{"id":9556,"date":"2013-02-17T09:30:06","date_gmt":"2013-02-17T09:30:06","guid":{"rendered":"http:\/\/www.ch.imperial.ac.uk\/rzepa\/blog\/?p=9556"},"modified":"2013-02-17T09:30:06","modified_gmt":"2013-02-17T09:30:06","slug":"linking-numbers-and-twist-and-writhe-components-for-two-extended-porphyrins","status":"publish","type":"post","link":"https:\/\/www.rzepa.net\/blog\/?p=9556","title":{"rendered":"Linking numbers, and twist and writhe components for two extended porphyrins."},"content":{"rendered":"
\n

My last comment as\u00a0appended to the previous post<\/a>\u00a0promised to analyse two so-called extended porphyrins for their topological descriptors. I start with the\u00a0C\u00e3lug\u00e3reanu\/Fuller<\/a> theorem\u00a0 which decomposes the topology of a space curve into two components, its twist (Tw) and its writhe (Wr, this latter being the extent to which coiling of the central curve has relieved local twisting) and establishes a topological invariant called the linking number<\/a>[1]<\/a><\/span><\/p>\n

\u00a0Lk = Tw + Wr\u00a0<\/strong><\/p>\n\n\n\n
\n
\"Click

HIYTAL. Click for 3D.<\/p><\/div>\n<\/td>\n

\n
\"SELQUW.

SELQUW. Click for 3D.<\/p><\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Visual inspection of the models above (I really do encourage you to click on the images to load the 3D coordinates) reveals that HIYTAL[2]<\/a><\/span> has a major coil that forms one and a half helical turns in a clockwise direction, and a loop connecting the ends of the coil which forms a half-helical turn in an anti-clockwise direction. SELQUW[3]<\/a><\/span>\u00a0has a major coil comprising one and half helical turns in an anti-clockwise direction and a connecting loop which also coils anti-clockwise. So the former sustains a total of one<\/strong>\u00a0full (clockwise) helical turn and the latter two<\/strong>\u00a0full (anti-clockwise) helical turns.<\/p>\n

The nomenclature for helical molecules<\/a> includes a chiral descriptor P (for a positive helical turn) and M (for a negative helical turn). What such a descriptor does not do is quantify the total number of helices describing the topology. So I suggest we use instead the linking number Lk. Instead of P and M, we have positive and negative integers (in units of 2\u03c0) providing this quantitative information.<\/p>\n

The linking number analysis for these two molecules comes out as below.\u2021<\/sup> I have multiplied the linking number unit from 2\u03c0 to 1\u03c0 for a reason that I will explain shortly:<\/p>\n\n\n\n\n\n
\u00a0<\/td>\n\u03c0-electrons<\/td>\nLk<\/td>\nTw<\/td>\nWr<\/td>\n\u0394r<\/sub> (meso)<\/td>\n<\/tr>\n
SELQUW<\/td>\n56=4n<\/td>\n-4<\/td>\n-1.34\u00a0<\/td>\n-2.66<\/td>\n0.048<\/td>\n<\/tr>\n
HIYTAL<\/td>\n62=4n+2<\/td>\n+2\u00a0<\/td>\n+0.46<\/td>\n+1.54<\/td>\n0.045<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

You can see that the linking numbers (and their signs) correspond exactly to the visual analysis of the helical turns above. My reason for including the factor of 2 is that it enables us to make a further link to the H\u00fcckel aromaticity rule:<\/p>\n

    \n
  1. Cyclic conjugated systems are aromatic<\/strong> if they contain 4n+2 \u03c0-electrons and have an even or zero linking number (in units of 1\u03c0).\u00a0<\/span><\/li>\n
  2. Cyclic conjugated systems are aromatic<\/strong> if they contain 4n\u00a0\u03c0-electrons and have an odd linking number (in units of 1\u03c0).\u00a0<\/li>\n
  3. Cyclic conjugated systems are anti-aromatic<\/strong> if they contain 4n \u03c0-electrons and have an even or zero linking number (in units of 1\u03c0).\u00a0<\/li>\n
  4. Cyclic conjugated systems are anti-aromatic<\/strong> if they contain 4n+2 \u03c0-electrons and have an odd linking number (in units of 1\u03c0).\u00a0<\/li>\n<\/ol>\n

    By these rules, SELQUW contains (by the shortest path) 56 \u03c0-electrons, belongs to the 4n electron rule (n=14) and hence is formally anti-aromatic (rule 3 above). HIYTAL has a path of 62-electrons, belongs to the 4n+2 rule (n=15) and hence is formally aromatic (rule 1 above).\u00a0<\/p>\n

    For systems with so many (correlated) electrons, it is probably tenuous to make a connection between the bond-length alternation at the meso position and the aromaticity (or lack of it). I comment only that HIYTAL converts more of the coiling into writhing of the central curve than does SELQUW, and this destroys less\u00a0\u03c0-\u03c0 overlap by reducing the overall degree of twisting. I might also speculate that nevertheless a modest degree of twisting may impact upon the intrinsic distortivity of\u00a0\u03c0-electrons in cyclically conjugated systems (such as that in benzene[4]<\/a><\/span>), as noted in this earlier post<\/a>. Such effects may make the interpretation of bond-alternation in such helical systems more difficult.<\/p>\n

    \n

    \u2021 A program for calculating these components can be found here<\/a>. For a fun-packed journey through linking numbers and the association with valentine cards, go see this post<\/a> here!<\/sup><\/p>\n

    References<\/h2>\n
    1. S.M. Rappaport, and H.S. Rzepa, \"Intrinsically Chiral Aromaticity. Rules Incorporating Linking Number, Twist, and Writhe for Higher-Twist M\u00f6bius Annulenes\", Journal of the American Chemical Society<\/i>, vol. 130, pp. 7613-7619, 2008. https:\/\/doi.org\/10.1021\/ja710438j<\/a>\n\n<\/li>\n
    2. S. Shimizu, W. Cho, J. Sessler, H. Shinokubo, and A. Osuka, \"<i>meso<\/i>\u2010Aryl Substituted Rubyrin and Its Higher Homologues: Structural Characterization and Chemical Properties\", Chemistry \u2013 A European Journal<\/i>, vol. 14, pp. 2668-2678, 2008. https:\/\/doi.org\/10.1002\/chem.200701909<\/a>\n\n<\/li>\n
    3. S. Shimizu, N. Aratani, and A. Osuka, \"<i>meso<\/i>\u2010Trifluoromethyl\u2010Substituted Expanded Porphyrins\", Chemistry \u2013 A European Journal<\/i>, vol. 12, pp. 4909-4918, 2006. https:\/\/doi.org\/10.1002\/chem.200600158<\/a>\n\n<\/li>\n
    4. S. Shaik, A. Shurki, D. Danovich, and P.C. Hiberty, \"A Different Story of \u03c0-DelocalizationThe Distortivity of \u03c0-Electrons and Its Chemical Manifestations\", Chemical Reviews<\/i>, vol. 101, pp. 1501-1540, 2001. https:\/\/doi.org\/10.1021\/cr990363l<\/a>\n\n<\/li>\n<\/ol>\n\n<\/div> ","protected":false},"excerpt":{"rendered":"

      My last comment as\u00a0appended to the previous post\u00a0promised to analyse two so-called extended porphyrins for their topological descriptors. I start with the\u00a0C\u00e3lug\u00e3reanu\/Fuller theorem\u00a0 which decomposes the topology of a space curve into two components, its twist (Tw) and its writhe (Wr, this latter being the extent to which coiling of the central curve has relieved […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[6],"tags":[996,964],"class_list":["post-9556","post","type-post","status-publish","format-standard","hentry","category-interesting-chemistry","tag-conjugated-systems","tag-helical-systems"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p1gPyz-2u8","jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.rzepa.net\/blog\/index.php?rest_route=\/wp\/v2\/posts\/9556","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.rzepa.net\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.rzepa.net\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.rzepa.net\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.rzepa.net\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=9556"}],"version-history":[{"count":0,"href":"https:\/\/www.rzepa.net\/blog\/index.php?rest_route=\/wp\/v2\/posts\/9556\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.rzepa.net\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9556"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.rzepa.net\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9556"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.rzepa.net\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9556"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}