Combinatorial rules of icosahedral capsid growth(14)

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

TheinspectionofTableVIleadstothefollowingsimplerules:

1)FortheconstructionofacapsidwithgiventriangularnumberToneneedsexactlyTdi erentproteins(oratleast,Tdi erenttypesofstickingsides)-becausethea nitymatrixhasthedimensionT×T,aseasilyseenintheexamples.

2)Byde nition,the“a nitymatrices”haveonlyonenon-zeroitemineachrowandineachcolumn;moreover,theyaresymmetric.Thismeansthatallcapsidproteintypes(ormoreexactly,alldi erentstickingsides)encounteredinacompleteicosahedronappearwiththesamefrequency:60timeseach.Thiscanbemosteasilyseenforthep-typeformingapentamer:thereare12ofthemineachpentamer,andthereare12pentamersinanyicosahedralcapsid.Buttheneachpstickstoana,andexclusivelytoit:therefore,theremustbealso12a-typeproteinsinthecompletecapsid,andsoon,foreachdi erentprotein.Thismeansthatallthesubunitsthatassembleinpentamersandhexamerslateronhavetobeproducedatexactlythesamerateinordertooptimizecapsidproduction.

3)Thecapsomerscomposingagivencapsidshouldbeproducedatdi erentrates,withaverysimplerule:foreverydozenofpentamers,oneshouldhave60maximallydiversi edhexamersofthetype(abcdef),(andofeachdi erenttype,likethe(nanbncndnenf)intheT=9capsid);then30hexamersofeach(ababab)type;and20hexamersofthe(abcabc)type.ThisrulecanbeeasilyseenuponinspectionofFigures4 8.

4)Inordertoknowhowmany(andofwhichkinds)hexamersshouldbeused,theT-numbershouldbepartitionedinto1+therest,the”1”stayingfortheuniquetypeofpentamers’side,while“therest”mustbedecomposedintoasumofnumbers2,3or6,accordingtothesimplefactorsof6.ThisisshowninthelastcolumnofTableI:weseethatT=3=1+2;T=4=1+3;thenthenextcasesdecomposesasT=7=1+6;T=9=1+6+2(bytheway,withthisschemeinmindthereisnopointintryingtobuildacapsidwith1+6+3=10because10cannotbefoundamongthetriangularnumbers!).

5)Thereisaclear”evolutionary”patterninthelasttable-meaningthateverynext(bigger)typeofcapsidusesthepreviousconstruction,justaddingaminimalamountofnovelty:anditisclearmostofthetimewhichkindofnewhexameronemustadd,justlookingatthedi erencesbetween

14