M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 60
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Starting from leaf “0,” leaf “8” will bein eclipsed orientation to it. In order to reach leaf “8” from leaf “0,”(a)(b)(c)Figure 8-12. (a) The scattered leaf arrangement (phyllotaxis) of Plantago media(drawing by Ferenc Lantos, Pécs). Fibonacci numbers of spirals in the patterns of;(b) Scales of a pinecone; and (c) A cactus in Hawaii (b and c, photographs by theauthors).8.4. Rods, Spirals, and Similarity Symmetry385the stem has to be circled three times. The ratio of the two numbers,viz., 3/8, tells us that a new leaf occurs at each three/eighths part ofthe circumference of the stem. The ratio 3/8 is characteristic in phyllotaxis, as are 1/2, 1/3, 2/5, and even 5/13.
Very little is known aboutthe origin of phyllotaxis. What has been noted a long time ago is thatthe numbers occurring in these characteristic ratios, viz.,1, 1, 2, 3, 5, 8, 13, . . .are members of the so-called Fibonacci series, in which each consecutive number is the sum of the previous two. Fibonacci numbers canbe observed also in the numbers of the spirals of the scales of pinecones as viewed from below, displaying 13 left-bound and 8 rightbound spirals of scales as in Figure 8-12b. Left-bound and right-boundspirals in strictly Fibonacci numbers are found in other plants as well.The plate of seeds of the sunflower appears as if a compressed scattered arrangement around the stem.
Another example is the spiralsof cactus thorns in Figure 8-12c. It is a striking observation that thecontinuation of the ratios of the characteristic leaf arrangements eventually leads to a famous irrational number, 0.381966. . ., expressingthe golden mean!An important application of one-dimensional space groups is forpolymeric molecules in chemistry. Figure 8-13 illustrates the structureand symmetry elements of an extended polyethylene molecule.
Thetranslation, or identity period, is shown, which is the distance betweentwo carbon atoms separated by a third one. However, any portion withthis length may be selected as the identity period along the polymericchain. The translational symmetry of polyethylene is characterized bythis identity period.The discovery of stereoregular organic polymers dates back to themid-1950s. First, Karl Ziegler used his catalysts to produce such polymers under mild conditions, but they were not highly ordered structurally. This problem was solved by Giulio Natta, who—in the wordsof the Swedish presenter of their Nobel Prizes—broke the monopolyof nature over making stereoregular polymers [14].
The structuralaspects had great importance for practical applications that skyrocketed following Ziegler and Natta’s works. There is a schematic representation of the configuration of isotactic, syndiotactic, and atacticvinyl polymers in Figure 8-14, which was shown in Natta’s Nobel3868 Space-Group SymmetriesFigure 8-13. Top: The structure and translation period of the polyethylene chainmolecule; Bottom: Symmetry elements of the polyethylene chain molecule.Figure 8-14. Schematic representation of the structures of isotactic (top); syndiotactic (middle); and atactic (bottom) vinyl polymers [15].8.4. Rods, Spirals, and Similarity Symmetry387lecture and has been quoted in many other publications ever since.The labels to characterize the configurations were carefully chosen,using Greek words.
Isos is the same and tasso means to put in order,hence “isotactic”; syn dyo means every two and combined with reference to order yields “syndiotactic”; finally “atactic” means lackingorder. The isotactic structure can be imagined as achieved by repetition through simple translation, whereas the syndiotactic structureby repetition through glide reflection.
Usually, these polymers take ahelical structure whose determination was greatly aided by the factthat, by then, there was a lot of information available about the structure of the alpha-helix of proteins as well as about the double helix ofnucleic acids.Biological macromolecules are often distinguished by their helicalstructures to which one-dimensional space-group symmetries areapplicable. Figure 8-15a shows Linus Pauling’s sketch of a polypeptide chain, which he drew while he was looking for the structureof alpha-keratin.
When he decided to fold the paper, he arrived atthe alpha-helix. The solution may have come in a sudden moment,(a)(b)Figure 8-15. (a) Linus Pauling’s sketch of the polypeptide chain in 1948. Thealpha-helix came together eventually when he folded the paper along the creases[16]; (b) Computer-drawing of the alpha-helix (courtesy of Ilya Yanov, Jackson,Mississippi).3888 Space-Group SymmetriesFigure 8-16. Helical segments; Left: isotactic poly-4-methylpentene having 3.5monomeric units per pitch; Right: alpha-helix having 3.7 amino acid residues perpitch [17].but Pauling had been working on the problem on and off for almosttwo decades. Figure 8-15b also depicts an alpha-helix structure as acomputer drawing.
A segment of one of Natta’s polymers and that ofalpha-helix are given in Figure 8-16. The structure of alpha-helix isaccomplished by intramolecular hydrogen bonds.When James Watson and Francis Crick reported their suggestionfor the structure of deoxyribonucleic acid (DNA) [18], it had important novel features.
One was that it had two helical chains, each coilingaround the same axis, but having opposite direction. The two helicescomplement each other, which is a simple consequence of the twofoldsymmetry of the double helix with the axis of twofold rotation beingperpendicular to the axis of the molecule. The other novel feature wasthe manner in which the two chains are held together by the purineand pyrimidine bases.
They wrote: [the bases] “are joined in pairs,as a single base from one chain being hydrogen-bonded to a singlebase from the other chain, so that the two lie side by side with identical z-coordinates. One of the pair must be purine and the other apyrimidine for bonding to occur” [19]. A little later they mention that8.4. Rods, Spirals, and Similarity Symmetry389“. . .if the sequence of bases on one chain is given, then the sequenceon the other chain is automatically determined” [20]. Thus, symmetryand complementarity appear most beautifully in this model. The paperculminates in a final remark that sounds like a symmetry descriptionof a simple rule to generate a pattern: “It has not escaped our noticethat the specific pairing we have postulated immediately suggests apossible copying mechanism for the genetic material” [21].A diagrammatic sketch of the double helix by Odile Crick illustrates this article, whose harmony and proportions have remainedunsurpassed.
It is also interesting that when Crick decided to erect ametallic sculpture above the entrance to his Cambride home, he chosea single helix rather than a double helix (Figure 8-17). The reason mayhave been that the realization of the helical structure of biologicalmacromolecules was a most important milestone. It was also Crickand his two colleagues who worked out the theory of diffraction ofthe polypeptide (single) helix [22], which was then applicable to thedescription of diffraction by any helical structure. In their study, Crickand his associates assumed a structure that was based on Pauling’salpha-helix.Figure 8-17.
Francis Crick’s Golden Helix structure above the entrance to theCricks’ former home in Cambridge, UK. (Photograph by the authors).3908 Space-Group SymmetriesWe have mentioned above the long quest by Pauling for the structure of alpha keratin. Symmetry considerations helped him greatlyin finding the solution. At one point he remembered a mathematical theorem that referred to a general operation that converts anasymmetric object into another asymmetric object.
The asymmetricobject might be an amino acid and the operation was a rotation–translation—that is, a rotation around an axis combined with a translation along the axis—and repetition of this operation produces a helix.This put Pauling onto the right course [23]. Artistic representations ofthe double helix are shown in Figure 8-18. The double helix is heldtogether by the hydrogen bonds of the base pairs in between the twohelices. This is depicted in one of the artistic expressions of the doublehelix (Figure 8-18c).It has been a question of contention whether, and to what extent,Erwin Chargaff’s findings about the 1:1 ratios of purine and pyrimidine bases in the DNA molecules of diverse organisms helped Watsonand Crick’s discovery [24]. For our discussion it is instructive tonote the importance and relevance of Chargaff’s observation.
Whathe did was nothing less than the discovery of a pattern where thereseemed to be none. Looking at the raw data and their scatter on thepurine and pyrimidine contents of DNA from various organisms, itis understandable that Chargaff felt “a great reluctance to accept suchFigure 8-18. Artistic representations of the double helix; (a) Spirals Time—TimeSpirals by Charles A. Jencks; (b and c) Sculpture of the double helix in the lobbyof the Watson School; both sculptures are at Cold Spring Harbor Laboratory, ColdSpring Harbor, New York (photographs by the authors).8.4. Rods, Spirals, and Similarity Symmetry391regularities” [25]. However, at the end, he did, and communicated the1:1 ratios.The structure and assembly of the tobacco mosaic virus (TMV) isan interesting example of helical symmetry.
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