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1301 Calculation of Molecular Volumes and Areas for Structures Of __ ~~ ~~ - turc; instcad, thcy arc rcitcratcd in hierarchic tashion ranging from thc For the purpose of this discussion we shall conccntrntc on thc dcfini- whole protein monomer through supcrscconditty s(ructurcs down to indi- tion and calculation of certain geometrical volumes and areas that can bc vidual helices and slrmds. derived from high-resolution structural data. No attempt will be made to The concept of a domain is subject to reinterpretation when consid- give a critical discussion of the use (or potential misuse) of these values. ered in the broader context of its molecular hierarchy. The conventional , .._ The following basic input data are required: large domains cited in the literature (near the top of the hierarchy) are seen to be structural composites, with subparts that are domains in their 1. The list of Cartesian coordinates of the atom centers. own right. Thus, the large, spatially distinct chain segments that can be 2. One or more lists of attributes for each atom: resolved within a protein mayno1 represent strictly autonomous units. a. Assigned van der Waals radii for each atom (required in most but may arise instead in consequence of a concluding step in thc scqucn- procedures). tial folding process. b. Assignedcovalent radius Tor each atom (uscdepends on thc volume algorithm selected). c. Covalent connectivity (needed if b is used, and also lo assemble Acknowledgments atoms into reasonable packing groups) X-Ray coordinafes were provided by the Urookhaven Dak~ U;lnk.!v The work was sup- 3. Choice of radius for a spherical probe. ported by PHD Grant 29458 and by an NIH Research Career Development hwt~rd. In the presently used procedures no allowance is made for errors or fluctuations in the coordinates. The structure is simply a colleclion of 29 F. C. Bernstein, T. G. Koetzle, G. J. B. Williams. E. F. Meyer. Jr.. M. D. Brice. J. R. Rogers, 0. Kennard. T. Shimanouchi. and M. Tasumi. J. Md. Bi~l.112, S3S (1977). points in three-dimensional space, Sensitivity to error, if required. is esli- mated by repeating the calculation with appropriately altered coordinate lists and comparing the results. None of the algorithms place any restric- tions on the position of the atom cenlers. The van der Wttuls Envelope 1301 Calculation of Molecular Volumes and Areas for Because of the diffuse radial distribution of the electron density sur- Structures of Known Geometry rounding any atomic center, the apparent position of the surface of a By FREDERIC M.RICHARDS molecule will depend on the technique used to examine it. For chemically bonded atoms the distribution is not spherically symmetric nor are the properties of such atoms isotropic. In spite of all this the use of the hard Introduction sphere model has a venerablehistory and an enviable record in explaining At the macroscopic level the concepts of area and volume are quite a variety of different observable properties. As applied specifically to clear, and the methods of measurement straightforward in principle. At proteins, the work of G. N. Ramachandran and colleagues has provided the level of individual molecules the definitions become less obvious, much of our present thinking about permissible peptide chain conforma- tion.' Different approaches using more realistic models, complex mathc- When different methods of measurement are used, it is not clcar that area and volume are single valued characteristics. For a one-component sys- I matics, and even quantum mechanical approximations have improvedthe tem of known molecular weight,the mass densityis sufficient to unambig- details but have not altered the basic outline provided by the hard sphere uously define the mean volume per molecule. For a systemof two or more approximation. The steepnessof the repulsive term in the potential func- components, thermodynamic parameters. the partial molar volumes, can ..,-.x tion for nonbonded interactions is responsible for the success of "hard" be defined uniquely, but these values bear no necessary simple relation to in the hard sphere. the actual physical volumes of the individual components which, in turn, In spite of the general success of the hard sphere approximation, the may not be uniform throughout the mixture. The conccpts of total rnolec- van der Waals envelope ot'a molecule is not unique and is defined differ- ular area are even more elusive (see below). ' G. N. Ramach;rndran and U. Sasisckharan. Ah. ProrCJijr Chm. 23, 2R4 (IW). Copyrighi 0 19115 by Academic I'rcu. Inc. hlETHODS IN ENZYMOL.OCY. VOL. I15 All righis of reprwluclion in rny rorm rcrcrvcd. ti. SOME LISTSOF VAN DER WAALS RADIIFOR SELECTEDGROUPS OF ATOMS \ DunfieId ('I ClI.l, Richmond Gelin and GelinRichmond L ee and Shrake and and and Nemelhy and and and Shrake and Lee Symbol DesignationRichardshRupley'SymbolBondi' Richard9Chothia"Richards' Karplus' PI d.' -CHI Aliphatic. methyl 2.0 I .x0 2.0 2.0 I.87 1.9 1.95 2.13 -CHr Aliphatic. methyl 1.0 I.80 2.0 1.0 I .87 1.9 1.90 2.23 XH- Aliphatic. CH - I.70 2.0 2.0 I .87 1.9 1.8.5 2.x 2 CH Aromatic. CH - I.no I.85 j I .76 1.7 1.90 2.10 >C= Trigonal or aromatic I .74 I.80 1.5i1.85 I .7 1.76 I .7 1.80 1.85 -NH; Amino, protonated - I.x0 I .5 2.0 1 .so .7 1.75 -NH2 Amino or amide I .75 I.80 I.5 - I.65 I .7 1.70 - X (OM) Amide (N or 0 unknown) 1.75 I.5L 1.5 I.6 I .65 I .7 >NH Peptide. NH or N I .6S I.E I.4 I .7 I .6S I .7 1.65 1.75 (N) =O oxygenCarbonyl I .5 I.80 1.4 I.4 I.40 I .4 1-60 1.56 -OH Alcoholic hydroxyl - I.80 I .4 I .6 I .40 I .4 1.70 -OM Carboxyl oxygen - I .x0 I .89 I .5 I .40 I .4 1.60 1.62 -SH Sulfhydryl - I .x0 1.85 - I .85 I .x 1 .w -S- Thioether or -S-S- I .80 I.x I .85 1.8 1.90 2.08 A. Bondi. "Molecular Crystals. Liquids and Glasses." Wiley, New York. 1968. Radii assigned on the basis of observed Packing in condensed phases. * Lee and Richards.?? Values adapled from A. Bondi. J. flr.vs. Chem. 68, 441 (19641. I. Shrake and Rupley.:h Values taken from L. C. Pauling. "The Nature of the Chemical Bond." 3rd d. Cornell Univ. Press. 1Ih;tc;l. New York. 1960. Richards.h Minor modification and extension of Bondi (1968) set (see footnote a. ;Ibove). Rationale not given. ' Chothia.1' From packing in amino acid cryslal slruclurcs. Personal comrnunicalion from T. Koelzle quOted. ' Richmond and Richards.!' No rationale given for values used. Gelin and Karplus.4 Origin of values not specified. '3 Dunfield ('1 u/,' &tailed description of deconvolution of molecular crystal energies. V;llues represen1 rmc-half of Ihe heav!--:htom W'i'r;'. tion at the minimum of [he Lennard-Jones 6-12 po1enli;d function> for symmelric;d in1cr;lclionh. ' G. Ncmclhy. M. S. Pottle. and H. A. Shrraga. J. fhys. C/rcwr. 8'7. 18x5 (19831. I See original paper. 444 ANALYSIS OF STRUCTURE [301 [301 MOLECUI-AR VOLUMES AND AREAS 44 5 The Limiting Polyhedron P \ Consider a collection of point atoms where the specific atom. i, is strrroundcd by il group of' ncighbors j. X., 1. .... Thc limiting pc~lyhcdron will consist of;\ sct ol'intcrsccting planes pcrpcndicularto thc inlcr;tlomic vcclors drawn from i to.i. k. I. .... Each plane containing a facc of thc polyhedron may be identified by a single index, j,k. etc. The edges of the polyhedron are identified by pairs of indices from the planes whose inter- section produces the edge jk, kl, etc. The vertices of the polyhcdron require three indices, jl k. I, which can be takcn eithcr from thc lhrec planes whose intersection gives the vertex or alternately from the unique set of indices embedded in the index pairs which represent the threc lines which also intersect to define the same vertex (see Fig. 2). There are at least two somewhat different approaches to sclccting thc FIG. I. An example of the Voronoi construction in two dimensions. The heavy lines limiting polyhedron. In the first, possible vertices are both generated and outline the limiting polygons. The middle weigh1 lines are infer;ktomic vectors between the selected iteratively on the basis of their position with respect to a develop- atom centers represented by dots. The fine lines are the perpendicular bisectors of the vectors. The circled points hbeled A ;md U we intcricir points surrounclud hy unique closcd ing set of planes which ultimately represents the faces of the final poly- polygons. The point labeled C is on the surface of the set of points and ;I closed polygon hedron. The intersections of the planes represenling the faces yields thc cannot be defined, (Adapted from Richards.9 final list of vertices. In the second, an array of possible vertices. one for each set of four atom centers,which include i. is generated on the basis of discussed in this chapter are very sensitive to the values of the van der defined distance criteria. This list is then checked to eliminate all vertices Waals radii cho\sen, and especially to their ratios. When reporting area or volume calculations, the assumed radii should be carefully listed. Volume \mn nlnoI For an infinite set of arbitrary points in space, a geometrical procedure was introduced by Voroni in 1908.' which divides up space with a unique volume assigned to each point.
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