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Team:HZAU-China/testmath
From 2014.igem.org
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\begin{equation} | \begin{equation} | ||
P(\text{binding})=\frac{[Pro']}{n}=\frac{[TF]^n}{K_D+[TF]^n} | P(\text{binding})=\frac{[Pro']}{n}=\frac{[TF]^n}{K_D+[TF]^n} | ||
| + | \end{equation} | ||
| + | </p> | ||
| + | <p> | ||
| + | If the transcription factor is a activator, the gene is transcribed at the maximal transcription rate $\beta_1$ when the promoter is bound by the transcription factor. If the transcription factor is a repressor, the gene is transcribed at the maximal transcription rate $\beta_1$ when the promoter is free from the transcription factor. | ||
| + | |||
| + | In deterministic model, we use Hill function to describe the rate of production, which is $\beta_1$ times its occurring probability. | ||
| + | |||
| + | For transcription activation, the maximal transcription rate will occur if the activator bind to the promoter, so the Hill function is | ||
| + | \begin{equation} | ||
| + | f(x)=\beta_1\frac{[TF]^n}{K^n+[TF]^n}. | ||
| + | \end{equation} | ||
| + | For transcription repression, the maximal transcription rate will occur if the repressor doesn't bind to the promoter, so the Hill function is | ||
| + | \begin{equation} | ||
| + | f(x)=\beta_1\frac{K^n}{K^n+[TF]^n}. | ||
\end{equation} | \end{equation} | ||
</p> | </p> | ||
Revision as of 10:47, 11 October 2014
Our project aims to engineer cells that can process information to adapt to the environment as we expected, so we need to yield quantitative predictions of gene behaviors. Mathematic model becomes a powerful tool here. It not only helps us analyze the phenomena, but also contributes to the design of genetic circuits. Our models can be divided into several parts. Firstly, we will introduce the biological processes in our design and translate them into mathematic language. Then we will make some comparisons to explain why we designed circuits this way. Next, we will present the simulation of deterministic model and stochastic model. Parameter sensitivity analysis will also be made here. Finally, we will talk about the design principle of the rewirable circuit using ODE sets with matrices. We hope that any other researches can get insight from the mathematic model and make the rewirable circuit widely used. \section{Biological processes}
In this part, we will list the important biological processes and explain how the related molecules work, and then we can describe them using some equations. Because the timescales of many processes are separated, we use a quasi-steady-state approximation (QSSA) to reduce the number of dimensions in the systems in most cases. However, if we focus on some processes more than the final steady state, we may use other approximations such like prefactor method (Bennett et al., 2007). \subsection{Transcription and translation}
According to the Central Dogma, some DNAs can be transcribed into RNAs and some RNAs can be translated into proteins. These procedures sometimes can be regulated by some molecules like transcription factors or non-coding RNAs. The interactions among them can be understood by chemical reactions. We use reaction (1) to depict the interaction between transcription factor $TF$ (sometimes it forms a polymer, we omit this step) and inducible promoter $Pro$. \begin{equation} K_D=\frac{k_{-1}}{k_1}=\frac{[TF]^n\cdot [Pro]}{[Pro']}. \end{equation}
The equilibrium dissociation constant $K_D$ for the reaction can be calculated. \begin{equation} K_D=\frac{k_{-1}}{k_1}=\frac{[TF]^n\cdot [Pro]}{[Pro']}. \end{equation}
We assume that the DNA copy number is a constant n, \begin{equation} [Pro]+[Pro']=n. \end{equation} Then the $Pro'$ proportion depends on the concentration of $TF$, and this proportion also be regarded as the probability of the binding events. \begin{equation} P(\text{binding})=\frac{[Pro']}{n}=\frac{[TF]^n}{K_D+[TF]^n} \end{equation}
If the transcription factor is a activator, the gene is transcribed at the maximal transcription rate $\beta_1$ when the promoter is bound by the transcription factor. If the transcription factor is a repressor, the gene is transcribed at the maximal transcription rate $\beta_1$ when the promoter is free from the transcription factor. In deterministic model, we use Hill function to describe the rate of production, which is $\beta_1$ times its occurring probability. For transcription activation, the maximal transcription rate will occur if the activator bind to the promoter, so the Hill function is \begin{equation} f(x)=\beta_1\frac{[TF]^n}{K^n+[TF]^n}. \end{equation} For transcription repression, the maximal transcription rate will occur if the repressor doesn't bind to the promoter, so the Hill function is \begin{equation} f(x)=\beta_1\frac{K^n}{K^n+[TF]^n}. \end{equation}
