Math 277B, Spring 2011
Topics in Mathematics and BiochemistryBiophysics
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:
 Partial differential equations and dynamical systems models and computation of diffusion process;
 Numerical methods for elliptic interface problems for continuum dielectric models of biomolecular interactions;
 Geometric and field based surface motion for biomolecular solvation and cell dynamics;
 Stochastic process and computation of Brownian motion of biomolecules; and
 The FokkerPlanck 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, steadystate diffusion equations. Variational techniques.
EulerLagrange equations. Examples: The AllenCahn functional. The PoissonBoltzmann
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, nonuniqueness of solutions,
Newton's iteration and its convergence, ellptic interface problems  two equivalent
formulations.
 Pages 20  27.
The PoissonBoltzmann equation. Background. Basics of the equation.
Special cases: linearized PBE and sinh PBE.
Meanfield electrostatic freeenergy 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 PoissonBoltmann equation continued. Convexity. Bounds of equilibrium concentrations.
The existence and uniqueness of the solution to the boundaryvalue problem of the
PoissonBoltzmann equation.
 Pages 41  44.
The PoissonBoltzmann theory does not predict the wallmediated
likecharge attraction.
 Pages 45  52.
Include size effects  uniform or nonuniform sizes  in a meanfield model of
electrostatics. Some mathematical results: uniform boundes of concentrations, existence and
uniqueness, convexity. Implicit Boltzmann distributions. Numerical methods
for the case of nonuniform 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 freeenergy functionals. Surface variations.
Some examples. General approach.
 Pages 77  81.
General description of the PoissonNernstPlanck (PNP) system with appliction to biomolecular
systems. Fluxes. Noflux boundary conditions and the Boltzmann distributions.
A reduced PNP sysetm. Semianalytical and numerical sotitions to the
reduced PNP system and reaction rates.
 Pages 82  93
Langevin dynamics  an example, FluctuationDissipation Theorem,
Some background of stochastic processes, Brownian motion, Ito stochastic differential
equations. The FokkerPlanck equation. Examples. Derivation.
 Pages 94 106.
A variational principle of the FokkerPlanck equation.
Steepest descent, examples. Time discretization of heat equation and a variational principle.
Main result on a variational principle of the FokkerPlanck equation. Wasserstein metric. Some properties and some examples.
References
Continuum Electrostatics

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 J. Che, J. Dzubiella, B. Li, and J. A. McCammon. Electrostatic free energy and its
variations in implicit solvent models. J. Phys. Chem. B, 112:30583069, 2008.
(Preprint in PDF,
published version in PDF at the journal website.)
 H.B. Cheng, L.T. Cheng, and B. Li, Yukawafield approximation
of electrostatic free energy and dielectric boundary force, 2011 (submitted).
(Preprint in PDF)

M. E. Davis and J. A. McCammon. Electrostatics in biomolecular structure and
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(Published version in PDF 
permission and copyright of SIAM.)

B. Li. Continuum electrostatics for ionic solutions with nonuniform ionic sizes.
Nonlinearity, 22:811833, 2009.
(Preprint in PDF,
published version in PDF at the journal website.)
 B. Li, X. Cheng, and Z. Zhang,
Dielectric boundary force in molecular solvation with the PoissonBoltzmann free energy:
A shape derivative approach, 2011 (submitted).
(Preprint in PDF)

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S. Zhou, Z. Wang, and B. Li, MeanFiled description of ionic size effect with
nonuniform ionic sizes: A numerical approach, 2011 (submitted).
(Preprint in PDF)
Variational Implicit Solvation
 J. Che, J. Dzubiella, B. Li, and J. A. McCammon. Electrostatic free energy and its
variations in implicit solvent models. J. Phys. Chem. B, 112:30583069, 2008.
(Preprint in PDF,
published version in PDF at the journal website.)
 H.B. Cheng, L.T. Cheng, and B. Li, Yukawafield approximation
of electrostatic free energy and dielectric boundary force, 2011 (submitted).
(Preprint in PDF)

L.T. Cheng, J. Dzubiella, J. A. McCammon, and B. Li,
Application of the levelset 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.)
 L.T. Cheng, B. Li, and Z. Wang,
LevelSet minimization of potential controlled Hadwiger valuations
for molecular solvation, J. Comput. Phys., 229:84978510, 2010.
(Preprint in PDF)

L.T. Cheng, Z. Wang, P. Setny, J. Dzubiella, B. Li, and J. A. McCammon,
Interfaces and hydrophobic interactions in receptorligand systems:
A levelset variational implicit solvent approach, J. Chem. Phys., 131:144102, 2009.
(Preprint in PDF)
 L.T. Cheng, Y. Xie, J. Dzubiella, J. A. McCammon, J. Che, and B. Li,
Coupling the levelset method with molecular mechanics for variational
implicit solvation of nonpolar molecules, J. Chem. Theory Comput., 5:257266, 2009.
(Preprint in PDF,
published version in PDF at the journal website.)
 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.
 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.
 B. Li. Minimization of electrostatic free energy and the PoissonBoltzmann equation
for molecular solvation with implicit solvent. SIAM J. Math. Anal., 40:25362566, 2009.
(Published version in PDF 
permission and copyright of SIAM.)
 B. Li, X. Cheng, and Z. Zhang,
Dielectric boundary force in molecular solvation with the PoissonBoltzmann free energy:
A shape derivative approach, 2011 (submitted).
(Preprint in PDF)
 B. Li and J. Shopple, An interfacefitted finite element level set method
with application to solidification and solvation, Commun. Comput. Phys.,
10:3256, 2011.
(Preprint in PDF)
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P. Setny, Z. Wang, L.T. Cheng, B. Li, J. A. McCammon, and J. Dzubliella,
Dewettingcontrolled binding of ligands to hydrophobic pockets,
Phys. Rev. Lett., 103:187801, 2009.
(Preprint in PDF)
Langevin Dynamics, FokkerPlanck Equation, and PoissonNernstPlanck Equation, with
Applications to Biomolecular Interactions
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system and determination of reaction rates, Physica A, 389:13291345, 2010.
(Preprint in PDF)
 B. Lu and Y. C. Zhou, PoissonNernstPlanck equations for simulating biomolecular
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and diffusionreaction rates, Biophys. J., 100:24752485, 2011.
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Last updated by Bo Li on May 19, 2011.
