The building game: from enumerative combinatorics to conformational diffusion. (English) Zbl 1347.52011
Summary: We study a discrete attachment model for the self-assembly of polyhedra called the building game. We investigate two distinct aspects of the model: (i) enumerative combinatorics of the intermediate states and (ii) a notion of Brownian motion for the polyhedral linkage defined by each intermediate that we term conformational diffusion. The combinatorial configuration space of the model is computed for the Platonic, Archimedean, and Catalan solids of up to 30 faces, and several novel enumerative results are generated. These represent the most exhaustive computations of this nature to date. We further extend the building game to include geometric information. The combinatorial structure of each intermediate yields a systems of constraints specifying a polyhedral linkage and its moduli space. We use a random walk to simulate a reflected Brownian motion in each moduli space. Empirical statistics of the random walk may be used to define the rates of transition for a Markov process modeling the process of self-assembly.
MSC:
| 52B05 | Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) |
| 70B15 | Kinematics of mechanisms and robots |
| 92C40 | Biochemistry, molecular biology |
| 92E10 | Molecular structure (graph-theoretic methods, methods of differential topology, etc.) |
| 60D05 | Geometric probability and stochastic geometry |
Keywords:
self-assembly; polyhedra; molecular cages; polyhedral linkages; Brownian motion on manifoldsSoftware:
OEISReferences:
| [1] | Baker, J.E.: Limiting positions of a Bricard linkage and their possible relevance to the cyclohexane molecule. Mech. Mach. Theory 21, 253-260 (1986) · doi:10.1016/0094-114X(86)90101-1 |
| [2] | Bhatia, D., Mehtab, S., Krishnan, R., Indi, S.S., Basu, A., Krishnan, Y.: Icosahedral DNA nanocapsules by modular assembly. Angew. Chem. Int. Ed. 48, 4134-4137 (2009) · doi:10.1002/anie.200806000 |
| [3] | Caspar, D., Klug, A.: Physical principles in the construction of regular viruses. In: Cold Spring Harbor Symposium in Qantitative Biology, vol. 27, pp. 1-27 (1962) |
| [4] | Cox, D., Little, J., O’Shea, D.: Ideals, varieties, and algorithms. In: Undergraduate Texts in Mathematics. Springer, New York (1992). doi:10.1007/978-1-4757-2181-2 · Zbl 0756.13017 |
| [5] | Crick, F.H., Watson, J.D.: Structure of small viruses. Nature, 177, 473-475 (1956) |
| [6] | Douglas, S.M., Dietz, H., Liedl, T., Högberg, B., Graf, F., Shih, W.M.: Self-assembly of DNA into nanoscale three-dimensional shapes. Nature 459, 414-418 (2009) · doi:10.1038/nature08016 |
| [7] | Endres, D., Miyahara, M., Moisant, P., Zlotnick, A.: A reaction landscape identifies the intermediates critical for self-assembly of virus capsids and other polyhedral structures. Protein Sci. 14, 1518-1525 (2005) · doi:10.1110/ps.041314405 |
| [8] | Galletti, C., Fanghella, P.: Single-loop kinematotropic mechanisms. Mech. Mach. Theory 36, 743-761 (2001) · Zbl 1140.70330 · doi:10.1016/S0094-114X(01)00002-7 |
| [9] | Holmes-Cerfon, M., Gortler, S., Brenner, M.: A geometrical approach to computing free-energy landscapes from short-ranged potentials. Proc. Natl. Acad. Sci. 110, E5-E14 (2013) · doi:10.1073/pnas.1211720110 |
| [10] | Hsu, P.: Brownian motion and Riemannian geometry. Contemp. Math. 73, 95-104 (1988) · Zbl 0658.58044 · doi:10.1090/conm/073/954633 |
| [11] | Johnson, D.C.: Combinatorial and Geometric Structure in the Self-Assembly of Polyhedra, PhD thesis, Brown University (May 2015) · Zbl 0082.37602 |
| [12] | Johnson, D.C., Menon, G.: Supporting Data: The Building Game: From Enumerative Combinatorics to Conformational Diffusion (2016). https://repository.library.brown.edu/studio/item/bdr:589790/ · Zbl 1347.52011 |
| [13] | Jørgensen, E.: The central limit problem for geodesic random walks. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32, 1-64 (1975) · Zbl 0292.60103 · doi:10.1007/BF00533088 |
| [14] | Kaplan, R., Klobusicky, J., Pandey, S., Gracias, D.H., Menon, G.: Building polyhedra by self-folding: theory and experiment. Artif. Life 20, 409-439 (2014) · doi:10.1162/ARTL_a_00144 |
| [15] | Kroto, H., Heath, J., O’Brien, S., Curl, R., Smalley, R.: \[{C}_{60}\] C60: Buckminsterfullerene. Nature 318, 162-163 (1985) · doi:10.1038/318162a0 |
| [16] | Li, D., Zhou, W., Landskron, K., Sato, S., Kiely, C., Fujita, M., Liu, T.: Viral-capsid-type vesicle-like structures assembled from \[m_{12}l_{24}\] m12l24 metalorganic hybrid nanocages. Angew. Chem. Int. Ed. 50, 5182-5187 (2011) · doi:10.1002/anie.201007829 |
| [17] | Li, X.H., Menon, G.: Numerical solution of dyson brownian motion and a sampling scheme for invariant matrix ensembles. J. Stat. Phys. 153, 801-812 (2013) · doi:10.1007/s10955-013-0858-x |
| [18] | Liu, Y., Hu, C., Comotti, A., Ward, M.: Supramolecular Archimedean cages assembled with 72 hydrogen bonds. Science 333, 436-440 (2011) · doi:10.1126/science.1204369 |
| [19] | Mezzadri, F.: How to generate random matrices from the classical compact groups. Notices Amer. Math. Soc. 54, 592-604 (2007) · Zbl 1156.22004 |
| [20] | Möller, T.: A fast triangle-triangle intersection test. J. Graph. Tools 2, 25-30 (1997) · doi:10.1080/10867651.1997.10487472 |
| [21] | Pandey, S., Ewing, M., Kunas, A., Ngyen, S., Gracias, D., Menon, G.: Algorithmic design of self-folding polyhedra. Proc. Natl. Acad. Sci. 108, 19885-19890 (2011) · doi:10.1073/pnas.1110857108 |
| [22] | Pandey, S., Johnson, D., Kaplan, R., Klobusicky, J., Menon, G., Gracias, D.H.: Self-assembly of mesoscale isomers: the role of pathways and degrees of freedom. PLoS One 9, e108960 (2014) · doi:10.1371/journal.pone.0108960 |
| [23] | Rotman, J.: An Introduction to the Theory of Groups, 4th edn. Springer, New York (1995) · Zbl 0810.20001 · doi:10.1007/978-1-4612-4176-8 |
| [24] | Rotman, J.J.: An Introduction to the Theory of Groups, Volume 148 of Graduate Texts in Mathematic, 4th edn. Springer, New York (1995) · Zbl 0810.20001 · doi:10.1007/978-1-4612-4176-8 |
| [25] | Rudin, M.E.: An unshellable triangulation of a tetrahedron. Bull. Am. Math. Soc. 64, 90-91 (1958) · Zbl 0082.37602 · doi:10.1090/S0002-9904-1958-10168-8 |
| [26] | Russell, E.R., Menon, G.: Energy landscapes for the self-assembly of supramolecular polyhedra. J. Nonlinear Sci. (2016). doi:10.1007/s00332-016-9286-9 · Zbl 1342.52012 |
| [27] | Sloane, N.: The On-line Encyclopedia of Integer Sequences. arXiv:math.CO/0312448 (2003) · Zbl 1044.11108 |
| [28] | Sun, Q., Iwasa, J., Ogawa, D., Ishido, Y., Sato, S., Ozeki, T., Sei, Y., Yamaguchi, K., Fujita, M.: Self-assembled \[{M}_{24}{L}_{48}\] M24L48 polyhedra and their sharp structural switch upon subtle ligand variation. Science 328, 1144-1147 (2010) · Zbl 1355.92151 · doi:10.1126/science.1188605 |
| [29] | Thurston, W.P., Weeks, J.R.: The mathematics of three-dimensional manifolds. Sci. Am. 265, 108-121 (1984) · doi:10.1038/scientificamerican0784-108 |
| [30] | Wales, D.: Closed-shell structures and the building game. Chem. Phys. Lett. 141, 478-484 (1987) · doi:10.1016/0009-2614(87)85064-9 |
| [31] | Whiteley, W.: How to describe or design a polyhedron. J. Intell. Robot. Syst. 11, 135-160 (1994) · Zbl 0823.52010 · doi:10.1007/BF01258299 |
| [32] | Wohlhart, K.: Kinematotropic linkages. In: Lenarcic, J., Parenti-Castelli, V. (eds.) Recent Advances in Robot Kinematics, pp. 359-368. Springer, Berlin (1996) |
| [33] | Zlotnick, A.: An equilibrium model of the self assembly of polyhedral protein complexes. J. Mol. Biol. 241, 59-67 (1994) · doi:10.1006/jmbi.1994.1473 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
