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ON THE PERFECT HEXAGONAL PACKING OF TUBES E. L. Starostin Max-Planck-Institut f ¨ ur Physik komplexer Systeme, Dresden, Germany E-mail: [email protected] Abstract. In most cases the hexagonal packing of fibrous structures extremizes the energy of interaction between strands. If the strands are not straight, then it is still possible to form a perfect hexatic bundle. Conditions under which the perfect hexag- onal packing of curved tubular structures may exist are formulated. Of particular in- terest are closed bundles like DNA toroids or spools. The closure or return con- straints of the bundle result in an allow- able group of automorphisms of the cross- sectional hexagonal lattice. The structure of this group is explored. Examples of an open helical-like and closed toroidal- like bundles are presented. A possible implication on a condensation of DNA in toroids and its packing inside a viral cap- sid is briefly discussed [1]. Introduction It is known that the densest packing of in- finite straight cylinders is hexagonal when all their axes are parallel [2]. It evi- dently corresponds to hexagonal packing of disks in a plane. The hexagonal pack- ing of tubular objects occurs in numerous instances at nano to macro scale. Among examples there are nanotubes [3], high density columnar hexatic liquid crystalline DNA mesophases [4] and others. In most cases, this packing extremizes the inter- action energy between filaments. Geo- metrically, it means that all pairs of neigh- bouring axes are located at constant dis- tance to each other. In some instances, the filaments are not straight. Then, the natural question arises of whether it is still possible to reach the same maximal density of packing. If yes, then the second question can be formu- lated as: what is the set of configurations of infinite (or closed) tubes that have the maximal density? By a tube (or a tubu- lar neighbourhood) here we understand the set of all points in space whose dis- tance from the smooth axial curve does not exceed the constant thickness radius. We can set the scale by fixing this ra- dius to 1. Moreover we will assume that the global curvature of the axes is less or equal to 1. Thus, the tubes cannot over- lap, but they are perfectly flexible. It will be shown in the following that the densest packing class includes curvilinear axes, which should be relatively parallel. This im- plies that an arbitrary small twist of one axis around another immediately destroys the hexagonal packing. A complicated structure arises when the tubes are in contact with themselves. An important example is a condensation of DNA in toroids [5, 6, 7] or a DNA arrange- ment inside of viral capsids [8, 9, 10]. a b These pictures are taken from [7]. They show cryoelectron micrographs of DNA toroids with the plane of the toroid (col- umn a) approximately coplanar with the microscope image plane (top-view) and (column b) orthogonal to the microscope image plane (edge-view). Hexagonal packing of DNA is clearly seen on the right figure (A). Unconstrained tube packing We start with consideration of a perfect tube of some length with axis r 0 (s), s be- ing the arclength parametrization. Let the tube be in a continuous contact with the maximal allowed number of other tubes of the same thickness. This number equals 6 [11], thus it may be said that the tubes are hexagonally packed. Denote the axes of the neighbouring tubes by r j (s), j = 1,..., 6. We can choose the same para- metrization for all the tubes such that for every s the points r j (s), j =1,..., 6 are the closest to the central axis r 0 (s) and they lie in the vertices of a regular trian- gular lattice. The vector field m j 0 (s) ≡ r j (s) - r 0 (s) is relatively parallel [12]. It implies that there is no twist of vectors m j 0 (s) about the central axis. We can add more lay- ers of the tubes in the same manner as first six tubes. Proceeding this way will al- low us to build a bundle of parallel tubes that fill up some domain in space. The hexagonal packing provides the maximal density in this domain. Let us now obtain an equation that gov- erns the position of the neighbouring tube for a given central axis. We omit the index j for clarity. Since kmk = const(= 2), we can write dm ds = ω × m, (1) and the vector ω may be represented as ω = ω 1 m + ω 2 T × m [11], where we de- note by T = dr ds the tangent to the central axis. By definition, the vector m connects the closest points on two curves, which im- plies m · T =0. Differentiating this equa- tion and further substitute eq. (1) for dm ds , we come to ω 2 m 2 = m · dT ds , or, with the help of the Serret-Frenet equations, ω 2 m 2 = κm · N. where N is the principal normal to r(s) and κ the curvature of this curve. Finally, the equation for the orien- tation vector m can be given the form dm ds = -κ(m · N)T. (2) This is the main equation that describes the arrangement of tubes in the hexago- nally packed bundle. Remark. In the continuum limit, the elastic Frank-Oseen energy density e = 1 2 K 1 [∇· T] 2 + 1 2 K 2 [T · (∇× T)] 2 + 1 2 K 3 [T × (∇× T)] 2 reduces to the term 1 2 K 3 κ 2 , with K 3 being the bending rigidity. –2 –1 0 1 2 –2 0 2 0 2 4 6 8 10 12 EXAMPLE: a regular helical curve: r(s)= (cos as, sin as, √ 1 - a 2 s), 0 ≤ a ≤ 1. Equa- tion (2) transforms to the system dξ ds = aη, dη ds = a(a 2 - 1)ξ, dm z ds = a 2 p 1 - a 2 ξ, (3) and the first two components of the vector m =(m x ,m y ,m z ) are expressed as m x = ξ cos as - η sin as, m y = ξ sin as + η cos as. The explicit solution of eq. (3) is easy to find: ξ = c 1 cos τs + c 2 sin τs, η = p 1 - a 2 (c 2 cos τs - c 1 sin τs), m z = a(c 1 sin τs - c 2 cos τs), where τ = a √ 1 - a 2 is the torsion and m 2 = c 2 1 + c 2 2 . The figure shows a bun- dle of six tubes arranged at constant dis- tance from the central tube (shown in blue) and from the neighbours. At every section which is orthogonal to their axes the crossing points form the hexagonal lattice. The tubes are shown thinner to ease representation. Cycled bundles In this work, particular attention is given to closed bundles of hexagonally packed tubes. The closedness condition imposes a strict constraint on the whole structure. Indeed, take an orthogonal cross-section of the bundle. Then, we study the map- ping of the 2D hexagonal lattice in the cross-section onto itself. The automor- phisms that preserve both the distances and the connectivity form a discrete infi- nite group. a b c d e f The automorphism group of the lattice is finitely generated by the following set of transformations: 1. an identity map, 2. translations along the lattice vectors e 1 and e 2 (fig. d), 3. a rotation through 1 3 π around the origin (figs. a,b,c), 4. a rotation through 2 3 π around 1 3 (e 1 + e 2 ) (fig. e), 5. a rotation through π around 1 2 e 1 (fig. f). This allows us to characterize all possible closed hexagonally packed bundles: the writhing number [13] of each axis that real- izes the mapping should equal n/6, where n is integer. Here are the examples: The perfectly packed bundle made up of two closed tubes (cf. case (a)). The core (dark) makes one turn and the second tube winds six times. The colour varia- tion codes the arclength. The tubes are shown thinner to ease representation. The perfectly packed bundle that corre- sponds to case (f). Under certain conditions, the DNA toroids may deform taking on a warped shape [14]. Generally, this deformation affects the interstrand distances and the interaction energy between strands. This effect may influence the twist-bend insta- bility of the DNA condensates [15]. A perfectly packed structure may have its overall shape that closely resembles the warped DNA toroids. Left. The perfectly packed bundle made up of four closed tubes (cf. case (e)). The tubes are shown thinner to ease repre- sentation. Right. The DNA bundle looks like twisted skein of yarn (an electron mi- crograph from [14]). Concluding remarks It is shown that curvilinear tubular fila- ments may be packed in the most com- pact way in some spatial domain. Such a packing extremizes the interaction energy between neighbouring filaments and it is only possible if the bundle is twist-free. An equation is derived that governs the relative positions of the tubes in the hexagonal bundle. Particular attention is given to cycled arrangements of the tubes. The corre- sponding automorphisms of the hexago- nal lattice in the cross-section are classi- fied. The cycled structures are only possible if the writhe of the central axis is n/6, n an in- teger. The automorphism group structure forbids a cycled hexagonally packed bun- dle made up with a single filament: frus- tration is inevitable (cf. [16]). The results are of a universal nature and are applicable to various fibrous struc- tures, in particular, to a condensation of DNA in toroids as well as to its packing inside the viral capsids. References [1] Starostin, E. L. (2004) On the perfect hexagonal packing of tubes. Preprint mpi-pks/0410005, 7 p. [2] Bezdek, A & Kuperberg, W. (1990) Mathematika 37, 74–80. [3] Thess, A, et al. (1996) Science 273, 483–487. [4]Livolant, F, et al. (1989) Nature 339, 724–726. [5]Evdokimov, Y. M, et al. (1972) FEBS Lett. 23, 180–184. [6] Golan, R, et al. (1999) Biochemistry 38, 14069–14076. [7] Hud, N. V & Downing, K. H. (2001) Proc. Natl. Acad. Sci. USA 98, 14925–14930. [8]Earnshaw, W. C & Harrison, S. C. (1977) Nature 268, 598–602. [9] Lepault, J, et al. (1987) The EMBO J. 6, 1507–1512. [10] Cerritelli, M. E, et al. Steven, A. C. (1997) Cell 91, 271–280. [11] Starostin, E. L. (2004) On the num- ber of tubes touching a sphere or a tube. Preprint mpi-pks/0408004, 17 p. To appear in Geometriae Dedicata. [12] Bishop, R. L. (1975) Amer. Math. Month. 82, 246–251. [13]Fuller, F. B. (1971) Proc. Natl. Acad. Sci. USA 68, 815–819. [14] Earnshaw, W. C, et al. (1978) Cell 14, 559–568. [15] Kuli ´ c, I. M, Andrienko, D, & Deserno, M. (2004) Europhys. Lett. 67, 418– 424. [16] Park, S. Y, Harries, D, & Gelbart, W. M. (1998) Biophys. J. 75, 714–720.

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Page 1: TUBES OF CKING A P GONAL HEXA PERFECT THE ONucesest/JSMD2004_poster.pdf · image plane (top-vie w) and (column b) or thogonal to the microscope image plane (edge-vie w). He xagonal

ONTHEPERFECTHEXAGONALPACKINGOFTUBESE.L.Starostin

Max-Planck-InstitutfurPhysikkomplexerSysteme,Dresden,GermanyE-mail:[email protected]

Abstract.Inmostcasesthehexagonalpackingoffibrousstructuresextremizestheenergyofinteractionbetweenstrands.Ifthestrandsarenotstraight,thenitisstillpossibletoformaperfecthexaticbundle.Conditionsunderwhichtheperfecthexag-onalpackingofcurvedtubularstructuresmayexistareformulated.Ofparticularin-terestareclosedbundleslikeDNAtoroidsorspools.Theclosureorreturncon-

straintsofthebundleresultinanallow-ablegroupofautomorphismsofthecross-sectionalhexagonallattice.Thestructureofthisgroupisexplored.Examplesofanopenhelical-likeandclosedtoroidal-likebundlesarepresented.ApossibleimplicationonacondensationofDNAintoroidsanditspackinginsideaviralcap-sidisbrieflydiscussed[1].

IntroductionItisknownthatthedensestpackingofin-finitestraightcylindersishexagonalwhenalltheiraxesareparallel[2].Itevi-dentlycorrespondstohexagonalpackingofdisksinaplane.Thehexagonalpack-ingoftubularobjectsoccursinnumerousinstancesatnanotomacroscale.Amongexamplestherearenanotubes[3],highdensitycolumnarhexaticliquidcrystallineDNAmesophases[4]andothers.Inmostcases,thispackingextremizestheinter-actionenergybetweenfilaments.Geo-metrically,itmeansthatallpairsofneigh-bouringaxesarelocatedatconstantdis-tancetoeachother.

Insomeinstances,thefilamentsarenotstraight.Then,thenaturalquestionarisesofwhetheritisstillpossibletoreachthesamemaximaldensityofpacking.Ifyes,thenthesecondquestioncanbeformu-latedas:whatisthesetofconfigurationsofinfinite(orclosed)tubesthathavethemaximaldensity?Byatube(oratubu-larneighbourhood)hereweunderstandthesetofallpointsinspacewhosedis-tancefromthesmoothaxialcurvedoesnotexceedtheconstantthicknessradius.Wecansetthescalebyfixingthisra-diusto1.Moreoverwewillassumethattheglobalcurvatureoftheaxesislessorequalto1.Thus,thetubescannotover-lap,buttheyareperfectlyflexible.Itwill

beshowninthefollowingthatthedensestpackingclassincludescurvilinearaxes,whichshouldberelativelyparallel.Thisim-pliesthatanarbitrarysmalltwistofoneaxisaroundanotherimmediatelydestroysthehexagonalpacking.

Acomplicatedstructureariseswhenthetubesareincontactwiththemselves.AnimportantexampleisacondensationofDNAintoroids[5,6,7]oraDNAarrange-mentinsideofviralcapsids[8,9,10].

ab

Thesepicturesaretakenfrom[7].TheyshowcryoelectronmicrographsofDNAtoroidswiththeplaneofthetoroid(col-umna)approximatelycoplanarwiththemicroscopeimageplane(top-view)and(columnb)orthogonaltothemicroscopeimageplane(edge-view).HexagonalpackingofDNAisclearlyseenontherightfigure(A).

UnconstrainedtubepackingWestartwithconsiderationofaperfecttubeofsomelengthwithaxisr0(s),sbe-ingthearclengthparametrization.Letthetubebeinacontinuouscontactwiththemaximalallowednumberofothertubesofthesamethickness.Thisnumberequals6[11],thusitmaybesaidthatthetubesarehexagonallypacked.Denotetheaxesoftheneighbouringtubesbyrj(s),j=1,...,6.Wecanchoosethesamepara-metrizationforallthetubessuchthatforeverysthepointsrj(s),j=1,...,6aretheclosesttothecentralaxisr0(s)andtheylieintheverticesofaregulartrian-gularlattice.

Thevectorfieldmj0(s)≡rj(s)−r0(s)isrelativelyparallel[12].Itimpliesthatthereisnotwistofvectorsmj0(s)about

thecentralaxis.Wecanaddmorelay-ersofthetubesinthesamemannerasfirstsixtubes.Proceedingthiswaywillal-lowustobuildabundleofparalleltubesthatfillupsomedomaininspace.Thehexagonalpackingprovidesthemaximaldensityinthisdomain.

Letusnowobtainanequationthatgov-ernsthepositionoftheneighbouringtubeforagivencentralaxis.Weomittheindexjforclarity.Since‖m‖=const(=2),wecanwritedm

ds=ω×m,(1)

andthevectorωmayberepresentedasω=ω1m+ω2T×m[11],wherewede-notebyT=dr

dsthetangenttothecentralaxis.

Bydefinition,thevectormconnectstheclosestpointsontwocurves,whichim-pliesm·T=0.Differentiatingthisequa-tionandfurthersubstituteeq.(1)fordm

ds,wecometoω2m2=m·dT

ds,or,withthehelpoftheSerret-Frenetequations,ω2m2=κm·N.whereNistheprincipalnormaltor(s)andκthecurvatureofthis

curve.Finally,theequationfortheorien-tationvectormcanbegiventheform

dm

ds=−κ(m·N)T.(2)

Thisisthemainequationthatdescribesthearrangementoftubesinthehexago-nallypackedbundle.

Remark.Inthecontinuumlimit,theelasticFrank-Oseenenergydensity

e=1

2K1[∇·T]2+

1

2K2[T·(∇×T)]2+

1

2K3[T×(∇×T)]2

reducestotheterm12K3κ2,withK3beingthebendingrigidity.

–2 –1 0 1 2

–20

2

0

2

4

6

8

10

12

EXAMPLE:aregularhelicalcurve:r(s)=(cosas,sinas,

√1−a2s),0≤a≤1.Equa-

tion(2)transformstothesystemdξ

ds=aη,

dη

ds=a(a2−1)ξ,

dmz

ds=a2√

1−a2ξ,(3)andthefirsttwocomponentsofthevectorm=(mx,my,mz)areexpressedasmx=ξcosas−ηsinas,my=ξsinas+ηcosas.Theexplicitsolutionofeq.(3)iseasytofind:

ξ=c1cosτs+c2sinτs,

η=√

1−a2(c2cosτs−c1sinτs),

mz=a(c1sinτs−c2cosτs),whereτ=a

√1−a2isthetorsionand

m2=c21+c2

2.Thefigureshowsabun-dleofsixtubesarrangedatconstantdis-tancefromthecentraltube(showninblue)andfromtheneighbours.Ateverysectionwhichisorthogonaltotheiraxesthecrossingpointsformthehexagonallattice.Thetubesareshownthinnertoeaserepresentation.

CycledbundlesInthiswork,particularattentionisgiventoclosedbundlesofhexagonallypackedtubes.Theclosednessconditionimposesastrictconstraintonthewholestructure.Indeed,takeanorthogonalcross-sectionofthebundle.Then,westudythemap-pingofthe2Dhexagonallatticeinthecross-sectionontoitself.Theautomor-phismsthatpreserveboththedistancesandtheconnectivityformadiscreteinfi-nitegroup.

abc

def

Theautomorphismgroupofthelatticeisfinitelygeneratedbythefollowingsetoftransformations:1.anidentitymap,

2.translationsalongthelatticevectorse1ande2(fig.d),

3.arotationthrough13πaroundtheorigin

(figs.a,b,c),

4.arotationthrough23πaround

13(e1+e2)(fig.e),

5.arotationthroughπaround12e1(fig.f).

Thisallowsustocharacterizeallpossibleclosedhexagonallypackedbundles:thewrithingnumber[13]ofeachaxisthatreal-izesthemappingshouldequaln/6,wherenisinteger.Herearetheexamples:

Theperfectlypackedbundlemadeupoftwoclosedtubes(cf.case(a)).Thecore(dark)makesoneturnandthesecondtubewindssixtimes.Thecolourvaria-tioncodesthearclength.Thetubesareshownthinnertoeaserepresentation.

Theperfectlypackedbundlethatcorre-spondstocase(f).

Undercertainconditions,theDNAtoroidsmaydeformtakingonawarpedshape[14].Generally,thisdeformationaffectstheinterstranddistancesandtheinteractionenergybetweenstrands.Thiseffectmayinfluencethetwist-bendinsta-bilityoftheDNAcondensates[15].AperfectlypackedstructuremayhaveitsoverallshapethatcloselyresemblesthewarpedDNAtoroids.

Left.Theperfectlypackedbundlemadeupoffourclosedtubes(cf.case(e)).Thetubesareshownthinnertoeaserepre-sentation.Right.TheDNAbundlelooksliketwistedskeinofyarn(anelectronmi-crographfrom[14]).

ConcludingremarksItisshownthatcurvilineartubularfila-mentsmaybepackedinthemostcom-pactwayinsomespatialdomain.Suchapackingextremizestheinteractionenergybetweenneighbouringfilamentsanditisonlypossibleifthebundleistwist-free.

Anequationisderivedthatgovernstherelativepositionsofthetubesinthehexagonalbundle.

Particularattentionisgiventocycledarrangementsofthetubes.Thecorre-spondingautomorphismsofthehexago-

nallatticeinthecross-sectionareclassi-fied.

Thecycledstructuresareonlypossibleifthewritheofthecentralaxisisn/6,nanin-teger.Theautomorphismgroupstructureforbidsacycledhexagonallypackedbun-dlemadeupwithasinglefilament:frus-trationisinevitable(cf.[16]).

Theresultsareofauniversalnatureandareapplicabletovariousfibrousstruc-tures,inparticular,toacondensationofDNAintoroidsaswellastoitspackinginsidetheviralcapsids.

References

[1]Starostin,E.L.(2004)Ontheperfecthexagonalpackingoftubes.Preprintmpi-pks/0410005,7p.

[2]Bezdek,A&Kuperberg,W.(1990)Mathematika37,74–80.

[3]Thess,A,etal.(1996)Science273,483–487.

[4]Livolant,F,etal.(1989)Nature339,724–726.

[5]Evdokimov,Y.M,etal.(1972)FEBSLett.23,180–184.

[6]Golan,R,etal.(1999)Biochemistry38,14069–14076.

[7]Hud,N.V&Downing,K.H.(2001)Proc.Natl.Acad.Sci.USA98,14925–14930.

[8]Earnshaw,W.C&Harrison,S.C.(1977)Nature268,598–602.

[9]Lepault,J,etal.(1987)TheEMBOJ.6,1507–1512.

[10]Cerritelli,M.E,etal.Steven,A.C.(1997)Cell91,271–280.

[11]Starostin,E.L.(2004)Onthenum-beroftubestouchingasphereoratube.Preprintmpi-pks/0408004,17p.ToappearinGeometriaeDedicata.

[12]Bishop,R.L.(1975)Amer.Math.Month.82,246–251.

[13]Fuller,F.B.(1971)Proc.Natl.Acad.Sci.USA68,815–819.

[14]Earnshaw,W.C,etal.(1978)Cell14,559–568.

[15]Kulic,I.M,Andrienko,D,&Deserno,M.(2004)Europhys.Lett.67,418–424.

[16]Park,S.Y,Harries,D,&Gelbart,W.M.(1998)Biophys.J.75,714–720.

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