Modeling of protein conformational fluctuations in pKa predictions

A Gaussian-chain model for treating residual charge-charge interactions in the unfolded state of proteins

Characterization of the unfolded state is essential for understanding the protein folding problem. In the unfolded state, a protein molecule samples vastly different conformations. Here I present a simple theoretical method for treating residual charge-charge interactions in the unfolded state. The method is based on modeling an unfolded protein as a Gaussian chain. After sampling over all conformations, the electrostatic interaction energy between two charged residues (separated by l peptide bonds) is given by W = 332(6/pi)(1/2)[1 - pi(1/2)xexp(x(2))erfc(x)]/epsilond, where d = bl(1/2) + s and x = kappad/6(1/2). In unfolded barnase, the residual interactions lead to downward pK(a) shifts of approximately 0.33 unit, in agreement with experiment. pK(a) shifts in the unfolded state significantly affect pH dependence of protein folding stability, and the predicted effects agree very well with experimental results on barnase and four other proteins. For T4 lysozyme, the charge reversal mutation K147E is found to stabilize the unfolded state even more than the folded state (1.39 vs. 0.46 kcal/mol), leading to the experimentally observed result that the mutation is net destabilizing for the folding. The Gaussian-chain model provides a quantitative characterization of the unfolded state and may prove valuable for elucidating the energetic contributions to the stability of thermophilic proteins and the energy landscape of protein folding.

Calculation of protein ionization equilibria with conformational sampling: pK(a) of a model leucine zipper, GCN4 and barnase

The use of conformational ensembles provided by nuclear magnetic resonance (NMR) experiments or generated by molecular dynamics (MD) simulations has been regarded as a useful approach to account for protein motions in the context of pK(a) calculations, yet the idea has been tested occasionally. This is the first report of systematic comparison of pK(a) estimates computed from long multiple MD simulations and NMR ensembles. As model systems, a synthetic leucine zipper, the naturally occurring coiled coil GCN4, and barnase were used. A variety of conformational averaging and titration curve-averaging techniques, or combination thereof, was adopted and/or modified to investigate the effect of extensive global conformational sampling on the accuracy of pK(a) calculations. Clustering of coordinates is proposed as an approach to reduce the vast diversity of MD ensembles to a few structures representative of the average electrostatic properties of the system in solution. Remarkable improvement of the accuracy of pK(a) predictions was achieved by the use of multiple MD simulations. By using multiple trajectories the absolute error in pK(a) predictions for the model leucine zipper was reduced to as low as approximately 0.25 pK(a) units. The validity, advantages, and limitations of explicit conformational sampling by MD, compared with the use of an average structure and a high internal protein dielectric value as means to improve the accuracy of pK(a) calculations, are discussed.

Optimizing the hydrogen-bond network in Poisson-Boltzmann equation-based pK(a) calculations

pK(a) calculation methods that are based on finite difference solutions to the Poisson-Boltzmann equation (FDPB) require that energy calculations be performed for a large number of different protonation states of the protein. Normally, the differences between these protonation states are modeled by changing the charges on a few atoms, sometimes the differences are modeled by adding or removing hydrogens, and in a few cases the positions of these hydrogens are optimized locally. We present an FDPB-based pK(a) calculation method in which the hydrogen-bond network is globally optimized for every single protonation state used. This global optimization gives a significant improvement in the accuracy of calculated pK(a) values, especially for buried residues. It is also shown that large errors in calculated pK(a) values are often due to structural artifacts induced by crystal packing. Optimization of the force fields and parameters used in pK(a) calculations should therefore be performed with X-ray structures that are corrected for crystal artifacts.

Engineering the pH-optimum of a triglyceride lipase: from predictions based on electrostatic computations to experimental results

The optimisation of enzymes for particular purposes or conditions remains an important target in virtually all protein engineering endeavours. Here, we present a successful strategy for altering the pH-optimum of the triglyceride lipase cutinase from Fusarium solani pisi. The computed electrostatic pH-dependent potentials in the active site environment are correlated with the experimentally observed enzymatic activities. At pH-optimum a distinct negative potential is present in all the lipases and esterases that we studied so far. This has prompted us to propose the "The Electrostatic Catapult Model" as a model for product release after cleavage of the ester bond. The origin of the negative potential is associated with the titration status of specific residues in the vicinity of the active site cleft. In the case of cutinase, the role of Glu44 was systematically investigated by mutations into Ala and Lys. Also, the neighbouring Thr45 was mutated into Proline, with the aim of shifting the spatial location of Glu44. All the charge mutants displayed altered titration behaviour of active site electrostatic potentials. Typically, the substitution of the residue Glu44 pushes the onset of the active site negative potential towards more alkaline conditions. We, therefore, predicted more alkaline pH optima, and this was indeed the experimentally observed. Finally, it was found that the pH-dependent computed Coulombic energy displayed a strong correlation with the observed melting temperatures of native cutinase.

Modeling the effects of mutations on the free energy of the first electron transfer from QA- to QB in photosynthetic reaction centers

Numerical calculations of the free energy of the first electron transfer in genetically modified reaction centers from Rhodobacter (Rb.) sphaeroides and Rb. capsulatus were carried out from pH 5 to 11. The multiconformation continuum electrostatics (MCCE) method allows side chain, ligand, and water reorientation to be embedded in the calculations of the Boltzmann distribution of cofactor and amino acid ionization states. The mutation sites whose effects have been modeled are L212 and L213 (the L polypeptide) and two in the M polypeptide, M43(44) and M231(233) in Rb. capsulatus (Rb. sphaeroides). The results of the calculations were compared to the experimental data, and very good agreement was found especially at neutral pH. Each mutation removes or introduces ionizable residues, but the protein maintains a net charge close to that in native RCs through ionization changes in nearby residues. This reduces the effect of mutation and makes the changes in state free energy smaller than would be found in a rigid protein. The state energy of QA-QB and QAQB- states have contributions from interactions among the residues as well as with the quinone which is ionized. For example, removing L213Asp, located in the QB pocket, predominantly changes the free energy of the QA-QB state, where the Asp is ionized in native RCs rather than the QAQB- state, where it is neutral. Side chain, hydroxyl, and water rearrangements due to each of the mutations have also been calculated showing water occupancy changes during the QA- to QB electron transfer.

A pragmatic approach to structure based calculation of coupled proton and electron transfer in proteins

The coupled motion of electrons and protons occurs in many proteins. Using appropriate tools for calculation, the three-dimensional protein structure can show how each protein modulates the observed electron and proton transfer reactions. Some of the assumptions and limitations involved in calculations that rely on continuum electrostatics to calculate the energy of charges in proteins are outlined. Approaches that mix molecular mechanics and continuum electrostatics are described. Three examples of the analysis of reactions in photosynthetic reaction centers are given: comparison of the electrochemistry of hemes in different sites; analysis of the role of the protein in stabilizing the early charge separated state in photosynthesis; and calculation of the proton uptake and protein motion coupled to the electron transfer from the primary (Q(A)) to secondary (Q(B)) quinone. Different mechanisms for stabilizing intra-protein charged cofactors are highlighted in each reaction.

Tanford-Kirkwood electrostatics for protein modeling

Solvent plays a significant role in determining the electrostatic potential energy of proteins, most notably through its favorable interactions with charged residues and its screening of electrostatic interactions. These energetic contributions are frequently ignored in computational protein design and protein modeling methodologies because they are difficult to evaluate rapidly and accurately. To address this deficiency, we report a revised form of the original Tanford-Kirkwood continuum electrostatic model [Tanford, C. & Kirkwood, J. G. (1957) J. Am. Chem. Soc. 79, 5333-5339], which accounts for the effects of solvent polarization on charged atoms in proteins. The Tanford-Kirkwood model was modified to increase its speed and to improve its sensitivity to the details of protein structure. For the 37 electrostatic self-energies of the polar side-chains in bovine pancreatic trypsin inhibitor, and their 666 interaction energies, the modified Tanford-Kirkwood potential of mean force differs from a computationally intensive numerical potential (DelPhi) by root-mean-square errors of 0.6 kcal/mol and 0.08 kcal/mol, respectively. The Tanford-Kirkwood approach makes possible a realistic treatment of electrostatics in computationally demanding protein modeling calculations. For example, pH titration calculations for ovomucoid third domain that model polar side-chain relaxation (including >2 x 10(23) rotamer conformations of the protein) provide pKa values of unprecedented accuracy.

Calculated protein and proton motions coupled to electron transfer: electron transfer from QA- to QB in bacterial photosynthetic reaction centers

Reaction centers from Rhodobacter sphaeroides were subjected to Monte Carlo sampling to determine the Boltzmann distribution of side-chain ionization states and positions and buried water orientation and site occupancy. Changing the oxidation states of the bacteriochlorophyll dimer electron donor (P) and primary (QA) and secondary (QB) quinone electron acceptors allows preparation of the ground (all neutral), P+QA-, P+QB-, P0QA-, and P0QB- states. The calculated proton binding going from ground to other oxidation states and the free energy of electron transfer from QA-QB to form QAQB- (DeltaGAB) compare well with experiment from pH 5 to pH 11. At pH 7 DeltaGAB is measured as -65 meV and calculated to be -80 meV. With fixed protein positions as in standard electrostatic calculations, DeltaGAB is +170 meV. At pH 7 approximately 0.2 H+/protein is bound on QA reduction. On electron transfer to QB there is little additional proton uptake, but shifts in side chain protonation and position occur throughout the protein. Waters in channels leading from QB to the surface change site occupancy and orientation. A cluster of acids (GluL212, AspL210, and L213) and SerL223 near QB play important roles. A simplified view shows this cluster with a single negative charge (on AspL213 with a hydrogen bond to SerL233) in the ground state. In the QB- state the cluster still has one negative charge, now on the more distant AspL210. AspL213 and SerL223 move so SerL223 can hydrogen bond to QB-. These rearrangements plus other changes throughout the protein make the reaction energetically favorable.

Simplified methods for pKa and acid pH-dependent stability estimation in proteins: removing dielectric and counterion boundaries

Much computational research aimed at understanding ionizable group interactions in proteins has focused on numerical solutions of the Poisson-Boltzmann (PB) equation, incorporating protein exclusion zones for solvent and counterions in a continuum model. Poor agreement with measured pKas and pH-dependent stabilities for a (protein, solvent) relative dielectric boundary of (4,80) has lead to the adoption of an intermediate (20,80) boundary. It is now shown that a simple Debye-Huckel (DH) calculation, removing both the low dielectric and counterion exclusion regions associated with protein, is equally effective in general pKa calculations. However, a broad-based discrepancy to measured pH-dependent stabilities is maintained in the absence of ionizable group interactions in the unfolded state. A simple model is introduced for these interactions, with a significantly improved match to experiment that suggests a potential utility in predicting and analyzing the acid pH-dependence of protein stability. The methods are applied to the relative pH-dependent stabilities of the pore-forming domains of colicins A and N. The results relate generally to the well-known preponderance of surface ionizable groups with solvent-mediated interactions. Although numerical PB solutions do not currently have a significant advantage for overall pKa estimations, development based on consideration of microscopic solvation energetics in tandem with the continuum model could combine the large deltapKas of a subset of ionizable groups with the overall robustness of the DH model.

Improving the accuracy of protein pKa calculations: conformational averaging versus the average structure

Several methods for including the conformational flexibility of proteins in the calculation of titration curves are compared. The methods use the linearized Poisson-Boltzmann equation to calculate the electrostatic free energies of solvation and are applied to bovine pancreatic trypsin inhibitor (BPTI) and hen egg-white lysozyme (HEWL). An ensemble of conformations is generated by a molecular dynamics simulation of the proteins with explicit solvent. The average titration curve of the ensemble is calculated in three different ways: an average structure is used for the pKa calculation; the electrostatic interaction free energies are averaged and used for the pKa calculation; and the titration curve for each structure is calculated and the curves are averaged. The three averaging methods give very similar results and improve the pKa values to approximately the same degree. This suggests, in contrast to implications from other work, that the observed improvement of pKa values in the present studies is due not to averaging over an ensemble of structures, but rather to the generation of a single properly averaged structure for the pKa calculation.

Calculations of proton-binding thermodynamics in proteins

Computational models of proton binding can range from the chemically complex and statistically simple (as in the quantum calculations) to the chemically simple and statistically complex. Much progress has been made in the multiple-site titration problem. Calculations have improved with the inclusion of more flexibility in regard to both the geometry of the proton binding and the larger scale protein motions associated with titration. This article concentrated on the principles of current calculations, but did not attempt to survey their quantitative performance. This is (1) because such comparisons are given in the cited papers and (2) because continued developments in understanding conformational flexibility and interaction energies will be needed to develop robust methods with strong predictive power. Nevertheless, the advances achieved over the past few years should not be underestimated: serious calculations of protonation behavior and its coupling to conformational change can now be confidently pursued against a backdrop of increasing understanding of the strengths and limitations of such models. It is hoped that such theoretical advances will also spur renewed experimental interest in measuring both overall titration curves and individual pKa values or pKa shifts. Exploration of the shapes of individual titration curves (as measured by Hill coefficients and other parameters) would also be useful in assessing the accuracy of computations and in drawing connections to functional behavior.

The pH-induced release of iron from transferrin investigated with a continuum electrostatic model

A reduction in pH induces the release of iron from transferrin in a process that involves a conformational change in the protein from a closed to an open form. Experimental evidence suggests that there must be changes in the protonation states of certain, as yet not clearly identified, residues in the protein accompanying this conformational change. Such changes in protonation states of residues and the consequent changes in electrostatic interactions are assumed to play a large part in the mechanism of release of iron from transferrin. Using the x-ray crystal structures of human ferri- and apo-lactoferrin, we calculated the pKa values of the titratable residues in both the closed (iron-loaded) and open (iron-free) conformations with a continuum electrostatic model. With the knowledge of a residue's pKa value, its most probable protonation state at any specified pH may be determined. The preliminary results presented here are in good agreement with the experimental observation that the binding of ferric iron and the synergistic anion bicarbonate/carbonate results in the release of approximately three H+ ions. It is suggested that the release of these three H+ ions may be accounted for, in most part, by the deprotonation of the bicarbonate and residues Tyr-92, Lys-243, Lys-282, and Lys-285 together with the protonation of residues Asp-217 and Lys-277.