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Entropy Localization in Proteins Da-Wei Li, Scott A. Showalter,† and Rafael Bru¨schweiler* Chemical Sciences Laboratory, Department of Chemistry and Biochemistry and National High Magnetic Field Laboratory, Florida State UniVersity, Tallahassee, Florida 32306, United States ReceiVed: October 15, 2010
The configurational entropy of a protein is under physiological conditions a major contributor to the free energy. Its quantitative characterization is therefore an important step toward the understanding of protein function. The configurational entropy of the oncoprotein MDM2, whose determination is a challenge by experiment alone, is studied here by means of 0.4 µs molecular dynamics computer simulations in both the presence and absence of the p53-peptide ligand. By characterizing protein motions in dihedral angle space, it is found that the motional amplitudes change considerably upon ligand binding while correlations between dihedral angle motions are remarkably well conserved. This applies for backbone and side-chain dihedral angle pairs at both short- and long-range distance to the binding site. As a direct consequence, the change of the configurational entropy can be decomposed into a sum of local contributions. This significantly facilitates the understanding of the relationship between protein dynamics and thermodynamics, which is important, for example, in the context of protein-ligand and protein-protein interactions. The findings also have implications for the direct derivation of entropy changes from site-specific dynamics measurements as afforded by NMR spectroscopy. Introduction Most proteins adopt a well-defined three-dimensional structure, but in order to perform their function they undergo significant conformational dynamics. Such motions can play a direct mechanistic role,1 and at the same time, they can impact the stability of protein substates, such as ligand-bound vs free, through their effect on the configurational entropy.2 A complete description of protein dynamics is challenging due the large number of coordinates or degrees of freedom (DOF) that can move on a wide range of time scales. Moreover, the motions of different DOF are not necessarily independent; that is, they can be dynamically correlated, which has potentially important consequences for the configurational entropy. To unravel this complexity, computational studies are playing an important role in addition to experimental observations. The nature and amount of motional correlations critically depends on the choice of the DOF. For example, depending on the reference frame, the motion about a single dihedral angle located in the linker between two rigid domains can lead to highly collective changes of the Cartesian atomic coordinates within each domain, even though in dihedral angle space, dynamic (cross-)correlations are not present. This illustrates the complementarity of the description of protein dynamics in Cartesian versus internal coordinates.3 To study correlation effects, principal component analysis (PCA)4 is often applied to a molecular dynamics (MD) trajectory that decomposes the covariance matrix of a selected set of DOF into orthogonal eigenmodes, which can be treated analogously to normal modes for the estimation of the configurational entropy.5,6 PCA is usually performed in Cartesian coordinate space, but Cartesian coordinates are not optimal to represent curvilinear motion of atoms associated with large dihedral angle * Corresponding author. Phone: 850-644-1768. Fax: 850-644-8281. E-mail:
[email protected]. † Current address: Department of Chemistry, Pennsylvania State University, 104 Chemistry Building, University Park, PA 16802.
rotations. This makes the study of protein dynamics in dihedral angle space attractive because these soft DOF represent in compact form the dominant motions.7-9 Indeed, over recent years, entropy characterization in internal coordinates has gained traction.8,10,11 For example, Demchuk and co-workers12,13 have estimated the absolute internal configurational entropy of molecules with one or two degrees of freedom. Gilson and co-workers14 have shown that errors tend to be magnified in the quasi-harmonic approach when using Cartesian rather than internal coordinates, also because of the presence of mode dependencies.15 Dihedral angles have been shown to be well suited to describe the free energy surface of polypeptides,7,16 with the ability to identify free energy minima along with the transition paths among them. Recently, we analyzed dynamics correlations of dihedral angles in long molecular dynamics (MD) trajectories of two proteins, ubiquitin and calbindin D9k. Without exception, all mobile dihedral angle pairs with sizable dynamics correlations are found to be at short-range distance ( 72, S(1) is even more accurate than S(3).
We rationalize this behavior by the large number of correction terms in S(3) that grows with O(L3). Each of these terms carries a statistical error, and their net effect counteracts or even exceeds the benefit of the higher-order correction. Therefore, in the case of a large number of dihedral angles, L, encountered in mediumsized to large proteins, S(1) (which is equivalent to S1d) provides a good and stable estimate of the entropy change. A possible improvement of the S(2) or S(3) methods eliminates low correlation coefficient pairs, which allows a drastic reduction in the number of dihedral angle pairs and triples. For example, if only pairs with R2 > 0.1 are included, the error for the full length protein reduces to 0.65kB from 1.02kB of the original S(2), an error that is still higher than the one of 0.35kB in S(1). It is unclear in how far such as an approach is more widely applicable. Comparison with the Side-Chain Histogram Entropy Method. In previous sections, we found that most inter-residue side chain dihedral angle correlations are very weak, and sizable correlations are usually between dihedrals that belong to the same residue. Therefore, each side chain can be treated in good approximation separate from the others. Because of the relatively small number of dihedral angles per side chain, the configurational entropy can be determined using a multidimensional histogram method (see eq 18). To obtain bin populations with low statistical errors, it is critical that the bin size is sufficiently large (36° in this case). As a direct consequence, the resolution of the probability distribution is low and differences in the distributions below the resolution limit will go undetected. Figure 8 shows ∆Shist values (black symbols) for each side chain of MDM2, where ∆Shist is defined as Sapo - Sbound. For comparison, the results for S1d are also indicated (red symbols). The sum of the contributions from all residues is 1.69kB for ∆Shist; it is 1.77kB for S(1). This confirms the good accuracy of the S(1) method. When the ∆SQH contributions from the backbone dihedral angles are coadded to ∆Shist for the side-chain contributions, ∆S is 3.54kB, which compares quite well with 3.97kB (11% difference) from S1d for the backbone plus S(1) for the side chains. Discussion Considering that the configurational entropy is a key component of the free energy, the availability of a robust yet efficient computational method to estimate the configurational entropy difference between two different protein states is an important problem in computational biophysics. Over the years, methods have been proposed that use different types of approximations to address correlation effects5,8,12,13,39 Early methods mostly employed Cartesian coordinates, although geometric constraints imposed by stiff bond lengths and bond angles introduce strong
Entropy Localization in Proteins correlations between coordinates. For example, Cartesian treatment of the three hydrogen atoms in a methyl group shows that inclusion of second-order correlation effects is not sufficient for a correct entropy estimate.35 In recent years, methods that use internal coordinates, and in particular dihedral angles, have gained in popularity, since these DOF are uncorrelated or only weakly correlated. The assessment of the importance of correlation effects of internal coordinates on the configurational entropy of proteins critically depends on the statistical accuracy of these DOF.10 The availability of trajectories that reach into the hundreds of nanosecond range and beyond now provide the accuracy needed to address this question for an increasingly complete set of dihedral angles. In the present study, out of 350 dihedral angles, only 32 had to be excluded due to incomplete convergence. Some of the dihedral angles that were excluded may report on functionally interesting concerted motions. However, their transient behavior during the course of the trajectories introduces large statistical uncertainties, which precludes the quantitative determination of their entropic contributions. The use of significantly longer trajectories will help to overcome such limitations. The use of internal coordinates introduces a complication; namely, the Jacobian associated with the coordinate transformation becomes a function of the internal coordinates and must be taken into account explicitly in the entropy estimation. Go and Scheraga40 have proposed a computationally tractable form for the partition function for which the Jacobian of the transformation between Cartesian and internal coordinates is constant, depending only on the hard DOF (i.e., bond lengths and bond angles), allowing the estimation of the entropy difference between two states from the soft DOF (i.e., dihedral angles) plus an analytical contribution from the hard DOF. This approximation has been shown to be remarkably accurate in recent in silico studies of peptides41 and proteins.18 Dihedral angles can be dynamically correlated, although typically to a much smaller extent than their Cartesian counterparts. It is therefore important to know how correlated motions affect the entropy difference as a function of temperature18 or between two states representing a ligand-bound and an apo state as for the MDM2 protein system studied here. Our results suggest that when correlation effects between two states are largely conserved, their effect on the entropy difference is negligible. This can be shown analytically for the case of bivariate Gaussian fluctuations, and it is confirmed by numerical analysis of the MD trajectories. For MDM2, we find that the extent of correlation is well conserved between the bound and free states. Hence, the entropy difference, ∆S, is determined simply by the sum of contributions due to the changes of the fluctuation amplitudes of individual dihedral angles (∆S1d), irrespective of the amount of their (conserved) mutual correlations. Within the framework of thermodynamic perturbation theory, the decomposition of free energies and entropies into groups of different interactions has been demonstrated previously.36,42 Because a pair of dihedral angles can share the same interactions, the assignment of entropy to individual dihedral angles proposed here is conceptually different from the above type of decomposition. Complete neglect of correlations may seem severe, but the error on entropy estimation is typically small. This result is consistent with free energy calculations for both peptide and protein systems.18,41 It is interesting to ask why dihedral angle correlations are rather well conserved in the peptide binding process. The dynamics of the dihedral angles and their correlations is governed by a multitude of factors, such as the average 3D structure of the protein, local secondary structure, local
J. Phys. Chem. B, Vol. 114, No. 48, 2010 16043 contacts, and solvent effects. The overall structure of MDM2 is mostly unchanged in the binding process, whereas p53 binding perturbs the local packing near the binding site, an effect that leads to only small changes in other parts of the protein.28 If the topology markedly changes between two protein states, which is the case, for example, when one of the two states is unfolded, conservation of correlations is expected to break down, and hence, the entropy becomes nonlocal. In MDM2, the correlations among backbone dihedral angles experience, on average, smaller changes than do the side-chain dihedral angles. This might be explained by the fact that backbone dihedral angles are constrained from both their Nand C-termini and, therefore, are better protected against perturbations, whereas side chain dihedral angles are often less restrained toward the end and, thus, more susceptible to interactions with the environment. Even near the binding site, alterations in correlation strength are very moderate, and entropy changes due to correlation changes represent only a small contribution compared with the change of entropy that stems from the changes in motional amplitudes of individual dihedral angles. Taken together, the results of this study suggest that protein entropy and its changes can be viewed as a sum of localized contributions of individual DOF. Such a local view of the configurational entropy makes this central thermodynamic function more easily accessible, both conceptually and in relationship to site-resolved experimental data. In fact, such a localized view of entropy has direct implications for the thermodynamic interpretation of NMR relaxation measurements, converting changes in NMR order parameters into changes in the configurational entropy. Due to the local nature of the NMR information, the interpretation of the extracted NMR order parameters invariably assumes decomposition of the total entropy change into local contributions. Through detailed computation, the present study provides a quantitative theoretical justification. Acknowledgment. This work was supported by grant MCB0918362 of the National Science Foundation. References and Notes (1) Boehr, D. D.; McElheny, D.; Dyson, H. J.; Wright, P. E. Science 2006, 313, 1638–1642. (b) Mittermaier, A.; Kay, L. E. Science 2006, 312, 224–228. (c) Henzler-Wildman, K. A.; Lei, M.; Thai, V.; Kerns, S. J.; Karplus, M.; Kern, D. Nature 2007, 450, 913–916. Tsai, C. J.; Del Sol, A.; Nussinov, R. Mol. Biosyst. 2009, 5, 207–216. (2) Gruenberg, R.; Nilges, M.; Leckner, J. Structure 2006, 14, 683– 693. (3) Mendez, R.; Bastolla, U. Phys. ReV. Lett. 2010, 104, 228103. (4) Levy, R. M.; Karplus, M.; Kushick, J.; Perahia, D. Macromolecules 1984, 17, 1370–1374. (b) Garcia, A. E. Phys. ReV. Lett. 1992, 68, 2696– 2699. (c) Amadei, A.; Linssen, A. B. M.; Berendsen, H. J. C. Proteins 1993, 17, 412–425. (5) Karplus, M.; Kushick, J. N. Macromolecules 1981, 14, 325–332. (6) Missimer, J. H.; Steinmetz, M. O.; Baron, R.; Winkler, F. K.; Kammerer, R. A.; Daura, X.; van Gunsteren, W. F. Protein Sci. 2007, 16, 1349–1359. (7) Mu, Y. G.; Nguyen, P. H.; Stock, G. Proteins 2005, 58, 45–52. (8) Wang, J.; Bru¨schweiler, R. J. Chem. Theory Comput. 2006, 2, 18– 24. (9) Altis, A.; Otten, M.; Nguyen, P. H.; Hegger, R.; Stock, G. J. Chem. Phys. 2008, 128, 245102. (10) Killian, B. J.; Kravitz, J. Y.; Somani, S.; Dasgupta, P.; Pang, Y. P.; Gilson, M. K. J. Mol. Biol. 2009, 389, 315–335. (11) Chang, C. E. A.; Chen, W.; Gilson, M. K. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 1534–1539. (12) Hnizdo, V.; Fedorowicz, A.; Singh, H.; Demchuk, E. J. Comput. Chem. 2003, 24, 1172–1183. (13) Darian, E.; Hnizdo, V.; Fedorowicz, A.; Singh, H.; Demchuk, E. J. Comput. Chem. 2005, 26, 651–660.
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(14) Chang, C. E.; Chen, W.; Gilson, M. K. J. Chem. Theory Comput. 2005, 1, 1017–1028. (15) Prompers, J. J.; Bru¨schweiler, R. J. Phys. Chem. B 2000, 104, 11416–11424. (16) Maisuradze, G. G.; Leitner, D. M. Proteins 2007, 67, 569–578. (17) Li, D. W.; Meng, D.; Bru¨schweiler, R. J. Am. Chem. Soc. 2009, 131, 14610–14611. (18) Li, D. W.; Bru¨schweiler, R. Phys. ReV. Lett. 2009, 102, 118108. (19) Akke, M.; Bru¨schweiler, R.; Palmer, A. G. J. Am. Chem. Soc. 1993, 115, 9832–9833. (b) Yang, D.; Kay, L. E. J. Mol. Biol. 1996, 263, 369– 382. (c) Li, Z.; Raychaudhuri, S.; Wand, A. J. Protein Sci. 1996, 5, 2647– 2650. (20) Igumenova, T. I.; Frederick, K. K.; Wand, A. J. Chem. ReV. 2006, 106, 1672–1699. (21) Jarymowycz, V. A.; Stone, M. J. Chem. ReV. 2006, 106, 1624– 1671. (22) Wrabl, J. O.; Shortle, D.; Woolf, T. B. Proteins 2000, 38, 123– 133. (b) Prabhu, N. V.; Lee, A. L.; Wand, A. J.; Sharp, K. A. Biochemistry 2003, 42, 562–570. (c) Li, D. W.; Bru¨schweiler, R. J. Am. Chem. Soc. 2009, 131, 7226–7227. (23) Buck, M.; Bouguet-Bonnet, S.; Pastor, R. W.; MacKerell, A. D. Biophys. J. 2006, 90, L36–L38. (24) Hornak, V.; Abel, R.; Okur, A.; Strockbine, B.; Roitberg, A.; Simmerling, C. Proteins 2006, 65, 712–725. (25) Klepeis, J. L.; Lindorff-Larsen, K.; Dror, R. O.; Shaw, D. E. Curr. Opin. Struct. Biol. 2009, 19, 120–127. (26) (a) Levine, A. J. Cell 1997, 88, 323–331. (b) Romer, L.; Klein, C.; Dehner, A.; Kessler, H.; Buchner, J. Angew. Chem., Int. Ed. 2006, 45, 6440– 6460. (27) Cho, Y.; Gorina, S.; Jeffrey, P. D.; Pavletich, N. P. Science 1994, 265, 346–355. (b) Kussie, P. H.; Gorina, S.; Marechal, V.; Elenbaas, B.; Moreau, J.; Levine, A. J.; Pavletich, N. P. Science 1996, 274, 948–953. (28) Showalter, S. A.; Bru¨schweiler-Li, L.; Johnson, E.; Zhang, F.; Bru¨schweiler, R. J. Am. Chem. Soc. 2008, 130, 6472–6478. (29) Case, D. A.; Cheatham, T. E.; Darden, T.; Gohlke, H.; Luo, R.; Merz, K. M.; Onufriev, A.; Simmerling, C.; Wang, B.; Woods, R. J. J. Comput. Chem. 2005, 26, 1668–1688. (30) Showalter, S. A.; Bru¨schweiler, R. J. Am. Chem. Soc. 2007, 129, 4158–4159. (b) Showalter, S. A.; Johnson, E.; Rance, M.; Bru¨schweiler, R. J. Am. Chem. Soc. 2007, 129, 14146–14147.
Li et al. (31) Showalter, S. A.; Bru¨schweiler, R. J. Chem. Theory Comput. 2007, 3, 961–975. (32) Ryckaert, J. P.; Ciccotti, G.; Berendsen, H. J. C. J. Comput. Chem. 1977, 23, 327–341. (33) Grasberger, B. L.; Lu, T.; Schubert, C.; Parks, D. J.; Carver, T. E.; Koblish, H. K.; Cummings, M. D.; LaFrance, L. V.; Milkiewicz, K. L.; Calvo, R. R.; Maguire, D.; Lattanze, J.; Franks, C. F.; Zhao, S.; Ramachandren, K.; Bylebyl, G. R.; Zhang, M.; Manthey, C. L.; Petrella, E. C.; Pantoliano, M. W.; Deckman, I. C.; Spurlino, J. C.; Maroney, A. C.; Tomczuk, B. E.; Molloy, C. J.; Bone, R. F. J. Med. Chem. 2005, 48, 909– 912. (34) Bremi, T.; Bru¨schweiler, R.; Ernst, R. R. J. Am. Chem. Soc. 1997, 119, 4272–4284. (35) Killian, B. J.; Kravitz, J. Y.; Gilson, M. K. J. Chem. Phys. 2007, 127, 024107. (36) Brady, G. P.; Sharp, K. A. Curr. Opin. Struct. Biol. 1997, 7, 215– 221. (37) Trbovic, N.; Cho, J. H.; Abel, R.; Friesner, R. A.; Rance, M.; Palmer, A. G. J. Am. Chem. Soc. 2009, 131, 615–622. (38) Levy, R. M.; Karplus, M. Biopolymers 1979, 18, 2465–2495. (b) Johnson, E.; Showalter, S. A.; Bru¨schweiler, R. J. Phys. Chem. B 2008, 112, 6203–6210. (39) (a) Edholm, O.; Berendsen, H. J. C. Mol. Phys. 1984, 51, 1101– 1028. (b) Karplus, M.; Ichiye, T.; Pettitt, B. M. Biophys. J. 1987, 52, 1083– 1085. (c) Schlitter, J. Chem. Phys. Lett. 1993, 215, 617–621. (d) Schafer, H.; Mark, A. E.; van Gunsteren, W. F. J. Chem. Phys. 2000, 113, 7809– 7817. (e) Schafer, H.; Daura, X.; Mark, A. E.; van Gunsteren, W. F. Proteins 2001, 43, 45–56. (f) Cheluvaraja, S.; Meirovitch, H. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 9241–9246. (g) Cheluvaraja, S.; Meirovitch, H. J. Chem. Phys. 2006, 125, 024905. (h) Frederick, K. K.; Marlow, M. S.; Valentine, K. G.; Wand, A. J. Nature 2007, 448, 325–329. (i) Hnizdo, V.; Darian, E.; Fedorowicz, A.; Demchuk, E.; Li, S.; Singh, H. J. Comput. Chem. 2007, 28, 655–668. (j) Minh, D. D. L.; Hamelberg, D.; McCammon, A. J. Chem. Phys. 2007, 127, 154105. (k) Hnizdo, V.; Tan, J.; Killian, B. J.; Gilson, M. K. J. Comput. Chem. 2008, 29, 1605–1614. (40) Go, N.; Scheraga, H. A. Macromolecules 1976, 9, 535–542. (41) Li, D. W.; Khanlarzadeh, M.; Wang, J. B.; Huo, S.; Bru¨schweiler, R. J. Phys. Chem. B 2007, 111, 13807–13813. (42) Brady, G. P.; Szabo, A.; Sharp, K. A. J. Mol. Biol. 1996, 263, 123–125. (b) Boresch, S.; Karplus, M. J. Mol. Biol. 1995, 254, 801–807.
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