Polymer 43 %2002) 451±459 www.elsevier.com/locate/polymer Residue coordination in proteins conforms to the closest packing of spheresq Zerrin Bagcia, Robert L. Jerniganc, Ivet Bahara,b,* aChemical Engineering Department and Polymer Research Center, Bogazici University, Istanbul, Turkey bCenter ofComputational Biology & Bioinformatics, and Department ofMolecular Genetics & Biochemistry, School ofMedicine, University ofPittsburgh, Pittsburgh, PA 15213 USA cMolecular Structure Section, Laboratory ofExperimental and Computational Biology, Division ofBasic Sciences, National Cancer Institute, National Institutes ofHealth, Bethesda, MD 20892-5677, USA Abstract Coarse-grained protein structures have the unusual propertyof manifesting a greater regularitythan is evident when all atoms are considered. Here, we follow proteins at the level of one point per residue. We con®rm that lattices with large coordination numbers provide better ®ts to protein structures. But, underlying these protein structures, there is an intrinsic geometry that closely resembles the face- centered-cubic %fcc) lattice, in so far as the coordination angles observed in clusters of near neighboring residues are concerned. While the fcc lattice has 12 neighbors, the coordination number about anygiven residue in a protein is usuallysmaller; however, the neighbors are not distributed in a uniform, less dense way, but rather in a clustered dense way, occupying positions closely approximating those of a distorted fcc packing. This packing geometry is a direct manifestation of the hydrophobic effect. Surprisingly, speci®c residues are clustered with similar angular geometry, whether on the interior or on the exterior of a protein. q 2001 Published byElsevier Science Ltd. Keywords: Packing in proteins; Face-centered-cubic lattice; Uniform packing density 1. Introduction other than those between the nearest neighbor bonds, which are usuallyconsidered on Ramachandran plots. They Proteins are compact linear molecules with some regula- are stabilized byso-called higher order interactions. For rities in their polypeptide backbone. Foremost among these example, a-helices are stabilized bythe hydrogenbonds are two regular motifs commonlyobserved: a-helices and between the CyO and N±H groups of the respective b-strands. Studies on the regularities of the polypeptide residues i and i 1 4; while sheet formation stabilizes the backbone began with the pioneering work of Ramachandran b-strands bythe association of polar groups of segments and coworkers [1]. The occurrence of a-helices and even further along the sequence. Thus, non-bonded b-strands in native proteins could be attributed to the acces- interactions do playa major role in stabilizing secondary sibilityof the corresponding rotational angles seen in the structures. Ramachandran maps. Yet, the calculated energies of a The importance of non-bonded interactions in proteins single residue for the rotational angles do not completely goes far beyond the stabilization of secondary structures. account for the high frequencyof occurrence of these Proteins are unique in that each sequence of amino acids secondarystructures in folded proteins. These secondary selects one or sometimes a few three-dimensional structure structures are indeed stabilized signi®cantlybyinteractions and the non-bonded interactions are accepted to be the major determinant of the different folds taken bydifferent q This paper was originallysubmitted to Computational and Theoretical amino acid sequences. Polymer Science and received on 14 December 2000; received in revised Thus, in contrast to the statistical analysis of polymers in form on 28 February2001; accepted on 28 February2001. Following the which signi®cant information on conformational behavior incorporation of Computational and Theoretical Polymer Science into can be extracted from the rotational isomeric state distribu- Polymer, this paper was consequentlyaccepted for publication in Polymer. * Corresponding author. Address: School of Medicine, Universityof tion of individual %or pairwise dependent) bonds, analyzing Pittsburgh, Kaufmann Building, Suite 601, 3471 Fifth Avenue, Pittsburgh, proteins' conformational preferences necessitates a thor- PA 15213, USA. Tel.: 11-412-648-6671/190-212-2631540, ext. 2003; ough understanding of the preferences for the non-bonded fax: 11-412-648-6676/190-212-2575032. interactions of amino acids. Not surprisingly, a large E-mail addresses: [email protected] %I. Bahar), number of computational and theoretical studies have [email protected] %I. Bahar), [email protected] %I. Bahar). been directed at characterizing the empirical potentials for 0032-3862/02/$ - see front matter q 2001 Published byElsevier Science Ltd. PII: S0032-3861%01)00427-X 452 Z. Bagci et al. / Polymer 43 ;2002) 451±459 inter-residue interactions [2]. The Protein structure Data that connect everyother site on a tetrahedral lattice, and it %PDB) [3] where the atomic coordinates of all proteins has indeed been utilized for locating everyother backbone determined byX-raycrystallographyor NMR spectroscopy atom in polyethylene-like chains. Monte Carlo %MC) are deposited has been exploited as an important source for simulations of polyethylene melts performed on the 2nnd extracting information on inter-residue potentials. The lattice and reverse mapping of the equilibrium structures to essential approach is to use the so-called inverse Boltzmann full atomic continuous space yielded several properties, law, i.e. calculate effective potentials, or free energies, from including the cohesive energydensities, that were in close the observed frequencies of amino acid pairs, assuming agreement with experimental data [19]. that the examined PDB structures contain a complete In the present coarse-grained study, the virtual bond equilibrated set of residue pairs. model is adopted for modeling the backbone. The geometric Although signi®cant efforts have been devoted to modeling features of this model conform to those of an fcc lattice, inter-residue interaction potentials as a function of their spatial hence the special suitabilityof the fcc directions for describ- separation, much less attention has been given to the orienta- ing the polypeptide backbone. However, the tendency of tional preferences of amino acids or to the coordination residues to be coordinated in conformitywith the fcc directions of amino acids. Examinations of such angular geometryis stronger than the preference that would be preferences revealed [4±9], on the other hand, some non- imparted bythe geometric suitabilityfor the backbone randomness, which could signal the occurrence of some alone. Furthermore, this tendencyis discerned even when generic preferences for the packing of residues in proteins. bonded neighbors are excluded. The fact that the fcc pack- Clearly, the existence of some regularity in residue ing is the closest packing geometryfor identical spheres packing would be of great utilityfor reducing the computa- brings into consideration the possible drive for maximizing tional time and increasing the accuracyof conformational packing densityas a possible %origin for the appearance of searches. Presently, methods of bioinformatics can predict the fcc geometry. Besides, even the formation of helices in secondarystructure with up to 80% accuracy[10], and proteins and DNA has been recentlyshown to be a conse- signi®cant progress can be made in tertiarystructure quence of optimal packing requirement [20±21]. Clearly, prediction bycombining the tools for predicting secondary residue side chains differ in their size and shape. But at a structure with those ef®cientlydiscriminating between coarse-grained scale where such differences are ignored, an alternative non-bonded interaction geometries. intrinsic bias towards a universal geometrythat provides the A common approach for reducing the conformational space most ef®cient packing for identical spheres can be of macromolecules is to adopt coarse-grained lattice models. discerned, which suggests that the drive for optimizing Two major computational advantages of lattice models are packing densitymaybe stronger than previouslythought. to discretize the conformational space, and to be amenable It can be concluded that close packing, though indirectly, is to integer algorithms. Lattice models not onlyprovide an a result of hydrophobic effects. The reason is that hydro- ef®cient means of generating conformations and perturbing phobic interactions are generallynon-speci®c; theydo not them to investigate the dynamic characteristics, but also distinguish between different interactions geometries, and provide insights about the global behavior of molecular thus could conform to anygeometrythat would maximize systems that could not be explored in atomic detail [11]. the packing density. On the other hand, the interactions In the present work, the possible regularityin the between polar %or hydrophilic) groups require speci®c asso- association of non-bonded residues in folded proteins will ciation of functional groups, along speci®c directions, and be investigated. The representation of the coordination ofre- do not necessarilyconform to a uniform, symmetric packing sidues in terms of different lattice geometries will be geometry. The fact that a regular packing is approximated explored. It will be shown that a diversityof lattice geome- signals the importance of hydrophobic effects. tries, including simple lattices, such as simple cubic %sc), can be adopted for a satisfactorydescription