More Refined Continuum Methods Pages 513-520. Methods based on Poisson-Boltzmann Equation 2 r =...
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Transcript of More Refined Continuum Methods Pages 513-520. Methods based on Poisson-Boltzmann Equation 2 r =...
![Page 1: More Refined Continuum Methods Pages 513-520. Methods based on Poisson-Boltzmann Equation 2 r = [-4 r / Poisson Equation (9.56) If](https://reader036.fdocuments.us/reader036/viewer/2022081816/56649d0d5503460f949e1a2f/html5/thumbnails/1.jpg)
More Refined Continuum Methods
Pages 513-520
![Page 2: More Refined Continuum Methods Pages 513-520. Methods based on Poisson-Boltzmann Equation 2 r = [-4 r / Poisson Equation (9.56) If](https://reader036.fdocuments.us/reader036/viewer/2022081816/56649d0d5503460f949e1a2f/html5/thumbnails/2.jpg)
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)
![Page 3: More Refined Continuum Methods Pages 513-520. Methods based on Poisson-Boltzmann Equation 2 r = [-4 r / Poisson Equation (9.56) If](https://reader036.fdocuments.us/reader036/viewer/2022081816/56649d0d5503460f949e1a2f/html5/thumbnails/3.jpg)
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
![Page 4: More Refined Continuum Methods Pages 513-520. Methods based on Poisson-Boltzmann Equation 2 r = [-4 r / Poisson Equation (9.56) If](https://reader036.fdocuments.us/reader036/viewer/2022081816/56649d0d5503460f949e1a2f/html5/thumbnails/4.jpg)
. ..
.
.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)
![Page 5: More Refined Continuum Methods Pages 513-520. Methods based on Poisson-Boltzmann Equation 2 r = [-4 r / Poisson Equation (9.56) If](https://reader036.fdocuments.us/reader036/viewer/2022081816/56649d0d5503460f949e1a2f/html5/thumbnails/5.jpg)
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
![Page 6: More Refined Continuum Methods Pages 513-520. Methods based on Poisson-Boltzmann Equation 2 r = [-4 r / Poisson Equation (9.56) If](https://reader036.fdocuments.us/reader036/viewer/2022081816/56649d0d5503460f949e1a2f/html5/thumbnails/6.jpg)
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
![Page 7: More Refined Continuum Methods Pages 513-520. Methods based on Poisson-Boltzmann Equation 2 r = [-4 r / Poisson Equation (9.56) If](https://reader036.fdocuments.us/reader036/viewer/2022081816/56649d0d5503460f949e1a2f/html5/thumbnails/7.jpg)
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
![Page 8: More Refined Continuum Methods Pages 513-520. Methods based on Poisson-Boltzmann Equation 2 r = [-4 r / Poisson Equation (9.56) If](https://reader036.fdocuments.us/reader036/viewer/2022081816/56649d0d5503460f949e1a2f/html5/thumbnails/8.jpg)
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
![Page 9: More Refined Continuum Methods Pages 513-520. Methods based on Poisson-Boltzmann Equation 2 r = [-4 r / Poisson Equation (9.56) If](https://reader036.fdocuments.us/reader036/viewer/2022081816/56649d0d5503460f949e1a2f/html5/thumbnails/9.jpg)
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
![Page 10: More Refined Continuum Methods Pages 513-520. Methods based on Poisson-Boltzmann Equation 2 r = [-4 r / Poisson Equation (9.56) If](https://reader036.fdocuments.us/reader036/viewer/2022081816/56649d0d5503460f949e1a2f/html5/thumbnails/10.jpg)
Langevin Dipole Model
![Page 11: More Refined Continuum Methods Pages 513-520. Methods based on Poisson-Boltzmann Equation 2 r = [-4 r / Poisson Equation (9.56) If](https://reader036.fdocuments.us/reader036/viewer/2022081816/56649d0d5503460f949e1a2f/html5/thumbnails/11.jpg)
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