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Team:Todai-Tokyo/Modeling
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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. One of the defining characteristics is the duel role of the promotor. It could be bound with maximum of two lacI protien and one araC protien. When a araC is bound on it, it activates the translation of both araC and lacI, increasing lacI concentration while enhancing the possibility of lacI binding to the promotor as well. When promotor is bound with to lacI, it could transmute in to a looped form, in which araC is forbidden to bind. We will note the state of promotor as <math>P^{a/r}_{i,j}</math>, where i is representing the number of araC bound, while j represents that of lacI's.
According to [1], the reaction involved in the dynamics of the promotor state changes and 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}_{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>
and X is the total number of ssrA tags in the system. k_{a}(It is a tag in order to quicken the disintegration process of the proteins.) The parameters of above reaction is listed as below: <math>b_{a} = b_{r} = 0.36 min^{-1}, \alpha = 20, k_{a} = 0.533 min^{-1}, k_{r} = 0.0181min^{-1}, k_{-a} = k_{-r} = 1.8 min^{-1}, t_{a} = t_{r} = 90min^{-1}, d_{a} = d_{r} = 0.54 min^{-1}, k_{fa} = k_{fr} = 0.9 min^{-1} , k_{da} = k_{dr} = k_{t} = 0.018 min^{-1} molecules^{-1}, k_{−da} = k_{−dr} = k_{−t} = 0.00018 min^{-1}, k_{l} = 36 min^{−1} , k_{ul} = 0.18 min^{−1}, \gamma = 1080molecules/min, c= 0.1 molecules, \lambda = 2.5, and \epsilon = 0.2 </math>
Modeling and the Result
We set the initial state of the system to have only 75 <math>P^{a/r}_{0,0}</math> and every other compound is set to zero. We used the SimBiology toolbox in the MATLAB for the simulation, and the result is shown as the graph below.
From the graph we could see the oscillation pattern stabilized after the first peek. And the period of oscillation is around 50 min. Since this is the result of a deterministic method, it is no surprise that the peeks are perfectly identical. From the result we get, it is safe to predict that the actual system should exhibit the oscillation pattern similar to the graph above with some possible noise and unstable behavior.
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 ( and 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
