Combinatorial rules of icosahedral capsid growth(3)

A model of growth of icosahedral viral capsids is proposed. It takes into account the diversity of hexamers' compositions, leading to definite capsid size. We show that the observed yield of capsid production implies a very high level of self-organization

Thecommonpointisthepresenceofpentagonsandhexagonsintheresultingstructure,andthestricttopologicalrulesthatresultfromEuler’stheoremonconvexpolyhedra:V E+F=2,withVnumberofvertices,Enumberofedges,andFnumberoffaces.Fromthisonederivesthefactthatwhenonlypentagonalandhexagonalfacesareallowed,thenumberofpentagonsisalwaysN5=12,whilethehexagonnumberisN6=10(T 1).Contrarytothecaseoffullerenemolecules,whoseyieldfromthehotplasmaisinthebestcasenohigherthan10%oftotalmassofcarbonsooth,virusesusealmost100%ofpentamersandhexamersattheirdisposaltoformperfecticosahedralcapsidstructures,intowhichtheirDNAgeneticmaterialisdenselypackedoncethecapsidiscomplete.

Thismeansthattheinitialnucleationratioofpentamersversushexam-ersisveryclosetoits nalvalueincapsidsinordertominimizethewaste.Secondly,the nalsizeofthecapsidmustdependonparticularassemblyrules,whichcanbefairlywelldeducedfromthestatisticalweightsofvariousagglomerationsteps,foundbymaximizingthe nalproductionrate.Letusinvestigatetherulesthatde nethetypeandthesizeofcapsids,simultane-ouslyoptimizingtheproductionrate.

2Probabilisticanalysisofagglomeration

Thesimpleststochasticmodelofgrowthsuccessfullyappliedtofullerenefor-mation[11]isbasedontheassumptionthatthedominantagglomerationprocessesconsistofformingnewpolygonsinthecavitiesbetweentwopoly-gonsontheborderoftheexistingcluster,byadjoiningaC2oraC3moleculetoacavityfoundinaclusteralreadyformed.OneofsuchprocessesisshowninFig.1below.

Itisclearthattheresulting(666)clusteriswastedforfurtherfullereneformation,whereasa(656)clustercanbeusedinthenextagglomerationstep.Becausethe(666)clustersarealsoabsentin nalfullerenecage,itiseasytoseethatateachofconsecutiveagglomerationstepstheyieldof”proper”clusters,usefulforfurtherfullereneconstruction,isexactly1/2.Afterabout23to24stepsleadingfromtheinitialthree-polygonstructuretoanalmost nishedfullerenecagewith27to28(outoftotalof32)polygonsalreadyinplace,thetotalyieldwouldapproach2 24 10 8insteadofobserved10 1,i.e.10%!Thismeansthatthereisamechanismthatfavoursthecreationof“correct”structuresversusthe“wrong”ones,sothattheaverageyieldof

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