Numerical calculations of the pH of maximal protein stability The effect of the sequence composition and three-dimensional structure Emil Alexov Howard Hughes Medical Institute and Columbia University, Biochemistry Department, New York, USA A large number of proteins, found experimentally to have different optimum pH of maximal stability, were studied to reveal the basic principles of their preferenence for a par- ticular pH. The pH-dependent free energy of folding was modeled numerically as a function of pH as well as the net charge of the protein. The optimum pH was determined in the numerical calculations as the pH of the minimum free energy of folding. The experimental data for the pH of maximal stability (experimental optimum pH) was repro- ducible (rmsd ¼ 0.73). It was shown that the optimum pH results from two factors – amino acid composition and the organization of the titratable groups with the 3D structure. It was demonstrated that the optimum pH and isoelectric point could be quite different. In many cases, the optimum pH was found at a pH corresponding to a large net charge of the protein. At the same time, there was a tendency for proteins having acidic optimum pHs to have a base/acid ratio smaller than one and vice versa. The correlation between the optimum pH and base/acid ratio is significant if only buried groups are taken into account. It was shown that a protein that provides a favorable electrostatic environment for acids and disfavors the bases tends to have high optimum pH and vice versa. Keywords: electrostatics; pH stability; pK a ; optimum pH. The concentration of hydrogen ions (pH) is an important factor that affects protein function and stability in different locations in the cell and in the body [1]. Physiological pH varies in different organs in human body: the pH in the digestive tract ranges from 1.5 to 7.0, in the kidney it ranges from 4.5 to 8.0, and body liquids have a pH of 7.2–7.4 [2]. It was shown that the interstitial fluid of solid tumors have pH ¼ 6.5–6.8, which differs from the physiological pH of normal tissue and thus can be used for the design of pH selective drugs [3]. The structure and function of most macromolecules are influenced by pH, and most proteins operate optimally at a particular pH (optimum pH) [4]. On the basis of indirect measurements, it has been found that the intracellular pH usually ranges between 4.5 and 7.4 in different cells [5]. The organelles’ pH affects protein function and variation of pH away from normal could be responsible for drug resistance [6]. Lysosomal enzymes function best at the low pH of 5 found in lysosomes, whereas cytosolic enzymes function best at the close to neutral pH of 7.2 [1]. Experimental studies of pH-dependent properties [7–11] such as stability, solubility and activity, provide the benchmarks for numerical simulation. Experiments revealed that altho- ugh the net charge of ribonuclease Sa does affect the solubility, it does not affect the pH of maximal stability or activity [12]. Another experimental technique as acidic or basic denaturation [13–15] demonstrates the importance of electrostatic interactions on protein stability. pH-dependent phenomena have been extensively mode- led using numerical approaches [16–19]. A typical task is to compute the pK a s of ionizable groups [20–26], the isoelectric point [27,28] or the electrostatic potential distribution around the active site [29]. It was shown that activity of nine lipases correlates with the pH dependence of the electrostatic potential mapped on the molecular surface of the molecules [29]. pH dependence of unfolding energy was modeled extensively and the models reproduced reasonable the experimental denaturation free energy as a function of pH [19,30–36]. The success of the numerical protocol to compute the pH dependence of the free energy depends on the model of the unfolded state, the model of folded state and thus on the calculated pK a s. It is well recognized that the unfolded state is compact and native-like, but the magni- tude of the residual pairwise interactions and the desol- vation energies has been debated. Some of the studies found that any residual structure of the unfolded state has negligible effect on the calculated pH dependence of unfolding free energy [31], while others found the opposite [33–36]. It was estimated that the pK a s of the acidic groups in unfolded state are shifted by – 0.3 pK units in respect to the pK a s of model compounds. Although including the measured and simulated pK shifts into the model of unfolded state changes the pH dependence of the unfolding free energy, it most of the cases it does not change the pH of maximal stability [33–36]. Much more Correspondence to E. Alexov, Howard Hughes Medical Institute and Columbia University, Biochemistry Department, 630 W 168 Street, New York, NY 10032, USA. Fax: + 1 212 305 6926, Tel.: + 1 212 305 0265, E-mail: ea388@columbia.edu Abbreviations: MCCE, multi-conformation continuum electrostatic; SAS, solvent accessible surface. (Received 15 September 2003, accepted 11 November 2003) Eur. J. Biochem. 271, 173–185 (2004) Ó FEBS 2003 doi:10.1046/j.1432-1033.2003.03917.x important is the modeling of the folded state, where the errors of computing pK a s could be significantly larger than 0.3 units. Over the years it has been a continuous effort to develop methods for accurate pK a predictions [20,21]. These include empirical methods [37], macroscopic methods [38–41], finite difference Poisson–Boltzmann (FDPB)-based methods [20–22,42], FDPB and molecular dynamics [43–45], FDPB and molecular mechanics [25,46,47] and Warshel’s microscopic methods (e.g., [16,17]). The predicted pK a s were benchmarked against the experimental data and the average rmsd were found to vary from the best value of 0.5pK [38], to 0.7pK [48], to 0.83pK [25] and to 0.89 [22]. Multi-Conformation Con- tinuum Electrostatics (MCCE) [25] method was shown to be among the best pK a s predictors and it will be employed in this work. In the present work we compute the pH dependence of the free energy of folding and the net charge. The optimum pH was identified as the pH at which the free energy of folding has minimum. A large number of proteins having different optimum pH [49] were studied to find the effect of the amino acid composition and 3D structure on the optimum pH. Experimental procedures Methods Calculations were carried out using available 3D structures of selected proteins. A text search was performed on BRENDA database [49] in the field of ÔpH of stabilityÕ.Fol- lowing search strings were used: Ômaximal stabilityÕ, Ômaxi- mum stabilityÕ, Ôoptimal stabilityÕ, Ôoptimum stabilityÕ, Ôbest stabilityÕ, Ôhighest stabilityÕ and Ôgreatest stabilityÕ.This revealed 168 proteins with experimentally determined pHs of maximal stability. Then a search of the Protein Data Base (PDB) was performed to find available structures for these proteins. An attempt was made to select PDB structures of proteins from the same species as those used in the experiment (43 structures). Structures with missing residues were omitted as well as the structures of proteins participa- ting in large complexes resulting in the final set of 28 protein structures. The protein names, the PDB file names and the experimental pH of maximal stability are provided in Table 1. The source of the data is BRENDA database and thus the present study is limited to the proteins listed there. There will always be proteins with experimentally determined Table 1. Proteins and corresponding PDB [57] files used in the paper. The experimental optimum pH (pH of optimal stability) is taken from BRENDA website [49]. The calculated optimum pH (the pH of the minimum of free energy of folding) is given in the forth column. The difference is the calculated optimum pH minus the experimental number (fifth column). Bases/acid ratio for all ionizable groups is in sixth column, while the seventh shows the bases/acids ratio for 66% buried groups. The last three columns show the averaged intrinsic pK shift, the averaged pK a shift and the net charge of the folded protein at pH optimum, respectively. Protein pdb code Experimental optimum pH Calculated optimum pH Difference Base/acid ratio Buried base/acid ratio Averaged intrinsic pK shift Averaged pK a shift Net charge at optimum pH Dioxygenase 1b4u 8.0 8.0 0.0 0.94 1.33 0.08 ) 0.51 ) 3.0 Transferase 1f8x 6.5 5.0 ) 1.5 0.72 0.28 0.40 0.34 ) 5.5 Glutathione synthetase 1sga 8.0 7.5 ) 0.5 0.87 0.88 0.41 ) 0.58 ) 10.0 Isomerase 1b0z 6.0 6.0 0.0 1.02 0.90 0.05 ) 0.48 2.1 Coenzyme A 1bdo 6.5 7.0 0.5 0.67 1.50 0.22 0.03 ) 4.1 Dienelactone hydrolase 1din 7.0 6.5 ) 0.5 1.04 1.17 0.26 ) 0.36 ) 2.7 Dehydrogenase 1dpg 6.2 6.0 ) 0.2 0.79 1.05 0.38 ) 0.41 ) 13.0 Endothiapepsin 1gvx 4.15 4.0 ) 0.15 0.52 0.07 1.45 2.06 6.5 Dehydratase 1aw5 9.0 9.0 0.0 1.07 0.85 0.17 ) 0.48 ) 6.8 Cathepsin B 1huc 5.15 5.0 ) 0.15 0.90 0.73 1.28 0.11 5.8 Alginate lyase 1hv6 7.0 7.0 0.0 1.17 0.93 0.63 ) 0.72 2.7 Xylanase 1igo 5.5 6.5 1.0 1.41 1.00 0.60 ) 0.74 7.3 Hydrolase 1iun 7.5 7.0 ) 0.0 0.86 1.50 0.11 ) 1.15 ) 1.1 Aspartic protease 1j71 4.15 3.0 ) 1.15 0.54 0.33 0.98 1.32 9.4 Aldolase 1jcj 8.5 8.5 0.0 0.97 0.54 0.55 ) 0.19 ) 5.1 L -Asparaginase 1jsl 8.5 7.0 ) 1.5 1.17 1.85 ) 0.12 ) 0.83 ) 0.1 Amylase 1lop 5.9 6.0 0.1 0.81 1.00 0.33 ) 0.42 ) 8.2 c-Glutamil hydrolase 1l9x 7.0 7.5 0.5 1.19 0.77 0.45 ) 0.02 2.8 Mutase 1m1b 7.0 6.0 ) 1.0 0.95 0.86 0.25 0.13 ) 3.2 Methapyrogatechase 1mpy 7.7 7.0 ) 0.7 1.0 1.33 0.11 ) 1.35 ) 12.0 Pyrovate oxidase 1pow 5.7 6.0 0.3 0.91 0.78 0.60 ) 0.51 ) 2.0 Chitosanase 1qgi 6.0 6.5 0.5 1.09 0.54 0.29 ) 0.31 5.0 Xylose isomerase 1qt1 8.0 8.0 0.0 0.84 1.50 0.24 ) 0.30 ) 16.0 Pyruvate decarboxylase 1zpd 6.0 7.0 1.0 1.02 0.83 0.47 ) 0.24 3.8 Acid a-amylase 2aaa 4.9 4.0 ) 0.90 0.51 0.64 1.53 1.48 ) 1.7 Formate dehydrogenase 2nac 5.6 7.0 1.40 1.11 1.42 0.06 ) 1.1 2.4 Phosphorylase 2tpt 6.0 5.0 ) 1.0 0.91 0.93 0.38 ) 0.34 ) 3.8 b-Amylase 5bca 5.5 5.0 ) 0.5 1.07 0.91 0.19 ) 0.13 15.1 174 E. Alexov (Eur. J. Biochem. 271) Ó FEBS 2003 optimum pH that were not in the database, and therefore are not modeled in the paper. However, an additional four well studied proteins were used to benchmark the method in broad pH range and to compare the effect of mutations. Free energy and net charge of unfolded state The unfolded state is modeled as a chain of noninteracting amino acids (the possibility of residual interactions in the unfolded state is discussed at the end of the discussion section). Thus, the free energy of ionizable groups (pH- dependent free energy) is calculated as [31]: DG unf ¼ÀkT lnðZ unf Þ ¼ÀkT X N iÀ1 lnf1 þ exp½À2:3cðiÞðpH À pK sol ðiÞÞg ð1Þ where k is the Boltzmann constant, T is the temperature in Kelvin degrees, N is the number of ionizable groups, c(i)is1 for bases, )1foracids,pK sol (i) is the standard pK a value in solution of group ÔiÕ (e.g., [47]), pH is the pH of the solution and N is the number of ionizable residues. Z unf is the partition function of unfolded state and DG unf is the free energy of unfolded state. The reference state of zero free energy is defined as state of all groups in their neutral forms [31]. The net charge is calculated using the standard formula that comes from Henderson–Hasselbalch equation: q unf ¼ X N i¼1 10 ÀcðiÞðpHÀpK sol ðiÞÞ 1 þ 10 ÀcðiÞðpHÀpK sol ðiÞÞ cðiÞð2Þ where c(i) ¼ )1 or +1 in the case of acid or base, respectively. Free energy and net charge of the folded state The pH-dependent free energy of the folded state is calculated using the 3D structure of proteins listed in Table 1. The 3D structure comprises N ionizable groups (the same number as in the unfolded state) and L polar groups. Each of them might have several alternative side- chain rotamers [50], or alternative polar proton positions [47]. In addition, ionizable groups are either ionized or neutral. All these alternatives are called ÔconformersÕ,being ionizational and positional conformers. There is no apriori information to indicate which conformer is most likely to exist at certain conditions of, for example, pH and salt concentration. Each microstate is comprised of one con- former per residue. The Monte Carlo method was used to estimate the probability of microstates. This procedure is called multi-conformation continuum electrostatics (MC CE) and it is described in more details elsewhere [25,47,50]. A brief summary of the MCCE method is provided in a later section. To find the free energy one should calculate the partition function for each of the proteins. Thus, one should construct all possible combinations of conformers. Because of the very large number of conformers (most of the cases more than 1000), the Monte Carlo method (Metropolis algorithm [51]) is used to find the probability of the microstates [20,47,50,52]. However, to construct the partition function one should know all microstate energies and to sum them up as exponents. Each microstate energy should be taken only once, which induces extra level of complexity. A special procedure is designed that collects the lowest microstate energies and that assures that each microstate is taken only once [50]. A microstate was considered to be unique if its energy differs by more than 0.001 kT from the energies of all previously generated states. A much more stringent procedure that compares the microstate composition would require significant computation time and therefore was not implemented. This results in a function that estimates the partition function. This effective partition function will not have the states with high energy (they are rejected by the Metropolis algorithm), but they have negligible effect [53]. In addition, the constructed partition function may not have all low energy microstates, because given microstate may not be generated in the Monte Carlo sampling or because two or more distinctive microstates may have identical or very similar energies. Bearing in mind all these possibilities, the effective partition function (Z fol )iscalculatedas[50]: Z fol ¼ X X fol n¼1 expðÀDG fol n =kTÞð3Þ where DG fol n is the energy of the microstate ÔnÕ and X fol is the number of microstates collected in Monte Carlo procedure. Then the free energy of ionizable and polar groups in folded state is: DG fol ¼ÀkT lnðZ fol Þð4Þ The occupancy of each conformer (q fol i ) [52] is calculated in the Metropolis algorithm and then used to calculate the net charge of the folded state: q fol ¼ X M i¼1 q fol i cðiÞð5Þ M is the total number of conformers. [Note that c(i)¼ 0 for non ionizable conformers.] Free energy of folding The pH-dependent free energy of folding is calculated as a difference between the free energy of folded and unfolded states: DDG folding ¼ DG fol À DG unf ð6Þ An alternative formula of calculating the pH dependence of the free energy of folding is [19,31,54,55]: DDG folding ¼ 2:3kT Z pH 2 pH 1 DqdpH ð7Þ where, pH 1 and pH 2 determine the pH interval and Dq is the change of the net charge of the protein from unfolded to folded state. Ó FEBS 2003 Calculating pH of maximal protein stability (Eur. J. Biochem. 271) 175 Computational method: MCCE method The basic principles of the method have been described elsewhere [47,50]. The MCCE [25] method allows us to find the equilibrated conformation and ionization states of protein side chains, buried waters, ions, and ligands. The method uses multiple preselected choices for atomic posi- tions and ionization states for many selected side chains and ligands. Then, electrostatic and nonelectrostatic energies are calculated, providing look-up tables of conformer self- energies and conformer–conformer pairwise interactions. Protein microstates are then constructed by choosing one conformer for each side chain and ligand. Monte Carlo sampling then uses each microstate energy to find each conformer’s probability. Thus, the MCCE procedure is divided into three stages: (a) selection of residues and generation of conformers; (b) calculation of energies and (c) Monte Carlo sampling. Selection of residues. The amino acids that are involved in strong electrostatic interactions (magnitude > 3.5 kT) are selected. They will be provided with extra side-chain rotamers to reduce the effects of possible imperfections of crystal structures. The reason is that a small change in their position might cause a significant change in the pairwise interactions [56]. The threshold of 3.5 kT is chosen based on extensive modeling of structures and fitting to experiment- ally determined quantities [25]. The selection is made by calculating the electrostatic interactions using the ori- ginal PDB [57] structure. The alternative side chains for these selected residues are built using a standard library of rotamers [58] and by adding an extra side chain position using a procedure developed in the Honig’s laboratory [59]. The backbone is kept rigid. Then the original structure and alternative side chains were provided with hydrogen atoms. Polar protons of the side chains are assigned by satisfying all hydrogen acceptors and avoiding all hydrogen donors [25]. Thus, every polar side chain and neutral forms of acids have alternative polar proton positions. Calculation of energies. The alternative side chains and polar proton positions determine the conformational space for a particular structure, and they are called ÔconformersÕ. The next step is to compute the energies of each conformer and to store them into look-up tables. Because of conformation flexibility, the energy is no longer only electrostatic in origin, but also has nonelec- trostatic component [47,50]. Electrostatic energies are calculated by DelPhi [60,61], using the PARSE [62] charge and radii set. Internal dielectric constant is 4 [63], while the solution dielectric constant is taken to be 80. The molecular surface is generated with a water probe of radius 1.4 A ˚ [64]. Ionic strength is 0.15 M and the linear Poisson–Boltzmann equation is used. Focusing technique [65] was employed to achieve a grid resolution of about two grids per A ˚ ngstrom. The M calculations, where M is the number of conformers, produce a vector of length M for reaction field energy DG rxn,i and an MxM array of the pairwise interactions between all possible conformers DG ij el . In addition, each conformer has pairwise electrostatic interactions with the backbone resulting in a vector of length M DG pol,i .The magnitude of the strong pairwise and backbone interactions is altered as described in [56]. Such a correction was shown to improve significantly the accuracy of the calcu- lated pK a s[25]. Having alternative side chains and polar hydrogen positions requires nonelectrostatic energy to be taken into account too. This energy is a constant in calculations that use a ÔrigidÕ protein structure (and therefore should not be calculated), but in MCCE plays important role discrim- inating alternative positional conformers. The non- electrostatic interactions for each conformer are the torsion energy, a self-energy term which is independent of the position of all other residues in the protein, and the pairwise Lennard–Jones interactions, both with por- tions of the protein that are held rigid, and with conformers of side chains that have different allowed posi- tions [25,47,50]. Thus, the microstate ÔnÕ pH-dependent free energy of folded state is [20,21,47,50]: DG fol n ¼ X M i¼1  2:3kTd n ðiÞ½cðiÞðpH À pK sol ðiÞÞ þ DpK int ÞðiÞ þ X M j¼iþ1 d n ðiÞd n ðjÞðG ij el þ G ij nonel Þ  ; DpK int ðiÞ¼DpK solv ðiÞþDpK dip ðiÞþDpK nonel ðiÞ ð8Þ where d n (i)is1ifith conformer is present in the nth microstate, M is the total number of conformers, DpK int (i) is the electrostatic and non electrostatic permanent energy contribution to the energy of conformer ÔiÕ (note that it does not contain interactions with polar groups), c(i)is1for bases, )1 for acids, and 0 for neutral groups, DpK solv (i)isthe change of solvation energy of group ÔiÕ, DpK dip (i)isthe electrostatic interactions with permanent charges, DpK nonel (i) is the nonelectrostatic energy with the rigid part of protein, G ij el and G ij nonel are the pairwise electrostatic and non electrostatic interactions, respectively, between con- former ÔiÕ and ÔjÕ. Monte Carlo sampling. TheMonteCarloalgorithmis used to estimate the occupancy (the probability) of each conformer at given pH. The convergence is considered successful if the average fluctuation of the occupancy is smaller than 0.01 [25]. The pH where the net charge of given titratable group is 0.5 is pK ½ . To adopt a common nomenclature, pK ½ will be referred as pK a throughout the text. Optimum pH, isoelectric point (pI) and bases/acids ratio The experimental pH of maximal stability for each of the proteins listed in Table 1 is taken from the website BRENDA [49]. The database does not always provide a single number for the optimum pH. If given protein is reported to be stable in a range of pHs, then the optimum pH is taken to be the middle of the pH range. The optimum pH in the numerical calculation is deter- mined as pH at which the free energy of folding has minimum. In the case that the free energy of folding has a 176 E. Alexov (Eur. J. Biochem. 271) Ó FEBS 2003 minimum in a pH interval, the optimum pH is the middle of the interval. The calculations were carried out in steps of DpH ¼ 1. Thus, the computational resolution of determin- ing the pH optimum was 0.5 pH units. The calculated and experimental pH intervals were not compared, because in many cases BRENDA database provides only the pH of optimal stability. In addition, in most cases the experimental pH interval of stability given in the BRENDA database does not provide information for the free energy change that the protein can tolerate and still be stable. Therefore it cannot be compared with the numerical results which provide only the pH dependence of the folding free energy. Some proteins may tolerate a free energy change of 10 kcalÆmol )1 and still be stable, while others became unstable upon a change of only a few kcalÆmol )1 . The calculated isoelectric point (pI) is the pH at which the net charge of folded state is equal to zero. There is practically no experimental data for the pI of the proteins listed in Table 1. The net charge at optimum pH is the calculated net charge of the folded protein at pH optimum. Base/acid ratio was calculated by counting all Asp and Glu residues as acids and all Arg, Lys and His residues as bases. In some cases, one or more acidic and/ or His residues was calculated to be neutral at a particular pH optimum, but they were still counted. The reason for this was to avoid the bias of the 3D structure and to calculate the base/acid ratio purely from the sequence. The given residue is counted as 66% buried if its solvent accessible surface (SAS) is one-third of the SAS in solution. Averaged intrinsic pK shifts were calculated as 1 N X N i¼1 ðpK int ðiÞÀpK sol ðiÞÞ and the averaged pK a sshiftas 1 N X N i¼1 ðpK a ðiÞÀpK sol ðiÞÞ Thus, a negative pK shift corresponds to conditions such that the protein stabilizes acids and destabilizes bases and vice versa. Arginines were not included in the calculations because their pK a s are calculated in many cases to be outside the calculated pH range. Results Origin of optimum pH The paper reports the pH dependence of the free energy of folding. Despite the differences among the calculated proteins, the results show that the pH-dependence profile of the free energy of folding is approximately bell-shaped and has a minimum at a certain pH, referred to through the paper as the optimum pH. To better understand the origin of the optimum pH, a particular case will be considered in details. Figure 1A shows the free energies of cathepsin B calculated in pH range 0–14. Three energies were computed: the free energy of the unfolded state (bottom line), the free energy of the folded state (middle line) and the free energy of folding (top curve). For the sake of convenience the free energies of the folded state and folding are scaled by an additive constants so to have the same magnitude as the free energy of the unfolded state at the pH of the extreme value (in this case pH ¼ 5). It improves the resolution of the graph without changing its interpretation, because the energies contain an undetermined constant (hydrophobic interactions, entropy change, van der Waals interactions and other pH-inde- pendent energies). Free energy of unfolded state. It can be seen (Fig. 1A) that the free energy of the unfolded state has a maximum value at pH ¼ 5 and it rapidly decreases at low and high pHs. Such a behavior can be easily understood given equation 1. At low pH, the pK sol of all acidic groups is higher than the current pH and thus they contribute negligible to the partition function. In contrast, all basic groups contribute significantly to the partition function. As the pH decreases, their contribution increases, making the free energy more negative. At medium pHs, all ionizable groups are ionized (except His and Tyr), but their effect on the free energy is quite small, because their pK sol areclosetothepH.This results in a maximum of the free energy corresponding to the least favorable state. At high pHs, the situation is reversed: all acidic groups have a major contribution to the partition function, while bases add very little. Thus, the free energy profile of the unfolded state is always a smooth curve (bell-shaped) with a maximum at a certain pH. The shape of the curve and the position of the maximum depend entirely upon the amino acid composition. Fig. 1. Cathepsin B pH-dependent properties. (A) Free energy; (B) net charge. Ó FEBS 2003 Calculating pH of maximal protein stability (Eur. J. Biochem. 271) 177 Free energy of folded state. Thefreeenergyofthefolded state behaves in a similar manner, but it changes less with the pH (Fig. 1A). Note that it has maximum at pH ¼ 6. The major difference occurs at low and high pHs where free energy of the folded state does not decrease as fast as for the unfolded state. The 3D structure adds to the microstate energy (Eqn 8) and to the partition function several new energy terms )DpK int (i) (that originates in part from the desolvation energy) and pairwise interactions G ij (a detailed discussion on the effect of desolvation and pairwise energies on the stability is given in [31]). If these two terms compensate each other, then Eqn 8 might be thought to reassemble the microstate energy formula of the unfolded state, Eqn 1. But there is an important difference: the amino acids are coupled through the pairwise interactions. The pairwise energies are a function of the ionization states. Thus, the de-ionization of a given group will cancel its pairwise interaction energies with the rest of the protein. The effect of the coupling can be easily understood at the extremes of pH. Consider a very low pH such that the pK a s of all acidic groups are higher than the current pH. At such pH all acids will be fully protonated and thus the bases (having their own desolvation penalty) will be left without favorable interactions. Thus the energy of the folded state will be less favorable (because of the desolvation energy and the lack on favorable interactions) than the energy of unfolded state. Free energy of folding. The pH dependence of the free energy of folding results from the difference of the above free energies (Fig. 1A). It always will have a minimum at certain pH (in principle it might have more than one minimum). This minimum may or may not coincide with the pH where the unfolded free energy has maximum. The folding free energy always has a bell shape, and it is unfavorable at low and high pHs as compared to the free energy at optimum pH. Net charge. An alternative way of addressing the same question is to compute the net charge of the protein (Fig. 1B). One can see that at the extremes of pH, the protein is highly charged. At low pH it has a huge net positive charge and at high pH a huge net negative charge. A straightforward conclusion could be made that acidic/ basic denaturation is caused by the repulsion forces among charges with the same type. However all these positive chargesatlowpHexistalsoatmediumpH,wherethe proteins are stable. The thing that is missing at low pH and causes acid denaturation is the favorable interactions with negatively charged groups. At low pH, bases are left without the support of acids, and they have to pay an energy penalty for their desolvation and unfavorable pairwise energies among themselves. Equation 7 provides an additional tool for determining the optimum pH. At the optimum pH, the curve of folding free energy must have an extremum, i.e. the curve must invert its pH behavior. At pH lower than the optimum pH, the free energy of folding should decrease with increasing the pH, then it should have a minimum at pH equal to the optimum pH, and then it should increase with further increase of the pH. Such behavior corresponds to a negative net charge difference between the folded and unfolded state at pH smaller than the optimum pH. As pH increases, the net charge difference should get smaller, and at the optimum pH, it should be zero. Further increase of the pH (above the optimum pH) should make the net charge difference a positive number. One can see in Fig. 1B that the net charge of folding follows such pattern and is zero at pH ¼ 5, where the free energy of folding has a minimum. General analysis of the optimum pH Comparison to experimental data. Although this paper focuses on the pH of maximal stability, it is useful to compare the calculated pH dependence of the folding free energy on a set of proteins subjected to extensive experi- mental measurements. Figure 2 plots the calculated and experimental pH dependence of the free energy of folding. The experimental data is taken from Fersht [66,67], Robertson [68] and Pace [10]. One can see that the calculated pH-dependent free energy agrees well with the experimental data. The most important conclusion for the aims of the paper is that the calculated pH dependence profile of the free energy of folding is similar to that of the experiment. The only exception is ribonuclease A where the calculated pH optimum is 8 while the experiment finds the best stability at pH ¼ 6. It should be noted that the calculated results are similar to the results reported by Elcock [33] and Zhou [36] in cases of idealized unfolded state. From the works of the above authors, as well as from Karshikoff laboratory [34], one can see that the residual interactions in unfolded state do not affect the pH optimum in majority of the studied cases. An additional possibility for comparison is offered by the mutant data. Table 2 shows the stability change of barnase caused by mutations of charged residues. The calculated numbers are the pK a shifts (in respect to the standard pK sol ) of each of these ionizable residues. Thus, the energy of the mutant residue is not taken into account in the numerical calculations. Even under such simplification, the calculated numbers are 0.84 kcalÆmol )1 rmsd from the experiment. Figure 3 compares the calculated optimum pH vs. experimental optimum pH for 28 proteins listed in Table 1. One can see that calculated values are in good agreement with experimental data. The slope of the fitting line is 0.93 and Pearson correlation coefficient is 0.86. The rmsd between calculated and experimentally determined opti- mum pHs is 0.73. The optimum pH ranges from 2 to 9 (4–9 experimentally) which provides a broad range of pHs to be compared. The origin of the optimum pH. The position of the optimum pH depends on the amino acid composition and on the organization of the amino acids within the 3D structure. To find which of these two factors dominates we plotted the calculated optimum pH of the free energy of folding vs. the pH at which the free energy of unfolded state has maximum (Fig. 4). The free energy of folding results from the difference of the free energy of folded and unfolded states. Thus, if the last two energies have the same pH dependence, the free energy of folding will be pH independ- ent. If both the free energy of unfolded and of folded state have similar shape and maximum at the same pH, then most likely the optimum pH will also be at this pH. If the curve of 178 E. Alexov (Eur. J. Biochem. 271) Ó FEBS 2003 the free energy of the folded state is steeper at basic pHs (or flatter at acidic pHs) compared to the free energy of the unfolded state, then the difference, i.e. the free energy of folding will have optimum pH shifted to the right pH scale. Such a phenomenon will occur if the protein stabilizes acids. Then the optimum pH will be higher than the pH of maximal free energy of unfolded state (points above the Table 2. Experimental and calculated effect of single mutants on the stability of barnase. Mutant Experiment (kcalÆmol )1 ) Calculation (kcalÆmol )1 ) D12A ) 0.95 ) 1.83 R69S, R69M ) 2.67, ) 2.24 ) 1.9 D75N ) 4.51 ) 2.92 R83Q ) 2.23 ) 4.07 D93N ) 4.17 ) 4.27 R110A ) 0.45 ) 2.17 Fig. 2. The calculated pH dependence of the free energy of folding (solid line) and experi- mental data (d). The ionic strength was selected to match experimental conditions: barnase (I ¼ 50 m M ), OMKTY3 (I ¼ 10 m M ), CI2 (I ¼ 50 m M ) and ribonuc- lease A (I ¼ 30 m M ). Fig. 3. The calculated optimum pH vs. the experimental optimum pH. The figure shows only 27 data points, because the calculated and experimental data for 1b4u and 1qt1 overlap. Fig. 4. The calculated optimum pH vs. the pH of maximal free energy of unfolded state. Only 19 points can be seen in the figure, because of an overlap, but all 28 points are taken into account in the calculation of the correlation coefficient. Ó FEBS 2003 Calculating pH of maximal protein stability (Eur. J. Biochem. 271) 179 diagonal). If the protein stabilizes bases (or destabilizes acids), then the optimum pH is lower than the pH of maximum of the free energy of unfolded state (point below the diagonal). The points lying on the diagonal represent cases for which the amino acid sequence dominates in determining the optimum pH. The points below the diagonal show proteins with pH optimum lower than the pH of maximum of the free energy of unfolded state. The points offset from the diagonal manifest the importance of the 3D structure. In each case where the 3D structure causes a shift of the solution pK a of ionizable groups, the stability changes [31,69]. If protein favors the charges, then the stability increases. From 28 proteins studied in the paper, nine lie on the main diagonal (tolerance 0.5pK units), while 19 are offset by more than of 0.5pK units. Thus, in 32% of the cases the amino acid composition is the dominant factor determining the optimum pH and in 68% of the cases, the 3D structure does. To check for possible correlation between the optimum pH and the pK shifts in respect to the standard pK sol ,they were plotted in Fig. 5. Two pK shifts were calculated: intrinsic pK which does not account for the interactions with ionizable and polar groups, and pK a shift which reflects the total energy change from solution to the protein for each ionizable group. In both cases the correlation with pH optimum exists, although the correlation coefficients are not very good. A positive pK shift corresponds to pK of acids and bases bigger that of model compounds and thus to electrostatic environment that disfavors acids and favors bases. The most acidic enzymes were found to use this strategy to lower their optimum pH (see the most right hand side of the Fig. 5). The most basic enzymes induce slight positive shift of the intrinsic pK, but adding the pairwise interactions turns the pK shift to a negative number. The enzymes between these two extremes do not induce large pK shift on average. It is well known that the pH dependence of the free energy is an integral of the net charge difference between folded and unfolded states over a particular pH interval (Equation 7) [31,55,70]. A negative net charge difference corresponds to a negative change of the free energy (the free energy gets more favorable as pH increases). Thus, if an acid has a pK a lower than the standard pK sol , it will titrate at lower pH in the folded state compared to unfolded. As a result, such a group will contribute to the net charge difference by a negative number. Conversely, a positive net charge difference corresponds to a positive free energy change, i.e. to a less favorable free energy of folding. This corresponds to pK a s higher than the standard pK sol .At optimum pH the net charge difference should be zero. At very low and at very high pHs, the free energy of folding is unfavorable, because either bases or acids are left without the support of the contra partners. Between these two extremes, the free energy of folding must have a minimum. Starting from very low pH to high pH, the first several ionization events will be the deprotonation of acids. Because these few acids are in the environment of the positive potential of bases, they have pK a s lower than of unfolded state and thus, the net charge difference between folded and unfolded states will be negative. Thus, the free energy of folding will decrease. If the protein does not support the acids, then the rest of acids will have pK a s higher than that of the unfolded state. This results to a positive net charge difference between the folded and unfolded state and increases the free energy of folding. Thus, the optimum pH will be at low pH. Conversely, if the protein favors the acids, then most of them will have pK a s lower than of unfolded state and the net charge difference between folded and unfolded states will be negative. Thus, the free energy of folding will keep decreasing with increasing pH. This will result in optimum pH shifted to higher pHs. The optimum pH is not uniquely determined by the ratio of basic to acidic groups. Figure 6A demonstrates that enzymes with quite different bases to acids ratio have similar optimum pH and that proteins with similar bases to acids ratio function at completely different pHs. At the same time, the trend is clearly seen. The proteins that function at low pH have fewer bases (low base to acid ratio), while the enzyme working at high pH have more bases than acids (see also Table 2). The Pearson correlation coefficient is less than 0.4, which demonstrates that the base/acid ratio is not the most important factor in determining the optimum pH. However, restricting the counting to buried amino acids only, one finds much better correlation (Fig. 6B). This improvement suggests that the pH optimum is mostly determined by the buried charged groups, but the correla- tion is still weak. The effect of the net charge on the stability of the proteins is demonstrated in Fig. 7A,B, where the optimum pH is plotted against the calculated isoelectric point (pI) and the net charge at optimum pH. At the isoelectric point the net charge of the protein is zero, i.e. there are equal number negative and positive charges. The graph shows that there is no correlation (Pearson coeffi- cient ¼ 0.09) between the isoelectric point and the opti- mumpH.Atthesametime,thecorrelationbetweenthe Fig. 5. The experimental optimum pH vs. the averaged pK shifts. (A) Averaged intrinsic pK a ; (B) averaged pK a s shift. 180 E. Alexov (Eur. J. Biochem. 271) Ó FEBS 2003 optimum pH and the net charge of folded state is not neglectable. The signal is weak, but there is a clear tendency for proteins with acidic optimum pH to be positively charged and for proteins with basic optimum pH to carry negative net charge. There are only a few proteins which do not have net charge at optimum pH. Discussion The study has shown that the pH of maximal stability can be calculated using the 3D structure of proteins. Twenty- eight different proteins were studied, most of them with undetectable sequence and structural similarity. The opti- mum pH varies from very acidic pH to very basic pH. Such a diversity provided a good test for the computational method (MCCE) used in the study. Relatively good agreement with the experimental data was achieved result- ing to correlation of 0.85 and rmsd ¼ 0.73. At the same time, as indicated in Fig. 3, there are three proteins with calculated optimum pH of about 1.5 pK units offset from the experimental value (see Table 1). The reason for such a discrepancy could be conformation changes that are not included in the model. In addition, all calculations were carried out at physiological salt concentration (I ¼ 0.15 M ), while the experimental conditions of measuring the opti- mum pH in many cases are not available. This may or may not be a source of significant error, because although the salt concentration strongly affects the pK a values in proteins [71,72] and in model compounds [73], it may not necessary affect the optimum pH [74]. At the same time, it is interesting to point out that the average rmsd of calculated to experimental pH optimum is 0.73, which is similar and slightly better than the average rmsd of pK a s calculations [25]. Two major factors determine the optimum pH, amino acid composition and 3D structure of the proteins. The relative importance of these two factors varies among the proteins. To test our conclusions, two proteins that have different optimum pH (acidic and basic) and are structurally superimposable will be discussed below. Figure 8A shows a structural alignment of acid a-amylase (pdb code 2aaa) and xylose isomerase (pdb code 1qt1). The first protein has acidic optimum pH (calculated optimum pH ¼ 4, experimental optimum pH ¼ 4.9), while the second has basic optimum pH (calculated and experi- mental optimum pH ¼ 8). The core structures of the proteins are well aligned (rmsd ¼ 5.0 A ˚ and PSD ¼ 1.47 [75]). The part of the sequence alignment generated from the structural superimposition is shown in Fig. 8B. The posi- tions that correspond to Arg or Lys residues in the xylose isomerase sequence and are aligned to nonbasic groups in acid a-amylase sequence are highlighted. One can see that 31 basic groups of xylose isomerase sequence are replaced by negative, polar or neutral groups in acid a-amylase sequence. There are only a few examples of the opposite case that are not shown in the figure. This results to base/ acid ratio of 0.51 for acid a-amylase and 0.84 for xylose isomerase. This difference in the amino acid composition results in a different pH dependence of the free energy of the unfolded state and thus demonstrates the effect of the amino acid composition on the optimum pH. From a structural point of view it is interesting to mention that most of the Fig. 7. The experimental optimum pH vs. the calculated isoelectric point (A) and the net charge at pH optimum (B). Fig. 6. The experimental optimum pH vs. the ratio of bases/acids. Twenty-seven data points can be seen, because of the overlap between 1qtl and 1b4u. (A) All amino acids; (B) buried amino acids. Ó FEBS 2003 Calculating pH of maximal protein stability (Eur. J. Biochem. 271) 181 extra basic groups within the xylose isomerase structure are not within the extra loop regions, but rather within the core structure (see Fig. 8A). This confirms the observation (Fig. 7B) that buried groups affect the optimum pH and an enzyme that has acidic optimum pH has low acid/base ratio. It remains to be shown that this is a general behavior of all enzymes operating at low pH. Three-dimensional structure of the protein plays an even more significant role than the sequence composition on the optimum pH (68% of the cases in this work). The ability of Fig. 8. Alignment of acid alpha-amylase (2aaa.pdb) and xylose isomerase (1qt1.pdb). (A) Structural and sequence alignments are carried out with GRASP 2 [79]. Structural alignment in ribbon representation: acid amylase backbone is shown in green and xylose isomerase in blue. The red patches show the positions of substitution of Arg/Lys to negative, polar or neutral groups from xylose isomerase to acid amylase (see Fig. 8B). (B) Sequence alignment from the structural superimposition: highlighted are the positions at which Arg/Lys in the xylose isomerase sequence are aligned to acid, polar or neutral groupinacida-amylase sequence. 182 E. Alexov (Eur. J. Biochem. 271) Ó FEBS 2003 [...]... engineering the surface charges of ribonuclease Sa [12] Increasing the net charge of the molecule does not change its pH of maximal stability, but changes the isoelectric point and increases solubility [12] Another strategy used to reduce the bias from the amino acid composition is to change pKas of ionizable groups in the protein If protein favors the negative charges on acidic groups, then the optimum pH. ..Ó FEBS 2003 Calculating pH of maximal protein stability (Eur J Biochem 271) 183 the proteins to reduce the bias of the amino acid sequence composition was shown by comparing the isoelectric point, the net charge and the optimum pH It was shown that for most proteins the optimum pH does not coincide with the pI and that the protein is most stable when it caries net charge This... pH- dependence curve is sensitive to the model of the unfolded state, the optimum pH does not depend significantly on it [33–36] The success of the modeling of the pH dependent free energy of folding critically depends of the accuracy of the calculated pKas of the ionizable groups Recent benchmarks of MCCE on 166 titratable groups resulted to an rmsd 0.83 pK as compared to the experimentally determined pKas... [10].) The modeling of the unfolded state would eventually require molecular dynamic runs [33] or some assumptions of the organization of the amino acids in unfolded state [34,36] or even an experimental determination of the pKas in model compounds [35,73] Our goal was to compute the pH at which the free energy of folding has minimum It was shown in the literature that while the shape of the pH- dependence... ionizable groups by the protein always increases protein stability It should be emphasized that this paper does not make an attempt to calculate the all of the details of pH dependence of the free energy of denaturation This will require an appropriate model of the unfolded state [7,66], which is believed to be compact and native-like (In addition, the denaturated state may not be the same in thermal, urea... (2000) pH dependence of stability of staphyloccocal nuclease: evidence of substantial electrostatic interactions in the denaturated state Biochemistry 39, 14292–14304 8 Pots, A., Jongh, H., Gruppen, H., Hessing, M & Voragen, A (1998) The pH dependence of the structural stability of patatin J Agric Food Chem 46, 2546–2553 9 Khurana, R., Hate, A., Nath, U & Udgaonkar, B (1995) pH dependence of the stability. .. modeling of the denaturated states of proteins allows accurate calculations of the pH dependence of protein stability J Mol Biol 294, 1051–1062 34 Kundrotas, P & Karshikoff, A (2002) Modeling of denaturated state for calculation of the electrostatic contribution to protein stability Prot Sci 11, 1681–1686 35 Tollinger, M., Crowhurst, K., Kay, L & Forman-Kay, J (2003) Site-specific contributions to the pH dependence... pH as compared to the pH at which unfolded free energy has maximum and vice versa (Fig 5) The same is valid for basic groups but the effect is less noticeable simply because their pKas are too high (except for histidines) It should be emphasized that one should distinguish between the amplitude of the free energy of folding and optimum pH As discussed in previous papers [31,69], the stabilization of. .. made to study the sensitivity of the results against different values of the dielectric constant Other parameters that were not tested include the charge set [76], the choice of molecular surface (van der Waals surface vs molecular surface) [56,77,78] and the effect of energy minimization of PDB structures [26] These will require a separate study In addition, it should be noted that the relatively... assumes that pKas of the protein as the same as in model compounds) will not work in this case, because it will result in pH- independent free energy of folding Despite of several failures, the presented methodology can predict the optimum pH with reasonable accuracy This information can be used to identify a possible cellular compartment or body organ where the protein may function Obviously a protein with . range of pHs, then the optimum pH is taken to be the middle of the pH range. The optimum pH in the numerical calculation is deter- mined as pH at which the. was determined in the numerical calculations as the pH of the minimum free energy of folding. The experimental data for the pH of maximal stability (experimental