Improved algorithms for symmetry analysis: structure preserving permutations

Abstract

We propose an improved algorithm for calculating Avnir’s continuous symmetry and chirality measures of molecules. These measures evaluate the deviation of a given structure from symmetry by calculating the distance between the structure and its nearest symmetric counterpart. Our new algorithm utilizes structural properties of the given molecule to increase the accuracy of the calculation and dramatically reduce the running time by up to tens orders of magnitude. Consequently, a wide variety of molecules of medium size with ca. 100 atoms and even more can be analyzed within seconds. Numerical evidence of the algorithm’s efficiency is presented for several families of molecules such as helicenes, porphyrins, dendrimers building blocks, fullerene and more. The ease and efficiency of the calculation make the continuous symmetry and chirality measures promising descriptors for integration in quantitative structure–activity relationship tools, as well as chemical databases and molecular visualization software.

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References

  1. 1.

    M. Petitjean, Entropy 5, 271–312 (2003)

    CAS  Article  Google Scholar 

  2. 2.

    P.G. Mezey, Mol. Phys. 104, 723–729 (2006)

    CAS  Article  Google Scholar 

  3. 3.

    P.G. Mezey, J. Math. Chem. 45, 544–549 (2009)

    CAS  Article  Google Scholar 

  4. 4.

    P.G. Mezey, K. Fukui, S. Arimoto, K. Taylor, Int. J. Quantum Chem. 66, 99–105 (1998)

    CAS  Article  Google Scholar 

  5. 5.

    G.M. Crippen, Curr. Comput. Aided Drug Des. 4, 259–264 (2008)

    CAS  Article  Google Scholar 

  6. 6.

    H. Zabrodsky, S. Peleg, D. Avnir, J. Am. Chem. Soc. 114, 7843–7851 (1992)

    CAS  Article  Google Scholar 

  7. 7.

    M. Pinsky, C. Dryzun, D. Casanova, P. Alemany, D. Avnir, J. Comput. Chem. 29, 2712–2721 (2008)

    CAS  Article  Google Scholar 

  8. 8.

    H. Zabrodsky, D. Avnir, J. Am. Chem. Soc. 117, 462–473 (1995)

    CAS  Article  Google Scholar 

  9. 9.

    S. Alvarez, P. Alemany, D. Avnir, Chem. Soc. Rev. 34, 313–326 (2005)

    CAS  Article  Google Scholar 

  10. 10.

    M. Pinsky, D. Avnir, Inorg. Chem. 37, 5575–5582 (1998)

    CAS  Article  Google Scholar 

  11. 11.

    S. Alvarez, Dalton Trans. 13, 2209–2233 (2005)

    Article  Google Scholar 

  12. 12.

    S. Alvarez, P. Alemany, D. Casanova, J. Cirera, M. Llunell, D. Avnir, Coord. Chem. Rev. 249, 1693–1708 (2005)

    CAS  Article  Google Scholar 

  13. 13.

    D. Yogev-Einot, D. Avnir, Tetrahedron Asymmetry 17, 2723–2725 (2006)

    CAS  Article  Google Scholar 

  14. 14.

    C. Dryzun, Y. Mastai, A. Shvalb, D. Avnir, J. Mater. Chem. 19, 2062–2069 (2009)

    CAS  Article  Google Scholar 

  15. 15.

    I. Tuvi-Arad, D. Avnir, J. Math. Chem. 47, 1274–1286 (2010)

    CAS  Article  Google Scholar 

  16. 16.

    I. Tuvi-Arad, D. Avnir, J. Org. Chem. 76, 4973–4979 (2011)

    CAS  Article  Google Scholar 

  17. 17.

    I. Tuvi-Arad, D. Avnir, Chem. Eur. J. 18, 10014–10020 (2012)

    CAS  Article  Google Scholar 

  18. 18.

    I. Tuvi-Arad, T. Rozgonyi, A. Stirling, J. Phys. Chem. A 117, 12726–12733 (2013)

    CAS  Article  Google Scholar 

  19. 19.

    I. Tuvi-Arad, A. Stirling, Isr. J. Chem. 56, 1067–1075 (2016)

    CAS  Article  Google Scholar 

  20. 20.

    M. Bonjack-Shterengartz, D. Avnir, Proteins Struct. Funct. Bioinf. 83, 722–734 (2015)

    CAS  Article  Google Scholar 

  21. 21.

    S. Keinan, D. Avnir, J. Am. Chem. Soc. 122, 4378–4384 (2000)

    CAS  Article  Google Scholar 

  22. 22.

    M.H. Jamroz, J.E. Rode, S. Ostrowski, P.F.J. Lipinski, J.C. Dobrowolski, J. Chem. Inf. Model. 52, 1462–1479 (2012)

    CAS  Article  Google Scholar 

  23. 23.

    D. Milner, S. Raz, H. Hel-Or, D. Keren, E. Nevo, Pattern Recogn. 40, 2237–2250 (2007)

    Article  Google Scholar 

  24. 24.

    I. Saragusti, I. Sharon, O. Katzenelson, D. Avnir, J. Archaeol. Sci. 25, 817–825 (1998)

    Article  Google Scholar 

  25. 25.

    R. Iovita, I. Tuvi-Arad, M.H. Moncel, J. Despriee, P. Voinchet, J.J. Bahain, Plos One 12, e0177063 (2017)

    Article  Google Scholar 

  26. 26.

    C. Dryzun, A. Zait, D. Avnir, J. Comput. Chem. 32, 2526–2538 (2011)

    CAS  Article  Google Scholar 

  27. 27.

    O. Katzenelson, J. Edelstein, D. Avnir, Tetrahedron Asymmetry 11, 2695–2704 (2000)

    CAS  Article  Google Scholar 

  28. 28.

    Molecular Operating Environment (MOE), Chemical Computing Group Inc, 1010 Sherbooke St. West, Suite #910, Montreal, QC, Canada, H3A 2R7 (2016)

  29. 29.

    C.R. Groom, I.J. Bruno, M.P. Lightfoot, S.C. Ward, Acta Cryst. B72, 171–179 (2016)

    Google Scholar 

  30. 30.

    J.-J. Zhang, J. Glaser, S.A. Gamboa, A. Lachgar, J. Chem. Crystallogr. 39, 1–8 (2009)

    Article  Google Scholar 

  31. 31.

    X.-L. Wang, D.-N. Liu, H.-Y. Lin, N. Han, G.-C. Liu, J. Inorg. Organomet. Polym Mater. 25, 671–679 (2015)

    CAS  Article  Google Scholar 

  32. 32.

    Z. Yi, X. Yu, W. Xia, L. Zhao, C. Yang, Q. Chen, X. Wang, X. Xu, X. Zhang, CrystEngComm 12, 242–249 (2010)

    CAS  Article  Google Scholar 

  33. 33.

    C. du Peloux, A. Dolbecq, P. Mialane, J. Marrot, F. Secheresse, Dalton Trans. 1259–1263 (2004). doi:10.1039/B401250J

  34. 34.

    H.-Y. Ma, L.-Z. Wu, H.-J. Pang, X. Meng, J. Peng, J. Mol. Struct. 967, 15–19 (2010)

    CAS  Article  Google Scholar 

  35. 35.

    M.P. Byrn, C.J. Curtis, Y. Hsiou, S.I. Khan, P.A. Sawin, S.K. Tendick, A. Terzis, C.E. Strouse, J. Am. Chem. Soc. 115, 9480–9497 (1993)

    CAS  Article  Google Scholar 

  36. 36.

    V.E. de Oliveira, C.C. Corrêa, C.B. Pinheiro, R. Diniz, L.F.C. de Oliveira, J. Mol. Struct. 995, 125–129 (2011)

    Article  Google Scholar 

  37. 37.

    P. Dastidar, I. Goldberg, Acta Crystallogr. Sect. C 52, 1976–1980 (1996)

    Article  Google Scholar 

  38. 38.

    S. McGill, V.N. Nesterov, S.L. Gould, Acta Crystallogr. Sect. E 69, m471 (2013)

    CAS  Article  Google Scholar 

  39. 39.

    L. Abbassi, Y.M. Chabre, N. Kottari, A.A. Arnold, S. Andre, J. Josserand, H.-J. Gabius, R. Roy, Polym. Chem. 6, 7666–7683 (2015)

    CAS  Article  Google Scholar 

  40. 40.

    W. C. Marsh, J. Trotter, J. Chem. Soc. A, 161–173 (1971). doi:10.1039/J19710000169

  41. 41.

    H.A. Alidağı, Ö.M. Gırgıç, Y. Zorlu, F. Hacıvelioğlu, S.Ü. Çelik, A. Bozkurt, A. Kılıç, S. Yeşilot, Polymer 54, 2250–2256 (2013)

    Article  Google Scholar 

  42. 42.

    H. R. Allcock, S. Al-Shali, D. C. Ngo, K. B. Visscher, M. Parvez, J. Chem. Soc. Dalton Trans. 3549–3559 (1996). doi:10.1039/DT9960003549

  43. 43.

    Y. Tümer, H. Bati, N. Çalişkan, Çd Yüksektepe, O. Büyükgüngör, Zeitschrift für anorganische und allgemeine Chemie 634, 597–599 (2008)

    Article  Google Scholar 

Download references

Acknowledgements

Supported by the Israel Science Foundation (Grant 411/15). We are sincerely grateful for fruitful discussions with Prof. David Avnir (The Hebrew University of Jerusalem). The programming of the new code was done by Itay Zandbank and Devora Witty (The scientific software company, Israel). We are thankful to Sagiv Barhoom (The Open University) for his help in programming and Yaffa Shalit (The Open University) for her help in testing the code. Researchers interested in using the CSM code are welcome to contact Dr. Tuvi-Arad.

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Correspondence to Gil Alon or Inbal Tuvi-Arad.

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This study was funded by the Israel Science Foundation (Grant Number 411/15).

Appendix: Proof of formulas (3) and (4)

Appendix: Proof of formulas (3) and (4)

For the sake of completeness, we present a rigorous proof of formulas (3) and (4). Given a permutation \(\pi \) of \(\{1,2,\ldots ,N\}\) satisfying \(\pi ^{n}=id\), we look at the minimum of \(\mathop {\sum }\limits _{k=1}^N {\left| {\mathbf{Q}_k -\mathbf{P}_k } \right| ^{2}} \) over all the \(\pi \)-symmetric structures \(\left\{ {\mathbf{P}_k } \right\} \). Let \(C_1 ,C_2 ,\ldots ,C_f \) be the cycles of \(\pi \), as in Eq. (8). Given the symmetry operation T, we have

$$\begin{aligned} \min \sum _{k=1}^N {\left| {\mathbf{Q}_k -\mathbf{P}_k } \right| ^{2}} =\min \sum _{j=1}^f {\sum _{k\in C_j } {\left| {\mathbf{Q}_k -\mathbf{P}_k } \right| ^{2}} } \end{aligned}$$
(13)

where the minimum is over all the structures \(\left\{ {\mathbf{P}_k } \right\} \) satisfying Eq. (2).

For each cycle \(C_j \) and \(k_0 \in C_j \), we have

$$\begin{aligned} C_j =\{\pi (k_0 ),\pi ^{2}(k_0 ),\ldots ,\pi ^{l}(k_0 )=k_0 \} \end{aligned}$$
(14)

where l is a divisor of n. We therefore have (as T is an isometry)

$$\begin{aligned} \sum _{k\in C_j } {\left| {\mathbf{Q}_k -\mathbf{P}_k } \right| ^{2}}= & {} \sum _{u=1}^l {\left| {\mathbf{Q}_{\pi ^{u}(k_0 )} -\mathbf{P}_{\pi ^{u}(k_0 )} } \right| ^{2}} \nonumber \\= & {} \sum _{u=1}^l {\left| {\mathbf{Q}_{\pi ^{u}(k_0 )} -T^{u}{} \mathbf{P}_{k_0 } } \right| ^{2}} =\sum _{u=1}^l {\left| {T^{-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} -\mathbf{P}_{k_0 } } \right| ^{2}} \end{aligned}$$
(15)

Therefore, the minimum in (13) can be found by minimizing the expression in (15) separately for each cycle, and summing over j.

It is a well known and easy to prove fact that given vectors \(\mathbf{x}_1 ,\ldots ,\mathbf{x}_m \), the vector \(\mathbf{x}\) which minimizes the sum \(\sum _{k=1}^m {\left| {\mathbf{x}_k -\mathbf{x}} \right| ^{2}} \) is the average, \(\mathbf{x}=\frac{1}{m}\mathop {\sum }\limits _{k=1}^m {\mathbf{x}_k } \). The minimum value is

$$\begin{aligned} \sum _{k=1}^m {\left| {\mathbf{x}_k -\mathbf{x}} \right| ^{2}}= & {} \sum _{k=1}^m {\left| {\mathbf{x}_k } \right| ^{2}} +m\left| \mathbf{x} \right| ^{2}-2\mathbf{x}\cdot \sum _{k=1}^m {\mathbf{x}_k } \nonumber \\= & {} \sum _{k=1}^m {\left| {\mathbf{x}_k } \right| ^{2}} -m\left| \mathbf{x} \right| ^{2}=\sum _{k=1}^m {\left| {\mathbf{x}_k } \right| ^{2}} -\frac{\sum _{k=1}^m {\mathbf{x}_k } \cdot \sum _{k=1}^m {\mathbf{x}_k } }{m} \nonumber \\= & {} \frac{1}{2m}\left( {2m\sum _{k=1}^m {\left| {\mathbf{x}_k } \right| ^{2}} -2\sum _{k=1}^m {\mathbf{x}_k } \sum _{k=1}^m {\mathbf{x}_k } } \right) \nonumber \\= & {} \frac{1}{2m}\sum _{k_1 =1}^m {\sum _{k_2 =1}^m {\left| {\mathbf{x}_{k_1 } -\mathbf{x}_{k_2 } } \right| ^{2}} } \end{aligned}$$
(16)

Therefore, the minimum of (15) is attained when

$$\begin{aligned} \mathbf{P}_{k_0 } =\frac{1}{l}\sum _{u=1}^l {T^{-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} } \end{aligned}$$
(17)

Since \(\pi ^{l}(k_0 )=k_0 \), and by (2), the summand in Eq. (17) is periodic in u with period l. Since l divides n, we conclude that

$$\begin{aligned} \mathbf{P}_{k_0 } =\frac{1}{n}\sum _{u=1}^n {T^{-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} } \end{aligned}$$
(18)

This proves Eq. (4).

By Eq. (16), the minimum value of Eq. (15) is

$$\begin{aligned} \min \sum _{k\in C_j } {\left| {\mathbf{Q}_k -\mathbf{P}_k } \right| ^{2}}= & {} \min \sum _{u=0}^{l-1} {\left| {T^{-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} -\mathbf{P}_{k_0 } } \right| ^{2}} \nonumber \\= & {} \frac{1}{2l}\sum _{u,v=0}^{l-1} {\left| {T^{-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} -T^{-v}{} \mathbf{Q}_{\pi ^{v}(k_0 )} } \right| ^{2}} \nonumber \\= & {} \frac{1}{2l}\sum _{u,v=0}^{l-1} {\left| {T^{v-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} -\mathbf{Q}_{\pi ^{v}(k_0 )} } \right| ^{2}} \nonumber \\= & {} \frac{1}{2l}\sum _{u,v=0}^{l-1} {\left| {T^{v-u}{} \mathbf{Q}_{\pi ^{u}(k_0 )} -\mathbf{Q}_{\pi ^{v-u}(\pi ^{u}(k_0 ))} } \right| ^{2}} \nonumber \\= & {} \frac{1}{2l}\sum _{i=1}^l {\sum _{k\in C_j } {\left| {T^{i}{} \mathbf{Q}_k -\mathbf{Q}_{\pi ^{i}(k)} } \right| ^{2}} } \nonumber \\= & {} \frac{1}{2n}\sum _{i=1}^n {\sum _{k\in C_j } {\left| {T^{i}{} \mathbf{Q}_k -\mathbf{Q}_{\pi ^{i}(k)} } \right| ^{2}} } \end{aligned}$$
(19)

Summing over j, we get (3).

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Alon, G., Tuvi-Arad, I. Improved algorithms for symmetry analysis: structure preserving permutations. J Math Chem 56, 193–212 (2018). https://doi.org/10.1007/s10910-017-0788-y

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Keywords

  • Chirality
  • Continuous symmetry measures
  • Graph automorphism
  • Permutations
  • QSAR descriptors

Mathematics Subject Classification

  • 92E10
  • 92-08
  • 05C85