# Dissociation constant

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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 Kd, which is the inverse of the association constant, or Ka. The strength of the binding interaction is inversely proportional to the Kd. Extremely tight-binding molecules such as antibodies and the their target exhibit Kd values in the picomolar range (10-12), while many drugs bind to their targets with Kd 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 Kd, that is,

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

The Kd, by definition, is

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

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

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

expanding the top terms yields

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

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

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

whose solution is:

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

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%