{x }}_{\xi }= & {} \left( \frac{(\tilde{C_1} - \tilde{C_2}) \rho ^{mG}_{off}[mGAP_{tot}] }{ 1 + \rho ^{mG}_{off}[mGAP_{tot}] }, \frac{(\tilde{C_1} - \tilde{C_2})}{ 1 + \rho ^{mG}_{off}[mGAP_{tot}]}, \right. \\&\left. \xi , \frac{\left( \tilde{C_2} - [mGAP_{tot}] -\xi ] \right) \rho ^{tG}_{on}\xi }{\rho ^{tG}_{off} + \rho ^{tG}_{on}\xi } ,0, \frac{ \left( \tilde{C_2} - [mGAP_{tot}] -\xi ] \right) \rho ^{tG}_{off}}{\rho ^{tG}_{off} + \rho ^{tG}_{on}\xi } , [mGAP_{tot}] \right. \Bigg ) \end{aligned}$$

\(\underline{\hbox {Case 3: }\widehat{[mG^*]}=0\hbox { and }\widehat{[tGEF^*]}=0}\)

From Eqs. 3.39 and 3.40, we have \(\widehat{[mG]}=0\) and \(\widehat{[tG^*]}=0\), respectively. Subtracting Eq. 3.38 from Eq. 3.37, in this case we get \(\widehat{[tGEF]} = \tilde{C_2} - \tilde{C_1} \ge 0\) only if \(\tilde{C_2} \ge \tilde{C_1} \). Now, from the conservation law given by Eq. 3.37, we obtain \(\widehat{[mGAP^*]} = \tilde{C_1} - \xi \), and thus, \(\widehat{[mGAP^*]} \in [0,[mGAP_{tot}]]\) only if \(\max (0,\tilde{C_1} - [mGAP_{tot}]) \le \xi \le \tilde{C_1}\). In this case, the \(\xi \)-dependent family of steady states is given by

$$\begin{aligned} \widehat{\mathbf{x }}_{\xi }= & {} \left( 0,0,\xi ,0,\tilde{C_2}-\tilde{C_1},0,\tilde{C_1} - \xi \right) . \end{aligned}$$

\(\underline{\hbox {Case 4: }\widehat{[tGEF]}=0\hbox { and }\widehat{[tGEF^*]} = 0}\)

Equation 3.40 gives \(\widehat{[tG^*]}=0\) and the conservation law given by Eq. 3.38 yields \(\widehat{[mGAP^*]} = \tilde{C_2} - \xi \). Now using Eq. 3.39 to write \(\widehat{[mG]} = \rho ^{mG}_{off} (\tilde{C_2} - \xi ) \widehat{[mG^*]} \), the conservation law given by Eq. 3.37 gives

$$\begin{aligned} \widehat{[mG^*]} = \frac{(\tilde{C_1} - \tilde{C_2})}{ 1+ \rho ^{mG}_{off}(\tilde{C_2} - \xi )} \quad \text {and} \quad \widehat{[mG]} = \frac{ \rho ^{mG}_{off} (\tilde{C_1} - \tilde{C_2}) (\tilde{C_2} - \xi ) }{ 1+ \rho ^{mG}_{off}(\tilde{C_2} - \xi )} \end{aligned}$$

and since \([mGAP^*] \in \left[ 0,[mGAP_{tot}]\right] \) and the steady states must be nonnegative, we must have

$$\begin{aligned} \max (0,\tilde{C_2} - [mGAP_{tot}]) \le \xi \le \tilde{C_2} \le \tilde{C_1}. \end{aligned}$$

The \(\xi \)-dependent family of steady states is therefore given by

$$\begin{aligned} \widehat{\mathbf{x }}_{\xi }= & {} \left( \frac{ \rho ^{mG}_{off} (\tilde{C_1} - \tilde{C_2}) (\tilde{C_2} - \xi ) }{ 1 + \rho ^{mG}_{off}(\tilde{C_2} - \xi ) }, \frac{(\tilde{C_1} - \tilde{C_2})}{ 1 + \rho ^{mG}_{off}(\tilde{C_2} - \xi )}, \xi ,0,0,0, \tilde{C_2}- \xi \right) . \end{aligned}$$

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Stolerman, L.M., Ghosh, P. & Rangamani, P. Stability Analysis of a Signaling Circuit with Dual Species of GTPase Switches. Bull Math Biol 83, 34 (2021). https://doi.org/10.1007/s11538-021-00864-w

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Keywords

  • GTPases
  • Biochemical switches
  • Network motifs
  • Stability analysis