Finite Difference Poisson-Boltzmann Equation Research Articles

Hemoglobin Bohr Effects: Atomic Origin of the Histidine Residue Contributions

The Bohr effect in hemoglobin, which refers to the dependence of the oxygen affinity on the pH, plays an important role in its cooperativity and physiological function. The dominant contribution to the Bohr effect arises from the difference in the pKa values of His residues of the unliganded (deoxy) and liganded (carbonmonoxy) structures. Using recent high resolution structures, the residue pKa values corresponding to the two structures are calculated. The method is based on determining the electrostatic interactions between residues in the protein, relative to those of the residue in solution, by use of the linearized finite difference Poisson-Boltzmann equation and Monte Carlo sampling of protonation states. Given that good agreement is obtained with the available experimental values for the contribution of His residues in HbA to the Bohr effect, the calculated results are used to determine the atomic origin of the pKa shift between deoxy and carbonmonoxy HbA. The contributions to the pKa shift calculated by means of the linear response approximation show that the salt bridge involving His146 plays an important role in the alkaline Bohr effect, as suggested by Perutz but that other interactions are significant as well. A corresponding analysis is made for the contribution of His143 to the acid Bohr effect for which there is no proposed explanation. The method used is summarized and the program by which it is implemented is described in the Appendix .

Highly efficient and exact method for parallelization of grid-based algorithms and its implementation in DelPhi.

The Gauss-Seidel (GS) method is a standard iterative numerical method widely used to solve a system of equations and, in general, is more efficient comparing to other iterative methods, such as the Jacobi method. However, standard implementation of the GS method restricts its utilization in parallel computing due to its requirement of using updated neighboring values (i.e., in current iteration) as soon as they are available. Here, we report an efficient and exact (not requiring assumptions) method to parallelize iterations and to reduce the computational time as a linear/nearly linear function of the number of processes or computing units. In contrast to other existing solutions, our method does not require any assumptions and is equally applicable for solving linear and nonlinear equations. This approach is implemented in the DelPhi program, which is a finite difference Poisson-Boltzmann equation solver to model electrostatics in molecular biology. This development makes the iterative procedure on obtaining the electrostatic potential distribution in the parallelized DelPhi several folds faster than that in the serial code. Further, we demonstrate the advantages of the new parallelized DelPhi by computing the electrostatic potential and the corresponding energies of large supramolecular structures.

Similar structures but different mechanisms: Prediction of FABPs–membrane interaction by electrostatic calculation

The role of fatty acid binding proteins as intracellular fatty acid transporters may require their direct interaction with membranes. In this way different mechanisms have been previously characterized through experimental studies suggesting different models for FABPs–membrane association, although the process in which the molecule adsorbs to the membrane remains to be elucidated. To estimate the importance of the electrostatic energy in the FABP–membrane interaction, we computationally modeled the interaction of different FABPs with both anionic and neutral membranes. Free Electrostatic Energy of Binding (dE), was computed using Finite Difference Poisson Boltzmann Equation (FDPB) method as implemented in APBS (Adaptive Poisson Boltzmann Solver). Based on the computational analysis, it is found that recruitment to membranes is facilitated by non-specific electrostatic interactions. Also energetic analysis can quantitatively differentiate among the mechanisms of membrane association proposed and determinate the most energetically favorable configuration for the membrane-associated states of different FABPs. This type of calculations could provide a starting point for further computational or experimental analysis.

Prediction of the most favorable configuration in the ACBP–membrane interaction based on electrostatic calculations

Acyl-CoA binding proteins (ACBPs) are highly conserved 10 kDa cytosolic proteins that bind medium- and long-chain acyl-CoA esters. They act as intracellular carriers of acyl-CoA and play a role in acyl-CoA metabolism, gene regulation, acyl-CoA-mediated cell signaling, transport-mediated lipid synthesis, membrane trafficking and also, ACBPs were indicated as a possible inhibitor of diazepam binding to the GABA-A receptor. To estimate the importance of the non-specific electrostatic energy in the ACBP–membrane interaction, we computationally modeled the interaction of HgACBP with both anionic and neutral membranes. To compute the Free Electrostatic Energy of Binding ( dE), we used the Finite Difference Poisson Boltzmann Equation (FDPB) method as implemented in APBS. In the most energetically favorable orientation, ACBP brings charged residues Lys18 and Lys50 and hydrophobic residues Met46 and Leu47 into membrane surface proximity. This conformation suggests that these four ACBP amino acids are most likely to play a leading role in the ACBP–membrane interaction and ligand intake. Thus, we propose that long range electrostatic forces are the first step in the interaction mechanism between ACBP and membranes.

Empirical free energy analysis of the engrailed Q50K homeodomain mutant-DNA complex and its mutants

Empirical free energy calculations were performed for the engrailed homeodomain mutant protein in complex with its consensus promoter. The program, CONGEN™, was used to generate the atomic coordinates, which were missing in the original crystal structure of the Q50K mutant protein. To generate the said atomic coordinates more accurately, the initial CONGEN-generated model was subjected to 300 ps of belly dynamics in AMBER™. Complex formation energies were calculated for the ‘wild-type’ complex as well as ten computer-generated mutants that parallel mutants/substitutions studied by Sauer's group for each of three models. The hydrophobic, electrostatic, and entropic contributions were calculated. The non-linearized finite difference Poisson–Boltzmann equation was solved for the complexes at the ionic strength under which dissociation constant measurements were taken by the aforementioned group. A good overall agreement existed between calculated and experimental ΔΔG estimates. Discrepancies between absolute experimental and calculated values for ΔΔG are hypothesized to be due to the missing conformational entropic contribution of the operator DNA molecule.

The configurational dependence of binding free energies: a Poisson-Boltzmann study of Neuraminidase inhibitors.

The linear finite difference Poisson-Boltzmann (FDPB) equation is applied to the calculation of the electrostatic binding free energies of a group of inhibitors to the Neuraminidase enzyme. An ensemble of enzyme-inhibitor complex conformations was generated using Monte Carlo simulations and the electrostatic binding free energies of subtly different configurations of the enzyme-inhibitor complexes were calculated. It was seen that the binding free energies calculated using FDPB depend strongly on the configuration of the complex taken from the ensemble. This configurational dependence was investigated in detail in the electrostatic hydration free energies of the inhibitors. Differences in hydration energies of up to 7 kcal mol(-1) were obtained for root mean square (RMS) structural deviations of only 0.5 A. To verify the result, the grid size and parameter dependence of the calculated hydration free energies were systematically investigated. This showed that the absolute hydration free energies calculated using the FDPB equation were very sensitive to the values of key parameters, but that the configurational dependence of the free energies was independent of the parameters chosen. Thus just as molecular mechanics energies are very sensitive to configuration, and single-structure values are not typically used to score binding free energies, single FDPB energies should be treated with the same caution.

Structural determinants of trypsin affinity and specificity for cationic inhibitors.

The binding free energies of four inhibitors to bovine beta-trypsin are calculated. The inhibitors use either ornithine, lysine, or arginine to bind to the S1 specificity site. The electrostatic contribution to binding free energy is calculated by solving the finite difference Poisson-Boltzmann equation, the contribution of nonpolar interactions is calculated using a free energy-surface area relationship and the loss of conformational entropy is estimated both for trypsin and ligand side chains. Binding free energy values are of a reasonable magnitude and the relative affinity of the four inhibitors for trypsin is correctly predicted. Electrostatic interactions are found to oppose binding in all cases. However, in the case of ornithine- and lysine-based inhibitors, the salt bridge formed between their charged group and the partially buried carboxylate of Asp189 is found to stabilize the complex. Our analysis reveals how the molecular architecture of the trypsin binding site results in highly specific recognition of substrates and inhibitors. Specifically, partially burying Asp189 in the inhibitor-free enzyme decreases the penalty for desolvation of this group upon complexation. Water molecules trapped in the binding interface further stabilize the buried ion pair, resulting in a favorable electrostatic contribution of the ion pair formed with ornithine and lysine side chains. Moreover, all side chains that form the trypsin specificity site are partially buried, and hence, relatively immobile in the inhibitor-free state, thus reducing the entropic cost of complexation. The implications of the results for the general problem of recognition and binding are considered. A novel finding in this regard is that like charged molecules can have electrostatic contributions to binding that are more favorable than oppositely charged molecules due to enhanced interactions with the solvent in the highly charged complex that is formed.

Non-random ionic-charge distribution responsible for the structural stability and molecular recognition of proteins

The `ionic-charge shuffling method' is presented to generate a complete set of electrostatic mutants for a natural protein where ionic charges on the molecular surface of the template protein are exhaustively interchanged with each other. Total Coulomb interaction energies are evaluated for all of the mutants by numerically solving the finite difference Poisson-Boltzmann equation and their distribution in the ensemble is obtained. This method has been applied to five natural proteins to reveal that they have a significantly lower Coulomb energy than the average over the ensemble of their mutants. It is also shown that these natural proteins have a significantly larger and smaller number of pairs of attractive and repulsive ionic groups, respectively, than those expected for their randomly shuffled ensemble: They have been `designed' through molecular evolution so that a pair of ionic charges with opposite signs may have a higher tendency to be located close to each other, while a pair with the same sign are away from each other.

The electrostatic contribution to DNA base-stacking interactions.

Base-stacking and phosphate-phosphate interactions in B-DNA are studied using the finite difference Poisson-Boltzmann equation. Interaction energies and dielectric constants are calculated and compared to the predictions of simple dielectric models. No extant simple dielectric model adequately describes phosphate-phosphate interactions. Electrostatic effects contribute negligibly to the sequence and conformational dependence of base-stacking interactions. Electrostatic base-stacking interactions can be adequately modeled using the Hingerty screening function. The repulsive and dispersive Lennard-Jones interactions dominate the dependence of the stacking interactions on roll, tilt, twist, and propellor. The Lennard-Jones stacking energy in ideal B-DNA is found to be essentially independent of sequence.