Physical Chemistry and Soft Matter, Wageningen University and Research, Stippe-neng 4, 6708 WE Wageningen, Netherlands.*These authors contributed equally to this work.†Corresponding author. Email: [email protected]
van Galen et al., Sci. Adv. 2019;5 : eaaw3353 11 October 2019
Allosteric pathway selection in templated assemblyMartijn van Galen*, Ruben Higler*, Joris Sprakel†
Assembling large numbers of molecular building blocks into functional nanostructures is no trivial task. It relies onguiding building blocks through complex energy landscapes shaped by synergistic and antagonistic supramolecularinteractions. In nature, the use of molecular templates is a potent strategy to navigate the process to thedesired structure with high fidelity. Yet, nature’s templating strategy remains to be fully exploited in man-madenanomaterials. Designing effective template-guided self-assembling systems can only be realized through preciseinsight into how the chemical design of building blocks and the resulting balance of repulsive and attractive forcesgive rise to pathway selection and suppression of trapped states. We develop a minimal model to unravel the kineticpathways and pathway selection of the templated assembly of molecular building blocks on a template. We showhow allosteric activation of the associative interactions can suppress undesired solution-aggregation pathways andgives rise to a true template-assembly path.
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INTRODUCTIONDespite its vast complexity, nature is able to construct functional nano-structures from small biomolecular building blockswith an astonishingprecision and fidelity. Virtually all functional biological structures thatform the basis of life, including protein complexes, lipid bilayers, andDNA double helix, self-assemble spontaneously from precisely definedbuilding blocks guided by well-balanced supramolecular interactions(1). It is remarkable that the self-assembly of complex structures indense and crowded surroundings occurs with such a high degree of fidel-ity, as it requires a large number of molecular building blocks to cometogether in a precise order and orientation. High-precision molecularassembly is governed by a delicate balance between repulsive and at-tractive supramolecular forces, excited by thermal fluctuations, whichprevent the formation of kinetically trapped states and structural poly-morphism (2–4). Driven by thermal motion, self-assembly is a chaoticprocess consisting of numerous docking, folding, and reorganizationsteps, occurring in a complex energy landscape that is rich in local energyminima and activation barriers.
Nature has developed a number of strategies to navigate assemblyprocesses through this complex landscape. One particularly potentstrategy is the use of a molecular template as a blueprint to guide theassembly process. The use of a template can greatly enhance both theefficiency and the fidelity of the assembly process: The template canact as a molecular staging area, forcing the building blocks to assemblein specificmorphologies that are otherwise not possible (5). Templatescan also catalyze the assembly process by attracting building blocks,increasing their effective local concentration with respect to the bulksolution. A prominent example of templated assembly is the assemblyof viral capsids, inwhich the organization of the coat proteins is guidedby the nucleic acid polymer it will ultimately encapsulate (6, 7). Otherexamples include the self-organization of light-sensitive rhodopsinproteins on retinal membranes (8, 9) and septin assemblies on actinfibers, which control cytoskeletal structure and dynamics (10).
The effectiveness of templated assembly as a control mechanism innature has also triggered the interest in using molecular templating togain control over the assembly of molecular building blocks into syn-thetic nanostructures (11), such as in template-guided covalent and
noncovalent polymerization (12, 13); the use of carbon nanotubesor DNA origami to template the assembly of proteins, polymers, ornanoparticles (14–16); or the creation of bioengineered artificial virusesfor gene delivery (17, 18).
The final states of templated assembly processes have been studiedin a variety of biological and synthetic systems, yet the underlying kineticpathways are still poorly understood. Unraveling these pathways iscrucial not only to understand how nature can create intricate andadaptable structures with high fidelity but also as a design principleto gain control over kinetic pathways to increase the fidelity with whichsynthetic nanostructure self-assembles. Several analyticalmodels for thekinetic pathways of templated assembly have been proposed (7, 19, 20).A prominent example is the kinetic zipper model for the rod-shapedtobacco mosaic virus, in which the templating process is modeled asa thermodynamic process of monodirectional elongation that startsfrom a predefined nucleation site and the kinetics are steered by theinteractions between the building blocks (20). Coarse-grained molec-ular dynamics simulations have also been used to follow templatedassembly in time (21, 22). Although both analytical approaches andcoarse-grained models successfully describe the template-assemblykinetics, many describe specific cases of templated assembly, suchas predesigned capsid shapes belonging to specific types of viruses.Unraveling the generic design rules of templated assembly will require amore generic model, capable of being tuned to predict a wide variety oftemplate-assembling architectures to address the question that is centralto this paper: Which design requirements must a system obey to effi-ciently and effectively guide a multitude of building blocks to a prede-signed templated structure?
In this paper, we present aminimal simulationmodel to capture theessential and generic features of templated assembly inmacromolecularsystems.Our simulations reveal that assembly occurs along twoprimarykinetic pathways, occurring either through aggregation of buildingblocks in solution followed by template binding or by true templatedassembly in which the template is the staging area for the subsequentorganization of the building blocks into a functional structure.Whilethe relative balance between these two pathways can be tuned by thechemical design of the system, the aggregation pathway remains im-portant under all conditions if no further precautions are taken. Thisimplies that without means of pathway selection, solution aggregationand subsequent kinetic trapping is an important pitfall in templatingsystems.We show how allosteric activation of lateral interactions canalmost completely suppress the aggregation pathway and lead to full
and true template-guided assembly.We combine our simulationswith akinetic reaction model to disentangle the complex process into a set ofprimary supramolecular reactions. This predictive and minimal modelallows a rational design of new templated assembly strategies to gotoward high-fidelity structuring of synthetic nanomaterials usingmolecular blueprinting.
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RESULTSIn essence, templated assembly requires two primary ingredients(Fig. 1A). First, a multivalent template, such as a polymer chain withsupramolecular binding capabilities, must be present onto whichmultiple biomolecular building blocks can bind. Second, we need nu-merous building blocks, here denoted as the “assemblers,” whichbind this template and assemble into the final structure. These buildingblocks typically feature three domains: a domain to dock to the template(domain D), a domain to provide lateral associative interactions be-tween template-bound molecules (domain A), and a stability domain(domain S) that provides a means to moderate associative forces, solu-bilize the final structure, and regulate its size and shape. This function-ality triad is found in many biological building blocks and variousbioinspired systems (7, 12, 17). In our work, we model the assemblersas short flexible chains that feature these three domains in a consecutivefashion along the chain (Fig. 1A). The repulsive and attractive inter-actions between assemblers and between assembler and template areencoded by simple potentials, as outlined in Fig. 1B.
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Within this minimal model, we can identify several possible states(Fig. 1C). Initially, assemblers are free in solution (state F) from whichthey have the possibility of diffusing toward a template and bindingto form a docked state (state D). Attractive interactions between as-semblers, mediated by their assembly domain, can lead to the associa-tion of assembles in solution in the absence of a template, leading to afreely diffusing assembled aggregate (state FA). Last, the scenario ofinterest is that lateral association of multiple assemblers is mediatedby the binding and accumulation on the template, leading to the desiredfinal product of a docked and assembled state (state DA).
The transition from one state to the next is governed by a complexset of reversible supramolecular reactions; even with two molecularspecies and four possible states, this gives rise to a vast number of kineticpathways from the initially dissolved state to the final architecture. Wecan describe the dynamical transitions between the four states in thisconceptual model as a supramolecular reaction network (Fig. 1C) thatis governed by the following set of differential equations
d½F�dt
¼ �kFFA½F�2 � kFD½F� : xþ kFAF½FA� þ kDF½D� ð1Þ
d½D�dt
¼ kFD : x½F� þ kDAD½DA� � ðkDDA þ kDFÞ½D� ð2Þ
d½FA�dt
¼ kFFA½F�2 � ðkFAF þ kFADA : xÞ½FA� ð3Þ
d½DA�dt
¼ �kDAD½DA� þ kDDA½D� þ kFADA : x½FA� ð4Þ
x ¼ 1� ½D� þ ½DA�½T� ð5Þ
In these equations, [F], [FA], [D], and [DA] represent the numberdensities of assemblers in each of the four states. [T] and x are thenumber density of template positions and the fraction of unoccupiedtemplate positions, respectively. The rate constant of each transitionis denoted as kxy, with x as the initial state and y as the final state, e.g.,kD − DA describes the rate of template-guided lateral association,while kDA − D governs the reverse process of dissociation along thetemplate. For the sake of simplicity, we have chosen to neglect severaltransitions, which are not likely to occur. The cross-transitions FA→Dand DA → F involve the simultaneous detachment or attachment ofmultiple assemblers from the template coupled with simultaneous(de-)aggregation. As this requires many supramolecular reactionsto occur within a short time period, it is not likely to contribute signif-icantly to the process. In addition, the transition DA→ FA is negligiblein most cases as the multivalency of an assembled state gives rise to astrong cooperativity suppression of simultaneous detachment. We de-scribe the underlying assumptions and their validity in more detail inthe Supplementary Materials.
Within this complex reaction network, two main pathways lead thesystem toward the final docked-assembled (DA) state: (Pathway I: F→D→DA)Assemblers first bind the template, after which they laterallyassemble, the pathway of true templated assembly, or (Pathway II: F→FA → DA) by first forming aggregates in solution, after which theaggregate recruits a template with its exposed docking domains. Whilethe first pathway is the route envisioned in designing a templated
Fig. 1. Overviewof our template-assembling systemand kineticmodel. (A) Struc-ture of assembler units and template as proposed in our model and used in the
simulations. Assemblers consist of a part responsible for template binding, thedocking domain (D), an assembly domain (A) capable of providing lateral attractionsbetween assemblers, and a purely repulsive stability domain (S). We use a templateconsisting of 90 beads, T90, and assemblers made from one docking domain, fiveassembly domains, and four stability domains (D1A5S4). (B) Schematic overview ofall pair interactions used in our simulations. Note that the remaining repulsive WCApotentials besides USS are not shown. (C) Suggested assembly states and pathways,including the rate constants defined in the kinetic model. Gray arrows indicate tran-sitions that were ignored in our model.
assembly system, pathway II is often relevant and can lead to aggregatesthat are kinetically trapped and can steer the system to a nonequilibriumand dysfunctional state. It is our aim to understand how the design ofthe system and its interactions can be used to gain pathway selectivityand, ideally, to completely suppress any pathway, such as F → FA →DA, that can lead to kinetic trapping.
As the reaction network sketched above is governed by a plethoraof unknown rate constants, we use a molecular model based onBrownian dynamics simulations to understand how the reactionnetwork responds to changes in the molecular properties of the system.The simulation procedure is described in detail in Materials andMethods. We model the assembler and template as flexible polymersaccording to the Kremer-Grest bead-spring model (23). Our assemblerspecies consists of three types of monomers, designed to match thecorresponding stability (S), assembly (A), and docking (D) domains.The template is modeled as a longer Kremer-Grest polymer consist-ing of 90 template (T) beads. The attractive interactions between theassembly domains, interactions AA, are described by an attractiveLennard-Jones pair potentialUAA, in which the strength of the interac-tions is set by the energy scale EAA. The binding interactions betweenthe docking domains and the template, interaction DT, are modeledwith an inverse Gaussian pair potential UDT, controlled by the energyscale EDT. This choice ensures that each template position can only binda single dockingmonomer such that the stoichiometry of the binding iswell controlled. The stoichiometry of assembler and template bindingsites is a crucial aspect of virtually all templated assembly systems, e.g.,the RNA template encapsulated by the tobaccomosaic virus has a limitednumber of negative charges that can be compensated by positivelycharged capsid proteins, leading to an exact preferred binding stoi-chiometry. While minimal, our model captures this important fea-ture. All other interactions are described by short-ranged repulsiveWeeks-Chandler-Andersen (WCA) potentials. The stability domainsS only have repulsive interactions, both with themselves and all otherdomains, and thus act as steric groups that balance the associativeinteractions in the system.
Pathway selection by balancing supramolecular interactionsOf the two primary pathways from free molecules in solution to aself-assembled nanostructure, pathway I is preferred. This pathmakes full use of the template as a coordinating species of assemblyand, unlike pathway II, is less likely to result in kinetically trappedstates, which can severely delay the assembly process and result instructural polymorphism. A mixture of both pathways would leadto a polymorphous end product. The central question is thus: Howcan we tune the design of the system to steer the assembly processtoward pathway I, F → D → DA?
Our choice for the interaction potentials conveniently allows usto regulate the assembler-assembler and assembler-template withtwo control dials: EAA and EDT, respectively. To elucidate their effectson the kinetic pathways, we perform Brownian dynamics simulationsin which we vary EAA and EDT independently. As there are multiplesupramolecular forces at play, templated assembly systems undergo alarge number of elementary supramolecular reaction steps. As a result,the assembly process is characterized by many transient intermediatestates (24).
To classify these, we assign to each assembler in our system one ofthe four states illustrated in Fig. 1. For selected time points in thesimulation, we determine what fraction f of the assemblers is presentin each of these different states; for details on the categorization, we refer
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to Materials andMethods. The formation of the final DA state emergesin time as a sigmoidal curve in the resulting kinetic diagrams (Fig. 2A).Such a sigmoidal growth of the nanostructures toward their finalarchitecture is typical for templated assembly and observed previouslyin both simulations and experiments (25–27). This is usually interpretedby the existence of three distinct phases. First, there is a slow initiationphase, during which intermediate structures are formed, and which isgoverned by diffusion-limited kinetics. Second, the assembly acceleratesdue to the cooperativity of the process. Third, the system approachesa steady state as the concentrations of free assemblers in solution de-creases. In our results, the initiation coincides with a small number ofassemblers that bind individually onto the template, forming a docked(D) fraction (Fig. 2A). Before the formation of the DA structure, we ob-serve the formation of an aggregate fraction in the FA state, which laterdecreases as the aggregates dock and transition into a DA structure. Inaddition, cooperativity in the template binding, mediated by the lateralinteractions of the assembly domains (A), emerges in our simulationsthrough the formation of binding nuclei that grow in time to encapsu-late the template (see the SupplementaryMaterials). The initial lag timein the sigmoidal kinetic curve could well be related to detachment ki-netics of individual assemblers to the template. The formation of apacked assembled structure on the template requires a stable nucleusof docked assemblers to be formed first. If the detachment rate ofindividually docked assemblers is high, one can expect a long initiallag time before such a nucleus is formed. These initial lag times due
Fig. 2. Monitoring templated self-assembly over time. (A) Typical kinetic dia-gram showing the evolution of the fraction of assemblers, f, in each of the fourstates over time. Simulation performed at EAA = 1.0 kBT and EDT = 17 kBT. (B to D)The general picture of simulations of the assembly process starts with a homo-geneous field of assembler units and a single template (purple beads) (B); aftersome time, there appear free-assembled assembler species in solution, and thereare clumps of assemblers on the template—the changes in template morphologyas a direct result of assembled assemblers is visible (C). At the end of thesimulation, most assemblers will have attached to the template and fully covera highly deformed template (D). All simulations are performed in a box with pe-riodic boundary conditions.
to reversible attachment and detachment kinetics have been previ-ously observed in colloidal self-assembly with weak attractive inter-actions (28).
In our simulations, we increase the binding energy of the dockingdomain to the template, EDT, from 5 to 21 kBT, at a constant assembly-assembly domain attraction strength of EAA = 1.0 kBT. Kinetic diagramsfor four representative values of EDT (others shown in the Supplemen-tary Materials) are shown in Fig. 3 (A to D). As we increase EDT, weobserve an increase in the fractions of assemblers in the docked andDA states. At higher values of EDT, we observe a depletion of thefree-assembled (FA) fraction at later time scales as the aggregates startdocking onto the template. The maximum observed fraction of as-semblers in the FA states remains constant at 0.3, indicating that theformation of freely assembled structures in solution remains un-affected by EDT. This is expected, as this is driven solely by the lateralinteractions set by EAA. At weak template binding strengths, theaggregated template-free state in solution is the stable final product,while stronger interactions with the template lead to template recruit-ment of these aggregates and transform them into DA state throughundesired pathway II, F → FA → DA.
We can quantify the influence of template binding strength on thetemplated assembly process, for all values of EDT, by fitting our kineticdiagrams to our analytical reaction equationmodel.We solve the kineticreaction model using the Runge-Kutta method and fit it to the simula-tion data using a simulated annealing algorithm, as described in moredetail in Materials and Methods. We find that the simplified reactionmodel describes the simulation data with rather high precision (seelines in Fig. 3). Thus, while the reactionmodel in Eq. 1 is approximate,it appears to capture the governing states and pathways. From thequantitative mapping of the model onto the simulation results, we gainaccess to the rate constants that govern the primary supramolecularreaction steps.
We find that the rate constants kFFA (F→ FA) and kFAF (FA→ F)remain within the same order of magnitude (Fig. 4A), confirming ourobservation that the formation of FA structures in solution is un-affected by the assembler-template interactions. As EDT increases from
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5 to 20kBT, the undocking rate kDF (D→ F) decreases bymore than sixorders of magnitude (Fig. 4B). Between EDT 8 and 18 kBT, this rateconstant appears to decrease exponentially, which is in agreementwithArrhenius behavior expected for a thermally activated process. By con-trast, the forward docking rate kFD is virtually constant as a function ofEDT, as the F→D transition is rate-limited by diffusion. Notably, forrate constants kFD, kDF, kDDA, and kDAD, we find high standard de-viations and fluctuating rate constants for EDT between 5 and 10 kBT(Fig. 4, B and C). In this regime, the fractions of assemblers in thedocked and DA states are very low (see Fig. 3) such that the fit givesresults with a very low confidence. As EDT > 10 kBT, both kDDA andkDAD become insensitive to EDT, as expected. Last, we note that therate constant kFADA (FA→DA), describing the transition where largeaggregates in solution bind a templatemolecule in one go, which is thetemplate recruitment of aggregates asmentioned before, shows strongscatter. This is due to the simplification in ourmodel that this reactionstep is controlled by a single rate constant; in reality, this process isdiffusion-limited and thus depends on the size of the aggregates,which will show time evolution and distribution. While for the sakeof simplicity, we chose here not to incorporate this complexity in ourmodel, this could, in principle, be done by introducing a diffusion-limited aggregation kernel into the kinetic rate equations (29).
Upon increasing EDT, the equilibrium of the reversible reactionF→D is shifted toward the right, while that of F→ FAandD→DA islargely unaffected. As a result, the fraction of assemblers that freelydocks onto the template is accelerated, while the number of assem-blers intermediately trapped in the FA state remains constant. Fromthese observations, we can extract a first design rule: Increasing theassembler-template attraction can be used to accelerate the favorableassembly pathway I, with respect to pathway II that occurs throughsolution aggregation. Thus, while this is a promising route towardpathway selectivity, we find that, given the parameter space exploredhere, a complete suppression of pathway II is not possible by tuningthe template binding strength alone.
Because the FA state that we would like to suppress is mediatedby the AA interaction, we turn our attention to pathway selection by
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Fig. 3. Tuning interaction strength steers the assembly kinetics. Kinetic diagrams are shown for a series of simulations with varying assembler-template attractionstrengths (EDT) (A to D) and varying assembly-assembly (lateral associative) attraction (EAA) (E to H). We show the evolution of the fraction of assemblers f in each of thefour assembly states over time t/tB. Black lines represent the optimal solutions found for the kinetic model. Plotted for EDT = 5, 10, 15, and 20 kBT (A to D) and EAA = 0.4,0.8, 1.2, and 1.8 kBT (E to H).
tuningEAA.Weperforma series of simulationswith increasing assembler-assembler interaction strengths from EAA = 0.2 kBT to EAA = 2.4 kBT atstrong assembler-template interactions, EDT = 17 kBT (Fig. 3, E to H); fullkinetic diagram can again be found in the Supplementary Materials. Inaddition, we compare the results from Brownian dynamics simulationswith our kinetic reaction network model to evaluate how pathway se-lectivity can be attained. Between EAA = 0.2 kBT and EAA = 1.0 kBT,which correspond to overall lateral pair interactions strengths of 1 to5 kBT as our model features five monomers in each A domain, oursimulation results are well described by our model. Over the courseof this series, we observe increasing fractions of assemblers in the FAstate, corresponding to the formation of more assembler aggregates.This trend is visible in the rate constants we obtain from our kineticmodel, where the rate constant kFAF, which describes the dissociationof assemblers from the FA state, decreases by more than an order ofmagnitude (Fig. 4E). At these moderate assembly-assembly attractionstrengths, we observe an increase in the amount of assemblers tightlypacked on the template (DA) and a corresponding reduction in theamount loosely packed on the template (D). This trend highlights thenecessity of lateral interactions between assemblers for tight packing ofassemblers on the template and indicates that, while EAA steers theassembly process mainly via pathway II, it also contributes positivelyto the second stage of pathway I, where docked units assemble furtheron the template via lateral attractions.
Stronger assembler-assembler attraction also causes adverse effects,as it promotes the formation of aggregates in the FA state that only slowlydock onto the template, as this process is diffusion-limited and is thusinversely proportional to the aggregation number of the solution clusters(Fig. 3G). The aggregates become more pronounced as we increase theassembly-assembly attraction to EAA values upward of 1.0 kBT. Underthese conditions, our kinetic diagrams show a much slower assemblyprocess. We observe large transient fractions of assemblers in the FA
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state that transition toward the DA state in a stepwise fashion (Fig.3H), where every step corresponds to the docking event of a cluster.The formation of these large aggregates can also be observed in thesimulation itself (movie S1).
At the point where aggregates start to dominate the kinetics, EAA >1.2 kBT, the assembly process is no longer captured by ourmodel, as theassumption that the FA → DA transition is a first-order process thatrelies on the single rate constant kFADAno longer holds:Our simulationsshow a range of aggregate species with varying sizes and hence varyingmobilities.
From these results, we conclude that increasing EAA has a two-faced effect on the assembly process:On the onehand, it leads to adensercapsid around the template, as templated assembly through pathway I ispromoted by strong lateral interactions of docked assemblers, while, onthe other hand, promoting the formation of solution aggregates thatslow down the assembly process and promote the relative occurrenceof undesirable pathway II. This leads us to a second design rule: Whilelateral interactions are essential to create templated assembly and adensecohesive structure, it must be moderated to avoid excessive aggregationin solution that could lead to kinetic trapping and polymorphism.
However, strong interaction energies do not necessarily lead totrapped intermediate states, depending on the interaction being tailored.While strong assembler-assembler interactions lead to aggregation insolution and, consequently, the breakdown of the kinetic model, strongassembler-template interactions, in fact, push the system toward thepreferred pathway (pathway I) without inducing nontemplated aggre-gates. This effect occurs even though detailed balance is broken, as theassembler undocking rate (kDF) is multiple orders of magnitude lowerthan the assembler docking rate (kFD) at high EDT (Fig. 4B).
We note, in passing, that other factors besides these interaction en-ergies might play a role in tuning the pathways of templated assembly.One example is template flexibility. Besides the simulations discussed
Fig. 4. Evolution of rate constants over varying interaction strengths. Rate constants found for the kinetic model are shown over a series of simulations withincreasing assembler-template interaction strength (EDT) (A to D) and increasing assembly-assembly attraction strength (EAA) (E to H). Data points are the average valueof the 10 best fits found by the simulated annealing optimization algorithm. We performed each simulation five times and have plotted results for all replicates as anindicator of the spread between identical simulations. The gray shaded area in (E) to (H) represents the values for EAA ( ≥ 1.2 kBT), where we find substantial aggregatesin bulk and our model starts to loose accuracy.
here, which contain a flexible template, we have also run simulations inwhich the template is entirely rigid, as shown in the SupplementaryMaterials. If template folding is entirely suppressed, this inevitably leadsto nanostructure geometries that are dictated by the template, ratherthan by the assemblers. This implies that tuning template flexibilitybetween the two extremes of entirely flexible and rigid might result in arange of potential structures. Our model system could be readily adapt-ed to include a bond-bending potential tomimic semiflexible polymers,which makes the effect of template flexibility an interesting avenue forfuture study.
Allosteric control of templating pathwaysFrom the results discussed in the previous section, it is clear that, whilesome selectivity in pathways can be obtained by tailoring the balance ofsupramolecular interactions in the system, complete suppression of theundesired kinetic pathway II is not possible. The two-sided nature of theassembler-assembler interaction strength means that the range of EAAenergies that can be usedwhen designing effective templating systems isrestricted. Interaction strengths should promote lateral interactions andthus promote dense packings on the template but should also aim tominimize the aggregation of assemblers in bulk. Nature has developedclever design strategies, e.g., in templated assembly ofmany RNA viralcapsids, to circumvent these limitations (30, 31). One suchmechanismis allostery, where capsid proteins switch conformation as they bind tothe RNA template that activates their lateral interactions (32, 33). Thus,the assembler-assembler interactions only become active upon bindingtheir template. This strategy allows high attraction strengths betweenthe assemblers to ensure a dense and cohesive nanostructure, withoutthe side effect of aggregation in solution. Despite the widespread use ofallostery in natural systems, it remains to be harnessed to gain pathwayselectivity in synthetic templated assembly systems. To explore how allo-steric control could be used to select the kinetic pathway of interest, weintroduce a coarse-grained allosteric mechanism into our simulations.
We model allostery as a binary switch that toggles the attractiveinteractions between the assembly domains from off to on, depending
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onwhether assemblers are docked to the template (Fig. 5A). During thesimulation, we track for each assembler if it is in solution (F and FAstates) or docked to the template (D and DA states). If an assembleris in solution and the allostericmechanism is not activated, the assemblydomains are repulsive bymeans of aWCApotential. Upon docking, theallosteric switch changes the identity of these monomers in theassembler from repulsive to attractive.
To evaluate the effectiveness of allostery as a selection mechanismfor the desired assembly pathway, we compare the formation rate ofthe DA fraction in the presence and absence of allostery at varyingassembler-assembler attraction strengths (Fig. 5B). At weak assembler-assembler attractions (EAA = 1.0 kBT), we observe little difference be-tween simulations with and without allostery. However, at greater EAAof 1.4 and 1.8 kBT, allostery greatly reduces the time required for as-sembly by suppressing the formation of slow and potentially kineticallytrapped solution aggregates. We no longer find the stepwise assemblythat occurs in the absence of allostery, through the template recruitmentof large aggregates, but rather observe a smooth assembly process thatoccurs exclusively through the desired pathway I.
To understand how this works, we visualize the assembly fromsimulation snapshots. In the absence of allostery, we observe manyaggregate docking events for strong assembler-assembler attractions(Fig. 6A). These events characteristically show the initial docking of asingle assembler within the aggregate, followed by a structural reorga-nization that results in docking of the remainder of the assemblers. Ifallostery is introduced, the formation of these aggregates is completelysuppressed. This effect is visible as we compare simulations in the ab-sence and presence of allostery (movies S1 and S2, respectively). Insteadof aggregate docking, we observe many assembler-recruitment events,where assemblers bind to assemblers already docked onto the templateand subsequently reorganize to dock onto the template themselves(Fig. 6B). These recruitment events lead to a transient semi–FA state,as can be observed in the kinetic diagrams of the simulations performedwith allostery (fig. S5). We performed a cluster analysis to differentiatebetween those assemblers in the true FA state and those that are
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Fig. 5. Allosteric switching accelerates templated self-assembly. (A) Mechanism of allostery defined in the simulations. (B) Kinetic diagrams showing the fraction ofassemblers (f) in the docked-assembled (DA) state over time, with allostery both turned on (green) and off (red) for the three values of EAA. Simulations were carried outat EDT = 17 kBT.
indirectly attached to the template (section S4).Using this cluster analysis,we find that, for our simulations with allostery, all assemblers in theFA state are indirectly attached to the template, and no freely diffusingaggregates are present (fig. S8). In contrast, in our simulations withoutallostery, we do identify large fractions of assemblers that are part ofaggregates (fig. S8). Allostery, as used by nature, is thus an exquisitestrategy for pathway selectivity in templated assembly.
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DISCUSSIONIn this work, we presented a minimal simulation model to capture theessential and generic features of templated self-assembly. Being a sim-plified model, we have chosen to work with relatively unspecific poten-tials. As such, one may expect the outcomes to be generic in nature.Although using more specific potentials with different shapes wouldaffect the precise values of the rate constants in ourmodel, the overallpicture remains, allowing us to extract several important and novelinsights. Our results shed new light on the pathways involved in tem-plated assembly and how these pathways can be controlled throughchemical design. We have revealed how tuning the balance betweenthe interactions between the multivalent template and the buildingblocks that it binds can be used to gain some level of control over thepathways in which the system evolves. However, the lateral interac-tions between building blocks, which are a necessity to create a cooper-ative assembly process that leads to a dense and cohesive nanostructure,also invariably lead to the formation of solution aggregates that slowdown the process, introducing kinetically trapped states and structuralpolymorphism.We have shownhownature’s use of allosteric activationis ideally suited to suppress all undesired pathways and lead to excellentpathway selectivity in the assembly process. To date, no synthetic reali-zation for allosteric control of templated assembly exists, and severalbuilding blocks to realize synthetic allostery have been developed(34, 35), which may be used to go toward truly biomimetic templatingstrategies to create nanostructures with high fidelity using our designmodel as a rational basis. Moreover, the design model we have intro-duced here for a flexible one-dimensional template could be readilyadapted to explore how other design factors in templated assembly,
van Galen et al., Sci. Adv. 2019;5 : eaaw3353 11 October 2019
such as the template flexibility and dimensionality, assembler geom-etry, stoichiometry, and overall reactant concentrations, could steerthe kinetic pathways and resulting end products. This would makefor a rational design basis to create synthetic systems in whichdesired nanostructures can be formed through templated assemblywith high fidelity.
MATERIALS AND METHODSSimulationsWe performed Brownian dynamics simulations using the HOOMD-blue v2.2.0 package (36, 37). Quantities are expressed in reduced units,in terms of the monomer diameter s, the monomer self-diffusion timetB, and kBT as the characteristic energy scale. All simulations containtwo molecular species: The first is a single template consisting of 90template domains (T90), and the second species are 96 assemblerscontaining solubility domains (S), assembly domains (A), and a dockingdomain (D) in conformation S4A5D1. All connecting bonds weredefined according to the Kremer-Grest bead-spring model, in whichneighboring particles are connected by a FENE spring, defined as (23)
UFENEðrÞ ¼ � 12kr20 ln 1� r
r0
� �2 !
þ VWCAðrÞ ð6Þ
where k= 30 kBT/s2 is the attractive force strength, r denotes the center-
to-center distance between two bonded domains, r0 = 1.5s is the bondsize parameter, and VWCA(r) is a repulsive WCA potential
UWCAðrÞ ¼4e
sr
� �12� s
r
� �6� �þ e r < 2
16s
0 r ≥ 2
16s
8>>><>>>:
ð7Þ
Here, e = 1 kBT is the characteristic energy and r = 21/6 s is the inter-particle distance where the potential is zero.
Fig. 6. Two observed assembly mechanisms. (A) Simulation snapshot of a recruitment event of an individual assembler by one already docked on to the template.(B) Snapshot of a collective docking event of an aggregate of assemblers. These simulations have been performed on a static and stretched template as an aid to visualclarity. All other data in this paper use a fully flexible template as described in Materials and Methods.
For the assembled-assembler attraction, we used the Lennard-Jonespair potential
UAA ¼ 4EAAsrAA
� �12
� srAA
� �6" #
ð8Þ
with −EAA as the attraction strength of the Lennard-Jones potential andrAA as the center-to-center distance between the assembly domains.
For the interaction between the docking domain and the template,we used an inverted Gaussian pair potential described by
UDT ¼ �EDT : SðrÞ : exp � 12
rDT0:25s
� �2� �ð9Þ
Here, rDT is the docking template domain interparticle distance and−EDT is the minimum of the potential at r = 0s. S(r) is a smoothingfunction that ensures that the pair potential transitions smoothly to 0at large values of r. S(r) operates from ron = 0.4s to rcut = 0.5s as follows
The interactions between all remaining particle pairs are describedby the repulsive WCA potential (Eq. 7).
Simulationswere performed in the canonical ensemble by integrat-ing the overdamped Langevin equation using time steps of 10−4 tB(38). In our simulations with a flexible template, assemblers were initial-ly positioned on a cubic lattice within the simulation box of dimensions25 × 25 × 25 s, with the template localized in a spiral conformation inthe center. Before the measurement, the initial conformations wereequilibrated for a period of 103 tB, where both the A-A Lennard-Jonesand the D-T inverse Gaussian potentials were replaced by a WCApotential, to suppress attractive interactions and allow the system toreach a randomized configuration. Consecutively, simulations werecarried out for a period of 5 × 104 tB. Simulation snapshots were storedevery 0.05 tB for the initial 10 tB of simulation time and 0.5 tB for theremainder to capture processes that occur on both short and long timescales with a sufficiently high time resolution.
We incorporated allostery in our simulations with the use of acallback function. Every 0.1tB, the positions of the docking domainsand template positions were compared. If the docking domain of anassembler is not within a distance of 0.5s from a template position,its assembly domains are turned inactive, replacing the attractiveLennard-Jones pair potential governed by EAA by a purely repulsive,hard sphere–like, WCA pair potential. If the docking domain of anassembler is within this target distance of a template position, this pro-cess is reversed. Simulations with allostery were performed on a flex-ible template, with EDT = 17 kBT (Fig. 6B).
AnalysisWedetermined the fraction of assemblers in each of the four states (free,free assembled, docked, and docked assembled) over time by assessingfor each snapshot whether each assembler was docked to the template,attached to other assemblers, or both. Assemblers were counted as
van Galen et al., Sci. Adv. 2019;5 : eaaw3353 11 October 2019
docked when they were within 0.5s distance from a template domain.Similarly, they were counted as assembled when at least four pairs of Adomains with a neighboring assembler were within a distance of 1.3s.The criterion of four domainswas chosen to distinguish between assem-blers that were firmly assembled and those that were coincidentally ineach other’s proximity but not firmly bonded. The datawere temporallyaveraged in 2000 logarithmic spaced bins along the entire length of thesimulation.
Solving the kinetic model and fit algorithmWe solved the kinetic model numerically using the Runge-Kuttamethod, with step size dt= 0.01 tB for the initial 10 tB of the simulationand dt = 0.25 tB for the remainder. Parameters for the kinetic modelwere found by minimizing the sum of squared residuals (R) betweenthe kinetic diagrams and the kinetic model for each binned data point
where n = 2000 equals the number of logarithmic spaced binned datapoints, [FA]i is the number density of assemblers in the FA state atdata point i, and fFA, i is the concentration as obtained from the ki-netic model at this data point. R was minimized using a simulatedannealing fit algorithm with an exponential multiplicative coolingschedule as described by Kirkpatrick and coworkers (39). For everyrun of the algorithm, we performed 500 cooling steps, starting froman initial effective temperature T0 = 10, with a cooling factor of a =0.97. At every cooling step, the seven fit parameters (kFFA, kFAF, kFD,kDF, kDDA, kDAD, and kFADA) were changed randomly using theNumPyrandom number generator. As the simulated annealing algorithm pro-ceeds to lower temperatures, the tolerance for unfavorable steps de-creases, lowering the acceptance rate. To ensure that sufficientmeaningful parameter adjustments were made, we continuouslychanged the parameter adjustment step size to keep the acceptance rateat 0.5. Our use of seven fit parameters resulted in a complex goodness-of-fit landscape with many possible local minima. To find the globalminimum in this landscape, we repeated the simulated annealingalgorithm 100 times starting from randomized initial parametervalues. Average parameter values and standard deviations were com-puted for the best 10 fits obtained in this way. Corresponding residuals arepresented in section S5.
SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/10/eaaw3353/DC1Section S1. Assumptions made in the kinetic modelSection S2. Cooperativity without allosterySection S3. Overview of all kinetic diagramsSection S4. Differentiating between aggregates and recruited assemblersSection S5. Residuals obtained for the simulated annealing fit algorithmFig. S1. Determining the reaction order of free assembly.Fig. S2. Template occupancy profiles for simulations with a rigid template.Fig. S3. Overview of kinetic diagrams at varying assembler-template interaction strengthswithout allostery.Fig. S4. Overview of kinetic diagrams at varying assembler-assembler interaction strengthswithout allostery.Fig. S5. Overview of kinetic diagrams at varying assembler-assembler interaction strengthsincluding allostery.
Fig. S6. Schematic representation of two free-assembled (FA) states.Fig. S7. Overview of kinetic diagrams at varying assembler-assembler interaction strengthswithout allostery, differentiating between two FA states.Fig. S8. Overview of kinetic diagrams at varying assembler-assembler interaction strengthsincluding allostery, differentiating between two FA states.Fig. S9. Histograms of fit residuals at varying assembler-template interaction strength.Fig. S10. Histograms of fit residuals at varying assembler-assembler interaction strength.Movie S1. Time lapse of a simulation in the absence of allostery.Movie S2. Time lapse of a simulation including allostery.References (40–46)
on Decem
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AcknowledgmentsFunding: This work is part of the VIDI research program with project number 723.016.001,which was financed by the Netherlands Organization for Scientific Research (NWO). Authorcontributions: J.S. and R.H. conceived the project. M.v.G. performed the simulations. M.v.G.and R.H. performed the data analysis. All authors discussed the data and their interpretationand co-wrote the manuscript. Competing interests: The authors declare that they have nocompeting interests. Data and materials availability: All data needed to evaluate theconclusions in the paper are present in the paper and/or the Supplementary Materials.Additional data related to this paper may be requested from the authors.
Submitted 30 January 2019Accepted 14 September 2019Published 11 October 201910.1126/sciadv.aaw3353
Citation: M. van Galen, R. Higler, J. Sprakel, Allosteric pathway selection in templatedassembly. Sci. Adv. 5, eaaw3353 (2019).
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