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| Home :: KinExA Systems |
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| Kd Mathematics |
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The mathematical steps involved in relating fluorescent signals from uncomplexed reactant
to the equilibrium (affinity) constant.
We begin with a reversible biomolecular reaction:
At equilibrium, the rate of AB complex formation is equal to the rate of dissociation, or:
(1) kon[B][A] = koff[AB]
Where:
kon = forward rate constant
[B] = receptor (antibody) binding site concentration
[A] = ligand (antigen) binding site concentration
koff = reverse rate constant
[AB] = concentration of complex
The equilibrium dissociation constant, Kd, is defined as koff/kon, therefore:
(2) 
Where:
Kd = koff/kon
From conservation of mass:
(3) [A0] = [A] + [AB]
Where:
[A0] = Total ligand (antigen) binding site concentration
and:
(4) [B0] = [B] + [AB]
Where:
[B0] = Total receptor (antibody) binding site concentration
Using equations (3) and (4) to express equation (2) in terms of [B0], [B], Kd, and [A0], and rearranging yields:
(5) 
The physically realizable solution to the quadratic equation is the positive root:
(6) 
Which simplifies to:
(7) 
The instrument signal is linearly related to [B] (Blake et. al, 1997), and therefore follows equation (8):
(8) 
Where:
Signal = instrument signal for [B]
Sig100% = signal from [B0]
Sig0% = signal from [B] = 0
(non- specific binding or NSB)
Solving equation (8) for [B], substituting [B] into equation (7), and solving for signal gives:
(9) 
Equation (9) gives the instrument signal as a function of [A0], [B0], Kd, Sig100%, Sig0%, and is the basic equation used in the equilibrium analysis files.
Experimental data is in the form of instrument signals measured at various [A0]'s. The KinExA data analysis package uses equation (9) to determine theoretical signals for each [A0], and then calculates the error between the theoretical signal and the corresponding instrument signal. An iterative least squares approach is used to find an optimal solution for the unknowns, [B0], Kd, Sig0%, and Sig100%. The optimal solution values are displayed and used to create a theoretical signal vs. [A0] curve which is plotted with the experimental data.
For the dual curve analysis, two experiments using different [B0] values are performed, and both sets of experimental data are analyzed simultaneously. Sig0% and Sig100% values are treated separately for each data set and [B0] maintains a fixed dilution factor between the two sets of data. The Kd value is a system constant, and therefore maintains the same value for both curves. The data analysis package uses an iterative least squares approach to find an optimal dual curve solution for Kd, and optimal separate curve solutions for [B0], Sig0% and Sig100%. The optimal solution values are given and used to create two theoretical signal vs. [A0] curves which are plotted with the experimental data.
The 95% confidence interval for the calculated Kd value is determined by varying the Kd parameter and re-optimizing the remaining variables at each point. The 95% confidence interval for [B0] is determined by varying the [B0] value and re-optimizing the remaining variables at each point.' |
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