Post on 18-Dec-2015
Chem 388: Molecular Dynamics and Molecular Modeling
Topics
• Poisson Equation
• Poisson-Boltzmann Equation
• Finite Difference Method
• Born Model
• Generalized Born
• Hydrophobic effect
• Free Energy Component Analysis
Chem 388: Molecular Dynamics and Molecular Modeling
Electrostatics in Molecular Biophysics
• Electrostatic interactions are very long-ranged• Most of the common biological macromolecules
are highly charged so we cannot simply ignore electrostatic interactions!
• The cytoplasm of cells contains a relatively high concentration of dissolved ions such as Na+, Ca2+ and Cl-
Chem 388: Molecular Dynamics and Molecular Modeling
Electrostatic Steering
AChE and Fasciculin 2 bind with electrostatically-steered, diffusion-controlled kinetics.
Chem 388: Molecular Dynamics and Molecular Modeling
Coulomb Equation
Electrostatic Potential
arg
1
ch es
ii
qr
r r
Electrostatic work required to bring a charge q’ to the point rq’ in the potential
qW q r
Chem 388: Molecular Dynamics and Molecular Modeling
Distance Dependent Dielectric Function
2 2 22
; 1.2; 80; 1.76
i
i
r e
sr s
Local Dielectric Environment of B-DNA in Solution: Results from a 14 ns Molecular Dynamics Trajectory M. A. Young, B. Jayaram, and D. L. Beveridge J. Phys. Chem. B, 102 (39), 7666 -7669, 1998
Chem 388: Molecular Dynamics and Molecular Modeling
Dielectric ScreeningPhysical Basis of Dielectric Screening : Polarization
Dipoles & PolarizationInduced – Electronic ~2
Permanent – Orientational ~80
Reaction Field
Chem 388: Molecular Dynamics and Molecular Modeling
Molecule in Solution
Solute: Low dielectric region with fixed partial charges p ~2-4
Solvent: High dielectric region with unlocalized charges p ~80
Presence of dielectric boundaries presents the effects of an induced surface charge
Chem 388: Molecular Dynamics and Molecular Modeling
Poisson Equation
4E r r E r r where
- differential operator , ,x y z
If a dielectric medium screens the field
4r E r r
4r r r
r
r
r
Poisson Equation
Relative permittivity
Electrostatic potential
Charge density
Chem 388: Molecular Dynamics and Molecular Modeling
Effect of Dissolved Electrolytes
Mobile ions are distributed according to the Boltzmann statistics. Mean local concentration (c(r))of ions relative to bulk concentration (cbulk)
expbulki i ic r c q r RT
1
expN
bulkions i i i
i
r q c q r RT
Ion Distribution
Chem 388: Molecular Dynamics and Molecular Modeling
Poisson-Boltzmann Equation
4r r r
Poisson Equation
Charge Distribution
prot ionsr r r
1
4 expN
bulksolute i i i
i
r r r q c q r RT
Poisson-Boltzmann Equation
Chem 388: Molecular Dynamics and Molecular Modeling
Linear Poisson-Boltzmann Equation
For 1r RT
2
1 1 1
expN N N
bulk bulk bulki i i i i i i
i i i
rq c q r RT c q c q
RT
1
0N
bulki i
i
c q
For a charge balanced system
2
1
4N
bulksolute i i
i
r r r q c r RT
24 soluter r r r r 2
2
0
8 i
B
q I
k T
where 2
1
1
2
Nbulki i
i
I c q
1
4 expN
bulksolute i i i
i
r r r q c q r RT
Chem 388: Molecular Dynamics and Molecular Modeling
Finite Difference Approximation
0
0 2 2
4i i
i
qh
h N
N=1 for linear equation
2 40 01 3! 5! ..N for nonlinear
Chem 388: Molecular Dynamics and Molecular Modeling
Features of Continuum Model
• Location and magnitude of charges• Surface features of solute and solvent boundary• Difference in dielectric across the boundary• Ionic strength
Weakness:• Ignores molecular nature of solvent• Finite size of ions• Ion-ion correlation effects
Chem 388: Molecular Dynamics and Molecular Modeling
Total Electrostatic Energy
ext coul self crossi i i i
total ext owni i i
1 1
2 2ext total own
E i i i i ii i
G q q
1
2t g totalE i i
i
G q
1,80 ,80 ,1t g t gE E s E sG G G
Chem 388: Molecular Dynamics and Molecular Modeling
Applications of FDPB Methods
• Solvation energy
• pKa Shifts
• Binding energies
• Conformational analysis
• Effect of ionic strength on binding
Chem 388: Molecular Dynamics and Molecular Modeling
Solvation Energy
sol np el
np vdW cav
np
G G G
G G G
G a bS
Solvation thermodynamics of amino acidsS. B. Dixit, B. Jayaram et al.J. Chem. Soc., Faraday Trans., 1997, 93(6), 1105-1113
Chem 388: Molecular Dynamics and Molecular Modeling
Accessible Surface Area
*
sol i ii
h
G b S
G SASA c
Image source: http://www.netsci.org/Science/Compchem/feature14.html
Chem 388: Molecular Dynamics and Molecular Modeling
Born Model of Solvation
2
0 2 1
1 1 1
4 2el
qG
D D
2
1 1 1
0.52 2 2 2
1 1166 1 166 1
; ; 4
n n ni j i
poli j iGB i
j i
DGB ij ij ij i j ij ij
q q qG
f
f r e D r
Generalized Born Equation
A modification of the generalized Born theoryB. Jayaram, Y. Liu and D. L. BeveridgeJ. Chem. Phys., 109, 1465-1471.
Chem 388: Molecular Dynamics and Molecular Modeling
pKa Calculation
0
2.303m
i ipK
Study the effect of site-directed mutations on change in pK value
0 Potential at site I due to the original group
Potential at site I due to the original groupm
i -1 or 1 for an acidic or basic group
Chem 388: Molecular Dynamics and Molecular Modeling
Free Energy of Conformational Change
( ) 1, 1,conf B A B Asol sol FF FFG III IV G G G G
Chem 388: Molecular Dynamics and Molecular Modeling
Free Energy of Binding
bind salt elec npG G G G
Macroscopic Models of Aqueous SolutionsBarry Honig, Kim Sharp and An-Suei YangJ. Phys. Chem. 1993,97, 1101-1109