Team:UT-Tokyo/Counter
From 2014.igem.org
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<p>First of all, we constructed the deterministic model to estimate the behavior of the counter. In this model, chemical reactions are discribed as differential equations and concentration of reaction product can be calculated by those of reactants. This model is intutive, simple and hence popular to estimate the result of experiment. We could therefore get some parameters for modelling of counter from previous works.[→ parameter]</p> | <p>First of all, we constructed the deterministic model to estimate the behavior of the counter. In this model, chemical reactions are discribed as differential equations and concentration of reaction product can be calculated by those of reactants. This model is intutive, simple and hence popular to estimate the result of experiment. We could therefore get some parameters for modelling of counter from previous works.[→ parameter]</p> | ||
<p>We had simplified the counstruction of mathematical model before described time evolution in which concentrations of mRNAs and proteins change as differential equations. First, we regarded that the reaction between taRNA(transactivating RNA) and crRNA(cis-repressor RNA) in riboregulator is much faster than that of transcription or translation and equilibrium reaction. This diminution of parameters enable us to use the equilibrium constant as a parameter and prevent us from overfitting when we adapt this model to raw data.</p> | <p>We had simplified the counstruction of mathematical model before described time evolution in which concentrations of mRNAs and proteins change as differential equations. First, we regarded that the reaction between taRNA(transactivating RNA) and crRNA(cis-repressor RNA) in riboregulator is much faster than that of transcription or translation and equilibrium reaction. This diminution of parameters enable us to use the equilibrium constant as a parameter and prevent us from overfitting when we adapt this model to raw data.</p> | ||
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<p>We decided to describe mRNAs and the coupling of taRNA and crRNA as stated above. Subscript mean coding sequence of its mRNA. We regarded that the affinity of one riboregulators which the counter had was equal to that of the other. The dissociation constant of equilibrium reaction was therefore shown as following.</p> | <p>We decided to describe mRNAs and the coupling of taRNA and crRNA as stated above. Subscript mean coding sequence of its mRNA. We regarded that the affinity of one riboregulators which the counter had was equal to that of the other. The dissociation constant of equilibrium reaction was therefore shown as following.</p> | ||
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<p>Using dissociation constant, concentrations of reaction products such as [mcr<sub>cr-σ</sub>] could be discribed as function of those of taRNA and mRNA of σ and GFP. We put X, A and B as the total quantity of taRNA, sigma and GFP.</p> | <p>Using dissociation constant, concentrations of reaction products such as [mcr<sub>cr-σ</sub>] could be discribed as function of those of taRNA and mRNA of σ and GFP. We put X, A and B as the total quantity of taRNA, sigma and GFP.</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/0/0b/Ono_%283%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/0/0b/Ono_%283%29.png" height="50px" class = "math" /> |
- | <img src = "https://static.igem.org/mediawiki/2014/0/04/Ono_%284%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/0/04/Ono_%284%29.png" height="50px" class = "math" /> |
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<p>Using these equations((3)-(7)) and equilibrium constant, concentrations of binding taRNA or not mRNA coding sigma and GFP were discribed as following. These are all of simplifications.</p> | <p>Using these equations((3)-(7)) and equilibrium constant, concentrations of binding taRNA or not mRNA coding sigma and GFP were discribed as following. These are all of simplifications.</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/7/71/Ono_%286%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/7/71/Ono_%286%29.png" height="50px" class = "math" /> |
- | <img src = "https://static.igem.org/mediawiki/2014/6/67/Ono_%287%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/6/67/Ono_%287%29.png" height="50px" class = "math" /> |
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<p>Finally, we built up differential equations about concentrations of reaction products including mRNA of sigma which has no riboregulator. (It makes positive feedback loop.) We hypothesized relationship between promoter and the amount of transcriptional product increasing per unit time. The amount is in proportion to the number of promoter if the promoter expressed constitutively and is determined by Hill equation if the inducer controled its promoter. We also hypothesized propotional connection between decomposition amount of mRNA and protein and concentration of that. Some of used parameters were cited from references.[1]~[6]</p> | <p>Finally, we built up differential equations about concentrations of reaction products including mRNA of sigma which has no riboregulator. (It makes positive feedback loop.) We hypothesized relationship between promoter and the amount of transcriptional product increasing per unit time. The amount is in proportion to the number of promoter if the promoter expressed constitutively and is determined by Hill equation if the inducer controled its promoter. We also hypothesized propotional connection between decomposition amount of mRNA and protein and concentration of that. Some of used parameters were cited from references.[1]~[6]</p> | ||
<p>We aimed to determine parameters about sigma through experiment and used provisonal parameter deter- mined in reference to other promotor.</p> | <p>We aimed to determine parameters about sigma through experiment and used provisonal parameter deter- mined in reference to other promotor.</p> | ||
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<p>The amount of sigma mRNA transcribed in positive feedback loop and that of anti sigma mRNA transcribed by IPTG induction to reset the counterwere described as following.</p> | <p>The amount of sigma mRNA transcribed in positive feedback loop and that of anti sigma mRNA transcribed by IPTG induction to reset the counterwere described as following.</p> | ||
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<p>In our project, IPTG induction was aimed at enough production of anti-sigma to reset the counter and the sensitivity of lac promoter was not our main interest. Therefore, we used simple equation,(15) to describe how lac promoter behave. <I>P<sub>lac</sub></I> depend on the concentration of IPTG but we regarded it as a fixed number in this modeling.</p> | <p>In our project, IPTG induction was aimed at enough production of anti-sigma to reset the counter and the sensitivity of lac promoter was not our main interest. Therefore, we used simple equation,(15) to describe how lac promoter behave. <I>P<sub>lac</sub></I> depend on the concentration of IPTG but we regarded it as a fixed number in this modeling.</p> | ||
<p>Taking into account that translation coincide with transcription in prokaryotes, we hypothesized linear relationship between transcriptional product and the amount of translational product increasing per unit time and that this relationship does not depend on the kind of translational product. We also hypothesized that anti-sigma combine with sigma and form inert matter, and the reaction velocity of that is proportional to product of these.</p> | <p>Taking into account that translation coincide with transcription in prokaryotes, we hypothesized linear relationship between transcriptional product and the amount of translational product increasing per unit time and that this relationship does not depend on the kind of translational product. We also hypothesized that anti-sigma combine with sigma and form inert matter, and the reaction velocity of that is proportional to product of these.</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/0/0d/Ono_%2816%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/0/0d/Ono_%2816%29.png" height="50px" class = "math" /> |
- | <img src = "https://static.igem.org/mediawiki/2014/8/86/Ono_%2817%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/8/86/Ono_%2817%29.png" height="50px" class = "math" /> |
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<p>Using above-mentioned differential equations, we simulated behavior of the counter by Euler's method.</p> | <p>Using above-mentioned differential equations, we simulated behavior of the counter by Euler's method.</p> | ||
<h3>Parameter</h3> | <h3>Parameter</h3> | ||
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<p>If there are a lot of molecules, modeling usually uses ordinary differntial equations, but some in vivo reactions involve only a few molecules. For example, transcription involves the cell's genomic DNA which is one copy or plasmids which are about 200 copies \cite{plasmid1}\cite{plasmid2} in a cell of <I> Escherichia coli</I>. The average size of a cell of <I>E. coli</I> is about 1.0 * 10^{-15}[L]\cite{volume}, so the concentration of DNA is about 1.7[nM] and the concentration of plasmids is about 200 times of it. This is obviously weak. Reactions like this are well affected by fluctuations due to the reactants's limited copy numbers. So, we need to take this fluctuations into our modeling which is derived from stochastic methods. We also introduce delay effect.</p> | <p>If there are a lot of molecules, modeling usually uses ordinary differntial equations, but some in vivo reactions involve only a few molecules. For example, transcription involves the cell's genomic DNA which is one copy or plasmids which are about 200 copies \cite{plasmid1}\cite{plasmid2} in a cell of <I> Escherichia coli</I>. The average size of a cell of <I>E. coli</I> is about 1.0 * 10^{-15}[L]\cite{volume}, so the concentration of DNA is about 1.7[nM] and the concentration of plasmids is about 200 times of it. This is obviously weak. Reactions like this are well affected by fluctuations due to the reactants's limited copy numbers. So, we need to take this fluctuations into our modeling which is derived from stochastic methods. We also introduce delay effect.</p> | ||
<p>First we explain about the Gillespie algorithm which is often used in stochastic simulations. In the Gillespie algorithm, we treated not the concentration of molecules but the number of them. Reactions are also viewed as descrete, essentially instantaneous physical events. What we have to determine when using the Gillespie algorithm is (1) when the next reaction is going to occur and (2) which type of the reaction it will be. Looking more closely at the Gillespie algorithm by the next set of reaction formulas:</p> | <p>First we explain about the Gillespie algorithm which is often used in stochastic simulations. In the Gillespie algorithm, we treated not the concentration of molecules but the number of them. Reactions are also viewed as descrete, essentially instantaneous physical events. What we have to determine when using the Gillespie algorithm is (1) when the next reaction is going to occur and (2) which type of the reaction it will be. Looking more closely at the Gillespie algorithm by the next set of reaction formulas:</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/8/8e/Ono_%2819%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/8/8e/Ono_%2819%29.png" height="50px" class = "math" /> |
<p>Let n<sub>1</sub>, n<sub>2</sub>, and n<sub>3</sub> denote the respective copy number of the components X<sub>1</sub>, X<sub>2</sub>, and X<sub>3</sub>. Notice that they are all integer. First we have to determine how easily each reactions could happen. It depends on the number of components copied. In stochatic simulations, we often determine the paremeter called stochastic rate constant, which is often written as "c''. We assume that each possible combinations of reactant molecules have the same probability c per unit time to react. In other words, c * dt gives the probability that a particular combination of reactant molecules will react in a short time interval [t,t+dt). We call the stochastic rate constant of a reaction j, c<sub>j</sub>. Considering the all combinations of reactant molecules, the probability that the reaction 0 occur in [t,t+dt) is c*n<sub>1</sun>*n<sub>2</sub>. We now define the propensity function as the function of which product with dt gives the probability that a particular reaction will occur in the next infinitesimal time dt, which is often written as "a''. Later on, the propensity function of a reaction j is a<sub>j</sub>. Following the equation:</p> | <p>Let n<sub>1</sub>, n<sub>2</sub>, and n<sub>3</sub> denote the respective copy number of the components X<sub>1</sub>, X<sub>2</sub>, and X<sub>3</sub>. Notice that they are all integer. First we have to determine how easily each reactions could happen. It depends on the number of components copied. In stochatic simulations, we often determine the paremeter called stochastic rate constant, which is often written as "c''. We assume that each possible combinations of reactant molecules have the same probability c per unit time to react. In other words, c * dt gives the probability that a particular combination of reactant molecules will react in a short time interval [t,t+dt). We call the stochastic rate constant of a reaction j, c<sub>j</sub>. Considering the all combinations of reactant molecules, the probability that the reaction 0 occur in [t,t+dt) is c*n<sub>1</sun>*n<sub>2</sub>. We now define the propensity function as the function of which product with dt gives the probability that a particular reaction will occur in the next infinitesimal time dt, which is often written as "a''. Later on, the propensity function of a reaction j is a<sub>j</sub>. Following the equation:</p> | ||
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<p>Notice that c<sub>j</sub> is invariant parameter, but a<sub>j</sub> changes as the state changes. In the same way, a<sub>1</sub> = c<sub>1</sub>*n<sub>3</sub>.</p> | <p>Notice that c<sub>j</sub> is invariant parameter, but a<sub>j</sub> changes as the state changes. In the same way, a<sub>1</sub> = c<sub>1</sub>*n<sub>3</sub>.</p> | ||
<p>First we answer the question (1) when is the next reaction going to occur? Now, to simplify the situation we assume the situation that only the reaction 0 occurs. Set the time as 0, and define P(t) as the probability that the reaction 0 doesn't occur in [0,t). Then from the definition of a,we obtain the equation; P(t+dt) = P(t)*(1-a*dt). (Because the probability that the reaction 0 doesn't occur in [0,t+dt) is the product of the probability that the reaction 0 doesn't occur in [0,t) with the probability that the reaction 0 doesn't occur in [t,t+dt).) Using P(t+dt) = P(t) + dP(t)/dt, we get :</p> | <p>First we answer the question (1) when is the next reaction going to occur? Now, to simplify the situation we assume the situation that only the reaction 0 occurs. Set the time as 0, and define P(t) as the probability that the reaction 0 doesn't occur in [0,t). Then from the definition of a,we obtain the equation; P(t+dt) = P(t)*(1-a*dt). (Because the probability that the reaction 0 doesn't occur in [0,t+dt) is the product of the probability that the reaction 0 doesn't occur in [0,t) with the probability that the reaction 0 doesn't occur in [t,t+dt).) Using P(t+dt) = P(t) + dP(t)/dt, we get :</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/d/d5/Ono_%2821%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/d/d5/Ono_%2821%29.png" height="50px" class = "math" /> |
<p>Because the probability that the reaction0 doesn't occur in a 0 second interval is zero; P(0)=1. Solving the above ordinary differential eqaution we get :</p> | <p>Because the probability that the reaction0 doesn't occur in a 0 second interval is zero; P(0)=1. Solving the above ordinary differential eqaution we get :</p> | ||
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<p> If r<sub>1</sub> is a uniform number from [0,1], the time of the next reaction should be determined by solving P(t) = r<sub>1</sub>. Using (2), we get t = -a<sub>0</sub>/log r<sub>1</sub>.</p> | <p> If r<sub>1</sub> is a uniform number from [0,1], the time of the next reaction should be determined by solving P(t) = r<sub>1</sub>. Using (2), we get t = -a<sub>0</sub>/log r<sub>1</sub>.</p> | ||
<p>Now we suppose there is N types of reactions. Let a<sub>1</sub>,a<sub>2</sub>,…,a<sub>N</sub> denote the respective propensity function of reaction 1,2,…,N. From previous method;</p> | <p>Now we suppose there is N types of reactions. Let a<sub>1</sub>,a<sub>2</sub>,…,a<sub>N</sub> denote the respective propensity function of reaction 1,2,…,N. From previous method;</p> | ||
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<p>Let dt be so small that we can ignore the term of higher than two orders of dt. The equation(3) becomes:</p> | <p>Let dt be so small that we can ignore the term of higher than two orders of dt. The equation(3) becomes:</p> | ||
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<p>Solving (4) (a = \sum_{j=1}^{N}a_{j}):</p> | <p>Solving (4) (a = \sum_{j=1}^{N}a_{j}):</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/8/81/Ono_%2825%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/8/81/Ono_%2825%29.png" height="50px" class = "math" /> |
<p>Setting $\tau$ as the time of the next reaction, we get:</p> | <p>Setting $\tau$ as the time of the next reaction, we get:</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/3/3d/Ono_%2826%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/3/3d/Ono_%2826%29.png" height="50px" class = "math" /> |
<p>Second we answer the question (2) what types of the reaction will it be? We determined the time of the next reaction, so what we have left to do is to determine what kind of reaction occurs. Some people may feel queer, but in the Gillespie algorithm, first the time of next reaction will be determined, and second the kind of reaction will be determined. It is natural to determine that the probability that the reaction j occurs is a<sub>j</sub>/a. If r<sub>2</sub> is a uniform number from [0,1], j is the only number that meets below in equations:</p> | <p>Second we answer the question (2) what types of the reaction will it be? We determined the time of the next reaction, so what we have left to do is to determine what kind of reaction occurs. Some people may feel queer, but in the Gillespie algorithm, first the time of next reaction will be determined, and second the kind of reaction will be determined. It is natural to determine that the probability that the reaction j occurs is a<sub>j</sub>/a. If r<sub>2</sub> is a uniform number from [0,1], j is the only number that meets below in equations:</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/1/1a/Ono_%2827%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/1/1a/Ono_%2827%29.png" height="50px" class = "math" /> |
<p>In the case a<sub>0</sub> ≧ a * r<sub>2</sub>, the reaction that occured is reaction 0.</p> | <p>In the case a<sub>0</sub> ≧ a * r<sub>2</sub>, the reaction that occured is reaction 0.</p> | ||
<p>Now we can run the Gillespie algorithm by following the next steps.(t<sub>MAX</sub> is the finish time of the simulation.)<br />1.Initialize the system at t = 0 with initial numbers of molecules for each spices, n<sub>0</sub>,… ,n<sub>s</sub><br />2.For each j = 0,1,…,r, calculate a<sub>j</sub>(n) based on the current state n using (21)<br />3.Calculate the exit rate a(n) = \sum_{j=0}^{r} a_{j}(n).<br />4.Compute a sample tau of the time until the next time using (27)<br />5.Update the time t = t + tau<br />6.Compute a sample j of the reaction index using (28)<br />7.Update the state n according to the reaction j.<br />8.If $t < t<sub>MAX</sub>, return to Step 2</p> | <p>Now we can run the Gillespie algorithm by following the next steps.(t<sub>MAX</sub> is the finish time of the simulation.)<br />1.Initialize the system at t = 0 with initial numbers of molecules for each spices, n<sub>0</sub>,… ,n<sub>s</sub><br />2.For each j = 0,1,…,r, calculate a<sub>j</sub>(n) based on the current state n using (21)<br />3.Calculate the exit rate a(n) = \sum_{j=0}^{r} a_{j}(n).<br />4.Compute a sample tau of the time until the next time using (27)<br />5.Update the time t = t + tau<br />6.Compute a sample j of the reaction index using (28)<br />7.Update the state n according to the reaction j.<br />8.If $t < t<sub>MAX</sub>, return to Step 2</p> | ||
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<p>However, these results should not be taken to imply that the mathematical forms of the propensity functions are just heuristic extrapolations. The propensity functions are grounded in molecular physics, and the formulas of deterministic chemical kinetics are approximate consequences of the formulas of stochastic chemical kinetics, not the other way around.</p> | <p>However, these results should not be taken to imply that the mathematical forms of the propensity functions are just heuristic extrapolations. The propensity functions are grounded in molecular physics, and the formulas of deterministic chemical kinetics are approximate consequences of the formulas of stochastic chemical kinetics, not the other way around.</p> | ||
<p>The Gillespie algorithm is so clear and useful that it is often used. However, this algorithm is not suitable for describing transcriptions and translations beacuse they are very slow and complex reactions involving many kinds of reactant molecules. If we treat transcription from plasmids as one reaction, assuming the copy number of plasmids as 200, then the propensity function a equals to the stochastic rate constant multiplied by 200 (200*c). So it will take about one of a two hundred times of an average transcription time to finish one transcription. Of course, in the time scale of average transcription time it is not a big problem, but this may not be good for simulating, like in our project, the system that uses the time for transcriptions and translations cannot be shortened. We introduce time-delay into the Gillespie algorithm based on \cite{delay1}$\sim$\cite{delay3}. The mathematical correctness of this algorithm is proved in \cite{delay3}. Time-delay means treating reactions as following:</p> | <p>The Gillespie algorithm is so clear and useful that it is often used. However, this algorithm is not suitable for describing transcriptions and translations beacuse they are very slow and complex reactions involving many kinds of reactant molecules. If we treat transcription from plasmids as one reaction, assuming the copy number of plasmids as 200, then the propensity function a equals to the stochastic rate constant multiplied by 200 (200*c). So it will take about one of a two hundred times of an average transcription time to finish one transcription. Of course, in the time scale of average transcription time it is not a big problem, but this may not be good for simulating, like in our project, the system that uses the time for transcriptions and translations cannot be shortened. We introduce time-delay into the Gillespie algorithm based on \cite{delay1}$\sim$\cite{delay3}. The mathematical correctness of this algorithm is proved in \cite{delay3}. Time-delay means treating reactions as following:</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/4/43/Ono_%2828%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/4/43/Ono_%2828%29.png" height="50px" class = "math" /> |
<p>Furthermore, transcriptions and translations are too complex to list up all of the reactions step by step. Therfore it is better to treat them as time-delay than reaction formulas.</p> | <p>Furthermore, transcriptions and translations are too complex to list up all of the reactions step by step. Therfore it is better to treat them as time-delay than reaction formulas.</p> | ||
<p>Now we begin to model our project, sigma Re-counter. In our model, there are only three reactions: transcription, translation, and an association and disassociation of crRNA and taRNA. We introduce time-delay into only transcription and translation. Then, we explain how we treat these three reactions in general.</p> | <p>Now we begin to model our project, sigma Re-counter. In our model, there are only three reactions: transcription, translation, and an association and disassociation of crRNA and taRNA. We introduce time-delay into only transcription and translation. Then, we explain how we treat these three reactions in general.</p> | ||
<p>First we explain transcription's model\cite{stochastic}. When the RNA polymerase binds to the promoter region, first they take the RNAP・promoter close complex. At this state, the complex can disociate. But with a certain probability, the close complex turn to the open complex which doesn't disociate. After the RNA polymerase and the promoter region take the open complex, a transcription starts. Then the reaction formula of transcription can be described as following's reactions:</p> | <p>First we explain transcription's model\cite{stochastic}. When the RNA polymerase binds to the promoter region, first they take the RNAP・promoter close complex. At this state, the complex can disociate. But with a certain probability, the close complex turn to the open complex which doesn't disociate. After the RNA polymerase and the promoter region take the open complex, a transcription starts. Then the reaction formula of transcription can be described as following's reactions:</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/c/cd/Ono_%2829%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/c/cd/Ono_%2829%29.png" height="50px" class = "math" /> |
<p>combining reaction3' and reaction3'', we get:</p> | <p>combining reaction3' and reaction3'', we get:</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/d/dd/Ono_%2830%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/d/dd/Ono_%2830%29.png" height="50px" class = "math" /> |
<p>Second, we refer to the translational model [8]. Similary to the transcrptional model we model as following;</p> | <p>Second, we refer to the translational model [8]. Similary to the transcrptional model we model as following;</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/c/c1/Ono_%2831%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/c/c1/Ono_%2831%29.png" height="50px" class = "math" /> |
<p>combining reaction2' and reaction2'', we get:</p> | <p>combining reaction2' and reaction2'', we get:</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/2/26/Ono_%2832%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/2/26/Ono_%2832%29.png" height="50px" class = "math" /> |
<p>Last, the model of association and disassociation of crRNA and taRNA is a reversible reaction. So we model as following:</p> | <p>Last, the model of association and disassociation of crRNA and taRNA is a reversible reaction. So we model as following:</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/3/30/Ono_%2833%29.png" class = "math" /> | + | <img src = "https://static.igem.org/mediawiki/2014/3/30/Ono_%2833%29.png" height="50px" class = "math" /> |
<p>We can conclude that reaction formulas of our model are as follows:</p> | <p>We can conclude that reaction formulas of our model are as follows:</p> | ||
<h3>Parameter</h3> | <h3>Parameter</h3> | ||
Line 2,065: | Line 2,065: | ||
</div> | </div> | ||
<div id = "Modeling-4"> | <div id = "Modeling-4"> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/6/6f/Sub_implementation.png" class = "contTitle" /> | + | <img src = "https://static.igem.org/mediawiki/2014/6/6f/Sub_implementation.png" height="50px" class = "contTitle" /> |
<p>In this section we have discussed the improved models of the σ-recounter.</p> | <p>In this section we have discussed the improved models of the σ-recounter.</p> | ||
<p>First, we modeled the triple σ-recounter, the expansion of the double counter. Below is the construct of the triple re-counter.</p> | <p>First, we modeled the triple σ-recounter, the expansion of the double counter. Below is the construct of the triple re-counter.</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/5/59/Ono_3count_construct.png" class = "figure" /> | + | <img src = "https://static.igem.org/mediawiki/2014/5/59/Ono_3count_construct.png" height="50px" class = "figure" /> |
<p>The explanation on this construct is available <a href="Javascript:loadContent('Project-block','Project-3')">here</a>. The reaction formulas were established just like as the above-mentioned deterministic model. The result of the modeling of the triple recounter:</p> | <p>The explanation on this construct is available <a href="Javascript:loadContent('Project-block','Project-3')">here</a>. The reaction formulas were established just like as the above-mentioned deterministic model. The result of the modeling of the triple recounter:</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/3/3b/Ono_3count_result.png" class = "figure" /> | + | <img src = "https://static.igem.org/mediawiki/2014/3/3b/Ono_3count_result.png" height="50px" class = "figure" /> |
<p>The unit of vertical axis is [nM], and that of the horizontal axis is [sec].</p> | <p>The unit of vertical axis is [nM], and that of the horizontal axis is [sec].</p> | ||
<p>Fig 3count result is the result of the modeling of the triple recounter. Although there seems to be a few leak expression, the count is precisely conducted. Here we did not model resetting, because it is obvious from its orthogonality that resetting will be precisely conducted if the pulse length is long enough.</p> | <p>Fig 3count result is the result of the modeling of the triple recounter. Although there seems to be a few leak expression, the count is precisely conducted. Here we did not model resetting, because it is obvious from its orthogonality that resetting will be precisely conducted if the pulse length is long enough.</p> | ||
<p>Second, we thought of genetic circuits that would not be affected by the pulse length of the arabinose induction. The current σ Re-counter depends much on pulse length; when the pulse length is too long, it would count 2 or more (if there is). (Non-improved version)</p> | <p>Second, we thought of genetic circuits that would not be affected by the pulse length of the arabinose induction. The current σ Re-counter depends much on pulse length; when the pulse length is too long, it would count 2 or more (if there is). (Non-improved version)</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/4/4e/Ono_implementation_failure.png" class = "figure" /> | + | <img src = "https://static.igem.org/mediawiki/2014/4/4e/Ono_implementation_failure.png" height="50px" class = "figure" /> |
<p>induction time: 20000-40000, 60000-80000</p> | <p>induction time: 20000-40000, 60000-80000</p> | ||
<p>If the induction is too long, there will be no difference in the first induction and the second induction; that is, it has no function of counting.</p> | <p>If the induction is too long, there will be no difference in the first induction and the second induction; that is, it has no function of counting.</p> | ||
<p>However, by improving this construct a little, our counter would not count more than 1 by a single pulse, as long as the pulse length is long enough (longer than tau<sub>0</sub>) for it to count. The figure shown below is the improved constructs.</p> | <p>However, by improving this construct a little, our counter would not count more than 1 by a single pulse, as long as the pulse length is long enough (longer than tau<sub>0</sub>) for it to count. The figure shown below is the improved constructs.</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/c/cc/Ono_implementation_construct.png" class = "figure" /> | + | <img src = "https://static.igem.org/mediawiki/2014/c/cc/Ono_implementation_construct.png" height="50px" class = "figure" /> |
<p>X and Y are substances that bind together to activate <I>P<sub>X&Y</sub></I> promoter.</p> | <p>X and Y are substances that bind together to activate <I>P<sub>X&Y</sub></I> promoter.</p> | ||
<p>Before arabinose is induced, <I>P<sub>Tet</sub></I> and <I>P<sub>const</sub></I> express Y and crRBS-sigma. When the arabinose is induced (for longer than time τ0), <I>P<sub>BAD</sub></I> becomes activated and TetR and X are expressed. X binds to Y and the transcription of taRNA from <I>P<sub>X&Y</sub></I> occur, which leads to counting. At that time, expression of Y is repressed by TetR and the amount of Y decreases exponentially. Thus, <I>P<sub>X&Y</sub></I> is again repressed, the amount of taRNA decreases, and the counter never counts more than 1. You might be afraid that <I>P<sub>X&Y</sub></I> also begins transcription of taRNA when the induction ends; however, supposed degradation of X is faster than that of TetR, it will not occur. When the induction ends, X first degrades while still a lot of TetR remain and Y is not abundant. Since <I>P<sub>Tet</sub></I> has a simoidal transcriptional response, the production rate of Y will change little even if the concentration of TetR decrease a little. When TetR degrades so much that it finishes repression of Y, most of X have already decomposed, and <I>P<sub>X&Y</sub></I> will not be activated to begin transcription of taRNA.</p> | <p>Before arabinose is induced, <I>P<sub>Tet</sub></I> and <I>P<sub>const</sub></I> express Y and crRBS-sigma. When the arabinose is induced (for longer than time τ0), <I>P<sub>BAD</sub></I> becomes activated and TetR and X are expressed. X binds to Y and the transcription of taRNA from <I>P<sub>X&Y</sub></I> occur, which leads to counting. At that time, expression of Y is repressed by TetR and the amount of Y decreases exponentially. Thus, <I>P<sub>X&Y</sub></I> is again repressed, the amount of taRNA decreases, and the counter never counts more than 1. You might be afraid that <I>P<sub>X&Y</sub></I> also begins transcription of taRNA when the induction ends; however, supposed degradation of X is faster than that of TetR, it will not occur. When the induction ends, X first degrades while still a lot of TetR remain and Y is not abundant. Since <I>P<sub>Tet</sub></I> has a simoidal transcriptional response, the production rate of Y will change little even if the concentration of TetR decrease a little. When TetR degrades so much that it finishes repression of Y, most of X have already decomposed, and <I>P<sub>X&Y</sub></I> will not be activated to begin transcription of taRNA.</p> | ||
<p>We modeled this construct to test if it can be realized. We did not modeled resetting this time, either.</p> | <p>We modeled this construct to test if it can be realized. We did not modeled resetting this time, either.</p> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/d/d0/Ono_implementation_result.png" class = "figure" /> | + | <img src = "https://static.igem.org/mediawiki/2014/d/d0/Ono_implementation_result.png" height="50px" class = "figure" /> |
<p>The inductions were modeled to be conducted just the same as non-improved version. Although pulse length is long, counts are precisely done. Thus, theoretically, the counter independent of the pulse length is suggested to be available. Only thing we have to do is to research for the substances that satisfy these conditions!</p> | <p>The inductions were modeled to be conducted just the same as non-improved version. Although pulse length is long, counts are precisely done. Thus, theoretically, the counter independent of the pulse length is suggested to be available. Only thing we have to do is to research for the substances that satisfy these conditions!</p> | ||
</div> | </div> | ||
<div id = "Modeling-5"> | <div id = "Modeling-5"> | ||
- | <img src = "https://static.igem.org/mediawiki/2014/7/76/Sub_guideformodeling.png" class = "contTitle" /> | + | <img src = "https://static.igem.org/mediawiki/2014/7/76/Sub_guideformodeling.png" height="50px" class = "contTitle" /> |
<h3>What is modeling</h3> | <h3>What is modeling</h3> | ||
<p>The model should be sufficiently detailed and precise so that it can in principle be used to simulate the bevavior of the system on a computer. </p> | <p>The model should be sufficiently detailed and precise so that it can in principle be used to simulate the bevavior of the system on a computer. </p> | ||
Line 2,098: | Line 2,098: | ||
<h3>Stochastic Approaches</h3> | <h3>Stochastic Approaches</h3> | ||
<p>The most common formulation of stochastic models for biochemical networks is the chemical master equation (CME). The analytical nature of the early stochastic approaches was highly complicated and, in some cases, intractable so that they received little attention in the biochemical community. Later, the situation changed with the increasing computational power of modern computers. And finally Gillespie presented an ground-breaking algorithm for numerically generating sample trajectories of the abundances of chemical species in chemical reaction networks. The so-called "stochastic simulation algorithm," or "Gillespie algorithm," can easily be implemented in any programming or scripting language that has a pseudorandom number generator. Several software packages implementing the algorithm have been developed. Differernt stochastic approaches and their interrelationchips are depicted in Figure.</p> | <p>The most common formulation of stochastic models for biochemical networks is the chemical master equation (CME). The analytical nature of the early stochastic approaches was highly complicated and, in some cases, intractable so that they received little attention in the biochemical community. Later, the situation changed with the increasing computational power of modern computers. And finally Gillespie presented an ground-breaking algorithm for numerically generating sample trajectories of the abundances of chemical species in chemical reaction networks. The so-called "stochastic simulation algorithm," or "Gillespie algorithm," can easily be implemented in any programming or scripting language that has a pseudorandom number generator. Several software packages implementing the algorithm have been developed. Differernt stochastic approaches and their interrelationchips are depicted in Figure.</p> | ||
- | <img src="https://static.igem.org/mediawiki/2014/e/e8/Chart.png" class="figure" /> | + | <img src="https://static.igem.org/mediawiki/2014/e/e8/Chart.png" height="50px" class="figure" /> |
<p>For large biochemical systems, with many species and reactions, stochasitc simulations (based on the original Gillespie algorithm) become computationally demanding. Recent years have seen a large interest in improving the efficiency/speed of stochastic simulations by modification/approximation of the original Gillespie algorithm. These improvements include the "next reaction" method of Gibson and Bruck, the "τ-leap" method and its various improvements and generalizations and the "maximal time step method", which combines the next rection and the τ-leap methods.</p> | <p>For large biochemical systems, with many species and reactions, stochasitc simulations (based on the original Gillespie algorithm) become computationally demanding. Recent years have seen a large interest in improving the efficiency/speed of stochastic simulations by modification/approximation of the original Gillespie algorithm. These improvements include the "next reaction" method of Gibson and Bruck, the "τ-leap" method and its various improvements and generalizations and the "maximal time step method", which combines the next rection and the τ-leap methods.</p> | ||
<p>While stochastic simulations sre a practical way to realize the CME, analytical approxinmations offer more insihgts into the influence of noise on cell function. Formally, the CME is a continuous-time discrete-state Markov process. For gaining intuitive insight and a quick characteriztion of fluctuations in biochemical networks, the CME is usually approximated analytically in different ways, including the frequently used chemical Langevin equation (CLE), the linear noise approximation (LNA), and the two-moment approximation (2MA).</p> | <p>While stochastic simulations sre a practical way to realize the CME, analytical approxinmations offer more insihgts into the influence of noise on cell function. Formally, the CME is a continuous-time discrete-state Markov process. For gaining intuitive insight and a quick characteriztion of fluctuations in biochemical networks, the CME is usually approximated analytically in different ways, including the frequently used chemical Langevin equation (CLE), the linear noise approximation (LNA), and the two-moment approximation (2MA).</p> |