Topics in Math and Biochem-Biophysics (MBB) - Course Web Site

Math 277B, Spring 2011
Topics in Mathematics and Biochemistry-Biophysics

Instructor: Bo Li

Time: 3:00 pm - 3:50 pm, Mondays, Wednesdays, and Fridays
Room: AP&M 7421

Course Announcement

Biological systems are perhaps among the most fascinating and complex systems in life and science. Recently there have been growing interests and initial success of developing rigorous mathematical theories and novel numerical methods for understanding the basic principles and solving concrete problems of biochemistry and biophysics. Tremendous challenging and opportunities have emerged in such development and applications.

This research oriented course will focus on a few selected topics on mathematical and computational aspects of biochemistry and biophysics. These topics include but are not limited to:

  1. Partial differential equations and dynamical systems models and computation of diffusion process;
  2. Numerical methods for elliptic interface problems for continuum dielectric models of biomolecular interactions;
  3. Geometric and field based surface motion for biomolecular solvation and cell dynamics;
  4. Stochastic process and computation of Brownian motion of biomolecules; and
  5. The Fokker-Planck equation for nonequilibrium biological processes such as protein folding.

This course is designed for graduate students of mathematics, computational science, biochemistry and biophysics, and bioengineeering who are interested in the related interdisciplinary researches on mathematical biological sciences. No knowledge of biology, biochemistry and biophysics is required. No textbooks will be used. Sometimes lecture notes will be distributed. Research projects will be introduced and discussed.

Lecture Notes (Password Required) with Brief Descriptions

  • Pages 1 - 10. Outline of the course. Nonlinear, steady-state diffusion equations. Variational techniques. Euler-Lagrange equations. Examples: The Allen-Cahn functional. The Poisson-Boltzmann equation and related functional. Variational structures of nonlinear diffusion problems for gene expressions, etc.

  • Pages 11 - 19. Some remarks about nonlinear problems: boundary conditions, solution regularity, non-uniqueness of solutions, Newton's iteration and its convergence, ellptic interface problems - two equivalent formulations.

  • Pages 20 - 27. The Poisson-Boltzmann equation. Background. Basics of the equation. Special cases: linearized PBE and sinh PBE. Mean-field electrostatic free-energy functional of ionic concentrations. Derivation of the Boltzmann distributions. Special cases with analytical formulas of solutions: one charged wall, two parallel charged walls, Born's calculation extended, etc.

  • Pages 28 - 40. The Poisson-Boltmann equation continued. Convexity. Bounds of equilibrium concentrations. The existence and uniqueness of the solution to the boundary-value problem of the Poisson-Boltzmann equation.

  • Pages 41 - 44. The Poisson-Boltzmann theory does not predict the wall-mediated like-charge attraction.

  • Pages 45 - 52. Include size effects - uniform or non-uniform sizes - in a mean-field model of electrostatics. Some mathematical results: uniform boundes of concentrations, existence and uniqueness, convexity. Implicit Boltzmann distributions. Numerical methods for the case of non-uniform sizes.

  • Pages 53 - 61. Dielectric boundary forces: a shape derivative approach. Definition and some formulas.

  • Pages 61 - 71. Review of differential geometry. First and second fundamental forms. Definition of the mean and Gaussian curvatures. Special parameterizations of surfaces.

  • Pages 72 - 76. Hadwiger valuations and Helfrich free-energy functionals. Surface variations. Some examples. General approach.

  • Pages 77 - 81. General description of the Poisson-Nernst-Planck (PNP) system with appliction to biomolecular systems. Fluxes. No-flux boundary conditions and the Boltzmann distributions. A reduced PNP sysetm. Semi-analytical and numerical sotitions to the reduced PNP system and reaction rates.

  • Pages 82 - 93 Langevin dynamics - an example, Fluctuation-Dissipation Theorem, Some background of stochastic processes, Brownian motion, Ito stochastic differential equations. The Fokker-Planck equation. Examples. Derivation.

  • Pages 94- 106. A variational principle of the Fokker-Planck equation. Steepest descent, examples. Time discretization of heat equation and a variational principle. Main result on a variational principle of the Fokker-Planck equation. Wasserstein metric. Some properties and some examples.

References

  • Continuum Electrostatics

    1. D. Andelman. Electrostatic properties of membranes: The Poisson-Boltzmann theory. In R. Lipowsky and E. Sackmann, editors, Handbook of Biological Physics, volume 1, pages 603-642. Elsevier, 1995.
    2. I. Borukhov, D. Andelman, and H. Orland. Steric effects in electrolytes: A modified Poisson-Boltzmann equation. Phys. Rev. Lett., 79:435-438, 1997.
    3. J. Che, J. Dzubiella, B. Li, and J. A. McCammon. Electrostatic free energy and its variations in implicit solvent models. J. Phys. Chem. B, 112:3058-3069, 2008. (Preprint in PDF, published version in PDF at the journal website.)
    4. H.-B. Cheng, L.-T. Cheng, and B. Li, Yukawa-field approximation of electrostatic free energy and dielectric boundary force, 2011 (submitted). (Preprint in PDF)
    5. M. E. Davis and J. A. McCammon. Electrostatics in biomolecular structure and dynamics. Chem. Rev., 90:509-521, 1990.
    6. F. Fixman. The Poisson-Boltzmann equation and its application to polyelecrolytes. J. Chem. Phys., 70:4995-5005, 1979.
    7. V. Kralj-Iglic and A. Iglic. A simple statistical mechanical approach to the free energy of the electric double layer including the excluded volume effect. J. Phys. II (France), 6:477-491, 1996.
    8. B. Li. Minimization of electrostatic free energy and the Poisson-Boltzmann equation for molecular solvation with implicit solvent. SIAM J. Math. Anal., 40:2536-2566, 2009. (Published version in PDF - permission and copyright of SIAM.)
    9. B. Li. Continuum electrostatics for ionic solutions with nonuniform ionic sizes. Non-linearity, 22:811-833, 2009. (Preprint in PDF, published version in PDF at the journal website.)
    10. B. Li, X. Cheng, and Z. Zhang, Dielectric boundary force in molecular solvation with the Poisson-Boltzmann free energy: A shape derivative approach, 2011 (submitted). (Preprint in PDF)
    11. J. C. Neu. Wall-mediated forces between like-charged bodies in an electrolyte. Phys. Rev. Lett., 82:1072-1074, 1999.
    12. J. E. Sader and D. Y. C. Chan. Long-range electrostatic attractions between identically charged particles in confined geometries: An unresolved problem. J. Colloid Interf. Sci., 213:268-269, 1999.
    13. K. A. Sharp and B. Honig. Calculating total electrostatic energies with the nonlinear Poisson-Boltzmann equation. J. Phys. Chem., 94:7684-7692, 1990.
    14. K. A. Sharp and B. Honig. Electrostatic interactions in macromolecules: Theory and applications. Annu. Rev. Biophys. Biophys. Chem., 19:301-332, 1990.
    15. G. Tresset. Generalized Poisson-Fermi formalism for investigating size correlation effects with multiple ions. Phys. Rev. E, 78:061506, 2008.
    16. E. Trizac and J.-L. Raimbault. Long-range electrostatic interactions between like-charged colloids: Steric and confinement effects. Phys. Rev. E, 60:6530-6533, 1999.
    17. H.-X. Zhou, Macromolecular electrostatic energy within the nonlinear Poisson-Boltzmann equation, J. Chem. Phys., 100:3125-3162, 1994.
    18. S. Zhou, Z. Wang, and B. Li, Mean-Filed description of ionic size effect with non-uniform ionic sizes: A numerical approach, 2011 (submitted). (Preprint in PDF)

  • Variational Implicit Solvation

    1. J. Che, J. Dzubiella, B. Li, and J. A. McCammon. Electrostatic free energy and its variations in implicit solvent models. J. Phys. Chem. B, 112:3058-3069, 2008. (Preprint in PDF, published version in PDF at the journal website.)
    2. H.-B. Cheng, L.-T. Cheng, and B. Li, Yukawa-field approximation of electrostatic free energy and dielectric boundary force, 2011 (submitted). (Preprint in PDF)
    3. L.-T. Cheng, J. Dzubiella, J. A. McCammon, and B. Li, Application of the level-set method to the implicit solvation of nonpolar molecules, J. Chem. Phys., 127:084503, 2007. (Preprint in PDF, published version in PDF at the journal website.)
    4. L.-T. Cheng, B. Li, and Z. Wang, Level-Set minimization of potential controlled Hadwiger valuations for molecular solvation, J. Comput. Phys., 229:8497-8510, 2010. (Preprint in PDF)
    5. L.-T. Cheng, Z. Wang, P. Setny, J. Dzubiella, B. Li, and J. A. McCammon, Interfaces and hydrophobic interactions in receptor-ligand systems: A level-set variational implicit solvent approach, J. Chem. Phys., 131:144102, 2009. (Preprint in PDF)
    6. L.-T. Cheng, Y. Xie, J. Dzubiella, J. A. McCammon, J. Che, and B. Li, Coupling the level-set method with molecular mechanics for variational implicit solvation of nonpolar molecules, J. Chem. Theory Comput., 5:257-266, 2009. (Preprint in PDF, published version in PDF at the journal website.)
    7. J. Dzubiella, J. M. J. Swanson, and J. A. McCammon, Coupling hydrophobicity, dispersion, and electrostatics in continuum solvent models, Phys. Rev. Lett., 96:087802, 2006.
    8. J. Dzubiella, J. M. J. Swanson, and J. A. McCammon, Coupling nonpolar and polar solvation free energies in implicit solvent models, 124:084905, J. Chem. Phys., 2006.
    9. B. Li. Minimization of electrostatic free energy and the Poisson-Boltzmann equation for molecular solvation with implicit solvent. SIAM J. Math. Anal., 40:2536-2566, 2009. (Published version in PDF - permission and copyright of SIAM.)
    10. B. Li, X. Cheng, and Z. Zhang, Dielectric boundary force in molecular solvation with the Poisson-Boltzmann free energy: A shape derivative approach, 2011 (submitted). (Preprint in PDF)
    11. B. Li and J. Shopple, An interface-fitted finite element level set method with application to solidification and solvation, Commun. Comput. Phys., 10:32-56, 2011. (Preprint in PDF)
    12. B. Roux and T. Simonson, Implicit solvent models, Biophys. Chem., 78:1-20, 1999.
    13. P. Setny, Z. Wang, L.-T. Cheng, B. Li, J. A. McCammon, and J. Dzubliella, Dewetting-controlled binding of ligands to hydrophobic pockets, Phys. Rev. Lett., 103:187801, 2009. (Preprint in PDF)

  • Langevin Dynamics, Fokker-Planck Equation, and Poisson-Nernst-Planck Equation, with Applications to Biomolecular Interactions

    1. R. B. Best and G. Hummer, Diffusive model of protein folding dynamics with Kramers turnover in rate, Phys. Rev. Lett., 96:228104, 2006.
    2. R. B. Best and G. Hummer, Coordinate-dependent diffusion in protein folding, Proc. Nat. Sci. Acad., 107:1088-1093, 2010.
    3. J. D. Bryngelson and P. G. Wolynes, Intermediates and barrier crossing in a random energy model (with applications to protein folding), J. Phys. Chem. 63:6902-6915, 1989.
    4. S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys., 15:1-89, 1943.
    5. D. L. Ermak, and J. A. McCammon, Brownian dynamics with hydrodynamic interactions, J. Chem. Phys., 69:1352-1360, 1978.
    6. C. Gardiner, Stochastic Methods, 4th ed., Springer, 2009.
    7. R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29:1-17, 1998.
    8. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., Graduate Texts in Mathematics, Springer, 1998.
    9. A. I. Khinchin, Mathematical Foundations of Statistical Mechanics, Dover, 1949.
    10. H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica VII:284-304, 1940.
    11. H.-H. Kuo, Introduction to Stochastic Integration, Springer, 2006.
    12. B. Li, B. Lu, Z. Wang, and J. A. McCammon, Solutions to a reduced Poisson-Nernst-Planck system and determination of reaction rates, Physica A, 389:1329-1345, 2010. (Preprint in PDF)
    13. B. Lu and Y. C. Zhou, Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes II: Size effects on ionic distributions and diffusion-reaction rates, Biophys. J., 100:2475-2485, 2011.
    14. R. M. Mazo, Brownian Motion: Fluctuations, Dynamics and Applications, Oxford University Press, 2002.
    15. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, 2nd ed., Springer, 1996.
    16. Z. Schuss, Singular perturbation methods in Stochastic differential equations of mathematical physics, SIAM J. Review, 22:119-155, 1980.
    17. Z. Schuss, Theory and Applications of Stochastic Processes: An Analytical Approach, Springer, 2009.
    18. Z. Schuss, B. Nadler, and R. S. Eisenberg, Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model, Phys. Rev. E, 64, 036116, 2001.
    19. N. D. Socci, J. N. Onuchi, and P. G. Wolynes, Diffusive dynamics of the reaction coordinate for protein folding funnels, J. Chem. Phys., 104:5860-5868, 1996.
    20. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed., Elsevier, 2007.
    21. R. Zwanzig, Nonequiilbrium Statistical Mechanics, Oxford University Press, 2001.

    Last updated by Bo Li on May 19, 2011.