Computational biochemistry ferenc Bogár György Ferency



Download 2.94 Mb.
Page14/36
Date02.05.2018
Size2.94 Mb.
#47263
1   ...   10   11   12   13   14   15   16   17   ...   36

UHBD [20]: The electrostatic interactions can be calculated by LPBE or the full NPBE. The potential of the mean force is approximated by the electrostatic energy.with neclecting the non-electrostatics and the effect of ions. The most active part of the interactions can be refined by increasing the resolution of the meshes.

APBS [21]: The method is for the evaluation of the electrostatic properties for a wide range of length scale (tens to millions atoms).

MEAD [22]: The method solves the PBE and optionally calculates Brownian dynamics pKa of protein sidechains.

ZAP [23]: The algorithm calculates (i) an electrostatic potential field in and around a small- or biomacro- molecule. (ii) calculates solvation energy for a single molecule or a group of small molecules, (iii) estimates the binding affinity of a ligand bound to a particular enzyme, (iv) predicts pKa for residues within a protein.

The numerical solution of the electrostatics can be performed by (i) finite difference with considering the neighbouring points (FD), (ii) boundary element (BE) method by using analytical solutions obtained in terms of Green’s functions, (iii) finite element (FEM) method is adaptive multilevel approach. It uses tetrahedral elements in the mesh, the dielectric discontinuity is smoothed [20]. The first method is fast with Cartesian mesh and demands low memory, but the resolution of the solution is poor and non-adaptive. The second method smaller numerically and only applicable for linear problems. FEM is highly adaptive and fast [24].



The electrostatic potential map on the van der Waals surface of the Barnase-Barstar protein complex can be seen in Figure 3.7.

Figure 3.7. Electrostatic potential on the solvent accessible surface of Barnase-Barstar protein (PDB Id.: 2BRS without structural water) calculated by Delphi (red: negative, blue: positive, white: neutral).



Figure 3.8. Electrostatic potential on the van der Waals surface of Barnase-Barstar protein association at different distances from each other (PDB Id.: 2brs without structural water) calculated by Delphi (red: negative, blue: positive, white: neutral), BarnBar_H: complex, BarnBar_5_H: distance between mass centres is 5 Ǻ, BarnBar_10_H: between mass centres is 10 Ǻ, BarnBar_15_H: between mass centres is 15 Ǻ, BarnBar_20_H: between mass centres is 20 Ǻ.



Figure 3.8. describes the change in electrostatic potential at different distances between the centre of masses.

7.1. Solution of LPBE



The PBE is usually applied to a one to one salt solution and the PBE becomes Eq. 3.26




(3.26)

(3.29) where κ2=2z2eFc0/(kTε0), z is the charge of the ion, F is the Faraday constant, T is the temperature, k is the Boltzmann constant, c0 is the concentration. Eq. 3.26 is valid for (i) ions with spherical field (Debye-Hückel theory), (ii) ions near a charged plane. The linear PBE (see later) becomes in spherical coordinates (3.27)




(3.27)

The solution is (3.28)




(3.28)

where A is a constant. A can be given for two boundary conditions [25].

8. Langevine and Brownian dynamics



The mathematical basis of molecular motion in a solvent was developed by Paul Langevin. The model of the solvation media is implicit solvation. The basic expression of the Langevin Dynamics (LD) is described in Eq. 3.29 with canonical ensemble. It is a stochastic equation with N particles (Σi=1i = N, the total number of molecules). It does not consider the hydrophobic effects and the electrostatic screening.




(3.29)

where φ(ri) is the particle interaction potential. The first expression is the particle interaction force. The second and the third expressions in the right side are the frictional and the random force, respectively. The frictional force represents the viscosity of the solution, the random force reepresents the thermal motion of the solvent molecules. R(t) is a delta correlated stationary Gaussian process Eq. 3.30-3.31:




(3.30)






(3.31)

where δ is the Dirac delta. γ is the friction constant. The greater the γ, the larger the viscosity is. Generally, 5-20 ps-1 is chosen in the simulation. The integration time (see Chapter 5) is not a real time. The simulation can give us information on the folding of peptides and small proteins in solvents. The main problem that the real solvent-solute interaction can not be described by this method.




(3.32)

Eq. 3.29 and Eq. 3.32 can be solved by the methods of solution applied in molecular dynamics (MD) (see Chapter 5).

Langevin and Brownian dynamics are good methods for studying folding, association of peptides, proteins and protein-ligands.

Brownian dynamics can be used for the calculation of diffusion constants of proteins and the motion of proteins on nanoscale metal particles with an interface SDA developed by Wade et al. [26].

9. Summary

The electrostatic properties of extended structures, solvation free energies, association energies and the properties of proteins (and other biomolecules) on metal (gold) surface are possible by the solution of Poisson-Boltzmann, Linear Poisson-Boltzmann and Tanford-Kirkwood equations. The folding of peptides, association of proteins can be studied by Langevine or Brownian dynamics.

10. References



  1. P. Kukic, J. E. Nielsen, Electrostatics in proteins and protein-ligand complexes, Future Med. Chem. 2(4), 647-666(2010).

  2. X. Hao, A. Varshney, “Efficient Solution of Poisson–Boltzmann Equation for Electrostatics of Large Molecules”, High Performance Computing Symposium 71–76 (2004).

  3. P. A. Atkins, Physical Chemistry, 6th Edition, Oxford, UK, 2004.

  4. M. K. Gilson, Introduction to electrostatics, with molecular applications. www.gilsonlab.umbi.umd.edu.

  5. K. A. Sharp, B. Honig, Electrostatic interactions in macromolecules: theory and applications. Annu. Rev. Biophys. Chem. 19, 301-332(1990).

  6. X. Shi, P. Koehl, The Geometry Behind Numerical Solvers of the Poisson-Boltzmann Equation Comm. in Comp. Phys. 3(5), 1032-1050(2008).

  7. B. Z. Lu, Y. C. Zhou, M. J. Holst, J. A. McCammon, Recent Progress in Numerical Methods for the Poisson-Boltzmann Equation in Biophysical Applications. Comm. in Comp. Phys. 3(5) 973-1009(2008).

  8. S. S. Kuo, M. D. Altman, J. P. Bardhan, R. Tidor, J. K. White, Fast Methods for Simulation of Biomolecule Electrostatic, Computer Aided Design, 2002. ICCAD 2002. IEEE/ACM International Conference, Date of Conference: 10-14 Nov. 2002.

  9. M. Holst, F. Saied, Multigrid Solution of the Poisson-Boltzmann Equation, J. Comput. Chem., 14, 105-113(1993).

  10. N. Perrin, Probabilistic Interpretation for the Nonlinear Poisson-Boltzmann Equation in Molecular Dynamics, ESAIM: Proceedings, 35, 174-183, March 2012.

  11. F. Fogolari, P. Zuccato, G. Esposito, P.Viglino, Biomolecular Electrostatics with the Linearized Poisson-Boltzmann Equation, Biophys. J. 76, 1-16 (1999).

  12. J. H. Chaudhry, Stephen D. Bond, Luke N. Olson, Finite Element Approximation to a Finite-Size Modified Poisson-Boltzmann Equation, J. Sci. Comp. 47(3), 347-364(2011).

  13. A. Abrashkin, D. Andelman, H. Orland, Dipolar Poisson-Boltzmann equation: ions and dipoles close to charge interfaces, Phys. Rev. Lett. 99(7), 077801(2007).

  14. B. Jayaram, D. L. Beveridge, Tanford-Kirkwood Theory for Concentric Dielectric Continua: Application to Dimethylphosphate, Biopolymers, 27, 617-627(1988).

  15. F. L. B. Da Silva, B. Jönsson, R. Penfold, A Critical Investigation of the Tanford-Kirkwood Scheme by means of Monte Carlo Simulations, Protein Sci. 10, 14151425(2001).

  16. J. J. Havranek, P. B. Harbury, Tanford-Kirkwood electrostatics for protein modeling. Proc. Natl. Acad. Sci. USA 96, 11145-11150(1999).

  17. M. Schnieders, J. W. Ponder, Polarizable Atomic Multipole Solutes in a Generalized Kirkwood Continuum, J. Chem. Theory Comput. 3, 2083-2097(2007).

  18. C. Tanford, J. G. Kirkwood, Theory of Protein Titration Curves. I. General Equations for Impenetrable Spheres. J. Am. Chem. Soc., 79 (20), 5333–5339(1957).

  19. B.Honig and A.Nicholls. Classical electrostatics in biology and chemistry. Science. 268, 1144-1149 (1995).

  20. J. D. Madura, J. M. Briggs, R. C. Wade, M. E. Davis, B. A. Luty, A. Ilin, J. Antosiewicz, M. K. Gilson, B. Bagheri, L. R. Scott and J. A. McCammon, Electrostatics and diffusion of molecules in solution: simulations with the University of Houston Brownian Dynamics program, Comp. Phys. Comm. 91, 57-95 (1995).

  21. N. A. Baker, D. Sept, S. Joseph, M. J. Holst, J. A. McCammon, Electrostatics of nanosystems: application to microtubules and the ribosome. Proc. Natl. Acad. Sci. USA 98, 10037-10041(2001).

  22. a) D. Bashford, K. Gerwert, Electrostatic Calculations of the pKa Values of Ionizable Groups in Bacteriorhodopsin, J Mol Biol vol. 224, 473-486 (1992).. b) J. L. Chen, L. Noodleman, D. A. Case, D. Bashford, Incorporating Solvation Effects Into Density Functional Electronic Structure Calculations, J. Phys. Chem., 98 (43), 11059–11068(1994). c) D. Bashford, Y. Ishikawa,  R. R. Oldehoeft, J. V. W. Reynders, M. Tholburn, An Object-Oriented Programming Suite for  Electrostatic Effects in Biological Molecules  Eds, Scientific Computing in Object-Oriented Parallel  Environments, volume 1343 of Lecture Notes in Computer Science, pages  233-240, Berlin, 1997.

  23. J. A. Grant, B. T. Pickup, A. Nicholls A, A smooth permittivity function for Poisson-Boltzmann solvation methods. J. Comput. Chem., 22, 608-640(2001).

  24. J.-P. Hsu, B.-T. Liu, Exact Solution to the Linearized Poisson-Boltzmann Equation for Spheroidal Surfaces, J. Coll. And Interface Sci. 175, 785-788(1996).

  25. X. Cheng, Implicit Solvation Models, Introduction to Molecular Biophysics, ÚT/ORNL Center for Molecular Biophysics, 2008.

  26. R. R. Gabdouline, R. C. Wade, Brownian Dynamics Simulation of Protein-Protein Diffusional Encounter, METHODS: A Comparision to Methods in Enzymology, 14, 329-341(1998).

11. Further Readings

  1. R. Leach, Molecular Modelling, Principles and Applications, 2nd Edition, Prentice Halls, pp. 603-608, Pearson Education Limited, 2001.

  2. B.Z. Lu, Y. C. Zhou, M. J. Holst, J. A. McCammon, Recent Progress in Numerical Methods for the Poisson-Boltzmann Equation in Biophysical Application, Review Article, Comm. in Comp. Phys. 8(5), 973-1009(2008).

  3. M. J. Holst, The Poisson-Boltzmann Equation. Analysis and Multilevel Numerical Solution. Monograph based on the Ph.D. Thesis below). Applied Mathematics and CRPC, California Institute of Technology, 1994.

  4. M. Holst, N. Baker, and F. Wang, Adaptive Multilevel Finite Element Solution of the Poisson-Boltzmann Equation I: Algorithms and Examples. J. Comput. Chem., 21, 1319-1342(2000).

  5. J. P. Bardhan, Numerical Solution of Boundary Integral Equations for Molecular Electrostatics, J. Chem. Phys. 130(9), 094102(2009).

  6. P. Grochowski, J. Tylska, Continuum molecular electrostatics, salt effects, and countarion binding – a review of the Poisson-Boltzmann theory and its modifications, Biopolymers, 89(2), 93-113(2008).

  7. B. Kraczek, Solving the Poisson-Boltzmann Equation, ICES Multi-scale Group meeting, Sept. 29, 2008.

  8. W.Rocchia, E.Alexov, and B.Honig. Extending the Applicability of the Nonlinear Poisson-Boltzmann Equation: Multiple Dielectric Constants and Multivalent Ions. J. Phys. Chem. B 105(28), 6507-6514(2001).

  9. B. L. Tembe, J. A. McCammon, Ligand-Receptor Interactions, Comp. Chem. 8(4), 281-283(1984).

  10. N. Baker, M. Holst, F. Wang, Adaptive multilevel finite element solution of the Poisson–Boltzmann equation II. Refinement at solvent-accessible surfaces in biomolecular systems, J. Comp. Chem. 21, 1343–1352(2000).

  11. M. T. Neves-Petersen, S. B. Petersen, Protein electrostatics: a review of the equations and methods used to model electrostatic equations in biomolecules-applications in biotechnology, Biotechnol. Annu. Rev. 9, 315-395(2003).

12. Questions

  1. What is the interaction between two point charges in a media with ε dielectric constant?

  2. Please, write the Coulomb’s law.

  3. The Poisson equation in a media with constant permittivity.

  4. The Poisson equation in a media with variable permittivity.

  5. The Boltzmann distribution.

  6. The Poisson-Boltzmann equation

13. Glossary


Download 2.94 Mb.

Share with your friends:
1   ...   10   11   12   13   14   15   16   17   ...   36



The database is protected by copyright ©ininet.org 2024
send message
    Main page
basic knowledge
better understanding