Use of Polar Maps in Conformational Analysis James P. OunswMth and Larry Weiler' University of British Columbia, Vancouver. BC, Canada V6T 1Y6 Recently there has been a resurgence of interest in the chemistrv of laree rines. Much of this interest stems from the novei and u~usua~compounds that have been found to contain laree rines, includine the macrolide antibiotics and crown eth& ( 1 ) : These r o ~ p o u n d shave attracted a great deal of attention from the synthetic chemist as well. Often it has been found that the chemistry or the physical properties of these macrocyclic compounds depends on the conformation ofthe lnrge&s (2).~articulariyelegant usesof ronformational cuntrnl in the rhemistry of medium and large rings hnverome from the lal~orat(myof M . C. Still (8).The conformatimal analysis of large rings can be rather complex ( 2 6 ) . Freauentlv this comnlexitv leads to difficulties in internreting the physical or chemical properties of m a c r o c y c ~sysi~ tems. We would like to sueeest a relativelv " simole nrocedure to identify different or, e&ally as important, similar conformations of large rings. To illustrate the difficulty, consider I and I1 which are perspective drawings of cyclotetradecane and cyclotetradecanone. &
Figure 1. Numbering of the atoms (a) and the torsional angles (b) in cyclotetradecane.
.
Figure 2. Newman projections for bonds in conformationI of cane: (a) band 6, (b) bond 10, (c)band 6 from the "bacw'.
These drawings fail to show any similarity between the conformations of the two compounds. However. I1 can he rotated into a second perspeciive where some similarities to I hecome apparent. This type of three-dimensional manipulation is a challenge, even to modern computers, and it is error-prone and time-ronsuming if left to an individual with a set bfmolecuIar models. A methud that quickly and unequivocally yields the conformation of the cyclic system in-a manner that makes comparisons with other conformations clear and simple would he exceedingly useful. Such a method depends on the torsional angles of the bonds within the ring. An analysis of the sien of the torsional aneles has been anolied .. to the conform&ional analysis of s m k and medium ring systems ( 4 ) and to an interpretation of the stereoselectivity in the reactions of small and medium ring compounds (5). When this is combined with the use of dihedral maps (6).one has a simple and powerful method to compare the conformations of any ring. The complete set of torsional angles formed by the ring atoms will always uniquely define the conformation of the ring. This procedure is particularly effective when applied to large ring systems as illustrated below. First, we shall review the definitions and conventions associated with the use of torsional angles, and then propose a small numher of additional conventions to use in rings. This
'
Author to whom correspondence may be addressed. *Primes are used in numbering the atoms of the ring to avoid confusion with the boldface unprimed numbers for the torsional angles. 568
Journal of Chemical Education
cyclotetrade-
discussion will be limited to 14-member rings; however, the concepts are applicable to any ring system. Endocycilc Torslonal Angles Consider the unsuhstituted cyclotetradecane. We shall onlv consider the endocvclic torsional aneles. - . that is, those toriiona~angles involvLg the carbon atoms of thk ring. When this procedure is extended to heterocvclic rines. we still only consider the endocyclic t o r ~ i o n a l a n ~ l eif s . the atoms are numbered as shown in Figure l a , then the torsional angle between the planes of atoms 14'-1'-2' and 1'-2'-3' is defined as The torsional angle mav be obtained bv examining a molecular model of cyclotetradecane, or perhaps from X-ray crystallographic data or from molecular mechanics calculations. The atoms are arbitrarily numbered in a clockwise fashion in Figure la, hut this does not affect the subsequent analysis. Order and Sign of Torslonal Angles Usine the above convention the torsional angles are numbered as shown in Figure l h (care must he exercised to avoid confusion with the numbering of the atoms). The magnitude and sign of the torsional angle is determined by looking a t a Newman ~roiectionalone the hond. with the lower numbered atom ciosest to the observer-except for the last angle in the ring. This is illustrated for angle 6 of I in Figure 2a. The torsional angle is positive, if the direction of rotation from the front plane to the rear plane is clockwise. Similarly the torsional angle is negative if the direction of rotation from the front plane to the rear plane is counterclockwise, as shown for hond 10 of I in Figure 2h.
and it is difficult to compare two conformations which are numbered from different starting points. It has been recognized that pictorial representations of the signs of torsional angles contain symmetry elements of the molecule (2, 4, 5). Such a representation of the chair and boat conformation of cyclohexane is illustrated in I11 and IV, and the [3434] conformation I of cyclotetradecane as in V.
Figure 3.Newmsn projections far two trans butane bonds in conformation I of cyclotetradecane: (a) band 4, (b) bond 8.
Table 1. Torsional Angles of the [3434] Contormatlon I of Cyclotetradecane Bond
Torsional Angle
1 2 3 4 5 6
-180 60 60 -180 60 60 -180 180 -60 -60 180 -60 -60
7 8 9 10 11 12 13
For small rings usually only the sign of the torsional angle is required to determine uniquely the symmetry of a conformation. Clearly that is not so with large rings which may have several torsional angles with the same sign but very different magnitudes, for example honds 6 and 8 in the above conformation of cyclotetradecane (I). If the signs and magnitude of the torsional angles are added to the planar drawings above to give VI for cyclotetradecane, we see additional information.
I t should be noted that the torsional angle is the same regardless of whether it is viewed from the "front" or the "back", as is illustrated for hond 6 in Figure 2c. Therefore, the sign and magnitude of the torsional andes are i n d e ~ e n dent of the clockwise or counterclockwise ;urnhering of the atoms in Figure la. Torslonal Angles of f180° An ambiguity arises for bonds with a trans butane arrangement, for example, bonds 4 and 8 in I. As illustrated in Figure 3, these torsional anales can be i180°. The convention we suggest in this case is to consider the remaining atoms of the ring and define the endocyclic torsional angle as the angle that is formed within the ring. Thus the torsional angle 4 is -180' and the torsional anele 8 is +180°. I t should be noted that torsionalangles of +18@ and -180" are equivalent, and torsional angles of +I75 and -175 are very nearly equal-a bending of only 10° converts one into the other. When experimentally determined torsional angles obtained from X-ray data or calculated values from molecular mechanics calculations are used, it is important to recognize this near equivalency.
Although informative, these depictions are difficult to work out-for example, identifying the "corner" atoms in these rectaneular re~resentations.T o avoid some of the problems associated with adding the torsional angles to planar re~resentations,we looked for a eraohic disnlav . . of torsionalmglevs hond. One possiblity wkch~,econsideredwas the "linear grnph" of rhe torwnal anale versus bond shown in Figure 41~;mmetry elements a r e n o t obvious from this
Polar Maps of Torslonal Angles There are several methods to tabulate or display torsional angles. One possibility is a simple table, such as Table 1for the 134341 conformation (I) of c~clotetradecane.~ However, any molecular symmetry is not obvious in this tabulation,
For adiscussion of the [abcd] nomenclature of large ring conformations. see ref 2b.c.
Figure 4. Linear graph of torsional angle vs. band forthe [3434]conformation i of cyclotetradecane.
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Number 7 July 1987
589
figure. Patterns emerge which are suggestive, hut i t is difficult to compare two canformations. Figure 5 gives a second reoresentation of the 134341 conformation of cvclotetradecane, hut with a different origin. Wv w r e not confident that this method w ~ ~alwavr l d allow us to identifv identical conformations. However, abolar map of torsio"a1 angle versus hond overcomes all of the shortcomings - in the above methods. T o our knowledge the first use of a polar map of torsional angles was that of Ogura et al. in their illustration of the conformation of oleandomycin from the X-ray data (6).This group also used polar maps to describe the conformation of other 14-member macrolide antibiotics (7). Generation of Polar Maps of Torslonai Angles In order to prepare a polar map, the torsional angles of a compound in a particular conformation (obtained from an inspection of a molecular model, from X-ray data, or from molecular mechanics calculations) are plotted on a graph consisting of concentric circles which represent the dihedral angles. For convenience we show only the circles for the dihedral angles of On,f60°, f120°, and f180°. The torsional angle for each endocyclic hond is plotted and numbered as shown in the examnles given below. Inversion of the sign i f all of the torsional angles results in a map of the mirror image conformation. Thus enantiomers will have polar maps that are mirror images of one another (Fig. 6). In the case of cyclotetradecane these enantiomers are identical. However, this may not necessarily he the case for substituted cyclotetradecanes or for other conformations of cyclotetradecane. In generating a polar map we must remember that torsional aneles close to 180" are nearlv. eaual . to torsional aneles closeio -180'. Therefore, when comparing the polar maps of two conformations. we must consider those maps in which the torsional angles ciose to 180" are replaced by their mirror imaee angle. ~ j h epo'iar map of the [3434] conformation of cyclotetradecane is shown in Figure 6. The polar map of the closely related 15344)conformation is shown inFigure 7. It is easy to differentiate these two conformations from the shape and symmetry of the two polar maps in spite of the fact that the two conformations differ only a t the torsional angles 5, 6, and 7. Indeed translation of atoms C6' and C7' interconvert these two confc, mations. A superposition of these two maps readily illustrates the similarities and differences between the two conformations. We can obtain several of the symmetry elements in the [3434] conformation in Figure 6. A C2 axis runs through the torsional angles 4 and 11, i.e. through the middle of the bonds between C4'-C5' and Cl1'-C12'. The sign of the torsional angle is the same after this symmetry operation or for any C , axis. The molecule I has a center of symmetry. The sign of the torsional angle is inverted on the operation of a center of symmetry or for any S. axis, including a mirror
plane. The operation of the molecular center of symmetry can be seen in the middle of the polar map (Fig. 6). The mirror d a n e which runs throueh atoms C1' and C8'is not as obvious in the polar map. In general, it is more difficult to locate a S. axis in oolar mans because of the sign inversion. Note the 'cpaxis through Carbons 3' and 10' i f the [3344] conformation.
.~
Examples of Complex Polar Maps The polar map of any one conformation is a unique descri~tionof that conformation. Thus the conformation of a c o ~ p l e xmolecule can he determined by comparison of its rnapa,ith that ot'n known cunforn~ationalthc unsuhstituted molecule for example. This is illustrared for the polar map or' the bromoacetate VII
The X-rav structure of VII was disordered and the molecule existed as-a 1:l mixture of two conformations in the solid (8). The torsional aneles - for these two conformations are -eiven in Table 2. The polar maps of these two conformations are shown in Figure 8. These maps can be compared to the maps for the ideal [3434] conformation (Fig. 6a) and for the ideal [3344]
Figure 6. Polar map of the torsional angler of the cyclotetradecane: (a)I. (b) enantiamer 1'.
134341 conformation I of
Figure 7. Polar map of the torsional angles of the 133441 conformation of cyciotetradecane,assuming only trans and gauche butane angles. 570
Journal of Chemical Education
Table 2.
Torsional Angles of t h e T w o Conformations of t h e cis Bromoacetate VII from t h e X-ray D a t a (ref
[3434]
8).
[3344]
Bond
Torsional Angle
Bond
Torsional Angle
1 2 3 4 5 6 7 8
-144 65 56 165O 77 76 -164 159 -64 -69 170 -66 -94 176
1 2 3 4 5 6 7 8 9 10 11 12 13 14
-144 65 56 -171 163 -69 -81 -177b -64 -69 170 -66 -94 176
9 10 11 12 13 14
a1 angles 5.6, and 7. This follows from the observation that the X-ray structure is disordered a t C6' and C7'. The polar map of the ideal 134341 conformation can he superimposed on the polar map of the X-ray structure to determine where the observed conformation deviates from the ideal. This is facilitated if one of the polar maps is plotted on a transparency. Also one polar map can be rotated relative to the second, so i t does not matter if different origins are used in numbering the bonds in the maps. Pattern recognition is the key and chemists are adept at this type of analysis. In Fiaure 9. the ideal 134341. nolar . man is superimp&d on the ~ k l n map r for ihr (34311 ronti,rmn;ion ot'\'11 from the X-ravdnta. Th(~onlvsirnifirnntdifference in these maps is at bonds 1 and 13. These bonds contain the lactone functional group and the methyl-substituted carbon. Molecular mechanics calculations on the trimethyl lactone VIII
'metasional angles rspated in reference 8sre fa me snantiomsr. ~hssevaiuso in
me table are obtained by simply changing me sign of me angles reponed in reference 8 This illustratesthe ease of imerconverting e~ntiorners. 'The slgn of this angle is changed in me polar map.
conformation (Fig. 7) of cyclotetradecane. I t is clear from this analysis that the two X-ray conformations of VII are the [3434] and the [3344] conformation. This facile and unambiguous identification of conformations should he contrasted with examination of tables of torsional angles or examination of perspective drawings, neither of which allowed us to identifv these conformations immediatelv and unamhieuously. 1n addition, the two polar maps in Figure 8 show t i a t the only significant differences in the maps occur a t torsion-
suggested that the lowest energy conformation has the torsional angles given in Table 3. The polar map for this conformation is illustrated in Figure 10. This pattern is immediately recognized as the [3434] conformation of the 14-member ring. NMR analysis of VIII is consistent with conformation IX for this molecule in solution. Torsional Angles of t h e Lowest Conformation of 3.3Dimethyl-13-tetradecanolide(VIII) l r o m M M 2 Calculations
T a b l e 3.
Bond
(01
Ibl
Torsional Angle
1
17P
2 3 4 5 6 7
72
8 9 10
H Figure 8. Polar maps of lhe torsional angles of lhe two conformations of bromoacetate VII: (a) 134341 conformation. (b) [3344] conformation,
12 13
64 -174 57 53 -176 175 -54 -55 178 -67 -68
*mesign of this angle is changed in the polar map.
Figure 9. Superposition of the polar maps of me ideal [3434] conformation (doUed lines), and the [3434] conformation of VII from the X-ray data (solid lines).
Figure 10. Polar map of the torsional angles of Vlll from Table 3.
Volume 64 Number 7
July 1987
571
Table 4.
Torsional Angles of Methyl lsosartortuate (X) from the X-ray Data (ref 9) Ring A
Ring B Torsional Angle
Bond
Torsional Angle
Bond
1 2 3 4 5 6
56 100 178 73 -100 -179
21 22 23 24 25
-127 - 179. -82 -69 157
26
-71
Figure 11. Polar maps of the twsional angles ofthe two 14-member rings in methyl isosanortuate (X): (a) ring A. (b) ring B. The dotted lines in (b) show the ideal [3434] conformation.
*me*Ian m this angle is chaw&
in the posr map.
Recently, the X-ray structure of methyl isosartortuate (X)
of the fused cyclohexene and the tetrahydrofuran rings. T h e polar map of ring A suggests that this ring does not adopt any regular conformation. The structure is highly disordered in the region C5'-Car, which makes up a portion of ring A. In summary, the use of polar maps generated from molecular mechanics calculations, inspection of molecular models, or from X-rav crvstalloera~hicdata can he used to determine the conformkionsof iing systems. The method is particularly powerful in the conformational analvsis of laree ~. rings.
-
Acknowledgment We are grateful to J. Trotter and J. Clardy for the X-ray data, to Paul Phillips for the programming for the generation of the polar maps, and to the Natural Sciences and Engineering Research Council for financial support of this work and for a post-graduate fellowship to JPO. Literature Clted
has been published (9). This molecule has two 14-member carhocyclic rings. The torsional angles for these rings are given in Table 4 and the polar maps for the two rings are illustrated in Figure 11. For the polar map in Figure l l b , the signs of the torsional angles associated with bonds 22 and 29 were inverted. From these polar maps we see that ring B closely approximates the [3434] conformation. The largest deviations from the ideal conformation occurs in the region
5971. 2. la) Dale. J. Angew. Chem. lnf. Ed. Engl. 1966.5, 1MN: (b) Dale, J. Top. Stersochsm. 1976.9.199: ic) Dale. J. Act. Cham. Scond. 1973.27,1115. 3. See for example: (a) Still, W. C.: Novaek, V. J. J. Am. Chem. Sac. 19&1.108,1148: (b) Still,W.C.;Murats,S.:Reui~l.G.:Yoahihsra,K. J.Am.Chem.Soc. 1983,105.625:ic) Stili,W.C.:Gslynker,I.J . A m Chem.Soe. 1982,104,177CldJStill, W.C.:Galynker. I.Tetrahedron 19P' 97 Q'1Qq 4. is1 Bucourt. R. Too. Stweochem. 1974,8. 159: IbJDe Clercq, P. J., Tefmhedron 1984, 40,WZBsnd xf&r.cestl rerein. 5. la) Toromanoff. E. Tetrahadm" 1980,36,2809: IbJDe Clercq, P.J. Tetrahedron, 1981, 37, 4277 6. Owra,H.;Furuhata,K.: Harads.Y.: 1itaka.Y.J Am. Chem.Soc. 1978,100,6733. 7. Owra,H.;Furuhats, K.; Kuwano, H.; Suruki, M. Tetrahedron 198i,37(Suppl. 1). 165. 8. Ponnu~wsmy,M. N.;Trotter, J. Acfo Crysi. 1984 C41, 1109. 9. Jingyu,S.; Kanghou, L.;Tangsheng, P.; Cun-heng. H.:Clmdy, J. J. Am. Chem. Soe. 1986,108, 177.
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Anna Louise Hoffman Award for Outstanding Achievement In Graduate Research The Anna Louise Hoffman Award for Outstanding Achievement in Graduate Research, sponsored by Iota Sigma Pi, a professional honorary society for women in chemistry, has been awarded to Jo Ann Canich of Texas A & M University. Canich has been working with A. F. Cotton on oxidative addition chemistry of Group V and VI metal dimers and has a 4.0 average. Prior to attending Texas A & M, she received both a BS and MS from Portland State University in Oregon with Gary Gard. She is amember of ACS, Phi Kappa Phi, PhiBetaUpailon, American Institute of Chemists, and Iota Sigma Pi and isapast recipient of both theSherman Schaffer Award of the Portland Section of ACS and the A. E. Martell Graduate Student Professmnal Development Award.
572
Journal of Chemical Education
