# pKa Calculation

### PB-Based methods

First finite-difference PB (FDPB) solver for irregular shape was constructed by Bashford and Karplus(Bashford-1990). Neilsen et al optimize the hydrogen-bond network in protein for PB-based pKa calculations.(Nielsen-2001)

$\alg pK'_a = pKa + \frac{1}{RT \ln 10}[\Delta G_p (A,HA) - \Delta G_s (A,HA)]\ealg$ ,其中s代表溶液状态,p代表蛋白状态.

• Born Formula
The Born Formula can calculate the a charged atom with a lower dielectric constant $\varepsilon_{int}$ immersed in a continuum media with a higher dielectric constant $\varepsilon_{ext}$. It can be used to calculate the solvation energy and check the accuracy of PB solver.

In the formula, $e=1.602176565\times 10^{-19}C$, $\varepsilon_0=8.8541878176\times 10^{-12}F/m$ (permittivity of free space), $k=1.38\times 10^{-23}J/K, T=297.33K, NA=6.022\times 10^{23}, cal=4.184 J$, $\varepsilon_{int} \varepsilon_{ext}$: interior dielectric constant and exterior dielectric constant.

For example, Q=10e, $\varepsilon_{int}=4.0$, $\varepsilon_{ext}=80.0$, r=1 A, energy is -6673.71kT; $\varepsilon_{int}=20.0$, $\varepsilon_{ext}=80.0$, energy is -1024.255kT

A python script to calculate the Coulombic energy from pqr file:

###### FILE: pqr2col.py
#! /usr/bin/env python
# -*- coding: utf8 -*-

# Author: Hom, Date: 2015.6.17
# To calculate the coulombic energy from pqr file.
# Usage: python pqr2col.py input.pqr [indi]
# Default interior dielectric is set to 1.

import os,sys
from math import *

if (__name__ == '__main__'):
if (not(len(sys.argv) == 2 or len(sys.argv) == 3)):
print "Please assign the pqr file."
input()
exit()
indi=1.0
if (len(sys.argv) == 3):
indi=float(sys.argv[2])
fname=sys.argv[1]
fnamelist=os.path.splitext(fname)
fr=open(fname)
atomlist=[];#[[x,y,z,charge],..]
for line in fr:
items=line.split()
if (items[0]=="ATOM" or items[0]=="HETATM"):
atomlist.append([items[5],items[6],items[7],items[8]])
totalatoms=len(atomlist);
sum=0.0;
for i in range(totalatoms):
for j in range(i+1,totalatoms):
x1=float(atomlist[i][0])
y1=float(atomlist[i][1])
z1=float(atomlist[i][2])
x2=float(atomlist[j][0])
y2=float(atomlist[j][1])
z2=float(atomlist[j][2])
r=sqrt((x1-x2)**2+(y1-y2)**2+(z1-z2)**2)
sum=sum+(float(atomlist[i][3])*float(atomlist[j][3])*332.06364261/(r*indi))
print "Total coulombic energy is "+str(sum)+" kcal/mol"
#end main


Ref: (Li-2012)

### Empirical/Scoring function Based

• PROPKA
Empirical pKa predictors based on physical description of the desolvation and dielectric response for the protein. Most update 3.0 reference: (Olsson-2011, Søndergaard-2011)

• Rosetta pKa
Consider side-chain ﬂexibility and use new scoring function incorporating a Coulomb electrostatic potential and optimizing the solvation reference energies for pKa calculations. (Kilambi-2012).

## Reference

1. Donald Bashford and Martin Karplus. pKa’s of Ionizable Groups in Proteins: Atomic Detail from a Continuum Electrostatic Model. Biochemistry 1990, 29, 10219-10225. ref
2. An-Suei Yang, M. R. Gunner, Rosemary Sampogna, Kim Sharp, and Barry Honig. On the Calculation of pKas in Proteins. PROTEINS: Structure, Function, and Genetics 1993, 15, 252-265. ref
3. Jens E. Nielsen and Gerrit Vriend. Optimizing the Hydrogen-Bond Network in Poisson–Boltzmann Equation-Based pKa Calculations. PROTEINS: Structure, Function, and Genetics 2001, 43, 403–412. ref
4. Sunhwan Jo, Miklos Vargyas, Judit Vasko-Szedlar, Benoît Roux and Wonpil Im. PBEQ-Solver for online visualization of electrostatic potential of biomolecules. Nucleic Acids Research, 2008, 36, W270–W275. ref
5. Mats H. M. Olsson, Chresten R. Søndergaard, Michal Rostkowski, and Jan H. Jensen. PROPKA3: Consistent Treatment of Internal and Surface Residues in Empirical pKa Predictions. J. Chem. Theory Comput. 2011, 7, 525–537. ref
6. Chresten R. Søndergaard, Mats H. M. Olsson, Michaz Rostkowski, and Jan H. Jensen. Improved Treatment of Ligands and Coupling Effects in Empirical Calculation and Rationalization of pKa Values. J. Chem. Theory Comput. 2011, 7, 2284–2295. ref
7. Krishna Praneeth Kilambi and Jeffrey J. Gray. Rapid Calculation of Protein pKa Values Using Rosetta. Biophysical Journal. 2012, 103, 587–595.ref
8. Lin Li, Chuan Li, Subhra Sarkar, Jie Zhang, Shawn Witham, Zhe Zhang, Lin Wang, Nicholas Smith, Marharyta Petukh and Emil Alexov. DelPhi: a comprehensive suite for DelPhi software and associated resources. BMC Biophysics 2012, 5:9. ref

◆ 本文地址: http://platinhom.github.io/2015/06/15/pKa-Calculation/, 转载请注明 ◆

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Source 类别: CompCB