More Refined Continuum Methods Pages

Methods based on Poisson-Boltzmann Equation  2  r  = [-4  r  /  Poisson Equation (9.56) If  varies with position, then .  r  r  = -4  r  (9.57) When mobile ions are present, n(r) = N exp(-V(r)/k B T) (9.58)

Methods based on Poisson-Boltzmann Equation .  r  r  -  / sinh[  r  ] = -4  r  (9.59) where  /  Debye-Huckel Inverse Length .  r  r  -  /  r  [1+  r  2 /6 +  r  4 /120 …] = -4  r  (9.61) . E  r  r  -  /  r  = -4  r  Linearised Poisson-Boltzmann Equation

..... h  1  2  3  4  1  3  2  4 q 0  0  0 = ∑  1   q 0 /h ∑  1 +  /  f   0 ) 0 f  0 ) = 1 (linear case), f  0 ) = [1+  r  2 /6 +  r  4 /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 O 2 - focused on a specific region of +ve electrostatic potential

Solvation Free Energy Using FDPB   sol = 1/2∑   (  i 80 -  i 1 ) (9.63)  s  m  m  m  m  m  m  m  m  m  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 = K 0 +K 1 a 12 + K 2 a 12 2 ; K i : depend on the volume of the solvent molecule, a 12 : average of the diameters of the solvent molecule and the spherical solute molecule

Very Simple Solvation Models   sol = ∑a i S i (9.66) S i : 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 E i exp{C  0 |E i |/k B T} + exp{-C  0 |E i |/k B T} 1 |Ei||Ei| exp{C  0 |E i |/k B T} - exp{-C  0 |E i |/k B T} C  0 |E i |/k B T …..(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∑  . E i (9.54) 0 EiEi 0 = Field due to the solute charges alone