Self-limited self-assembly of chiral filaments

0.49 MB·English

by  Yasheng Yang

Self-limitedself-assembly ofchiral filaments
Yasheng Yang, Robert B. Meyer, and Michael F. Hagan
Department of Physics, Brandeis University, Waltham, MA, 02454∗
(Dated:January26,2010)
Theassemblyoffilamentousbundleswithcontrolleddiametersiscommoninbiologicalsystemsanddesir-
ableforthedevelopmentofnanomaterials. Wediscussdynamicalsimulationsandfreeenergycalculationson
patchysphereswithchiralpairinteractionsthatspontaneouslyassembleintofilamentousbundles.Thechirality
frustrateslong-rangecrystalorderbyintroducingtwistbetweeninteractingsubunits.Forsomerangesofsystem
parameters thisconstraint leads tobundles with afinitediameter asthe equilibrium state, and inother cases
frustrationisrelievedbytheformationofdefects.Whilesomeself-limitedstructurescanbemodeledastwisted
0
filamentsarrangedwithlocalhexagonalsymmetry,otherstructuresaresurprisingintheircomplexity.
1
0
2
Filamentous bundles assembled from protein subunits are writethefreeenergyforatwistedbundlecomprisedofn fil-
f
n essentialstructuralandregulatorycomponentsofcellsandtis- amentsandnsubunitsasG(n,n)=n(g (n)−G(1,1))+
f rod f
a sues. Forexample,filamentousactin,microtubules,andinter- G (n,n) with g the subunit chemical potential in the
J cap f rod
mediatefilamentsassembleanddisassembletocreateastrong cylindrical‘body’ofthebundleandGcaptheexcesschemical
6
butdynamiccytoskeleton,fibrogensubunitsassembleintofib- potentialofthesubunitsinthe‘caps’ateitherend. G and
2 cap
rin fibers and networks to form blood clots (e.g. [1, 2]) and g are independent of length for long bundles, but depend
rod
] sicklehemoglobinassemblesintofibersthatimpairredblood onthebundlediameter,ornf,sincesubunitsatdifferentradii
t cell function (e.g. [3–8]). In vivo and in vitro studies sug- experiencedifferentenvironments. For a solutionwith fixed
f
o gestthatfiberswithfinitediametersarethestablemorphology total subunitdensity c , minimizingthe solution free energy
T
s forfibrin[1,2]underavarietyofconditions,while7-double- densityG = ρ(n,n)G(n,n)−TSwiththemixing
mat. bsturtanthdebfourncdelsesthoaftlsiimckitlefilhaemmeongtlgorboiwntahreremmeatiansutanbclleea[r5.,T7h,e8o]-, eunntirtospizyeSgtoi=tve−sPkthBenP,lnafnw,noffρnm,fnafslsna(ρct(inof,nnrfe)sσu3l)t fwoirtheqσutihliebrsiuubm-
reticalcalculations[5,9–12]havesuggestedthatfinitebundle bundledensities
-
d diametersarethethermodynamicallyfavoredstatefortwisted
n bundlesassembled from chiral subunits. These calculations, ρ(n,n)=exp[β(µn−G(n,n))] (1)
f f
o however,assume specificpackingsofprotofilamentswithout
c defects, which could relieve strain and thereby enable un- with β = 1/kBT and the monomer chemical potential µ =
[ boundedgrowth. Theobjectiveofthisarticleistodetermine, G(1,1)+ kBT ln(ρ(1,1)σ3). The onset of spontaneous as-
1 withoutassumptionsaboutassemblypathwaysorassemblage sembly is identified with the concentration at which half of
v geometries,ifchiralitycanresultinstablebundleswithfinite the subunits are assembled into bundles with the other half
0 diameters. We construct a model subunit with simple pair- freeinsolution,whichisgivenbyby[18]ccsc ≈exp[β(grod−
8 wise chiral interactions that drive assembly into filamentous G(1,1))]. ByminimizingGtotwithrespecttonandnfweob-
7
bundles, and combine umbrella sampling [13] and forward tain that bundlesfollow an exponentialdistribution of mass-
4
. fluxsampling[14]toexplorethestructuresthatspontaneously averagedlengthsP(l)∼le−l/hli,withl=n/nfandthemean
1 assemble for varying degrees of chirality. The simulations length hli ≃ [cTexp(Gcap)]1/2. However, if there is a bun-
00 demonstrate that chirality can result in regular self-limited dlediameter(n∗f)thatminimizesgrod,thedistributionwillbe
1 bundlesforarangeofinteractionstrengths,butthatstronger sharply peaked aboutn∗f when hli ≫ 1, and growth will be
: interactions enable defects which give rise to branched net- self-limited. Thus, webeginbymeasuringgrod as afunction
v worksorirregularbundles. ofnf.
Xi Drawingconclusionsaboutself-limitedgrowthinamacro- We note that in Ref. [15] filamentous bundle formation
is described in terms of the theory of linear polymerization
r scopic system fromsimulationswith a finite numberof sub-
a [19,20]andbundlingoflinearpolymers.Thatanalysiswould
units is challenging–one must distinguish between simula-
be complicated for our modelsince the subunit binding free
tions in which growth terminates due to physicalconstraints
energydependscruciallyonthenumberofbundledfilaments
fromthoseinwhichthesystemrunsoutofsubunits[15]. To
duetothetwistimposedbychirality.
overcomethislimitation,wesimulatethegrandcanonicalen-
SubunitModel. Weconsidersphericalsubunitsofdiame-
semble(µVT),inwhichgrowthcannotterminatebecauseof
ter σ that are endowedwith a polarorientationunit vectorpˆ
subunitdepletion,sincethesystemiscoupledtoanunlimited
andhave azimuthalsymmetry,( Fig. 1). Subunitshavepair-
bath of free subunits. Before discussing the simulations, we
wiseinteractionsthatdrivenorth-to-southpolealignmentinto
considertheconditionsforself-limitedfilamentousassembly
filamentsandweakequator-to-equatorinteractionsthatdrive
atfixedtotalsubunitconcentration(theNVT ensemble).
bundlingoffilaments.Theinteractionbetweentwosubunitsi
We build onthe theoryfor cylindricalmicelles[16–18] to
andj isgivenby
V(i,j)=V (r )+V (cosθ,d )
h ij p ij
∗Electronicaddress:[email protected] +Vp(cosθ,dji)+Ve(~rij,pˆi,pˆj) (2)
2
where~r = ~r −~r is the interparticledisplacement, r =
ij j i ij
|~r |, cosθ = pˆ ·pˆ gives the angle between the two polar
ij i j
directions, and d = |(~r −σpˆ /2)−(~r +σpˆ/2)| is the
ij j j i i
distance between poles. Excluded volume is imposed by a
hard-sphereinteraction
∞ r <σ
V (r)= . (3)
h (cid:26)0 r ≥σ
Thepole-poleinteractionisgivenby
V (cosθ,d)=−ǫ H(d −d)f(θ,θ ) (4)
p p p max FIG.1: Subunit model. (left)Thepole-poleinteraction. d isthe
ij
distance between the corresponding poles, and θ is the angle be-
withH(x)theHeavisidestepfunction,ǫ thepole-poleinter-
p tween pˆi and pˆj. (right) The equatorial interaction. ϕ is the di-
actionstrength, d the distance tolerance,and θ the angle
p max hedralanglebetweentheplane(~rij,pˆi)andtheplane(~rij,pˆj). For
tolerance.Parallelalignmentisdrivenby all simulations reported inthis work, the polar interaction strength
f(y,y )= exp(−y2/ym2ax) y <ymax (5) iasndǫpth=e 1d4isktaBnTc0e, tahnedlaatnegrlaelitnotleerraacntcioenpsatrraemngettherissaǫree=dp2.=95k0B.1Tσ0,,
max (cid:26) 0 y ≥ymax de = 0.1σ, βmax = 1, φmax = θmax = 0.25 with angles in ra-
dians. The temperature is T = T0, except for the simulation of
Theequatorialinteractionisgivenby ϕskew = 0,forwhichT =1.05T0. TheGCbathchemicalpotential
isµ = k Tln0.01. ForthismodelG(1,1) = k Tln8π2 dueto
b B B
V (~r ,pˆ,pˆ )=−ǫ H(d −r )H(β −β ) rotationalentropy.
e ij i j e e ij max ij
H(β −β )f(ϕ−ϕ ,ϕ ) (6)
max ji skew max
whereǫ istheequatorialinteractionstrength,d istheequa- rapidriseinfreeenergyatsmallnduringwhichshortstruc-
e e
torialdistancetolerance,andthewidthoftheequatorialinter- tureswithnf =3,4,and5filamentsappearsuccessively,fol-
action bandis set by βmax with cosβij = |~rij ·pˆi|/rij. The lowedbythecriticalnucleuswithnf =7andn≈25subunits
final factor in Eq. 6 measures the degree of twist, with ϕ as (Fig. 2A. The free energy is unfavorable below this size be-
the dihedral angle between the plane (~r ,pˆ) and the plane causethe majorityofsubunitshaveunsatisfiedlateraland/or
ij i
(~r ,pˆ )(Fig.1),whichcanbecalculatedfromthefollowing polar contacts (see Cap Free Energybelow). After reaching
ij j
relations,with~q =~r ×pˆ thecriticalnucleus,thebundlegrowslengthwiseinbothdirec-
ij ij i
tions, while maintainingthe same structure, and the free en-
sinϕ= (~qij ×~qji)·~rij, cosϕ= ~qij ·~qji (7) ergydecreaseslinearly.Asshownbelow,thenf =7structure
correspondstotheoptimalbundlestructureforϕ = 0.38
|~q ||~q ||r | |~q ||~q | skew
ij ji ij ij ji
and thus further lateral growth is unfavorable. The slope of
The degreeof chirality is dictated by the preferredskew an- thefreeenergyinthisregioncorrespondstothechemicalpo-
gle ϕskew, with ϕmax as the tolerance for deviationsfrom the tentialgrod(nf =7).
preferredskew. While the exponential distribution of filament lengths is
Simulations. WeexploreassemblywithMonteCarlo,with derived above for the NVT ensemble, in the µVT ensemble
random translations and rigid body rotations of subunits ac- thebundlewillcontinuetogrowlengthwiseindefinitely(pro-
cepted according to the Metropolis criterion[13]. We focus videdthatgrod < −µb). Toevaluatethefreeenergyoflateral
on parameters for which nucleation is a rare event and bun- growth,weimposedhardsphericalboundaryconditionswith
dles grow by addition of monomers, and thus we use only adiameterD =44σ,whichislargeenoughthatbundleprop-
single particle moves. To ensure that self-limited growth is ertiesareindependentofD. Uponreachingtheboundary,the
notaresultofsubunitdepletion,wesamplethegrandcanon- bundlegrowsbyincreasingitsdiameter,whichresultsinthe
ical ensemble by coupling the system to a subunit bath at increasingfreeenergyatlargeninFig.2.
chemical potential µ with subunit insertion/deletion moves Cap free energy. The cap free energyis calculated using
b
[13]. Since there are large nucleation barriers, we employ Gcap(n,nf)=G(n,nf)−n(grod(nf)−g1),withgrodobtained
umbrella sampling simulations[13] to calculate the free en- fromtheslopeofthelinearregimeinthefreeenergy(orfrom
ergy. We use the total bundle size n as the reaction coor- Fig. 3 below). As shown in Fig. 2, G rises rapidly un-
cap
dinate, and measure the probability p(n,n) that a particu- til saturating at the critical nucleus with G ≈ 37k T for
f cap B
lar subunit is in a bundle of size n, using a series of win- ϕ = 0.38. Thisvalueissimilartocapenergiesmeasured
skew
dowsinwhichhardwallsconstrainthesizeofthatclusterto forcylindricalmicelles[18] andcorrespondstoa largeaver-
a range of n. The free energy is then obtained from Eq. 1 agebundlelengthinthe canonicalensemble; usingthemea-
withG(n,n) = −k T ln[ρ(n,n)σ3]+µ ,withρ(n,n) = suredg andG (n,n)we solvedEq.1 to obtaina mass-
f B f b f rod cap f
p(n,n)/n[21,22]. averaged bundle size of about 108 subunits at the CSC. Al-
f
The free energyfor ϕ = 0.38 and snapshotsof repre- thoughthemagnitudeofG dependsonthestrengthofthe
skew cap
sentativebundleconfigurationsareshowninFig.2. Weseea polarbonds(ǫ = 14k T),wefindthatrobustbundleforma-
p B
3
40 -3
A Cap energy
30 -4 ϕ =0.38
skew
20 n)f
10 G(n,nf)-nµbath g(rod -5 ϕskew=0.32
0 B C -6 ϕskew=0.26
-10 -7
0 100 200 300 400 3451+1+1+2+3+3+4+1+1+1+1+1+
566889156666
Bundle size n + 0+++++
1 19111
0 222
++
47
FIG.2: . (left)FreeenergyGandcapfreeenergyGcap asfunctions Bundle structures
ofthenumber ofsubunitsnwithϕ = 0.38obtainedfromum-
skew
brellasampling. Theriseinfreeenergyatlargenoccurswhenthe FIG. 3: (left) Subunit free energy as a function of filament struc-
bundlereachesthehardsystemboundaryandbeginstogrowathird turefor threeskew angles fromthe constrained umbrella sampling
layer.(right)Structurescorrespondingtotheindicatedpointsonthe (CFNUS). Structures are labeled with the number of filaments in
freeenergyplot. eachlayer startingfromthecenter. Resultsareshownonlyfor the
lowestfreeenergymorphologyateachvalueofn.(right)Snapshots
f
fromCFNUSsimulations,shownincross-sectionview,illustrateop-
tion requires ǫp ≫ ǫe, implying that large cap free energies timalbundlemorphologiesforsomevaluesofnf.
andhencelargeaveragefilamentlengthsaregeneral.
Similar results were obtained for lower skew angles
ϕ = 0.32 and ϕ = 0.25 at low n, but convergence to the central filament; tilt increases with layer number for
skew skew
inthefreeenergycalculationwasquestionablebecausetran- preferred skew angles ϕ above a critical value. Due to
skew
sitionsbetweendifferentvaluesofn wererareatlargen. We tilt, filaments trace a curved path around the bundle, which
f
overcamethislimitationasfollows. requires unfavorable bending of polar bonds. Furthermore,
Self-limited bundle diameters depend on preferred completeformationoflateralbondsbetweenlayers, requires
skew. Intheumbrellasamplingsimulationsthesystemadopts thattheexteriorfilamentsstretchwhiletheinteriorfilaments
thenumberoffilamentsn∗thatminimizesGforagivenn. To compress. With each additional layer, the degree of exten-
f
determine the dependenceof the subunit free energy g on sionandcompressionincreases.Theseeffectsoverwhelmthe
rod
n, we performed additional sets of ‘constant filament num- energeticbenefitofaddinganadditionallayer attheoptimal
f
ber’umbrellasampling(CFNUS)simulationsinwhichn and bundlediametern∗. Sincethemagnitudeofthetiltincreases
f f
n are constrained. A small structurewith nf filamentsis ex- with ϕskew, n∗f decreases with increasing ϕskew (Fig. 3). In
tractedfromanumbrellasamplingsimulation,andsubjected agreement with this explanation, the chemical potential for
toasimulationinwhichanymovethatchangesnfisrejected. ϕskew = 0 (correspondingto achiral interactions) does NOT
Specifically,moveswhichcausethenumberofsubunitsinany showaminimum,andasshownbelow(Fig.4b)bundleswith
filament to differ by more than 3 are rejected and the mean ϕ = 0 have unboundedlateral growth. This observation
skew
of each filament along the bundle axis must remain within confirms that chirality is the reason for self-limited bundle
2σ of the bundle center. These additionalrequirementscon- sizesinthismodel.
straintheconfigurationsofthecapandhenceaffectG ,but Interestingly, the optimal bundle morphology for a given
cap
do notaffectg in long bundles, which is determinedfrom numberoffilamentschangeswithn.AsshowninFig.3b,the
rod f
g (n) = (∂G(n,n)/∂n) at largen (wheng becomes centrallayerofthebundlecanvarybetween1and4protofil-
rod f f nf rod
independent of n). This procedure is repeated for all com- aments, and usually corresponds to the structure that maxi-
monlyobservedmorphologieswithagivenn. mizes rotational symmetry. While these are the lowest free
f
The chemical potential grod calculated from the CFNUS energystructuresateachvalueofnf amongthosetakenfrom
simulations is shown as a function of bundle size for three unrestrainedumbrella simulations, we cannotrule out lower
preferredskewanglesinFig.3. Ineachcase,thereisanopti- freeenergymorphologiesthatwedidnottest.
malbundlediameter,ornumberoffilamentsn∗. Althoughthe Dynamics. Having shown that self-limited bundles are
f
minimaappearshallow,thelargeaveragebundlelengthscal- the equilibrium state for our model chiral subunits, we now
culatedaboveensure thatthe free energiesfor differentbun- demonstrate that they are also the kinetically selected state.
dle morphologies differ by many k T. Note that n∗ = 7 Since bundle formation is not accessible by straightforward
B f
for ϕ = 0.38, in agreement with the unrestrained um- dynamical simulations due to the large nucleation barrier
skew
brella sampling. Furthermore, the slope of the free energy (Fig.2),we usedforwardfluxsampling(FFS)[14]toobtain
inthelinearregionofFig.2givesg = −4.75(viaEq.1), an unbiased ensembleof assembly trajectories. We used the
rod
whichmatchesthechemicalpotentialdeterminedfornf = 7 bundlesizenastheorderparameterandperformedFFSuntil
inFig.3,showingthatthetwoprotocolsagree. bundlesreachedasizelargerthanthecriticalnucleus(itisnot
Theexistenceofanoptimaldiametercanbeunderstoodas necessarythattheorderparameterbeagoodreactioncoordi-
follows.Addinglayersdecreasesenergy,sincesubunitsinthe nate,althoughabadorderparametercaninhibitconvergence).
outermost layer have unsatisfied lateral contacts. However, Atthispoint,FFSwasnolongerneeded,andwecontinuedthe
the preferredskew ϕ causes filamentsto tilt with respect simulations with straightforward dynamic Monte Carlo; nu-
skew
4
104 independentofnf,inagreementwiththefreeenergycalcula-
tions. Although the bundle grows with hexagonalorder, we
Ne 103 note that pentagonal defects become trapped within the as-
z
Si semblage,particularlyduringrapidgrowththatoccursunder
e
undl102 strongerinteractions.
B
N ∝ t2 Stronginteractionsleadtobranchednetworks. Bundles
101106 107 108 withnf >n∗f tendtoformdefectsthatrelievestrain. Insome
t [sweeps] cases these defects serve as nucleation points for branching,
whichenablesfurthergrowthuntilthebranchreachesitsop-
FIG.4: (a)BundlesizeasafunctionofMonteCarlo(MC)sweeps timal diameter and a system boundary. As shown in Fig. 5,
forthreetrajectorieswithϕskew = 0andT = 1.05T0. Trajectories each branch maintains a finite radius. Additional branching
wereinitiatedusingforwardfluxsampling(FFS)asdescribedinthe
andbundlegrowthleadtoabranchednetwork(seeRef.[19]
text,dynamicsareshownafterFFSended. Thesolidlineindicates
forfurtherdiscussionofbranchedbundles). Wealsoobserve
thescalingn(t)∼t2.(b)Asnapshotfromonetrajectoryin(a).
branchinginthe canonicalensemblesimulationswith strong
interactions. While thissimple modelisnotintendedtorep-
resent a particular molecule and crowding affects assembly
athighdensity[6],wenotethatfibrinclotsarecomposedof
branchednetworksoffibrinbundleswithuniformbundleradii
(e.g.[2]).
Inconclusion,wesimulatedtheassemblyofsubunitswith
a simple potential that drives the formation of filamentous
bundles. For moderate interaction strengths, the local pack-
FIG. 5: A branched bundle formed in a dynamical trajectory with ing constraints that arise due to chirality cause assembly to
ϕ =0.32. terminate at a finite bundle diameter, while bundles propa-
skew
gate easily in length. Stronger interactions, however, lead
to defectswhich enable the formationof multiply connected
cleatedbundlesreadilygrowin lengthuntiltheyreachedthe branched networks. The optimal morphologies of assem-
imposedboundary(D = 32σ), butlateralgrowthterminates bledbundleswithdifferentnumbersoffilamentshavediffer-
at n = 7, in agreement with the equilibrium calculations. ent symmetries. The simulation results indicate that sponta-
f
ThedynamicnucleationpathwaysobservedwithFFSclosely neously assembled structures can deviate significantly from
followtheminimumfreeenergypathwayobservedwithum- regularhexagonalbundles, and thus it is importantto evalu-
brellasampling(Fig.2b),indicatingthatstructuresbelowthe ateassemblybehaviorwithdynamicalalgorithmsthatdonot
nucleussizeachieverelativeequilibriumquicklyincompari- impose particular assembly pathways or morphologies. The
sontothenucleationtime[23,24]. approachwehaveadoptedtoevaluateself-limitedgrowthina
In contrast, assembly trajectories for ϕ = 0 (Fig. 4) finite-sizedsimulationcouldbeusedtounderstandspecificbi-
skew
demonstrate unbounded lateral growth, and after reaching a ologicalmolecules;forexample,apatchy-spheremodelcould
size of n ≈ 100 scale as n(t) ∼ t2, with t the number of be constructed from atomic-resolution structures of sickle
Monte Carlo sweeps. This scaling is consistent with growth hemoglobininordertounderstandtheeffectsofchiralityand
dominated by subunit addition to the bundle body, and g sphere-packingonhemoglobinfilamentassembly[3–5].
rod
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