Assumption
1. In order to write the equations down simply, we assume that all the chemical reaction rates are proportional to the concentration of each reagent (e.g. for the reaction: A+B+C→D+E,the forward rate r+=k+[A][B][C]).
2.For every substances produced by biobricks, we assume that their production rate =φ[mRNA],
[mRNA]= the concentration of the promoted biobrick
φ= the result of multiplication of rate constant, coefficient of correction (since a biobrick is different from a reagtant), a dimension T-1
Equations & Solutions
According to the picture, we can write down 3 equations as follows:
P.S. φ= the result of multiplication of rate constant, coefficient of correction (since a promoter is different from a reagtant), a dimension 𝑇−1
By solving these 3 equations, the solution expressed by φ、k and [Pa] are as follows:
When the concentration of each activated promoter reaches to each of their steady state, then we can simplify the equations as follows:
Besides, since limt→∞(1-1/ekt) = 1, satisfying the definition of the horizontal asymptotes. And d(1-1/ekt)/dt=ke-kt>0 (t∈[0,∞)), so it is a strictly increasing function.
So, this is a strictly increasing and convergent function with an upper bound 1.
Then the result is that the extremum of the concentration is:
Degradation Rate Constant Calculation
As for the other variable written in the solutions, the degradation rate constant, can also be solved with differential equations. Since the degradation rate is an “order one” reaction, the equation can be written as follow:
dM/dt= -kdM
Then, after solving the equation and substituting the boundary conditions
(t = 0⇒M = M0), the the solution is:
According to the project 2008 iGEM KULeuven and 2014 iGEM Edinburgh had done, both GFP-LVA and RFP-LVA degrades to half of the amount within 50 to 60 minutes, so we assume that cjblue is the same. The RFP and BFP reference are as follow (the latter degrades to half of the amount about 50 minutes while the former does about 3 hours). So we can get
From these degradation rate constants and the relation between concentration and time, the “[cjblue],[RFP],[BFP]-t Diagram” is as follow:
According to this simulation diagram, we can know that cjblue and BFP increase faster—coming to 90% of maximum only takes about 3 hours. As for RFP, it takes about 5 hours to reach 70% of maximum, which is also acceptable.
Through mathematical modeling, when observing the sicker changing to a specific color, we can calculate the ratio of each kind of chromoprotein by quantifying it.
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