The Impact of ConformationalFluctuations on Self-Assembly:Cooperative Aggregation of ArchaealChaperonin ProteinsStephen Whitelam,†,‡ Carl Rogers,§ Andrea Pasqua,| Chad Paavola,⊥Jonathan Trent,⊥ and Phillip L. Geissler*,|,#

Systems Biology Centre, UniVersity of Warwick, CoVentry CV4 7AL, U.K., BiophysicsGraduate Group, UniVersity of California at Berkeley, Berkeley, California 94720,Department of Chemistry, UniVersity of California at Berkeley, Berkeley, California 94720,Bioengineering Branch, NASA Ames Research Center, Mail Stop 239-15, Moffett Field,California 94035, and Materials Sciences, Physical Biosciences, and Chemical SciencesDiVisions, Lawrence Berkeley National Laboratory, Berkeley, California, 94720

Received September 25, 2008; Revised Manuscript Received November 19, 2008

ABSTRACT

Protein complexes called rosettasomes self-assemble in solution to form large-scale filamentous and planar structures. The relative abundanceof these aggregates varies abruptly with environmental conditions and sample composition. Our simulations of a model of patchy nanoparticlescan reproduce this sharp crossover, but only if particles are allowed to switch between two internal states favoring different geometries oflocal binding. These results demonstrate how local conformational adaptivity can fundamentally influence the cooperativity of pattern-formingdynamics.

Group II chaperonin complexes from the hyperthermophilicarchaeon Sulfolobus, called rosettasomes, spontaneouslyassemble in vitro into micrometer-scale structures.1,2 Tworemarkably different architectures can result from thisprocess: bundles of filamentous chains of roughly sphericalrosettasome units bound pole-to-pole, and stacks of hexago-nally packed sheets of units bound equator-to-equator. Suchmaterials separately display great promise for nanotechnol-ogy. Sheets and strings of mutant chaperonins have beenused to build ordered 2d arrays of quantum dots2 andmetallized conducting nanowires.3 Filaments may also fulfilla biological role as components of the archaeal cytoskeleton.4

The dual nature of rosettasome assembly offers furtherpromise for materials science: the prospect of switchingbetween the assembly of different structures by controllingonly environmental conditions is very attractive. Conceptu-

ally, dual assembly poses new questions regarding thestatistical dynamics of organized growth: How do multiplemodes of assembly cooperate or interfere? Are hybridstructures an unavoidable end point when forces favoringdifferent structural motifs combine?

Emulating the complex self-assembly properties of roset-tasomes with synthetic particles requires interactions of a“patchy” nature. It may seem sufficient to situate a stronglysticky region at each particle’s equator and a distinct kindof stickiness at its poles. Several computer simulationstudies5-15 and a few ambitious experiments have adoptedan analogous perspective, attempting to design specificaggregate geometries through anisotropy of microscopicforces. However, component design based solely on structuralconsiderations rarely results in successful assembly. Thepicture emerging from these studies is one of intensecompetition between thermodynamics and kinetics: even forsimple patchy nanoparticles, stochastic growth dynamics andthe difficulties of defect annealing condition successfulassembly on the fine-tuning of nanoparticle attractionstrengths and binding geometry specificities.

In this paper we use theoretical models based on thesephysical perspectives to explain the dual nature of rosetta-some assembly. We draw attention to a striking qualitative

* Author to whom correspondence should be addressed. Email: geissler@berkeley.edu.

† Systems Biology Centre, University of Warwick.‡ Current address: Molecular Foundry, Lawrence Berkeley National

Laboratory, Berkeley, CA 94720.§ Biophysics Graduate Group, University of California at Berkeley.| Department of Chemistry, University of California at Berkeley.⊥ Bioengineering Branch, NASA Ames Research Center.# Materials Sciences, Physical Biosciences, and Chemical Sciences

Divisions, Lawrence Berkeley National Laboratory.

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10.1021/nl8029306 CCC: $40.75 2009 American Chemical SocietyPublished on Web 12/12/2008

observation from experiments, which suggests rosettasomeaggregation to be a binary rather than a hybrid process:structures intermediate in character between bundled stringsand few-layered sheets are not seen in micrographs. Togetherwith the observation that planar and filamentous structuressometimes appear in the same sample, this fact suggests aprecipitous transition in aggregate geometry, from stringliketo sheetlike, as control parameters (temperature, concentra-tion, protein amino acid sequence, etc.) are changed. Bycontrast, our simulations of collections of patchy nanopar-ticles give rise to products that display a gradual transforma-tion with model parameters from stringlike to sheetlikecharacter. We resolve this discrepancy by considering inaddition a conformational flexibility of assembling units. Thekinetic effects of this simple local detail are global in bothspace and time. Although our examination of this cooperativedynamical phenomenon is inspired by the behavior of aspecific protein complex, its implications are quite general:endowing nanoscopic self-assembling components with asimple degree of conformational flexibility permits modesof assembly unattainable to components of fixed conforma-tion.16

The precise interactions between two rosettasome units,each comprising about 105 atoms, are enormously compli-cated. Detailed molecular dynamics simulations might ac-curately reproduce these solvent-mediated forces, but suchan approach could scarcely be applied over time scales ofmilliseconds or longer that give rise to the higher-orderstructures considered here. Therefore, we treat free energiesof interaction in a schematic way, considering for each pairof units only the dependence on a handful of geometricparameters, as sketched in Figure 1. Our effective potentialsare constructed with attention to a few key physical factsand ideas: each unit excludes a roughly spherical volume ofradius a ) 8.5 nm; attractive interactions extend overdistances small compared to a and vary weakly with rotationof a unit about its axis;23 and potentials of mean forcestabilize bindings pole-to-pole and equator-to-equator, asindicated by high-resolution microscopy.1

In detail, our model consists of N nonoverlapping spheresof radius a, each representing a single rosettasome unit. Theattraction upair between units i and j possesses polar andequatorial contributions, both depending on the interunitseparation vector rij and unit orientation vectors ni and nj

upair(rij, ni, nj))-εeqυeq(rij, ni, nj)- εpolVpol(rij, ni, nj) (1)

The strengths εpol and εeq of these two interactions, relativeto that of typical thermal excitations kBT, constitute the mostimportant control parameters of our study. In simulationsthat permit units to change conformation, these parametersbecome in effect fluctuating variables.

The short-range of rosettasomes’ interactions relative totheir size motivates our simplified choice of the potential’sdistance dependence: the model functions Vpol and Veq vanishfor separations greater than 2a + a/4, yielding an attractionrange a/4 ≈ 2 nm consistent with Debye screening atexperimental ionic strengths. Below this cutoff distance, thesefunctions vary only with particle orientations, and do soaccording to the equations

υeq ) C1(�ij;σalign)C0(θi;σeq)C0(θj;σeq) (2)

υpol ) C1(�ij;σalign)C1(θi;σpol)C1(θj;σpol) (3)

Equations 2 and 3 describe in mathematical terms thepatchiness sketched in Figure 1. The effective size of polarand equatorial patches is determined by inverse specificitiesσalign, σeq, and σpol; these parametrize a function CR(ψ; σ) ≡e-(cosψ-R)2/σ2

that rewards the alignment of angles ψ and cos-1

R within a tolerance σ. The function CR(ψ; σ) ≡ CR(ψ; σ)+ C-R(ψ; σ) is made symmetric to preserve the up-downequivalence of rosettasome units suggested by crystalstructures of homologous proteins. We set σalign ) σeq ) 0.3and σp ) 0.12 (see Supporting Information for a discussionof model parameters).

We report below on long trajectories of this model system,corresponding to roughly tens of milliseconds of evolutionfor an experimental system. Dynamics were propagated usinga virtual-move Monte Carlo procedure11 designed to mimicthe aggregation kinetics of strongly associating nanopar-ticles.24 We initiated each simulation by selecting particleorientations and positions at random (while respecting stericconstraints) within a cubic box of length L. Periodic boundaryconditions were imposed in imitation of bulk surroundings.

We have explored the assembly dynamics of units withfixed conformation as a function of εeq and εpol. For manysets of values for these parameters, we ran trajectories of N) 800 units in a box of length 60a. Units therefore occupied1.5% of the box volume, corresponding to a proteinconcentration of 10.9 mg/mL, typical of experimentalsamples (1-20 mg/mL). Results of these numerous trajec-tories are summarized by the kinetic phase diagram in Figure2, which reports both on the occurrence of assembly (judgedby the presence of clusters comprising 20 or more units)25

and on resulting geometries.We define an order parameter

S ≡ 16

(6- neq)-12

(2- npol) (4)

to quantify the morphology of each aggregate. For a givencluster above the cutoff size of 20 units, neq estimates thenumber of equatorial neighbors per particle (computed bysumming υeq over all interacting pairs and dividing by thenumber of particles); npol is calculated in an analogousmanner. S takes the value +1 for an infinitely long filament,S ) -1 for an infinitely extended close-packed sheet, and

Figure 1. Geometry for our model of associating rosettasomes.Angular variables are related to orientation vectors by ni·nj ) cosφij and ni·rij/|rij| ) cos θi.

Nano Lett., Vol. 9, No. 1, 2009 293

intermediate values for structures of finite size and forgeometric character intermediate between sheets and strings.Any aggregate having S > 1/4 is identified as a “string”,while any aggregate for which S < -1/4 is classed as a“sheet”. We consider aggregates satisfying -1/4 e S e 1/4to be of ambiguous character.

Kinetic accessibility is a crucial issue in the assembly ofsheetlike structures. Although sheets are thermodynamicallystable under many of the conditions we have considered,the range of εeq in which planar aggregates grow in anordered fashion is small (see Supporting Information). Theregion εeq g 7kBT is dominated by kinetic frustration, whilethe region εeq ) 6kBT, from εpol ) 0 to εpol ) 4kBT issupercooled: sheets are thermodynamically stable but nucle-ate too slowly to appear in the simulation times considered.

The kinetic phase diagram and illustrated configurationsindicate a propensity for hybrid assembly when both equato-rial and polar attractions are appreciable. Under no conditionsdo structures of distinct geometry coexist at the samethermodynamic state. Instead, we observe a continuousvariation with model parameters from sheetlike to stringlikeassembly. At some values of (εeq, εpol), the two modes ofassembly clearly hinder one another, either by sequesteringmaterial or by leading the ordered growth of nascent clustersastray. At other parameter values these modes cooperate,generating well-ordered hybrid structures that interpolatesmoothly between layered sheets and bundled strings.

This smooth evolution is expressly not what is observedin experiments. Instead, laboratory measurements suggest avery sharp crossover from sheets to strings with changingenvironmental conditions. The absence of an intermediateregime of hybrid assembly is highlighted by the observationof behavior akin to phase coexistence. Transmission electronmicroscopy (TEM) images in Figure 3 demonstrate thesimultaneous occurrence of sheets and strings. The discrep-ancy with simulation results is striking. Accord betweenexperiments and our patchy nanoparticle model is unlikelyto be achieved by revising our choices of interaction strengths

and specificities: we have performed many additionalsimulations in which we varied component concentrations,system sizes, binding tolerances, and energy scales and failedto observe sheet-string coexistence in all cases.

Why does the model we have presented fail to producecoexisting superstructure types? We cannot rule out thepossibility that fine details of intermolecular forces are keyto capturing this emergent behavior. We argue, however, thatthe qualitative contrast between binary assembly reminiscentof a phase transition and hybrid aggregation that variessmoothly with model parameters points to the neglect of amore fundamental feature. We propose that this missingingredient is the conformational flexibility of each rosetta-some unit.

Previous work has established as many as three differentconformations for the S. Shibatae rosettasome.17,18 Motivatedby these studies and by our findings above, we haveaugmented our model of patchy nanoparticles by allowingeach particle the freedom to change between two differentconformations. One of these conformations is biased towardforming sheetlike geometries. When particles adopt this state,we refer to them as planons. Particles assuming the alternateconformation, which we dub stringons, have a preferencefor filamentous self-assembly. We will see that the abilityof particles to fluctuate between these identities indeedgreatly sharpens the transition between assembled structuresand can effect coexistence of sheets and filaments.

These preferences conferred by internal degrees of freedomappear in our model through variations in binding energies.In all the calculations we will describe, the equatorialcoupling strength εeq between two adjacent planons is set to6.5kBT, sufficient to ensure that a collection of particles

Figure 2. Kinetic phase diagram for units of fixed conformation.We identify regions in which assembled structures are stringlike,e.g., (a), sheetlike, e.g., (d), or of ambiguous character, e.g., (c).Kinetically frustrated structures, e.g., (b), are found with increasingabundance as we increase the potency of the equatorial coupling.

Figure 3. TEM images taken from a single sample of self-assembling rosettasomes demonstrate the coexistence of super-structures with distinctly different architectures: (a) a close-packedsheet (spanning the whole image but partially obscured by overlaymaterial) and (b) bundled strings. For materials and methods, seeSupporting Information.

294 Nano Lett., Vol. 9, No. 1, 2009

constrained to their planon states can form large-scale orderedsheets. To ensure the possibility of filamentous growth wechoose the strength εpol of polar interactions between twoadjacent stringons to be 16kBT (these choices are discussedin detail in Supporting Information). We also assign minorcouplings of varying strengths: these consist of polarinteractions between planons, equatorial interactions betweenstringons, and interspecies interactions. These are sum-marized in Table 1.

In the course of model assembly dynamics, we permitmonomer conformations to fluctuate at random, subject tobiases from interaction energies and from an intrinsic freeenergy difference between planon and stringon states. Eachunit possesses an internal energy uinternal, which we set to-kBT ln δ for stringons and 0 for planons. The parameter δthus represents the ratio of stringons to planons in a sampleof noninteracting rosettasome units. Just as internal statesbias propensity for structure formation, however, so ag-gregation can bias conformational populations significantlyaway from ideal statistics.

Algorithmically, we attempted a change of each unit’sconformation on average once every Mchange ) 1000 MonteCarlo sweeps, irrespective of that unit’s position, orientation,or environment. By accepting such changes with probability

pchange )min[1, exp(-�∆E- �∆uinternal)] (5)

we established a basic rate ωchange ) ω0(1 + δ)/Mchange ≈105 s-1 for conformational fluctuations of isolated units thatis consistent with estimates for similar large-scale rearrange-ments in related proteins.19,26 Here ω0

-1 is the duration of asingle MC sweep (involving on average one proposed moveper particle). On the basis of the diffusivity expected fromStokes law for a sphere of radius 8.5 nm in liquid water, weestimate that ω0 ≈ 108 s-1. In eq 4, ∆E and ∆uinternal are thechanges in interaction energy and internal energy resultingfrom a proposed conformation change, respectively. Whilethe attempt rate for a unit’s conformation change does notdepend on that unit’s environment, the acceptance rate forthe change takes account of the energy of interaction betweenthe unit and its neighbors and so biases the conformation ofan interacting unit with respect to that of its isolatedcounterpart.

We computed assembly trajectories for N ) 1500 con-formationally flexible particles in a box of length 64a. Byexamining several values of δ and of the minor couplings,we varied the strength of interactions between planons andstringons, as well as the degree of distinction between them.Units’ initial identities were randomly assigned as stringonswith probability δ/(1 + δ), or alternatively as planons withprobability 1/(1 + δ).

We focus first on the case of vanishing minor couplings.Here, stringons and planons are orientationally ambivalenttoward one another, interacting only through sterics butintimately linked through interconversion. In Figure 4 weshow a time-ordered series of snapshots taken from a singletrajectory featuring this extreme example of conformationalchange. Allowing unit identities to fluctuate can indeed yieldat late times a coexistence of structures possessing sheetlikeand stringlike symmetry. Interestingly, in order to observesuch coexistence, we must bias the internal energy of unitsstrongly in favor of planons; here δ ) 0.05, correspondingto an internal bias of 3kBT. The necessity for this imbalanceoriginates in an unambiguous physical mechanism: As one-dimensional structures, strings lack a nucleation barrierand thus form much more rapidly than sheets, which mustawait the appearance of a critical nucleus. This observationrecalls the rule of Stranski and Totomanow,20 who proposedthat the crystal phase that forms most readily is the oneconfronted by the smallest free energy barrier to its nucle-ation. We have quantified the free energy barrier ∆G‡ to sheetnucleation at various planon concentrations using umbrellasampling simulations (see Supporting Information). At thehighest density, corresponding to the initial state of anassembly trajectory, this barrier is little larger than energiestypical of thermal fluctuations, so that sheet initiation is onlyweakly impeded. However, string formation quickly seques-ters units, in effect diluting the pool of monomers availablefor sheet formation. Our umbrella sampling results show thatremoving 60% of the available units raises ∆Gq by severalkBT, slowing nucleation more than 100-fold. This dynamicelevation of the nucleation barrier necessitates a large biasin internal energy in favor of planons (small δ) in order toobtain coexistence. These considerations underscore thefundamentally nonequilibrium nature of assembly within ourmodel: when fully relaxed (at inaccessibly long times), thissystem would possess almost no strings or stringons.

Equipping units with nonzero minor couplings introducesinterspecies attractions, which, as one might expect, facilitatethe formation of structures of a composite character, namely,stacked sheets and bundled filaments. The minor couplingsalso control interspecies energetic interactions, which influ-

Table 1. Stringon and Planon Interaction Strengths.a

interacting species εeq εpol

planon-planon 6.5kBT εpol(min)

stringon-planon εeq(min) εpol

(min)

stringon-stringon εeq(min) 16kBT

a We consider fixed major couplings and different sets of minor couplings.The specific case εeq

(min) ) εpol(min) ) 0 corresponds to planons and stringons

interacting only through hard core constraints. We have considered minorequatorial attractions as strong as εeq

(min) ) 4kBT and minor polar attractionsas strong as 5kBT. In the limit εpol

(min) ) 16kBT, εeq(min) ) 6.5kBT, the

distinction between planons and stringons vanishes. Figure 4. Time sequence of the assembly of conformation-changingunits. Here, units of different types interact only through intercon-version and sterics (εeq

(min) ) 0 ) εpol(min)). The internal energy bias

parameter is δ ) 0.05, so that isolated planons (blue) are stronglyfavored thermodynamically over stringons (yellow). However, stringformation is facile, leading to an increase in the number of stringonunits with time. Parameters are such that we observe coexistencebetween large-scale assembled structures of the two distinctsymmetries. Time is measured in millions of Monte Carlo movesper particle.

Nano Lett., Vol. 9, No. 1, 2009 295

ence the ease of unit interconversion. We find that the kineticbias toward string formation identified previously is quali-tatively unchanged, but interactions between species modifyquantitatively the rate at which structures of the two speciesproliferate (see Supporting Information). In Figure 5 we showqualitatively the effect of minor couplings by presentingsnapshots from long trajectories generated using nine dif-ferent sets of parameters (representing a variety of thermo-dynamic states in the real system). These states differ in theinternal energies of conformational states (set by δ) and intheir minor energetic couplings.

Cooperative binding as a mechanism for precipitousresponse is a textbook notion of biochemistry: interactionswith one ligand heighten affinity for additional bindingevents, establishing positive feedback and enabling sharpswitching behavior. Our results illustrate the dramatic kineticimplications of this kind of cooperativity. The moleculardynamics of self-assembly enriches traditional cooperativityscenarios in several interesting ways. The growth of extendedstructures, for example, introduces spatially heterogeneouseffects of binding and leads to the emergence of “mi-crophases” of distinct response types. In addition, kineticbarriers to the formation of distinct structures can dramati-cally alter the threshold conditions at which switching occurs,imbuing the resulting behavior with a fundamentally non-equilibrium character. Further, cooperative binding can drivethe formation of composite assemblies through fusion ofwell-defined superstructures (see Supporting Information).

In the context of rosettasome self-assembly, our resultssupport a picture in which internal conformational fluctua-

tions bias superstructure nucleation, thereby inducing thesimultaneous growth of structures with distinct symmetries.Rosettasome allostery is a known phenomenon, whilesuperstructure coexistence is a striking feature of recentexperiments involving the wild-type protein and its geneti-cally engineered variants. While the function of manyproteins depends on self-organization at the nanometer scale,the emergence of long-range order with a Variable outcomeis exceptional. Such a capability might be of considerableuse to a hyperthermophilic organism seeking to respond toextremes of heat by promoting, for instance, the rapidassembly of membrane-stabilizing scaffolding (biases forparticular rosettasome conformations might be mediated bychanges in their protein composition, known to vary inresponse to environmental cues21). From the perspective ofnanoscience, mimicking such cooperativity offers a plausibleroute to adaptive assembly, whereby subtle changes inenvironment can engender a dramatic change in the assemblyproducts of a system.

Acknowledgment. This work was supported by theDirector, Office of Science, Office of Basic Energy Sciences,Materials Sciences and Engineering Division, of the U.S.Department of Energy under Contract No. DE-AC02-05CH11231. S.W. was supported by the BioSim EuropeanUnion Network of Excellence (LSHB-CT-2004-005137).C.R. was supported by a NIH Molecular Biophysics TrainingGrant (T32GM08295). Computing facilities were providedin part by the Centre for Scientific Computing at theUniversity of Warwick with support from the ScienceResearch Investment Fund. We used VMD22 to rendersimulation configurations.

Supporting Information Available: Experimental pro-cedures for transmission electron microscopy and additionaldiscussion of the computational model, as referenced in thetext. This material is available free of charge via the Internetat http://pubs.acs.org.

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Figure 5. Long-time conformations for self-assembly of conforma-tion-changing units at nine different thermodynamic states. Columnscontain states at fixed values of δ; rows contain states at fixed valuesof the minor couplings. (a) Nonattracting species (εeq

(min) ) 0 )εpol

(min)); (b) conditions under which planons have a tendency tostack (εeq

(min) ) 0, εpol(min) ) 3kBT); and (c) strongly attracting species

(εeq(min) ) 4kBT, εpol

(min) ) 5kBT). Assemblies with hybrid characterappear as minor couplings are increased, but the fundamental kineticbias in favor of string formation remains.

296 Nano Lett., Vol. 9, No. 1, 2009

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33–38.(23) Rosettasomes are 9-fold symmetric but are observed to pack in

hexagonal lattices incommensurate with such symmetry. We assumetherefore that equatorial interactions vary weakly with longitude.

(24) We drew translation magnitudes from a uniform distribution withmaximum 0.5 a and drew rotation angles from a uniform distributionwith maximum 22.5°. We scaled collective translation acceptance ratesby the reciprocal of an approximate hydrodynamic radius of themoving cluster11 and scaled acceptance rates for rotations by the cubeof the same factor.

(25) The “no assembly” cutoff of size 20 means that we ignore assemblyin regions of the phase diagram where short filaments form, but thisneglect does not change in an important way the trends we identify.

(26) The results we present are qualitatively unchanged by varying Mchange

from 1 to 1000. This robustness suggests that assembly behavior isnot strongly sensitive to changes in the rate of conformationalfluctuations, provided it outpaces the rate of basic structure formation.

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