View source for Dissociation constant

{{Short description|Chemical property}}
In [[chemistry]], [[biochemistry]], and [[pharmacology]], a '''dissociation constant''' (''K''<sub>D</sub>) is a specific type of [[equilibrium constant]] that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a [[Complex (chemistry)|complex]] falls apart into its component [[molecule]]s, or when a [[salt (chemistry)|salt]] splits up into its component [[ion]]s.  The dissociation constant is the [[multiplicative inverse|inverse]] of the [[association constant]]. In the special case of salts, the dissociation constant can also be called an [[ionization constant]].<ref>{{Cite web|date=2015-08-09|title=Dissociation Constant|url=https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Equilibria/Chemical_Equilibria/Dissociation_Constant|access-date=2020-10-26|website=Chemistry LibreTexts|language=en}}</ref><ref>Bioanalytical Chemistry Textbook De Gruyter 2021 https://doi.org/10.1515/9783110589160-206</ref> For a general reaction:

:<chem>
 A_\mathit{x} B_\mathit{y} <=> \mathit{x} A{} + \mathit{y} B
</chem>

in which a complex <math chem>\ce{A}_x \ce{B}_y</math> breaks down into ''x''&thinsp;A subunits and ''y''&thinsp;B subunits, the dissociation constant is defined as

:<math chem="">
 K_\mathrm{D} = \frac{[\ce A]^x [\ce B]^y}{[\ce A_x \ce B_y]}
</math>

where [A], [B], and [A<sub>''x''</sub>&thinsp;B<sub>''y''</sub>] are the equilibrium concentrations of A, B, and the complex A<sub>''x''</sub>&thinsp;B<sub>''y''</sub>, 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''<sub>D</sub> has a simple physical interpretation: when [A] = ''K''<sub>D</sub>, then [B] = [AB] or, equivalently, <math chem="">\tfrac {[\ce{AB}]}{{[\ce B]} + [\ce{AB}]} = \tfrac{1}{2}</math>.  That is, ''K''<sub>D</sub>, 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.{{citation needed|date=June 2016}}  It is useful as a quick description of the binding of a substance, in the same way that [[EC50|EC<sub>50</sub>]] and [[IC50|IC<sub>50</sub>]] 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 a free molecules, either [A] or [B].<ref name=Bisswanger2008>{{cite book 
| last = Bisswanger 
| first = Hans 
| year = 2008 
| title = Enzyme Kinetics: Principles and Methods 
| pages = 302 
| url = http://www.wiley-vch.de/books/sample/3527319573_c01.pdf 
| isbn = 978-3-527-31957-2 
| publisher = Wiley-VCH 
| location = Weinheim}}</ref>
In principle, the total amounts of molecule [A]<sub>0</sub> and [B]<sub>0</sub> added to the reaction are known.
They separate into free and bound components according to the mass conservation principle:

:<math chem>\begin{align}
  \ce{[A]_0} &= \ce{{[A]} + [AB]} \\
  \ce{[B]_0} &= \ce{{[B]} + [AB]}
\end{align}</math>

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,

:<math chem="">
 [\ce A]_0 = K_\mathrm{D} \frac{[\ce{AB}]}{[\ce B]} + [\ce{AB}]
</math>

This yields the concentration of the complex related to the concentration of either one of the free molecules
:<math chem="">
 \ce{[AB]} = \frac\ce{[A]_0 [B]}{K_\mathrm{D} + [\ce B]} = \frac\ce{[B]_0 [A]}{K_\mathrm{D} + [\ce A]}
</math>

===Macromolecules with identical independent binding sites===

Many biological proteins and enzymes can possess more than one binding site.<ref name=Bisswanger2008/>
Usually, when a [[ligand]] {{math|L}} binds with a macromolecule {{math|M}}, it can influence binding kinetics of other ligands {{math|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 {{mvar|n}} subunits are identical, symmetric and that they possess only a single binding site. Then the concentration of bound ligands <chem>[L]_{bound}</chem> becomes

:<math chem="">
 \ce{[L]}_\text{bound} = \frac{n\ce{[M]}_0 \ce{[L]}}{K_\mathrm{D} + \ce{[L]}}
</math>

In this case, <math chem>\ce{[L]}_\text{bound} \neq \ce{[LM]}</math>, but comprises all partially saturated forms of the macromolecule:

:<math chem>
 \ce{[L]}_\text{bound} = \ce{[LM]} + \ce{2[L_2 M]} + \ce{3[L_3 M]} + \ldots + n \ce{[L_\mathit{n} M]}  
</math>

where the saturation occurs stepwise

:<math chem>\begin{align}
                  \ce{{[L]} + [M]} &\ce{{} <=> {[LM]}}             & K'_1 &= \frac\ce{[L][M]}{[LM]}              & \ce{[LM]}       &= \frac\ce{[L][M]}{K'_1} \\
                 \ce{{[L]} + [LM]} &\ce{{} <=> {[L2 M]}}           & K'_2 &= \frac\ce{[L][LM]}{[L_2 M]}          & \ce{[L_2 M]}    &= \frac\ce{[L]^2[M]}{K'_1 K'_2} \\
               \ce{{[L]} + [L2 M]} &\ce{{} <=> {[L3 M]}}           & K'_3 &= \frac\ce{[L][L_2 M]}{[L_3 M]}       & \ce{[L_3 M]}    &= \frac\ce{[L]^3[M]}{K'_1 K'_2 K'_3} \\
                                   & \vdots                        &      & \vdots                               &                 & \vdots \\
 \ce{{[L]} + [L_\mathit{n - 1} M]} &\ce{{} <=> {[L_\mathit{n} M]}} & K'_n &= \frac\ce{[L][L_{n - 1} M]}{[L_n M]} & [\ce L_n \ce M] &= \frac{[\ce L]^n[\ce M]}{K'_1 K'_2 K'_3 \cdots K'_n}
\end{align}</math>

For the derivation of the general binding equation a saturation function <math chem>r</math> is defined as the quotient from the portion of bound ligand to the total
amount of the macromolecule:

:<math chem>
  r = \frac\ce{[L]_{bound}}\ce{[M]_0} 
    = \frac\ce{{[LM]} + {2[L_2 M]} + {3[L_3 M]} + ... + \mathit n[L_\mathit{n} M]}\ce{{[M]} + {[LM]} + {[L_2 M]} + {[L_3 M]} + ... + [L_\mathit{n} M]}
    = \frac{\sum_{i=1}^n \left( \frac{i [\ce L]^i}{\prod_{j=1}^i K_j'} \right) }{1 + \sum_{i=1}^n \left( \frac{[\ce L]^i}{\prod_{j=1}^i K_j'} \right)}
</math>
''K′<sub>n</sub>'' 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′''<sub>1</sub> describes a ligand being bound to any of the three binding sites. In this example, ''K′''<sub>2</sub> describes two molecules being bound and ''K′<sub>3</sub>'' 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''<sub>D</sub>. In our example, ''K′''<sub>1</sub> 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′''<sub>2</sub> 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′''<sub>3</sub> 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′''<sub>1</sub>, ''K′''<sub>2</sub> and ''K′''<sub>3</sub>) is not equal. This can be understood intuitively for our example of three possible binding sites. ''K′''<sub>1</sub> 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′''<sub>1</sub> would therefore be three times smaller than the individual ''K''<sub>D</sub>. ''K′''<sub>2</sub> describes the reaction from three states (one ligand bound) to three states (two ligands bound); therefore, ''K′''<sub>2</sub> would be equal to ''K''<sub>D</sub>. ''K′''<sub>3</sub> describes the reaction from three states (two ligands bound) to one state (three ligands bound); hence, the apparent dissociation constant ''K′''<sub>3</sub> is three times bigger than the microscopic dissociation constant ''K''<sub>D</sub>.
The general relationship between both types of dissociation constants for ''n'' binding sites is

:<math chem="">
 K_i'  = K_\mathrm{D} \frac{i}{n - i + 1}
</math>

Hence, the ratio of bound ligand to macromolecules becomes

:<math chem="">
  r = \frac{\sum_{i=1}^n i \left( \prod_{j=1}^i \frac{n - j + 1}{j} \right) \left( \frac\ce{[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 \binom{n}{i} \left( \frac{[L]}{K_\mathrm{D}} \right)^i }{1 + \sum_{i=1}^n \binom{n}{i} \left( \frac\ce{[L]}{K_\mathrm{D}} \right)^i}
</math>

where <math chem>\binom{n}{i} = \frac{n!}{(n - i)!i!}</math> is the [[binomial coefficient]].
Then the first equation is proved by applying the binomial rule
:<math chem="">
  r = \frac{n \left( \frac\ce{[L]}{K_\mathrm{D}} \right) \left(1 + \frac\ce{[L]}{K_\mathrm{D}} \right)^{n - 1} }{\left(1 + \frac\ce{[L]}{K_\mathrm{D}} \right)^n}
    = \frac{n \left( \frac\ce{[L]}{K_\mathrm{D}} \right) }{\left(1 + \frac\ce{[L]}{K_\mathrm{D}} \right)}
    = \frac{n [\ce L]}{K_\mathrm{D} + [\ce L]}
    = \frac\ce{[L]_{bound}}\ce{[M]_0}
</math>

==Protein–ligand binding==<!-- This section is linked from [[Antibody]] and [[MDMA]] -->
{{Main|Receptor–ligand kinetics}}
The dissociation constant is commonly used to describe the [[Chemical affinity|affinity]] between a [[Ligand (biochemistry)|ligand]] <chem>L</chem> (such as a [[drug]]) and a [[protein]] <chem>P</chem>; i.e., how tightly a ligand binds to a particular protein.  Ligand–protein affinities are influenced by [[non-covalent| non-covalent intermolecular interactions]] between the two molecules such as [[hydrogen bond]]ing, [[electrostatic| electrostatic interactions]], [[hydrophobic]] and [[van der Waals force]]s. Affinities can also be affected by high concentrations of other macromolecules, which causes [[macromolecular crowding]].<ref>{{Cite journal
 | last1 = Zhou   | first1 = H.
 | last2 = Rivas  | first2 = G.
 | last3 = Minton | first3 = A.
 | title = Macromolecular crowding and confinement: biochemical, biophysical, and potential physiological consequences
 | journal = Annual Review of Biophysics
 | volume = 37
 | pages = 375–397
 | year = 2008
 | pmid = 18573087
 | doi = 10.1146/annurev.biophys.37.032807.125817
 | pmc = 2826134 
}}</ref><ref>{{Cite journal
 | last1 = Minton | first1 = A. P.
 | title = The influence of macromolecular crowding and macromolecular confinement on biochemical reactions in physiological media
 | journal = The Journal of Biological Chemistry
 | volume = 276
 | issue = 14
 | pages = 10577–10580
 | year = 2001
 | pmid = 11279227
 | doi = 10.1074/jbc.R100005200   
| url = http://www.jbc.org/content/276/14/10577.full.pdf
 | doi-access = free
 }}</ref>

The formation of a [[Protein–ligand complex|ligand–protein complex]] <chem>LP</chem> can be described by a two-state process

:<chem>
 L + P <=> LP
</chem>

the corresponding dissociation constant is defined

:<math chem="">
 K_\mathrm{D} = \frac{\left[ \ce{L} \right] \left[ \ce{P} \right]}{\left[ \ce{LP} \right]}
</math>

where <chem>[P], [L]</chem>, and <chem>[LP]</chem> represent [[molar concentration]]s of the protein, ligand, and protein–ligand complex, respectively.

The dissociation constant has [[Molar concentration|molar]] units (M) and corresponds to the ligand concentration <chem>[L]</chem> at which half of the proteins are occupied at equilibrium,<ref>{{Cite journal|last1=Björkelund|first1=Hanna|last2=Gedda|first2=Lars|last3=Andersson|first3=Karl|date=2011-01-31|title=Comparing the Epidermal Growth Factor Interaction with Four Different Cell Lines: Intriguing Effects Imply Strong Dependency of Cellular Context|journal=PLOS ONE|volume=6|issue=1|pages=e16536|doi=10.1371/journal.pone.0016536|pmid=21304974|issn=1932-6203|bibcode=2011PLoSO...616536B|pmc=3031572|doi-access=free}}</ref> i.e., the concentration of ligand at which the concentration of protein with ligand bound <chem>[LP]</chem> equals the concentration of protein with no ligand bound <chem>[P]</chem>.  The smaller the dissociation constant, the more tightly bound the ligand is, or the higher the affinity between ligand and protein.  For example, a ligand with a nanomolar (nM) dissociation constant binds more tightly to a particular protein than a ligand with a micromolar (μM) dissociation constant.

Sub-picomolar dissociation constants as a result of non-covalent binding interactions between two molecules are rare.  Nevertheless, there are some important exceptions.  [[Biotin]] and [[avidin]] bind with a dissociation constant of roughly 10<sup>−15</sup> M = 1 fM = 0.000001 nM.<ref>{{Cite journal
 | last1 = Livnah  | first1 = O.
 | last2 = Bayer   | first2 = E.
 | last3 = Wilchek | first3 = M.
 | last4 = Sussman | first4 = J.
 | year = 1993
 | journal = Proceedings of the National Academy of Sciences of the United States of America
 | pages = 5076–5080
 | pmid = 8506353
 | pmc = 46657
 | doi = 10.1073/pnas.90.11.5076
 | volume = 90
 | title = Three-dimensional structures of avidin and the avidin-biotin complex
 | issue = 11
 |bibcode = 1993PNAS...90.5076L
| doi-access = free
 }}</ref>
[[Ribonuclease inhibitor]] proteins may also bind to [[ribonuclease]] with a similar 10<sup>−15</sup> M affinity.<ref>{{Cite journal
 | last1 = Johnson     | first1 = R.
 | last2 = Mccoy       | first2 = J.
 | last3 = Bingman     | first3 = C.
 | last4 = Phillips Gn | first4 = J.
 | last5 = Raines      | first5 = R.
 | title = Inhibition of human pancreatic ribonuclease by the human ribonuclease inhibitor protein
 | journal = Journal of Molecular Biology
 | volume = 368
 | issue = 2
 | pages = 434–449
 | year = 2007
 | pmid = 17350650
 | doi = 10.1016/j.jmb.2007.02.005
 | pmc = 1993901 
}}</ref>  

The dissociation constant for a particular ligand–protein interaction can change with solution conditions (e.g., [[temperature]], [[pH]] and salt concentration). The effect of different solution conditions is to effectively modify the strength of any [[non-covalent|intermolecular interactions]] holding a particular ligand–protein complex together.

Drugs can produce harmful side effects through interactions with proteins for which they were not meant to or designed to interact. Therefore, much pharmaceutical research is aimed at designing drugs that bind to only their target proteins (negative design) with high affinity (typically 0.1–10 nM) or at improving the affinity between a particular drug and its ''[[in-vivo|in vivo]]'' protein target (positive design).

===Antibodies===
In the specific case of antibodies (Ab) binding to antigen (Ag), usually the term '''affinity constant''' refers to the association constant.
:<chem>
 Ab + Ag <=> AbAg 
</chem>

:<math chem="">
 K_\mathrm{A} = \frac{\left[ \ce{AbAg} \right]}{\left[ \ce{Ab} \right] \left[ \ce{Ag} \right]} = \frac{1}{K_\mathrm{D}} 
</math>

This [[chemical equilibrium]] is also the ratio of the on-rate (''k''<sub>forward</sub> or ''k''<sub>a</sub>) and off-rate (''k''<sub>back</sub> or ''k''<sub>d</sub>) constants. Two antibodies can have the same affinity, but one may have both a high on- and off-rate constant, while the other may have both a low on- and off-rate constant.

:<math chem="">
 K_A = \frac{k_\text{forward}}{k_\text{back}}  = \frac{\mbox{on-rate}}{\mbox{off-rate}}
</math>

== Acid–base reactions ==
{{Acids and bases}}
{{Main|Acid dissociation constant}}
For the [[deprotonation]] of [[acid]]s, ''K'' is known as ''K''<sub>a</sub>, the  [[acid dissociation constant]].  Strong acids, such as [[sulfuric acid|sulfuric]] or [[phosphoric acid]], have large dissociation constants; weak acids, such as [[acetic acid]], have small dissociation constants.

The symbol ''K''<sub>a</sub>, used for the acid dissociation constant, can lead to confusion with the [[association constant]], and it may be necessary to see the reaction or the equilibrium expression to know which is meant.

Acid dissociation constants are sometimes expressed by p''K''<sub>a</sub>, which is defined by

:<math chem="">
 \text{p}K_\text{a} = -\log_{10}{K_\mathrm{a}}
</math>

This <math chem>\mathrm{p}K</math> notation is seen in other contexts as well; it is mainly used for [[covalent]] dissociations (i.e., reactions in which chemical bonds are made or broken) since such dissociation constants can vary greatly.

A molecule can have several acid dissociation constants. In this regard, that is depending on the number of the protons they can give up, we define ''monoprotic'', ''diprotic'' and ''triprotic'' [[acid#Polyprotic acids|acids]].  The first (e.g., acetic acid or [[ammonium]]) have only one dissociable group, the second (e.g., [[carbonic acid]], [[bicarbonate]], [[glycine]]) have two dissociable groups and the third (e.g., phosphoric acid) have three dissociable groups. In the case of multiple p''K'' values they are designated by indices: p''K''<sub>1</sub>, p''K''<sub>2</sub>, p''K''<sub>3</sub> and so on. For amino acids, the p''K''<sub>1</sub> constant refers to its [[carboxyl]] (–COOH) group, p''K''<sub>2</sub> refers to its [[amino]] (–NH<sub>2</sub>) group and the p''K''<sub>3</sub> is the p''K'' value of its [[side chain]].

:<math chem="">\begin{align}
      \ce{H3 B} &\ce{{} <=> {H+} + {H2 B^-}}   & K_1 &= \ce{[H+] . [H2 B^-] \over [H3 B]}      & \mathrm{p}K_1 &= -\log K_1 \\
    \ce{H2 B^-} &\ce{{} <=> {H+} + {H B^{2-}}} & K_2 &= \ce{[H+] . [H B ^{2-}] \over [H2 B^-]} & \mathrm{p}K_2 &= -\log K_2 \\
  \ce{H B^{-2}} &\ce{{} <=> {H+} + {B^{3-}}}   & K_3 &= \ce{[H+] . [B^{3-}] \over [H B^{2-}]}  & \mathrm{p}K_3 &= -\log K_3 
\end{align}</math>

==Dissociation constant of water==
{{Main|Self-ionization of water}}
The dissociation constant of [[water]] is denoted ''K''<sub>w</sub>:

:<math chem>K_\mathrm{w} = [\ce{H}^+] [\ce{OH}^-]</math>

The concentration of water, [H<sub>2</sub>O], is omitted by convention, which means that the value of ''K''<sub>w</sub> differs from the value of ''K''<sub>eq</sub> that would be computed using that concentration.

The value of ''K''<sub>w</sub> varies with temperature, as shown in the table below.  This variation must be taken into account when making precise measurements of quantities such as pH.

:{| class="wikitable" style="text-align:center;"
|-
! Water temperature
! ''K''<sub>w</sub>
! p''K''<sub>w</sub><ref>{{cite journal | last1 = Bandura | first1 = Andrei V. | last2 = Lvov | first2 = Serguei N. | year = 2006 | title = The Ionization Constant of Water over Wide Ranges of Temperature and Density | journal = Journal of Physical and Chemical Reference Data | volume = 35 | issue = 1 | pages = 15–30 | doi = 10.1063/1.1928231 | url = https://www.nist.gov/data/PDFfiles/jpcrd696.pdf | bibcode = 2006JPCRD..35...15B | access-date = 2017-07-13 | archive-date = 2013-05-12 | archive-url = https://web.archive.org/web/20130512174613/http://www.nist.gov/data/PDFfiles/jpcrd696.pdf | url-status = dead }}</ref>
|-
|{{0|00}}0&nbsp;°C
|{{0}}0.112{{e|-14}}
|14.95
|-
|{{0}}25&nbsp;°C
|{{0}}1.023{{e|-14}}
|13.99
|-
|{{0}}50&nbsp;°C
|{{0}}5.495{{e|-14}}
|13.26
|-
|{{0}}75&nbsp;°C
|19.95{{0}}{{e|-14}}
|12.70
|-
|100&nbsp;°C
|56.23{{0}}{{e|-14}}
|12.25
|}

== See also ==
* [[Acid]]
* [[Equilibrium constant]]
* [[Ki Database|''K''<sub>i</sub> Database]]
* [[Competitive inhibition]]
* [[pH]]
* [[Scatchard plot]]
* [[Ligand binding]]
* [[Avidity#Antibody-antigen interaction|Avidity]]

==References==
{{Reflist}}

{{Chemical equilibria}}
{{Pharmacology}}
{{Authority control}}

[[Category:Equilibrium chemistry]]
[[Category:Enzyme kinetics]]