Dissociation constant
In chemistry, biochemistry, and pharmacology, a dissociation constant (Kd{displaystyle K_{d}}
For a general reaction:
- AxBy↽−−⇀xA+yB{displaystyle {ce {A_{mathit {x}}B_{mathit {y}}<=>{mathit {x}}A{}+{mathit {y}}B}}}
in which a complex AxBy{displaystyle {ce {A}}_{x}{ce {B}}_{y}}
- Kd=[A]x[B]y[AxBy]{displaystyle K_{d}={frac {[{ce {A}}]^{x}[{ce {B}}]^{y}}{[{ce {A}}_{x}{ce {B}}_{y}]}}}
![{displaystyle K_{d}={frac {[{ce {A}}]^{x}[{ce {B}}]^{y}}{[{ce {A}}_{x}{ce {B}}_{y}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/491f39be9c3e2983be35d734b07b06706f12a2b3)
where [A], [B], and [AxBy] are the equilibrium concentrations of A, B, and the complex AxBy, 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, Kd has a simple physical interpretation: when [A]=Kd{displaystyle [{ce {A}}]=K_{d}}![{displaystyle [{ce {A}}]=K_{d}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15e90773fb0b7aff63d33eb59d6bc93208132584)
![{displaystyle [{ce {B}}]=[{ce {AB}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7a65fdb22d6431cbd7c98446959be3ee5c00f52)
![{displaystyle {tfrac {[{ce {AB}}]}{{[{ce {B}}]}+[{ce {AB}}]}}={tfrac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28addb6a76851f6b1035e5c68ba2e816b86ec48d)
Contents
1 Concentration of bound molecules
1.1 Molecules with one binding site
1.2 Macromolecules with identical independent binding sites
2 Protein-ligand binding
2.1 Antibodies
3 Acid–base reactions
4 Dissociation constant of water
5 See also
6 References
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].[1]
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:
- [A]0=[A]+[AB][B]0=[B]+[AB]{displaystyle {begin{aligned}{ce {[A]_0}}&={ce {{[A]}+ [AB]}}\{ce {[B]_0}}&={ce {{[B]}+ [AB]}}end{aligned}}}
![{displaystyle {begin{aligned}{ce {[A]_0}}&={ce {{[A]}+ [AB]}}\{ce {[B]_0}}&={ce {{[B]}+ [AB]}}end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7836162d1b0e3622e3d8e115fe2294be41011ff3)
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,
- [A]0=Kd[AB][B]+[AB]{displaystyle [{{ce {A}}}]_{0}=K_{d}{frac {[{{ce {AB}}}]}{[{{ce {B}}}]}}+[{{ce {AB}}}]}
![{displaystyle [{ce {A}}]_{0}=K_{d}{frac {[{ce {AB}}]}{[{ce {B}}]}}+[{ce {AB}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a112c4113975672bd679acdb4a5e1e7695ed1b90)
This yields the concentration of the complex related to the concentration of either one of the free molecules
- [AB]=[A]0[B]Kd+[B]=[B]0[A]Kd+[A]{displaystyle {{ce {[AB]}}}={frac {{ce {[A]_{0}[B]}}}{K_{d}+[{{ce {B}}}]}}={frac {{ce {[B]_{0}[A]}}}{K_{d}+[{{ce {A}}}]}}}
![{displaystyle {ce {[AB]}}={frac {ce {[A]_{0}[B]}}{K_{d}+[{ce {B}}]}}={frac {ce {[B]_{0}[A]}}{K_{d}+[{ce {A}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0796bbf31a1ac58f4edf6c6c6fa7f585ced78625)
Macromolecules with identical independent binding sites
Many biological proteins and enzymes can possess more than one binding site.[1]
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 one single binding site. Then, the concentration of bound ligands [L]bound{displaystyle {{ce {[L]_{bound}}}}}![{displaystyle {ce {[L]_{bound}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/139547b41fe0f169d090d17df376a3faed288f2e)
- [L]bound=n[M]0[L]Kd+[L]{displaystyle {ce {[L]}}_{text{bound}}={frac {n{ce {[M]}}_{0}{ce {[L]}}}{K_{d}+{ce {[L]}}}}}
![{displaystyle {ce {[L]}}_{text{bound}}={frac {n{ce {[M]}}_{0}{ce {[L]}}}{K_{d}+{ce {[L]}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85941905b594adf62e3c693b0d248cc2a7ed1a89)
In this case, [L]bound≠[LM]{displaystyle {ce {[L]}}_{text{bound}}neq {ce {[LM]}}}![{displaystyle {ce {[L]}}_{text{bound}}neq {ce {[LM]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aad16ddd46e8b75335ded50dec9365e31d025df)
- [L]bound=[LM]+2[L2M]+3[L3M]+…+n[LnM]{displaystyle {ce {[L]}}_{text{bound}}={ce {[LM]}}+{ce {2[L_{2}M]}}+{ce {3[L_{3}M]}}+ldots +n{ce {[L_{mathit {n}}M]}}}
![{displaystyle {ce {[L]}}_{text{bound}}={ce {[LM]}}+{ce {2[L_{2}M]}}+{ce {3[L_{3}M]}}+ldots +n{ce {[L_{mathit {n}}M]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2142e9f8236bf9ceede4a0a76c7f255b0efd11a3)
where the saturation occurs stepwise
- [L]+[M]↽−−⇀[LM]K1′=[L][M][LM][LM]=[L][M]K1′[L]+[LM]↽−−⇀[L2M]K2′=[L][LM][L2M][L2M]=[L]2[M]K1′K2′[L]+[L2M]↽−−⇀[L3M]K3′=[L][L2M][L3M][L3M]=[L]3[M]K1′K2′K3′⋮⋮⋮[L]+[Ln−1M]↽−−⇀[LnM]Kn′=[L][Ln−1M][LnM][LnM]=[L]n[M]K1′K2′K3′⋯Kn′{displaystyle {begin{aligned}{ce {{[L]}+[M]}}&{ce {{}<=>{[LM]}}}&K'_{1}&={frac {ce {[L][M]}}{[LM]}}&{ce {[LM]}}&={frac {ce {[L][M]}}{K'_{1}}}\{ce {{[L]}+[LM]}}&{ce {{}<=>{[L2M]}}}&K'_{2}&={frac {ce {[L][LM]}}{[L_{2}M]}}&{ce {[L_{2}M]}}&={frac {ce {[L]^{2}[M]}}{K'_{1}K'_{2}}}\{ce {{[L]}+[L2M]}}&{ce {{}<=>{[L3M]}}}&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{aligned}}}
![{displaystyle {begin{aligned}{ce {{[L]}+[M]}}&{ce {{}<=>{[LM]}}}&K'_{1}&={frac {ce {[L][M]}}{[LM]}}&{ce {[LM]}}&={frac {ce {[L][M]}}{K'_{1}}}\{ce {{[L]}+[LM]}}&{ce {{}<=>{[L2M]}}}&K'_{2}&={frac {ce {[L][LM]}}{[L_{2}M]}}&{ce {[L_{2}M]}}&={frac {ce {[L]^{2}[M]}}{K'_{1}K'_{2}}}\{ce {{[L]}+[L2M]}}&{ce {{}<=>{[L3M]}}}&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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8d8a0056fee40398183175c1057fb022dc18d9)
For the derivation of the general binding equation a saturation function r{displaystyle r}
amount of the macromolecule:
- r=[L]bound[M]0=[LM]+2[L2M]+3[L3M]+⋅⋅⋅+n[LnM][M]+[LM]+[L2M]+[L3M]+⋅⋅⋅+[LnM]=∑i=1n(i[L]i∏j=1iKj′)1+∑i=1n([L]i∏j=1iKj′){displaystyle 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)}}}
![{displaystyle 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)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8efa29182db762b71be1a091e241848587661937)
Even if all microscopic dissociation constants[clarification needed] are identical, they differ from the macroscopic ones and there are differences between each binding step.[clarification needed]
The general relationship between both types of dissociation constants for n binding sites is
- Ki′=Kdin−i+1{displaystyle K_{i}'=K_{d}{frac {i}{n-i+1}}}
Hence, the ratio of bound ligand to macromolecules becomes
- r=∑i=1ni(∏j=1in−j+1j)([L]Kd)i1+∑i=1n(∏j=1in−j+1j)([L]Kd)i=∑i=1ni(ni)([L]Kd)i1+∑i=1n(ni)([L]Kd)i{displaystyle r={frac {sum _{i=1}^{n}ileft(prod _{j=1}^{i}{frac {n-j+1}{j}}right)left({frac {{ce {[L]}}}{K_{d}}}right)^{i}}{1+sum _{i=1}^{n}left(prod _{j=1}^{i}{frac {n-j+1}{j}}right)left({frac {[L]}{K_{d}}}right)^{i}}}={frac {sum _{i=1}^{n}i{binom {n}{i}}left({frac {[L]}{K_{d}}}right)^{i}}{1+sum _{i=1}^{n}{binom {n}{i}}left({frac {{ce {[L]}}}{K_{d}}}right)^{i}}}}
![{displaystyle r={frac {sum _{i=1}^{n}ileft(prod _{j=1}^{i}{frac {n-j+1}{j}}right)left({frac {{ce {[L]}}}{K_{d}}}right)^{i}}{1+sum _{i=1}^{n}left(prod _{j=1}^{i}{frac {n-j+1}{j}}right)left({frac {[L]}{K_{d}}}right)^{i}}}={frac {sum _{i=1}^{n}i{binom {n}{i}}left({frac {[L]}{K_{d}}}right)^{i}}{1+sum _{i=1}^{n}{binom {n}{i}}left({frac {{ce {[L]}}}{K_{d}}}right)^{i}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f41b5017ab1fd0c9ef8f08405195b59387cd9d)
where (ni)=n!(n−i)!i!{displaystyle {binom {n}{i}}={frac {n!}{(n-i)!i!}}}
Then, the first equation is proved by applying the binomial rule
- r=n([L]Kd)(1+[L]Kd)n−1(1+[L]Kd)n=n([L]Kd)(1+[L]Kd)=n[L]Kd+[L]=[L]bound[M]0{displaystyle r={frac {nleft({frac {{ce {[L]}}}{K_{d}}}right)left(1+{frac {{ce {[L]}}}{K_{d}}}right)^{n-1}}{left(1+{frac {{ce {[L]}}}{K_{d}}}right)^{n}}}={frac {nleft({frac {{ce {[L]}}}{K_{d}}}right)}{left(1+{frac {{ce {[L]}}}{K_{d}}}right)}}={frac {n[{{ce {L}}}]}{K_{d}+[{{ce {L}}}]}}={frac {{ce {[L]_{bound}}}}{{ce {[M]_{0}}}}}}
![{displaystyle r={frac {nleft({frac {ce {[L]}}{K_{d}}}right)left(1+{frac {ce {[L]}}{K_{d}}}right)^{n-1}}{left(1+{frac {ce {[L]}}{K_{d}}}right)^{n}}}={frac {nleft({frac {ce {[L]}}{K_{d}}}right)}{left(1+{frac {ce {[L]}}{K_{d}}}right)}}={frac {n[{ce {L}}]}{K_{d}+[{ce {L}}]}}={frac {ce {[L]_{bound}}}{ce {[M]_{0}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4930fc4967270e115944ffb7eb24b93de6615915)
Protein-ligand binding
The dissociation constant is commonly used to describe the affinity between a ligand L{displaystyle {{ce {L}}}}
The formation of a ligand-protein complex LP{displaystyle {{ce {LP}}}}
- L+P↽−−⇀LP{displaystyle {ce {L + P <=> LP}}}
the corresponding dissociation constant is defined
- Kd=[L][P][LP]{displaystyle K_{d}={frac {left[{{ce {L}}}right]left[{{ce {P}}}right]}{left[{{ce {LP}}}right]}}}
![{displaystyle K_{d}={frac {left[{ce {L}}right]left[{ce {P}}right]}{left[{ce {LP}}right]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00ea381e95fd298f4d60f186f9106f412af58e17)
where [P],[L]{displaystyle {ce {[P], [L]}}}![{displaystyle {ce {[P], [L]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66dafbae31bda58e9b609d05d328006fb92bec8d)
![{displaystyle {ce {[LP]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf0a1624e5788f2a6e8a2319528de629fc9dd75)
The dissociation constant has molar units (M) and corresponds to the ligand concentration [L]{displaystyle {{ce {[L]}}}}![{displaystyle {ce {[L]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d168a8fcf5a74047be127a23620e6c9a5534c1)
![{displaystyle {ce {[P]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1396543ba4ae6401e8207fd5de9e164af1dfbcea)
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 10−15 M = 1 fM = 0.000001 nM.[5]Ribonuclease inhibitor proteins may also bind to ribonuclease with a similar 10−15 M affinity.[6]
The dissociation constant for a particular ligand-protein interaction can change significantly with solution conditions (e.g., temperature, pH and salt concentration). The effect of different solution conditions is to effectively modify the strength of any 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 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.
- Ab+Ag↽−−⇀AbAg{displaystyle {ce {Ab + Ag <=> AbAg}}}
- Ka=[AbAg][Ab][Ag]=1Kd{displaystyle K_{a}={frac {left[{{ce {AbAg}}}right]}{left[{{ce {Ab}}}right]left[{{ce {Ag}}}right]}}={frac {1}{K_{d}}}}
![{displaystyle K_{a}={frac {left[{ce {AbAg}}right]}{left[{ce {Ab}}right]left[{ce {Ag}}right]}}={frac {1}{K_{d}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47048a99e79ca78ee25d27947e888114d644119b)
This chemical equilibrium is also the ratio of the on-rate (kforward) and off-rate (kback) 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.
- Ka=kforwardkback=on-rateoff-rate{displaystyle K_{a}={frac {k_{text{forward}}}{k_{text{back}}}}={frac {mbox{on-rate}}{mbox{off-rate}}}}
Acid–base reactions
| Acids and Bases |
|---|
|
Acid types |
|
Base types |
|
For the deprotonation of acids, K is known as Ka, the acid dissociation constant. Stronger acids, for example sulfuric or phosphoric acid, have larger dissociation constants; weaker acids, like acetic acid, have smaller dissociation constants.
(The symbol Ka{displaystyle K_{a}}
Acid dissociation constants are sometimes expressed by pKa{displaystyle pK_{a}}
- pKa=−log10Ka{displaystyle {text{p}}K_{a}=-log _{10}{K_{a}}}
This pK{displaystyle mathrm {p} K}
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 acids. The first (e.g., acetic acid or ammonium) have only one dissociable group, the second (carbonic acid, bicarbonate, glycine) have two dissociable groups and the third (e.g., phosphoric acid) have three dissociable groups. In the case of multiple pK values they are designated by indices: pK1, pK2, pK3 and so on. For amino acids, the pK1 constant refers to its carboxyl (-COOH) group, pK2 refers to its amino (-NH3) group and the pK3 is the pK value of its side chain.
- H3B↽−−⇀H++H2B−K1=[H+]⋅[H2B−][H3B]pK1=−logK1H2B−↽−−⇀H++HB−2K2=[H+]⋅[HB−2][H2B−]pK2=−logK2HB−2↽−−⇀H++B−3K3=[H+]⋅[B−3][HB−2]pK3=−logK3{displaystyle {begin{aligned}{ce {H3B}}&{ce {{}<=>{H+}+{H2B^{-}}}}&K_{1}&={ce {[H+].[H2B^{-}] over [H3B]}}&mathrm {p} K_{1}&=-log K_{1}\{ce {H2B^{-}}}&{ce {{}<=>{H+}+{HB^{-2}}}}&K_{2}&={ce {[H+].[HB^{-2}] over [H2B^{-}]}}&mathrm {p} K_{2}&=-log K_{2}\{ce {HB^{-2}}}&{ce {{}<=>{H+}+{B^{-3}}}}&K_{3}&={ce {[H+].[B^{-3}] over [HB^{-2}]}}&mathrm {p} K_{3}&=-log K_{3}end{aligned}}}
![{displaystyle {begin{aligned}{ce {H3B}}&{ce {{}<=>{H+}+{H2B^{-}}}}&K_{1}&={ce {[H+].[H2B^{-}] over [H3B]}}&mathrm {p} K_{1}&=-log K_{1}\{ce {H2B^{-}}}&{ce {{}<=>{H+}+{HB^{-2}}}}&K_{2}&={ce {[H+].[HB^{-2}] over [H2B^{-}]}}&mathrm {p} K_{2}&=-log K_{2}\{ce {HB^{-2}}}&{ce {{}<=>{H+}+{B^{-3}}}}&K_{3}&={ce {[H+].[B^{-3}] over [HB^{-2}]}}&mathrm {p} K_{3}&=-log K_{3}end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e3bd986d2966c3121c319a35fca3875725b1253)
Dissociation constant of water
The dissociation constant of water is denoted Kw:
- Kw=[H+][OH−]{displaystyle K_{w}=[{{ce {H}}}^{+}][{{ce {OH}}}^{-}]}
![{displaystyle K_{w}=[{ce {H}}^{+}][{ce {OH}}^{-}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc7f05d11899a1629f1b75107ef51f1e0fa475e)
The concentration of water H2O{displaystyle {ce {H2O}}}
The value of Kw 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.
| Water temperature | Kw / 10−14 | pKw[7] |
|---|---|---|
| 0 °C | 0.112 | 14.95 |
| 25 °C | 1.023 | 13.99 |
| 50 °C | 5.495 | 13.26 |
| 75 °C | 19.95 | 12.70 |
| 100 °C | 56.23 | 12.25 |
See also
- Acid
- Equilibrium constant
Ki Database- Competitive inhibition
- pH
- Scatchard plot
- Ligand binding
References
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^ Zhou, H.; Rivas, G.; Minton, A. (2008). "Macromolecular crowding and confinement: biochemical, biophysical, and potential physiological consequences". Annual Review of Biophysics. 37: 375–397. doi:10.1146/annurev.biophys.37.032807.125817. PMC 2826134. PMID 18573087.
^ Minton, A. P. (2001). "The influence of macromolecular crowding and macromolecular confinement on biochemical reactions in physiological media" (PDF). The Journal of Biological Chemistry. 276 (14): 10577–10580. doi:10.1074/jbc.R100005200. PMID 11279227.
^ Björkelund, Hanna; Gedda, Lars; Andersson, Karl (2011-01-31). "Comparing the Epidermal Growth Factor Interaction with Four Different Cell Lines: Intriguing Effects Imply Strong Dependency of Cellular Context". PLOS ONE. 6 (1): e16536. Bibcode:2011PLoSO...616536B. doi:10.1371/journal.pone.0016536. ISSN 1932-6203.
^ Livnah, O.; Bayer, E.; Wilchek, M.; Sussman, J. (1993). "Three-dimensional structures of avidin and the avidin-biotin complex". Proceedings of the National Academy of Sciences of the United States of America. 90 (11): 5076–5080. Bibcode:1993PNAS...90.5076L. doi:10.1073/pnas.90.11.5076. PMC 46657. PMID 8506353.
^ Johnson, R.; Mccoy, J.; Bingman, C.; Phillips Gn, J.; Raines, R. (2007). "Inhibition of human pancreatic ribonuclease by the human ribonuclease inhibitor protein". Journal of Molecular Biology. 368 (2): 434–449. doi:10.1016/j.jmb.2007.02.005. PMC 1993901. PMID 17350650.
^ Bandura, Andrei V.; Lvov, Serguei N. (2006). "The Ionization Constant of Water over Wide Ranges of Temperature and Density" (PDF). Journal of Physical and Chemical Reference Data. 35 (1): 15–30. Bibcode:2006JPCRD..35...15B. doi:10.1063/1.1928231.
