J. Phys. Chem. B 2007, 111, 5483-5486
5483
Approximate Values for Force Constant and Wave Number Associated with a Low-Frequency Concerted Motion in Proteins Can Be Evaluated by a Comparison of X-ray Structures Antonello Merlino,† Filomena Sica,†,‡ and Lelio Mazzarella*,†,‡ Dipartimento di Chimica, UniVersita` degli Studi di Napoli “Federico II”, Via Cynthia, 80126 Napoli, Italy, and Istituto di Biostrutture e Bioimmagini, CNR, Via Mezzocannone 6, 80134 Napoli, Italy ReceiVed: February 19, 2007; In Final Form: March 5, 2007
Low-frequency internal motions in protein molecules play a key role in biological functions. A direct relationship between low-frequency motions and enzymatic activity has been suggested for bovine pancreatic ribonuclease (RNase A). The flexibility-function relationship in this enzyme has been attributed to a subtle and concerted breathing motion of the β-sheet regions occurring upon substrate binding and release. Here, we calculate an approximate value for the force constant and the wave number of the low-frequency β-sheet breathing motion of RNase A, by using the Boltzmann hypothesis on a set of data derived from a simple conventional structural superimposition of an unusual large number of X-ray structures available for the protein. The results agree with previous observations and with theoretical predictions on the basis of normal-mode analysis. To the best of our knowledge, this is the first example in which the wave number and the force constant of a low-frequency concerted motion in a protein are directly derived from X-ray structures.
Introduction The importance of the dynamic aspects of the behavior of biological macromolecules is now widely recognized. To determine the features of protein internal flexibility, a variety of different experimental techniques are used, like those detecting 15N order parameters derived from NMR experiments or those based on fluorescence quenching measurements or on Raman spectroscopy studies. These experimental approaches are often complemented by theoretical studies, such as molecular dynamics and Monte Carlo simulations. X-ray crystallography can also provide important information about the dynamic aspects of a molecular structure. Some information about protein flexibility comes from X-ray crystallography in the form of atomic mean square displacement or B-factors. A less direct but possibly more general approach to extracting dynamic information from crystallographic data is based on comparison of structures of the same molecule determined from different crystal forms or in different environmental conditions. The analysis of these multiple forms has revealed low-energy conformational variations or other manifestations of protein flexibility that are biologically relevant.1-3 Collective motions that involve several neighboring residues moving in a concerted fashion occur in proteins and play an important role in biological process.4 Experimental observations of low-frequency vibrations associated with these motions have been collected by using neutron diffusion scattering5,6 and Raman techniques.7 The Raman spectrum of lysozyme shows peaks at 25 and 75 cm-1.8 Similar Raman peaks are seen in R-chimotripsin at about 30 cm-1.7 In spectra of other proteins, peaks are observed in the range from 14 to 36 cm-1. Normal-mode analysis has also been used to study collective motions in biomolecules. The modes * To whom correspondence should be addressed. Fax: +39-081-674090; e-mail:
[email protected]. † Universita ` degli Studi di Napoli “Federico II”. ‡ Istituto di Biostrutture e Bioimmagini.
relevant to the localized large amplitude torsion displacement of domain or subdomain collective motions appear to be located in the frequency 1-200 cm-1.9 Vibrations of β-sheet region occur between 5 and 30 cm-1.10 The hinge-bending modes derived for BPTI and mouse epidermal growth factor correspond to a frequency of 4.4 and 4.1 cm-1, respectively.9 The lowest frequency for bovine pancreatic ribonuclease (RNase A) and crambin is between 2 cm-1 and 5 cm-1.9 Cryo-crystallographic experiments on RNase A gave perhaps the first really experimental evidence of the intimate relationship between protein flexibility and function.11 The crystal structure of RNase A is formed by three R-helices and two large antiparallel β-sheets, which form a V-shaped motif with one of the three helices as a hinge.12,13 The two antiparallel β-sheet regions (V1 and V2) are made up of residues 61-63, 71-75, 105-111, 116-124 and residues 42-46, 82-87, 96-101, respectively. The flexibility-function relationship in RNase A has been recently attributed to subtle and concerted motions of V1 and V2 occurring upon substrate binding and release in this enzyme.14 Similar changes of the tertiary structure of RNase A have been observed in other members of pancreatic-like ribonucleases.15-17 By using essential dynamics analyses, we have also shown that this collective β-sheet motion is an intrinsic dynamic property of the ligand-free RNase A.15,18 Recently, it has been suggested that this motion should fall in the MHz/ GHz range.19 The use of bovine pancreatic ribonuclease as a model system in several fundamental studies in enzymology and structural biology has led to the determination of a large number of structures in several different experimental conditions. Taking advantage of the availability of an unusually high number of crystal structures of RNase A (132 models), we calculate the constant force and the wave number of the β-sheet breathing motion of this protein. Our study is based on the idea that it is possible to combine structural variations observed in the
10.1021/jp071399h CCC: $37.00 © 2007 American Chemical Society Published on Web 04/13/2007
5484 J. Phys. Chem. B, Vol. 111, No. 19, 2007
Merlino et al.
Figure 2. Histogram for the θ distribution in the database of 132 RNase A structures from the Protein Data Bank. A step ) 1° was chosen for the histogram. Figure 1. CR atom drawing of a structural ensemble of bovine pancreatic ribonuclease superimposed on the V1 β-sheet region. The rotation angle θ required to best fit V2, after the superimposition of V1, is also shown.
different experimentally determined RNase A structures into a formal description of its β-sheet breathing motion. Each structure provides a snapshot of the protein movement with a characteristic relative orientation of the V1 and V2 region (Figure 1). A distribution of the experimentally observed structures will tend to be concentrated around the minimum of the potential energy surface with a Boltzmann-type distribution (Boltzmann hypothesis). Experimental Section Database. The RNase A structure files used in this study come from the Protein Data Bank (PDB).20 At the time of the study, the PDB contains 135 X-ray structures of RNase A. In our analyses, only three structures have been eliminated (pdb code 1c0b, 1c0c, 1rha) because of the drastic conditions in which RNase A structure has been studied (crystals severely desiccated for 2.5 and 4 days and low humidity crystal form of the enzyme). This provides a data set of 132 structures. Overall, there are a total of seven different space groups in which RNase A has been crystallized. The resolution of the X-ray structures ranges from 0.87 to 2.8 Å, while the R-factor values range from 0.12 to 0.25. The pH values range from 4.3 to 8.8, and the temperature of the data collections range from 98 to 384 K. In particular, 37 structures have been determined at a temperature of 100 K, eight between 100 K and 280 K, two between 320 and 340 K, and the others at room temperature. In 112 structures, RNase A is complexed with substrate or product analogues, inhibitors, metals, and different ions. Twenty-one structures contain at least one sequence change. With the exception of three structures, the estimated crystal solvent content in the analyzed X-ray models is always higher than 41%. Force Constant and Wave Number Calculations. The V-shaped β-sheet region of RNase A was divided in two arms (V1 and V2), according to the definition reported by Vitagliano et al. (14). Separate superimpositions of V1 or V2 arms of all the analyzed RNase A structures yield very low root-mean-
square deviation (RMSD) (0.1 and 0.2 Å for V1 and V2, respectively), thus showing that the structure of the individual arms is rigidly preserved. To characterize the difference in the spatial position of the two arms in the structures, we compared each RNase A structure with a reference model using a stepwise superimposition method. Specifically, we determine the rotation angle (θ) required to best fit V2, after the superimposition of V1 (Figure 1). Applying the θ rotation, the RMSD of V2 is reduced on average from 1.0 Å to 0.2 Å. Moreover, the polar coordinates phi and psi of the axis used for the rotation cluster within a very narrow range. The mean value of the θ distribution, which depends on the arbitrarily chosen reference structure, is -3.8°, and the standard deviation is 2.4°. The distribution of θ values was then reported in histogram form (Figure 2), and the “energy values” (∆E/RT) were derived using the Boltzmann equation ni/n0 ) e-∆Ε/RT, where ni is the number of data points in state i, R is the gas constant, T is the temperature, and ∆E is the energy difference between the two energy states. The position of θ maximum has been taken performing a fitting of the distribution. The histogram was built using a one degree step, which was considered to be a good compromise to have a representative sampling of the distribution function with a reasonable number of protein structures within each interval. Moreover, using a 0.5° step, the calculated k did not change significantly. The energy values derived from the θ distribution were fitted with a Morse-like potential U(θ) ) d[1 - e-f(θ-r)]2, where d, f, and r are three adjustable parameters. The energy profile and the fitting curve are plotted in Figure 3 with the minimum translated at the origin. The force constant k was calculated as the second derivative of the fitting curve. A similar value was obtained using the harmonic approximation (data not shown). The wave number was then calculated directly from k using a value of 3.4 Kgmol-1 for the reduced mass of the protein and a value of 300 K for the temperature. Although the last value was somewhat arbitrarily chosen, by choosing a value within a 200-300 K interval, the order of magnitude of the calculated frequency parameters, which was indeed the aim of this approach, does not change. The structure of RNase A identified with the pdb code 1jvt (B chain) and determined in our laboratory was used as a reference.21
Values of Low-Frequency Motion in Proteins
Figure 3. Energy well for the β-sheet breathing motion of RNase A, derived from the θ distribution and converted to “energies” using the Boltzmann relationship ni/n0 ) e-∆/kBT, where ni is the number of the observation in the state i, kB is the Boltzmann factor, and T is set to 300 K. The values used for the fitting are colored in red. The points are fitted using a Morse-like potential. θ′ is the θ angle, translated a few degrees to impose the minimum at the origin.
J. Phys. Chem. B, Vol. 111, No. 19, 2007 5485 using a a Morse-like anharmonic potential (Figure 3) to derive the force constant. Interestingly, similar results were obtained calculating the approximate value for k, using a simple harmonic oscillator, although in this last case, the fitting of experimental data is less satisfactory (data not shown). Our analysis indicates that the breathing motion of RNase A should have a force constant in the range of 0.2-0.3 Kcal mol-1 deg-2 (about 1-1.5 kJ mol-1 deg-2 or 5 Nm-1) and a wave number of about 5 cm-1. These values are in good agreement with the predicted frequency and the expected force constant associated with this motion. In fact, weak force constants (from 0.03 to 20 Nm-1) are expected for global collective motions with a large number of atoms.5,6 The calculated value for the force constant associated with the breathing motion of RNase A is comparable to that describing the hinge-bending motion between the A-B and C-D dimers in the tetrameric structure of human S-adenosylhomocysteine hydrolase (0.17 Kcal mol-1 deg-2)22 and to that determined for lysozyme hinge bending (0.13 Kcal mol-1 deg-2).23 Furthermore, a concerted motion of the β-sheet region involving an opening and closing of the RNase A active site cleft has been predicted to have a frequency of 2.4 cm-1 by normal-mode analyses calculations.9 Indeed, normal-mode analysis conducted on RNase A reveals that this protein has 12 normal modes between 0 and 10 cm-1.
Results
Discussion
To characterize the differences in the spatial position of the V1 and V2 regions in the RNase A structures reported in the Protein Data Bank, we evaluated the rotation angle (θ) required to best fit V2, after the superimposition of V1, for each RNase A X-ray structure with respect to a reference structure (see Experimental Section). For each structure, the root-meansquare deviation (RMSD) of V1 (RMSD_V1), and of V2 after the superimposition of V1 (RMSD_V2i) and after the rotation of the θ angle required to best-fit V2 (RMSD_V2f), and the value of θ were calculated. In all the analyzed structures, the values of RMSD_V1 and RMSD_V2f are both very low and are close to 0.2 ( 0.1 Å. This means that the structure of the V1 and V2 regions is strictly preserved. The θ rotation decreases the RMSD for V2 from 1.0 ( 0.4 Å (RMSD_V2i) to 0.2 ( 0.1 Å (RMSD_V2f). The standard deviation of the distribution is 2.4°. In almost all the analyzed structures, the orientation of the rotation axis is reasonably well preserved. The θ distribution is shown in histogram form in Figure 2. On the assumption that experimentally observed distribution of RNase A structures could provide a collection of snapshots of the protein movement, these data were substituted into a Boltzmann-type distribution and ∆E/RT values were calculated. This is based on the hypothesis that packing contacts in different crystal forms and different experimental conditions have on average random effects on the molecular structure3 and that the energy associated with intermolecular interactions in crystals is generally smaller than that involved in the native conformational changes of the protein. The ∆E/RT values as a function of the rotation angle θ were then used to obtain a picture of the potential energy associated with the breathing motion of RNase A (Figure 3). In this drawing, the position of the minimum, which depends on the arbitrarily chosen reference structure, was translated at the origin. The θ angle distribution shown in Figure 2 and the calculated values of the energy reported in Figure 3 indicate the expected anharmonicity of the breathing motion. Thus, we fitted the curve
A surprising number of different interactions and structural features in proteins display Boltzmann-like behavior.24 Example of distributions that conform to the Boltzmann hypothesis include hydrogen bonds,25 ion pairs,26 occurrence of residues in secondary structures,27 buried-exposed distribution of side chains,28 size distribution of cavities28 and distribution of side chain conformations,29 occurrence of cis and trans prolines,30 and so on. When free-energy values were estimated using the Boltzmann hypothesis on these data, good correlation with values measured by conventional physical methods has been found. Finkelstein et al. have presented a plausible rationalization of why protein structures at different levels can be described by Boltzmann law,28 but why the appropriate value of the temperature used to describe the distribution of protein interactions and structural parameters corresponding to 300 K remains unexplained.29 Although the application of Boltzmann distribution for the evaluation of quantitative energy-related relationship from statistical analysis is debated,31 using Boltzmann law without a justification is common.29 Here, we have used the Boltzmann hypothesis on a set of data derived from 132 crystal structures of RNase A. We believe that the experimentally observed distribution of RNase A structures could be related to that of a thermal equilibrium ensemble. In this respect, we have previously demonstrated that the conformational space sampled by available X-ray structures of RNase A well agrees with that calculated on the basis of a molecular dynamics study performed at the constant temperature of 300 K.15 Moreover, although our data are based on an ensemble of a limited number of protein structures, the calculated values for the force constant and the wave number associated with the breathing motion of RNase A are in good agreement with previous observations and with theoretical predictions on the basis of normal-mode analysis. These findings support the recent hypothesis that even a modest number of experimentally determined protein structures capture a representative subset of the true native state ensemble32 and once again confirm the validity of the Boltzmann hypothesis.
5486 J. Phys. Chem. B, Vol. 111, No. 19, 2007 Acknowledgment. The work was financially supported by Ministero dell’Istruzione, dell’Universita` e della Ricerca (FIRB RBNE03B8KK) and by Regione Campania (Legge 5). The authors are grateful to Marco Grassi for performing data collection and to Giuseppe Graziano (Dipartimento di Scienze Biologiche ed Ambientali, Universita` del Sannio, Benevento, Italy) for a critical reading of the manuscript. References and Notes (1) Zoete, V.; Michielin, O.; Karplus, M. J. Mol. Biol. 2002, 315, 21. (2) van Aalten, D. M.; Conn, D. A.; de Groot, B. L.; Berendsen, H. J.; Findlay, J. B.; Amadei, A. Biophys. J. 1997, 73, 2891. (3) Zhang, X. J.; Wozniak, J. A.; Matthews, B. W. J. Mol. Biol. 1995, 250, 527. (4) Hayward, S. Proteins: Struct., Funct., Genet 1999, 36, 425. (5) Zaccai, G. Science 2000, 288, 1604. (6) Bicout, D. J.; Zaccai, G. Biophys. J. 2001, 80, 1115. (7) Brown, K. G.; Erfurth, S. C.; Small, E. W.; Peticolas, W. L. Proc Natl. Acad Sci. U.S.A. 1972, 69, 1467. (8) Bartunik, H. D.; Jolles, P.; Berthou, J.; Dianoux, A. J. Biopolymers 1982, 21, 43. (9) Levitt, M.; Sander, C.; Stern, P. S. J. Mol. Biol. 1985, 181, 423. (10) Cao, Z. W.; Chen, X.; Chen, Y. Z. J. Mol. Graphics Modell. 2003, 21, 309. (11) Rasmussen, B. F.; Stock, A. M.; Ringe, D.; Petsko, G. A. Nature 1992, 357, 423. (12) Kartha, G.; Bello, J.; Harker, D. Nature 1967, 213, 862. (13) Berisio, R.; Sica, F.; Lamzin, V. S.; Wilson, K. S.; Zagari, A.; Mazzarella, L. Acta Crystallogr, Sect. D 2002, 58, 441. (14) Vitagliano, L.; Merlino, A.; Zagari, A.; Mazzarella, L. Proteins 2002, 46, 97.
Merlino et al. (15) Merlino, A.; Vitagliano, L.; Ceruso, M. A.; Mazzarella, L. Proteins 2003, 53, 101. (16) Vitagliano, L.; Adinolfi, S.; Riccio, A.; Sica, F.; Zagari, A.; Mazzarella, L. Protein Sci. 1998, 7, 1691. (17) Sica, F.; Di Fiore, A.; Merlino, A.; Mazzarella, L. J. Biol. Chem. 2004, 279, 36753. (18) Merlino, A.; Vitagliano, L.; Ceruso, M. A.; Di Nola, A.; Mazzarella, L. Biopolymers 2002, 65, 274. (19) Oleinikova, A.; Sasisanker, P.; Weingartner, H. J. Phys. Chem. 2004, 108, 8467. (20) Berman, H. M.; Bhat, T. N.; Bourne, P. E.; Feng, Z.; Gilliland, G.; Weissig, H.; Westbrook, J. Nat. Struct. Biol. 2000, 7 (Suppl. 7), 957. (21) Vitagliano, L.; Merlino, A.; Zagari, A.; Mazzarella, L. Protein Sci. 2000, 9, 1217. (22) Wang, M.; Borchardt, R. T.; Schowen, R. L.; Kuczera, K. Biochemistry 2005, 44, 7228. (23) Brooks, C. L., III; Karplus, M.; Pettitt, B. M. The Proteins: A theoretical perspectiVe of dynamics, structure and thermodynamics; Wiley and Sons: New York, 1988. (24) Shortle, D. Protein Sci. 2003, 12, 1298. (25) Sippl, M. J. J. Comput.-Aided Mol. Des. 1993, 7, 473. (26) Bryant, S. H.; Lawrence, C. E. Proteins 1991, 9, 108. (27) Chou, P. Y.; Fasman, G. D. AdV. Enzymol. Relat. Areas Mol. Biol. 1978, 47, 45. (28) Finkelstein, A. V.; Badretdinov, A.; Gutin, A. M. Proteins 1995, 23, 142. (29) Butterfoss, G. L.; Hermans, J. Protein Sci. 2003, 12, 2719. (30) MacArthur, M. W.; Thornton, J. M. J. Mol. Biol. 1991, 218, 397. (31) Burgi, H. B.; Dunitz, J. D. Acta Crystallogr., Sect. B 1988, 44, 445. (32) Best, R. B.; Lindorff-Larsen, K.; DePristo, M. A.; Vendruscolo, M. Proc Natl. Acad. Sci. U.S.A. 2006, 103, 10901.
