Dissociation constant

${\displaystyle {\text{A}}_{\mathit{\text{x}}}^{}{\text{B}}_{\mathit{\text{y}}}^{}\stackrel{-\rightharpoonup }{\leftharpoondown -}\mathit{\text{x}}\text{A}+\mathit{\text{y}}\text{B}}$

in which a complex ${\displaystyle {\text{A}}_x{\text{B}}_y}$ breaks down into x A subunits and y B subunits, the dissociation constant is defined as

${\displaystyle K_{\mathrm{D}}=\frac{[\text{A}{]}^x[\text{B}{]}^y}{[{\text{A}}_x{\text{B}}_y]}}$

where [A], [B], and [A_x B_y] are the equilibrium concentrations of A, B, and the complex A_x B_y, respectively.

One reason for the popularity of the dissociation constant in biochemistry and pharmacology is that in the frequently encountered case where x = y = 1, K_D has a simple physical interpretation: when [A] = K_D, then [B] = [AB] or, equivalently, ${\displaystyle \frac{[\text{AB}]}{[\text{B}]+[\text{AB}]}=\frac{1}{2}}$. That is, K_D, which has the dimensions of concentration, equals the concentration of free A at which half of the total molecules of B are associated with A. This simple interpretation does not apply for higher values of x or y. It also presumes the absence of competing reactions, though the derivation can be extended to explicitly allow for and describe competitive binding. It is useful as a quick description of the binding of a substance, in the same way that EC₅₀ and IC₅₀ describe the biological activities of substances.

Concentration of bound molecules

Molecules with one binding site

Experimentally, the concentration of the molecule complex [AB] is obtained indirectly from the measurement of the concentration of free molecules, either [A] or [B]. In principle, the total amounts of molecule [A]₀ and [B]₀ added to the reaction are known. They separate into free and bound components according to the mass conservation principle:

${\displaystyle \begin{array}{cc}{[\text{A}]}_0^{}& =[\text{A}]+[\text{AB}]\\ {[\text{B}]}_0^{}& =[\text{B}]+[\text{AB}]\end{array}}$

To track the concentration of the complex [AB], one substitutes the concentration of the free molecules ([A] or [B]), of the respective conservation equations, by the definition of the dissociation constant,

${\displaystyle [\text{A}{]}_0=K_{\mathrm{D}}\frac{[\text{AB}]}{[\text{B}]}+[\text{AB}]}$

This yields the concentration of the complex related to the concentration of either one of the free molecules

${\displaystyle [\text{AB}]=\frac{{[\text{A}]}_0^{}[\text{B}]}{K_{\mathrm{D}}+[\text{B}]}=\frac{{[\text{B}]}_0^{}[\text{A}]}{K_{\mathrm{D}}+[\text{A}]}}$

Macromolecules with identical independent binding sites

Many biological proteins and enzymes can possess more than one binding site. Usually, when a ligand L binds with a macromolecule M, it can influence binding kinetics of other ligands L binding to the macromolecule. A simplified mechanism can be formulated if the affinity of all binding sites can be considered independent of the number of ligands bound to the macromolecule. This is valid for macromolecules composed of more than one, mostly identical, subunits. It can be then assumed that each of these n subunits are identical, symmetric and that they possess only a single binding site. Then the concentration of bound ligands ${\displaystyle {[\text{L}]}_{\text{bound}}^{}}$ becomes

${\displaystyle {[\text{L}]}_{\text{bound}}=\frac{n{[\text{M}]}_0[\text{L}]}{K_{\mathrm{D}}+[\text{L}]}}$

In this case, ${\displaystyle {[\text{L}]}_{\text{bound}}\ne [\text{LM}]}$, but comprises all partially saturated forms of the macromolecule:

${\displaystyle {[\text{L}]}_{\text{bound}}=[\text{LM}]+2[{\text{L}}_2^{}\text{M}]+3[{\text{L}}_3^{}\text{M}]+\dots +n[{\text{L}}_{\mathit{\text{n}}}^{}\text{M}]}$

where the saturation occurs stepwise

${\displaystyle \begin{array}{cccccc}[\text{L}]+[\text{M}]& \stackrel{-\rightharpoonup }{\leftharpoondown -}[\text{LM}]& K_1^{\prime }& =\frac{[\text{L}][\text{M}]}{[LM]}& [\text{LM}]& =\frac{[\text{L}][\text{M}]}{K_1^{\prime }}\\ [\text{L}]+[\text{LM}]& \stackrel{-\rightharpoonup }{\leftharpoondown -}[{\text{L}}_2^{}\text{M}]& K_2^{\prime }& =\frac{[\text{L}][\text{LM}]}{[L_{2}M]}& [{\text{L}}_2^{}\text{M}]& =\frac{{[\text{L}]}^2[\text{M}]}{K_1^{\prime }{K}_2^{\prime }}\\ [\text{L}]+[{\text{L}}_2^{}\text{M}]& \stackrel{-\rightharpoonup }{\leftharpoondown -}[{\text{L}}_3^{}\text{M}]& K_3^{\prime }& =\frac{[\text{L}][{\text{L}}_2^{}\text{M}]}{[L_{3}M]}& [{\text{L}}_3^{}\text{M}]& =\frac{{[\text{L}]}^3[\text{M}]}{K_1^{\prime }{K}_2^{\prime }{K}_3^{\prime }}\\ & ⋮& & ⋮& & ⋮\\ [\text{L}]+[{\text{L}}_{\mathit{\text{n}}\mathit{-}\mathit{1}}^{}\text{M}]& \stackrel{-\rightharpoonup }{\leftharpoondown -}[{\text{L}}_{\mathit{\text{n}}}^{}\text{M}]& K_n^{\prime }& =\frac{[\text{L}][{\text{L}}_{\text{n}-1}^{}\text{M}]}{[L_{n}M]}& [{\text{L}}_{\text{n}}^{}\text{M}]& =\frac{{[\text{L}]}^{\text{n}}[\text{M}]}{K_1^{\prime }{K}_2^{\prime }{K}_3^{\prime }\cdots K_n^{\prime }}\end{array}}$

For the derivation of the general binding equation a saturation function ${\displaystyle r}$ is defined as the quotient from the portion of bound ligand to the total amount of the macromolecule:

${\displaystyle r=\frac{{[\text{L}]}_{\text{bound}}^{}}{{[\text{M}]}_0^{}}=\frac{[\text{LM}]+2[{\text{L}}_2^{}\text{M}]+3[{\text{L}}_3^{}\text{M}]+\cdot \cdot \cdot +\mathit{\text{n}}[{\text{L}}_{\mathit{\text{n}}}^{}\text{M}]}{[\text{M}]+[\text{LM}]+[{\text{L}}_2^{}\text{M}]+[{\text{L}}_3^{}\text{M}]+\cdot \cdot \cdot +[{\text{L}}_{\mathit{\text{n}}}^{}\text{M}]}=\frac{\sum _{i=1}^n\left(\frac{i[\text{L}{]}^i}{\prod _{j=1}^i{K}_j^{\prime }}\right)}{1+\sum _{i=1}^n\left(\frac{[\text{L}{]}^i}{\prod _{j=1}^i{K}_j^{\prime }}\right)}}$

K′_n are so-called macroscopic or apparent dissociation constants and can result from multiple individual reactions. For example, if a macromolecule M has three binding sites, K′₁ describes a ligand being bound to any of the three binding sites. In this example, K′₂ describes two molecules being bound and K′₃ three molecules being bound to the macromolecule. The microscopic or individual dissociation constant describes the equilibrium of ligands binding to specific binding sites. Because we assume identical binding sites with no cooperativity, the microscopic dissociation constant must be equal for every binding site and can be abbreviated simply as K_D. In our example, K′₁ is the amalgamation of a ligand binding to either of the three possible binding sites (I, II and III), hence three microscopic dissociation constants and three distinct states of the ligand–macromolecule complex. For K′₂ there are six different microscopic dissociation constants (I–II, I–III, II–I, II–III, III–I, III–II) but only three distinct states (it does not matter whether you bind pocket I first and then II or II first and then I). For K′₃ there are three different dissociation constants — there are only three possibilities for which pocket is filled last (I, II or III) — and one state (I–II–III).

Even when the microscopic dissociation constant is the same for each individual binding event, the macroscopic outcome (K′₁, K′₂ and K′₃) is not equal. This can be understood intuitively for our example of three possible binding sites. K′₁ describes the reaction from one state (no ligand bound) to three states (one ligand bound to either of the three binding sides). The apparent K′₁ would therefore be three times smaller than the individual K_D. K′₂ describes the reaction from three states (one ligand bound) to three states (two ligands bound); therefore, K′₂ would be equal to K_D. K′₃ describes the reaction from three states (two ligands bound) to one state (three ligands bound); hence, the apparent dissociation constant K′₃ is three times bigger than the microscopic dissociation constant K_D. The general relationship between both types of dissociation constants for n binding sites is

${\displaystyle K_i^{\prime }=K_{\mathrm{D}}\frac{i}{n-i+1}}$

Hence, the ratio of bound ligand to macromolecules becomes

${\displaystyle r=\frac{\sum _{i=1}^{n}i\left(\prod _{j=1}^i\frac{n-j+1}{j}\right){\left(\frac{[\text{L}]}{K_{\mathrm{D}}}\right)}^i}{1+\sum _{i=1}^n\left(\prod _{j=1}^i\frac{n-j+1}{j}\right){\left(\frac{[L]}{K_{\mathrm{D}}}\right)}^i}=\frac{\sum _{i=1}^{n}i(\genfrac{}{}{0ex}{}{n}{i}){\left(\frac{[L]}{K_{\mathrm{D}}}\right)}^i}{1+\sum _{i=1}^n(\genfrac{}{}{0ex}{}{n}{i}){\left(\frac{[\text{L}]}{K_{\mathrm{D}}}\right)}^i}}$

where ${\displaystyle (\genfrac{}{}{0ex}{}{n}{i})=\frac{n!}{(n-i)!i!}}$ is the binomial coefficient. Then the first equation is proved by applying the binomial rule

${\displaystyle r=\frac{n\left(\frac{[\text{L}]}{K_{\mathrm{D}}}\right){\left(1+\frac{[\text{L}]}{K_{\mathrm{D}}}\right)}^{n-1}}{{\left(1+\frac{[\text{L}]}{K_{\mathrm{D}}}\right)}^n}=\frac{n\left(\frac{[\text{L}]}{K_{\mathrm{D}}}\right)}{\left(1+\frac{[\text{L}]}{K_{\mathrm{D}}}\right)}=\frac{n[\text{L}]}{K_{\mathrm{D}}+[\text{L}]}=\frac{{[\text{L}]}_{\text{bound}}^{}}{{[\text{M}]}_0^{}}}$