# Stability analysis of a reduced transcription-translation model of RNA polymerase and ribosomes

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## The full closed reversible Michaelis-Menten system

In 1913, Michaelis and Menten studied the kinetics of a simple enzymatic reaction involving a single enzyme. Consider the reaction consisting of a substrate S, an enzyme E and a product P. Michaelis and Menten proposed the following description and equations (we refer to  and ). The enzyme forms a transitory complex C before returning to its original form, giving product P from substrate S. The reactions are described by: S + E k1 k􀀀1 C k2* P + E.
However, (see ), in principle, all reactions catalyzed by enzymes are reversible (similarly to chemical reactions), and this fact could play a prominent role in biochemistry. So considering the last step of the above model as a reversible reaction, the reaction scheme becomes: S + E k1 k􀀀1 C k2 k􀀀2 P + E.

### Global stability of the equilibrium using Lyapunov function

In the following we study the global stability of this equilibrium. The Jacobian of the system (2.2.2) matrix is J(s; c; p) This matrix is compartmental. We recall in the appendix the denition of a compartmental matrix, used to describe compartmental systems . Remember that c < E0, therefore the diagonal elements of J(s; c; p) are negative and the o-diagonal elements nonnegative. Moreover, the last proposition (A.3.3) is also veried because the system is closed. Therefore the Jacobian matrix is compartmental. The graph of the Jacobian matrix is given in gure 2.2.1, therefore it is irreducible (see appendix A.4). Then the following theorem applies: Remark 2.1. In the case that c = E0, the Jacobian matrix is not irreducible any more. But we showed above that this point is never attained. Theorem 2.2. Proposition 5 in  Let M(s; c; p) = s + c + p the (xed) total concentration of the closed system. If the Jacobian matrix of the system is irreducible and compartmental, then for any M0 > 0, hyperplane H0 = f(s; c; p) 2 <3 + : M(s; c; p) = M0 > 0g is forward invariant and contains a unique globally asymptotically stable equilibrium in H0. We obtain the proposition: Proposition 2.3. In the invariant hyperplane M0 = s+c+p, the equilibrium of (2.2.2) is unique and globally asymptotically stable.

#### Global stability using monotone systems theory

We proved using a Lyapunov function (see above) that the equilibrium (if it exist, u < k2E0), is globally asymptotically stable. We want now to show the stability with another method, with a basin of attraction as large as possible with fullls the constraints on the variables (they are nonnegative and c < E0). To do so, we use a simple property of cooperative (or monotone) systems. The theory of monotone systems has strong links with compartmental systems, but it is more general. The basic facts and the theorem we need are recalled in the appendix.
Proof. Monotone systems conserve the partial ordering of two solutions; if there is a point greater than the equilibrium point (with respect to the usual partial ordering) where all the derivatives are nonpositive, then the trajectory issued from this point is always decreasing, and converges toward the equilibrium point. Similarly, if there is a point smaller than the equilibrium where the derivatives are nonnegative, then the solution from this point increases until the equilibrium. The whole hyperrectangle built with these two points is invariant, and all the trajectories initiating in this rectangle converge toward the equilibrium.

The switch between two regimes: the equilibrium exists and do not exist

To see what happen if there is a switch between these two regimes (equilibrium exists or not), we chose a varying input u(t), having two values corresponding to the existence, or not, of an equilibrium. By varying the frequency of commutation between the two inputs, we may obtain a limit cycle which is globally stable (see Fig.2.3.4). The solution could also becomes unbounded (see Fig.2.3.5) in the case that the time interval when there is no equilibrium is longer than the time interval when there is equilibrium. These results are obtained by numerical simulations, but we think that some proofs are possible.

Remerciements
Abstract
Articles
Contents
1 Modelisations des reactions metaboliques et reseaux de regulations genetiques
1.1 Introduction
1.2 Principe de regulation cellulaire
1.2.2 Le dogme central de la biologie cellulaire
1.2.3 La transcription de l’ADN en ARN messager
1.2.4 La traduction de l’ARN messager en proteine
1.3 La modelisation des reactions enzymatiques
1.4 L’equation de vitesse du mecanisme irreversible de Michaelis-Menten .
1.5 Reseaux de regulation genetique
1.5.1 Les dierentes types de modelisation
1.5.2 Approche de modelisation par equations dierentielles
1.5.3 Modelisation par equations dierentielles continue
1.5.3.1 Exemple d’un modele classique d’expression genetique
1.5.3.2 Exemple d’un modele d’un reseau genetique oscillant
1.6 Modelisation par equations dierentielles anes par morceaux
1.6.1 Les points focaux
1.6.2 Solution au sens de Filippov au niveau des surfaces discontinues .
1.6.3 Point d’equilibre au sens de Filippov
1.6.4 Stabilite de l’equilibre au sens de Filippov
1.7 Conclusion
2 Stability Analysis of Michaelis-Menten models and reduction of coupled metabolic-genetic systems.
2.1 Introduction
2.2 The full closed reversible Michaelis-Menten system
2.2.1 Equilibrium
2.2.2 Global stability of the equilibrium using Lyapunov function
2.3 The full open reversible Michaelis-Menten system
2.3.1 Equilibrium
2.3.2 Global stability of the equilibrium
2.3.2.1 Global stability using a Lyapunov function
2.3.2.2 Global stability using monotone systems theory
2.3.3 The switch between two regimes: the equilibrium exists and do not exist
2.4 The closed enzymatic chain
2.5 Open enzymatic chain with one input and output
2.6 Coupling metabolic and genetic systems
2.7 Time-scale reduction of metabolic-genetic systems
2.7.1 Product inhibits the production of enzymes
2.7.2 Product activates the production of enzymes
2.7.3 Activation with the substrate of the enzyme
2.8 Conclusion
3 Transcription translation coupled models for gene expression with in puts.
3.1 Introduction
3.2 The gene transcription model
3.2.1 The « closed » transcription model
3.2.2 Description of the model
3.2.3 Equilibrium
3.2.4 Global stability of the equilibrium of the closed transcription model
3.2.5 The « open » transcription model
3.2.7 Equations of the model
3.2.8 Equilibrium
3.2.9 Global stability of the equilibrium of the open transcription model
3.3 The closed translation model
3.4 A coupled transcription-translation model
3.4.1 Equations of the model
3.4.2 Stability of the coupled model
3.5 Conclusion
4 Reduction and stability analysis of a transcription-translation model of RNA polymerase
4.1 Introduction
4.2 The coupled transcription-translation model of RNA polymerase
4.2.1 Description of the model
4.2.2 Full equations
4.3 Time-scale reduction (Fast-Slow Behavior)
4.3.1 Parameters values for the coupled transcription translation models of RNA polymerase
4.3.2 Separate the full system into \fast » and \slow » variables
4.3.3 Fast-Slow subsystems of the full system
4.4 Checking conditions of Tikhonov’s Theorem for the fast subsystems
4.5 Applying Tikhonov’s Theorem
4.6 The Reduced System
4.7 Simulations of the full and the reduced system
4.8 Dynamical study of the Reduced System
4.8.1 Equilibria of the reduced system
4.8.2 Study of the stability of equilibria
4.8.2.1 Stability of the equilibrium (0; 0) in the case that it is the unique equilibrium of the system
4.8.2.2 Stability in the case that two equilibria exist, which are (0; 0), and a unique, positive (z; q)
4.9 Generalization with qualitative functions
4.10 Generalization: stability of monotone system with a decreasing Jacobian matrix
4.11 Conclusions
5 Stability analysis of a reduced transcription-translation model of RNA polymerase and ribosomes
5.1 Introduction
5.2 A more general model with ribosome synthesis
5.3 Applying Tikhonov’s Theorem
5.4 The reduced system
5.5 Dynamical study of the reduced system
5.5.1 Equilibria of the reduced system
5.5.2 Stability of the equilibria
5.5.2.1 Stability of the origin
5.5.2.2 Stability of the second equilibrium which is near to the origin
5.5.2.3 Stability of the third equilibrium which is far from the origin
5.5.3 The global stability of the equilibria
5.6 Conclusions
6 Control of small genetic networks systems
6.1 Introduction
6.2 The genetic oscillator
6.3 The genetic oscillator with control: rst approach
6.4 The genetic oscillator with control: second approach
6.5 Qualitative control for a negative ODE loop
6.6 Conclusion
7 Conclusions et perspectives
Appendix
A Monotone And Compartmental Systems
A.1 Monotone Systems
A.2 Invariance of the Positive Orthant
A.3 Matrices and Compartmental systems
A.4 Theorem on monotone and concave systems
B The Frobenius-Perron theorem
B.1 The Frobenius-Perron theorem
C Tikhonov’s Theorem
C.1 Tikhonov’s Theorem 
Bibliography

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