Team:UT-Tokyo/Counter

From 2014.igem.org

(Difference between revisions)
Line 1,965: Line 1,965:
<img src = "https://static.igem.org/mediawiki/2014/8/89/Sub_overview.png" class = "contTitle" />
<img src = "https://static.igem.org/mediawiki/2014/8/89/Sub_overview.png" class = "contTitle" />
<p>Modeling is an attempt to describe, in a precise way, an understanding of the elements of a system of interest, their states, and their interactions with other elements.</p>
<p>Modeling is an attempt to describe, in a precise way, an understanding of the elements of a system of interest, their states, and their interactions with other elements.</p>
-
<p>The purpose of our modeling team is to peel back the layer of appearance of the device to reveal it's underlying nature. We tried to improve the device, cooperating with the experiment team. To achieve our goal, we have developed three fundamental themes. These three themes divide the modeling part into three parts. At the beginning, we con?rmed whether our circuit realizes a reaction:this for part 1. Next, we adjusted the parts and the conditions, for the device to reproduce a satisfactory value suitable for naming the device as a counter:this for part 2. Finally, we discussed what would be appropriate modeling, frequent issue to attack, in order to ?nd the best strategy of modeling and wrote how we constructed our model:this for part 3.</p>
+
<p>The purpose of our modeling team is to peel back the layer of appearance of the device to reveal it's underlying nature. We tried to improve the device, cooperating with the experiment team. To achieve our goal, we have developed three fundamental themes. These three themes divide the modeling part into three parts. At the beginning, we confirmed whether our circuit realizes a reaction:this for part 1. Next, we adjusted the parts and the conditions, for the device to reproduce a satisfactory value suitable for naming the device as a counter:this for part 2. Finally, we discussed what would be appropriate modeling, frequent issue to attack, in order to find the best strategy of modeling and wrote how we constructed our model: this for part 3.</p>
<p>In Part1(Deterministic Model,Stochastic Model), we approached the problem in two ways.</p>
<p>In Part1(Deterministic Model,Stochastic Model), we approached the problem in two ways.</p>
-
<p>・Deteministic model:In this model,chemical reactions are discribed as differential equations and concentration of reaction product can be calu- culated by those of reactants. This model is intutive, simple and hence popular to estimate the result of experiment.</p>
+
<p>・Deteministic model:In this model,chemical reactions are discribed as differential equations and concentration of reaction products can be calculated by those of reactants. This model is intutive, simple and hence popular to estimate the results of experiment.</p>
<p>・Stochastic model:The most common formulation of stochastic models for biochemical networks is the chemical master equation (CME). We used Gillepie Algorithm to solve CME.</p>
<p>・Stochastic model:The most common formulation of stochastic models for biochemical networks is the chemical master equation (CME). We used Gillepie Algorithm to solve CME.</p>
-
<p>In Part2(Result), changing measured values of gene copy numbers, strength of pConst, sequence of taRNA and etc. in silico, we estimated in which combination of values the counter outputs a sufficient amount of data.</p>
+
<p>In Part2(Result), changing measured values of gene copy numbers, strength of <I>P<sub>Const</sub></I>, sequence of taRNA and etc. in silico, we estimated in which combination of values the counter outputs a sufficient amount of data.</p>
<p>In Part3(Guide for Modeling), what is modeling, aims of modeling and differernt stochastic approaches and their interrelationShips</p>
<p>In Part3(Guide for Modeling), what is modeling, aims of modeling and differernt stochastic approaches and their interrelationShips</p>
</div>
</div>
<div id = "Modeling-2">
<div id = "Modeling-2">
<img src = "https://static.igem.org/mediawiki/2014/9/9e/Sub_deterministic.png" class = "contTitle" />
<img src = "https://static.igem.org/mediawiki/2014/9/9e/Sub_deterministic.png" class = "contTitle" />
-
<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 calu- culated 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>
+
<h3>Formulation of the Model</h3>
-
<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 over ?tting when we adapt this model to raw data.</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>
<img src = "https://static.igem.org/mediawiki/2014/9/9b/Ono_%281%29.png" class = "math" />
<img src = "https://static.igem.org/mediawiki/2014/9/9b/Ono_%281%29.png" class = "math" />
<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>
Line 1,983: Line 1,984:
<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" class = "math" />
<img src = "https://static.igem.org/mediawiki/2014/0/05/Ono_%285%29.png" class = "math" />
<img src = "https://static.igem.org/mediawiki/2014/0/05/Ono_%285%29.png" class = "math" />
-
<p>Using these equations((3)-(7)) and equilibrium constant, concentrations of binding taRNA or not mRNA coding σ 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" 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" class = "math" />
Line 2,003: Line 2,004:
<img src = "https://static.igem.org/mediawiki/2014/d/d2/Ono_%2818%29.png" class = "math" />
<img src = "https://static.igem.org/mediawiki/2014/d/d2/Ono_%2818%29.png" class = "math" />
<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>
-
<p>We explained the parameters of the deterministic model. PoPS (promoter per second) is 0.03\cite{promoter}, so its promoter activity is $0.03/6.0*10^{23}\cdot 1.0\cdot 10^{-15}$[M], 0.051[nM/sec]. The switch point and hill coefficients of <I>P<sub>BAD</sub></I> is writen in \cite{pBAD1}. RPU (relative promoter unit) is $\frac{5}{60}\cdot1.7$[nM]. We set the RPU of pLac as 2 when induced. We don't consider the leak expression from pLac.</p>
+
<h3>Parameter</h3>
-
<p>The average half life of mRNA is 2-5 min\cite{Uri}, so we set the degradation rate of mRNA as 0.020[/sec]. The half life of GFP is $\infty$\cite{GFP}, so we set the degradation of GFP as 0.0[sec]. The degradation rate of sigma factor[2] is fast. So we set as 0.0001[/sec]. The equilibrium constant of the equations (1)(2) is 80.0[nM]\cite{taRNA}. The number of plasmids copied is 100$\sim$300\cite{plasmid1}\cite{plasmid2} , so we set as 200. The number of ribosomes on a mRNA is about 20 and the time for a ribosome to translate is about 2 minute, so we set the translational rate as 1.43[/sec].</p>
+
<p>We explain how we determined the parameters of the deterministic model. PoPS (promoter per second) of <I>P<sub>Const</sub></I> is 0.03¥cite{promoter}, so its promoter activity is 0.03/(6.0*10^{23}*1.010^{-15})[M] = 0.051[nM/sec]. The switch point and hill coefficients of <I>P<sub>BAD</sub></I> is writen in ¥cite{pBAD1}. PoPS of <I>P<sub>BAD</sub></I> is 5/60¥cite{pBAD1} , so its RPU (relative promoter unit) is (5/60)/(0.03) = 2.78. We set the RPU of <I>P<sub>lac</sub></I> as 2 when induced. We don't consider the leak expression from <I>P<sub>lac</sub></I>.</p>
 +
<p>The average half life of mRNA is 2-5 min¥cite{Uri}, so we set the degradation rate of mRNA as 0.010[/sec]. The half life of GFP is infinite¥cite{GFP}, so we set the degradation of GFP as 0.0[sec]. The degradation rate of sigma factor[2](reference) is fast. So we set as 0.0001[/sec]. The degradation rate of anti-sigma is unknown, so we set as 6.0*10^{-6}, the average degradation rate of protin(reference). The equilibrium constant of the equations (1)(2) is 80.0[nM]¥cite{taRNA}. The reaction rate of the association of sigma and anti-sigma is unknown. We assumed this reaction is fas so we set as 10.0[/M sec]. The number of plasmids copied is 100~300¥cite{plasmid1}¥cite{plasmid2} , so we set as 200. The number of ribosomes on a mRNA is about 20(reference) and the time for a ribosome to translate is about 2 minute(reference), so we set the translational rate as 20/120 = 0.167[/sec].</p>
<p>The summary of the parameters of this model is given in Table 1.</p>
<p>The summary of the parameters of this model is given in Table 1.</p>
 +
<h3>Result</h3>
<br />
<br />
<img src = "https://static.igem.org/mediawiki/2014/5/51/Ono_2count_result.png" class = "figure" />
<img src = "https://static.igem.org/mediawiki/2014/5/51/Ono_2count_result.png" class = "figure" />
Line 2,016: Line 2,019:
<div id = "Modeling-3">
<div id = "Modeling-3">
<img src = "https://static.igem.org/mediawiki/2014/9/9c/Sub_stochastic.png" class = "contTitle" />
<img src = "https://static.igem.org/mediawiki/2014/9/9c/Sub_stochastic.png" class = "contTitle" />
-
<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 $1.0 \cdot 10^{-15}$[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>
+
<h3>Formulation of the Model</h3>
 +
<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" 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, $c \cdot\mathrm{dt}$ 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 $c_{j}$. Considering the all combinations of reactant molecules, the probability that the reaction 0 occur in [t,t+dt) is $c_{0}\cdot n_{1} \cdot n_{2}$. 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>
<img src = "https://static.igem.org/mediawiki/2014/9/97/Ono_%2820%29.png" class = "math" />
<img src = "https://static.igem.org/mediawiki/2014/9/97/Ono_%2820%29.png" class = "math" />
-
<p>Notice that c<sub>j</sub> is invariant parameter, but a<sub>j</sub> changes as the state changes. In the same way, $a_{1} = c_{1} \cdot n_{3}$.</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)$\cdot$(1$-$a$\cdot$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) = $\displaystyle \mathrm{P(t)} + \frac{d\mathrm{P(t)}}{\mathrm{dt}} \cdot \mathrm{dt}$, 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" 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>
<img src = "https://static.igem.org/mediawiki/2014/3/3f/Ono_%2822%29.png" class = "math" />
<img src = "https://static.igem.org/mediawiki/2014/3/3f/Ono_%2822%29.png" class = "math" />
-
<p> If $r_{1}$ is a uniform number from [0,1], the time of the next reaction should be determined by solving P(t) = $r_{1}$. Using (2), we get t = $\displaystyle -\frac{a_{0}}{\mathrm{log}r_{1}}$.</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>
<img src = "https://static.igem.org/mediawiki/2014/c/c9/Ono_%2823%29.png" class = "math" />
<img src = "https://static.igem.org/mediawiki/2014/c/c9/Ono_%2823%29.png" class = "math" />
-
<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>
<img src = "https://static.igem.org/mediawiki/2014/f/f6/Ono_%2824%29.png" class = "math" />
<img src = "https://static.igem.org/mediawiki/2014/f/f6/Ono_%2824%29.png" class = "math" />
-
<p>Solving (4) ($\displaystyle a = \sum_{j=1}^{N}a_{j}$);</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" 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" 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 occured. 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 $\displaystyle \frac{a_{j}}{a}$. If $r_{2}$ is a uniform number from [0,1], j is the only number that meets below inequations;</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" class = "math" />
-
<p>In the case $a_{0} \geq a \cdot r_{2}$, 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.($t_{MAX}$ 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 $\displaystyle 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 $t < t_{MAX}$, 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>
-
<p>Stochastic rate constant can be determined by the parameters we used in the deterministic model (if we modeled the reaction in the determinsitic model) . If there are a lot of reactant molecules, stochastic simulations have to show similar results as those of determinisitic simulations. For this reason, stochastic rate constant, $c$, can be calculated from the chemical reaction rate constant, $k$. See \cite{gillespie1} if you want to know the deriving process. Here we just write the result.</p>
+
<p>Stochastic rate constant can be determined by the parameters we used in the deterministic model (if we modeled the reaction in the determinsitic model) . If there are a lot of reactant molecules, stochastic simulations have to show similar results as those of determinisitic simulations. For this reason, stochastic rate constant, c, can be calculated from the chemical reaction rate constant, k. See \cite{gillespie1} if you want to know the deriving process. Here we just write the result.</p>
-
<p>For a unimolecular reaction, $c$ numerically equals to $k$, whereas for a bimolecular reaction, $c$ equals to $\displaystyle \frac{k}{N_{A}V}$ if the species of the reactant molecules are different, or $\displaystyle \frac{2k}{N_{A}V}$ if they are the same. $V$ is the volume of the system and $N_{A}$ is the Avogadro's constant.</p>
+
<p>For a unimolecular reaction, c numerically equals to k, whereas for a bimolecular reaction, c equals to k/N<sub>A</sub>V if the species of the reactant molecules are different, or 2k/N<sub>A</sub>V if they are the same.V is the volume of the system and N<sub>A</sub> is the Avogadro's constant.</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>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 rightness 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" 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>
Line 2,057: Line 2,061:
<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" 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>
 +
The summary of the parameters of this model is given in Table 3.
 +
<h3>Result</h3>
 +
Result.
</div>
</div>
<div id = "Modeling-4">
<div id = "Modeling-4">
Line 2,067: Line 2,075:
<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" 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 $\tau_{0}$) 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" 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>
Line 2,099: Line 2,107:
<h3>Stochastic Formulation and Markov Process</h3>
<h3>Stochastic Formulation and Markov Process</h3>
<p>Since the occurrence of reactions involves discrete and random events at the microscopic level, it is impossible to deterministically predict the progress of recations interms of the macroscopic variables (obsevables) N(t) and Z(t). To acount for this uncertainty, one of the observables N()Z()</p>
<p>Since the occurrence of reactions involves discrete and random events at the microscopic level, it is impossible to deterministically predict the progress of recations interms of the macroscopic variables (obsevables) N(t) and Z(t). To acount for this uncertainty, one of the observables N()Z()</p>
-
<p>Our goal is to determine how the process N(t) of copy numbers evolves in time. Starting at time t=0 from some initial state N(0), every sample path of the process remains in state N(0) for a random amount of time W\_1 until the occurrence of a reaction takes process to a new state N(W\_1); it remains in state N(W\_1) for another random amount of time W\_2 until the occurrence of another reaction takes the process to a new state N(W\_1+W\_2), and so on. In other words, the time-dependent copy number N(t) is a jump process.</p>
+
<p>Our goal is to determine how the process N(t) of copy numbers evolves in time. Starting at time t=0 from some initial state N(0), every sample path of the process remains in state N(0) for a random amount of time W_1 until the occurrence of a reaction takes process to a new state N(W_1); it remains in state N(W_1) for another random amount of time W_2 until the occurrence of another reaction takes the process to a new state N(W_1+W_2), and so on. In other words, the time-dependent copy number N(t) is a jump process.</p>
-
<p>The stochasitc process N(t) is characterized by a collection of state probabilities and transition probabilities. The state probability P(n,t)=Pr[N(t)=n] is the probability that the process N(t) is the state n at a time t. The transition probability Pr[N(t\_0+t)=n|N(t\_0)=m] is the conditional probability that process N(t) has moved from state m to state n during the time interval [t\_0,t\_0+t]. The analysis of a stochastic process becomes greatly simplified when the above transition probability depends on (i) the starting state m but not on the states before time t\_0 and (ii) the interval length t but not on the. Property (i) is the well-known Markov process. The process holding property (ii) is said to be homogeneous process.</p>
+
<p>The stochasitc process N(t) is characterized by a collection of state probabilities and transition probabilities. The state probability P(n,t)=Pr[N(t)=n] is the probability that the process N(t) is the state n at a time t. The transition probability Pr[N(t_0+t)=n|N(t_0)=m] is the conditional probability that process N(t) has moved from state m to state n during the time interval [t_0,t_0+t]. The analysis of a stochastic process becomes greatly simplified when the above transition probability depends on (i) the starting state m but not on the states before time t_0 and (ii) the interval length t but not on the. Property (i) is the well-known Markov process. The process holding property (ii) is said to be homogeneous process.</p>
-
<p>[1]D.J.Wilkinson.Stochastic Modelling for Systems Biology.Mathematical \& Computational Biology. Chapman \& Hall/CRC, London, Apr. 2006. ISBN 1584885408</p>
+
<p>[1]D.J.Wilkinson.Stochastic Modelling for Systems Biology.Mathematical & Computational Biology. Chapman & Hall/CRC, London, Apr. 2006. ISBN 1584885408</p>
-
<p>[2]Mukhtar Ullah \& Olaf Wolkenhauer Stochastic Approaches for Systems Biology.</p>
+
<p>[2]Mukhtar Ullah & Olaf Wolkenhauer Stochastic Approaches for Systems Biology.</p>
<h3>References</h3>
<h3>References</h3>
<p>[1] Uri Alon『An introductio to Systems Biology: Design Principles od Biological Circuits』</p>
<p>[1] Uri Alon『An introductio to Systems Biology: Design Principles od Biological Circuits』</p>
Line 2,118: Line 2,126:
<p>[13] D.J.Wilkinson.Stochastic Modelling for Systems Biology.Mathematical & Computational Biology.Chapman & Hall/CRC, London, Apr. 2006. ISBN 1584885408</p>
<p>[13] D.J.Wilkinson.Stochastic Modelling for Systems Biology.Mathematical & Computational Biology.Chapman & Hall/CRC, London, Apr. 2006. ISBN 1584885408</p>
<p>[14] Mukhtar Ullah & Olaf Wolkenhauer Stochastic Approaches for Systems Biology.</p>
<p>[14] Mukhtar Ullah & Olaf Wolkenhauer Stochastic Approaches for Systems Biology.</p>
-
<p>[15] Part:BBa I13453 <http://parts.igem.org/Part:BBa_I13453> ( We ?nally accessed on 2014/8/20)</p>
+
<p>[15] Part:BBa I13453 <http://parts.igem.org/Part:BBa_I13453> ( We finally accessed on 2014/8/20)</p>
<p>[16] iGEM Kyoto 2010 <https://2010.igem.org/Team:Kyoto/Project/Goal_A></p>
<p>[16] iGEM Kyoto 2010 <https://2010.igem.org/Team:Kyoto/Project/Goal_A></p>
-
<p>[17] pSB1A2 <http://parts.igem.org/Part:pSB1A2> ( We ?nally accessed on 2014/8/20)</p>
+
<p>[17] pSB1A2 <http://parts.igem.org/Part:pSB1A2> ( We finally accessed on 2014/8/20)</p>
-
<p>[18] pSB1C3 <http://parts.igem.org/Part:pSB1C3> ( We ?nally accessed on 2014/8/20)</p>
+
<p>[18] pSB1C3 <http://parts.igem.org/Part:pSB1C3> ( We finally accessed on 2014/8/20)</p>
</div>
</div>
</div>
</div>

Revision as of 02:26, 14 October 2014