Chem 388: Molecular Dynamics and Molecular Modeling

Continuum Electrostatics AndMM-PBSA

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

Molecular Surface Potential

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

Chem 388: Molecular Dynamics and Molecular Modeling

Thermodynamics of Protein-DNA BindingJayaram et al. (1999) J. Comp. Phys. 151, 333.