{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,4]],"date-time":"2024-06-04T10:40:10Z","timestamp":1717497610375},"reference-count":80,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2016,1,21]],"date-time":"2016-01-21T00:00:00Z","timestamp":1453334400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000930","name":"NSF","doi-asserted-by":"publisher","award":["CCF-1117695"],"id":[{"id":"10.13039\/501100000930","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"Many remarkably robust, rapid and spontaneous self-assembly phenomena occurring in nature can be modeled geometrically, starting from a collection of rigid bunches of spheres. This paper highlights the role of symmetry in sphere-based assembly processes. Since spheres within bunches could be identical and bunches could be identical, as well, the underlying symmetry groups could be of large order that grows with the number of participating spheres and bunches. Thus, understanding symmetries and associated isomorphism classes of microstates that correspond to various types of macrostates can significantly increase efficiency and accuracy, i.e., reduce the notorious complexity of computing entropy and free energy, as well as paths and kinetics, in high dimensional configuration spaces. In addition, a precise understanding of symmetries is crucial for giving provable guarantees of algorithmic accuracy and efficiency, as well as accuracy vs. efficiency trade-offs in such computations. In particular, this may aid in predicting crucial assembly-driving interactions. This is a primarily expository paper that develops a novel, original framework for dealing with symmetries in configuration spaces of assembling spheres, with the following goals. (1) We give new, formal definitions of various concepts relevant to the sphere-based assembly setting that occur in previous work and, in turn, formal definitions of their relevant symmetry groups leading to the main theorem concerning their symmetries. These previously-developed concepts include, for example: (i) assembly configuration spaces; (ii) stratification of assembly configuration space into configurational regions defined by active constraint graphs; (iii) paths through the configurational regions; and (iv) coarse assembly pathways. (2) We then demonstrate the new symmetry concepts to compute the sizes and numbers of orbits in two example settings appearing in previous work. (3) Finally, we give formal statements of a variety of open problems and challenges using the new conceptual definitions.<\/jats:p>","DOI":"10.3390\/sym8010005","type":"journal-article","created":{"date-parts":[[2016,1,22]],"date-time":"2016-01-22T16:36:13Z","timestamp":1453480573000},"page":"5","source":"Crossref","is-referenced-by-count":4,"title":["Symmetry in Sphere-Based Assembly Configuration Spaces"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"http:\/\/orcid.org\/0000-0003-1031-1403","authenticated-orcid":false,"given":"Meera","family":"Sitharam","sequence":"first","affiliation":[{"name":"Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL 32601, USA"}]},{"given":"Andrew","family":"Vince","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Florida, Gainesville, FL 32601, USA"}]},{"ORCID":"http:\/\/orcid.org\/0000-0001-8724-4216","authenticated-orcid":false,"given":"Menghan","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL 32601, USA"}]},{"given":"Mikl\u00f3s","family":"B\u00f3na","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Florida, Gainesville, FL 32601, USA"}]}],"member":"1968","published-online":{"date-parts":[[2016,1,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4233","DOI":"10.1039\/a706221d","article-title":"Structural consequences of the range of the interatomic potential a menagerie of clusters","volume":"93","author":"Doye","year":"1997","journal-title":"J. Chem. Soc. Faraday Trans."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1002\/(SICI)1097-0134(19990501)35:2<133::AID-PROT1>3.0.CO;2-N","article-title":"Effective energy function for proteins in solution","volume":"35","author":"Lazaridis","year":"1999","journal-title":"Proteins"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1002\/prot.10410","article-title":"Effective energy function for proteins in lipid membranes","volume":"52","author":"Lazaridis","year":"2003","journal-title":"Proteins"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2900","DOI":"10.1016\/S0006-3495(03)74712-2","article-title":"An implicit membrane generalized Born theory for the study of structure, stability, and interactions of membrane proteins","volume":"85","author":"Im","year":"2003","journal-title":"Biophys. J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1021\/ma50003a019","article-title":"Method for estimating the configurational entropy of macromolecules","volume":"14","author":"Karplus","year":"1981","journal-title":"Macromolecules"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"6289","DOI":"10.1063\/1.1401821","article-title":"On the calculation of entropy from covariance matrices of the atomic fluctuations","volume":"115","author":"Andricioaei","year":"2001","journal-title":"J. Chem. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1002\/jcc.20589","article-title":"Nearest-neighbor nonparametric method for estimating the configurational entropy of complex molecules","volume":"28","author":"Hnizdo","year":"2007","journal-title":"J. Comput. Chem."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1605","DOI":"10.1002\/jcc.20919","article-title":"Efficient calculation of configurational entropy from molecular simulations by combining the mutual-information expansion and nearest-neighbor methods","volume":"29","author":"Hnizdo","year":"2008","journal-title":"J. Comput. Chem."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Hensen, U., Lange, O.F., and Grubm\u00fcller, H. (2010). Estimating Absolute Configurational Entropies of Macromolecules: The Minimally Coupled Subspace Approach. PLoS ONE, 5.","DOI":"10.1371\/journal.pone.0009179"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Killian, B.J., Yundenfreund Kravitz, J., and Gilson, M.K. (2007). Extraction of configurational entropy from molecular simulations via an expansion approximation. J. Chem. Phys., 127.","DOI":"10.1063\/1.2746329"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1609","DOI":"10.1021\/jp963817g","article-title":"Mining Minima: Direct Computation of Conformational Free Energy","volume":"101","author":"Head","year":"1997","journal-title":"J. Phys. Chem. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/B978-0-12-381270-4.00004-4","article-title":"Chapter Four\u2014Modeling Loop Entropy","volume":"Volume 487","author":"Johnson","year":"2011","journal-title":"Computer Methods, Part C"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2891","DOI":"10.1021\/jp2068123","article-title":"Efficient calculation of molecular configurational entropies using an information theoretic approximation","volume":"116","author":"King","year":"2012","journal-title":"J. Phys. Chem. B"},{"key":"ref_14","unstructured":"Wales, D.J. (2003). Energy Landscapes: Applications to Clusters, Biomolecules and Glasses, Cambridge University Press."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"6896","DOI":"10.1063\/1.478595","article-title":"The double-funnel energy landscape of the 38-atom Lennard-Jones cluster","volume":"110","author":"Doye","year":"1999","journal-title":"J. Chem. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3965","DOI":"10.1039\/c3cp44332a","article-title":"Symmetrisation schemes for global optimization of atomic clusters","volume":"15","author":"Oakley","year":"2013","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cplett.2013.07.066","article-title":"Surveying a complex potential energy landscape: Overcoming broken ergodicity using basin-sampling","volume":"584","author":"Wales","year":"2013","journal-title":"Chem. Phys. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"10717","DOI":"10.1039\/C4NR02670E","article-title":"Energy landscapes of planar colloidal clusters","volume":"6","author":"Morgan","year":"2014","journal-title":"Nanoscale"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"5159","DOI":"10.1021\/ct3004589","article-title":"A Local Rigid Body Framework for Global Optimization of Biomolecules","volume":"8","author":"Kusumaatmaja","year":"2012","journal-title":"J. Chem. Theory Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4026","DOI":"10.1021\/ct400403y","article-title":"Exploring Energy Landscapes: Metrics, Pathways, and Normal-Mode Analysis for Rigid-Body Molecules","volume":"9","author":"Kusumaatmaja","year":"2013","journal-title":"J. Chem. Theory Comput."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Baxter, R. (1968). Percus-Yevick equation for hard spheres with surface adhesion. J. Chem. Phys., 49.","DOI":"10.1063\/1.1670482"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1203","DOI":"10.1007\/BF01030007","article-title":"Sticky spheres and related systems","volume":"63","author":"Stell","year":"1991","journal-title":"J. Stat. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Miller, M., and Frenkel, D. (2003). Competition of percolation and phase separation in a fluid of adhesive hard spheres. Phys. Rev. Lett., 90.","DOI":"10.1103\/PhysRevLett.90.135702"},{"key":"ref_24","first-page":"E5","article-title":"A geometrical approach to computing free-energy landscapes from short-ranged potentials","volume":"110","author":"Gortler","year":"2013","journal-title":"Proc. Natl. Acad. Sci."},{"key":"ref_25","unstructured":"Ozkan, A., and Sitharam, M. (2011, January 23\u201325). EASAL: Efficient Atlasing and Search of Assembly Landscapes. Proceedings of the BiCoB Symposium, New Orleans, LA, USA."},{"key":"ref_26","unstructured":"Ozkan, A., Pence, J., Peters, J., and Sitharam, M. (2012). EASAL: Theory and Algorithms for Efficient Atlasing and Search of Assembly Landscapes, in preparation."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Arkus, N., Manoharan, V.N., and Brenner, M.P. (2009). Minimal energy clusters of hard spheres with short range attractions. Phys. Rev. Lett., 103.","DOI":"10.1103\/PhysRevLett.103.118303"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2491","DOI":"10.1002\/cphc.201000233","article-title":"Energy Landscapes of Clusters Bound by Short-Ranged Potentials","volume":"11","author":"Wales","year":"2010","journal-title":"ChemPhysChem"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3280","DOI":"10.1039\/c0sm01526a","article-title":"Free energy landscapes for colloidal crystal assembly","volume":"7","author":"Bevan","year":"2011","journal-title":"Soft Matter"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1085","DOI":"10.1039\/C1NR10679A","article-title":"Energy landscapes of colloidal clusters: Thermodynamics and rearrangement mechanisms","volume":"4","author":"Calvo","year":"2012","journal-title":"Nanoscale"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"16015","DOI":"10.1021\/la303894s","article-title":"Nucleation in short-range attractive colloids: ordering and symmetry of clusters","volume":"28","author":"Khan","year":"2012","journal-title":"Langmuir ACS J. Surf. Coll."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Hoy, R.S., Harwayne-Gidansky, J., and O\u2019Hern, C.S. (2012). Structure of finite sphere packings via exact enumeration: Implications for colloidal crystal nucleation. Phys. Rev. E, 85.","DOI":"10.1103\/PhysRevE.85.051403"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Hoy, R.S. (2015). Structure and dynamics of model colloidal clusters with short-range attractions. Phys. Rev. E, 91.","DOI":"10.1103\/PhysRevE.91.012303"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Martin, S., Thompson, A., Coutsias, E.A., and Watson, J.P. (2010). Topology of cyclo-octane energy landscape. J. Chem. Phys., 132.","DOI":"10.1063\/1.3445267"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3464","DOI":"10.1002\/jcc.21931","article-title":"Randomized tree construction algorithm to explore energy landscapes","volume":"32","author":"Jaillet","year":"2011","journal-title":"J. Comput. Chem."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2170","DOI":"10.1002\/jcc.20733","article-title":"Complete maps of molecular-loop conformational spaces","volume":"28","author":"Porta","year":"2007","journal-title":"J. Computat. Chem."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1089\/10665270252935395","article-title":"Using motion planning to study protein folding pathways","volume":"9","author":"Amato","year":"2002","journal-title":"J. Comput. Biol."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Gfeller, D., De Lachapelle, D.M., De Los Rios, P., Caldarelli, G., and Rao, F. (2007). Uncovering the topology of configuration space networks. Phys. Rev. E Stat. Nonlinear Soft Matter Phys., 76.","DOI":"10.1103\/PhysRevE.76.026113"},{"key":"ref_39","unstructured":"Varadhan, G., Kim, Y.J., Krishnan, S., and Manocha, D. (2006). Topology preserving approximation of free configuration space. Robotics, 3041\u20133048."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1808","DOI":"10.3390\/ijms10041808","article-title":"Uncovering the Properties of Energy-Weighted Conformation Space Networks with a Hydrophobic-Hydrophilic Model","volume":"10","author":"Lai","year":"2009","journal-title":"Int. J. Mol. Sci."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Prada-Gracia, D., G\u00f3mez-Gardenes, J., Echenique, P., and Falo, F. (2009). Exploring the Free Energy Landscape: From Dynamics to Networks and Back. PLoS Comput. Biol., 5.","DOI":"10.1371\/journal.pcbi.1000415"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Yao, Y., Sun, J., Huang, X., Bowman, G.R., Singh, G., Lesnick, M., Guibas, L.J., Pande, V.S., and Carlsson, G. (2009). Topological methods for exploring low-density states in biomolecular folding pathways. J. Chem. Phys., 130.","DOI":"10.1063\/1.3103496"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Nagl, M., Schurr, A., and Munch, M. (1999). Planning Geometric Constraint Decompositions Via Graph Transformations. AGTIVE \u201999 (Graph Transformations with Industrial Relevance, Springer.","DOI":"10.1007\/3-540-45104-8"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1006\/jsco.2000.0402","article-title":"Decomposition of geometric constraints systems, Part I: Performance measures","volume":"31","author":"Hoffmann","year":"2001","journal-title":"J. Symb. Comput."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1006\/jsco.2000.0403","article-title":"Decomposition of geometric constraints systems, Part II: New algorithms","volume":"31","author":"Hoffmann","year":"2001","journal-title":"J. Symb. Comput."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1232","DOI":"10.1089\/cmb.2006.13.1232","article-title":"Modeling virus assembly using geometric constraints and tensegrity: Avoiding dynamics","volume":"13","author":"Sitharam","year":"2006","journal-title":"J. Comput. Biol."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1414","DOI":"10.1242\/jcs.064931","article-title":"Stepwise evolution of the centriole-assembly pathway","volume":"123","author":"Machado","year":"2010","journal-title":"J. Cell Sci."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1080\/01442350600676921","article-title":"Energy landscapes: Calculating pathways and rates","volume":"25","author":"Wales","year":"2006","journal-title":"Int. Rev. Phys. Chem."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"726","DOI":"10.1007\/s11538-010-9606-4","article-title":"Enumeration of viral capsid assembly pathways: Tree orbits under permutation group action","volume":"73","author":"Sitharam","year":"2011","journal-title":"Bull. Math. Biol."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Bunker, P.R., and Jensen, P. (2004). Fundamentals of Molecular Symmetry, CRC Press.","DOI":"10.1201\/9781315273334"},{"key":"ref_51","unstructured":"Cotton, F.A. (2008). Chemical Applications of Group Theory, John Wiley & Sons."},{"key":"ref_52","unstructured":"Bonchev, D., and Rouvray, D. (1995). Chemical Group Theory: Techniques and Applications, Taylor & Francis."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Kerber, A., Laue, R., Meringer, M., R\u00fccker, C., and Schymanski, E. (2013). Mathematical Chemistry and Chemoinformatics: Structure Generation, Elucidation and Quantitative Structure-Property Relationships, Walter de Gruyter.","DOI":"10.1515\/9783110254075"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"P\u00f3lya, G., and Read, R.C. (1987). Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds, Springer-Verlag New York, Inc.","DOI":"10.1007\/978-1-4612-4664-0"},{"key":"ref_55","unstructured":"Altmann, S.L. (1977). Induced Representations in Crystals and Molecules, Academic Press."},{"key":"ref_56","unstructured":"Hahn, T., Shmueli, U., Wilson, A.J.C., and Prince, E. (2005). International Tables for Crystallography, D. Reidel Publishing Company."},{"key":"ref_57","unstructured":"Bunker, P.R., and Jensen, P. (1998). Molecular Symmetry and Spectroscopy, NRC Research Press."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"4572","DOI":"10.1063\/1.442625","article-title":"Generating functions for the nuclear spin statistics of nonrigid molecules","volume":"75","author":"Balasubramanian","year":"1981","journal-title":"J. Chem. Phys."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1007\/s00454-009-9160-8","article-title":"Characterizing Graphs with Convex Cayley Configuration Spaces","volume":"43","author":"Sitharam","year":"2010","journal-title":"Discret. Comput. Geom."},{"key":"ref_60","unstructured":"EASAL video. Available online: http:\/\/www.cise.ufl.edu\/\u223csitharam\/EASALvideo.mpg."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Graver, J.E., Servatius, B., and Servatius, H. (1993). Combinatorial Rigidity, Graduate Studies in Math, AMS.","DOI":"10.1090\/gsm\/002"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1080\/17486700802168411","article-title":"The influence of symmetry on the probability of assembly pathways for icosahedral viral shells","volume":"9","author":"Sitharam","year":"2008","journal-title":"Comput. Math. Methods Med."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/978-3-319-21362-0_9","article-title":"On Flattenability of Graphs","volume":"Volume 9201","author":"Botana","year":"2015","journal-title":"Automated Deduction in Geometry"},{"key":"ref_64","unstructured":"Cheng, J. (2013). Towards Combinatorial Characterizations and Algorithms for Bar-And-Joint Independence and Rigidity in 3D and Higher Dimensions. [Ph.D Thesis, University of Florida]."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s00454-006-1284-5","article-title":"Realizability of Graphs","volume":"37","author":"Connelly","year":"2007","journal-title":"Discret. Computat. Geom."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Wales, D. (2015). Perspective: Insight into reaction coordinates and dynamics from the potential energy landscape. J. Chem. Phys., 142.","DOI":"10.1063\/1.4916307"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"2626","DOI":"10.1016\/S0006-3495(98)77708-2","article-title":"Local rules simulation of the kinetics of virus capsid self-assembly","volume":"75","author":"Schwartz","year":"1998","journal-title":"Biophys. J."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"7732","DOI":"10.1073\/pnas.91.16.7732","article-title":"Local rule-based theory of virus shell assembly","volume":"91","author":"Berger","year":"1994","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_69","unstructured":"Berger, B., and Shor, P. (1994). On the Mathematics of Virus Shell Assembly, Massachusetts Institute of Technology. Technical Report."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/S0166-218X(00)00187-6","article-title":"Local rules switching mechanism for viral shell geometry","volume":"104","author":"Berger","year":"2000","journal-title":"Discret. Appl. Math."},{"key":"ref_71","unstructured":"Schwartz, R., Prevelige, P., and Berger, B. (1998). Local Rules Modeling of Nucleation-Limited Virus Capsid Assembly, Massachusetts Institute of Technology. Technical Report, MIT-LCS-TM-584."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1016\/S0006-3495(98)77813-0","article-title":"Energetics of quasiequivalence: Computational analysis of protein-protein interactions in icosahedral viruses","volume":"74","author":"Reddy","year":"1998","journal-title":"Biophysical"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1006\/jmbi.1994.1473","article-title":"To build a virus capsid: An equilibrium model of the self assembly of polyhedral protein complexes","volume":"241","author":"Zlotnick","year":"1994","journal-title":"J. Mol. Biol."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"2559","DOI":"10.1016\/S0006-3495(93)81313-4","article-title":"Pattern formation in icosahedral virus capsids: The papova viruses and nudaurelia capensis \u03b2 virus","volume":"65","author":"Marzec","year":"1993","journal-title":"Biophysical"},{"key":"ref_75","first-page":"231","article-title":"Supramolecular self-assembly: Molecular dynamics modeling of polyhedral shell formation","volume":"121\u2013122","author":"Rapaport","year":"1998","journal-title":"Compt. Phys. Commun."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1006\/jmbi.1997.1068","article-title":"Quasi-equivalent viruses: A paradigm for protein assemblies","volume":"269","author":"Johnson","year":"1997","journal-title":"J. Mol. Biol."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1016\/j.jtbi.2006.04.023","article-title":"Master equation approach to the assembly of viral capsids","volume":"242","author":"Keef","year":"2006","journal-title":"J. Theor. Biol."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1080\/17486700802168379","article-title":"Pathway complexity of model virus capsid assembly systems","volume":"9","author":"Misra","year":"2008","journal-title":"Comput. Math. Methods Med."},{"key":"ref_79","doi-asserted-by":"crossref","unstructured":"Vince, A., and B\u00f3na, M. (2012). The Number of Ways to Assemble a Graph. Electron. J. Comb., 19.","DOI":"10.37236\/2644"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/s10867-013-9313-0","article-title":"Packaging signals in single-stranded RNA viruses: Nature\u2019s alternative to a purely electrostatic assembly mechanism","volume":"39","author":"Stockley","year":"2013","journal-title":"J. Biol. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/8\/1\/5\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,4]],"date-time":"2024-06-04T10:09:23Z","timestamp":1717495763000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/8\/1\/5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1,21]]},"references-count":80,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,1]]}},"alternative-id":["sym8010005"],"URL":"https:\/\/doi.org\/10.3390\/sym8010005","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,1,21]]}}}
