Team:UT-Tokyo/Counter/Project

From 2014.igem.org

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<img src = "https://static.igem.org/mediawiki/2014/a/af/Sub_application.png" class = "contTitle" />
<img src = "https://static.igem.org/mediawiki/2014/a/af/Sub_application.png" class = "contTitle" />
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<p>The genetic circuit we constructed can be considered as an automaton. In a previous study[1], many orthogonal sigma and anti-sigma have been reported. (what “orthogonal” means here is to make almost no crosstalk to other promoters or sigma factors) Thus, an automaton that have many states can be constructed.</p>
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<p>     As described above, the genetic circuit we constructed can be considered as an automaton. In a previous study[1], many sigma and anti-sigma that regulate transcription without crosstalk have been reported. Thus, an automaton that has many states can be constructed. Furthermore, though in this project reset is transition from other states to state 0, more general system that is capable of changing one state to any other states is possible. Thus more general automaton by genetic circuits may be possible. Examples of an automaton are such as biocomputer or lifegame etc. Bringing this concept from algorithmic world into synthetic biology is our challenge. Though in our project input is a single short intermittent signal, a more general circuit that responds to more general input is possible to be considered by integrating additional circuits into our circuit.</p>
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<p>Examples of an automaton are such as biocomputer, SUDOKU or lifegame etc. Bringing this concept from algorithmic world into synthetic biology was our challenge. Though in our project input is a single short intermittent signal ,a more general circuit that responds to more general input is possible to be considered by integrating additional circuits into our circuit.</p>
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<p>     For example, the change of balance between 2 substances itself can be considered as a input. Considering substances A and B, this additional circuit is possible:</p>
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<p>For example, the change of balnce between 2 elements itself can be considered as a input. Considering elements A and B, this additional circuit is possible:</p>
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<p>pA-repressorB-reporterA-activatorX-pX-repressorA</p>
<p>pA-repressorB-reporterA-activatorX-pX-repressorA</p>
<p>pB-repressorA-reporterB-activatorY-pY-repressorB</p>
<p>pB-repressorA-reporterB-activatorY-pY-repressorB</p>
<img src ="https://static.igem.org/mediawiki/2014/b/ba/Sumi_AB.png" class = "figure" />
<img src ="https://static.igem.org/mediawiki/2014/b/ba/Sumi_AB.png" class = "figure" />
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<p>This additional circuit essentially contains toggle switch structure.Therefore, change of dominance relation between A and B can be converted into a pulse with transition from a stable point to another which can be used as input in our circuit. Consequently, this additional circuit is thought to be an AD converter and if elementB is constant this can trace elementA by regarding reporterA as a spokesman.If A is industrial waste, this circuit can be a monitor of a milieu. </p>
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<p>     Here, substances A/B activate promter A/B. This additional circuit essentially contains toggle switch structure with delay negative feedback loop. For example, when substance A become dominant against substance B, the toggle switch amplifies the dominance of promoter A and the following negative feedback suppress the dominance. Consequently, this additional circuit is expected to convert the change of the dominance to a pulse expression of reporter protein. Therefore, this additional circuit can expand the range of input. As this circuit can be a monitor of a milieu if A is industrial waste, our genetic circuit can be applied to much wider range of problems.</p>
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<p>As described above, our genetic circuit can be applied to much wider range of problems.</p>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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" />
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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 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>
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<h3>Formulation of the Model</h3>
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<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>
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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>
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<img src = "https://static.igem.org/mediawiki/2014/9/9b/Ono_%281%29.png" class = "math" />
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<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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<img src = "https://static.igem.org/mediawiki/2014/9/9b/Ono_%281%29.png"  height="50px" 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>
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<img src = "https://static.igem.org/mediawiki/2014/7/74/Ono_%282%29.png" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/7/74/Ono_%282%29.png" height="100px" class = "math" />
<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>
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<img src = "https://static.igem.org/mediawiki/2014/0/0b/Ono_%283%29.png" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/0/0b/Ono_%283%29.png" height="50px" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/0/05/Ono_%285%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 σ and GFP were discribed as following. These are all of simplifications.</p>
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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>
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<img src = "https://static.igem.org/mediawiki/2014/7/71/Ono_%286%29.png" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/a/aa/Ono_%2810%29.png" height="50px" class = "math" />
<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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<img src = "https://static.igem.org/mediawiki/2014/e/e5/Ono_%2811%29.png" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/e/e5/Ono_%2811%29.png" height="50px" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/3/31/Ono_%2812%29.png" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/3/31/Ono_%2812%29.png" height="50px" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/7/7b/Ono_%2813%29.png" height="50px" class = "math" />
<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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<img src = "https://static.igem.org/mediawiki/2014/0/0d/Ono_%2814%29.png" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/0/0d/Ono_%2814%29.png" height="50px" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/1/1b/Ono_%2815%29.png" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/1/1b/Ono_%2815%29.png" height="50px" class = "math" />
<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>
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<img src = "https://static.igem.org/mediawiki/2014/0/0d/Ono_%2816%29.png" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/0/0d/Ono_%2816%29.png" height="50px" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/8/86/Ono_%2817%29.png" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/d/d2/Ono_%2818%29.png" height="50px" 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>
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<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>
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<h3>Parameter</h3>
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<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>
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<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>
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<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>
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<h3>Result</h3>
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<img src = "https://static.igem.org/mediawiki/2014/5/51/Ono_2count_result.png" class = "figure" />
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<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" />
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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 $1.0 \cdot 10^{-15}$[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>
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<h3>Formulation of the Model</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>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>
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<img src = "https://static.igem.org/mediawiki/2014/8/8e/Ono_%2819%29.png" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/8/8e/Ono_%2819%29.png" height="50px" class = "math" />
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<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>
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<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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<img src = "https://static.igem.org/mediawiki/2014/9/97/Ono_%2820%29.png" class = "math" />
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<img src = "https://static.igem.org/mediawiki/2014/9/97/Ono_%2820%29.png" height="50px" class = "math" />
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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, $a_{1} = c_{1} \cdot n_{3}$.</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>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>
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<img src = "https://static.igem.org/mediawiki/2014/d/d5/Ono_%2821%29.png" class = "math" />
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<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>
-
<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" height="50px" 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>
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<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" height="50px" 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" height="50px" 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" 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 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" height="50px" 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" 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>
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<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>
 +
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">
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<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" />
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<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 $\tau_{0}$) 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" 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">
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<img src = "https://static.igem.org/mediawiki/2014/7/76/Sub_guideformodeling.png" class = "contTitle" />
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<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 2,092: 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>
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<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>
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<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>
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<div id = "Achieve-block" style = "display:none;">
<div id = "Achieve-block" style = "display:none;">
<div id = "Achieve-1">
<div id = "Achieve-1">
-
</div>
 
-
</div>
 
-
<!-- Attribution -->
 
-
<div id = "Attribution-block" style = "display:none;">
 
-
<div id = "Attribution-1">
 
-
<img src="https://static.igem.org/mediawiki/2014/4/44/Sub_attribution.png" class = "contTitle" />
 
-
<p>Our activities involved in iGEM were all conducted by undergaduates alone!!<br>Based on fund-raisings and public relations, all team members had a lot of brainstoming, investigations and discussions in order to select project carefully. And we conducted experiments and FINALLY saw results.<br>We also designed and composed all publish tools, and of course, polished all scripts and presentation by ourselves.</p>
 
-
<h3>Project</h3>
 
-
<p>All we had a lot of brainstorming, investigations and discussion when we decide our projects.<br><b>Yoichi Irie</b>(Team Leader)<br />σ-ReCounter:<br><b>Shunsuke Sumi</b>(idea)<br><b>So Nakashima</b>(idea and comformation)<br><b>Takefumi Yoshikawa</b>(comformation)<br><br>CTCD:<br><b>Masayuki Osawa</b>(conformation)<br><b>Shigetaka Kobari</b>(investigation and conformation)<br><b>Shunsuke Sumi</b>(conformation)<br><b>Senkei Hyo</b>(investigation)<br><b>Yoshihiko Tomofuji</b>(conformation)<br><b>Yshiki Okesaku</b>(investigation)</p>
 
-
<h3>Experiment</h3>
 
-
<p>Lab. Leader:<br><b>Takefumi Yoshikawa</b>(construction, assay;σ-ReCounter)<br><br>Lab. Members:<br><b>Atsuki Ito</b>(construction)<br><b>Hajime Takemura</b>(construction)<br><b>Keisuke Tsukada</b>(construction)<br><b>Kentaro Tara</b>(construction)<br><b>Kento Nakamura</b>(construction, assay;σ-ReCounter)<br><b>Naruki Yoshikawa</b>(construction)<br><b>Nobuhiro Hiura</b>(construction)<br><b>Shigetaka Kobari</b>(assay;CTCD)<br><b>So Nakashima</b>(construction, Assay;σ-ReCounter)<br><b>Yoshihiko Tomofuji</b>(assay;CTCD)<br><b>Yuto Yamanaka</b>(construction)</p>
 
-
<h3>Modeling</h3>
 
-
<p><b>Keisuke Tsukada, Kentaro Tara, Manabu Nishiura, Masaki Ono</b></p>
 
-
<h3>Web</h3>
 
-
<p>Almost all members wrote drafts of our team wiki.<br><b>Cristian David</b>(check our English)<br><b>Hiroki Tsuboi</b>(team website, implementation of our team wiki)</p>
 
-
<h3>App</h3>
 
-
<p><b>Naruki Yoshikawa</b></p>
 
-
<h3>Design</h3>
 
-
<p><b>Cristian David</b>(parker design)<br><b>Yoshiki Okesaku</b>(all design, all figure)</p>
 
-
<h3>Presentation</h3>
 
-
<p><b>Kento Nakamura, Manabu Nishiura, Masato Ishikawa, Yumeno Koga, Yuto Yamanaka</b></p>
 
-
<h3>Poster</h3>
 
-
<br>
 
-
<h3>Public Relation</h3>
 
-
<p><b>Ding Yuewen, Keisuke Tsukada</b></p>
 
-
<h3>Adviser</h3>
 
-
<p><b>Kota Tosimitsu</b></p>
 
-
</div>
 
-
<div id = "Attribution-2">
 
-
<img src="https://static.igem.org/mediawiki/2014/4/44/Sub_Ackno.png" class="contTitle" />
 
-
</div>
 
-
<div id = "Attribution-3">
 
-
<img src="https://static.igem.org/mediawiki/2014/3/38/Sub_sponsors.png" class="contTitle" />
 
-
<img src="https://static.igem.org/mediawiki/2014/7/72/Promega.png" class="sponsor">
 
-
<p><b>Promega KK.</b>for chamical reagents</p>
 
-
<p><b>Teiyukai, Faculty of Engineering, The University of Tokyo</b>for fund</p>
 
-
<p><b>Integrated DNA Technologies MBL</b></p>
 
-
<img src="https://static.igem.org/mediawiki/2014/3/30/Cosmobio.png" class="sponsor">
 
-
<p><b>COSMO BIO Co., Ltd.</b>for fund<br></p>
 
-
<img src="https://static.igem.org/mediawiki/2014/1/17/Liveanest.png" class="sponsor">
 
-
<p><b>Leave a Nest Co., Ltd.(Hiroyuki Takahashi)</b>for advice for public relations & introduction of Promega KK.</p>
 
</div>
</div>
</div>
</div>
Line 2,225: Line 2,190:
<p>b) Does your project currently include any design features to reduce risks? Or, if you did all the future work to make your project grow into a popular product, would you plan to design any new features to minimize risks? (For example: auxotrophic chassis, physical containment, etc.) Such features are not required for an iGEM project, but many teams choose to explore them.</p>
<p>b) Does your project currently include any design features to reduce risks? Or, if you did all the future work to make your project grow into a popular product, would you plan to design any new features to minimize risks? (For example: auxotrophic chassis, physical containment, etc.) Such features are not required for an iGEM project, but many teams choose to explore them.</p>
<p>In the future study of sigma-Recounter project, in addition to the reset system, we intend to increase the number of nodes and to enable one state to move to any other states. Therefore, if our project is used to express a toxin to defeat pest or bacteria, you can prepare an anti-toxin node or a reset system for mistakenly expressing the toxin.</p>
<p>In the future study of sigma-Recounter project, in addition to the reset system, we intend to increase the number of nodes and to enable one state to move to any other states. Therefore, if our project is used to express a toxin to defeat pest or bacteria, you can prepare an anti-toxin node or a reset system for mistakenly expressing the toxin.</p>
 +
</div>
 +
</div>
 +
<!-- Attribution -->
 +
<div id = "Attribution-block" style = "display:none;">
 +
<div id = "Attribution-1">
 +
<img src="https://static.igem.org/mediawiki/2014/4/44/Sub_attribution.png" class = "contTitle" />
 +
<p>Our activities involved in iGEM were all conducted by undergaduates alone!!<br>Based on fund-raisings and public relations, all team members had a lot of brainstoming, investigations and discussions in order to select project carefully. And we conducted experiments and FINALLY saw results.<br>We also designed and composed all publish tools, and of course, polished all scripts and presentation by ourselves.</p>
 +
<h3>Project</h3>
 +
<p>All we had a lot of brainstorming, investigations and discussion when we decide our projects.<br><b>Yoichi Irie</b>(Team Leader)<br />σ-ReCounter:<br><b>Shunsuke Sumi</b>(idea)<br><b>So Nakashima</b>(idea and comformation)<br><b>Takefumi Yoshikawa</b>(comformation)<br><br>CTCD:<br><b>Masayuki Osawa</b>(conformation)<br><b>Shigetaka Kobari</b>(investigation and conformation)<br><b>Shunsuke Sumi</b>(conformation)<br><b>Senkei Hyo</b>(investigation)<br><b>Yoshihiko Tomofuji</b>(conformation)<br><b>Yshiki Okesaku</b>(investigation)</p>
 +
<h3>Experiment</h3>
 +
<p>Lab. Leader:<br><b>Takefumi Yoshikawa</b>(construction, assay;σ-ReCounter)<br><br>Lab. Members:<br><b>Atsuki Ito</b>(construction)<br><b>Hajime Takemura</b>(construction)<br><b>Keisuke Tsukada</b>(construction)<br><b>Kentaro Tara</b>(construction)<br><b>Kento Nakamura</b>(construction, assay;σ-ReCounter)<br><b>Naruki Yoshikawa</b>(construction)<br><b>Nobuhiro Hiura</b>(construction)<br><b>Shigetaka Kobari</b>(assay;CTCD)<br><b>So Nakashima</b>(construction, Assay;σ-ReCounter)<br><b>Yoshihiko Tomofuji</b>(assay;CTCD)<br><b>Yuto Yamanaka</b>(construction)</p>
 +
<h3>Modeling</h3>
 +
<p><b>Keisuke Tsukada, Kentaro Tara, Manabu Nishiura, Masaki Ono</b></p>
 +
<h3>Web</h3>
 +
<p>Almost all members wrote drafts of our team wiki.<br><b>Cristian David</b>(check our English)<br><b>Hiroki Tsuboi</b>(team website, implementation of our team wiki)</p>
 +
<h3>App</h3>
 +
<p><b>Naruki Yoshikawa</b></p>
 +
<h3>Design</h3>
 +
<p><b>Cristian David</b>(parker design)<br><b>Yoshiki Okesaku</b>(all design, all figure)</p>
 +
<h3>Presentation</h3>
 +
<p><b>Kento Nakamura, Manabu Nishiura, Masato Ishikawa, Yumeno Koga, Yuto Yamanaka</b></p>
 +
<h3>Poster</h3>
 +
<br>
 +
<h3>Public Relation</h3>
 +
<p><b>Ding Yuewen, Keisuke Tsukada</b></p>
 +
<h3>Adviser</h3>
 +
<p><b>Kota Tosimitsu</b></p>
 +
</div>
 +
<div id = "Attribution-2">
 +
<img src="https://static.igem.org/mediawiki/2014/4/44/Sub_Ackno.png" class="contTitle" />
 +
</div>
 +
<div id = "Attribution-3">
 +
<img src="https://static.igem.org/mediawiki/2014/3/38/Sub_sponsors.png" class="contTitle" />
 +
<img src="https://static.igem.org/mediawiki/2014/7/72/Promega.png" class="sponsor">
 +
<p><b>Promega KK.</b>for chamical reagents</p>
 +
<p><b>Teiyukai, Faculty of Engineering, The University of Tokyo</b>for fund</p>
 +
<p><b>Integrated DNA Technologies MBL</b></p>
 +
<img src="https://static.igem.org/mediawiki/2014/3/30/Cosmobio.png" class="sponsor">
 +
<p><b>COSMO BIO Co., Ltd.</b>for fund<br></p>
 +
<img src="https://static.igem.org/mediawiki/2014/1/17/Liveanest.png" class="sponsor">
 +
<p><b>Leave a Nest Co., Ltd.(Hiroyuki Takahashi)</b>for advice for public relations & introduction of Promega KK.</p>
</div>
</div>
</div>
</div>
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<p>Synthetic biology is not well-known and thought to be unfamiliar to ordinary people. To introduce synthetic biology into public, an easy entrance such as playing game is effective. We hope that players will get interested in synthetic biology by enjoying this game.</p>
<p>Synthetic biology is not well-known and thought to be unfamiliar to ordinary people. To introduce synthetic biology into public, an easy entrance such as playing game is effective. We hope that players will get interested in synthetic biology by enjoying this game.</p>
<p>You can download this game from Google Play.</p>
<p>You can download this game from Google Play.</p>
-
<h3>Lectures to general public</h3>
+
<img src="https://static.igem.org/mediawiki/2014/8/81/Screenshot1.png" class = "figure" style="width:380px;float:left;" />
 +
<img src="https://static.igem.org/mediawiki/2014/c/ce/Screenshot2.png" class = "figure" style="width:380px;float:left;" />
 +
<h3 style="clear:both;">Lectures to general public</h3>
<h4>School festivals</h4>
<h4>School festivals</h4>
<p>The university of Tokyo has two school festivals per year. The May festival is held in May and the Komaba festival was held in November.We explained iGEM and synthetic biology briefly and introduce our project to audience. We invited other iGEM teams in Japan to May festival.  We offered precious opportunities that Japanese iGEM teams meet through May festivals.</p>
<p>The university of Tokyo has two school festivals per year. The May festival is held in May and the Komaba festival was held in November.We explained iGEM and synthetic biology briefly and introduce our project to audience. We invited other iGEM teams in Japan to May festival.  We offered precious opportunities that Japanese iGEM teams meet through May festivals.</p>
-
<h4>Techno-Edge</h4>
+
<img src="https://static.igem.org/mediawiki/2014/3/39/Festival1.jpg" class = "figure" style="width:380px;float:left;" />
 +
<img src="https://static.igem.org/mediawiki/2014/6/6a/Festival2.JPG" class = "figure" style="width:380px;float:left;" />
 +
<h4 style="clear:both;">Techno-Edge</h4>
<p>Techno-Edge is the event which the department of technology of university of Tokyo held. The purpose of this event was to appeal department of technology to junior high or high school students.Many laboratories and academic circles such as Robotech took part in this event. iGEM UT-Tokyo also participated in it to appeal synthetic biology.</p>
<p>Techno-Edge is the event which the department of technology of university of Tokyo held. The purpose of this event was to appeal department of technology to junior high or high school students.Many laboratories and academic circles such as Robotech took part in this event. iGEM UT-Tokyo also participated in it to appeal synthetic biology.</p>
<p>Not only high school and junior high, but also primary school students came to our booth and were interested in our explanation.</p>
<p>Not only high school and junior high, but also primary school students came to our booth and were interested in our explanation.</p>
 +
<img src="https://static.igem.org/mediawiki/2014/2/27/Technoedge.JPG" class = "figure" />
<h4>Presentation</h4>
<h4>Presentation</h4>
<p>This year, iGEM Nagahama invited us to the genetics society of Japan, and we participated in it. We made an oral presentation workshop of synthetic biology and took part in poster session. There were many iGEM teams, and we could advertise iGEM in academic world. Moreover, specialists gave advice to us, and we were inspired by professors of synthetic biology.</p>
<p>This year, iGEM Nagahama invited us to the genetics society of Japan, and we participated in it. We made an oral presentation workshop of synthetic biology and took part in poster session. There were many iGEM teams, and we could advertise iGEM in academic world. Moreover, specialists gave advice to us, and we were inspired by professors of synthetic biology.</p>
<p>We also joined in Japanese Society for Cell Synthesis Research.</p>
<p>We also joined in Japanese Society for Cell Synthesis Research.</p>
-
<h4>A cram school</h4>
+
<img src="https://static.igem.org/mediawiki/2014/6/68/Presentation1.jpg" class = "figure" style="width:380px;float:left;" />
 +
<img src="https://static.igem.org/mediawiki/2014/4/4e/Presentation2.jpg" class = "figure" style="width:380px;float:left;" />
 +
<h4 style="clear:both;">A cram school</h4>
<p>We held a seminar in which we explained synthetic biology and iGEM for high school students at a cram school in Komaba.</p>
<p>We held a seminar in which we explained synthetic biology and iGEM for high school students at a cram school in Komaba.</p>
</div>
</div>
Line 2,267: Line 2,280:
<p>In March, iGEM Kyoto held iGEM-Japan West meeting. In this meeting, we shared each team's project and advised each other. This meeting was very useful because we could find our idea's weak point.</p>
<p>In March, iGEM Kyoto held iGEM-Japan West meeting. In this meeting, we shared each team's project and advised each other. This meeting was very useful because we could find our idea's weak point.</p>
<p>In August, iGEM TMU-Tokyo held iGEM-Japan East meeting, and we made presentations about our projects and criticized each other. Thanks to these meetings, we developed quality of our projects.</p>
<p>In August, iGEM TMU-Tokyo held iGEM-Japan East meeting, and we made presentations about our projects and criticized each other. Thanks to these meetings, we developed quality of our projects.</p>
 +
<img src="https://static.igem.org/mediawiki/2014/b/bb/Igemjapan1.jpg" class = "figure" style="width:380px;float:left;" />
 +
<img src="https://static.igem.org/mediawiki/2014/8/8b/Igemjapan2.jpg" class = "figure" style="width:380px;float:left;" />
 +
<p style="clear:both;"></p>
</div>
</div>
</div>
</div>
Line 2,619: Line 2,635:
</div>
</div>
<div class = "member">
<div class = "member">
-
<p class = "name">YUTO NAKAYAMA</p>
+
<p class = "name">YUTO YAMANAKA</p>
<img src = "https://static.igem.org/mediawiki/2014/2/26/Yamanaka_comb.png" class ="member-detail" />
<img src = "https://static.igem.org/mediawiki/2014/2/26/Yamanaka_comb.png" class ="member-detail" />
<dl class = "profile">
<dl class = "profile">
<dt>Name</dt>
<dt>Name</dt>
-
<dd>Yuto Nakayama</dd>
+
<dd>Yuto Yamanaka</dd>
<dt>Belong to</dt>
<dt>Belong to</dt>
-
<dd>e</dd>
+
<dd>Department of Electrical Engineering and Bioscience, School of Advanced Science and Engineering, Waseda University</dd>
<dt>Job</dt>
<dt>Job</dt>
-
<dd></dd>
+
<dd>Experimenter, Presenter</dd>
</dl>
</dl>
-
<p class ="comment">I want to be a doctor.</p>
+
<p class ="comment">I like Falcon tube.</p>
</div>
</div>
</div>
</div>

Revision as of 05:46, 16 October 2014