J. M. Eckert University of Sydney Sydney, Australia
Miibius Molecules
If a narrow sheet of paper is given a half-twist about its long axis and then joined into a band by fastening the short edges, a figure is obtained which has only one surface and only one edge.
'The one-sidedness of the band is demonstrated most clearly in the imaginative woodcut by Dutch artist Maurits Escher (Fig. l ) in which an ant is seen to crawl from any point on the sheet to any other point without having to cross an edge.
Figure 1. "Mobius Strip il", a woodcut by Maurits Escher. Copyright M C. Escher Foundation. The Hague.
Mobius Strips
Twisted bands are called Mobius strips after the German mathematician and astronomer, August Mobius, who first described them in 1858. They have their practical uses. For example, patents have been granted for Mobius strip conveyor belts designed to wear equally over the whole surface. There is also a Mobius filmstrip which records sound on hoth "sides" and a tape-recorder which by using twisted tape runs for twice as long as it would if the tape was untwisted ( I ) . Mobius strips, however, are probably most appreciated by children who for generations have been intrigued by the variety of figures that result from cutting the hands in half down their long axis. If a strip with one half-twist is bisected, the result is not two bands but rather a single larger band which turns out to be a Mobius strip with four half-twists. Bisecting a Mobius strip with two half-twists yields a pair of rings that are interlocked like the links in a chain while one with three half-twists gives a trefoil which is a ring with a knot in it. Frisch and Wasserman, in 1961, discussed the possibility of Mobius strips in chemical systems (2). They pointed out that a double-stranded molecule with cross-links constitutes a roughly rectangular surface which would produce a molecular Mobius strip if the two ends of each strand were tojoin'after a half-twist about the long axis.
Frisch and Wasserman noted the feasibility of doublestranded Mobius molecules containing more than one half-twist and sneculated about the svnthesis from these of other systems'(such as interlocked rings and trefoils) by cleavage of the cross-links between the strands.' Circular DNA Molecules do exist which consist of pairs of cross-linked strands. Deoxyribonucleic acid (DNA) is an example. The strands in a molecule of DNA are composed of repeating base-sugar-phosphate units (nucleotides) and are held to-
' Mass spectroscopic evidence has been presented for the formation of interlocked rings via Mabius strip intermediates in the metathesisreaction of cyelododecene (12, 13). 458
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Figure 2. An electron micrograph of DNA from the polyoma virus. W. Stoeckanius, the Rockefeller Institute.
gether in p a i n by hydrogen bonds between specific complementary bases. Further, DNA molecules can be circular. Naturally occurring circular duplex DNA molecules were first reported in 1963 when they were found to he present in polyoma virus particles (3, 4) and in the intracellular replicating form of the 4x174 bacteriophage (5, 6). An electron micrograph of polyoma DNA, obtained by W. Stoeckenius (4), is shown in Figure 2. Two-stranded circular molecules are not necessarily molecular Mobius strips. To generate Mobius topology, the two strands have to wind round each other (one or more times) before ring closure and also, the circle must be fully closed. There can be no breaks in either strand (Fig. 3). Intertwined strands are a feature of the double-helical structure of DNA proposed by Watson and Crick in 1953 (7). Given that each turn in a Watson-Crick helix contains 10 base pairs and that the average weight of a sodium nucleotide residue is 332, it is possible to calculate from the measured molecular weight of a samnle of circular DNA how many revolutions one strand must make about the other to generate the Watson-Crick structure. For example, circular DNA from the polyoma virus (Fig. 2) has a molecular weight of 3.1 X lo6. The number of nucleotide nairs in the molecule is therefore (3.1 x 106)/f2 x 332) or ;oughly 4700 and the required number df ;evolutions, 4700110 or 470 (8).
The other condition, that there are no breaks in either strand, is satisfied by one of two circular forms of polyoma DNA which have been identified (9). The major component of polyoma DNA (form I) consists of fully closed duplex circles. These have a compact twisted appearance in electron micrographs. It has been suggested. that the final ring closure occurs before winding of the two strands into the Watson-Crick structure is complete. The molecule achieves a stable structure by twisting into a coil of coils orsuperhelix (9). A minor component of polyoma DNA (form 11) is derived from form I by the introduction of single-strand breaks. The nicked duplex circles exhibit little or no supercoiling because there is a site for rotation of the helix in the complementary strand opposite the break.
Conclusions The fully closed circular form (I) of polyoma DNA possesses Mobius topology. This unusual structural feature accounts for many of its special physico-chemical properties. In alkali (pH 12.51, for example, molecules of form I adopt a highly compact duplex structure. The strands do not separate. In the same solvent, the strands of a nicked circle (form n) separate completely (9). The behavior of form I molecules in "strand-separating" solvents such as alkali (pH 12.5) reflects the fact that the strands in a molecular Mobius strip are topologically constrained and cannot separate unless at least one of them breaks. Circular duplex DNA molecules have now been isolated from a number of bacteria and viruses, as well as from the mitochondria of a wide variety of higher organisms. They are the subject of an extensive literature and are treated in some excellent reviews (10, 11). Literature Cited
Figure 3. A Mobius molecule. Tne stranas of tne auplex circle are crosslinked. intertwined and unbroken. The cross-links are not shown in this drawing.
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Volume 50, Number 7,July 1973
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