Theoretical study of competitive systems for the modelling of biological switches 

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Study of the feedback effect from miR-200/ZEB onto miR34/SNAIL

We study (Fig 8) the effect of feedback from miR-200/ZEB to miR34/SNAIL, modeled by H􀀀 Z;34 in (4). The absence of feedback does not affect the system hysteresis, and preserves the tristability for medium values of I.
This shows that we can neglect the feedback from miR-200/ZEB to miR-34/SNAIL, without impacting either the tristability of the sytem or the hysteresis. This allows us to study the miR-34/SNAIL circuit and the miR-200/ZEB circuit independently, by considering S as a driving signal in the second circuit.

From the ODE with three variables to the ODE with two variables

The purpose of this section is to reduce the miR-34/SNAIL and the miR-200/ZEB models which are ODEs with three variables, to ODEs with two variabes.
We consider again the MBC model, and we assume that mRNA quickly reaches equilibrium before substantial changes of the miR and the protein level. This is biologically relevant, because the intrinsic degradation rate of the mRNA is about five to ten times faster than that of the miR and the protein. Thus, we consider that the second equation of the ODE (MBC) is at equilibrium. Hence, by defining: m() := gm Ym() + km .

Theoretical study of competitive systems for the modelling of biological switches

The purpose of this section is to study competitive systems that model toggle switches. We have focused on competitive systems of the form:
( x_ = gxH1(y) 􀀀 kxx y_ = gyH2(x) 􀀀 kyy ; (8) where gx; gy > 0; kx; ky > 0, and H1 and H2 are two decreasing Hill functions with parameters n1; 1; z01 and n2; 2; z02 respectively. We recall that Hill functions are functions of the shape H : R+ ! R+ z 7! 1 + ( z z0 )n 1 + ( z z0 )n .
The purpose of this second section is to answer the following questions:
Can system (8) have more than three equilibrium point? Can it have more than two stable equilibria?
For which parameters is the system bistable? For which parameters is the system monostable? The results of our research are stated in the following two theorems:
Theorem. Let H be a decreasing Hill function, with parameters, n; ; z0; and let us consider the following symmetric ODE.

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Table of contents :

Introduction
1 Numerical study of a model for the reversible Epithelial–Mesenchymal Transition 
1.1 Model Description
1.1.1 miR-34/SNAIL circuit
1.1.2 miR-200/ZEB circuit
1.1.3 Coupled circuit
1.2 System simplification
1.2.1 Study of the feedback effect from miR-200/ZEB onto miR34/SNAIL
1.2.2 From the ODE with three variables to the ODE with two variables
2 Theoretical study of competitive systems for the modelling of biological switches 
2.1 Existence, uniqueness and behaviour of solutions
2.2 Study of a simple case: a symmetric system with Hill functions
2.2.1 General results for symmetric systems
2.2.2 Symmetric systems with Hill functions
2.3 Some results in the asymmetric case, with Hill functions
2.3.1 General results for asymmetric systems
2.3.2 Asymmetric systems with Hill functions
Discussion
Appendix
References 

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