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Basic Bistable

Introduction

To start with the model analysis, we show the reactions in which our plasmid is involved into the ‘E. Coli’ bacteria, and their kinetic laws. Due to the two expressions, each one must have the same rate law. Here we are showing a simplified analysis for every response:

We have the expression of two different proteins, expressions that are regulated by following reactions:

TRANSCRIPTION: RNAp + Promoter → RNAp + Promoter + mRNA

The mRNA production will be described using Michaelis-Menten kinetics.

TRANSLATION: mRNA + Ribosomes → mRNA + Ribosomes + Repressor

The Repressor production will be also modeled using Michaelis-Menten kinetics.

REPRESSION: Promoter + n · Repressor → Promoter_Repressor_n

Finally we use a Hill kinetics model for the transcription inhibition, which offers an approximation to the cooperative action.

Diagram

Here we can see all the species involved in the model for the bistable.

Equations

We define the reactions and the parameters involved in the system.

The following list shows a summary of the species:

Species:

• promoter1
• promoter2
• mRNA1
• mRNA2
• repressor1/r1
• repressor2
• RNAp
• ribosomes

Parameters:

• Km : Michaelis constant of transcription
• Kmu: Michaelis constant of translation
• δ: mRNA rate of degradation
• λ: mRNA maximum rate of synthesis
• k: Repressor rate of synthesis
• Kiu: Dissociation constant of repression
• γ: Cooperativity of inhibition by the repressor

Before describing the reactions’ equations, we describe the assumptions made:

• Repression is modeled as an inhibition of the mRNA synthesis. Instead of considering the repression as a different process, we include this effect into the Michaelis-Menten equation as a modifier of the Michaelis constant Km, assuming that the protein binds competitively with the RNAp to the promoter site, and knowing that the repressor may form multimers.
• Due to the high values of the RNAp and ribosomes concentration versus promoter concentration, we can assume that the Michaelis-Menten rate law may change into a mass action rate law. Although this assumption is valid for the transcription process, the rate constant changes his value according to the repressor level.

Transcription + Repression 1:

RNAp + promoter1 + repressor1 → RNAp + promoter1 + repressor1 + mRNA1

Transcription + Repression 2:

RNAp + promoter2 + repressor2 → RNAp + promoter2 + repressor2 + mRNA2

Tanslation 1:

Translation 2:

mRNA1/mRNA2 Degradation:

mRNA1 → null

mRNA2 → null

Repressor1/Repressor2 Degradation:

repressor1 → null

repressor2 → null

Simulations

When we initially consider the same concentrations of each species, if we let the system evolve the simulation offers the following result:

The repressor2 wins the competition: this effect is due to the stronger expression of the repressor2

But if we establish a higher initial concentration to the repressor1 mRNA:

Now, repressor1 wins. We can see that the maximum level reached by repressor1 is lower.

With these simulations we can see how the system is able to work on two different points.