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[CoPB],
[CoPB]= 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 (Figure 1), 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:
