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In [[computational biology]], '''protein pKa calculations''' are used to estimate the [[acid dissociation constant|pKa values]] of [[amino acid]]s as they exist within [[protein]]s. These calculations complement the pKa values reported for amino acids in their free state, and are used frequently within the fields of [[molecular modeling]], [[structural bioinformatics]], and [[computational biology]].

==Amino acid pKa values==
[[Acid dissociation constant|pKa values]] of amino acid [[side chain]]s play an important role in defining the pH-dependent characteristics of a protein. The pH-dependence of the activity displayed by [[enzyme]]s and the pH-dependence of [[protein stability]], for example, are properties that are determined by the pKa values of amino acid side chains.

The pKa values of an amino acid side chain in solution is typically inferred from the pKa values of model compounds (compounds that are similar to the side chains of amino acids). (''See [[Amino acid]] for the pKa values of all amino acid side chains inferred in such a way.) The table below lists the model pKa values that are normally used in a protein pKa calculation.

{|
! Amino Acid
! pKa
|-
| Asp (D)
| 3.9
|-
| Glu (E)
| 4.3
|-
| Arg (R)
| 12.0
|-
| Lys (K)
| 10.5
|-
| His (H)
| 6.08
|-
| Cys (C)
| 8.28 (-SH)
|-
| Tyr (Y)
| 10.1
|}

==The effect of the protein environment==
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 pKa value of the amino acid side chain, and pKa calculation methods generally calculate the effect of the protein environment on the model pKa value of an amino acid side chain<ref name="Bashford2004">Bashford (2004) Front Biosci. vol. 9 pp. 1082–99 [http://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 [http://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 [http://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 [http://dx.doi.org/10.1039/C1MB05170A doi 10.1039/C1MB05170A ]</ref>
.

Typically the effects of the protein environment on the amino acid pKa 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 pKa value to give the intrinsic pKa value. The pH-dependent effects cannot be added in the same straight-forward way and have to be accounted for using Boltzmann summation, Tanford-Roxby iterations or other methods.

The interplay of the intrinsic pKa values of a system with the electrostatic interaction energies between titratable groups can produce quite spectacular effects such as non-Henderson-Hasselbalch [[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 [http://dx.doi.org/10.1021/bi002740q doi 10.1021/bi002740q]</ref> [http://enzyme.ucd.ie/main/index.php/PKaTool pKaTool] provides an easy interactive and instructive way of playing around with these effects.

The image below shows a theoretical system consisting of three acidic residues. One group is displaying a back-titration event (blue group).
[[Image:Back titration.jpg|thumb|Coupled system consisting of three acids]]

==pKa calculation methods==
Several software packages and webserver are available for the calculation of protein pKa values. See links below or [http://enzyme.ucd.ie/Science/pKa/Software#other this table]

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

The [http://biophysics.cs.vt.edu/H++/ H++ web server], the [http://enzyme.ucd.ie/pKD pKD webserver], [http://www.sci.ccny.cuny.edu/~mcce/ MCCE], [http://agknapp.chemie.fu-berlin.de/karlsberg Karlsberg+], [http://www.itqb.unl.pt/labs/molecular-simulation/in-house-software/programs-for-electron-and-proton-titration-using-poisson-boltzmann-and-monte-carlo-methods PETIT] and [http://www.bisb.uni-bayreuth.de/People/ullmannt/parts/gmct-gcem.html GMCT] use the FDPB method to compute pKa values of amino acid side chains.

FDPB-based methods calculate the change in the pKa 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 pKa 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 pKa values of ionizable residues have been developed by [http://dx.doi.org/10.1002/prot.20660 Li, Robertson, and Jensen]. These rules form the basis for the [http://propka.ki.ku.dk web-accessible] program called PROPKA for rapid predictions of pKa values.

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

[[Molecular dynamics]] methods of calculating pKa 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 [http://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 [http://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 [http://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
pKa values than approaches based on the [[Poisson-Boltzmann equation]].
Limited conformational flexibility can also be realized within a
continuum electrostatics approach, e.g., for considering multiple
aminoacid 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 pKa values from titration curves or free energy calculations===

From the [[titration curve]] of protonatable group, one can read the so-called {{math| p<var>K</var>a1/2}}
which is equal to the pH value where the group is half-protonated.
The {{math| p<var>K</var>a1/2}} is equal to the Henderson-Hasselbalch pKa {{math| p<var>K</var>aHH}}
if the titration curve follows the [[Henderson-Hasselbalch equation]].<ref name="Ullmann2003">Ullmann (2003) J. Phys. Chem. B vol 107 pp. 1263–71 [http://dx.doi.org/10.1021/jp026454v doi 10.1021/jp026454v].</ref>
Most pKa calculation methods silently assume that all titration curves are Henderson-Hasselbalch shaped, and pKa values in pKa calculation programs are therefore often determined in this way.
In the general case of multiple interacting protonatable sites, the {{math| p<var>K</var>a1/2}} value is not thermodynamically meaningful.
In contrast, the Henderson-Hasselbalch pKa 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 {{math| <<var>x</var>>(pH)}} which can be read from its titration curve

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

Titration curves can be computed within a continuum electrostatics approach with
formally exact but more elaborate analytical or 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 [http://dx.doi.org/10.1002/jcc.22919 doi 10.1002/jcc.22919]</ref>
are
[[Metropolis Monte Carlo|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 [http://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 inter-cluster interactions.<ref name="TanfordRoxby1972">Tanford and Roxby (1972) Biochemistry vol 11 pp. 2192–2198 [http://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 [http://dx.doi.org/10.1021/j100176a093 doi 10.1021/j100176a093]</ref><ref name="Gilson1993">Gilson (1993) Proteins vol 15 pp. 266–82 [http://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 [http://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 [http://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 {{math| <<var>x</var>>}} 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 [http://dx.doi.org/10.1016/S0006-3495(95)80406-6 doi 10.1016/S0006-3495(95)80406-6]</ref> [[free-energy perturbation]],<ref name="Zwanzig1954">Zwanzig (1954) J. Chem. Phys. vol 22 pp. 1420–1426 [http://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 [http://dx.doi.org/10.1021/jp1093838 doi 10.1021/jp1093838]</ref> [[thermodynamic integration]],<ref name="Kirkwood1935">Kirkwood (1935) J. Chem. Phys. vol 2 pp. 300–313 [http://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 [http://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 [http://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 [http://dx.doi.org/10.1103/PhysRevE.56.5018 doi 10.1103/PhysRevE.56.5018]</ref>
or the [[Bennett acceptance ratio]] method.<ref name="Bennett1976">Bennett (1976) J. Comput. Phys. vol 22 pp. 245–268 [http://dx.doi.org/10.1016/0021-9991(76)90078-4 doi 10.1016/0021-9991(76)90078-4]</ref>

Note that the {{math| p<var>K</var>aHH}} 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 [http://dx.doi.org/10.1021/jp908926w doi 10.1021/jp908926w].</ref>

This dependence is small for weakly interacting groups like well solvated amino acid sidechains 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 [http://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 [http://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 [http://dx.doi.org/10.1021/jp204644h doi 10.1021/jp204644h]</ref>

===References===
{{Reflist}}

==Software For Protein pKa Calculations==
* [http://www.accelrys.com/products/dstudio/ AccelrysPKA] Accelrys CHARMm based pKa calculation
* [http://biophysics.cs.vt.edu/H++/ H++] Poisson-Boltzmann based pKa calculations
* [http://www.sci.ccny.cuny.edu/~mcce/ MCCE] Multi-Conformation Continuum Electrostatics
* [http://agknapp.chemie.fu-berlin.de/karlsberg Karlsberg+] pKa computation with multiple pH adapted conformations
* [http://www.itqb.unl.pt/labs/molecular-simulation/in-house-software/ PETIT] Proton and Electron TITration
* [http://www.bisb.uni-bayreuth.de/People/ullmannt/parts/gmct-gcem.html GMCT] Generalized Monte Carlo Titration
* [http://enzyme.ucd.ie/main/index.php/PKD pKD server] pKa calculations and pKa value re-design
* [http://propka.ki.ku.dk/ PROPKA] Empirical calculation of pKa values

===Analysis and teaching software===
* [http://enzyme.ucd.ie/main/index.php/PKaTool pKaTool] interactive graphical tool for analysing systems of titratable groups
* [http://biophysics.cs.vt.edu/H++ H++ webserver]

[[Category:Protein methods]]
[[Category:Equilibrium chemistry]]

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en>SmackBot: remove Erik9bot category,outdated, tag and general fixes