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Polygrammal Symmetries in Bio-Macromolecules: Heptagonal Poly D(As4t) . Poly D(As4t) and Heptameric -Hemolysin

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Polygrammal Symmetries in Bio-Macromolecules: Heptagonal Poly D(As4t) . Poly D(As4t) and Heptameric -Hemolysin Polygrammal Symmetries in Bio-macromolecules: Heptagonal Poly d(As4T) . poly d(As4T) and Heptameric -Hemolysin A. Janner * Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen The Netherlands Polygrammal symmetries, which are scale-rotation transformations of star polygons, are considered in the heptagonal case. It is shown how one can get a faithful integral 6-dimensional representation leading to a crystallographic approach for structural prop- erties of single molecules with a 7-fold point symmetry. Two bio-macromolecules have been selected in order to demonstrate how these general concepts can be applied: the left-handed Z-DNA form of the nucleic acid poly d(As4T) . poly d(As4T) which has the line group symmetry 7622, and the heptameric transmembrane pore protein ¡ -hemolysin. Their molecular forms are consistent with the crystallographic restrictions imposed on the combination of scaling transformations with 7-fold rotations. This all is presented in a 2-dimensional description which is natural because of the axial symmetry. 1. Introduction ture [4,6]. In a previous paper [1] a unifying general crystallography has The interest for the 7-fold case arose from the very nice axial been de®ned, characterized by the possibility of point groups of view of ¡ -hemolysin appearing in the cover picture of the book in®nite order. Crystallography then becomes applicable even to on Macromolecular Structures 1997 [7] which shows a nearly single molecules, allowing to give a crystallographic interpre- perfect 7-fold scale-rotation symmetry of the heptamer. In or- tation of structural relations which extend those commonly de- der to distinguish between law and accident, nucleic acids have noted as 'non-crystallographic' in protein crystallography. In been considered as an alternative molecular case. the same paper it is pointed out, that molecular crystallography Nucleic acids occur as polynucleotides in different helical can be approached in a fairly similar way as for quasicrystals. conformations, with a variety of axial symmetries. A full list of Alan Mackay, to whom this paper is dedicated on the occasion polynucleotides can be found in a historical survey of S. Arnott of his 75th anniversary, has given a pioneering and fundamen- in the Oxford Handbook of Nucleic Acids edited by Neidle [8]. tal contributionto the problem of the relations between structure There, only one case is reported with 7-fold axial symmetry: and symmetry in quasicrystals. For this occasion it seemed to the Z-DNA form of the left-handed poly d(As4T) . poly d(As4T) me a good idea to focus the attention to systems having a 7-fold which, therefore, has been selected together with ¡ -hemolysin. point group symmetry. A third case, the double chaperonin (a larger and a smaller one) Seven-fold self-similar quasicrystals, where crystallographic complexed with ADP and denoted as GroEl-GroES-(ADP)7 will point groups of in®nite order arise naturally, have been consid- be discussed elsewhere. The analysis of the molecular forms ered by various authors (see for some references [2-4]), but this is limited here to the prismatic forms of the backbone formed ®eld has not been developed because such quasicrystals have not by the C £ 's in the protein and disregarding hydrogen in the been observed in nature. Observed in nature are, however, bio- nucleic acid. The prismatic forms can be viewed along the macromolecules with a rotation symmetry of order 7 and it is 7-fold axis, allowing a 2-dimensional description of their 3- challenging to investigate the relevance of these point groups dimensional scale-rotational properties. in the molecular case. A ®rst attempt in this direction has been Many of the concepts raised so far are certainly non familiar made for two hexagonal DNA's in terms of scaling properties of to most readers. Before developing the subject, a visualization molecular forms re¯ecting an underlying point group of in®nite of what one is talking about is, therefore, important. Polygrams order leaving the hexagonal lattice invariant [5]. In this paper and star polygons are ®rst introduced, because these geometrical evidence is also given of similar behavior in B-DNA and in A- objects allow a simple approach to the symmetry of the various DNA, which correspond to the 10-fold and the 11-fold case, re- molecular forms. A star polygon denoted by the Schla¯iÈ symbol spectively. ¤ n/m ¥ is obtained from a regular polygon by connecting each of It is essential to verify whether these scaling properties occur the n vertices with its mth-subsequent one [9]. The regular poly- ¤ ¥ in nature as a rule rather than as an exception. From the point of gon itself represents the trivial case n ¦ 1 . A polygram consists view of molecular crystallography this implies molecular forms of star polygons. Known since more than two thousand years are ¤ ¤ ¥ with scaling factors critically related to the given point group the pentagram 5/2 ¥ and the hexagram 6/2 . In the 7-fold case ¤ ¥ symmetry. Typically, one should ®nd ¢ , the golden mean, in the in addition to the regular heptagon 7/1 , there are the two star ¤ ¤ ¥ 5-fold case, but this factor should be totally absent in the heptag- heptagons 7/2 ¥ and 7/3 (Fig. 1). The other possible ones co- onal case, because of the underlying number theoretical struc- incide with one of these. § c 0000 International Union of Crystallography 1 {7/2} GLU71 LYS147 {7/3} Figure 1 Figure 2 The two regular heptagonal star polygons and their Schla¯iÈ symbol Axial view of the heptameric protein -hemolysin with a prismatic en- ¡ 7/2 ¡ and 7/3 . closing form. The two delimiting heptagons, with vertices at the pro- jected positions of the residues GLU71 and LYS147, de®ne the external boundary and the central hole, respectively. They are related by a scal- ing transformation with factor 3 0 2469 which obeys to the same crystallographic restriction as in the star heptagon 7/3 ¡ . The morphological role of these star polygons consists in re- lating regular heptagons of various size characterizing a molec- ular form. In particular, the central hole can very often be ob- tained from the external polygonal boundary enclosing the bio- macromolecule by star polygons, as it has already been shown in the hexagonal case [5]. The structure of poly d(As4T) . poly d(As4T) projected along the helical axis is dominated by relations expressible in terms ¤ ¥ In the case of ¡ -hemolysin the external boundary is a regu- of 7/2 star heptagons. The central molecular hole (due to lar heptagon with vertices at the positions of the residue GLU71 s4T, where s4T indicates that the oxygen in the position 4 of the in the cap domain of the seven chains, whereas the vertices of thymine is replaced by a sulfur atom) is so small with respect the central hole can be de®ned by the positions of the residue to the external heptagonal boundary, that at this stage no reli- LYS147 of the transmembrane stem which has the structure of a able scaling factor can be derived from polygrammal relations. ¤ ¥ right-handed ¢ -barrel. These boundaries are related by a 7/3 Therefore, only the molecular form of the A-subsystem is pre- star heptagon (Fig. 2). This implies that the two heptagons have sented here (Fig. 3). One sees that in this case the central hole 2 the same orientation and that the axial distances of GLU71 and is scaled with respect to the external heptagon by a factor £ 1 § 2 § £ ¤ ¦ ¦ ¦ ¤ © © © © LYS147 are in the ratio 1 : £ determined by the symmetry of with 1 cos cos 0 6920 . The minus sign im- ¨ ¨ ¤ 7 7 § 3 § £¥¤ ¦ ¦ ¦ ¤ © © © © 7/3 ¥ , with cos cos 0 2469 plies that the two heptagons are in reverse orientation. ¨ 7 ¨ 7 2 IUCr macros version 2.0 5: 2001/06/20 sion, in the present case, is 6. ¦ ¢ ¢ Γ ¦ K0 ¥ K0 GL 6 Z (2) ¨§¦ ¨ where ¥ denotes group isomorphism. This representation can be obtained by applying R and m to six of the seven radial vec- ¢ © © ©¨¢ tors a1 ¢ a2 a7 of the heptagon. The orientation chosen is with a7 along the positive direction of the x-axis, and with the other Γ ¤ ¢ ¢ © © ©¨¢ ¥ ak as basis a for the representation : a ¤ a1 a2 a6 . This basis is linearly dependent over the reals R, but linearly indepen- dent over the rational integers Z. The components of these vec- ¤ ¢ ¥ tors with respect to an orthonormal basis e ¤ e1 e2 , with e1 and e2 in the direction of the x- and y-axis, respectively, are given by: 2 ¦ ¦ © ¢ ¦ © ¢ © ¤ ¤ ¢ ¢ © © © ak ¤ a0 cos k sin k and k 1 2 6 (3) ¨ ¨ ¨ 7 The representation matrices of the generators are then given by: 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 ¢ ¤ R ¦ a (4) ¨ 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 © ¤ m ¦ a (5) ¨ 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 ¦ Figure 3 By considering R ¦ a and m a as orthogonal transformations ¨ ¨ Axial view of the A-subsystem of the nucleic acid R and m in a 6-dimensional Euclidean space (the so-called poly d(As4T) . poly d(As4T) in a Z-DNA conformation. As in s s the previous ®gure, the external heptagon (delimited by the sugar) superspace) one gets a formulation equivalent to a higher- and the central hole (due to the adenine) are in a a scaling relation dimensional crystallography.
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