Explicit solvation models

Chapter 5.  pK_A Calculations of Biologically Active Molecules

(Tamás Körtvélyesi)

Keywords : pK_A shift of aminoacid side-chains in peptides and proteins, prediction the protonation (pK_A) of side chains in peptides and proteins.

What is described here? The biologically active molecules are working in solution which has pH and ionic strength (I). The pH and ionic strength (I) have influence on the charges in the side chains of the amino acids of peptides and proteins. The charges and stability of DNA are also sensitive on the pH and I. There are two main methods to predict the pK_A : (i) empirical calculation of the pK_A shift in peptides and proteins, (ii) non linear Poisson-Boltzmann equation (NPBE), linear Poisson-Boltzmann equation (LPBE) calculations with finite different method and the solution of the Tanford-Kirkwood equations (TKEs) combined with Monte Carlo methods to obtain the shift of the pK_A in the molecules.

What is it used for? The prediction of pK_A is important to know the protonation state of the side chains in peptides and proteins to predict the electrostatic interactions between the peptides, proteins and ligand molecules in the possible association processes.

What is needed? The basic knowledge of the structure, intra- and intermolecular interactions between molecules are important. Important also, the knowledge of the introduction to physical chemistry. The basic analytical chemistry knowledge on the acidity is also necessary.

1. Introduction

The pK_A of the side chains in peptides or proteins of aminoacids at a given pH are significant in the folding, in the formation of the structures and the working of protein-protein and protein-ligand binding. These protonation states affect the structure and stability of the peptides and proteins and also the binding mode of the pocket in these molecules. The pK_A values in peptides and proteins can be derived from the protonation constants of aminoacids alone.

The peptides and proteins fold in solvent and stabilize the 3D structure on the basis of the effect in the side chain charges (protonation). The titratable/hydrophylic amino acids are on the surface of the water/protein interface. In the core of the peptide/protein, hydrophobic aminoacids are burried. The pK_A values of the side chains in aminoacids are summerized in Table 5.1. (Henderson-Hasselbach equation must be checked in Lit. [1], Eq. 6.1)

Table 5.1. The pK_A values of side chains in individual aminoacids

Asp	3,9
Glu	4,1
His	6,0
Cis	8,4
Tyr	10,5
Lys	10,5
Arg	12,5

The charged side chains are in interction with each other and with the backbone atoms by point charge-point charge, point charge-dipole and dipole-dipole, etc. interactions. The effect on the pK_A values of the side chains depends on (i) pH and (ii) independent on pH. The latter influences are the desolvatation, interactions with the constant charges and dipoles. It means that the pK_A values change in peptides/proteins related to the individual aminoacids. The pH dependent part can be determined by Tanford-Roxby iteration or other methods (see later). With the determination of the pK_A values the titration curves of peptides/proteins can be calculated (Henderson-Hasselbalch titration curves if all the charged side chains behave on the basis of Henderson-Hasselbalch, etc.)

2. Empirical Methods

One of the empirical method which has no potential calculations is Propka [2-5]. The method is very fast. Version 1.0 [2] predicts pK_A of protein on the basis of the charge groups distances and empirical formulas. In Version 2.0 [3] a new function was developed to predict the effect of non protein molecules (ligands) on pK_A of the side chain protonation of protein and the shift of pK of ionizable groups on ligand. Ligands and the aminoacids with charged side chain far from the binding site can contribute protonation/de protonation. The small change in sequence can shift the pK_A. The parameters and the empirical rules were changed in the new version [4]. This method describe more precisely the desolvatation and dielectric response of the proteins. The classification of aminoacids was modified to internal aminoacids and amino acids on the surface. The method is precise to pK_A of Asp and Glu. The newest method (Version 3.1 [5]) includes the effect of the ligand ont he protonation state of the binding site and the pK shift of the ionizable nonprotein ligands. It makes possible th calculate the shift in multiligand complexes and non covalent coupled ligands which were not possible previously. The database was renewed and can be extended flexibly. A GUI was developed for the simple usage of the method [6]. The method is implemented in a web server [7].

Figure 5_1. HSP90 N-terminal structures: apo structure (1yes) and structure without the ligand (geldanomycine) (1yet)

Figure 5.2. The protonation of HSP90 N-terminal structures: apo structure (1yes) and structure without the ligand calculated by propKa 3.(geldanomycine) (1yet)

3.  Solvation of Poisson-Boltzmann Equation (PBE) and the Tanford-Kirkwod Equations (TKE) Coupled with Monte Carlo Methods

The application of the solution of Poisson-Boltzmann Equation (PBE)

The shift in pK_A in peptides/proteins depend on two factors: (i) the electrostatic environment of the charged (protonated/deprotonated) side chains of the amino acid, and the embedding of the side chain. These factors are influenced by the geometry of the molecules and vica versa.

Some methods are based on the solution of PBE mainly by FDPB (finite difference Poisson-Boltzmann method) or LPDF (linear Poisson-Boltzmann method) (see Chapter 3). It includes the modifications of the electrostatic environment int he protein. Some web servers are installed: H++ web server [8], pK_D web server [9], az MCCE [10] and the Karlsberg+ FDPB [11] method. The Tanford-Kirkwood equation is solved in Macrodox [12]. The latter method is suitable for Brownian dynamics, too.

The FDPB-based methods support the pK_A shift of the amino acid side chains with the difference of the totally solvated and in-protein condition. It is necessarry to give the effective dielectric constant in protein and int he bulk solvent. The previous value is between 2 to 20 (see Chapter 3).

The H++ server predicts the pK_A values of the ionizable side chains and automatically extend/delete the protons [8,13]. The input file has to be in pdb format, the output will be in pdbq, pqr format, pdb format or amber format. The theoretical background is summerized in Lit. [14]. We can calculate the isoelectronic points, titration curves and the protonation microstates. The titration curve of the whole protein and separately, the charged groups can be obtained.

On the basis of the LPBE solution and the structural relaxation of H-atoms and salt bridges the pK_A values are calculated [11,15]. The effective dielectric constant in the protein is 4.0. The method is capable to predict at different pHs the conformation and the position of H-atoms. The pH dependent conformations at different protonation states are calculated by Monte Carlo simulations. LPBE calculations were performed by TAPBS algorithm.

Karlsberg+ can calculate with optimization the electrostatic energies of the conformers of the proteins. The H-atoms ont he surface of the protein and the salt bridges are calculated at three pHs (low, middle and high) [15]. The pdb file of the protein is necessarry.

Macrodox method

Macrodox [12] is a command line pakage on Linux and Windows XP environment The algorithm uses the solution of Tanford-Kirkwood-eqation. In this calculation we can obtain the protonation/deprotonation states which depends on pH and ionic strength, change in temperature. The effective dielectric constant of the solvent, The internal dielectric constantat ionic strength of the media, pH can be changed. The command „titrate” starts the calculations. Protein is considered as a substance with low effective dielectric constant (we do not know precisley on the real effective constant in the internal part of the protein which is in the solvent with high effective dielectric constant. The effect of the buried ionazable side chains (burial factor) and the interactions between these side chains is important. A good example is the binding pocket in BACE (1fkn), where the Asp int he pocket is protonated which has an important effect on binding molecules (see the binding pocket in Figure 5.3). Another effect is the position of loop which depends on pH. The lower the pH is the loop is more open. The local pH in the cells is not the physiological pH, sometimes more or less of this value.

Figure 5.3. Binding pocket in .BACE (1fkn)

The difference maximum in the Barnbar-Barnase protein complexes at 0, 5, 10, 15 and 20 Ǻ distances between the mass centres (see Chapter 3) calculated by different methods are summarized in Table 5.2. Mainly, Tyr and His have the largest difference in the calculations. His has two tautomers and one positively charged protonated form. The environment parameters were 298.15 K, effective dielectric constant is 78.3, the internal effective dielectric constant is 4.0, the ionic strength was 0.1 M, the effective radius of the protein is 20.50 Ǻ.

Table 5.2. Largest difference maximum in the Barnbar-Barnase protein complexes at 0, 5, 10, 15 and 20 Ǻ distances between the mass centres (see Chapter 3) calculated by different methods without ligands.