Structure‐Based pKa Calculations Using Continuum Electrostatics Methods

Carolyn A. Fitch1, Bertrand García‐Moreno E.1

1 Johns Hopkins University, Baltimore, Maryland
Publication Name:  Current Protocols in Bioinformatics
Unit Number:  Unit 8.11
DOI:  10.1002/0471250953.bi0811s16
Online Posting Date:  January, 2007
GO TO THE FULL TEXT: PDF or HTML at Wiley Online Library

Abstract

Electrostatic free energy is useful for correlating structure with function in proteins in which ionizable groups play essential functional roles. To this end, the pKa values of ionizable groups must be known and their molecular determinants must be understood. Structure‐based calculations of electrostatic energies and pKa values are necessary for this purpose. This unit describes protocols for pKa calculations with continuum electrostatics methods based on the numerical solution of the linearized Poisson‐Boltzmann equation by the method of finite differences. Critical discussion of key parameters, approximations, and shortcomings of these methods is included. Two protocols are described for calculations with methods modified empirically to maximize agreement between measured and calculated pKa values. Applied judiciously, these methods can contribute useful and novel insight into properties of surface ionizable groups in proteins.

Keywords: pKa calculations; continuum electrostatics; finite difference; Poisson‐Boltzmann; UBHD

     
 
GO TO THE FULL PROTOCOL:
PDF or HTML at Wiley Online Library

Table of Contents

  • Basic Protocol 1: Calculating pKa Values Using the FDPB Method and the Single‐Site Charge Model (FDPB/SS)
  • Alternate Protocol 1: Calculating pKa Values Using the FDPB Method and the Full Charge Model (FDPB/F)
  • Guidelines for Understanding Results
  • Commentary
  • Literature Cited
  • Figures
  • Tables
     
 
GO TO THE FULL PROTOCOL:
PDF or HTML at Wiley Online Library

Materials

GO TO THE FULL PROTOCOL:
PDF or HTML at Wiley Online Library

Figures

Videos

Literature Cited

   Alexov, E. 2003. Role of the protein side‐chain fluctuations on the strength of pair‐wise electrostatic interactions: Comparing experimental with computed pK(a)s. Proteins 50:94‐103.
   Alexov, E.G., and Gunner, M.R. 1997. Incorporating protein conformational flexibility into the calculation of pH‐dependent protein properties. Biophys. J. 74:2075‐2093.
   Alexov, E.G. and Gunner, M.R. 1999. Calculated protein and proton motions coupled to electron transfer: Electron transfer from QA‐ to QB in bacterial photosynthetic reaction centers. Biochemistry 38:8253‐8270.
   Antosiewicz, J., McCammon, J.A., and Gilson, M.K. 1994. Prediction of pH‐dependent properties of proteins. J. Mol. Biol. 238:415‐436.
   Antosiewicz, J., Briggs, J.M., Elcock, A.H., Gilson, M.K., and McCammon, A. 1996a. Computing ionization states of proteins with a detailed charge model. J. Comput. Chem. 17:1633‐1644.
   Antosiewicz, J., McCammon, A.J., and Gilson, M.K. 1996b. The determinants of pKas in proteins. Biochemistry 35:7819‐7833.
   Archontis, G. and Simonson, T. 2005. Proton binding to proteins: A free‐energy component analysis using a dielectric continuum model. Biophys. J. 88:3888‐3904.
   Baker, N., Holst, M., and Wang, F. 2000. Adaptive multilevel finite element solution of the Poisson‐Boltzmann equation II. Refinement at solvent‐accessible surfaces in biomolecular systems. J. Comput. Chem. 21:1343‐1352.
   Baptista, A.M. and Soares, C.M. 2001. Some theoretical and computational aspects of the inclusion of proton isomerism in the protonation equilibrium of proteins. J. Phys. Chem. B 105:293‐309.
   Bashford, D. and Gerwert, K. 1992. Electrostatic calculations of the pKa values of ionizable groups in bacteriorhodopsin. J. Mol. Biol. 224:473‐486.
   Bashford, D. and Karplus, M. 1990. pKa's of ionizable groups in proteins: Atomic detail from a continuum electrostatic model. Biochemistry 29:10219‐10225.
   Bashford, D. and Karplus, M. 1991. Multiple‐site titration curves of proteins: An analysis of exact and approximate methods for their calculation. J. Phys. Chem. 95:9556‐9561.
   Bashford, D., Case, D., Dalvit, C., Tennant, L., and Wright, P. 1993. Electrostatic calculations of side‐chain pKa values in myoglobin and comparison with NMR data for histidines. Biochemistry 32:8045‐8056.
   Beroza, P. and Case, D.A. 1996. Including side chain flexibility in continuum electrostatic calculations of protein titration. J. Phys. Chem. 100:20156‐20163.
   Beroza, P. and Case, D.A. 1998. Methods to address the change in conformation resulting from ionization process OR fluctuations inherent at a particular pH or in a particular structure. Methods Enzymol. 295:170‐189.
   Beroza, D., Fredkin, D.R., Okamura, M.Y., and Feher, G. 1991. Protonation of interacting residues in a protein by a Monte Carlo method: Application to lysozyme and the photosynthetic reaction center of Rhodobacter sphaeroides. Proc. Natl. Acad. Sci. U.S.A. 88:5804‐5808.
   Brooks, B.R., Bruccoleri, R.E., Olafson, B.D., States, D.J., Swaminathan, S., and Karplus, M. 1983. CHARMM: A Program for macromolecular energy, minimization, and dynamics calculations. J. Comput. Chem. 4:187‐217.
   Davis, M.E., Madura, J.D., Luty, B.A., and McCammon, J.A. 1991. Electrostatics and diffusion of molecules in solution‐Simulations with the University of Houston Brownian Dynamics program. Comp. Phys. Commun. 62:187‐197.
   Demchuk, E. and Wade, R.C. 1996. Improving the continuum dielectric approach to calculating pKa s of ionizable groups in proteins. J. Phys. Chem. 100:17373‐17387.
   Demchuk, E., Genick, U.K., Woo, T.T., Getzoff, E.D., and Bashford, D. 2000. Protonation states and pH titration in the photocycle of photoactive yellow protein. Biochemistry 39:1100‐1113.
   Dillet, V., Dyson, H.J., and Bashford, D. 1998. Calculations of electrostatic interactions and pK(a)s in the active site of Escherichia coli thioredoxin. Biochemistry 37:10298‐10306.
   Dimitrov, R.A. and Crichton, R.R. 1997. Self‐consistent field approach to protein structure and stability .1. pH dependence of electrostatic contribution. Proteins 27:576‐596.
   Dwyer, J.J., Gittis, A.G., Karp, D.A., Lattman, E.E., Spencer, D.S., Stites, W.E., and García‐Moreno E., B. 2000. High apparent dielectric constants in the interior of a protein reflect water penetration. Biophys. J. 79:1610‐1620.
   Elcock, A.H. 1999. Realistic modeling of the denatured states of proteins allows accurate calculations of the pH dependence of protein stability. J. Mol. Biol. 294:1051‐1062.
   Fitch, C.A., Karp, D.A., Lee, K.K., Stites, W.E., Lattman, E.E., and García‐Moreno E., B. 2002. Experimental pKa values of buried residues: analysis with continuum methods and role of water penetration. Biophys. J. 82:3289‐3304.
   Fitch, C.A., Whitten, S.T., Hilser, V.J., and García‐Moreno E., B. 2005. Molecular mechanism of pH‐driven conformational transitions of proteins: Insights from continuum electrostatics calculations of acid unfolding. Proteins 63:113‐126.
   Garcia‐Moreno E., B. and Fitch, C.A. 2004. Structural interpretation of pH and salt‐dependent processes in proteins with computational methods. In Energetics Of Biological Macromolecules, Pt E. (M. J.M. Holt, M.L. Johnson, and G.K. Ackers, eds.) pp. 20‐51. Academic Press Inc., San Diego.
   Georgescu, R.E., Alexov, E.G., and Gunner, M.R. 2002. Combining conformational flexibility and continuum electrostatics for calculating pK(a)s in proteins. Biophys. J. 83:1731‐1748.
   Gibas, C.J. and Subramaniam, S. 1996. Explicit solvent models in protein pKa calculations. Biophys. J. 71:138‐147.
   Gilson, M.K. 1993. Multiple‐site titration and molecular modeling: Two rapid methods for computing energies and forces for ionizable groups in proteins. Proteins 15:266‐282.
   Gilson, M.K. 1995. Theory of electrostatic interactions in macromolecules. Curr. Biol. 5:216‐223.
   Gilson, M.K., Sharp, K.A., and Honig, B.H. 1988. Calculating the electrostatic potential of molecules in solution‐Method and error assessment. J. Comput. Chem. 9:327‐335.
   Gorfe, A.A., Ferrara, P., Caflisch, A., Marti, D.N., Bosshard, H.R., and Jelesarov, I. 2002. Calculation of protein ionization equilibria with conformational sampling: pK(a) of a model leucine zipper, GCN4 and barnase. Proteins 46:41‐60.
   Harvey, S.C. and Hoekstra, P. 1972. Dielectric relaxation spectra of water adsorbed on lysozyme. J. Phys. Chem. 76:2987‐2994.
   Havranek, J.J. and Harbury, P.B. 1999. Tanford‐Kirkwood electrostatics for protein modeling. Proc. Natl. Acad. Sci. U.S.A. 96:11145‐11150.
   Holst, M., Baker, N., and Wang, F. 2000. Adaptive multilevel finite element solution of the Poisson‐Boltzmann equation I. Algorithms and examples. J. Comput. Chem. 21:1319‐1342.
   Jorgensen, W.L. and Tirado‐Rives, J. 1988. The OPLS potential functions for proteins. Energy minimizations for crystals of cyclic peptides and crambin. J. Am. Chem. Soc. 110:1657‐1666.
   Karshikoff, A. 1995. A simple algorithm for the calculation of multiple‐site titration curves. Protein Eng. 8:243‐248.
   Khare, D., Alexander P., Antosiewicz J., Bryan P., Gilson M., and Orban J. 1997. pKa measurements from nuclear magnetic resonance for the B1 and B2 immunoglobin G‐binding domains of protein G: Comparison with calculated values for nuclear magnetic resonance and x‐ray structures. Biochemistry 36:3580‐3589.
   Klapper, I., Hagstrom, R., Fine, R., Sharp, K., and Honig, B. 1986. Focussing of electric fields in the active site of Cu‐Zn superoxide dismutase: Effects of ionic strength and amino‐acid modification. Proteins 1:47‐59.
   Koumanov, A., Spitzner, N., Rüterjans, H., and Karshikoff, A.D. 2001. Ionization properties of titratable groups in ribonuclease T1 II. Electrostatic analysis. Eur. Biophy. J. 30:198‐206.
   Koumanov, A., Ruterjans, H., and Karshikoff, A. 2002. Continuum electrostatic analysis of irregular ionization and proton allocation in proteins. Proteins 46:85‐96.
   Krishtalik, L.I., Kuznetsov, A.M., and Mertz, E.L. 1997. Electrostatics of proteins: Description in terms of two dielectric constants simultaneously. Proteins 28:174‐182.
   Kuhlman, B., Luisi, D., Young, P., and Raleigh, D. 1999. pKa values and the pH dependent stability of the N‐terminal domain of L9 as probes of electrostatic interactions in the denatured state: Differentiation between local and nonlocal interactions. Biochemistry 38:4896‐4903.
   Kundrotas, P.J. and Karshikoff, A. 2002. Modeling of denatured state for calculation of the electrostatic contribution to protein stability. Prot. Sci. 11:1681‐1686.
   Lee, K.K., Fitch, C.A., Lecomte, J.T.J., and Garcia‐Moreno E., B. 2002. Electrostatic effects in highly charged proteins: Salt sensitivity of pKa values of histidines in staphylococcal nuclease. Biochemistry 41:5656‐5667.
   Li, H., Robertson, A.D., and Jensen, J.H. 2005. Very fast empirical prediction and rationalization of protein pKa values. Proteins 61:704‐721.
   Linderstrøm‐Lang, K. 1924. On the ionization of proteins. C R Trav. Lab. Carlsberg 15:1‐29.
   MacKerell, A.D., Bashford, D., Bellott, M., Dunbrack, R.L., Evanseck, J.D., Field, M.J., Fischer, S., Gao, J., Guo, H., Ha, S., Joseph‐McCarthy, D., Kuchnir, L., Kuczera, K., Lau, F.T.K., Mattos, C., Michnick, S., Ngo, T., Nguyen, D.T., Prodhom, B., Reiher, W.E., Roux, B., Schlenkrich, M., Smith, J.C., Stote, R., Straub, J., Watanabe, M., Wiorkiewicz‐Kuczera, J., Yin, D., and Karplus, M. 1998. All‐atom empirical potential for molecular modeling and dynamics studies of proteins. J. Phys. Chem. B 102:3586‐3616.
   Madura, J.D., Briggs, J.M., Wade, R.C., Davis, M.E., Luty, B.A., Ilin, A., Antosiewicz, J., Gilson, M.K., Bagheri, B., Scott, L.R., and McCammon, J.A. 1995. Electrostatics and diffusion of molecules in solution‐Simulations with the University of Houston Brownian Dynamics program. Comp. Phys. Commun. 91:57‐95.
   Matthew, J.B., Gurd, F.R.N., García‐Moreno E., B., Flanagan, M.A., March, K.L., and Shire, S.J. 1985. pH‐dependent properties in proteins. CRC Crit. Rev. Biochem. 18:91‐197.
   Mehler, E.L. and Guarnieri, F. 1999. A self‐consistent, microenvironment modulated screened coulomb potential approximation to calculate pH‐dependent electrostatic effects in proteins. Biophys. J. 75:3‐22.
   Neria, E., Fischer, S., and Karplus, M. 1996. Simulation of activation free energies in molecular systems. J. Chem. Phys. 105:1902‐1921.
   Nielsen, J.E. and Vriend, G. 2001. Optimizing the hydrogen‐bond network in Poisson‐Boltzmann equation‐based pK(a) calculations. Proteins 43:403‐412.
   Nielsen, J.E., Andersen, K.V., Honig, B., Hooft, R.W.W., Klebe, G., Vriend, G., and Wade, R.C. 1999. Improving macromolecular electrostatics calculations. Prot. Eng. 12:657‐662.
   Oberoi, H. and Allewell, N.M. 1993. Multigrid solution of the nonlinear Poisson‐Boltzmann Equation and calculation of titration curves. Biophys. J. 65:48‐55.
   Ondrechen, M.J., Clifton, J.G., and Ringe, D. 2001. THEMATICS: A simple computational predictor of enzyme structure from function. Proc. Natl. Acad. Sci. U.S.A. 98:12473‐12478.
   Richards, F.M. 1977. Areas, volumes, packing, and protein structure. Annu. Rev. Biophys. Bioeng. 6:151‐176.
   Schaefer, M., Van Vlijmen, H.W.T., and Karplus, M. 1998. Electrostatic contributions to molecular free energies in solution. In Advances In Protein Chemistry, Vol 51. (E. Di Cera, D.E. Eisenberg, and F.M. Richards, eds.) pp. 1‐57. Academic Press Inc., San Diego.
   Schaefer, M., Bartels, C., Leclerc, F., and Karplus, M. 2001. Effective atom volumes for implicit solvent models: Comparison between Voronoi volumes and minimum fluctuation volumes. J. Comput. Chem. 22:1857‐1879.
   Scharnagl, C., Raupp‐Kossmann, R., and Fischer, S.F. 1999. Molecular basis for pH sensitivity and proton transfer in green fluorescent protein: Protonation and conformational substates from electrostatic calculations. Biophys. J. 77:1839‐1857.
   Schutz, C.N. and Warshel, A. 2001. What are the dielectric “constants” of proteins and how to validate electrostatic models? Protein 44:400‐417.
   Sham, Y.Y., Chu, Z.T., and Warshel, A. 1997. Consistent calculations of pKa's of ionizable residues in proteins: Semi‐microscopic and microscopic approaches. J. Phys. Chem. B 101:4458‐4472.
   Sham, Y.Y., Muegge, I., and Warshel, A. 1998. The effect of protein relaxation on charge‐charge interactions and dielectric constants of proteins. Biophys. J. 74:1744‐1753.
   Simonson, T. 2003. Electrostatics and dynamics of proteins. Reports On Progress In Physics 66:737‐787.
   Simonson, T. and Perahia, D. 1995. Internal and interfacial dielectric‐properties of Cytochrome‐C from molecular‐dynamics in aqueous‐solution. Proc. Natl. Acad. Sci. U.S.A. 92:1082‐1086.
   Simonson, T., Archontis, G., and Karplus, M. 1999. A Poisson‐Boltzmann study of charge insertion in an enzyme active site: The effect of dielectric relaxation. J. Phys. Chem. B 103:6142‐6156.
   Simonson, T., Carlsson, J., and Case, D.A. 2004. Proton binding to proteins: pK(a) calculations with explicit and implicit solvent models. J. Am. Chem. Soc. 126:4167‐4180.
   Sitkoff, D., Sharp, K.A., and Honig, B. 1994. Accurate calculation of hydration free energies using macroscopic solvent models. J. Phys. Chem. 98:1978‐1988.
   Soares, T.A., Lins, R.D., Straatsma, T.P., and Briggs, J.M. 2002. Internal dynamics and ionization states of the macrophage migration inhibitory factor: Comparison between wild‐type and mutant forms. Biopolymers 65:313‐323.
   Spassov, V.Z., Luecke, H., Gerwert, K., and Bashford, D. 2001. pK(a) calculations suggest storage of an excess proton in a hydrogen‐bonded water network in bacteriorhodopsin. J. Mol. Biol. 312:203‐219.
   Tanford, C. 1950. Preparation and properties of serum and plasma proteins. XXIII. Hydrogen ion equilibria in native and modified human serum albumins. J. Am. Chem. Soc. 72:441‐451.
   Tanford, C. 1957. Theory of protein titration curves II. Calculations for simple models at low ionic strength. J. Am. Chem. Soc. 79:5340‐5347.
   Tanford, C. and Roxby, R. 1972. Interpretation of protein titration curves: Application to lysozyme. Biochemistry 11:2192‐2198.
   Teixeira, V.H., Cunha, C.A., Machuqueiro, M., Oliveira, A.S.F., Victor, B.L., Soares, C.M., and Baptista, A.A. 2005. On the use of different dielectric constants for computing individual and pairwise terms in Poisson‐Boltzmann studies of protein ionization equilibrium. J. Phys. Chem. B 109:14691‐14706.
   Trylska, H., Antosiewicz, J., Geller, M., Hodge, C.N., Klabe, R.M., Head, M.S., and Gilson, M.K. 1999. Thermodynamic linkage between the binding of protons and inhibitors to HIV‐1 protease. Prot. Sci. 8:180‐195.
   Ullmann, G.M. and Knapp, E.W. 1999. Electrostatic models for computing protonation and redox equilibria in proteins. Eur. Biophys. J. 28:533‐551.
   van Vlijmen, H.W.T., Schaefer, M., and Karplus, M. 1998. Improving the accuracy of protein pKa calculations: Conformational averaging versus the average structure. Proteins 33:145‐158.
   Vijayakumar, M. and Zhou, H.‐X. 2001. Salt bridges stabilize the folded structure of barnase. J. Phys. Chem. B 105:7334‐7340.
   Voges, D. and Karshikoff, A. 1998. A model of a local dielectric constant in proteins. J. Chem. Phys. 108:2219‐2227.
   Warshel, A. 1981. Calculations of enzymatic‐reactions ‐ calculations of pKa, proton‐transfer reactions, and general acid catalysis reactions in enzymes. Biochemistry 20:3167‐3177.
   Warshel, A. and Aqvist, J. 1991. Electrostatic Energy and Macromolecular Function. Annu. Rev. Biophys. Biophys. Chem. 20:267‐298.
   Warshel, A. and Levitt, M. 1976. Theoretical studies of enzymic reactions‐Dielectric, electrostatic and steric stabilization of carbonium‐ion in reaction of lysozyme. J. Mol. Biol. 103:227‐249.
   Warshel, A. and Papazyan, A. 1998. Electrostatic effects in macromolecules: Fundamental concepts and practical modeling. Curr. Opin. Struct. Biol. 8:211‐217.
   Warshel, A. and Russell, S.T. 1984. Calculations of electrostatic interactions in biological‐systems and in solutions. Q. Rev. Biophys. 17:283‐422.
   Warshel, A., Naray‐Szabo, G., Sussman, F., and Hwang, J.K. 1989. How do serine proteases really work. Biochemistry 28:3629‐3637.
   Warwicker, J. 1994. Improved continuum electrostatic modeling in proteins, with comparison to experiment. J. Mol. Biol. 236:887‐903.
   Warwicker, J. 1997. Improving pKa calculations with consideration of hydration entropy. Prot. Eng. 10:809‐814.
   Warwicker, J. 1999. Simplified methods for pK(a) and acid pH‐dependent stability estimation in proteins: Removing dielectric and counterion boundaries. Prot. Sci. 8:418‐425.
   Warwicker, J. 2004. Improved pK(a) calculations through flexibility based sampling of a water‐dominated interaction scheme. Prot. Sci. 13:2793‐2805.
   Warwicker, J. and Watson, H.C. 1982. Calculation of the electric potential in the active site cleft due to a‐helix dipoles. J. Mol. Biol. 157:671‐679.
   Whitten, S.T. and García‐Moreno E., B. 2000. pH dependence of stability of staphylococcal nuclease: Evidence of substantial electrostatic interactions in the denatured state. Biochemistry 39:14292‐14304.
   Wlodek, S.T., Antosiewicz, J., and McCammon, J.A. 1997. Prediction of titration properties of structures of a protein derived from molecular dynamics trajectories. Prot. Sci. 6:373‐382.
   Yang, A.‐S., Gunner, M.R., Sampogna, R., Sharp, K., and Honig, B. 1993. On the calculation of pKas in proteins. Proteins 15:252‐265.
   You, T. and Bashford, D. 1995. Conformation and hydrogen ion titration of proteins: A continuum electrostatic model with conformational flexibility. Biophys. J. 69:1721‐1733.
   Zhou, H.‐X. 2002. A Gaussian‐chain model for treating residual charge‐charge interactions in the unfolded state of proteins. Proc. Natl. Acad. Sci. U.S.A. 99:3569‐3574.
   Zhou, H.‐X. 2003. Direct test of the Gaussian‐chain model for treating residual charge‐charge interactions in the unfolded state of proteins. J. Am. Chem. Soc. 125:2060‐2061.
   Zhou, H.‐X. and Vijayakumar, M. 1997. Modeling of protein conformational fluctuations in pKa predictions. J. Mol. Biol. 267:1002‐1011.
Key References
   Davis et al., 1991. See above.
  The above references contain a description of methodology for FDPB calculations with UHBD.
   Madura et al., 1995. See above.
  The above references contain reviews of PB and FDPB methods.
   Antosiewicz et al., 1996a. See above.
  The above references contain in depth discussions of problems of protein reorganization.
   Simonson, 2003. See above.
   Ullman and Knapp, 1999. See above.
   Garcia‐Moreno and Fitch, 2004. See above.
   Archontis and Simonson, 2005. See above.
   Sham et al., 1997. See above.
   Simonson et al., 1999. See above.
   Schutz and Warshel, 2001. See above.
   Simonson et al., 2004. See above.
Internet Resources
   See Table .
  Prof. Jens Nielsen's website discusses many aspects of pKa calculations. Tools are available at this website for calculations and analysis of H+ titration curves.
   http://enzyme.ucd.ie/Science/pKa/pKa_introduction)
  Explanation of MCCE method
   http://honiglab.cpmc.columbia.edu/mcce/mcce.html
GO TO THE FULL PROTOCOL:
PDF or HTML at Wiley Online Library