Calculation of Proteins' Total Side-Chain Torsional Entropy And
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doi:10.1016/j.jmb.2009.05.068 J. Mol. Biol. (2009) 391, 484–497 Available online at www.sciencedirect.com Calculation of Proteins’ Total Side-Chain Torsional Entropy and Its Influence on Protein–Ligand Interactions Kateri H. DuBay and Phillip L. Geissler⁎ Department of Chemistry, Despite the high density within a typical protein fold, the ensemble of University of California at sterically permissible side-chain repackings is vast. Here, we examine the Berkeley, Berkeley, CA 94720, extent of this variability that survives energetic biases due to van der Waals USA interactions, hydrogen bonding, salt bridges, and solvation. Monte Carlo simulations of an atomistic model exhibit thermal fluctuations among a Chemical Sciences Division, diverse set of side-chain arrangements, even with the peptide backbone Lawrence Berkeley National Lab, fixed in its crystallographic conformation. We have quantified the torsional Berkeley, CA 94720, USA entropy of this native-state ensemble, relative to that of a noninteracting Physical Biosciences Division, reference system, for 12 small proteins. The reduction in entropy per Lawrence Berkeley National Lab, rotatable bond due to each kind of interaction is remarkably consistent Berkeley, CA 94720, USA across this set of molecules. To assess the biophysical importance of these fluctuations, we have estimated side-chain entropy contributions Received 12 January 2009; to the binding affinity of several peptide ligands with calmodulin. received in revised form Calculations for our fixed-backbone model correlate very well with 20 May 2009; experimentally determined binding entropies over a range spanning more accepted 22 May 2009 than 80 kJ/(mol·308 K). Available online © 2009 Elsevier Ltd. All rights reserved. 28 May 2009 Keywords: side-chain entropy; configurational entropy; side-chain fluctuations; Edited by M. Levitt protein–ligand binding; protein thermodynamics Introduction niques in particular resolve fluctuations at the level of single bond vectors in both the backbones and – Native protein conformations are extremely dense, side-chains of folded proteins.3 5 The Lipari–Szabo with packing fractions comparable to those of order parameters6 they determine, which increase 1 2 2 organic crystals. This observation motivated in from Saxis =0 to Saxis =1 as rotational motion early studies of protein structure and dynamics a becomes restricted, report on the range of picose- jigsaw-puzzle notion, in which amino acid side cond to nanosecond dynamics for backbone amide chains of a folded structure become fixed in a unique and side-chain methyl groups. Computational stu- spatial arrangement by steric interactions with their dies suggest that order parameters lower than 0.8 neighbors. Still today, many computational proce- point to transitions between multiple rotameric dures that explore side-chain packing strive to states in addition to the inevitable vibrations about identify a single native configuration.2 optimal torsional angles.7,8 The three-dimensional structure of a protein, how- Side-chain methyl group order parameters often lie b 2 b 3 ever, can fluctuate considerably. Large-scale mo- in the range 0.2 Saxis 0.8, indicating extensive tions involve partial or full unfolding and backbone exploration of different rotameric states. These results hinge motions, but subtle structural variation on are corroborated by dipolar coupling measurements, smaller scales has been highlighted by several suggesting that side chains substantially populate experimental measurements. NMR relaxation tech- different rotameric states within the ensemble of folded configurations.9,10 Evidence for alternative side-chain conformations has even been found in *Corresponding author. E-mail address: electron density maps from crystallography expe- [email protected]. riments.11 The data accumulating from such studies Abbreviations used: MC, Monte Carlo; MD, molecular paint a consistent picture: Residual side-chain fluc- dynamics; LJ, Lennard–Jones; SB, salt bridge; HB, tuations in the native-state ensemble are distributed hydrogen bond; IS, implicit solvent; SASA, solvent- heterogeneously throughout the protein; side-chain accessible surface area; CaM, calmodulin; PYP, bond vectors fluctuate more significantly than do photoactive yellow protein; WL, Wang–Landau. those along the backbone; and the entropy associated 0022-2836/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. Calculation of Side-Chain Torsional Entropy 485 with such fluctuations is likely to be a significant in several systems, entropy changes figure promi- player in protein thermodynamics.4,12 nently in tuning protein binding affinities.22,23 In the Computational studies focusing on geometric case of stromelysin 1 binding to the N-terminal do- aspects of side-chain packing have reconciled the main of tissue inhibitor of metalloproteinases 1, they evidence for significant torsional fluctuations with even overcome a substantially unfavorable enthalpy constraints due to steric interactions in a dense of binding.24 Implicating the involvement of side environment.13 Much as in a dense liquid, volume chains in these phenomena, entropies inferred from exclusion reduces the diversity of accessible config- NMR order parameters correlate strongly with calo- urations greatly, but by no means completely. rimetrically determined binding entropies for cal- Nearly 1020 distinct side-chain conformations were modulin (CaM) and several peptide ligands.12 We determined to satisfy hard-core constraints in a 125- find even better agreement between binding en- residue protein with native backbone structure.13 To tropy measurements and calculated values based on what degree non-steric interactions further reduce the methods we have developed. This comparison is this variability is not at all clear a priori. Populating discussed in detail in Results and Discussion. even a very small fraction of the geometrically acceptable arrangements would be sufficient to Model allow for significant contributions to free energies of folding and ligand binding. In developing a theoretical approach, we are Mean field theories,14 various interpretations of guided by the notion that side-chain rearrangements – molecular dynamics (MD) simulations,15 18 and within a protein's native state are not strongly me- – several Monte Carlo (MC) approaches13,19 21 have diated by motions of the peptide backbone. Physi- all been used to estimate the residual entropy of side- cally, we expect that once the molecule has folded, it is chain rotations in folded proteins. Each of subject to global constraints of high packing fraction these approaches, however, is limited by underlying that vary little with small-amplitude backbone fluc- approximations or formidable practical challenges. tuations. Empirically, we note that correlations Mean field approaches, by definition, do not account observed between backbone NMR order parameters, 2 2 for a complete range of thermal fluctuations; straight- S , and their associated side-chain parameters, Saxis, forward MD simulations can explore only rearrange- are weak.25 Following Kussell et al.,13 we thus adopt a ments that occur on computationally accessible time model in which the peptide backbone is fixed in its scales. MC methods are similarly hindered by crystallographically determined conformation. As a sampling difficulties intrinsic to such tightly packed result, applications of our methods are limited to systems. As a compromise, entropies are sometimes proteins whose native structures have been deter- calculated separately for single residues or small mined with high resolution. groups of neighboring residues while keeping other The sole degrees of freedom in our calculations are residues fixed.19 Studies that do confront the full dihedral angles χ for rotatable side-chain bonds with combinatorial problem, allowing all side chains to heavy-atom (i.e., non-hydrogen) substituents. Other rotate simultaneously, have neglected potentially variables are known to influence side-chain entropy,15 important contributions from intra-rotameric mo- but torsional entropy alone is thought to provide a tions20 or have considered geometric effects indepen- good approximation.19 Natural amino acids possess dent of non-steric interactions.13,21 no more than a handful of such dihedral degrees of In this article, we present a new approach for freedom. Alanine, for example, has none, while lysine estimating side-chain torsional entropy. Building on and arginine possess the largest number (four). As in a algorithms developed by Kussell et al.,13 our calcula- simple molecule such as propane, local bonding tions are enabled by enhanced MC methods and a energetics bias such angles to lie in one of typically schematic treatment of forces due to sterics, van der three ranges. For classification purposes, we consider Waals interactions, hydrogen bonding, salt bridges these ranges as discrete rotameric states, each with an (SBs), and solvation. Through this combination, we ideal angular value θ. We do, however, permit achieve thorough sampling of thermal fluctuations, deviations from these ideal angles, ϕ=χ−θ.Weand incorporate fully coupled rotations of all residues, others have found them to be essential for accom- and address a comprehensive set of physical interac- modating tightly packed rearrangements.13,26 The tions. Model outlines our approach and the physical intrinsic energetic penalty Edihedrals limiting such perspectives underlying it. Results and Discussion fluctuations