# 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 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 K_{d} of an interaction, and the initial concentrations of the interacting molecules, the amount of complex can be calculated.

## [edit] Biomolecular Definition

Given two molecules, **A** and **B**, with initial molar concentrations **[A] _{0}** and

**[B]**, that form a reversible binding complex

_{0}**AB**, having a certain dissociation constant

**K**, that is,

_{d}

The K_{d}, by definition, is

Using the facts that [*A*] = [*A*]_{0} − [*A**B*] and [*B*] = [*B*]_{0} − [*A**B*] gives

expanding the top terms yields

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

whose solution is:

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

## [edit] 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 K_{d}.

[A]/K_{d} |
[B]/K_{d} |
%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% |

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