Abstract
This paper uses combinatorics and group theory to answer questions about the assembly of icosahedral viral shells. Although the geometric structure of the capsid (shell) is fairly well understood in terms of its constituent subunits, the assembly process is not. For the purpose of this paper, the capsid is modeled by a polyhedron whose facets represent the monomers. The assembly process is modeled by a rooted tree, the leaves representing the facets of the polyhedron, the root representing the assembled polyhedron, and the internal vertices representing intermediate stages of assembly (subsets of facets). Besides its virological motivation, the enumeration of orbits of trees under the action of a finite group is of independent mathematical interest. If G is a finite group acting on a finite set X, then there is a natural induced action of G on the set \(\mathcal{T}_{X}\) of trees whose leaves are bijectively labeled by the elements of X. If G acts simply on X, then |X|:=|X n |=n⋅|G|, where n is the number of G-orbits in X. The basic combinatorial results in this paper are (1) a formula for the number of orbits of each size in the action of G on \(\mathcal{T}_{X_{n}}\), for every n, and (2) a simple algorithm to find the stabilizer of a tree \(\tau\in\mathcal{T} _{X}\) in G that runs in linear time and does not need memory in addition to its input tree. These results help to clarify the effect of symmetry on the probability and number of assembly pathways for icosahedral viral capsids, and more generally for any finite, symmetric macromolecular assembly.
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References
Agbandje-McKenna, M., Llamas-Saiz, A. L., Wang, F., Tattersall, P., & Rossmann, M. G. (1998). Functional implications of the structure of the murine parvovirus, minute virus of mice. Structure, 6, 1369–1381.
Berger, B., & Shor, P. W. (1995). Local rules switching mechanism for viral shell geometry, Technical report, MIT-LCS-TM-527.
Bóna, M., & Sitharam, M. (2008). Influence of symmetry on probabilities of icosahedral viral assembly pathways. In Stockley, P. & Twarock, R. (Eds.), Computational and Mathematical Methods in Medicine: Special issue on Mathematical Virology.
Berger, B., Shor, P., King, J., Muir, D., Schwartz, R., & Tucker-Kellogg, L. (1994). Local rule-based theory of virus shell assembly. Proc. Natl. Acad. Sci. USA, 91, 7732–7736.
Brinkmann, G., & Dress, A. (1997). A constructive enumeration of fullerenes. J. Algorithms, 23, 345–358.
Caspar, D., & Klug, A. (1962). Physical principles in the construction of regular viruses. Cold Spring Harbor Symp. Quant. Biol., 27, 1–24.
Compton, K. J. (1989). A logical approach to asymptotic combinatorics II: monadic second-order properties. J. Comb. Theory, Ser. A, 50(1), 110–131.
Day, W. H. E. (1985). Optimal algorithms for comparing trees with labeled leaves. J. Classif., 2(1), 7–26.
Deza, M., & Dutour, M. (2005). Zigzag structures of simple two-faced polyhedra. Comb. Probab. Comput., 14(1–2), 31–57.
Deza, M., Dutour, M., & Fowler, P. W. (2004). Zigzags, railroads, and knots in fullerenes. J. Chem. Inf. Comput. Sci., 44, 1282–1293.
Gawron, P., Nekrashevich, V. V., & Sushchanskii, V. I. (1999). Conjugacy classes of the automorphism group of a tree. Math. Notes, 65(6), 787–790.
Johnson, J. E., & Speir, J. A. (1997). Quasi-equivalent viruses: a paradigm for protein assemblies. J. Mol. Biol., 269, 665–675.
Klin, M. H. (1973). On the number of graphs for which a given permutation group is the automorphism group (Russian). English translation: Kibernetika, 5, 892–870.
Marzec, C. J., & Day, L. A. (1993). Pattern formation in icosahedral virus capsids: the papova viruses and nudaurelia capensis β virus. Biophys. J., 65, 2559–2577.
Rapaport, D., Johnson, J., & Skolnick, J. (1999). Supramolecular self-assembly: molecular dynamics modeling of polyhedral shell formation. Comput. Phys. Commun., 121–122, 231–235.
Reddy, V. S., Giesing, H. A., Morton, R. T., Kumar, A., Post, C. B., Brooks, C. L., & Johnson, J. E. (1998). Energetics of quasiequivalence: computational analysis of protein-protein interactions in icosahedral viruses. Biophys. J., 74, 546–558.
Seress, Á. (2003). Permutation Group Algorithms. Cambridge: Cambridge University Press.
Sitharam, M., & Agbandje-McKenna, M. (2006). Modeling virus assembly using geometric constraints and tensegrity: avoiding dynamics. J. Bioinform. Comput. Biol., 13(6), 1232–1265.
Sitharam, M., & Bóna, M. (2004). Combinatorial enumeration of macromolecular assembly pathways. In Proceedings of the International Conference on Bioinformatics and Applications. Singapore: World Scientific.
Stanley, R. (1999). Enumerative Combinatorics (Vol. 2). Cambridge: Cambridge University Press.
Valiente, G. (2002). Algorithms on Trees and Graphs. New York: Springer.
van Lint, J. H., & Wilson, R. M. (2006). A Course in Combinatorics. Cambridge: Cambridge University Press.
Wagner, S. G. (2006). On an identity for the cycle indices of rooted tree automorphism groups. Electron. J. Comb., 13, 450–456.
Woods, A. R. (1998). Coloring rules for finite trees and probabilities of monadic second order sentences. Random Struct. Algorithms, 10(4), 453–485.
Zlotnick, A., Aldrich, R., Johnson, J. M., Ceres, P., & Young, M. J. (2000). Mechanisms of capsid assembly for an icosahedral plant virus. J. Virol., 277, 450–456.
Zlotnick, A. (1994). To build a virus capsid: an equilibrium model of the self assembly of polyhedral protein complexes. J. Mol. Biol., 241, 59–67.
Zlotnick, A., Johnson, J. M., Wingfield, P. W., Stahl, S. J., & Endres, D. (1999). A theoretical model successfully identifies features of hepatitis b virus capsid assembly. Biochemistry, 38, 14644–14652.
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M. Bóna was supported in part by NSF grant DMS0714912.
M. Sitharam was supported in part by NSF grant DMS0714912.
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Bóna, M., Sitharam, M. & Vince, A. Enumeration of Viral Capsid Assembly Pathways: Tree Orbits Under Permutation Group Action. Bull Math Biol 73, 726–753 (2011). https://doi.org/10.1007/s11538-010-9606-4
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DOI: https://doi.org/10.1007/s11538-010-9606-4