Bound Fraction &Unbound Fraction

By defining a simple equation where P is the promoter region and X is the ligand that can bind with P, which is an activator or a repressor:

The factional saturation of the promoter is denoted by Ps , which is the fraction of occupied binding sites:

At equilibrium of this reaction, the rate of forward reaction and that of backward reaction are the same:

Then we rearrange this equation and get:

Where kd is the dissociation constant of this binding event. Now we substitute equation 1.3 into equation 1.1 and hence obtain:

Assuming that the sum of the bound and the unbound equals one, the unbound fraction Pu is thus:

Bound fraction is considered for activation in gene regulation and unbound fraction for repressive regulation.

Modelling Equations

The chemical AHL (A) is required by LuxR to activate transcription. The active form for LuxR it consists of a dimer of the complex LuxR-AHL ((Y:A)2). Thus we can consider the production of this dimer as an elementary reaction: 2A+2Y --> (Y:A)2, whose rate depends upon A, the concentration of AHL, and the concentration of unbound LuxR monomer, Runbound.

Where rsynthesis is the rate of synthesis of the dimer, k1 is the binding coefficient RT is the total concentration of LuxR monomer, and R*(t) is the concentration of AHL-LuxR dimer And k2R*(t) is simply the rate of degradation of the dimer, where k2 is the degradation coefficient of AHL-LuxR.

For any given mRNA:

Where m(t) is the concentration of mRNA, k0 is the amount of mRNA produced, δm is the degradation rate of mRNA. For PhIF protein translated from the given mRNA:

Where P(t) is the concentration of PhIF protein, k3 is the translation rate, and δp is the degradation rate of PhIF protein Then substitute equation 1.4 into the calculation of k0 (for the activation of Plux promoter by AHL-LuxR correspond to bound fraction):

Where a is the transcription rate, k2 is the equivalent of kd in equation 1.4 and 1.5.

For sfGFP protein translated from the given mRNA:

Where G(t) is the concentration of sfGFP protein, k3 is the translation rate, and δg is the degradation rate of PhIF protein. Then substitute equation 1.5 into the calculation of k0 in this case (for the repression of PPhIF promoter by PhIF protein correspond to the unbound fraction):

Where a is the transcription rate, k2 is the equivalent of kd in equation 1.4 and 1.5.

Methods

Results

Discussion