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Team:UPO-Sevilla/Project/Basic Flip Flop/Modeling/Basic Bistable
From 2011.igem.org
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| - | <h1>Basic | + | <h1>Basic Flip-Flop</h1> |
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<h2>Introduction</h2> | <h2>Introduction</h2> | ||
| + | <p>Firstly, we have developed a mathematical model of the evolution of the species involved in the basic flip-flop. This model is composed of a set of differential equations that relate the change of rate of the concentrarions of the different substances with the concentrations of rest of substances. These models have been then implemented and simulated using MATLAB and MATLAB's Simbiology.</p> | ||
<p>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:</p> | <p>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:</p> | ||
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| + | <p>The reactions considered, and the rates of change of the produced species are listed here:</p> | ||
<p><strong>Transcription + Repression 1:</strong></p> | <p><strong>Transcription + Repression 1:</strong></p> | ||
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<p><strong>Tanslation 1:</strong></p> | <p><strong>Tanslation 1:</strong></p> | ||
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<p><strong>Translation 2:</strong></p> | <p><strong>Translation 2:</strong></p> | ||
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<img src="https://static.igem.org/mediawiki/2011/c/c5/UPO-BBEq6.png" alt="Repressors Degradation Rates" /> | <img src="https://static.igem.org/mediawiki/2011/c/c5/UPO-BBEq6.png" alt="Repressors Degradation Rates" /> | ||
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| + | <p>You can download the full Simbiology model of the Basic Flip-Flop here:</p> | ||
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| + | <a href="https://static.igem.org/mediawiki/2011/9/9c/Basic_bistable.sbproj.zip" title="Basic Flip-Flop"><img style="margin: 0em 0em 1em 0em" class="imgcenter" width="200px" src="https://static.igem.org/mediawiki/2011/9/97/UPOSevillaDownloadIcon.png" title="Download Basic Flip-Flop"></a> | ||
<h2>Simulations</h2> | <h2>Simulations</h2> | ||
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<img src="https://static.igem.org/mediawiki/2011/5/51/BISTABLE.jpg" alt="Repressors Degradation Rates" /> | <img src="https://static.igem.org/mediawiki/2011/5/51/BISTABLE.jpg" alt="Repressors Degradation Rates" /> | ||
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| + | <p>Once the basic operation is described, next we consider the inclusions of actuators to modify the state of the flip-flop, in the <a href="/Team:UPO-Sevilla/Project/Basic_Flip_Flop/Modeling/Toggle_Switch">Toggle Switch</a> section.</p> | ||
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Latest revision as of 22:03, 27 October 2011
Basic Flip-Flop
Introduction
Firstly, we have developed a mathematical model of the evolution of the species involved in the basic flip-flop. This model is composed of a set of differential equations that relate the change of rate of the concentrarions of the different substances with the concentrations of rest of substances. These models have been then implemented and simulated using MATLAB and MATLAB's Simbiology.
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_Repressorn
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.
The reactions considered, and the rates of change of the produced species are listed here:
Transcription + Repression 1:
RNAp + promoter1 + repressor1 → RNAp + promoter1 + repressor1 + mRNA1
Transcription + Repression 2:
RNAp + promoter2 + repressor2 → RNAp + promoter2 + repressor2 + mRNA2
Tanslation 1:
mRNA1 + ribosomes → mRNA1 + ribosomes + repressor2
Translation 2:
mRNA2 + ribosomes → mRNA2 + ribosomes + repressor1
mRNA1/mRNA2 Degradation:
mRNA1 → null
mRNA2 → null
Repressor1/Repressor2 Degradation:
repressor1 → null
repressor2 → null
You can download the full Simbiology model of the Basic Flip-Flop here:
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.
Once the basic operation is described, next we consider the inclusions of actuators to modify the state of the flip-flop, in the Toggle Switch section.
Sponsors
