A mathematical view on the decoupled sites representation

The decoupled sites representation (DSR) is a theoretical instrument which allows to regard complex pH titration curves of biomolecules with several interacting proton binding sites as composition of isolated, non-interacting sites, each with a standard Henderson–Hasselbalch titration curve. In this work, we present the mathematical framework in which the DSR is embedded and give mathematical proofs for several statements in the periphery of the DSR. These proofs also identify exceptions. To apply the DSR to any molecule, it is necessary to extend the set of binding energies from ${\mathbb{R}}$ to a stripe within ${\mathbb{C}}$ . An important observation in this context is that even positive interaction energies (repulsion) between the binding sites will not guarantee real binding energies in the decoupled system, at least if the molecule has more than four proton binding sites. Moreover, we show that for a given overall titration curve it is not only possible to find a corresponding system with an interaction energy of zero but with any arbitrary fix interaction energy. This result also effects practical work as it shows that for any given titration curve, there is an infinite number of corresponding hypothetical molecules. Furthermore, this implies that—using a common definition of cooperative binding on the level of interaction energies—a meaningful measure of cooperativity between the binding sites cannot be defined solely on the basis of the overall titration. Consequently, all measures of cooperativity based on the overall binding curve do not measure the type of cooperativity commonly defined on the basis of interaction energies. Understanding the DSR mathematically provides the basis of transferring the DSR to biomolecules with different types of interacting ligands, such as protons and electrons, which play an important role within electron transport chains like in photosynthesis.     

Curves of ligand binding. The use of hyperbolic functions for expressing titration curves

1. The dependences of the concentrations of the non-ligated, uni-ligated and bi-ligated forms of a molecule that binds two molecules of ligand are expressed as functions of the logarithm of free ligand concentration by means of hyperbolic functions. Expressions are also given for the saturation both of an individual site and of the molecule as a whole. This form of expression allows derivation of the following points. 2. The sharpness of bell-shaped curves of concentration of the uni-ligated form is analysed in terms of the heights of their points of inflexion; these can rise to 1/ radical2 of the curve. 3. A single group can exhibit a doubly sigmoid saturation curve if this group and another have comparable affinities for a ligand, and if ligand binding at one of them diminishes the affinity at the other. If the molecular pK values pK(1) and pK(2) for the first and second molecules of ligand are called pK*+/-logm, so that K*(2)=K(1)K(2) and m(2)=K(1)/K(2), then the doubly sigmoid curve can be represented by the sum of two independent one-site saturation curves, in general of unequal height, of pK values pK*+/-log(1/2)[m+ radical(m(2)-4)]. The error in such representation is small either if the mutual interaction between the groups (i.e. m) is large, or if the groups have very similar affinities for the ligand. 4. The sum of two one-site saturation curves, again of pK values of pK*+/-log(1/2)[m+ radical(m(2)-4)] but of equal heights, gives a precise value for the total saturation, provided that the binding of one molecule does not promote the binding of a second, i.e. providing that m>/=2. Hence determinations of saturation cannot distinguish interacting and possibly identical sites from independent and different ones.     

Ionization of tyrosine residues in horse-heart ferricytochrome c and its guanidinated and acetylated-guanidinated derivatives

Spectrophotometric titration curves were obtained at 242 nm for native and fully guanidinated horse-heart ferricytochrome c. The cytochrome c data were fit over the pH range 9-12 (I = 0.35) by a theoretical curve with pK' values of 10.35 and 11.70. The slope of the experimental data increases sharply above pH 12.5 suggesting that two tyrosine residues with pK' values greater than 12.5 are exposed by conformation change. The guanidinated cytochrome c data after correction for the alkaline spin-state transition were fit over the entire pH range 9-13.6 (I = 0.35) by a theoretical curve with pK' values 10.37, 10.78, 11.50, and 13.60. These results along with viscosity measurements indicate that the unfolding transition occurs at higher pH in the guanidinated derivative. N-Acetylimidazole was used to acetylate specific tyrosyl groups of guanidinated cytochrome c. Assignments of acetylated tyrosine residues were confirmed by peptide mapping of 14C-labelled derivatives. Spectrophotometric titrations with rapid data acquisition of two monoacetylated derivatives allowed assignments of pK'1 (10.37) to Tyr-67 and pK'4 (13.60) to Tyr-97. The basis for the large differences in acidity and chemical reactivity of these two residues is not obvious from the crystallographic structure and may arise from differences in solvent access due to motions of the polypeptide chain.     

Continuum electrostatic analysis of irregular ionization and proton allocation in proteins.

Irregular (nonsigmoidal) ionization behavior of titratable groups in proteins is analyzed theoretically, using a computational algorithm designed to count explicitly for tautomers of titratable groups and different locations of polar hydrogens. On the basis of calculations for different model systems (acid-acid, base-base, acid-base pairs, and cluster of three strongly interacting groups), it is demonstrated that the pK values, extracted from nonsigmoidal titration curves by fitting to a sum of Henderson-Hasselbalch equations, do not describe the ionization equilibrium correctly. The conditions for observation of irregular titration curves are derived analytically for the case of arbitrary couple of interacting ionizable groups. A possible relation between irregularly shaped titration curves and tautomerization is also illustrated. The protonation-deprotonation equilibrium of Asp76 in ribonuclease T1 is shown to be coupled to dipole reorientation of a water molecule bound at the protein-solvent interface. This finding provides a new interpretation of the experimentally observed chemical shift of this residue.     

His ... Asp catalytic dyad of ribonuclease A: histidine pKa values in the wild-type, D121N, and D121A enzymes

Bovine pancreatic ribonuclease A (RNase A) has a conserved His ... Asp catalytic dyad in its active site. Structural analyses had indicated that Asp121 forms a hydrogen bond with His119, which serves as an acid during catalysis of RNA cleavage. The enzyme contains three other histidine residues including His12, which is also in the active site. Here, 1H-NMR spectra of wild-type RNase A and the D121N and D121A variants were analyzed thoroughly as a function of pH. The effect of replacing Asp121 on the microscopic pKa values of the histidine residues is modest: none change by more than 0.2 units. There is no evidence for the formation of a low-barrier hydrogen bond between His119 and either an aspartate or an asparagine residue at position 121. In the presence of the reaction product, uridine 3'-phosphate (3'-UMP), protonation of one active-site histidine residue favors protonation of the other. This finding is consistent with the phosphoryl group of 3'-UMP interacting more strongly with the two active-site histidine residues when both are protonated. Comparison of the titration curves of the unliganded enzyme with that obtained in the presence of different concentrations of 3'-UMP shows that a second molecule of 3'-UMP can bind to the enzyme. Together, the data indicate that the aspartate residue in the His ... Asp catalytic dyad of RNase A has a measurable but modest effect on the ionization of the adjacent histidine residue.     

Correlation proton magnetic resonance studies at 250 MHz of bovine pancreatic ribonuclease. I. Reinvestigation of the histidine peak assignments.

The deuterium exchange kinetics of the C(2) protons of the four histidine residues of native bovine pancreatic ribonuclease A have been followed at pH 6.5 and 8.0 by proton magnetic resonance spectroscopy (1H NMR). Comparison of the order of exchange of the histidine peaks with tritium exchange rates into individual histidine residues [Ohe, M., Matsuo, H., Sakiyama, F., and Narita, K. (1974), J. Biochem. (Tokyo) 75, 1197] supports the previous assignment of histidine NMR peaks H(1) and H(4) to histidine-105 and histidine-48 but requires reassignment of peaks H(2) and H(3) to histidine-119 and histidine-12, respectively. Ribonuclease A samples having differentially deuterated histidines have been used to verify the existence of crossover points in the histidine proton magnetic resonance titration curves and to observe the discontinuous titration curve of histidine-48. Proton magnetic resonance peaks have been assigned to the C(4) protons of the four histidine residues of ribonuclease A on the basis of their unit proton areas and by matching their titration shifts with the more readily visible C(2)-H peaks of the histidines. The pK' values derived from the C(4)-H data agree, within experimental limits, with those derived from C(2)-H data. The C(4)-H peaks were assigned to histidine-12, -48, -105, and -119 of ribonuclease A on the basis of their pH dependence, pK' values, shifts of their pK' values in the presence of inhibitor cytidine 3'-phosphate, and by comparison with the assignments of the histidine C(2)-H peaks above.     

Conformational and inframolecular studies of the protonation of adenophostin analogues lacking the adenine moiety

Four adenophostin analogues lacking the adenine moiety were subjected to 31P- and 1H-NMR titrations in order to determine the acid-base behaviour of the individual ionisable groups of the molecules and the complex interplay of intramolecular interactions resulting from the protonation process. For the two trisphosphorylated compounds, the curve pattern of the phosphorus nuclei corresponds to the superimposition of the titration curves of a monophosphorylated polyol and a polyol carrying two vicinal phosphates, suggesting that the two phosphate moieties behave independently. Also, the general shape of 1H-NMR titration curves of the studied compounds is very close to that of adenophostin A, indicating that the adenine moiety does not specifically interact with the phosphorylated sugar moieties. The curves show, however, that both trisphosphorylated compounds adopt slightly different preferential conformations which could contribute to explain the difference in their affinity for Ins(1,4,5)P3 receptor. Their macroscopic as well as the microscopic protonation constants are higher than those of adenophostin A, indicating that the adenine moiety plays a base-weakening effect on the phosphate groups. Further analysis of the microscopic protonation constants confirms that the compound whose conformation is the closest to that of adenophostin A also shows the highest biological activity. The two bisphosphorylated analogues studied behave very similarly, suggesting that the deletion of the hydroxymethyl group on the pentafuranosyl ring only weakly influences the protonation process of the phosphate groups that bear the glucopyranose moiety.     

Zymogen activation in serine proteinases. Proton magnetic resonance pH titration studies of the two histidines of bovine chymotrypsinogen A and chymotrypsin Aalpha.

Reversible unfolding of bovine chymotrypsinogen A in 2H2O either by heating at low pH or by exposure to 6 M guanidinium chloride results in the exchange of virtually all the nitrogen-bound hydrogens that give rise to low-field 1H NMR peaks, without significant exchange of the histidyl ring Cepsilon1 hydrogens. These preexchange procedures have enabled the resolution of two peaks, using 250-MHz correlation 1H NMR spectroscopy, that are attributed to the two histidyl residues of chymotrypsinogen A. Assignments of the Cepsilon1 hydrogen peaks to histidine-40 and -57 were based on comparison of the NMR titration curves of the native zymogen with those of the diisopropylphosphoryl derivative. Two histidyl Cepsilon1 H peaks were also resolved with solutions of preexchanged chymotrypsin Aalpha. The histidyl peaks of chymotrypsin Aalpha were assigned by comparison of NMR titration curves of the free enzyme with those of its complex with bovine pancreatic trypsin inhibitor (Kunitz). The NMR titration curves of histidine-57 in the zymogen and enzyme and histidine-40 in the zymogen exhibit two inflections; the additional inflections were assigned to interactions with neighboring carboxyl groups: aspartate-102 in the case of histidine-57 and aspartate-194 in the case of histidine-40 of the zymogen. In bovine chymotrypsinogen A in 2H2O at 31 degrees C, histidine-57 has a pK' of 7.3 and aspartate-102 a pK' of 1.4, and the histidine-40-aspartate-194 system exhibits inflections at pH 4.6 and 2.3. In bovine chymotrypsin Aalpha under the same conditions, the histidine-57-aspartate-102 system has pK' values of 6.1 and 2.8, and histidine-40 has a pK' of 7.2. The results suggest that the pK' of histidine-57 is higher than the pK' of aspartate-102 in both zymogen and enzyme. A significant difference exists in the structure and properties of the catalytic center between the zymogen and activated enzyme. In addition to the difference in pK' values, the chemical shift of histidine-57, which is highly abnormal in the zymogen (deshielded by 0.6 ppm), becomes normalized upon activation. These changes may explain part of the increase in the catalytic activity upon activation. The 1H NMR chemical shift of the Cepsilon1 H of histidine-57 in the chymotrypsin Aalpha-pancreatic trypsin inhibitor (Kunitz) complex is constant between pH 3 and 9 at a value similar to that of histidine-57 in the porcine trypsin-pancreatic trypsin inhibitor complex [Markley, J.L., and Porubcan, M. A. (1976), J. Mol. Biol. 102, 487--509], suggesting that the mechanisms of interaction are similar in the two complexes.     

Thermal pertubation techniques in characterizing ligand-macromolecule interactions: Theory and application to the proflavins-alpha-chymotrypsin system.

A thermal perturbation curve (TPC) is defined to be the derivative of the fractional degree of saturation, f, with respect to temperature, considered as a function of the natural logarithm of free ligand concentration, y. The theoretical framework for the use of such curves in the thermodynamic analysis of ligand binding to macromolecules is presented. The thermal perturbation curve either provides or complements the information obtained from the derivative binding isotherm ?f/?_y. For a single set of identical and independent sites the TPC is identical to the derivative binding isotherm. Analysis of such a curve directly yield ΔH⁰ and ΔG⁰ for the binding reaction. In actual experimental work, however, the TPC can only be approximated because of “self-buffering” effects relations between the parameter of the approximate curve and the thermodynamic quantities have been developed. This technique is applied to the proflavin-α-chymotryspin system to demonstrate its usefulness. The general features of thermal perturbation curves for cases of multiple sets of independent sites and cooperatively interacting sites have also been developed. The analysis of thermal perturbation curves in combination with other methods should provide a more powerful approach to the characterization of ligand-macromolecule interactions.     

Intracellular pH determination by a 31P-NMR technique. The second dissociation constant of phosphoric acid in a biological system

To evaluate the accuracy of pH determination by 31P-NMR, factors which influence the pK value of phosphate were appraised on the basis of the titration of 1 mM phosphate buffer solution. When the method is used for the determination of cytoplasmic pH, ionic strength is the major factor causing shifts of apparent pK (pK') value, and the magnitude of the shift can be predicted from the ionic strength calculated by means of the Debye-Hückel equation. Ions (Na+, K+, Mg2+, and Ca2+) and salivary protein affected the pK' value by 0.1 to 0.3 units in solution with a given ionic strength depending on the species of ion. The form of the titration curve varied with temperature. Based on these results, the value of 6.75 was obtained with the uncertainty of 0.12 for the intracellular pK' of frog muscle at 24 degrees C.     

Statistical criteria for the identification of protein active sites using Theoretical Microscopic Titration Curves

Theoretical Microscopic Titration Curves (THEMATICS) may be used to identify chemically important residues in active sites of enzymes by characteristic deviations from the normal, sigmoidal Henderson-Hasselbalch titration behavior. Clusters of such deviant residues in physical proximity constitute reliable predictors of the location of the active site. Originally the residues with deviant predicted behavior were identified by human observation of the computed titration curves. However, it is preferable to select the unusual residues by mathematically well-defined criteria, in order to reduce the chance of error, eliminate any possible biases, and substantially speed up the selection process. Here we present some simple statistical tests that constitute such selection criteria. The first derivatives of the predicted titration curves resemble distribution functions and are normalized. The moments of these first derivative functions are computed. It is shown that the third and fourth moments, measures of asymmetry and kurtosis, respectively, are good measures of the deviations from normal behavior. Results are presented for 44 different enzymes. Detailed results are given for 4 enzymes with 4 different types of chemistry: arginine kinase from Limulus polyphemus (horseshoe crab); beta-lactamase from Escherichia coli; glutamate racemase from Aquifex pyrophilus; and 3-isopropylmalate dehydrogenase from Thiobacillus ferrooxidans. The relationship between the statistical measures of nonsigmoidal behavior in the predicted titration curves and the catalytic activity of the residue is discussed.     

Theoretical investigation of the behavior of titratable groups in proteins.

This paper presents a theoretical analysis of the titration behavior of strongly interacting titratable residues in proteins. Strongly interacting titratable residues exist in many proteins such as for instance bacteriorhodopsin, cytochrome c oxidase, cytochrome bc(1), or the photosynthetic reaction center. Strong interaction between titratable groups can lead to irregular titration behavior. We analyze under which circumstances titration curves can become irregular. We demonstrate that conformational flexibility alone can not lead to irregular titration behavior. Strong interaction between titratable groups is a necessary, but not sufficient condition for irregular titration curves. In addition, the two interacting groups also need to titrate in the same pH-range. These two conditions together lead to irregular titration curves. The mutation of a single residue within a cluster of interacting titratable residues can influence the titration behavior of the other titratable residues in the cluster. We demonstrate this effect on a cluster of four interacting residues. This example underlines that mutational studies directed at identifying the role of a certain titratable residue in a cluster of interacting residues should always be accompanied by an analysis of the effect of the mutation on the titration behavior of the other residues.     

Relationship between electrostatics and redox function in human thioredoxin: characterization of pH titration shifts using two-dimensional homo- and heteronuclear NMR.

The electrostatic behavior of potentially titrating groups in reduced human thioredoxin was investigated using two-dimensional (2D) 1H and 15N nuclear magnetic resonance (NMR) spectroscopy. A total of 241 chemical shift titration curves were measured over the pH range of 2.1-10.6 from homonuclear 1H-1H Hartmann-Hahn (HOHAHA) and heteronuclear 1H-15N Overbodenhausen correlation spectra. Nonlinear least-squares fits of the data to simple relationships derived from the Henderson-Hasselbalch equation led to the determination of pKas for certain isolated ionizable groups, including the single histidine residue at position 43 (pKa = 5.5 +/- 0.1) and a number of aspartic and glutamic acid carboxylate groups. Many of the titration curves demonstrate complex behavior due to the effects of interacting titrating groups, the long range of electrostatic interactions through the protein interior, and, perhaps, pH-induced conformational changes on the chemical shifts. Unambiguous assignment of the pKas for most of the 38 potentially ionizing groups of human thioredoxin could therefore not be made. In addition, there was no clear evidence that Asp-26 titrates in a manner corresponding to that observed in the Escherichia coli protein [Dyson, H. J., Tennant, L. L., & Holmgren, A. (1991) Biochemistry 30, 4262-4268]. The pKas of the active site cysteines were measured, however, with Cys-32 having an anomalously low value of 6.3 +/- 0.1 and that of Cys-35 between 7.5 and 8.6. These pKas are in agreement with proposed mechanisms for redox catalysis of thioredoxin and previously measured pKas within the active site of E. coli thioredoxin [Kallis, G. B., & Holmgren, A. (1980) J. Biol. Chem. 255, 10261-10265]. The stabilization of a thiolate anion at physiological pH can be explained by the interaction of the S gamma of Cys-32 with the amide of Cys-35 observed in the previously determined high-resolution solution structure of reduced human thioredoxin [Forman-Kay, J. D., Clore, G. M., Wingfield, P. T., & Gronenborn, A. M. (1991) Biochemistry 30, 2685-2698].     

A simple clustering algorithm can be accurate enough for use in calculations of pKs in macromolecules

Structure and function of macromolecules depend critically on the ionization states of their acidic and basic groups. Most current structure-based theoretical methods that predict pK of ionizable groups in macromolecules include, as one of the key steps, a computation of the partition sum (Boltzmann average) over all possible protonation microstates. As the number of these microstates depends exponentially on the number of ionizable groups present in the molecule, direct computation of the sum is not realistically feasible for many typical proteins that may have tens or even hundreds of ionizable groups. We have tested a simple and robust approximate algorithm for computing these partition sums for macromolecules. The method subdivides the interacting sites into independent clusters, based upon the strength of site-site electrostatic interaction. The resulting partition function is factorizable into computationally manageable components. Two variants of the approach are presented and validated on a representative test set of 602 proteins, by comparing the pK(1/2) values computed by the proposed method with those obtained by the standard Monte Carlo approach used as a reference. With 95% confidence, the relative error introduced by the more accurate of the two methods is less than 0.25 pK units. The algorithms are one to two orders of magnitude faster than the Monte Carlo method, with the typical settings. A graphical representation is introduced that visualizes the clusters of strong site-site interactions in the context of the three-dimensional (3D) structure of the macromolecule, facilitating identification of functionally important clusters of ionizable groups; the approach is exemplified on two proteins, bacteriorhodopsin and myoglobin.     

Triple inhibitor titrations support the functionality of the dimeric character of mitochondrial ubiquinol-cytochrome c oxidoreductase.

The ubiquinol-2 or duroquinol oxidoreductase activity of mitochondrial ubiquinol-cytochrome c oxidoreductase was titrated with combinations of antimycin, myxothiazol and N,N'-dicyclohexylcarbodiimide (DCCD). A statistical model has been developed that can predict the activity of the complex treated with all possible combinations of these inhibitors. On the basis of the measured titration curves the model had to accommodate interaction between the two promoters of the complex. The titrations confirm that treatment with DCCD results in the modification of a certain site in one of the two promoters of the bc1 dimer, thereby blocking one antimycin A binding site without inhibiting electron transfer. Modification of both antimycin A binding sites of the dimer is apparently required for inhibition of electron transfer through the complex, just as modification of both myxothiazol-binding sites is required for full inhibition. The conclusion can be drawn that mitochondrial ubiquinol-cytochrome c oxidoreductase is a functional dimer, consisting of electrically interacting protomers.     

1H n.m.r. studies of insulin. Assignment of resonances and properties of tyrosine residues.

The assignment of the aromatic 1H n.m.r. resonances of the four tyrosine residues of bovine 2-zinc insulin is reported, based on double resonance techniques, use of Hahn spin echo pulse sequences and examination of specific derivatives nitrated at tyrosines A14 and A19 as well as des-(B26-B30)-insulin. Titration curves of the four tyrosine residues show that residues A14 and B16 have normal pK' values of 10.3-10.6 in solution, consistent with their accessibility to solvent in monomer and dimer in the crystal. Tyrosine residues A19 and B26 have pK' values of 11.4 and exhibit other features in their titration curves that are consistent with limited accessibility to solvent and a nonpolar environment. The meta protons of residues B16 and B26 both observe the titration of a nearby tyrosine residue, probably A19. Interpretation of the n.m.r. data obtained in solution is consistent with the crystallographic data for the monomer and dimer obtained on insulin crystals [Blundell, Dodson, Hodgkin & Mercola (1972) Adv. Protein Chem. 26, 279-402].     

Calcium binding to tryptic fragments of calmodulin

Fragments of scallop testis calmodulin were prepared by tryptic digestion. One peptide consisted of 75 amino acid residues from N-acetylalanine to lysine at position 75 (F12) and the other of 71 residues from aspartic acid at position 78 to C-terminal lysine (F34). Flow dialysis and equilibrium dialysis experiments revealed the existence of two Ca2+ binding sites in each fragment. Half-saturating concentrations of the Ca2+ titration curves were 11 microM for F12 and 3.2 microM for F34, and Hill coefficients were obtained as 1.14 and 1.84, respectively. The results indicate that the high-affinity sites for Ca2+ are located on the C-terminal region of the calmodulin. The sum of the two Ca2+ titration curves of F12 and F34 fits well to the curves of Ca2+ binding to intact calmodulin. This shows that the characteristic of Ca2+ bindings in intact calmodulin did not change after separation of the whole molecule into two domains, F12 and F34. The domains corresponding to F12 and F34 may exist independently from each other in the intact calmodulin molecule.     

Acidic dissociation constants of folic acid, dihydrofolic acid, and methotrexate.

The acidic dissociation constants in the range HO--1.5 to pH 7 of folic acid, dihydrofolic acid, methopterin (N(10)methylfolic acid), and methotrexate have been measured by potentiometric and spectrophotometric titrations. Assignment of these dissociations was made by comparison to model compounds, by proton magnetic resonance measurements, and by examination of associated ultraviolet absorbance changes. For folic acid, the dissociation constants are as follows: N(1), pK' 2.35; N(10), pK' 0.20; N(5), pK' greater than -1.5. For dihydrofolic acid: N(5), pK' 3.84; N(1), pK' 1.38; N(10), pK' 0.28. For methotrexate: N(1), pK' 5.71; gamma-carboxyl, pK' 4.70; alpha-carboxyl, pK' 3.36; N(10), pK' 0.50; N(5), boxyl, pK' 4.70; alpha-carboxyl, pK' 3.36; N(10), pK' 0.50; N(5) pK' greater than -1.5. For methopterin: acidic ionization of amide, pK' 7.68; gamma-carboxyl, pK' 4.62; N(1), pK' 2.40; N(10), pK; 0.36; N(5), pK' greater than -1.5. The pK' values were determined directly for the four compounds at 25 degrees near 0.1 ionic strength, or in 0.1 to 4 M HCl for pK ln 0.1 M NaCl.