Dissociation constant

In biochemistry, chemistry and physics, the binding interaction of two molecules that bind with each other, for example a protein and a DNA duplex, is often quantified in terms of a dissociation constant, abbreviated as K_d, which is the inverse of the association constant, or K_a. The strength of the binding interaction is inversely proportional to the K_d. Extremely tight-binding molecules such as antibodies and the their target exhibit K_d values in the picomolar range (10^-12), while many drugs bind to their targets with K_d values in the nanomolar (10^-9) to micromolar (10^-6) range. Given the Kd of an interaction, and the initial concentrations of the interacting molecules, the amount of complex can be calculated.

Biomolecular Definition

Given two molecules, A & B, with initial molar concentrations [A]₀ and [B]₀, that form a reversible binding complex AB, having a certain dissociation constant K_d, that is,

$\mathbf {A} +\mathbf {B} \leftrightarrows \mathbf {AB}$

The Kd, by definition, is

$\mathbf {K_{d}} ={\frac {\mathbf {[A]} \times \mathbf {[B]} }{\mathbf {[AB]} }}$

Using the facts that $[A]=[A]_{0}-[AB]$ and $[B]=[B]_{0}-[AB]$ gives

$\mathbf {K_{d}} ={\frac {(\mathbf {[A]_{0}} -\mathbf {[AB]} )\times (\mathbf {[B]_{0}} -\mathbf {[AB]} )}{\mathbf {[AB]} }}$

expanding the top terms yields

$\mathbf {K_{d}} ={\frac {\mathbf {[A]_{0}} \times \mathbf {[B]_{0}} -\mathbf {[A]_{0}} \times \mathbf {[AB]} -\mathbf {[B]_{0}} \times \mathbf {[AB]} +\mathbf {[AB]} \times \mathbf {[AB]} }{\mathbf {[AB]} }}$

Multiplying both sides by [AB] and rearranging gives a quadratic equation:

$\mathbf {[AB]^{2}} -\left(\mathbf {[A]_{0}} +\mathbf {[B]_{0}} +\mathbf {K_{d}} \right)\times \mathbf {[AB]} +\left(\mathbf {[A]_{0}} \times \mathbf {[B]_{0}} \right)=\mathbf {0}$

whose solution is:

$\mathbf {[AB]} ={\frac {(\mathbf {[A]_{0}} +\mathbf {[B]_{0}} +\mathbf {K_{d}} )+/-{\sqrt {(\mathbf {[A]_{0}} +\mathbf {[B]_{0}} +\mathbf {K_{d})^{2}} -\mathbf {4} \mathbf {[A]_{0}} \mathbf {[B]_{0}} }}}{\mathbf {2} }}$

Given the physical limitation that [AB] can not be greater than either [A]₀ or [B]₀ eliminates the solution in which the square root term is added to the first term.

Implications

An inspection of the resulting solution shown above illustrates that in order to have an appreciable amount of bound material, the interacting molecules must be present at concentrations of 1/100 to 100 times the dissociation constant, as demonstrated in the table below, in which the concentrations of A and B are expressed in units of Kd.

[A]/Kd	[B]/Kd	%B bound ([AB]/[B])*100
0.001	0.001	0%
0.01	0.01	1%
0.1	0.1	8%
1.0	1.0	38%
10	10	73%
100	100	90%
1000	1000	97%