Post on 18-Dec-2015
More Refined Continuum Methods
Pages 513-520
Methods based on Poisson-Boltzmann Equation
2r = [-4r/ Poisson Equation (9.56)
If varies with position, then. rr = -4r (9.57)
When mobile ions are present,n(r) = N exp(-V(r)/kBT) (9.58)
Methods based on Poisson-Boltzmann Equation
. rr - /sinh[r] = -4r (9.59)
where / Debye-Huckel Inverse Length
. rr - /r[1+ r2/6 + r4/120 …] = -4r (9.61)
. Err - / r = -4r
Linearised Poisson-Boltzmann Equation
. ..
.
.h
12
3
4
1
3
2
4
q0
0
0 =∑11 + 4q0/h
∑1 + / f0)0
f0) = 1 (linear case), f0) = [1+ r2/6 + r4/120 …] (non-linear case)
;
Finite Difference Poisson-Boltzmann Methods (FDPB)
Choice of Grid Size
Technique of focusing :(i) Series of calculations are performed - system occupying greater fraction of grid box at each step(ii) Boundary points in each new grid internal point from previous grid(iii) Better estimates of the potential values at the boundary obtained(iv) Accuracy of calculations improved
Applications of FDPB(i) Electrostatic potential around a protein using FDPB
differs significantly from uniform dielectric models
(ii) Provides explanation for association of two +vely charged species; eg. trypsin and trypsin inhibitor : region of -ve potential appears in the region where the inhibitor binds
(iii) Identifies “active site” regions in enzyme substrates; eg. Enzyme Cu-Zn superoxide dismutase, attack of O2
-
focused on a specific region of +ve electrostatic potential
Solvation Free Energy Using FDPB
sol = 1/2∑ ( i80 - i
1) (9.63)
s
m
m
m
m
m
m
m
mm
m
m
s
Non-electrostatic Contributions
cav + vdw = A+ b (9.64)
A : solvent accessible area;
b : parameters, taken from experimentally
determined free energies
In some applications,cav = K0 +K1a12 + K2a12
2;
Ki : depend on the volume of the solvent molecule,
a12 : average of the diameters of the solvent molecule
and the spherical solute molecule
Very Simple Solvation Models
sol = ∑aiSi (9.66)
Si : exposed solvent accesible surface area of atom “I”
Rough method
Advantage : Very rapid way of calculating solvation contribution
Langevin Dipole Model
The Langevin Dipole Model
= 0 Ei exp{C0 |Ei|/kBT} + exp{-C0 |Ei|/kBT} 1|Ei| exp{C0 |Ei|/kBT} - exp{-C0 |Ei|/kBT} C0 |Ei|/kBT
…..(9.53) = Size and direction of each dipole
0 = Dipole moment of a solvent molecule
C = Parameter representing the degree to which the dipoles resist reorientation
sol = -1/2∑ . Ei (9.54)0
Ei0 = Field due to the solute charges alone