Team:Todai-Tokyo/Modeling

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the notebook

Overview

As we described in the project overview of the bioclock project, the actual clock consists of two separable parts: a bio-oscillator and a lambda phage switch. In the section we will be modeling each of the parts and come up with an integrated model, which resembles the ultimate goal of the project. By making comparisons of the result of our model and the result of the original article which we based our simulation upon, we will be trying to examine how realistic is the model, and gain better understanding of how the oscillation process and, eventually, the integrated model are working.

Bio-Oscillator

Construction of the Model

In the natural world, there are quite a few system were known to act as an oscillator. The project started out by examining the behavior of Cyanobacteria. We learnt that the mutual activation and repression between several compounds resulted in the oscillation pattern of protein expression. We finally settled on creating a system utilizing lacI and araC as the mutual-interaction compound to construct a oscillator.

Although it is possible to trace the reactions in detail all the way from the transcription of mRNA and translation of the proteins, with the precise reaction rate, it will be redundant to do so. In order to maintain clarity, we simplify this process to focus only on the rate of translation. According to [1], the reaction involved in the dynamics of the reaction could be summarized as following:

<math>P^{a/r}_{0,j} + a_{2} \xrightleftharpoons [k_{-a}]{k_{a}} P^{a/r}_{1,j}</math> <math>P^{a/r}_{i,0} + r_{4} \xrightleftharpoons [k_{-r}]{2k_{r}}P^{a/r}_{i,1}</math> <math>P^{a/r}_{i,1} + r_{4} \xrightleftharpoons [2k_{-r}]{k_{r}} P^{a/r}_{i,2}</math> <math>P^{a/r}_{1,2} \xrightarrow {k_{l}} P^{a/r}_{L,2} + a_{2}</math> <math>P^{a/r}_{0,2} \xrightarrow {k_{l}} P^{a/r}_{L,2} </math> <math>P^{a/r}_{L,2} \xrightarrow {k_{ul}} P^{a/r}_{0,0} </math> <math>P^{a/r}_{0,0} \xrightarrow {b_{a/r}} P^{a/r}_{0,0} + m_{a/r}</math> <math>P^{a/r}_{1,0} \xrightarrow {\alpha b_{a/r}} P^{a/r}_{1,0} + m_{a/r}</math>

<math>m_{a} \xrightarrow {t_{a}} m_{a} + a_{uf}</math> <math>m_{r} \xrightarrow {t_{a}} m_{r} + r_{uf}</math> <math>a_{uf} \xrightarrow {k_{fa}} a</math> <math>r_{uf} \xrightarrow {k_{fr}} r</math> <math>a+a \xrightleftharpoons[k_{-da}]{k_{da}} + a_{2}</math> <math>r+r \xrightleftharpoons[k_{-dr}]{k_{dr}} + r_{2}</math> <math>r_{2}+r_{2} \xrightleftharpoons[k_{-t}]{k_{t}} + r_{4}</math> <math>m_{a/r} \xrightarrow {d_{a/r}} \empty</math> <math>a_{uf} \xrightarrow {\lambda f(X)}} \empty</math> <math>r_{uf} \xrightarrow {f(X)} \empty</math> <math>a \xrightarrow {\lambda f(X)}} \empty</math> <math>r \xrightarrow {f(X)}} \empty</math> <math>a_{2} \xrightarrow {\lambda f(X)}} \empty</math> <math>r_{2} \xrightarrow {f(X)} \empty </math>

<math>P^{a/r}_{1,j} \xrightarrow {f(X)} P^{a/r}_{0,j} </math> <math>P^{a/r}_{i,1} \xrightarrow {f(X)} P^{a/r}_{i,0} </math> <math>P^{a/r}_{i,2} \xrightarrow {2f(X)} P^{a/r}_{i,1} </math> <math>P^{a/r}_{L,2} \xrightarrow {2\epsilon f(X)} P^{a/r}_{L,1} </math> <math>P^{a/r}_{L,1} \xrightarrow {\epsilon f(X)} P^{a/r}_{L,0} </math>

where

<math>f(X) = \frac{\gamma}{c_{e}+X} </math>

Reference

[1] Jesse Stricker, Scott Cookson, Matthew R. Bennett, William H. Mather, Lev S. Tsimring and Jeff Hasty, 2008 A fast, robust and tunable synthetic gene oscillator (with Supplementary Information) Nature 07389 Vol 456: 516-519

[2] Adam Arkin, John Ross and Harley H. McAdams, 1998 Stochastic Kinetic Analysis of Developmental Pathway Bifurcation in Phage lambda-Infected Escherichia coli Cells Genetics 149: 1633–1648

[3] Shea, M. A., and G. K. Ackers, 1985 The OR control system of bacteriophage lambda: a physical-chemical model for gene regulation J. Mol. Biol. 181: 211–230

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