Team:Grenoble/Modeling/Amplification/ODE
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<h1>Preliminary</h1> | <h1>Preliminary</h1> | ||
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- | Indeed, we have these types of evolution for the biological elements. The ones involved only in quick reactions are most of the time in a steady state, and there jump from one steady state to an other has an infinite speed, which | + | Indeed, we have these types of evolution for the biological elements. The ones involved only in quick reactions are most of the time in a steady state, and there jump from one steady state to an other has an infinite speed, which does not interest us. |
</section> | </section> | ||
<section> | <section> | ||
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- | < | + | <div style="margin-left: 30px;"> |
- | </ | + | <a href="#1"><img src="https://static.igem.org/mediawiki/2012/4/49/1_mod.png" alt="" /> What is the sensitivity of our system?<br/> |
- | < | + | <a href="#2"><img src="https://static.igem.org/mediawiki/2012/1/1e/2_mod.png" alt="" /> What is the time response? |
- | </ | + | <br/> |
+ | <a href="#3"><img src="https://static.igem.org/mediawiki/2012/5/57/3_mod.png" alt="" /> What steady states will our system always reach? | ||
+ | </div> | ||
+ | |||
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</section> | </section> | ||
<section> | <section> | ||
<h1>The system</h1> | <h1>The system</h1> | ||
- | Here is the schema of the real system, in orange are the reactions which | + | </br> |
+ | </br> | ||
+ | <b>We give the explanations on all the parameters used here in the <a href="https://2012.igem.org/wiki/index.php?title=Team:Grenoble/Modeling/Amplification/Sensitivity">parameters part</a>.</b> | ||
+ | </br> | ||
+ | </br> | ||
+ | Here is the schema of the real system, in orange are the reactions which did not appear in the simplified system of the overview: | ||
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<center><img src="https://static.igem.org/mediawiki/2012/8/82/Schema_system_grenoble.png" alt="" /></center> | <center><img src="https://static.igem.org/mediawiki/2012/8/82/Schema_system_grenoble.png" alt="" /></center> | ||
</br> | </br> | ||
- | cAMP is the quorum sensing molecule. When we put some cAMP out of the system, it enters into the system. Then, it complexes with CRP to create (CRP-cAMP), which is the transcription factor of the gene arac. When some | + | cAMP is the quorum sensing molecule. When we put some cAMP out of the system, it enters into the system. Then, it complexes with CRP to create (CRP-cAMP), which is the transcription factor of the gene arac. When some AraC is created, it will complex with arabinose to create AraC*. AraC* is the active form of Arac. Arac* with (CRP-cAMP) are the transcription factors of the gene cya. Then when some protein of adenylate cyclase is produced, it will catalyze the production of cAMP. |
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where <img src="https://static.igem.org/mediawiki/2012/2/22/Eq71_grenoble.png" alt="" /> | where <img src="https://static.igem.org/mediawiki/2012/2/22/Eq71_grenoble.png" alt="" /> | ||
</br> | </br> | ||
- | We have <img src="https://static.igem.org/mediawiki/2012/f/fc/Eq7_grenoble.png" alt="" /> | + | We have : <br/> |
+ | <br/> | ||
+ | <center><img src="https://static.igem.org/mediawiki/2012/f/fc/Eq7_grenoble.png" alt="" /></center> | ||
</br> | </br> | ||
- | <img src="https://static.igem.org/mediawiki/2012/b/b7/Eq8_grenoble.png" alt="" /> | + | <center><img src="https://static.igem.org/mediawiki/2012/b/b7/Eq8_grenoble.png" alt="" /></center> |
</br> | </br> | ||
Eventually, we have to choose between the two solutions | Eventually, we have to choose between the two solutions | ||
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- | (CRP-cAMP) is the transcription factor of the gene | + | (CRP-cAMP) is the transcription factor of the gene <i>araC</i>. When it appears in the network, it activates the production of the protein AraC. This is modeled by a Hill function. In addition, there is some outflow linked to the promoter paraBAD, which is the promoter regulating <i>araC</i>, thus there is a basal production of AraC. We take into account this basal production, because we need to know if because of them our system will always be turned on, thus useless. AraC is also naturally degraded by the bacterium. Thus, we get as the equation of evolution of AraC concentration: |
- | concentration: | + | |
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- | Then, the protein | + | Then, the protein AraC complexes with arabinose to create AraC active, written AraC*. It is modeled by the following chemical equation: |
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- | We get the evolution of | + | We get the evolution of AraC* r<SUB>AraC*</SUB>: |
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- | With the | + | With the QSSA, we get: |
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- | In addition we have the conservation equation of | + | In addition we have the conservation equation of AraC: |
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- | Then, | + | Then, AraC* with (CRP-cAMP) are the transcription factors of of the gene ca. When they appear in the network the protein Ca is produced. The product of two hill functions models this. For the same reasons as for AraC we take into account the basal production of the adenylate cyclase. In addition it is degraded by the bacterium. |
Thus, we get the equation: | Thus, we get the equation: | ||
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- | The QSSA enables us to have r<SUB>cAMP</SUB>=0. Then, we have: | + | The QSSA enables us to have r<SUB>cAMP</SUB> = 0. Then, we have: |
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- | We | + | We do not take into account in the equations the growth of the bacterium. Indeed, the bacterium grows as long as it has some glucose. And, as long as there is some glucose, the bacterium will not use the arabinose. However, if the bacterium does not use the arabinose, the protein AraC can not be actived, and thus no adenylate cyclase is produced. The bacterium begins to use the arabinose when the whole glucose has disappeared. But it does not grow with the arabinose. |
Indeed, the biologists in order to check the “AND gate” behavior, the biologists built, see protocol <a href="https://2012.igem.org/Team:Grenoble/Biology/Protocols/AND_test">protocol "AND gate test"</a> . Here we give the biological graphs of the absorbance and the graph of the RFU in function of the time for arabinose and cAMP maximum: | Indeed, the biologists in order to check the “AND gate” behavior, the biologists built, see protocol <a href="https://2012.igem.org/Team:Grenoble/Biology/Protocols/AND_test">protocol "AND gate test"</a> . Here we give the biological graphs of the absorbance and the graph of the RFU in function of the time for arabinose and cAMP maximum: | ||
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- | We can observe that until 100 minutes the bacterium is growing (the absorbance increases), and we | + | We can observe that until 100 minutes the bacterium is growing (the absorbance increases), and we do not get any RFU signal. Then, the bacterium almost stops growing, and thus we begin to get a signal. |
</section> | </section> | ||
<section> | <section> | ||
- | <h1><img src="https://static.igem.org/mediawiki/2012/4/49/1_mod.png" alt="" /> Sensitivity </h1> | + | <h1 id="1"><img src="https://static.igem.org/mediawiki/2012/4/49/1_mod.png" alt="" /> Sensitivity </h1> |
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- | Even if we | + | Even if we do not know the exact value of all the parameters, we have enough information on them to be able to have a good evaluation of the sensitivity of our system. |
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- | To answer this question, we plot the evolution of the steady state of adenylate cyclase in function of [cAMP<SUB>out</SUB>], the initial concentration of cAMP, which is assumed to be constant. <a href="https://static.igem.org/mediawiki/2012/f/f1/Resol_Eq_cAMP_grenoble.zip">Here</a> are the scripts that enable us to plot the graphs. We solve the differential equations to get the steady state, because if we wanted to solve a set of equations we would have had to give an initial point. If we had given 0, matlab would have stayed at this point, and we | + | To answer this question, we plot the evolution of the steady state of adenylate cyclase in function of [cAMP<SUB>out</SUB>], the initial concentration of cAMP, which is assumed to be constant. <a href="https://static.igem.org/mediawiki/2012/f/f1/Resol_Eq_cAMP_grenoble.zip">Here</a> are the scripts that enable us to plot the graphs. We solve the differential equations to get the steady state, because if we wanted to solve a set of equations we would have had to give an initial point. If we had given 0, matlab would have stayed at this point, and we could not give another initial point without solving the equations. |
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- | The sensitivity of our system is 10<SUP>-6</SUP> mol | + | The sensitivity of our system is 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span> of initial cyclic AMP. When we introduced this quantity in the system, the bacteria will turn on. |
Then, the next question is to know when we are under this value in how much time we will be able to observe that one bacteria turned on. | Then, the next question is to know when we are under this value in how much time we will be able to observe that one bacteria turned on. | ||
</section> | </section> | ||
<section> | <section> | ||
- | <h1><img src="https://static.igem.org/mediawiki/2012/1/1e/2_mod.png" alt="" /> Temporal evolution</h1> | + | <h1 id="2"><img src="https://static.igem.org/mediawiki/2012/1/1e/2_mod.png" alt="" /> Temporal evolution</h1> |
</br> | </br> | ||
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- | To evaluate the time it will take to be able to detect a signal, we need to plot the evolution of the adenylate cyclase in the time for an initial concentration of cAMP<SUB>out</SUB> | + | To evaluate the time it will take to be able to detect a signal, we need to plot the evolution of the adenylate cyclase in the time for an initial concentration of cAMP<SUB>out</SUB> ≥ 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span>. We first give the graph with cAMP<SUB>out</SUB> = 10<SUP>-3</SUP> mol.L<span class="exposant">-1</span> : |
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- | Then, we want to see the behavior of the system around the threshold. We give the evolution of the adenylate cyclase in the time in function with cAMP<SUB>out</SUB>=10<SUP>-6</SUP> mol | + | Then, we want to see the behavior of the system around the threshold. We give the evolution of the adenylate cyclase in the time in function with cAMP<SUB>out</SUB> = 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span> : |
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- | Here, to be able to begin to detect a signal, we should wait around 1300 minutes. So even if our system can detect | + | Here, to be able to begin to detect a signal, we should wait around 1300 minutes. So even if our system can detect cAMP<SUB>out</SUB> = 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span>, we may not be able to say if it is a real detection or a false positive. We will be able to answer this question with the stochastic part. |
</br> | </br> | ||
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- | Then we give one temporal evolution of the adenylate cyclase for cAMP<SUB>out</SUB>=10<SUP>-8</SUP> mol | + | Then we give one temporal evolution of the adenylate cyclase for cAMP<SUB>out</SUB> = 10<SUP>-8</SUP> mol.L<span class="exposant">-1</span>. It is bellow the threshold, but because of the basal values, we want to see exactly what happens. |
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<b>Conclusion:</b> | <b>Conclusion:</b> | ||
The more cAMP<SUB>out</SUB> is introduced in the system, the fastest the answer is. The quickest answer would be 200 minutes to reach half of the maximal expression of ca enabling us to get a signal. | The more cAMP<SUB>out</SUB> is introduced in the system, the fastest the answer is. The quickest answer would be 200 minutes to reach half of the maximal expression of ca enabling us to get a signal. | ||
- | Because of the basal values, the adenylate cyclase is always expressed. Thus, we will make a steady state study of the system. This is useful seeing that with the sensitivity graph we | + | Because of the basal values, the adenylate cyclase is always expressed. Thus, we will make a steady state study of the system. This is useful seeing that with the sensitivity graph we could not see the low expression of adenylate cyclase and it is only in the temporal part that we could notice it, so we need a real study. |
- | Then, the stochastic part will be really important to be sure that there | + | Then, the stochastic part will be really important to be sure that there will not be too many false positives because of these basal values. |
</section> | </section> | ||
<section> | <section> | ||
- | <h1><img src="https://static.igem.org/mediawiki/2012/5/57/3_mod.png" alt="" />Steady state study</h1> | + | <h1 id="3"><img src="https://static.igem.org/mediawiki/2012/5/57/3_mod.png" alt="" />Steady state study</h1> |
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- | Here, we want to find the steady states of the system, | + | Here, we want to find the steady states of the system, to understand why our system works. At the steady states, the derivatives of our system are null: |
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- | Equation (1) gives us | + | Equation (1) gives us AraC in function of Ca. If we plot this function, we plot all the steady states of AraC, while Ca is varying. Equation (2) gives us Ca in function of AraC. If we plot this function, we plot all the steady states of Ca while AraC is varying. These functions are called the isoclines of the system. The intersections of these two graphs represent the steady states of the system. |
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- | <a href="https://static.igem.org/mediawiki/2012/9/9e/Steadys_state_study.zip">Here</a> you can find the scripts we worked with in this part. First, I give the isoclines with cAMP<SUB>init</SUB>=10<SUP>-5</SUP> mol | + | <a href="https://static.igem.org/mediawiki/2012/9/9e/Steadys_state_study.zip">Here</a> you can find the scripts we worked with in this part. First, I give the isoclines with cAMP<SUB>init</SUB> = 10<SUP>-5</SUP> mol.L<span class="exposant">-1</span>. |
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<center><img src="https://static.igem.org/mediawiki/2012/d/dc/Graphe7_ampli_grenoble.png" alt="" /></center> | <center><img src="https://static.igem.org/mediawiki/2012/d/dc/Graphe7_ampli_grenoble.png" alt="" /></center> | ||
</br> | </br> | ||
- | Isoclines of Ca and | + | Isoclines of Ca and AraC with cAMP<SUB>init</SUB> = 10<SUP>-5</SUP> mol.L<span class="exposant">-1</span>. |
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- | In black we have Ca isocline, in purple, | + | In black we have Ca isocline, in purple, AraC one, it will always be the same color code. We notice that there is only one steady state. It is high enough to enable us to get a signal. |
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- | Then I give the isoclines with cAMP<SUB>init</SUB>=10<SUP>-6</SUP> mol | + | Then I give the isoclines with cAMP<SUB>init</SUB> = 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span>. |
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<center><img src="https://static.igem.org/mediawiki/2012/4/49/Graphe8_ampli_grenoble.png" alt="" /></center> | <center><img src="https://static.igem.org/mediawiki/2012/4/49/Graphe8_ampli_grenoble.png" alt="" /></center> | ||
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- | Ca and | + | Ca and AraC isoclines with cAMP<SUB>init</SUB> = 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span>. |
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<center><img src="https://static.igem.org/mediawiki/2012/0/08/Graphe9_ampli_grenoble.png" alt="" /></center> | <center><img src="https://static.igem.org/mediawiki/2012/0/08/Graphe9_ampli_grenoble.png" alt="" /></center> | ||
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- | Ca and Arac isoclines with cAMP<SUB>init</SUB>=10<SUP>-6</SUP> mol | + | Ca and Arac isoclines with cAMP<SUB>init</SUB> = 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span>, zoom around 0. |
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- | Eventually, I give the isoclines with cAMP<SUB>init</SUB>=10<SUP>-7</SUP> mol | + | Eventually, I give the isoclines with cAMP<SUB>init</SUB> = 10<SUP>-7</SUP> mol.L<span class="exposant">-1</span>: |
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<center><img src="https://static.igem.org/mediawiki/2012/8/85/Graphe10_ampli_grenoble.png" alt="" /></center> | <center><img src="https://static.igem.org/mediawiki/2012/8/85/Graphe10_ampli_grenoble.png" alt="" /></center> | ||
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- | Ca and | + | Ca and AraC isoclines with cAMP<SUB>init</SUB> = 10<SUP>-7</SUP> mol.L<span class="exposant">-1</span>. |
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<center><img src="https://static.igem.org/mediawiki/2012/6/60/Graphe11_ampli_grenoble.png" alt="" /></center> | <center><img src="https://static.igem.org/mediawiki/2012/6/60/Graphe11_ampli_grenoble.png" alt="" /></center> | ||
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- | Ca and | + | Ca and AraC isoclines with cAMP<SUB>init</SUB> = 10<SUP>-7</SUP> mol.L<span class="exposant">-1</span>, zoom around 0. |
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- | To actually | + | To actually understand why our system works, we need to study the stability of our steady states. If they are stable when our system stops to one of these steady states, it will always stay there. It works like this: |
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- | - cAMP<SUB>init</SUB>=10<SUP>-5</SUP> mol | + | - cAMP<SUB>init</SUB> = 10<SUP>-5</SUP> mol.L<span class="exposant">-1</span> : |
</br> | </br> | ||
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- | + | AraC steady state = 0.167058129527727 10<SUP>-4</SUP> mol.L<span class="exposant">-1</span> | |
</br> | </br> | ||
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- | Ca seady states = 10<SUP>-6</SUP> | + | Ca seady states = 0.1837444563636 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span> |
</br> | </br> | ||
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- | λ<SUB>1</SUB>= -0.006000000912526 λ<SUB>2 </SUB>= -0.005763188664176 | + | λ<SUB>1</SUB> = -0.006000000912526 λ<SUB>2</SUB> = -0.005763188664176 |
</br> | </br> | ||
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- | - cAMP<SUB>init</SUB>=10<SUP>-6</SUP> mol | + | - cAMP<SUB>init</SUB> = 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span> : |
</br> | </br> | ||
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- | + | AraC steady state = 0.166879570344986 10<SUP>-4</SUP> mol.L<span class="exposant">-1</span> | |
</br> | </br> | ||
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- | Ca seady states = 10<SUP>-6</SUP> | + | Ca seady states = 0.1832826298080 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span> |
</br> | </br> | ||
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- | λ<SUB>1</SUB>= -0.006000000910603 λ<SUB>2</SUB>= -0.005745344108236 | + | λ<SUB>1</SUB>= -0.006000000910603   λ<SUB>2</SUB>= -0.005745344108236 |
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- | - cAMP<SUB>init</SUB>=10<SUP>-7</SUP> mol | + | - cAMP<SUB>init</SUB> = 10<SUP>-7</SUP> mol.L<span class="exposant">-1</span> : |
</br> | </br> | ||
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- | + | AraC steady state = 0.182361098919416 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span> | |
</br> | </br> | ||
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- | Ca seady states = 10<SUP>-9</SUP> | + | Ca seady states = 0.249177541683 10<SUP>-9</SUP> mol.L<span class="exposant">-1</span> |
</br> | </br> | ||
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- | λ<SUB>1</SUB>= -0.006000006994365 λ<SUB>2 </SUB>= -0.002117175391388 | + | λ<SUB>1</SUB>= -0.006000006994365   λ<SUB>2 </SUB>= -0.002117175391388 |
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<b>Conclusion:</b> | <b>Conclusion:</b> | ||
- | Now, we can be sure that our system | + | Now, we can be sure that our system will not always be turned on. In function of the quantity of initial cAMP, our system will stay at a low or a high steady state. |
</section> | </section> | ||
<section> | <section> | ||
<h1>Conclusion</h1> | <h1>Conclusion</h1> | ||
- | The sensitivity of our system is around 10<SUP>-7</SUP> mol | + | The sensitivity of our system is around 10<SUP>-7</SUP> mol.L<span class="exposant">-1</span>. To be able to know if it is satisfying, we need to link it with the signaling part. |
- | In addition, to know if our system is fast we need to link this part with the signaling too. | + | In addition, to know if our system is fast we need to link this part with the signaling too. That is what we are going to do in the next part. |
</section> | </section> | ||
</div> | </div> |
Latest revision as of 03:47, 27 September 2012