estimating the efficiency of self-assemblies

Estimating the Efficiency of Self-Assemblies

Corinne L.D. Gibb and Bruce C. Gibb*

Department of Chemistry, University of New Orleans, New Orleans, LA 70148, USA

Received 3 October 2000; accepted 3 October 2000

Abstract—An approach to the semi-quantification of self-assembly processes that lead to descrete molecular and supramolecularentities is presented. A number of reversible and irreversible assemblies from the literature illustrate how this probabilistic approachcan be used to: 1) Compare disparate assembly systems; 2) define what processes can and cannot be described as self-assembling.# 2001 Elsevier Science Ltd. All rights reserved.

Self-Assembly: A Recap

What is meant by the term ‘self-assembly’? There havebeen many attempts in the chemical literature to definethis term. For example, in 1990,1 Hamilton described aself-assembly process as, ‘‘the non-covalent interactionof two or more molecular subunits to form an aggregatewhose novel structure and properties are determined bythe nature and positioning of the components.’’ A shorttime later, Whitesides elaborated a similar description;‘‘the spontaneous assembly of molecules into struc-tured, stable, non-covalently joined aggregates.’’2 Morerecently, Lehn has proposed a more general definitionwhereby self-assembly is ‘‘the evolution towards spatialconfinement through spontaneous connection of a few/many components, resulting in the formation of dis-crete/extended entities at either the molecular, covalentor the supramolecular, non-covalent level.’’3 Distillingand combining these and other definitions leads to agenerally accepted, although rather vague definition,that self-assembly is the spontaneous assembly of sets ofcomparatively simple subunits (molecular or otherwise)into highly complex supramolecular or molecular spe-cies of defined structure.4 Being so broad, this termencompasses a myriad of chemical processes includingthe artificial systems addressed here, crystal growth (forcrystal engineering, see refs 5–8), and an innumerablenumber of biological process including biomineraliza-tion,9 DNA duplex formation,10 and protein quaternarystructure development (for viral capsid synthesis seeref. 11). So ubiquitous in nature, it is not surprising that

a number of approaches to sub-divide ‘‘self-assembly’’have been sought in an effort to more precisely definethe term. For example, one powerful demarcation is tonote that the whole paradigm of self-assembly can bequalified into two types; those that lead to the forma-tion of essentially infinite lattices, and those that lead tothe formation of discrete species. That said and done,defining self-assembly in this qualitative manner stillleads to confusion, and all to often the words ‘self-assembly’ has been used to describe processes that donot fit the aforementioned definition(s). In this paper,we introduce a (semi) quantitative description of self-assemblies which form discrete molecular or supramo-lecular products. The advantages of seeking the quanti-fication of self-assembly processes are manifold. In thefirst instance, quantifying assembly processes naturallyreinforces the qualification of ‘self-assembly’ by pre-cisely defining the boundary between processes that canbe described as involving self-assembling and those thatcannot. Equally as important, a standard method ofquantification also engenders a common frame of refer-ence to compare different assembly systems; and as willbe detailed below, our regioselectivitity based approachallows a truer comparison of efficiency than product yieldswhich offer little in the way of valuable information.Finally, quantifying assemblies from a regioselectiveperspective also provides insight into the intermediatesand pathways that may be present. Thus, useful infor-mation concerning the importance of the symmetry ofmolecular subunits or intermediates, and an appreci-ation of how assembly information is stored within asubunit, are also garnered by the approach.

As just alluded to, our quantification of self-assemblydescribed below is based on regiochemistry. Consequently,

1472-7862/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.PI I : S1472-7862(01 )00003-X

Journal of Supramolecular Chemistry 1 (2001) 39–52

*Corresponding author. Tel.: +1-504-280-6311; fax: +1-504-280-6860; e-mail: [email protected]

it requires an appreciation of how information is storedwithin a molecular subunit. We will therefore reviewthis topic in the following section, before going on todescribe a method of categorizing assemblies systemsand our general approach to quantification.

What Makes Up A Molecular Subunit?

A pictorial representation of the aforementioned defini-tion of self-assembly is shown in Scheme 1. In this car-toon, because the ends of each ‘molecular’ subunit are‘sticky,’ and because these pairs of sticky ends are at anangle of 72� with respect to each other, five such sub-units can come together to make the resulting torus-likeproduct. What is the structural information within thesesubunits that leads to the assembly? It must first berealized that not all parts of this ‘molecule’ contribute inthe same way to the assembly process. Some part of themolecule (the gray spheres in the case shown in Scheme1) may have little or nothing to do with the assemblyprocess, but perhaps are required for the product tofunction properly. Of the parts of a molecule which arenecessary for successful assembly we can distinguish twotypes: those parts of the molecule that drive the assem-bly, and those parts of a molecule that direct theassembly. Remaining for the moment with the cartoonin Scheme 1, the parts of the molecule that drive thisparticular assembly are the sticky ends. In a chemicalsense these are the moieties that contribute to the (bulkof the) free energy change associated with assembly.They are for example the hydrogen bond donor andacceptor groups of the self-complementary AADD (A,D=hydrogen bond acceptor and donor) motif 1

(Scheme 2) reported by Zimmerman.12 The moietiesthat drive assembly may be identical groups, they maybe closely related groups such as hydrogen bond donorand acceptors, or in more complex situations they maybe completely disparate in nature. In the latter categorywe can for example mention protein quaternary struc-ture which is usually driven by the whole gamut of non-covalent forces available to Nature. We have defined13

the term (and collective noun) ‘supramolecular motif’ todescribe those parts of a molecule that drive a self-assembly.14

Also important in a molecular subunit are the moietiesthat direct the assembly process. This is often (thoughcertainly by no means always) the bulk of the structure,whose features position and direct the components ofthe supramolecular motif in three-dimensional space. Inthe case of Scheme 1, the structural element in questionis simply the material that connects the two sticky ends.In the case of 1, the structural elements are the twofused six-membered rings that preorganize the AA partof the supramolecular motif, and the amide group andimide carbonyl that form an intramolecular hydrogenbond to organize the DD part of the motif. Theremainder of 1, i.e., the n-butyl group presumably serveslittle purpose in the assembly, but undoubtedly aids inthe solvation of the subunit. In summary, to a firstapproximation the information stored within a molecularsubunit can be broken down into the supramolecularmotif which drives the assembly, other atoms whichgeometrically constrain the motif into a specific 3-Dtopology, and an ancillary set of atoms or groups whichare not (at least primarily) involved with the self-assembly.

Scheme 1. A cartoon of a self-assembly. Each subunit has two sticky ends which are at an angle of 72� with respect to each other.

Scheme 2. The self-assembly of 1 to form a heterodimer.12

40 C. L. D. Gibb, B. C. Gibb / Journal of Supramolecular Chemistry 1 (2001) 39–52

Now What Will The Subunit Do?

The two aforementioned examples of self-assembly wereboth assumed to be perfect in the sense that only oneproduct was formed. However, in reality this is theexception rather than the rule. As a case in point, inactuality the assembly of the AADD motif 1 results inseven species observed by NMR.12 In general, then, amolecular subunit can potentially undergo a range ofreaction options depending on the precision of theassembly information contained within it (Fig. 1). Theworst case scenario arises when the molecular subunitcontains no assembly information, and as a result anessentially infinite number of products or random poly-mer is produced. At the opposite end of the spectrumwe have, in the words of Sanders,15 a molecule that hasa ‘‘predisposition’’ for a particular assembly. In otherwords, the subunit possesses very precise assemblyinformation encoded within its structure with the resultthat only one product is formed in essentially quantita-tive yield. The situation is therefore clear. A range ofreactions from the unfocused to the highly focused ispossible, with the degree of focus being dependant onthe geometrical information within the molecular sub-unit that organizes the supramolecular motif, theassembly environment (temperature solvent, etc.) andpossibly additional additives that increase the focus ofassembly (templates16�19). What is less clear is in whichsituations we can apply the term ‘self-assembly’ and inwhich it would be unwise to do so. Quite obviously theformation of, for example, a discrete decameric speciesin quantitative yield would be a wonderful example ofself-assembly, while the formation of random polymercould not readily be described in these terms. But wheredoes self-assembly start? If instead of a decamer weformed a tetramer and a hexamer is this still an assem-bly process? What about the situation where five dis-crete species are formed but each one in only 10% yield?To solve this dilemma what is required is a means toquantify whether the formation of a particular productis better or not than expected from purely probabilisticconsiderations. A means of doing so would not onlyaddress these questions, but also address questionsconcerning comparisons of different assemblies; if twoquite different assemblies both give a 70% yield of theirrespective products are they of equal efficiency? In thefollowing sections, we describe one common frame ofreference for analyzing assembly systems. We will firstdetail the ground work and the rules of the process,

before working through some previously documentedassemblies to illustrate the method at work.

The Ground Work

Entropic considerations indicate that the building of aproduct via self-assembly occurs primarily in a bi(su-pra)molecular fashion. Thus, in the first step of anassembly process two subunits can be envisioned tocome together to form an intermediate dimer. In thisand in all subsequent steps the information containedwithin the molecular subunits dictates how, out of themany choices available, the two species are to associatewith each other. In other words as each subunit is addedto the forming supramolecule a regioselective reaction isoccurring. As a straightforward example, Scheme 3shows the first step towards a theoretical tetramerizationwhere the spatial orientation of the amide groups thatcomprise the supramolecular motif induce a regioselec-tive association in this (and the following) step. Indeveloping a common frame of reference for molecularassemblies it is therefore necessary to take into accounthow many regioselective options are available to thesubunit as it becomes part of the growing supramole-cule. Consequently it is important to classify assemblysubunits, i.e., determine how many different type ofmolecular subunit are present in the assembly, and howmany (significant) interactions (covalent or non-covalent) each can make with the growing supramole-cule. We define the subunit with the highest number ofmoieties capable of forming interactions as the primarysubunit; the next most complex as the secondary sub-unit and so on all the way down to templates; speciesthat appear in the final product but do not form anyspecific bonding patterns with the bulk of the assembly.Once we have determined the number of subunits andthe number of ways they can interact, it is possible toclassify the particular type of assembly. For example, anassembly of 50 components to form a dodecahedron,20

in which the 20 primary subunits have three points ofinteraction, and the 30 secondary subunits have twopoints of interaction, can be classified as a [203+302]process. Table 1 summarizes the type of systems that wewill be analyzing to demonstrate our approach toquantifying assemblies.

In a normal reaction, say the formation of a carboncarbon bond, the efficiency of the process can easily be

Figure 1. The possible extremes of the spectrum of self-assembly.

C. L. D. Gibb, B. C. Gibb / Journal of Supramolecular Chemistry 1 (2001) 39–52 41

determined by measuring the yield of the reaction. Inthis way, the reaction of choice can be compared toother methods for carbon�carbon bond formationbecause all of these types of reactions involve the for-mation of the same number (1) and type (C�C) ofbonds. Unfortunately, as Table 1 demonstrates, thesame cannot be said with self-assembly processes; boththe number of moieties that comprise the supramole-cular motif, and the number of molecular subunits tak-ing part in the assembly are not the same for eachassembly. Therefore, quoting a yield for an assemblysystem tell us nothing about how discerning each regio-selective step in the processes is. In other words, yieldstell us what went right during the assembly, but not whatpotentially could have gone wrong with the process, and itis the difference between these that dictates the efficiencyof an assembly. As will be shown below, by determiningthe quotient of the measured yield of an assembly processand a theoretical yield that takes into account the regio-selective options available at each step, we can estimatethe efficiency of an assembly process. In addition, it alsobecomes possible to quantitatively define whether thesystem in question is a self-assembling one or not.

We have already discussed the possibility of bifurcatingthe paradigm of ‘self-assembly’ into those that lead todiscrete species and those that form infinite lattices.However, before we begin to explain our approach forquantifying self-assemblies we should remind the readerthat another option is to divide the field of into pro-cesses that use either reversible or irreversible mechan-isms. The general definition of self-assembly given in theintroduction refers to reversible systems which areunder thermodynamic control. Such systems are said tobe self-correcting because errors (assemblies that do notrepresent the global minimum) occurring during assem-bly are removed as the true global minimum of theassembly is established. However, once an assembly has

equilibrated it is often possible to covalently connect thesubunits in an irreversible, kinetically controlled man-ner. These systems have become known as non-correct-ing, or kinetic-based, self-assemblies.21 In general,because of their highly efficient nature, reversible self-assemblies have figured more prominently in the litera-ture. Ironically however, the high kinetic stability ofproducts arising from irreversible systems often resultsin these type of systems being quantified more accu-rately. For the inexperienced reader, therefore, it shouldbe pointed out that the roughly 50:50 split in the exam-ples given below is biased (by this ease of quantification)towards the less well studied irreversible systems.

The Approximations Of The Method

As described above, two pieces of data are required toevaluate the efficiency of an assembly: the actual yieldobtained for the reaction; and a theoretical yield thattakes into account the possible regioselective optionsavailable to the molecular subunits. It is of course thelatter of these two numbers that mainly concerns ushere. In the determination of a theoretical yield twoessential approximations must be made. In the firstinstance, the examination of possible assembly routes isconstrained to the possibilities that can arise between thenumber of subunits that appears in the product. Putanother way, to prevent an analysis into infinity thepossibility that molecules or random polymer largerthan the product are formed, must be assumed to bezero. This limits the analyses to a practical level. Theother approximation also restrain the analyses, andperhaps more importantly, levels the playing field for allanalyses. Thus we assume that assembly occurs betweensubunits which act as rigid-rotor models, and that bar-ring impossible molecular motions (such as moleculespassing through molecules) all intermolecular andintramolecular processes occur on a flat potential energysurface.22 Thus, no biasing because of the conforma-tional sampling of the subunit or supramolecular array,nor biasing because of activation energies arising fromthe building processes, are taken into account. Thesenatural biases which in actuality are what makes anassembly successful, are assumed to be taken intoaccount in the actual yield of the product. The theoreticalyield is therefore representative of a worst case scenariowhere the subunit is devoid of assembly information.

The Rules

The basic approach to the semi-quantitative analysis ofself-assemblies centers around the construction of a

Scheme 3. The first step towards the assembly of a tetrameric species.

Table 1. Types of assembly used to describe the quantification of

assemblies

Assembly Type

[2]Catanane formation30 [12+12+10]a

Bis-[2]catanane fromation33 [14+22+20]a

Carceplex formation34,35,38 [24+42 +10]a

Hemi-carcerand formation13,37 [24+42]Formation of a truncated tetrahedron39 [43+62]Hydrogen bond driven formation of aquasi-spherical host40

[44+10]a,b

Cubeoctahedra formation42 [83+122]

an0 denotes n template(s) present in the product which do not form anyspecific bonding pattern with the remainder of the (supra)molecule.bThis particular assembly could be classified as a [48] type system (seetext and ref 41).


standard probability tree which details both the suc-cessful and the non-successful pathways of assembly.This (assembly) tree leads to the calculation of a prob-ability (r) for forming the product and hence a theore-tical yield. Specifically the rules for determining thetheoretical yield and evaluating the assembly are:

1. Examine the product of interest and determine thecomponents that make up the product. In otherwords, characterize the assembly. It is essential torecognize the different types of subunit, how manyprimary/secondary (etc.) subunits there are, andhow many interactions they can each make.

2. With the assembly characterized (e.g., [24+42]) wenow introduce our approximation to constrain theanalysis to reasonable levels. The components areplaced inside a hypothetical ‘reaction box.’ We areto consider all the reaction possibilities that canoccur between these components but will assumethat no reaction can occur between the compo-nents in the box and species out with the artificialconfinement. This approximation is necessary for thetask at hand, but is actually becoming less significantas the increasing efficiency of assemblies increasinglyprecludes the formation of side products.The reactions between the isolated assembly

components are then considered. Following gen-eral entropic considerations the product is to bebuilt up in a step-wise manner adding one subunitat a time. Irrespective of whether a covalent ornon-covalent bond is made, each bond is assumedto be formed with an efficiency of 100%. Further-more, each bond formed is assumed to be irre-versible. Irrespective of the mechanism of theassembly process, there is no going back. Eitherjoining together molecular subunits is workingtowards the formation of product, or it is workingtowards the formation of non-product. With eachbond formed the possible reaction options of theresulting product are then considered. Eitheranother molecular subunit (from the reaction box)is added to the fray, or an internal coupling reac-tion is to occur. All of these intra- and inter-molecular reaction options are given an equallikelihood of occurring. For those reactions whichlead to a new intermediate which can theoreticallystill go on to form product, this process is repeateduntil the product is formed. When a reaction leadsto a species that does not have the correct struc-ture that particular line of assembly is terminated.The result is the formation of an assembly (prob-ability) tree which details the reaction optionsavailable to the intermediates as the product is builtin a step-wise manner. Subsequently the probabilityof the overall assembly is calculated and a theoreticalyield for the assembly product determined.We should perhaps mention here that secondary

subunits with two (identical) points of interaction([ax b2] type processes where b>2) do not add tothe regioselective options defined by the primarysubunit (although they do however contribute tothe overall shape of the product). In these casestherefore the secondary subunit need only be

considered as a linker or ‘glue’ that joins theprimary subunits together, and need not be speci-fically considered in the assembly tree.

3. The theoretical yield is then compared to theactual yield of the ‘isolated’ assembly product, andthe quotient of these two numbers (actual yield/theoretical yield) calculated. We define this num-ber as the assembly number (AN). With assemblyinformation contained within the molecular sub-units the actual yield will be somewhat larger thanthat determined by assuming all assembly reac-tions options are equally likely. Thus, the assemblynumber (AN) will be greater than one and theprocess can be (quantifiably) defined as a self-assembling one. The higher the efficiency of theassembly (the greater the actual yield and/or thelower the theoretical yield) the higher the assemblynumber. Thus process with large numbers of sub-units each capable of forming many interactions,that occur in high actual yields, will be efficientassemblies with relatively high ANs. For non-self-assembly processes the actual yield will be lessthan the theoretical yield and the AN will be lessthan one. The border between assembly and non-assembly processes can therefore be quantified asAN=1; the maximal value obtained when a pro-cess such as a simple one bond forming reaction(no regioselectivity, therefore probability r=1,theoretical yield=100%) occurs in 100% yield. Inthe following section we will illustrate the applica-tion of this approach to a number of examples ofself-assembly.

Some Examples From The Literature

For general reviews on self-assembly, see refs 16 and23–29.

[2]Catenane formation: a [12+12 +10] type assembly

We will start of with a relatively straightforward yetelegant example of kinetic controlled self-assembly; theformation of [2]catenane 2 (Scheme 4).30 This is anexample of a hetero self-assembly with the convergenceof two dissimilar molecular subunits, a bis-paraquatmoiety and a 1,4-bis(bromomethyl)benzene. A thirdcomponent, bis-paraphenylene-34-crown 10, acts as atemplate to focus the assembly, and in the process ofdoing so becomes mechanically trapped. With only twoidentical bonds to be formed between two identicalpyridyl nitrogen moieties and two Ar–CH2Br groupsthere is no question of regioselectivity during bondformation (beyond extended polymerization which isignored in this treatment). Thus, the assembly becomesa question of whether or not the crown ether is caughtby the ring formation or not. An elaborated probabilitytree is shown in Scheme 5. With each outcome givenequal probability of occuring (1/2) the probability offorming 2 is readily calculated as r=0.5, which corre-sponds to a theoretical yield of 50%. The actual yieldfor this reaction is 70%. Thus, an assembly number(ANirr irr=irreversible) of 70/50 or 1.4 can be calculated.What does AN[12+12+10]irr=1.4 mean? Put simply,


given the choice of associating with solvent or with theparaquat molecule, the crown ether ring preferentiallyassociates with the latter. In more detail, the supramo-lecular motifs in each subunit (catenanes and rotaxanesof these type are distinctive in so much that nearly all oftheir molecular structure contains assembly informa-tion) result in the formation of the favorable ‘acidic’’ C–H� � �O forces and p–p stacking interactions31,32 and asuccessful assembly process. If these forces were not inoperation then an AN of less than one would result andthe process would not be characterized as a self-assembly.

Bis[2]catenane formation: a [14+22+20] type assembly

Remaining with the topic of catenanes, our secondexample demonstrates how a more accurate descriptionof assembly can be garnered when assembly numbersare calculated. Consider the formation of bis-[2]cate-nane 3 shown in Scheme 6.33 The relevant assembly tree

is shown in Scheme 7. The formation of the first N�Cbond occurs in 100% (r=1) while in the second step thesecond nitrogen can be alkylated by three carbons andin all cases either the crown ether ring can be trappedwithin the forming ring or it is not. The probability istherefore r=1/6. With no regioselectivity to be con-cerned with when forming the third bond, again aprobability r=1 is observed. Finally, formation of thefourth bond can produce either the product or thecompound with one crown ether ring missing (r=1/2).Therefore the probability of forming 3 is 1�1/6�1�1/2=1/12. Alternatively we can quote a theoretical yield of8.33%. With an isolated yield of 31%, this correspondsto AN[14+22+20]irr=3.72. In other words although theformation of [2]catenane 2 is higher yielding than theformation of bis-[2]catenane 3, when we take intoaccount all the assembly options involved with the for-mation of 3, it is apparent that it is the latter assemblythat is more efficient.

Scheme 5. The assembly tree for the formation of [2] catenane 2.30 After formation of the first covalent bond (100%), either the system will lead tosuccesful assembly (S) or it will fail (F). Both possible assembly outcomes of the second reaction (denoted with a �) are given equal probability (1/2)of occurring.

Scheme 6. The self-assembly of five components to form bis-[2]catenane 3.33

Scheme 4. Formation of [2]catenane 2 by the assembly of a bis-paraquat moiety and bis-paraphenylene-34-crown 10.30


While comparing the assembly reactions that form 2and 3, we can also see that the later section of theassembly tree for the latter (Scheme 7) is identical to theassembly tree for catenane 2 (Scheme 5). This is ofcourse not surprising since essentially the same reactionsare occurring when 2 is formed or when the monocrownether intermediate reacts to form 3. However, the abilityto transplant relatively simple assembly trees into thoseof related, but more complex systems is a useful tool ifcare is taken that the transplant is indeed viable.

Carceplex formation: a [24+42+10] type assembly

Another prominent example of kinetic based self-assembly can be found with a family of compoundscalled cavitands.34 Given the correct functionality, thesebowl shaped compounds, e.g., 4, can assemble into car-ceplexes such as 5; closed shell molecules which perma-nently entrap guest molecules within their interiors(Scheme 8). The precise topology of the guest dictates

the focus of the self-assembly. For example, with highlycomplementary guest and cavity topology (guest equalspyrazine) the corresponding carceplex 5 is isolated in87% yield.35 How efficicient is this high yielding assem-bly? Construction of the relevant assembly tree is shownin Scheme 9. This is a good example for differentiatingbetween primary and secondary subunits. The second-ary subunits, the BrClCH2 bis-electrophiles, do notincrease the regioselective options at any particular step.As a result, we do not need to construct a more complexassembly tree that considers all intermediates possessingcavitand–OCH2Br moieties. We simply envision thesecondary subunits as acting as glue to stick togetherthe primary (cavitand) subunits. Returning to Scheme 9,there is no regioselectivity in the first linking of the twobowls (r=1). However, from the corresponding inter-mediate (shown), there are a total of nine possiblereactions, six of which go on to form non-product. Theremaining three reactions form intermediates whichbecause of geometrical constraints are, over the remaining

Scheme 7. Assembly tree for the self-assembly of bis-[2]catenane 3. Four covalent bonds are formed by four reactions [represented by a filled circle(�)] with probabilities r=1, 1/6, 1 and 1/2.

Scheme 8. Self-Assembly of cavitand 4 to form carceplex 5.34,35

Scheme 9. Assembly tree for the formation of cavitand 4. The two cavitands initially are connected by reaction with one BrClCH2. Reactionsbetween the three remaining hydroxy groups (labelled A/B/C and A0/B0/C0) are then considered.


two steps, guaranteed to go on to product (r=1). Thus,the probability of forming the cavitand is r=[1/1�1/9�1/1�1/1]�3=1/3, which corresponds to a theoreticalyield of 33%. Thus, an AN[24+42+10]irr=2.64 can becalculated for this process.36 In terms of efficiency thecarceplex reaction is somewhat mid-way between thetwo aformentioned catenane reactions.

Hemi-carcerand formation: [24+42] type assemblies

Having commented on the possibility to transplantparts of an assembly tree to another, we will finish offour discussion of kinetic based systems with a warningthat structurally related molecules can have disparateassembly trees. Consider for example the ‘dimerization’of 6 to form the corresponding hemi-carcerand.37 Thisassembly is similar to the previous type of assembly butno template is found in the product. Presumably,instead of being templated, the forming cavity is simplysolvated. In this situation, there is considerably moreflexibility in the assembly subunit and the more complexassembly tree shown in Scheme 10 must be applied.The resulting probability of product formation is r=[1/1�1/9�1/4�1/1]�6=1/6. Thus, the theoretical yieldis half that of the assembly of 4 (16.67%). In this case,the much lower yield of 32% (for the case where thelinker between the subunits is -CH2CH2-) correspondsto an AN[24+42]irr=1.92. A noticable trend in thesetypes of reactions38 is the decrease in efficiency of theassembly as increasingly larger linker groups are ‘squeezed’between the two hemi-spheres. Thus when the linking

reagent is a,a0-dibromo-o-xylene the yield decreases to17% and AN[24+42]irr=1.37 The actual yield is essen-tially the same as the worst case scenario theoreticalyield and the reaction cannot quantifiably be describedas a self-assembly. Finally, we note that the assemblytree in Scheme 10 can also be applied to the assembly ofrelated deep-cavity cavitands such as 7 recently synthe-sized in our group.13 With an isolated yield of 80% forthe corresponding ‘dimer’ (linker=-CH2-) a highly effi-cient AN[24+42]irr=4.80 can be calculated.

So far, we have limited the examples of assembly toirreversible systems. However, in terms of sheer num-bers, those assemblies that occur via a reversiblemechanism are much more prevalent. The greaterefficiency of these self-correcting processes means thatthe complexity of some of the systems that have beeninvestigated is relatively high. Correspondingly, theattendant assembly trees are often complex, and theresulting AN numbers (ANrev) are usually much higherthan those recorded for kinetic based assemblies.

The formation of a truncated tetrahedron: a [43+62]type assembly

We start with a tetramerization type process driven bythe formation of coordination bonds between pyridylnitrogens and palladium atoms (Scheme 11).39 Actually,in this case a total of 10 molecules are brought together[six dinitro(ethylenediamine)palladium(II) complexesand four 2,4,6-tri(4-pyridyl)-1,3,5-triazine ligands], but

Scheme 10. Assembly tree for the assembly of deep-cavity cavitand (DCCs) 6. The two DCCs initially are connected by reaction with one BrClCH2.Reactions between the three remaining hydroxy groups (labelled A/B/C and A0/B0/C0) are then considered.


the nature of the secondary subunits (two identicalmeans of interaction) allows us to consider them aspalladium ‘glue’ which holds the primary subunitstogether. An abbreviation of the resulting assembly treeis depicted in Scheme 12 (the full tree can be found inthe Appendix). From this tree, a theoretical yield of13.01% can be determined, which when combined with thequantative yield of the complex, corresponds to anAN[43+62]rev=7.68. Thus, the reversible nature of thisprocess leads to an assembly which is more efficient thanthe aforementioned kinetic based systems.

The hydrogen bond driven formation of a quasi-sphericalhost: a [44+10]rev type assembly

We now examine the tetramerization of 8 (Fig. 2)reported by the Rebek group.40 The supramolecularmotif which drive this assembly is comprised of sulfamideand imide groups which form a total of 16 hydrogenbonds between the four subunits. The abbreviatedassembly tree for this tetramerization is shown inScheme 13.41 The corresponding theoretical yield fromthis assembly is 0.553% which corresponds to anAN[44+10]rev=180.8. Why is this tetramerizationapparently so much more efficient than the previous?The complexity lies in the supramolecular motif of thesubunit. In this case there are four rather than threepossible points of interaction that each subunit can havewith other subunits. Furthermore, in terms of sym-metry, subunit 8 possesses overall Cs symmetry, whereasthe previous assembly utilized a much more symmetrical(D3h) subunit.

Cubeoctahedra formation: a [83+122] type assembly

Finally, as a demonstration of the power of reversibleassemblies, we turn to an assembly recently reported byStang et al.42 In this highly complex assembly a total of20 components are brought together (Scheme 14).Again, however, only eight of them offer any regiose-lective options during assembly, so the remaining 12(bidentate) molecules can be envisioned simply as ‘glue’which hold the subunits together. Nevertheless, thequantitative assembly of eight subunits into the corre-sponding cubeoctahedra is an exceptionally complexprocess. Thus, generation of the assembly tree (anabbreviation of which is shown in the Appendix) showsthat 62 intermediates are formed over the 12 stages orgenerations of assembly [1st (1), 2nd (1), 3rd (2), 4th (3),

Scheme 11. The coordination bond promoted tetramerization of 2,4,6-tri(4-pyridyl)-1,3,5-triazine.39

Scheme 12. An abbreviated assembly tree derived from the work from the Fujita group.39 Each step or junction represents a reaction (the additionof a new subunit or an internal reaction). The probability at each step is indicated as a fraction. The intermediates (seven) are numbered in par-enthesis.

Figure 2. Two representations of subunit 8. (a) Normal chemicaldrawing. (b) Side view of molecular model highlighting curvature ofmolecule (all H atoms omitted except amide protons used in forminghydrogen bonds).


5th (5), 6th (9), 7th (13), 8th (14) 9th (9), 10th (4), 11th(1) 12th (1, the product)]. The result is the largestassembly number which we have calculated to date.Thus a value of AN[83+122]rev=2557.76 can be calcu-lated from the probability r=3.909�10�4.

Problems Associated With The Approach

The last example admirably demonstrates that one ofthe problems with this approach, at least when calcu-lated by hand, is that it is limited to say six molecularsubunits possessing three or four points of interaction.With more subunits the calculation rapidly becomes toocomplex for routine determination. Some assemblies arealso impractical to calculate by hand because theyutilize two type of molecular subunits which each pos-sess greater than two binding points. For example, inthe ([68+84] resorcinarene assembly reported by Mac-Gillivray and Atwood43 the resorcinarenes (the primarysubunits) are ‘held together’ by water molecules (sec-ondary subunits) which either act as bis-hydrogen bonddonors, or as combined hydrogen bond donor/acceptor

species. In other words all 14 subunits involved with theassembly, the six resorcinarenes and eight water mole-cules, offer regioselective options (>2 identical interac-tions) during assembly. This example is beyondcalculation on the back of an envelope!

Conclusions

We have presented here a way of classifying self-assem-blies, and a means to defining a common frame ofreference to relate assemblies in general. The resulting(semi) quantification of assemblies is we believe a usefultool to complement the qualitative descriptions of selfassembly that presently exist. Thus, not only does theapproach allow one to demarcate between what may bedefined as an assembly system and those processes thatdo not fit the description, but it also allows comparisonsof assembly efficiency to be made between disparatesystems. Furthermore, the outlined approach also givesgood insight into both the different types of assemblythat can exist and the importance of symmetry andstructure of the subunits. It is therefore hoped that

Scheme 14. The assembly of a nano-scale cubeoctahedra.42

Scheme 13. An abbreviated assembly tree derived from the work from the Rebek group.40 Each step or junction represents a reaction (the additionof a new subunit or an internal reaction). The probability at each step is indicated as a fraction. The intermediates (13) are numbered in parenthesis.


arming the supramolecular community with such infor-mation can help to strengthen the field as we move intothe realm of nanotechnology from the ‘bottom up.’

Acknowledgements

This work was partially supported by a Research Inno-vation Award from the Research Corporation, and theDonors of the Petroleum Research Fund, administeredby the America Chemical Society.

Appendix

(1) Hints for Drawing Assembly Trees

With a little bit of practice the science of drawing anassembly tree is relatively straightforward. As men-tioned in the main text however, if more than 6 mole-cular subunits are brought together the tree doesbecome rather complex and a systematic approach isessential (see below). It is recommended that interestedparties practice with some simple examples first andperfect their technique.

Our approach starts with CPK models (or equivalent).It is not strictly necessary to have models of all the spe-cies that are in the ‘reaction box’ i.e., all those thatappear in the product. Generally it is only at the start ofthe assembly where dimensions are relatively small thatit should be checked that a particular reaction can indeedoccur. As the intermediates grow in size, so its inherentflexibility increases and models are not necessary to see iftwo particular moieties can reach each other.

With models on hand we begin to form the assemblytree. The tree derived from work by the Fujita’s group,a [43 +62] type assembly,39 is shown in Scheme A1. Wewill use this example to explain our approach to deter-mining the theoretical yield. Note that we have depictedthe triazine ligands as small equilateral triangles. ThePd ‘glue’ has been omitted but is used to join up theapexes of the triangles.

In general we have found it best to complete each levelor generation of assembly rather than following a par-ticular branch all the way to the product. In this wayeach intermediate drawn in a branch can be alignedaccording to its level. This is useful when working outthe final numbers (see below). Returning to Scheme A1,the first level of assembly is straight forward and pro-duces a ‘bow tie’ with four equivalent bonding points.Models indicate that A1 cannot bond with A2 (or like-wise between B1 and B2) but the A groups can reactwith the B groups, or a third subunit can be introducedto one of the four equivalent sites.

In relatively complex intermediates it may not be readilyapparent how many internal reaction are possible.Therefore we use a standard approach to determiningthe number in internal reactions possible in an inter-mediate:(1) Determine which bonding positions can be

grouped into degenerate or equivalent sets.(2) For each set of equivalent bonding positions

determine the number of internal reactions theycan undergo.

(3) Multiply the number of possible internal reactionsby the number of positions in the equivalent setand divide by two.

Scheme A1. The assembly tree derived from the tetramerization of 2,4,6-tri(4-pyridyl)-1,3,5-triazine ligands.


(4) Repeat steps (2) and (3) for each set of equivalentpositions. The total of these values is the numberof possible internal reactions.

As an example, each of the four equivalent groups ofthe ‘bow tie’ intermediate can react in two ways.Therefore the total number of possible reactions is(4 �2)/2 = 4. These are listed (reaction between A1B1

for example) on their respective branches of the assem-bly tree. External reactions are then considered. We canadd one group to four equivalent sites so there are fourpossible external reactions which again are labeled onindividual branches. In total the ‘bow tie’ can do one ofeight things (of equal probability). We try to write thesechoices down in such an order to make sure that reac-tion options leading to identical products are groupedtogether. Each group is marked with a curly bracketand the number of members in that group are noted atthe top of the bracket. Models indicate that all theinternal reactions of the ‘bow tie’ lead to non-productso these are terminated with an F (fail). For those reac-

tion options which were successful, their respective pro-ducts are drawn and models constructed. The process isnow repeated. There is only one product in the secondstage of assembly (three triangles in a chain) and againthe total number of internal reactions are calculated([4 �3]/2 + [1�4]/2 = 8) and tallied along with the totalnumber of (five) external reactions with the fourth andfinal subunit. There are therefore a total of thirteenpossible reaction options open to this intermediate ofwhich nine go on to produce (three) intermediates thatwill ultimately form product. These intermediates aredrawn and models are made. These steps are repeateduntil the product has been formed. In many cases bran-ches that have previously diverged will actually shareintermediates at later stages (symmetry is good!). Forexample in the Fujita assembly the fourth and fifthstages all share the same intermediate. Once the tree hasbeen completed the probability must be calculated. Theeasiest way to do this is again to work from up anddown a generation to produce columns of fractions. Therows are then read off by the multiplication of each

Scheme A2. AN abbreviated assembly tree from the work reported by the #Stang group.42


fraction belonging to each branch. When all brancheshave been tallied the sum of all branches gives theprobability for forming the compound. Referring againto Scheme A1, Table A1 can be readily constructed.

The probability of the first (upper most) branch is: 1/1� 4/8 � 1/13 � 1/1 � 4/5 � 1/1 = 2/65. The secondbranch is: 1/1 � 4/8 � 4/13 � 4/13 � 4/5 � 1/1 = 32/845. Note that in this case the first two and the last twofractions are common. Finally, the third branch can betallied as: 1/1 � 4/8 � 4/13 � 3/6 � 4/5 � 1/1 = 192/3120. Adding each branch (2/65 + 32/845 + 192/3120)= gives an overall probability of 22/169 and hence atheoretical yield of 13.01%. Thus AN[43 +62]rev =7.68.

(2) Assembly Tree Complexity Rapidly Increases; TheFormation of a Nano-Scale Cubeoctahedron

Recently Stang et al. reported the assembly of twentymolecules to form a nanoscale cube-octahedra.42

Although the assembly involves 20 components onlyeight of them offer any regioselective options duringassembly. Overall it can therefore be considered as anoctamerization type process with the twelve bidentateligands acting to ‘glue’ the subunits together. Never-theless a very high assembly number of AN[83 +122]rev= 2557.76 can be calculated for this process. Anabbreviation of the assembly tree derived from this sys-tem is shown in Scheme A2. The Scheme is abbreviatedby: (1) omitting the actual structures of the inter-mediates (instead the intermediates that can go on togive product have been numbered sequentially); (2) bystating the overall probability of forming the sub-sequent structures, rather than drawing out each reac-tion option. Consider for example intermediate 26(Level 7 intermediate sixth from the top). Of the fortytwo possible reactions (the sum of the internal reactionsand the addition of one extra subunit) with this species,eight products are formed. The probability of formingeach intermediate (whose identifying number is in par-enthesis) is: 2/42 (41), 4/42 (42), 2/42 (43), 2/42 (44), 2/42 (37), 1/42 (38), 2/42 (45), 4/42 (46). To complementthis data, Table A2 shows the probability of forming theproduct from each of the numbered intermediates.

Quite obviously this is a highly complex assembly andthis probability calculation probably represents the limitof the approach without the aid of a computer to gen-erate the assembly tree. This large assembly numbermay also indicate that if more complex assemblies areultimately calculated then perhaps a logarithmic scalefor the assembly number would be more appropriate.

References and Notes

1. Tecilla, P.; Dixon, R. P.; Slobodkin, G.; Alavi, D. S.; Wal-deck, D. H.; Hamilton, A. D. J. Am. Chem. Soc. 1990, 112,9408–9410.2. Whitesides, G. M.; Mathias, J. P.; Seto, C. T. Science 1991,254, 1312–1319.3. Lehn, J.-M. Supramolecular Chemistry; VCH: Weinheim,1995.4. In passing, it should be noted that this definition of self-assembly is similar to terms used outside the physical sciences.For example, in linguistics a self-organizing system has beendefined as, ‘‘one in which some kind of higher level patternemerges from the interactions of multiple simple componentswithout the benefits of a leader, controller of orchestrator’’Clark, A. Being There: Putting Brain, Body and World Toge-ther Again; MIT Press: Cambridge, MA, 1997.5. Desiraju, G. R. Acc. Chem. Res. 1996, 29, 441–449.6. Desiraju, G. R. Angew. Chem., Int. Ed. Engl. 1995, 34,2311–2327.7. Zaworotko, M. J. Chem. Soc. Rev. 1994, 283–288.8. Hosseine, M. W.; De Cian, A. Chem. Commun. 1998, 727–733.9. Cowan, J. A. Inorganic Biochemistry: An Introduction;Wiley-VCH: New York, 1997; Chapter 6 (and referencestherein).10. Stryer, L. Biochemistry; W.H. Freeman and Company:New York, 1996.11. Casper, D. L. D.; Klug, A. Symp. Quant. Biol. 1962, 27, 1–24.12. Corbin, P. S.; Zimmerman, S. C. J. Am. Chem. Soc. 1998,120, 9710–9711.

Table A2. Probability of forming the product from the respective

(numbered) intermediate.

Structure Probability toform product

Structure Probability toform product

1 3.90966E-04 32 0.0220150662 7.81933E-04 33 0.0514913663 0.001951081 34 0.0146321784 0.002053512 35 0.1079365085 0.005853243 36 0.1287284146 0.003213169 37 0.0864468867 0.00361439 38 0.1045567778 0.006511946 39 0.1938754589 0.018537523 40 0.23247863210 0.01262657 41 0.08714191811 0.008032923 42 0.04232804212 0.009047995 43 0.08667230213 0.019404208 44 0.12645815714 0.022137799 45 0.04711186215 0.02523397 46 0.03891828316 0.016941953 47 0.04711186217 0.0244558061 48 0.01904761918 0.022070295 49 0.419 0.018717409 50 0.15238095220 0.010465864 51 0.34871794921 0.012023228 52 0.34871794922 0.058559326 53 0.15238095223 0.067221322 54 0.25714285724 0.06047194 55 0.11111111125 0.039765248 56 0.06666666726 0.030885817 57 0.13333333327 0.045762228 58 0.828 0.087178882 59 0.33333333329 0.031073765 60 0.33333333330 0.045926286 61 0.33333333331 0.027130817 62 1

Probability=3.90966�10�4. Theoretical yield (%): 0.039096645.

Table A1. Probabilities derived from the assembly tree shown in

Scheme A1

1stLevel

2ndLevel

3rdLevel

4thLevel

5thLevel

6thLevel

Branch 1 1/1 4/8 1/13 1/1 4/5 1/1Branch 2 1/1 4/8 4/13 4/13 4/5 1/1Branch 3 1/1 4/8 4/13 3/6 4/5 1/1


13. Gibb, C. L. D.; Stevens, E. D.; Gibb, B. C. Chem. Com-mun. 2000, 363–364. In this paper we erroneously quoted anAN for the assembly of 4.04.14. More specific terms such as ‘hydrogen bonding motif’ (arepeating pattern of groups capable of forming hydrogenbonding) are commonly in use but a more general term isnecessary when different types of groups orchestrate an assembly.15. Rowan, S. J.; Hamilton, D. G.; Brady, P. A.; Sanders,J. K. M. J. Am. Chem. Soc. 1997, 119, 2578–2579.16. Lindsey, J. S. New J. Chem. 1991, 15, 153–180.17. Busch, D. H. J. Incl. Phenom. Mol. Recogn. Chem. 1992,12, 389–395.18. Anderson, S.; Anderson, H. L.; Sanders, J. K. M. Acc.Chem. Res. 1993, 26, 469–475.19. Chapman, R. G.; Sherman, J. C. Tetrahedron 1997, 53,15911–15945.20. Olenyuk, B.; Levin, M. D.; Whiteford, J. A.; Shield, J. E.;Stang, P. J. J. Am. Chem. Soc. 1999, 121, 10434–10435.21. Whitesides has pointed out that the term covalent (kinetic)self-assembly may be a misnomer since the process involvescovalent joining within molecular aggregates that have been(previously) organized by non-covalent self-assembly.2 Putanother way, the species involved with the irreversible steps donot contain any self-assembly information in their own rightbut ‘stitch up’ the supramolecular species after assembly hasoccurred. That said, the term ‘kinetic self-assembly’ hasgenerally been accepted by the field and we will follow thisconvention.22. CPK models (Koltun W. L. Biopolymers, 1965, 3, 665–679) are thus ideal for the analyses described in the followingsections.23. Lehn, J.-M., Atwood, J. L., Davis, J. E. D., MacNicol, D.D., Vogtle, F., Eds.; Comprehensive Supramolecular Chem-istry; Pergamon: New York, 1996.24. Stoddart, J. F. From enzyme mimics to molecular self-assembly processes. In Chirality in Drug Design and Synthesis;C. Brown, Ed.; Academic Press: San Diego, 1990, pp. 53–81.25. Conn, M. M.; Rebek, J., Jr. Chem. Rev. 1997, 97, 1647–1668.26. Fujita, M. Chem. Soc. Rev. 1998, 17, 417.27. MacGillivray, L. R.; Atwood, J. L. Angew. Chem., Int. Ed.Engl. 1999, 38, 1018–1033.

28. Caulder, D. L.; Raymond, K. N. Acc. Chem. Res. 1999,32, 975–982.29. Leininger, S.; Olenyuk, B.; Stang, P. J. Chem. Rev. 2000,100, 853–908.30. Ashton, P. R.; Goodnow, T. T.; Kaifer, A. E.; Red-dington, M. V.; Slawin, A. M. Z.; Spencer, N.; Stoddart, J. F.;Vicent, C.; Williams, D. J. Angew. Chem., Int. Ed. Engl 1989,28, 1396–1399.31. Houk, K. N.; Menzer, S.; Newton, S. P.; Raymo, F. M.;Stoddart, J. F.; Williams, D. J. J. Am. Chem. Soc. 1999, 121,1479–1487.32. Hansen, J. G.; Feeder, N.; Hamilton, D. G.; Gunter, M. J.;Becher, J.; Sanders, J. K. M. Org. Lett. 2000, 2, 449–452.33. Ashton, P. R.; Preece, J. A.; Stoddart, J. F.; Tolley, M. S.Synlett 1994, 789–792.34. Cram, D. J.; Cram, J. M. Container molecules and theirguests. In Monographs in Supramolecular Chemistry; FraserStoddart, J., Ed.; Royal Society of Chemistry, 1994.35. Chapman, R. G.; Chopra, N.; Cochien, E. D.; Sherman,J. C. J. Am. Chem. Soc. 1994, 116, 369–370.36. An earlier, erroneous calculation of an AN for this systemof 1.4513 serves as a warning that considerable care is requiredwhen determining all reaction possibilities in an assembly tree.37. von dem Bussche-Hunnefeld, C.; Buhring, D.; Knobler,C. B.; Cram, D. J. Chem. Commun. 1995, 1085–1087.38. Jasat, A.; Sherman, J. C. Chem. Rev. 1999, 99, 932–967.39. Fujita, M.; Oguro, D.; Mlyazawa, M.; Oka, H.; Yama-guchi, K.; Ogura, K. Nature 1995, 378, 469–471.40. Martın, T.; Obst, U.; Rebek, J., Jr. Science 1998, 281,1842–1845.41. In this subunit, we have made the assumption that theimide, N–H, hydrogen donor and the imide, C¼O, hydrogenbond acceptors (and the corresponding sulfamide moieties)work as pairs in forming the product. Therefore, each subunitis assumed to have four points of interaction in their supra-molecular motif rather than eight. We believe that this is aworkable approximation considering the effective molarity ofthe groups in question after the first hydrogen bond betweentwo pairs has been established.42. Olenyuk, B.; Whiteford, J. A.; Fechtenkotter, A.; Stang,P. J. Nature 1999, 398, 796–799.43. MacGillivray, L. R.; Atwood, J. L. Nature 1997, 389, 469–472.


estimating the efficiency of self-assemblies

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