Reactions
To simplify the calculations, we make the following assumptions:
(1)we assume that there are no external factors affecting LGG.
(2)To simplify the model, we consider the process of MazF being
transcribed
from mRNA to act on
the cell.
(3)It is assumed that glucose quickly binds to the binding site on the
promoter,
reaching equilibrium in a negligible amount of time.
Based on these assumptions, we have the following two separate reactions:
Reaction 1 describes the binding of Glu with the PT-acrp to form Glu-PT-acrp complex, where Glu
represents the concentration of the PT-acrp, Glu represents the concentration of glucose, kA1
represents the binding rate, k’A1 represents the dissociation rate, and GP represents the
concentration of the Glu-PT-acrp complex.
Reaction 2 describes the influence of Glu-PT-acrp on mRNA transcription and translation into
protein expression. kA2 represents the rate of the entire process, k’A3 represents the
degradation rate of the entire process, MazF represents the expression level of MazF, and MazF-
represents the degradation level of MazF.
Using the law of mass action, we can convert them into two ODE:
A1 relates the change in Glu and PT-acrp to the rate of change of GP
A2 relates the rate of change of MazF to the changes in GP
In A1, we want to obtain GP. Before that, we can eliminate PT-acrp by using the
relationship:
Free Promoter=Total Promoter – Bound Promoter
Which gives us:
To eliminate the PT-acrp to simplify the equation, and here PPG stands for the totality of
PT-acrp.
Based on assumption 3 we simplify the equations using the quasi-steady-state approximation QSSA.
[1]
Substituting this back into reaction 1 and using QSSA, [1] with the rate of change as 0, we get:
We let
Which gives us
Finally, by substituting the above result into reaction 2 and using QSSA [1], we get:
Which represents the equations of the MazF reactions model.
[1]: An SSA-based IR for Quantum Computing(https://arxiv.org/pdf/2109.02409v1)