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Free Energy Calculation Methods Used in Computer Simulations

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Molecular Dynamics Simulations in Statistical Physics: Theory and Applications

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Abstract

In this chapter, we will present the most advanced methods used in the calculation of free energy from the computer simulations. First, in this chapter, we will discuss the methods employed in molecular dynamics simulations using explicit solvent models, such as the thermodynamic free energy perturbation method, thermodynamic integration method, and slow growth method. Then, the implicit solvation models will be discussed using either Poisson-Boltzmann or Generalized Born approximation for treating the electrostatic interactions.

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References

  • Alfrey, T. Jr., Berg, P.W., Morawetz, H.: The counter ion distribution in solutions of rod-shaped polyelectrolytes. J. Polym. Sci. 7, 543–547 (1951)

    Article  ADS  Google Scholar 

  • Ashbaugh, H.S., Kaler, E.W., Paulaitis, M.E.: A universal surface area correlation for molecular hydrophobic phenomena. J. Am. Chem. Soc. 121, 9243–9244 (1999)

    Article  Google Scholar 

  • Born, M.: Volumen und Hydratationswärme der ionen. Z. Phys. 1, 45–48 (1920)

    Article  ADS  Google Scholar 

  • Bren, M., Florián, J., Mavri, J., Bren, U.: Do all pieces make a whole? Thiele cumulants and the free energy decomposition. Theor. Chem. Acc. 117, 535–540 (2007)

    Google Scholar 

  • Bren, U., Martínek, V., Florián, J.: Decomposition of the solvation free energis of Deoxyribonucleoside Triphosphates using the free energy peturbation method. J. Phys. Chem. B 110, 12782–12788 (2006)

    Article  Google Scholar 

  • Chapman, D.L.: A contribution to the theory of electrocapillarity. Phil. Mag. 25, 475–481 (1913)

    Article  MATH  Google Scholar 

  • Chen, J., Brooks III, C.L.: Implicit modelling of non polar solvation for simulating protein folding and conformational transitions. Phys. Chem. Chem. Phys. 10, 471–481 (2008)

    Article  Google Scholar 

  • Chothia, C.: Hydrophobic binding and accessible surface area in proteins. Nature 248, 338–339 (1974)

    Article  ADS  Google Scholar 

  • Connolly, M.L.: Solvent-accessible surfaces of proteins and nucleic acids. Science 221, 709 (1983a)

    Article  ADS  Google Scholar 

  • Connolly, M.L.: Analytical molecular surface calculation. J. Appl. Cryst. 16, 548–558 (1983b)

    Article  Google Scholar 

  • Connolly, M.L.: Molecular surface triangulation. J. Appl. Cryst. 18, 499 (1985)

    Article  Google Scholar 

  • Cramer, C.J., Truhlar, D.G.: Implicit solvation models: equilibria, structure, spectra, and dynamics. Chem. Rev. 99(8), 2161–2200 (1999)

    Article  Google Scholar 

  • Debye, P., Hückel, E.: Zur theorie der elektrolyte. Phys. Zeitschr. 24, 185–206 (1923)

    MATH  Google Scholar 

  • Derjaguin, B., Landau, L.: A theory of the stability of strongly charged lyophobic sold and the coalescence of strongly charged particles in electrolytic solution. Acta Phys. Chim. USSR 14, 633–662 (1941)

    Google Scholar 

  • Eisenberg, D., McLachlan, A.D.: Solvation energy in protein folding and binding. Nature 319, 199–203 (1986)

    Article  ADS  Google Scholar 

  • Fogolari, F., Briggs, J.M.: On variational approach to the Poisson-Boltzmann free energies. Chem. Phys. Lett. 281, 135–139 (1997)

    Article  ADS  Google Scholar 

  • Fogolari, F., Zuccato, P., Esposito, G., Viglino, P.: Biomolecular electrostatics with the linearised Poisson-Boltzmann equation. Biophys. J. 76, 1–16 (1999)

    Article  Google Scholar 

  • Fogolari, F., Brigo, A., Molinari, H.: The Poisson-Boltzmann equation for biomolecular electrostatics: a tool for structural biology. J. Mol. Recognit. 15, 377–392 (2002)

    Article  Google Scholar 

  • Fowler, R.H., Guggenheimer, E.A.: Statistical Thermodynamics. Cambridge University Press, Cambridge (1939)

    Google Scholar 

  • Frenkel, D., Smit, B.: Understanding Molecular Simulation from Algorithms to Applications. Academic, San Diego (2001). ISBN 9780122673511

    Google Scholar 

  • Gallicchio, E., Levy, R.M.: AGBNP: an analytic implicit solvent model suitable for molecular dynamics simulations and high-resolution modeling. J. Comput. Chem. 25, 479–499 (2004)

    Article  Google Scholar 

  • Gilson, M.K., Honig, B.: The dielectric constant of a folded protein. Biopolymers 25, 2097–2119 (1986)

    Article  Google Scholar 

  • Gilson, M.K., Honig, B.H.: Calculation of the total electrostatic energy of a macromolecular system: solvation energies, binding energies, and conformational analysis. Proteins 4, 7–18 (1988)

    Article  Google Scholar 

  • Gohlke, H., Kiel, C., Case, D.A.: Insights into protein-protein binding by binding free energy calculation and free energy decomposition for Ras-Raf and Ras-RaIGDS complexes. J. Mol. Biol. 330(4), 891–913 (2003)

    Article  Google Scholar 

  • Gouy, M.: Sur la constitution de la charge électrique a la surface d’un électrolyte. J. Phys. 9, 457–468 (1910)

    MATH  Google Scholar 

  • Gronwall, T.H., La Mer, V.K., Sandved, K.: Über den einfluss der sogenannten höheren glieder in der Debye-Hückelschen theorie der lösungen starker elektrolyte. Phys. Zeitschr. 29, 358–393 (1928)

    MATH  Google Scholar 

  • Grycuk, T.: Deficiency of the Coulomb-field approximation in the generalised Born model: an improved formula for form radii evaluation. J. Chem. Phys. 119, 4817–4826 (2003)

    Article  ADS  Google Scholar 

  • Hawkins, G.D., Cramer, C.J., Truhlar, D.G.: Pairwise solute descreening of solute charges from a dielectric medium. Chem. Phys. Lett. 246, 122–129 (1995)

    Article  ADS  Google Scholar 

  • Hawkins, G.D., Cramer, C.J., Truhlar, D.G.: Parametrized models of aqueous free energies of solvation based on pairwise descreening of solute atomic charges from a dielectric medium. J. Phys. Chem. 100, 19824–19839 (1996)

    Article  Google Scholar 

  • Jackson, J.D.: Classical Electrodynamics. Wiley, New York (1962)

    MATH  Google Scholar 

  • Katchalski, A.: Polyelectrolytes. Pure Appl. Chem. 26, 327–371 (1971)

    Article  Google Scholar 

  • Kirkwood, J.G.: On the theory of strong electrolyte solutions. J. Chem. Phys. 2, 767–781 (1934a)

    Article  ADS  MATH  Google Scholar 

  • Kirkwood, J.G.: Theory of solutions of molecules containing widely separated charges with special applications to zwitterions. J. Chem. Phys. 7, 351–361 (1934b)

    Article  ADS  MATH  Google Scholar 

  • Kollman, P.A., Massova, I., Reyes, C., Kuhn, B., Huo, S.H., Chong, L., Lee, M., Lee, T., Duan, Y., Wang, W., Donini, O., Cieplak, P., Srinivasan, J., Case, D.A., Cheatham, T.E.: Calculating structures and free energies of complex molecules: combining molecular mechanics and continuum models. Accounts Chem. Res. 33(12), 889–897 (2000)

    Article  Google Scholar 

  • Lazaridis, T., Karplus, M.: Effective energy function for proteins in solution. Proteins 35, 133–152 (1999)

    Article  Google Scholar 

  • Lee, B., Richards, F.M.: The interpretation of protein structures: estimation of static accessibility. J. Mol. Biol. 55, 379–400 (1971)

    Article  Google Scholar 

  • Lee, M.S., Salsbury, F.R. Jr., Brooks III, C.L.: Novel generalized Born methods. J. Chem. Phys. 116, 10606–10614 (2002)

    Article  ADS  Google Scholar 

  • Levy, R.M., Zhang, L.Y., Gallicchio, E., Felts, A.K.: On the nonpolar hydration free energy of proteins: surface area and continuum solvent models for the solute-solvent interaction energy. J. Am. Chem. Soc. 125, 9523–9530 (2003)

    Article  Google Scholar 

  • Lifson, S., Katchalski, A.: The electrostatic free energy of polyelectrolyte solutions. J. Polym. Sci. 13, 43–55 (1954)

    Article  ADS  Google Scholar 

  • Linderstrom-Lang, K.: On the ionisation of proteins. Compt. Rend. Trav. Lab. Carlsberg 15, 1–29 (1924)

    Google Scholar 

  • Luo, R., Moult, J., Gilson, M.K.: Dielectric screening treatment of electrostatic solvation. J. Phys. Chem. B 101, 11216–11236 (1997)

    Google Scholar 

  • Majeux, N., Scarsi, M., Apostolakis, J., Ehrhardt, C., Caflisch, A.: Exhaustive docking of molecular fragments with electrostatic solvation. Proteins 37, 88–105 (1999)

    Article  Google Scholar 

  • Manning, G.S.: The molecular theory of polyelectrolyte solutions with applications to the electrostatic properties of polynucleotides. Q. Rev. Biophys. 11, 179–246 (1978)

    Article  Google Scholar 

  • Marcus, R.A.: Calculation of thermodynamic properties of polyelectrolytes. J. Chem. Phys. 23, 1057–1068 (1955)

    Article  ADS  Google Scholar 

  • Massova, I., Kollman, P.A.: Combined molecular mechanical and continuum solvent approach (MM-PBSA/GBSA) to predict ligand binding. Persp. Drug Disc. Des. 18, 113–135 (2000)

    Article  Google Scholar 

  • Misra, V.K., Sharp, K.A., Friedman, R.A., Honig, B.: Salt effects on ligand-DNA binding. Minor groove antibiotics. J. Mol. Biol. 238, 245–263 (1994)

    Article  Google Scholar 

  • Nemethy, G., Scheraga, H.A.: The structure of water and hydrophobic bonding in proteins. III. The thermodynamic properties of hydrophobic bonds in proteins. J. Phys. Chem. 66, 1773–1789 (1962)

    Google Scholar 

  • Nozaki, Y., Tanford, C.: Examination of titration behaviour. Meth. Enzymol. 11, 715–734 (1967)

    Article  Google Scholar 

  • Onsager, L.: Theories of concentrated electrolytes. Chem. Rev. 13, 73–89 (1933)

    Article  Google Scholar 

  • Ooi, T., Oobatake, M., Nemethy, G., Scheraga, H.A.: Accessible surface area as a measure of the thermodynamic parameters of hydration of peptides. Proc. Natl. Acad. Sci. USA 84, 3086–3090 (1987)

    Article  ADS  Google Scholar 

  • Pearlman, D.A., Kollman, P.A.: A new method for carrying out free energy perturbation calculations: dynamically modified windows. J. Chem. Phys. 90, 2460–2470 (1989)

    Article  ADS  Google Scholar 

  • Petitjean, M.: On the analysis calculation of van der Waals surfaces and volumes: some numerical aspects. J. Comput. Chem. 15, 507–523 (1994)

    Article  MathSciNet  Google Scholar 

  • Pierotti, R.A.: A scaled particle theory of aqueous and nonaqueous solutions. Chem. Rev. 76, 717–726 (1976)

    Article  Google Scholar 

  • Raschke, T.M., Tsai, J., Levitt, M.: Quantification of the hydrophobic interaction by simulations of the aggregation of small hydrophobic solutes in water. Proc. Natl. Acad. Sci. USA 98, 5965–5969 (2001)

    Article  ADS  Google Scholar 

  • Reiner, E.S., Radke, C.J.: Variational approach to the electrostatic free energy in charged colloidal suspensions: general theory for open systems. J. Chem. Soc. Faraday Trans. 86, 3901–3912 (1990)

    Article  Google Scholar 

  • Richards, F.M.: Areas, volumes, packing, and protein structure. Annu. Rev. Biophys. Bioeng. 6, 151–176 (1977)

    Article  Google Scholar 

  • Richards, F.M.: Optical matching of physical models and electron density maps: early developments. Methods Enzymol. 115, 145–154 (1985)

    Article  Google Scholar 

  • Roux, B., Simonson, T.: The implicit solvent models. Biophys. Chem. 78, 1–20 (1999)

    Article  Google Scholar 

  • Schaefer, M., Froemmel, C.: A precise analytical method for calculating the electrostatic energy of macromolecules in aqueous solution. J. Mol. Biol. 216, 1045–1066 (1990)

    Article  Google Scholar 

  • Schutz, C.N., Warshel, A.: What are the dielectric “constants” of proteins and how to validate electrostatic models? Proteins Struct. Funct. Genet. 44, 400–417 (2001)

    Article  Google Scholar 

  • Sharp, K.A., Honig, B.: Calculating total electrostatic energies with non-linear Poisson-Boltzmann equation. J. Phys. Chem. 94, 7684–7692 (1990)

    Article  Google Scholar 

  • Sharp, K.A., Nicholis, A., Fine, R.F., Honig, B.: Reconciling the magnitude of the microscopic and macroscopic hydrophobic effects. Science 252, 106–109 (1991)

    Article  ADS  Google Scholar 

  • Simonson, T.: Dielectric constant of cytochrome c from simulations in water droplet including all electrostatic interactions. J. Am. Chem. Soc. 120, 4875–4876 (1998)

    Article  Google Scholar 

  • Simonson, T.: Macromolecular electrostatics: continuum models and their growing pains. Curr. Opin. Struct. Biol. 11, 243–252 (2001)

    Article  Google Scholar 

  • Simonson, T., Brunger, A.T.: Solvation free energies estimated from macroscopic continuum theory: an accuracy assessment. J. Phys. Chem. 98, 4683–4694 (1994)

    Article  Google Scholar 

  • Sitkoff, D., Sharp, K.A., Honig, B.: Accurate calculation of hydration free energies using macroscopic solvent models. J. Phys. Chem. 98, 1978–1988 (1994)

    Article  Google Scholar 

  • Srinivasan, J., Trevathan, M.W., Beroza, P., Case, D.A.: Application of a pairwise generalized Born model to proteins and nucleic acids: inclusion of salt effects. Theor. Chem. Acc. 101, 426–434 (1999)

    Article  Google Scholar 

  • Still, W.C., Tempczyk, A., Hawley, R.C., Hendrickson, T.: Semi-analytical treatment of solvation for molecular mechanics and dynamics. J. Am. Chem. Soc. 112, 6127–6129 (1990)

    Article  Google Scholar 

  • Tanford, C.: Interfacial free energy and the hydrophobic effect. Proc. Natl. Acad. Sci. USA 76, 4175–4176 (1979)

    Article  ADS  Google Scholar 

  • Tjong, H., Zhou, H.X.: GBr6: a parametrization-free, accurate, analytical generalised Born method. J. Phys. Chem. B 111, 3055–3061 (2007a)

    Article  Google Scholar 

  • Tjong, H., Zhou, H.X.: GBr6NL: a generalised Born method for accurately reproducing solvation energy of the nonlinear Poisson-Boltzmann equation. J. Chem. Phys. 126, 195102 (2007b)

    Article  ADS  Google Scholar 

  • Verwey, E.J.W., Overbeek, J.T.G.: Theory of the Stability of Lyophobic Colloids. Elsevier, Amsterdam (1948)

    Google Scholar 

  • Vorobjev, Y.N., Hermans, J.: SIMS: computation of a smooth invariant molecular surface. Biophys. J. 73, 722 (1997)

    Article  Google Scholar 

  • Wang, W., Reyes, O.D.C., Kollman, P.A.: Biomolecular simulations: recent developments in force fields, simulations of enzyme catalysis, protein-ligand, protein-protein, and protein-nucleic acid noncovalent interactions. Annu. Rev. Bioph. Biom. 30, 211–243 (2001)

    Article  Google Scholar 

  • Widom, B.: Some topics in the theory of fluids. J. Chem. Phys. 39, 2808–2812 (1963)

    Article  ADS  Google Scholar 

  • Zacharias, M.: Protein-protein docking with a reduced protein model accounting for side-chain flexibility. Protein Sci. 12, 1271 (2003)

    Article  Google Scholar 

  • Zhou, H.X.: Macromolecular electrostatic energy within the nonlinear Poisson-Boltzmann equation. J. Chem. Phys. 100, 3152–3162 (1994)

    Article  ADS  Google Scholar 

  • Zwanzig, R.W.: High-temperature equation of state by a perturbation method. 1. Nonpolar gases. J. Chem. Phys. 22, 1420–1426 (1954)

    Google Scholar 

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Kamberaj, H. (2020). Free Energy Calculation Methods Used in Computer Simulations. In: Molecular Dynamics Simulations in Statistical Physics: Theory and Applications. Scientific Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-35702-3_5

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