User:David E. Volk/Sandbox: Difference between revisions

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In [[biochemistry]], 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<sub>d</sub>'', which is the inverse of the [[association constant]], or K<sub>a</sub>.  The strength of the binding interaction is inversely proportional to the K<sub>d</sub>.  Extremely tight-binding molecules such as [[antibody|antibodies]] and the their target exhibit K<sub>d</sub> values in the picomolar range (10<sup>-12</sup>), while many drugs bind to their targets with K<sub>d</sub> values in the nanomolar (10<sup>-9</sup>) to micromolar (10<sup>-6</sup>) 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 [[molarity|molar]] concentrations '''[A]<sub>0</sub>''' and '''[B]<sub>0</sub>''', that form a reversible binding complex '''AB''', having a certain dissociation constant '''K<sub>d</sub>''', that is:
<math> A + B \stackrel{\textstyle \leftarrow}{\rightarrow}  AB </math>
The Kd, by definition, is:
<math> K_d = \frac{[A]*[B]}{[AB]} </math>
Using the facts that <math>[A] = [A]_0 - [AB]</math> and <math>[B] = [B]_0 - [AB]</math> gives
<math> K_d = \frac{([A]_0 - [AB])*([B]_0 - [AB])}{[AB]} </math>
expanding the top terms yields
<math> K_d = \frac{[A]_0 * [B]_0 - [A]_0 * [AB] - [B]_0 * [AB] + [AB]*[AB]}{[AB]} </math>
Multiplying both sides by [AB] and rearranging gives a quadratic equation:
<math> [AB]^2 - ([A]_0 + [B]_0 + K_d)*[AB] + ([A]_0 * [B]_0) = 0 </math>
whose solution is:
<math> [AB] = \frac{([A]_0 + [B]_0 + K_d) +/- \sqrt{([A]_0 + [B]_0 + K_d)^2 - 4 [A]_0 [B]_0}}{2} </math>
Given the physical limitation that [AB] can not be greater than either [A]<sub>0</sub> or [B]<sub>0</sub> 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.
{| class = "wikitable" align="center"
! [A]/Kd!![B]/Kd!!%B bound<br>([AB]/[B])*100
|- align="center"
|0.001||0.001||0%
|- align="center"
|0.01||0.01||1%
|- align="center"
|0.1||0.1||8%
|- align="center"
|1.0||1.0||38%
|- align="center"
|10||10||73%
|- align="center"
|100||100||90%
|- align="center"
|1000||1000||97%
|-
|}

Revision as of 12:08, 20 June 2009

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