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

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 properties 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 projected 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 relating regular heptagons of various size characterizing a molecular form. In particular, the central hole can very often be obtained 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:

¦ ¦ © ¢ ¦ © ¢ © ¤ ¤ ¢ ¢ © © ©

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

¨ 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. Indeed, Rs and ms leave invariant

which obeys a crystallographic restriction, as revealed by the star a 6-dimensional lattice Σ, spanned in the superspace by a basis

2 ¤

heptagons 7/2 ¡ inserted. The negative value of scaling factor , s s s s

¢ ¢ © © ©¨¢ ¥

1 a ¤ a1 a2 a6 , with metric tensor

with 1 0 692 , implies that the central hole and the external

boundary have a reverse orientation. s s

¢ ¤ ¤ ¢ gii ¤ 6 gik 1 for i k (6)

Following these preliminary observations, in the next section s

left invariant by R and m . The projection : a a maps the s s k k

an outline is given of a crystallographic characterization of star ¡ 6

Σ ¤ ¢ ¢ © © ©¨¢ ¤

lattice into the Z-module M a1 a2 a6 ¥ Z , of all heptagons. In the other two sections, the molecular forms of ¡ - integral linear combinations of these basis vectors. Accordingly, hemolysin and of poly d(As4T) . poly d(As4T) are analyzed fur- K0 leaves M invariant: K0 M ¤ M. ther. Additional morphological features (like the elliptic holes recognizable in Fig. 3) are discussed on the basis of crystallo- Table 1

graphic linear scalings, which are dilations (or contractions) in Character Table of the point group D7

a given direction only. ( 2 7

1 2 2 3 3 k

2. Crystallography of star heptagons Γ 1 1 1 1 1 1

The symmetry of a regular heptagon is given by the 2- Γ 2

1 1 1 1 -1

dimensional point group K0 ¤ 7mm generated by a rotation R 3

Γ 2 2 cos 2 cos 2 2 cos 3 0

of order 7 and a re¯ection m: 4

2 2 cos 2 2 cos 3 2 cos 0 £ 5

¡ Γ

7 2 2 2 2 cos 3 2 cos 2 cos 2 0

¢ ¤ ¤ ¦ ¤ ¤ ©

K0 ¤ R m R m Rm 1 (1) ¨ The frame of general crystallography adopted here [1] requires The relation between the two bases a and as can be derived a faithful integral representation Γ of K0, whose minimal dimen- from the decomposition of the representation Γ into irreducible

IUCr macros version 2.0 5: 2001/06/20

2 1 0 0 0 1

components. Looking at the character table of D7 7mm (Ta-

0 1 1 0 0 1

ble I) one ®nds:

1 1 1 1 0 1

¡£¢ ¡ ¦ ¦ ¤ ¢ 3 ¡£¢ 4 5 S a (15)

3 ¨

Γ ¤ Γ Γ Γ

(7) 1 0 1 1 1 1

1 0 0 1 1 0

where

1 0 0 0 1 2

¡ ¡

3 ¡ 4 2 5 3

Γ Γ Γ ¦ ¦ ¦ ¡ ¤ ¢ ¦ ¡ ¤ ¢ ¦ ¡ ¤ ¢

¦ R e R e R e where:

¨ ¨ ¨ ¨ ¨ ¨

¡ ¡

3 ¡ 4 5

Γ ¦ ¢ ¤ Γ ¦ ¢ ¤ Γ ¦ ¢ ¤ ¦

m e ¢ (8)

¨ ¨ ¨ ¨

¤ ©©¨ © ¤ © © © © ¢

£ 2 2 cos 4 cos 2 0 35689 and

¡ ¢

¤ ¨ © ¤ © © © © with and the two generators of the abstract group D7, as in £ 3 1 2 cos 0 24697 Table I, and e denoting the 2-dimensional orthonormal basis introduced above. This implies the following embedding of the These additional transformations allow to extend the original basis a in the 6-dimensional basis as: Euclidean point group K0 to in®nite order, in a way still leaving the Z-module M invariant:

s s

¦ ¢ ¢ ¤ ¦ ¢ ¢

a ¤ a a a a a a a ¡

1 2 3 4 1 5 ¡ ¨

1 ¨ 4

¤ ¤ ¢ ¢ ¤ ¢ ¢ ¢ ¢ ¤

¥ ¦ ¥§¦ ¦

s s K 7 2 R m S 1 K 7 3 R m S 2 S 3 (17)

¦ ¢ ¢ ¤ ¦ ¢ ¢

a ¤ a2 a4 a6 a a5 a3 a1 (9) ¨ 2 ¨ 5

s s

¦ ¢ ¢ ¤ ¦ ¢ ¢ ©

a ¤ a3 a6 a2 a a6 a5 a4 Note the relations: ¨

3 ¨ 6

¤ ¦ ¦ ¤ ¥§¦ ¥ ¦ © Making use of the 2-dimensional metrical relations between the S ¥§¦ S and S 3 S 1 S 2 (18) elements of the basis a (with indices taken modulo 7):

In a heptagram, however, the points at the intersection of the two

¤ ¤ ¥

2 different star polygons 7/2 ¥ and 7/3 have also to be consid-

ai ai ¤ k a cos k (10) 0 ¨ ered for characterizing its self-similar symmetry. These points s one veri®es that the basis a forms indeed the Rs-invariantmetric cannot be obtained by radial scaling from the heptagon consid- 2 a0 s

ered but have, nevertheless, also integral coordinates (in the a tensor 2 g . The orthogonal projection considered above from the superspace to the heptagonal plane, and which corresponds basis). The reason is that they follow from integral linear scal-

3 ¡

Γ ¦ Γ ¦

to the homomorphism K K , is given by: ings X Y along the x- and the y-axis, respectively, whose prod-

0 0

¨ ¨ uct is a radial scaling S with the same scaling factor . This

¤ ¢ ¤ ¢ ¢ © © © ¢ © ak ak k 1 2 6 (11) is a property that the radial scalings considered so far do not have. For ®xing the ideas, consider a point P with coordinates

The idea on which a molecular crystallography approach is

¤ ¦ ¢

P ¦ e x y . Then for these linear scaling transformations one

¨ ¨ based is that there are possibly hidden molecular structural re- has:

lations expressible as additional transformations leaving the Z-

¦ ¢ ¤ ¦ ¢ ¢ ¦ ¢ ¤ ¦ ¢ ¢ ¦ ¢ ¤ ¦ ¢ ©

module M invariant. In the present case this is indeed the case X x y x y Y x y x y S x y x y

¨ ¨ ¨ ¨ ¨ ¨ for the radial scaling symmetries of a self-similar star heptagon (19) (obtained from a starting regular heptagon by successive direct In particular, there are two sets of linear transformations ensur-

and inverse star constructions). ing the equivalence of the additional points with the previous

¤ ¥§¦

In the case 7/2 ¥ the radial scaling transformation S 1 : ones:

4 3 1 1 3 5

2 2 1 0 1 2

1 1 0 0 0 1

0 0 1 1 1 1

3 2 0 1 2 3

2 2 1 1 0 1

¤ ¢ ¦

X ¦ a (20) ¢ ¥§¦ ¦ ¤

S a (12) 4 ¨

1 ¨

3 2 1 0 2 3

1 0 1 1 2 2

1 1 1 1 0 0 1 0 0 0 1 1

2 1 0 1 2 2 5 3 1 1 3 4

has a negative scaling factor £ with 0 1 1 1 1 1

1 1 2 2 2 1

£ ¤ ¨ ©¥ © ¤ © © © © ¢ © ¤ ¦ ©

1 2 cos 2 cos 2 0 692021 2 7

1 1 2 2 3 2 1

¦ ¦ ¤ Y a ¢ (21)

(13) 4 ¨

1 2 3 2 2 1 ¤

In the case 7/3 ¥ there are two radial scaling transformations,

1 2 2 2 1 1 £

with scaling factors indicated by £ and , respectively

2 3 1 1 1 1 1 0

1 1 2 0 2 1 0 3 2 2 3 1

0 2 1 2 2 1 1 4 3 3 5 1

1 1 0 1 0 1 0 1 1 0 1 0

¦ ¤ ¢ ¦ ¤ ¢ ¥§¦

S ¥§¦ a (14) X a (22) ¨

2 ¨ 5

1 0 1 0 1 1 0 1 0 1 1 0

1 2 2 1 2 0 1 5 3 3 4 1

1 2 0 2 1 1 1 3 2 2 3 0

IUCr macros version 2.0 5: 2001/06/20

2 1 2 2 1 3

1 0 1 1 1 1

2 1 1 2 1 2

¦ ¤ ¢ Y ¥§¦ a (23)

5 ¨

2 1 2 1 1 2

1 1 1 1 0 1

3 1 2 2 1 2 £ with scaling factors £ 4 and 5, where:

Q 21 y

1 Q 12

¤ ¨ © © ¤ © © © ©¨¢

£ 4 3 4 cos 4 cos 2 0 38404 y 2

¤ © © ¤ © © © © £ 5 1 4 cos 8 cos 2 0 28620 (24) µ 4y 1 P 23 −µ −µ µ

1 2 3 1

¤ ¥ Note that X ¥ X and so also for Y . x P P P P 70 71 72 73 −µ These results allow to express all the intersection points of a 5 y 2 P heptagram as integral linear combination of the basis vectors and 32 to assign to each point a set of integral indices, which are the component of the corresponding position vectors expressed in the basis a:

¤ ¦ ¢ ¢ © © ©¨¢ ¤ ©

P ¦ a n1 n2 n6 with P nk ak (25)

¨ ¨

k ¡ 1

The followingconventions for labeling the heptagrammal points have been adopted:

¤ ¢ ¤ ¢ ¢ ¤ ¢ ¢ ¢ ¤ ¢ ¢ © ©¨¢ ¢ £¢ Pik R P0k Pih S ¦ Pi0 k h 1 2 3 i 1 2 7

h ¥

¥ 1 1 i

¤ ¢ ¤ ¢ ¤ ¢ ¤ ¢ © ¥ ¦ Q21 Y¦ 4 P23 Q12 Y 5 P32 Qi j R Q0 j j 1 2(26) Figure 4

Labeling convention of the intersections points in a heptagram consist-

¡ ¡

¥¤ ¦§¦ £ where £ denotes h taken with the appropriate sign. A rela- ing of the two star polygons 7/2 and 7/3 with examples of their h ¨

tion between scaling factors and indices is visualized in Fig. 4. mutual scaling relations. So P71 P72 and P73 are obtained from P70 by

k radial scaling transformations S ¨ i , with scaling factors 1 2 3,

The missing indices are easily computed by applying R ¦ a to ¨ respectively, whereas linear scalings Y ¨ along the y axis and with fac-

those indicated. Table II gives the result. j

tors 4 and 5, relate correspondingly Q21 with P23 and Q12 with P32.

Table 2

Indexed intersection points of the heptagram (see Fig. 4)

¡ ¡ 7/1 ¡ 7/2 7/3

P10 1 0 0 0 0 0 P11 2 0 2 1 1 2 P12 1 0 1Å 1 1 1Å P13 2Å 0 1Å 1Å 1Å 1Å P20 0 1 0 0 0 0 P21 2Å 0 2Å 0 1Å 1Å P22 1 2 1 0 2 2 P23 1 1Å 1 0 0 0 P30 0 0 1 0 0 0 P31 1 1Å 1 1Å 1 0 P32 2Å 1Å 0 1Å 2Å 0 P33 0 1 1Å 1 0 0 P40 0 0 0 1 0 0 P41 0 1 1Å 1 1Å 1 P42 0 2Å 1Å 0 1Å 2Å P43 0 0 1 1Å 1 0 P50 0 0 0 0 1 0 P51 1Å 1Å 0 2Å 0 2Å P52 2 2 0 1 2 1 P53 0 0 0 1 1Å 1 P60 0 0 0 0 0 1 P61 2 1 1 2 0 2 P62 1Å 1 1 1Å 0 1 P63 1Å 1Å 1Å 1Å 0 2Å

P70 1Å 1Å 1Å 1Å 1Å 1Å P71 2Å 0 1Å 1Å 0 2Å P72 1Å 2Å 0 0 2Å 1Å P73 2 1 1 1 1 2

Q11 0 1 0 1Å 1 0 Q21 0 0 1 0 1Å 1 Q31 1Å 1Å 1Å 0 1Å 2Å Q41 2 1 1 1 2 1 Q51 1Å 1 0 0 0 1 Q61 1Å 2Å 0 1Å 1Å 1Å Q71 1 0 1Å 1 0 0 Q12 1Å 1Å 1Å 0 2Å 1Å Q22 1 0 0 0 1 1Å Q32 1 2 1 1 1 2 Q42 2Å 1Å 0 1Å 1Å 1Å Q52 1 1Å 0 1 0 0 Q62 0 1 1Å 0 1 0 Q72 0 0 1 1Å 0 1

IUCr macros version 2.0 5: 2001/06/20 The intersection points Qi j of a heptagram represent one way 3. ¥ -Hemolysin to arrive at crystallographiclinear scalings, but this way does not reveal their characteristic structural role. In order to ®nd out the typical aspect, consider how the vertices of a regular heptagon

(chosen in the x-orientation) transform under the combination of

linear scalings to ®rst order X , Y and the 7-fold rotations of the point group K0 as in the double cosets K0X K0 and K0Y K0. The

image points of the ak one gets, all are on circles with centerCk at

the mean distance between ak and S ak, for S ¤ X Y . While

X gives rise to an off-center heptagon in the same x-orientation,

Y yields an heptagon in the reverse orientation -x. Fig. 5 illus-

¥ 1

¤ £ ¤ © © © © trates these properties for the case 4 2 60388 given above. The presence of off-center circles in the structure of a given molecular form strongly suggests the possible presence of linear scalings.

¡ -Hemolysin is a heptameric transmembrane pore protein. The data for the atomic coordinates have been taken from PDB 7AHL by Song et al.[10]. The structure consists of 7 chains la- beled by A to G. Before projecting the structure along the axis,

the original data have been oriented and centered using the po- ¢ sitions of the residue GLU71 in the chains A ¢ D F. The corresponding prismatic molecular form has already been presented in the introduction (Fig. 2). (x, λ y)

(x,y) ( λ x,y)

The projection of a single chain shows that the scale-rotation ¤

symmetry of the star polygon 7/3 ¥ , which characterizes the molecular form of the heptamer, is already predisposed in the folding of the monomer (Fig. 6). One sees that in addition to the residues 71 and 147 indicated in Fig. 2, several more (150, 141, 135, 121, and 115) also ®t fairly well with the intersection points of the star heptagon. The corresponding scale-rotation transformations, however, do not connect atomic positions, but only ver- Figure 5 tical lines (parallel to the central axis) through these positions. Off-center circles obtained by applying linear scalings with given scal- This does not exclude the possibility of a 3-dimensional point ing factor to the vertices of a regular heptagon. Indicated is how the

group relating these same positions. In the spirit of molecular

¡ ¢ ¢

x y -coordinates of these points are transformed by X and Y to

¤£ crystallography, one even expects the existence of such a group

¡ ¡ x y and x y , respectively. The remaining points follow from the 7-fold rotational symmetry (leading to conjugated linear scalings which would explain the molecular form observed, but this in-

k k k k £ 1

¢ vestigation has not yet been carried out. From the molecular

R X R and R Y R ). Here, the scaling factor

¢ ¢

¡ ¡ 2 60388 ensuresthat the matrices X a andY a are integral, so that form one ®nds some evidence for additional linear scaling, but all these points can be indexed by integers. not strong enough to be presented here.

IUCr macros version 2.0 5: 2001/06/20 thymine is not simply scaled by a power of £ 1. The heptagons

2 3

¢ £ ¢ £ scaled by £ 1 1 1, show, however, the expected compatibility with morphological features.

150 147

115 141

121 135

Figure 6

A protomer of -hemolysin is shown in a view along the central axis of

the heptamer and in relation with the star heptagon 7/3 ¡ ®tted at the position of the residue 71. The projected positions of a number of other residues (150, 147, 141, 135, 121 and 115) at points of the star heptagon are an indication that the protomer folds in a way predisposed not only Figure 7 to the 7-fold rotational symmetry (in the stem domain), but also to crys- Prismatic molecular form of the s4T-subsystem in tallographic scale-rotations which relate the stem domain with the cap poly d(As4T) . poly d(As4T) viewed along the helical axis. Shown domain, in a way still requiring further investigation. are successive internal boundaries scaled from the external heptagon

2 3

by 1 , respectively, where 1 0 6920 is the same factor 1 1

4 4 on which the star heptagon 7/2 ¡ is based and which also plays a 4. Poly d(As T) . poly d(As T) role in the A-subsystem (Fig. 3). In the present case, the central hole The left-handed double helix formed by the two strands of and the external boundary are not related by a scaling factor obeying crystallographic restrictions. (See also Fig. 10). poly d(As4T) . poly d(As4T) is characterized by a dinucleotide repeat with line group symmetry 7622 of which the details have This general behavior is con®rmed when combining the two been reported in Nature [11]. The corresponding data are listed subsystems in the molecular form of the dinucleotide shown in as structure No 16 in a survey by Chandrasekaran and Arnott, Fig. 8. As the external boundaries of the two subsystems are published in a volume of the Landolt-BorsteinÈ collection [12]. slightly different in size, the combination requires an adaptation. It is convenient to consider ®rst the molecular form of each The one of the A-subsystem gives the best ®tting for the whole, nucleotide subsystem separately. The form of the A-subsystem despite the fact that two of the phosphate oxygens of the s4T nu- has already been presented in the introduction(Fig. 3), that of the cleotide are no more inside the heptagon of the external bound- 4 ¤

s T-subsystem is shown in Fig. 7. Both share the absolute value ary. The morphological importance of the star heptagon 7/2 ¥

¤ © © © © ¥ of scaling factor £ 1 0 6920 of the star heptagon 7/2 , but is con®rmed. not always the corresponding negative sign. This implies that Looking at Fig. 8, one recognizes elliptically shaped holes the successively scaled heptagons may have the same or the re- tangent to the external heptagon at mid-edge positions. As al- verse orientation. While in the case of adenine the central hole ready pointed out in the introduction, this indicates the possible is already reached at the second contraction level, that of the presence of linear scalings which, as explained in section 2, can

IUCr macros version 2.0 5: 2001/06/20 give rise to off-center circular structures. In the present case, these linear scaling transformations to the mid-edges of the ex- the natural candidates are the two linear scaling transformations ternal heptagon, as shown in Fig. 9. Moreover, the central hole

¢ ¦ ¦ £ X ¦ 1 Y 1 whose product gives the radial scaling S 1 . These linear can also be described by a circle which is a factor 1 smaller than scalings have not been considered so far because not integral in the one inscribed in the heptagon. One could also start from the the heptagonal basis a. But the appearance of mid-edge heptag- central hole and arrive at the off-center hole observed by apply-

¥ 1

onal points with half-integer indices, suggests a corresponding ing the inverse radial scaling S ¦ ®rst, and then the correspond-

1 ¥ ¥ 1 1

centering of the 6-dimensional lattice, leading to half-integers ¢ ¦ ing linear dilation X ¦ Y in combination with the appropriate 1 1 in the entries of the linear scaling when expressed in what one 7-fold rotations.

could denote as the conventional basis a. One ®nds indeed: This last interpretation opens the door to a corresponding

¦ ¦ ¦ ¦ ¦ ¦

3 2 3 2 1 2 1 2 3 2 5 2 characterization of the molecular form of the s T-subsystem, as

¦ ¦

1 2 1 0 0 0 1 2 represented in Fig. 10. Starting from the heptagon of the cen-

¦ ¦ ¦ ¦

3 2 1 1 2 1 2 1 3 2 tral hole with vertices at a mean O2 position of diadically re-

¢ ¦ ¤ X ¦ a 4

1 ¨

¦ ¦ ¦ ¦ lated thymine molecules, one gets by a radial scaling S the

3 2 1 1 2 1 2 1 3 2 1

¦ ¦ 1 2 0 0 0 1 1 2 heptagon formed by the corresponding mean C1 positions, and

¥ 3

¦ ¦ ¦ ¦ ¦ ¦ 5 2 3 2 1 2 1 2 3 2 3 2 by a further S 1 transformation those corresponding to the C6

(27) ones. From this last heptagon, one obtains the circular off-center

¥ 2 2

¦ ¦ ¦ ¦ ¦ ¦ ¦

1 2 1 2 1 2 1 2 1 2 1 2 holes by the linear transformations X ¦ Y , squares of the di-

1 1

1 ¦ 2 0 1 1 1 1 2 lations given above. These off-center circles are thus related to

¦ ¢ ¦ ¢

¦ ¦ ¦ ¦

1 2 1 1 2 3 2 1 1 2 the central hole by elements X 1 Y 1 R and m of the point group,

¦ ¤ Y¦ a

1 ¨

¦ ¦ ¦

1 ¦ 2 1 3 2 1 2 1 1 2 but no simple relation could be found with the heptagon delim-

¦ ¦

1 2 1 1 1 0 1 2 iting the molecular form. The reason is perhaps that this hep-

¦ ¦ ¦ ¦ ¦ 1 ¦ 2 1 2 1 2 1 2 1 2 1 2 tagon does not represent the relevant molecular envelop for the

(28) s4T-subsystem. This would explain why in this case there is no

¤ © © © © © with positive scaling factor £ 1 0 6920 The off-center holes crystallographic scaling transformation relating the central hole of the A-subsystem are fairly well approximates by applying and the external heptagon, as it is the case for the A-subsystem.

Thanks are expressed to R. Chandrasekaran for kindly having made available to the author the structural data of Poly d(As4T) - . poly d(As4T) reported in Landolt-Borstein;È to R. de Gelder and to B. Souvignier for pertinent remarks and valuable suggestions. Stimulating discussions with C.W. Hilbers are gratefully acknowledged. The graphical program XPS used has been originally de- signed by J.M. Thijssen.

References

[1] Janner A.. Acta Cryst. 2001, A57, 378. [2] Nischke K.-P.; Danzer L.. Discrete Comp. Geom. 1996, 15, 221. (See also Materialien/Preprints LXXIX, University of Bielefeld, 1994). [3] Barache D.. Ph. D. Thesis, University of Paris 7; 1995. [4] Gazeau J.P.. In Symmetry and Structural Properties of Condensed Matter; Lulek T.; Florek W.; Walcerz S. Eds. World Scienti®c: Singapore, 1995; p. 369. [5] Janner A.. Crystal Engineering 2001, 4, 119. [6] Janner A.. In GROUP21 Physical Applications and Mathematical Aspects of Geometry, Groups and Algebras.; Doebner H.-D.; Scherer W.; Schulte C. Eds. World Scienti®c: Singapore, 1997; p. 949. [7] Hendrickson Wayne A.; WuthrichÈ Kurt. Macromolecular structures 1997. Atomic structures of biological macromolecules reported during 1996; Current Biology Ltd.: London, 1997. [8] Arnott S.. In Oxford Handbook of Nucleic Acid Structure; Neidle S. Ed. Oxford University Press: Oxford, 1998; p. 1. [9] Coxeter H.S.M.. Introduction to geometry; John Wiley: New York, 1961. [10] Song L.; Hobaugh M.R.; Shustak C.; Cheley S.; Bayley H.; Gouaux J.E.. Science 1996, 274, 1859. [11] Arnott S.; Chandrasekaran R.; Birdsall D.L.; Leslie A.G.W.; Ratliff R.L.. Nature 1980, 283, 743. [12] Chandrasekaran R.; Arnott S.. In Landolt-BorsteinÈ Numerical Data and Functional Relationships in Science and Technology (Group VII, Biophysics), Subvolume VII, 1b; Springer: Berlin, Heidelberg, 1989; p. 31.

IUCr macros version 2.0 5: 2001/06/20 1 µ µ 2 1 1

Figure 8 Molecular form adopted for the (As4T)-system of poly d(As4T) . poly d(As4T), with the sugar of the A-nucleotide delimiting the external bound-

ary, leaving outside oxygens of the phosphate of the s4T-nucleotide. Despite this mismatch, the overall structure is still compatible with internal

prismatic boundaries based on 7/2 ¡ star polygons.

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Figure 9

The elliptical holes visible in the molecular form of the A-subsystem (as shown in Fig. 3) are obtained as off-center circles from linear scalings

¨ ¨

X 1 Y 1 (and their R-conjugated ones) applied to the mid-edges of the external heptagon, in the same way as illustrate in Fig. 5, but now for 1

¨ 0 692 One sees that the central hole then follows by a further radial scaling transformation S 1 , and this is unexpected.

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Figure 10 The same linear scalings as in Fig. 9 allow to ®nd a better envelop for the s4T-subsystem than the regular heptagon. Now successive radial scalings

3 4 ¨ S ¨ 1 and S 1 relate this curved boundary to the internal circular ones through the C1 and the O2 positions of the thymine, respectively. The latter positions de®ne the central hole.

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