Transition between monostability and bistability of a genetic toggle switch in Escherichia coli

Jie Li; Weinian Zhang

doi:10.3934/dcdsb.2020007

Article Contents

2020, Volume 25, Issue 5: 1871-1894. Doi: 10.3934/dcdsb.2020007

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Transition between monostability and bistability of a genetic toggle switch in Escherichia coli

Jie Li and
Weinian Zhang^,

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

^* Corresponding author: Weinian Zhang
^* Corresponding author: Weinian Zhang

Received: December 31, 2018

Revised: June 30, 2019

Early access: December 2019

Published: May 2020

Supported by NSFC grants #11771307, #11831012 and #11821001

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Abstract

In this paper, we investigate a genetic toggle switch in Escherichia Coli, which models an artificial double-negative feedback loop with two mutually repressors. This model is a planar differential system with three parameters, one of which is an integer power $ n\ge1 $, in the case that repressors 1 and 2 multimerize with $ n $ and 1 subunits respectively and its equilibria are decided by a polynomial of degree $ n+1 $. Since one hardly solves such a polynomial equation, a known result on bistability was given by omitting some small terms under the assumption that the promoters are strong and the expression ratio between the ON state and the OFF state is large. In this paper, determining distribution of zeros qualitatively for the polynomial of high degree, we analytically discuss on the system without the assumption and completely give qualitative properties for all equilibria, which corrects the known result of bistability. Furthermore, we prove that there may occur in the system a codimension 2 bifurcation, called cusp bifurcation, which is a collision of two saddle-node bifurcations and manifests the transition between bistability and monostability. We exhibit the global dynamics of repressors in various cases by analyzing equilibria at infinity and proving nonexistence of closed orbits.

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Mathematics Subject Classification: Primary: 34C05, 37G10; Secondary: 34D23, 92D20.

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Figure 1. Genetic toggle switch

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Figure 3. One intersection

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Figure 4. Three intersections

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Figure 2. One intersection

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Figure 15. Critical conditions

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Figure 16. a = 2.8, b = 3

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Figure 17. a = 1.8, b = 3

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Figure 5. $ \Phi'(u_2)>0 $

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Figure 6. $ \Phi'(u_2) = 0 $

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Figure 7. $ \Phi'(u_2)<0 $

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Figure 8. (S1)

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Figure 9. (S2)

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Figure 10. (S3)

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Figure 11. (S4)

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Figure 12. (S5)

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Figure 14. Global phase portraits

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Figure 13. Invariant manifolds for systems (45) and (47)

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Table 1. Distribution of zeros $ u_{01} $ and $ u_{02} $ of $ \Phi' $

(C3-1-1)	$ 0<u_{01}<\zeta_-<\tilde{u}_2<\zeta_+<u_{02}<a $,
(C3-1-2)	$ 0<u_{01}<\zeta_-<\tilde{u}_2<u_{02}=\zeta_+<a $,
(C3-1-3)	$ 0<u_{01}<\zeta_-<\tilde{u}_2<u_{02}<\zeta_+<a $,
(C3-2-1)	$ 0<u_{01}=\zeta_-<\tilde{u}_2<u_{02}<\zeta_+<a $,
(C3-2-2)	$ 0<u_{01}=\zeta_-<\tilde{u}_2<u_{02}=\zeta_+<a $,
(C3-2-3)	$ 0<u_{01}=\zeta_-<\tilde{u}_2<\zeta_+<u_{02}<a $,
(C3-3-1)	$ 0<\zeta_-<u_{01}<\tilde{u}_2<\zeta_+<u_{02}<a $,
(C3-3-2)	$ 0<\zeta_-<u_{01}<\tilde{u}_2<u_{02}=\zeta_+<a $,
(C3-3-3)	$ 0<\zeta_-<u_{01}<\tilde{u}_2<u_{02}<\zeta_+<a $.

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