View source for Biology:Protein pKa calculations

{{DISPLAYTITLE:Protein p''K''<sub>a</sub> calculations}}
In [[Biology:Computational biology|computational biology]], '''protein p''K''<sub>a</sub> calculations''' are used to estimate the [[Chemistry:Acid dissociation constant|p''K''<sub>a</sub> values]] of [[Chemistry:Amino acid|amino acid]]s as they exist within [[Biology:Protein|protein]]s. These calculations complement the p''K''<sub>a</sub> values reported for amino acids in their free state, and are used frequently within the fields of molecular modeling, [[Biology:Structural bioinformatics|structural bioinformatics]], and [[Biology:Computational biology|computational biology]].

==Amino acid p''K''<sub>a</sub> values==
[[Chemistry:Acid dissociation constant|p''K''<sub>a</sub> values]] of amino acid [[Chemistry:Side chain|side chain]]s play an important role in defining the pH-dependent characteristics of a protein. The pH-dependence of the activity displayed by [[Biology:Enzyme|enzyme]]s and the pH-dependence of protein stability, for example, are properties that are determined by the p''K''<sub>a</sub> values of amino acid side chains.

The p''K''<sub>a</sub> values of an amino acid side chain in solution is typically inferred from the p''K''<sub>a</sub> values of model compounds (compounds that are similar to the side chains of amino acids). See [[Chemistry:Amino acid|Amino acid]] for the p''K''<sub>a</sub> values of all amino acid side chains inferred in such a way. There are also numerous experimental studies that have yielded such values, for example by use of NMR spectroscopy.

The table below lists the model p''K''<sub>a</sub> values that are often used in a protein p''K''<sub>a</sub> calculation, and contains a third column based on protein studies.<ref name="Hass2015">Hass and Mulder (2015) ''Annu. Rev. Biophys.'' vol 44 pp. 53–75 [https://dx.doi.org/10.1146/annurev-biophys-083012-130351 doi 10.1146/annurev-biophys-083012-130351].</ref>

{| class="wikitable sortable"
! Amino Acid
! p''K''<sub>a</sub>
! p''K''<sub>a</sub>
|-
| Asp (D)
| 3.9
| 4.0{{0}}
|-
| Glu (E)
| 4.3
| 4.4
|-
| Arg (R)
| 12.0
| 13.5
|-
| Lys (K)
| 10.5
| 10.4
|-
| His (H)
| 6.08
| 6.8
|-
| Cys (C) (–SH)
| 8.28
| 8.3
|-
| Tyr (Y)
| 10.1
| 9.6
|-
| N-term
|
| 8.0
|-
| C-term
|
| 3.6
|}

==The effect of the protein environment==
[[Image:Back titration.jpg|thumb|Coupled system consisting of three acids. The black curve shows a back-titration event]]
When a protein folds, the titratable amino acids in the protein are transferred from a solution-like environment to an environment determined by the 3-dimensional structure of the protein. For example, in an unfolded protein, an aspartic acid typically is in an environment which exposes the titratable side chain to water. When the protein folds, the aspartic acid could find itself buried deep in the protein interior with no exposure to solvent.

Furthermore, in the folded protein, the aspartic acid will be closer to other titratable groups in the protein and will also interact with permanent charges (e.g. ions) and dipoles in the protein. All of these effects alter the p''K''<sub>a</sub> value of the amino acid side chain, and p''K''<sub>a</sub> calculation methods generally calculate the effect of the protein environment on the model p''K''<sub>a</sub> value of an amino acid side chain.<ref name="Bashford2004">Bashford (2004) ''Front Biosci.'' vol. 9 pp. 1082–99 [https://dx.doi.org/10.2741/1187 doi 10.2741/1187]</ref><ref name="Gunner2006">Gunner et al. (2006) ''Biochim. Biophys. Acta'' vol. 1757 (8) pp. 942–68 [https://dx.doi.org/10.1016/j.bbabio.2006.06.005 doi 10.1016/j.bbabio.2006.06.005]</ref><ref name="Ullmann2008">Ullmann et al. (2008) ''Photosynth. Res.'' 97 vol. 112 pp. 33–55 [https://dx.doi.org/10.1007/s11120-008-9306-1 doi 10.1007/s11120-008-9306-1]</ref><ref name="Antosiewicz2011">Antosiewicz et al. (2011) ''Mol. BioSyst.'' vol. 7 pp. 2923–2949 [https://dx.doi.org/10.1039/C1MB05170A doi 10.1039/C1MB05170A ]</ref>

Typically, the effects of the protein environment on the amino acid p''K''<sub>a</sub> value are divided into pH-independent effects and pH-dependent effects. The pH-independent effects (desolvation, interactions with permanent charges and dipoles) are added to the model p''K''<sub>a</sub> value to give the intrinsic p''K''<sub>a</sub> value. The pH-dependent effects cannot be added in the same straightforward way and have to be accounted for using Boltzmann summation, Tanford–Roxby iterations or other methods.

The interplay of the intrinsic p''K''<sub>a</sub> values of a system with the electrostatic interaction energies between titratable groups can produce quite spectacular effects such as non-Henderson–Hasselbalch [[Chemistry:Titration curve|titration curve]]s and even back-titration effects.<ref name="Onufriev">A. Onufriev, D.A. Case and G. M. Ullmann (2001). ''Biochemistry'' 40: 3413–3419 [https://dx.doi.org/10.1021/bi002740q doi 10.1021/bi002740q]</ref>

The image on the right shows a theoretical system consisting of three acidic residues. One group is displaying a back-titration event (blue group).
==p''K''<sub>a</sub> calculation methods==
Several software packages and webserver are available for the calculation of protein p''K''<sub>a</sub> values.

===Using the Poisson–Boltzmann equation===
Some methods are based on solutions to the [[Physics:Poisson–Boltzmann equation|Poisson–Boltzmann equation]] (PBE), often referred to as FDPB-based methods (''FDPB'' stands for "[[Finite difference|finite difference]] Poisson–Boltzmann"). The PBE is a modification of [[Physics:Poisson's equation|Poisson's equation]] that incorporates a description of the effect of solvent ions on the electrostatic field around a molecule.

The [http://newbiophysics.cs.vt.edu/H++/ H++ web server],<ref>{{Cite web |title=H++ (web-based computational prediction of protonation states and pK of ionizable groups in macromolecules) |url=http://newbiophysics.cs.vt.edu/H++/hppdetails.php |access-date=2023-01-26 |website=newbiophysics.cs.vt.edu}}</ref> the [https://web.archive.org/web/20070728080556/http://enzyme.ucd.ie/pKD pKD webserver],<ref>{{Cite journal |last1=Tynan-Connolly |first1=B. M. |last2=Nielsen |first2=J. E. |date=2006-12-22 |title=Redesigning protein pKa values |journal=Protein Science |language=en |volume=16 |issue=2 |pages=239–249 |doi=10.1110/ps.062538707 |issn=0961-8368 |pmc=2203286 |pmid=17189477}}</ref> [https://gunnerlab.github.io/Stable-MCCE/ MCCE2], [http://agknapp.chemie.fu-berlin.de/karlsberg Karlsberg+],{{Dead link|date=January 2023}} [https://www.itqb.unl.pt/labs/molecular-simulation/in-house-software/ PETIT] and [https://rtullmann.de/parts/gmct-gcem.html GMCT] use the FDPB method to compute p''K''<sub>a</sub> values of amino acid side chains.

FDPB-based methods calculate the change in the p''K''<sub>a</sub> value of an amino acid side chain when that side chain is moved from a hypothetical fully solvated state to its position in the protein. To perform such a calculation, one needs theoretical methods that can calculate the effect of the protein interior on a p''K''<sub>a</sub> value, and knowledge of the pKa values of amino acid side chains in their fully solvated states.<ref name="Bashford2004"/><ref name="Gunner2006"/><ref name="Ullmann2008"/><ref name="Antosiewicz2011"/>

===Empirical methods===
A set of empirical rules relating the protein structure to the p''K''<sub>a</sub> values of ionizable residues have been developed by Li, Robertson, and Jensen.<ref>{{Cite journal |last1=Li |first1=Hui |last2=Robertson |first2=Andrew D. |last3=Jensen |first3=Jan H. |date=2005-10-17 |title=Very fast empirical prediction and rationalization of protein pKa values |url=https://onlinelibrary.wiley.com/doi/10.1002/prot.20660 |journal=Proteins: Structure, Function, and Bioinformatics |language=en |volume=61 |issue=4 |pages=704–721 |doi=10.1002/prot.20660|pmid=16231289 |s2cid=38196246 }}</ref> These rules form the basis for the [https://biolib.com/bio-utils/propka/ web-accessible] program called PROPKA for rapid predictions of p''K''<sub>a</sub> values. A recent empirical p''K''<sub>a</sub> prediction program was released by Tan KP ''et.al.'' with the online server [http://mspc.bii.a-star.edu.sg/tankp/ DEPTH web server].<ref>{{Cite journal |last1=Tan |first1=Kuan Pern |last2=Nguyen |first2=Thanh Binh |last3=Patel |first3=Siddharth |last4=Varadarajan |first4=Raghavan |last5=Madhusudhan |first5=M. S. |date=2013-07-01 |title=Depth: a web server to compute depth, cavity sizes, detect potential small-molecule ligand-binding cavities and predict the pKa of ionizable residues in proteins |url=http://academic.oup.com/nar/article/41/W1/W314/1111943/Depth-a-web-server-to-compute-depth-cavity-sizes |journal=Nucleic Acids Research |language=en |volume=41 |issue=W1 |pages=W314–W321 |doi=10.1093/nar/gkt503 |issn=1362-4962 |pmc=3692129 |pmid=23766289}}</ref>

===Molecular dynamics (MD)-based methods===

[[Physics:Molecular dynamics|Molecular dynamics]] methods of calculating p''K''<sub>a</sub> values make it possible to include full flexibility of the titrated molecule.<ref name="Donnini2011">Donnini et al. (2011) ''J. Chem. Theory Comp.'' vol 7 pp. 1962–78 [https://dx.doi.org/10.1021/ct200061r doi 10.1021/ct200061r].</ref><ref name="Wallace2011">Wallace et al. (2011) ''J. Chem. Theory Comp.'' vol 7 pp. 2617–2629 [https://dx.doi.org/10.1021/ct200146j doi 10.1021/ct200146j].</ref><ref name="Goh2012">Goh et al. (2012) ''J. Chem. Theory Comp.'' vol 8 pp. 36–46 [https://dx.doi.org/10.1021/ct2006314 doi 10.1021/ct2006314].</ref>

Molecular dynamics based methods are typically much more computationally expensive, and not necessarily more accurate, ways to predict p''K''<sub>a</sub> values than approaches based on the [[Physics:Poisson–Boltzmann equation|Poisson–Boltzmann equation]]. Limited conformational flexibility can also be realized within a continuum electrostatics approach, e.g., for considering multiple amino acid sidechain rotamers. In addition, current commonly used molecular force fields do not take electronic polarizability into account, which could be an important property in determining protonation energies.

===Determining p''K''<sub>a</sub> values from titration curves or free energy calculations===

From the [[Chemistry:Titration|titration]] of [[Chemistry:Protonation|protonatable]] group, one can read the so-called p''K''<sub>a</sub><sup>{{1/2}}</sup> which is equal to the pH value where the group is half-protonated (i.e. when 50% such groups would be protonated). The p''K''<sub>a</sub><sup>{{1/2}}</sup> is equal to the Henderson–Hasselbalch p''K''<sub>a</sub> (p''K''{{su|b=a|p=HH}}) if the titration curve follows the [[Henderson–Hasselbalch equation]].<ref name="Ullmann2003">Ullmann (2003) ''J. Phys. Chem. B'' vol 107 pp. 1263–71 [https://dx.doi.org/10.1021/jp026454v doi 10.1021/jp026454v].</ref> Most p''K''<sub>a</sub> calculation methods silently assume that all titration curves are Henderson–Hasselbalch shaped, and p''K''<sub>a</sub> values in p''K''<sub>a</sub> calculation programs are therefore often determined in this way. In the general case of multiple interacting protonatable sites, the p''K''<sub>a</sub><sup>{{1/2}}</sup> value is not thermodynamically meaningful. In contrast, the Henderson–Hasselbalch p''K''<sub>a</sub> value can be computed from the protonation free energy via

<math> \mathrm{p}K_{\mathrm{a}}^{\mathrm{HH}}(\mathrm{pH}) =
\mathrm{pH} - \frac{\Delta G^{\mathrm{prot}}(\mathrm{pH})}{\mathrm{RT} \ln10}</math>

and is thus in turn related to the protonation free energy of the site via

<math>\Delta G^{\mathrm{prot}}(\mathrm{pH}) = \mathrm{RT} \ln10 \; ( \mathrm{pH} - \mathrm{p}K_{\mathrm{a}}^{\mathrm{HH}} ) </math>

The protonation free energy can in principle be computed from the protonation probability of the group {{angle brackets|''x''}}(pH) which can be read from its titration curve

<math> \Delta G^{\mathrm{prot}}(\mathrm{pH}) = -\mathrm{RT}\ln\left[ \frac{\langle x \rangle}{1-\langle x \rangle} \right] </math>

Titration curves can be computed within a continuum electrostatics approach with formally exact but more elaborate analytical or [[Monte Carlo method|Monte Carlo (MC) methods]], or inexact but fast approximate methods. MC methods that have been used to compute titration curves<ref name="Ullmann2012">Ullmann et al. (2012) ''J. Comput. Chem.'' vol 33 pp. 887–900 [https://dx.doi.org/10.1002/jcc.22919 doi 10.1002/jcc.22919]</ref> are Metropolis MC<ref name="Metropolis1953">Metropolis et al. (1953) ''J. Chem. Phys.'' vol 23 pp. 1087–1092 [http://dx.doi.org/10.1063/1.1699114 doi 10.1063/1.1699114]</ref><ref name="Beroza1991">Beroza et al. (1991) ''Proc. Natl. Acad. Sci. USA'' vol 88 pp. 5804–5808 [https://dx.doi.org/10.1073/pnas.88.13.5804 doi 10.1073/pnas.88.13.5804]</ref> or [[Wang and Landau algorithm|Wang–Landau MC]].<ref name="WangLandau2001">Wang and Landau (2001) Phys. Rev. E vol 64 pp 056101 [http://dx.doi.org/10.1103/PhysRevE.64.056101 doi 10.1103/PhysRevE.64.056101]</ref> Approximate methods that use a mean-field approach for computing titration curves are the Tanford–Roxby method and hybrids of this method that combine an exact statistical mechanics treatment within clusters of strongly interacting sites with a mean-field treatment of intercluster interactions.<ref name="TanfordRoxby1972">Tanford and Roxby (1972) ''Biochemistry'' vol 11 pp. 2192–2198 [https://dx.doi.org/10.1021/bi00761a029 doi 10.1021/bi00761a029]</ref><ref name="Bashford1991">Bashford and Karplus (1991) ''J. Phys. Chem.'' vol 95 pp. 9556–61 [https://dx.doi.org/10.1021/j100176a093 doi 10.1021/j100176a093]</ref><ref name="Gilson1993">Gilson (1993) ''Proteins'' vol 15 pp. 266–82 [https://dx.doi.org/10.1002/prot.340150305 doi 10.1002/prot.340150305]</ref><ref name="Antosiewicz1994">Antosiewicz et al. (1994) ''J. Mol. Biol.'' vol 238 pp. 415–36 [https://dx.doi.org/10.1006/jmbi.1994.1301 doi 10.1006/jmbi.1994.1301]</ref><ref name="Spassov1999">Spassov and Bashford (1999) ''J. Comput. Chem.'' vol 20 pp. 1091–1111 [https://dx.doi.org/10.1002/(SICI)1096-987X(199908)20:11%3c1091::AID-JCC1%3e3.0.CO;2-3 doi 10.1002/(SICI)1096-987X(199908)20:11<1091::AID-JCC1>3.0.CO;2-3]</ref>

In practice, it can be difficult to obtain statistically converged and accurate protonation free energies from titration curves if {{angle brackets|''x''}} is close to a value of 1 or 0. In this case, one can use various free energy calculation methods to obtain the protonation free energy<ref name="Ullmann2012"/> such as biased Metropolis MC,<ref name="Beroza1995">Beroza et al. (1995) ''Biophys. J.'' vol 68 pp. 2233–2250 [https://dx.doi.org/10.1016/S0006-3495(95)80406-6 doi 10.1016/S0006-3495(95)80406-6]</ref> [[Chemistry:Free-energy perturbation|free-energy perturbation]],<ref name="Zwanzig1954">Zwanzig (1954) ''J. Chem. Phys.'' vol 22 pp. 1420–1426 [https://dx.doi.org/10.1063/1.1740409 doi 10.1063/1.1740409]</ref><ref name="Ullmann2011">Ullmann et al. 2011 ''J. Phys. Chem. B.'' vol 68 pp. 507–521 [https://dx.doi.org/10.1021/jp1093838 doi 10.1021/jp1093838]</ref> [[Chemistry:Thermodynamic integration|thermodynamic integration]],<ref name="Kirkwood1935">Kirkwood (1935) ''J. Chem. Phys.'' vol 2 pp. 300–313 [https://dx.doi.org/10.1063/1.1749657 doi 10.1063/1.1749657]</ref><ref name="Boresch2011a">Bruckner and Boresch (2011) ''J. Comput. Chem.'' vol 32 pp. 1303–1319 [https://dx.doi.org/10.1002/jcc.21713 doi 10.1002/jcc.21713]</ref><ref name="Boresch2011b">Bruckner and Boresch (2011) ''J. Comput. Chem.'' vol 32 pp. 1320–1333 [https://dx.doi.org/10.1002/jcc.21712 doi 10.1002/jcc.21712]</ref> the [[Jarzynski equality|non-equilibrium work method]]<ref name="Jarzynski1997">Jarzynski (1997)  ''Phys. Rev. E'' vol  pp. 2233–2250 [https://dx.doi.org/10.1103/PhysRevE.56.5018 doi 10.1103/PhysRevE.56.5018]</ref> or the [[Physics:Bennett acceptance ratio|Bennett acceptance ratio]] method.<ref name="Bennett1976">Bennett (1976) ''J. Comput. Phys.'' vol 22 pp. 245–268 [https://dx.doi.org/10.1016/0021-9991(76)90078-4 doi 10.1016/0021-9991(76)90078-4]</ref>
 
Note that the p''K''{{su|b=a|p=HH}} value does in general depend on the pH value.<ref name="Bombarda2010">Bombarda et al. (2010) ''J. Phys. Chem. B'' vol 114 pp. 1994–2003 [https://dx.doi.org/10.1021/jp908926w doi 10.1021/jp908926w].</ref>

This dependence is small for weakly interacting groups like well solvated amino acid side chains on the protein surface, but can be large for strongly interacting groups like those buried in enzyme active sites or integral membrane proteins.<ref name="Bashford1992">Bashford and Gerwert (1992) ''J. Mol. Biol.'' vol 224 pp. 473–86 [https://dx.doi.org/10.1016/0022-2836(92)91009-E doi 10.1016/0022-2836(92)91009-E]</ref><ref name="Spassov2001">Spassov et al. (2001) ''J. Mol. Biol.'' vol 312 pp. 203–19 [https://dx.doi.org/10.1006/jmbi.2001.4902 doi 10.1006/jmbi.2001.4902]</ref><ref name="Ullmann2011PaAz">Ullmann et al. (2011) ''J. Phys. Chem. B'' vol 115 pp. 10346–59 [https://dx.doi.org/10.1021/jp204644h doi 10.1021/jp204644h]</ref>

==References==
{{Reflist}}

==External links==
* [http://www.accelrys.com/products/dstudio/ AccelrysPKA]{{Dead link|date=January 2023}} — Accelrys CHARMm based p''K''<sub>a</sub> calculation
* [http://newbiophysics.cs.vt.edu/H++/ H++] — Poisson–Boltzmann based p''K''<sub>a</sub> calculations
* [https://gunnerlab.github.io/Stable-MCCE/ MCCE2] — Multi-Conformation Continuum Electrostatics (Version 2)
* [http://agknapp.chemie.fu-berlin.de/karlsberg Karlsberg+]{{Dead link|date=January 2023}} — p''K''<sub>a</sub> computation with multiple pH adapted conformations
* [http://www.itqb.unl.pt/labs/molecular-simulation/in-house-software/ PETIT] — Proton and Electron TITration
* [https://rtullmann.de/parts/gmct-gcem.html GMCT] — Generalized Monte Carlo Titration
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* [http://enzyme.ucd.ie/main/index.php/PKD pKD server] p''K''<sub>a</sub> calculations and p''K''<sub>a</sub> value redesign
* [http://propka.ki.ku.dk/ PROPKA] Empirical calculation of p''K''<sub>a</sub> values
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* [http://mspc.bii.a-star.edu.sg/tankp/ DEPTH web server] — Empirical calculation of p''K''<sub>a</sub> values using Residue Depth as a major feature
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===Analysis and teaching software===
* [http://enzyme.ucd.ie/main/index.php/PKaTool pKaTool] interactive graphical tool for analysing systems of titratable groups 
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[[Category:Protein methods]]
[[Category:Equilibrium chemistry]]

{{Sourceattribution|Protein pKa calculations}}