Topological landscapes of porous organic cages

We deﬁne a nomenclature for the classiﬁcation of porous organic cage molecules, enumerating the 20 most probable topologies, 12 of which have been synthetically realised to date. We then discuss the computational challenges en-countered when trying to predict the most likely topological outcomes from dynamic covalent chemistry (DCC) reactions of organic building blocks. This allows us to explore the extent to which comparing the internal energies of possible reaction outcomes is successful in predicting topology for a series of 10 different building block combinations.

Whilst framework materials can be related to underlying extended network topologies, the equivalent topologies for molecular materials include, but are not limited to, polyhedra such as Platonic and Archimedean solids, see Figure 1. Throughout this work, we use the term topology to refer to the underlying connectivity of molecular building blocks (BBs) in the molecule, which is unchanged upon any physical deformation. Whilst initially capsule-like topologies, with two end groups such as cavitands connected by multiple ditopic ligands to form a cavity, dominated, in the last decade a broader range of topologies have been synthetically realised. These polyhedra include tetrahedra, cubes, octahedra, square antiprisms and cuboctahedra, although capsules, tetrahedra and cubes are most commonly observed. A molecule with a single topology could adopt multiple different geometrical shapes, dependent upon factors such as the geometry of the component BBs and their conformational flexibility. For example, as shown in Figure 2, a molecule with an underlying tetrahedral topology could have the approximate geometrical shape of a tetrahedron or an octahedron, or related intermediate shapes, dependent upon the component BBs.
The growing interest in the field of organic cage molecules makes it timely to identify the potential topological possibilities for these materials and to establish a uniform classification system for them. As stated by Brunner in 1981, the "synthesis of new structures requires not only chemical skill but also some knowledge of the principal topological possibilities". 28 Hay has recently discussed design principles for metal-organic polygons and MOPs, emphasising that referring to these molecules by  An example of how two porous organic cages that both have an underlying topology of a tetrahedron adopt different geometric shapes, in one case maintaining a tetrahedron shape 26 (right hand side), and in the other adopting an octahedral shape (left hand side). 27 The underlying topologies are shown in purple and the geometric shapes in orange. Hydrogen atoms and multiple bonds are omitted for clarity throughout the text.. their shape can lead to multiple classifications for the same underlying topology, particularly for polyhedra that are facedirected, dependent on how you relate the metals and ligands to the vertices, edges or faces. 29 Instead, Hay proposes a nomenclature based upon the number of vertices, edges of faces involved in the MOP and its point group. We agree with this approach, and here lay out an extension of this nomenclature for wholly organic molecules.
We then examine which topologies have been synthetically reported and choose a set of molecules from each topological family described to computationally investigate the underlying factors that influence the observed topological forms. Firstly however, in the following section we discuss the factors that can influence the topology formed for a porous organic cage.

Topological control?
Currently, the 'design' of a particular topology in a new porous cage system is possible for fairly rigid BBs, although is not always successful, as there can be unexpected outcomes. There are examples of BBs designed to meet the appropriate geometrical requirements for, for example, forming a molecular cube in an [8+12] reaction of eight corner units (with angles of~90 • ) and 12 linear struts for the edge, 30,31 or forming a cuboctahedron via molecular panelling of tritopic and tetratopic BBs. 32 More commonly, once a particular polyhedral cage has been made, a family of related polyhedra can be synthesised via small variations in the functionality of one or the other of the BBs. However, even this can fail, for example we reported a change from a cyclohexane diamine to a cyclopentane diamine resulting in a larger [8+12] molecule, with an underlying topology of a cube, forming in place of a [4+6] molecule. 33 Worse still, whilst the [4+6] molecule formed a porous solid state material, with a BET surface area of >1300 m 2 g −1 , the [8+12] molecule was found to lack shape persistence in the absence of solvent and hence to form an amorphous, non-porous material. Thus, a very subtle difference in the reaction precursors had an enormous effect on the reaction outcome and therefore properties of the molecular cage material. Fujita describes this type of phenomena as "emergent behaviour" 34 and it is clear that this can also thwart attempts at topological design for organic cage materials. Further complicating the matter, there are some reports of kinetic trapping of products 35 or of solvent choice influencing topological outcome; an example from Liu and Warmuth, where they react a tetraformylcavitand with ethylene diamine in an imine condensation reaction, found that changing between tetrahydrofuran, chloroform and dicholoromethane as the solvent could change the dominant product from a [4+8] cage, to a [6+12] octahedron to a [8+16] square antiprism. 36 As the majority of organic cages are formed by reversible dynamic covalent chemistry (DCC), the product distribution should be thermodynamically controlled, with those molecules with the lowest free energy being the dominant reaction product. Of course, this depends upon the reaction mixture being able to reach this product, rather than being kinetically trapped into alternative products, which is known to have happened with alkyne metathesis formed cages 35 and for 2D imine molecular ladders, 37 and may become more common as the number of bond formation reactions in a structure increases. Alternative products could include a range of different size oligomeric products, a polymeric material, alternative molecules with different topologies, or catenated molecules. Whilst reports of catenanes for porous organic cages are so far rare, examples from Hasell et al. 38 and Zhang et al. 39 suggest that in some cases catenanes can be the thermodynamic product, rather than their monomeric equivalent, as a result of 'self-templating' driven by the introduction of intermolecular interactions such as π-π stacking and alkyl π interactions in the catenated form. In general, it is likely that, dependent on after how much time the product is isolated and characterised, different products might be observed as the mixture evolves. Whilst many organic cage molecules have been isolated in high yield (up to 100%), 40 there are reports of systems where a mixture of products have been isolated from a single solution, 36 presumably a greater energetic preference for a single topology being required for isolation of a single product.
We now consider factors that can influence the topological energy landscape and thus the likely topology and shape of the organic cage molecules formed for a set of BBs: Number of reactive end groups: Whereas two BBs that are ditopic can only form single cycles, larger numbers of end groups can form more complex, polycyclic topologies. Examples of common BBs used in the synthesis of organic cage molecules thus far are shown in Figure 3. For organic BBs, where high symmetry is typically sought in the BBs, at least with a C n symmetry axis with n equal to the number of reactive end groups, there are fewer options as n increases. This distinguishes organic cage molecules from MOPs, where higher coordination numbers at metal sites may be more easily accessed.
Stoichiometry of building blocks: The component BBs need to be combined in the correct stoichiometric ratio to access the targeted topology, for example a cube with 8 tritopic vertex BBs and 12 ditopic edge BBs is formed from a 2:3 reactant ratio. Providing an excess of one BB has been known to skew the topological landscape towards alternative products, where not all of the reactive end groups have formed covalent bonds. 32 Building block geometry and rigidity : The geometry of the BB and the arrangement and direction of the reactive end groups can play a vital role in forming a desired topology. The angle between the reactive end groups and the centre of at least one of the BB must obviously be less than 180 • for a chance of making a finite molecular, rather than periodic, structure. If one wanted to synthesise a molecule with a cubic shape and underlying cubic topology, then we might consider that we want tritopic ligands with (end group)-(centre of mass)-(end group) angles of 90 • to form a corner and ditopic ligands of (end group)-(centre of mass)-(end group) angles of 180 • to form the edges. Indeed, this approach has been successful for the design of the two molecular cubes reported thus far from both the Warmuth 30 and Beuerle 31

Fig. 3
Examples of ditopic (Di), tritopic (Tri) and tetratopic (Tet) BBs reported for the synthesis of organic cages. Starred ditopic precursors have been used for the synthesis or boronate cages, where two hydroxy groups are required for the formation of one boronate bond. Thus, we still label these precursors as ditopic rather than tetratopic. groups. For MOPs, it was reported that the angles need not be perfectly matched to the desired topology, with deviations of <10 • likely to be tolerated, and even larger deviations often able to be compensated by the summation of small adaptions from other BBs, although significant deviations can lead to structures with shapes more severely distorted from the base topological shape. 41 For MOPs, design rules for both topology and shape design through complementary end group vector selection in the BBs have been proposed, 8,41 however, recent analysis of crystal structures of MOPs by Young and Hay found that whilst sometimes this approach works, it does not do so in the majority of cases. 29 These failures were due to either deviation from the ideal bonding vector through conformational flexibility or, more often, due to a concerted rotation about symmetry axes in the molecule. 29 If lack of rigidity in MOP assemblies frustrates the simple geometric design of BBs, then this is even more problematic for organic cages that have less strongly directional bonding. In the successful cases of geometric BB design thus far, rigidity of the underlying BBs has assisted in this design simplification. However, rigidity of the precursors is not a requirement for a organic cage assembly and with many functions of these materials, such as host-guest response behaviour, often relying on dynamic motion of the final assembly, we conclude it is important not to rule out more flexible BBs as potential cage components. This does however significantly increase the design challenge, and we believe necessitates the application of computational modelling in order to predict the conformations of the resultant assemblies. Of course 'rigidity' and 'flexibility' are not absolutes; whilst some flexibility and motion of a component is inevitable, excessive flexibility in the linkers is linked to lower product yields 9 and increases the likelihood that the end assembly will not be shape persistent.
Entropy : following the second law of thermodynamics, every spontaneous process is associated with an increase in symmetry (dynamic or static) of the original system. 42 The entropy of symmetry can play a role in favouring structures with higher symmetry, for example disfavouring linear oligomers over high symmetry structures, such as the symmetric topologies based on Platonic or Archimedean solids discussed here. The entropy of symmetry can be correlated to the symmetry number (σ ) 43,44 and as previously reported by Skowronek et al., 45 these contributions range from 3 -8 kJ mol −1 from a C 3 to D 3 to T d to O h symmetry molecule. Whilst significant in driving the formation of symmetric finite molecules, it is however unlikely that these contributions would influence which molecular topology is formed. Likely more energetically significant, is that entropy can be expected to disfavour large molecules over the alternative outcome of multiple smaller assemblies, thus to synthesise large architectures, a significant driving force is required.

Solvent:
The non-covalent interactions between solvent molecules and cages can alter both the thermodynamics and the kinetics of cage formation process, potentially favouring the selective synthesis of a specific product. 46,47 Poor solubility of reaction intermediates could result in low-yielding synthesis or precipitation during the assembly process. 48 Warmuth and co-workers propose that the major contribution from solvent effects is played by the improper solvation of flexible BBs. 36 Solvent molecules enclosed in a cavity tend to interact with the capsule's internal surface for a prolonged time (e.g. from 10 −3 to 10 3 s), having enough time to establish a strong interaction with the BBs. 46 During this process, solvent molecules located in the inner cavity and in the large windows will favour specific geometrical conformations of the BBs, preferentially driving the assembly towards a specific topology. The complex role of solvent effects on cage formation can therefore be summarised as depending on three main factors: (a) the entropic cost associated with molecular reorganisation by collecting the n BBs from the solvent phase; (b) the energetics of BB -solvent interactions (e.g. hydrophobic effects) and (c) the amount of strain felt by the BB within the final assembled system. Due to the complex and very variable character of solvent effects, these interactions are very hard to quantify and to predict. In the work of Warmuth and co-workers, the differences in solvent effects estimated to explain the yields observed for different imine cage topologies in different solvents amounted to around~17 kJ mol −1 per imine bond.
Synthesis method: The majority of organic cage molecules have been synthesised with batch chemistry at high dilution and slow addition, however recently cage molecules have also been synthesised through continuous flow methodology. 49,50 With flow methodology, very short reaction times with less solvent were achieved, but with high purity and yield. These approaches and novel synthesis methods such as mechanochemistry 51 may also provide alternative synthesis routes to different topologies due to influencing the kinetics of the reaction and the reaction pathways.
Other synthesis variables: These variables include reaction temperature, concentration, rate of mixing, order of addition of precursors, use of catalysts, pH and template molecules.

Enumeration of organic cage topologies
In our enumeration of possible organic cage topologies, we restrict ourselves to the following criteria: (i) The topology must be a polycyclic cage molecule, with the potential for a 3D internal cavity, excluding macrocycles such as cyclodextrins, crown ethers, cucurbiturils and 2D topologies that are polygons such as triangles, squares, pentagons etc.; (ii) it must be possible to form the topology from a two-component reaction of BBs through DCC; (iii) combinations of components that are ditopic, tritopic and tetratopic only; (iv) we presume the BBs have a C n symmetry axis with n equal to the number of reactive end groups and (v) for topologies with larger numbers of components, we include only higher symmetry structures that are relatable to Platonic or Archimedean solids that have been reported for MOPs, such that they are more feasible targets for organic cage molecules.
These criteria, whilst not producing an exhaustive set of possible topologies, result in 20 topologies that are the most plausible organic cage structures and indeed, almost two-thirds have already been synthetically reported. Recently, mathematicians have developed an algorithm for generating maps of 'stable planar cages', where they did not restrict themselves to 2component systems and generated >400,000 unique maps. 52 The majority of these maps are too complex to be feasible with organic chemistry and thus our limited number of plausible topologies is more practical. However, the topologies we discuss can still be used to describe the underlying connectivity in a multicomponent system and indeed, multiple components may be an attractive design approach for synthesis of some of the lower symmetry topologies. Several examples of multiple component systems, [53][54][55][56] including those with orthogonal DCC formation reactions, for example both boronate and imine formation, 51,57 have been reported.
For our discussion of organic cage topologies, we introduce a new nomenclature, labelling each structure as: where X and Y are the two different component BBs that constitute the cage. X and Y are labelled Di if they are ditopic, Tri if they are tritopic and Tet if they are tetratopic. The first BB, X, has the highest number of reactive end groups and if the underlying topology relates to a polyhedra, it will lie at its vertices. The second BB, Y, can have a number of reactive end groups less than or equal to X. Where the number of reactive end groups for X and Y is equal, which is denoted X and which is denoted Y is arbitrary. The superscripts, m and n, denote the number of each BB incorporated in the topology for X and Y respectively. The majority of the time, X-type BBs are connected to other Xtype BBs through only one Y-type BB; in this case no subscript p will be given. However, if two X-type BBs are directly connected through links with two distinct Y-type BBs, then they have p = 2. The subscript, p, thus gives the number of double connections between BB pairs within a topology. These multiple links are also reported for periodic nets 2 and result in topologies with multiple ring sizes. For some of the smaller topologies with only two Xtype BBs, there is triple or quadruple linking of those BBs; in this case no subscript is given as there is no alternative connectivity for that topology. We finally suggest using the prefix c− in the case of catenated cages, equivalent to its usage as a prefix with interpenetrated periodic nets. 4 We will now discuss each of the topologies within the four families that consist of combinations of different connectivity BBs. The topologies and their features are summarised in Table 1 and experimentally reported topologies in Table 2.

Tritopic + Ditopic topology family
Tritopic and ditopic BBs can be combined in a 2:3 ratio to form molecular cages. The smallest topology for this family is the capsular Tri 2 Di 3 topology formed from a [2+3] reaction of two tritopic BBs linked with three ditopic BBs, as shown in Figure 4, where all the topologies of this family are reported. There are multiple synthetic reports of this topology; example references, including reference codes for structures in the Cambridge Crystallographic Database Centre (CCDC) are given in Table 2, these cages have reported functions in gas separation 71 and sensing. 24 This structure can have an intrinsic, shape-persistent cavity if the BBs are relatively rigid and has three cycles which form windows to access the central cavity, these all have ring size 4 (where 4 is the number of BBs that constitute the ring).
The Tri 4 Di 6 topology is formed from a [4+6] reaction and is related to a tetrahedron via an edge-directed assembly, with the tritopic BB placed on each of the vertices of the tetrahedron and the ditopic BB along the edges. The topology has four potential access windows and these all have a ring size of 6. Again, this is a very commonly reported topology in the literature, including systems from Cooper and co-workers 27 for molecular separations 19,20,22 and as porous liquids, 25 and from Mastalerz and co-workers for sensing. 23 This topology has also been reported for CC1 (Covalent Cage 1), CC2, and CC4 molecules in a triply-interlocked catenated form, which we label c-Tri 4 Di 6 . 38 A molecule of Tri 4 Di 6 topology, if shape-persistent, can adopt final shapes ranging anywhere between a tetrahedron at one extreme to an octahedron at the opposite extreme. Reported examples of Tri 4 Di 6 topology molecules with these shapes are also shown in Figure 2. For a tetrahedral shape to form, the tritopic BB could adopt a conformation to act as a capping vertex, with (end group)-(centre of mass)-(end group) angles of~60 • , and the ditopic BB being linear. For an octahedral shape, the tritopic BB could be planar with (end group)-(centre of mass)-(end group) angles of~120 • and the ditopic BB bent with an angle of~60 • . With consideration of conformational flexibility in the BBs of organic cages, these geometric requirements are not essential, as compensation in each of the BBs could still achieve a given shape. The transformation of a tetrahedral shape to an octahedral shape occurs through a process of truncation of the tetrahedron's vertices. In Figure  6, we show this transformation alongside a series of hypothetical porous organic cages of Tri 4 Di 6 topology, whose changing BBs show one way in which this range of shapes could be accessed. From an initial molecule with both the shape and topology of a tetrahedron, one can cause "flattening" of the corners of the molecule by swapping the starting tritopic BB with a planar tritopic precursor. Then, increasing the size of the planar tritopic BBs can lead to geometric shapes with higher degrees of truncation. Finally, an octahedral geometrical shape is obtained by increasing the "bending" the linear ditopic precursor. The key point is that all these molecules have a tetrahedral topology, despite adopting different geometrical shapes (refer back to Figure  2).
The Tri 4 2 Di 6 topology is also formed from a [4+6] reaction, but it contains two doubly connected tritopic BBs as well as two singularly connected BBs. The double connections result in two distinct window sizes, two rings of size 4 and two rings of size 8. This topology has not yet been reported for a DCC reaction, however, in 2014 Wang et al. reported the synthesis of this topology via a one-component alkene metathesis reaction. 70 The component BB has a C 3 symmetry axis and therefore reduced symmetry in the BB need not necessarily be employed to reach this topology if BB flexibility can account for this. This topology should be achievable through two-component DCC reactions.
The Tri 6 Di 9 topology is formed from a [6+9] reaction and is related to a triangular prism via an edge-directed assembly and has five potential access windows, two with ring size 6 and three with ring size 8. We do not believe this topology has been synthetically reported, it will require tritopic BBs that have two (end group)-(centre of mass)-(end group) angles of 90 • and one of 60 • , to form both triangular and square connectivity rings, or BBs that can compensate for the differing angles required. This topology may be more easily achieved with lower symmetry BBs.
The Tri 8 Di 12 topology is formed from a [8+12] reaction and is related to a cube via an edge-directed assembly and has 6 potential access windows, all with ring size 8. Multiple examples of this topology have been synthetically reported, although several lack shape-persistence. A catenated form of this topology, c-Tri 8 Di 12 has been reported by Zhang et al.. 39 This molecule can adopt final shapes ranging from a cube to the opposite extreme of a shape formed from planar tritopic BBs, reported examples of Tri 8 Di 12 topology molecules with these shapes are shown in Figure 7. For a cubic shape to form, the tritopic BB could adopt a conformation to act as a capping vertex, with (end group)-(centre of mass)-(end group) angles of~90 • , and the ditopic BB would then be linear. For the vertex-folded shape, the tritopic BB could be more planar with (end group)-(centre of mass)-(end group) angles of~120 • and the ditopic BB bent.
The final topology that we suggest for this family is a Tri 20 Di 30 molecule from a [20+30] reaction and is related to a dodecahe-    Top row: The range of geometrical shapes potentially accessible with a Tri 4 Di 6 topology, from tetrahedron (left) to octahedron (right). The evolution of the shape is expressed through successive truncations of the vertices of the tetrahedron. The octahedron has been rotated into the page relative to the other shapes to give a clearer perspective. Bottom row: stages of truncation reproduced in hypothetical porous organic cages. The initial cage with a tetrahedral shape is shown in grey, precursors that lead to increasing truncation highlighted in teal, and the final (completely truncated) cage with an octahedral shape is in teal. The cage molecules are not shown to scale.  dron via an edge-directed assembly and has twelve potential access windows, all with ring size 10. This topology has not been reported for an organic cage molecule, although it has been reported for a MOP. 73 It remains an alluring synthetic target, albeit very challenging due to potential issues with solubility of the intermediates and the need to have a thermodynamic driving force to a dodecahedron over other topologies or an infinite polymeric product. Furthermore, it will be particularly challenging for this large molecule to be shape-persistent. The most obvious way to design this topology is to have an (end group)-(centre of mass)-(end group) angle of~108 • in the tritopic BB, to match that of a dodecahedron's vertices, and to combine this with a linear ditopic BB. The required angle of~108 • is close to that of a tetrahedral carbon, thus tri-functionalised methane molecules, such as used

Tetratopic + Ditopic topology family
Tetratopic and ditopic BBs can be combined in a 2:4 ratio to form molecular cages. The smallest topology in this family is the capsular Tet 2 Di 4 topology, formed from a [2+4] reaction of two tetratopic end groups linked with three ditopic linkers, as shown in Figure 5, where all the topologies of this family are reported. The topology has four windows of ring size 4. There are multiple synthetic reports of this topology (see Table 2) and the topology can have an intrinsic, shape-persistent cavity if the BBs are relatively rigid.
The Tet 3 3 Di 6 topology is formed from a [3+6] reaction, containing three sets of doubly connected tetratopic BBs, so all have the same connectivity environment. The double connections result in two distinct window sizes, three rings of size 4 and two rings of size 6. This topology has been reported using a bifunctional ditopic ligand with a cyclisation reaction. 65 The Tet 4 2 Di 8 topology is formed from a [4+8] reaction and contains two double connections, such that each tetratopic BB is part of one double-link and two single-links. The double connections result in two distinct window sizes, two rings of size 4 and four rings of size 6. Although there are no crystal structures reported, Warmuth and co-workers have reported this structure with tetratopic cavitands and ethane diamines in a solvent of tetrahydrofuran and a distorted tetrahedral shape. 36 The Tet 4 4 Di 8 topology is also formed from a [4+8] reaction, but in this case there are four double connections, such that each tetratopic BB is doubly connected to its two neighbours. The topology has two distinct window sizes, four windows of ring size 4 and two of ring size 8. This structure has been reported by Warmuth and co-workers, again with tetratopic cavitands and ethane diamine, although even in solvent it forms a folded structure that does not contain an internal void. 66 The Tet 5 5 Di 10 topology is formed from a [5+10] reaction that also has all building blocks doubly connected to each of their neighbours, creating two distinct windows, four with ring size 5 and two with ring size 10. This topology has not been experimentally reported and would have a high likelihood of lacking an internal void, as with the Tet 4 4 Di 8 topology, for that reason we exclude larger topologies with this type of connectivity.
The Tet 6 Di 12 topology is formed from a [6+12] reaction and is the first of the (Tetratopic + Ditopic) family that can be related to a polyhedron. Through an edge-directed assembly it relates to an octahedron, where the tetratopic BBs are placed on the vertices and the ditopic BBs are placed along the edges. The topology has eight windows of ring size 6. Whilst there are no reported crystal structures, this topology has been reported by Warmuth and co-workers, from the same BBs as the Tet 4 2 Di 8 topology, but using chloroform as a solvent, rather than tetrahydrofuran. 36 A molecule with this topology could adopt final shapes ranging from an octahedron to a cube, as shown in Figure 8. For an octahedral shape to form, the tetratopic BB could adopt a conformation to act as a capping vertex, with (end group)-(centre of mass)-(end group) angles of~60 • , and the ditopic BB linear. For the cubic shape, the tetratopic BB could be more planar with (end group)-(centre of mass)-(end group) angles of~90 • and the ditopic BB  The Tet 8 Di 16 topology is formed from a [8+16] reaction and can be related to a square antiprism through an edge-directed assembly of tetratopic BBs on vertices and ditopic BBs on edges. The topology has eight windows of ring size 6 and two larger windows of size 8. There are no crystal structures for this topology, however it was reported by Warmuth and co-workers for the same building blocks as the Tet 4 4 Di 8 and Tet 6 Di 12 topologies, but using a dichloromethane as the solvent. 36 Molecular mechanics simulations suggested that their molecule would maintain a square prism shape that is approximately a spherical ring, as shown in Figure 9. 36 The Tet 16 Di 32 topology is formed from a [16+32] reaction and can be related to the Archimedean solid of a cuboctahedron through an edge-directed assembly of tetratopic BBs on vertices and ditopic BBs on edges. There are two window sizes, eight of size 6 and six of size 8. There are no synthetic reports of this topology. Finally, there is the Tet 24 Di 48 topology, formed from a [24+48] reaction that can be related to another Archimedean solid, the rhombicuboctahedron through edge-directed assembly. There are two window sizes, eight of size 6 and eighteen of size 8. There are also no synthetic reports of this topology, however, Fujita and co-workers have reported the synthesis of a MOP with this topology using square planar Pd 2+ as the tetratopic vertex and a dipyridylfuran ligand with an angle 127 • as the edge. 34 This structure therefore remains a synthetic design target for an organic cage molecule.

Tritopic + Tritopic topology family
Two different tritopic BBs can be combined in a 1:1 ratio to form molecular cages. The smallest topology for this family is Tri 1 Tri 1 , which consists of a [1+1] reaction, with one of each tritopic BB being connected directly to the other tritopic BBs, resulting in three windows of ring size 2. This is shown in Figure 10, where all the Tritopic + Tritopic topologies are shown. Provided the tritopic BBs have a concave shape, the resultant organic cage can have an intrinsic internal cavity. This type of topology has been reported by Kataoka et al. via a condensation reaction of a bowlshaped triboronic acid and a bowl-shaped tri-diol molecule, the resultant capsule was shown to encapsulate small guests. 67 The Tri 2 2 Tri 2 topology is formed from a [2+2] reaction, containing two of each type of tritopic BB, with each BB having one set of double connections and one single connection to the opposite building block. Therefore all BBs have the same connectivity environment and the molecule has two windows of ring size 2 and two of ring size 4. The Tri 3 2 Tri 3 topology is formed from a [3+3] reaction, with two pairs of BBs having double connections and the remainder having single connections. The molecule has two windows of size 2 and four of size 4. There have been no synthetic reports of either the Tri 2 2 Tri 2 or Tri 3 2 Tri 3 topology to our knowledge.
The Tri 4 Tri 4 topology is formed from a [4+4] reaction and can be related to a tetrahedron through a face-directed assembly, where one of the tritopic BBs is placed on the vertices and another on the faces of the Platonic solid. This type of face-directed assembly has been termed 'molecular panelling' when applied to the construction of MOPs. 74 All the BBs are singularly linked and there are six windows of ring size 4. This topology has been reported by both Warmuth and co-workers 75 and Cooper and co-workers. 68 A molecule with this topology could adopt final shapes ranging from a cube 75 to a tetrapod, 68 as shown in Figure 11. For a cubic shape to form, both tritopic BBs could have a similar conformation with (end group)-(centre of mass)-(end group) angles of~90 • , such that they form the eight corners of the cube between them. For a more tetrapodal shape, one tritopic BB could be planar, with angles of~120 • , whereas the other could have very narrow angles, acting as the end of each 'foot' of the tetrapod.

Tetratopic + Tritopic topology family
A tetratopic and tritopic BB can be combined in a 3:4 ratio to form molecular cages. We identify only one plausible organic cage topology for this system, Tet 6 Tri 8 , as shown in Figure 12. Alternative topologies with fewer components do not form an organic cage, rather open bowl topologies, or require interweaved connections, thus we exclude them here. The Tet 6 Tri 8 topology is formed from a [6+8] reaction that can be related to a rhombic dodecahedron by a face-directed assembly, where both tetratopic BBs and tritopic BBs are on the vertices of the polyhedron. This topology consists of twelve windows of ring size 4.  The Tet 6 Tri 8 topology has been reported by Warmuth and co-workers using a tetratopic cavitand and a triphenylamine (although no crystal structure was reported for this experiment) 32 and two examples using porphyrin building blocks and triamines by Hong et al. (for which crystal structures are available), the resultant molecules showed selectivity for small gases. 69 The range of shapes possible for the Tet 6 Tri 8 topology span from a rhombic dodecahedron 32 through an octahedron to a cube, 69 as shown in Figure 13. For a rhombic dodecahedron shape to form, the tritopic and tetratropic BBs should have (end group)-(centre of mass)-(end group) angles of~60 • and~90 • , respectively. For a cubic shape to form, the tetratopic BB could be planar, with angles of~90 • and the tritopic BB acting as a cube corner, with angles of~60 • .
In the remainder of this article, we will examine specific case studies for each of the four topology families discussed above. This will allow us to compare the possible reaction outcomes -the alternative topologies that could be formed -and to investigate to what extent computer simulations can assist in predicting the topology from knowledge of the BBs alone.

Methods
In this work, where we consider the topological outcome resulting from the reaction of a pair of BBs, we assemble the possible topologies and then conduct a thorough search of the potential energy landscape for low energy conformations, this is essential, Fig. 13 The range of shapes possible for a Tet 6 Tri 8 topology cage. Top: the Tet 6 Tri 8 cage 32 can assume the shape of a rhombic dodecahedron (middle) when considering both tetratopic and tritopic precursors as vertices of a solid. It has the shape of an octahedron (right) when only the tetratopic vertices are linked together. Bottom: the Tet 6 Tri 8 cage 69 can be seen as a rhombic dodecahedron (middle, as before). It assumes the shape of a cube (right) when only the tritopic precursors are linked together.
as without care, assumptions can lead to only local energy minimum conformations being found. 76 Either a conformer search calculation or a high temperature molecular dynamics simulation (MD) was performed to explore the potential energy surface for each molecule, and to locate the lowest energy conformations using the OPLS3 forcefield. 77 This conformational searching will typically take >1 week for a relatively small cage molecule. More flexible molecules can adopt a large number of different conformations; for example, Figure 14 shows an overlay of 395 conformations sampled within a 20 kJ mol −1 energy range of the lowest energy conformation during an MD simulation for Tet 6 Tri 8 -C (molecule described later in text). In this specific example, the flexibility of the cage molecule is mainly a result of the possible phenyl ring rotations in the tritopic BB. Of course, more rigid molecules are characterised by a small number of conformers; for example, the Tri 4 Di 6 topology of CC3 has only a single conformation within a 20 kJ mol −1 energy range.
In cases where the cage molecule is found to be non-shape persistent, with the lowest energy conformations lacking an internal void, we then proceed with our previously developed enhanced sampling technique to "inflate" the molecules and find the open conformations that lie higher on the potential energy landscape. 72 This is essential to compare the conformations that the topologies would be adopting in solution when formed. The time requirements of this procedure naturally increase with molecular size, but as an example, the Tet 8 Di 16 molecule described below required multiple simulations of several weeks duration. Once a collection of open conformations were obtained, we perform a further refinement with DFT calculations in order to obtain more reliable relative energies. Each conformation was geometry optimised with the PBE functional, 78 and then its energy value was refined further with a single point calculation with the meta-GGA M06-2X functional. 79 Following this, relative internal energies were compared for each family of cages. The use of DCC reactions allowed us to assume that the molecule with lowest relative energy should be the most synthetically accessible. When the comparison of relative energies was not possible due to molecules containing different species, formation energies were calculated instead, with the assumption that the molecule with lowest formation energy is the most likely to form. For full simulation details, please refer to the ESI.

Computational challenges
The described procedure allows us to compare the relative internal energies of the hypothetical topologies for a given assembly of BBs, giving an indication of the relative strain (or lack of it), should the BBs adopt those particular topologies. At the start of this paper, we outlined a large number of factors that could influence the topological outcome, some of which we are not able to consider here, based on computational limitations that we will now explain. However, our calculations allow us to uncover to what extent these relative internal energies can explain previous experimental observations.
Firstly, in terms of solvent effects, our previously reported "inflation" procedure 72 allows us to look at how solvent influences the conformations of cage molecules. This approach allows us to reproduce the scaffolding or "templating" effect of the solvent molecules, but does not explicitly model individual solvent interactions. To reproduce explicit solvent interactions, we would firstly have to greatly increase the number of atoms in our simulations by the necessity of including hundreds of solvent molecules; in previous work we found a single cage molecule to be solvated by ∼80 dichloromethane molecules (400 atoms) even in the solid state solvate structure. 33 Secondly, we would then need to sample the conformations of these molecules surrounding the cage molecule to consider the dynamic nature of the solvation. Combined, these simulations are not currently tractable for these systems. An alternative implicit solvation approach, where a dielectric constant is applied to reproduce the dielectric screening a molecule in solution experiences, does not reproduce the scaffolding effect of the solvent, and preliminary tests by ourselves have found this to make no difference to the relative energies of cage topologies. This issue of solvent effect remains a big challenge in the field and efficient approaches need to be developed for the modelling of such complex systems.
Ideally we would go beyond the relative internal energies from DFT calculations and compare free energies. This would include a consideration of the translational, rotational and vibrational entropy of the systems, which should tend to disfavour the formation of larger assemblies over smaller ones. Practically, this would require a frequency analysis for each molecular conformation, and this is computationally intractable on a routine basis even for the smaller cage systems. Indeed, in our previous work we had to resort to considering only fragments of a comparably small molecular cage for calculations of the free energy correction to guest binding energies to be tractable. 80 Whilst not the focus of our work, one can also consider the use of simulations to predict preferential catenane formation over monomeric forms of the topologies discussed. Catenanes could be expected to form as thermodynamic products if a combination of solvent interactions, steric compatibility of the BBs involved in the reaction, and the formation of stabilising interactions between the two monomeric units act as a thermodynamic driving force. The obvious starting point for simulations is to calculate the binding energy for a catenane pair by comparing the total energy to that of two gas space monomers. An unfavourable binding energy, for example resulting from steric clashes of the molecules in a catenated form, would suggest that the monomeric form would be preferred. However, favourable binding energies would need to be considered with caution, as these would indicate stabilisation of the catenane relative only to isolated gas phase monomers, without consideration of stabilisation of those monomers by either solvent interactions or intermolecular interactions gained via crystal packing. Thus, arguably, for a series of systems, binding energies might be best used to qualitatively rank a series of monomers in terms of likelihood of catenane preference over monomeric forms. We further note that for some systems there would be significant, and even potentially prohibitive, sampling requirements when looking for low energy catenated forms. These would arise from the requirement to consider both different mechanical interlocking arrangements and also configurational sampling for each interlocked form. Finally, reliable binding energies across a series of different systems are likely to require DFT calculations, rather than computationally cheaper FF calculations.

The Tritopic + Ditopic family: CC3
For the Tritopic + Ditopic family, we investigated the outcome of combining 1,3,5-triformylbenzene (1 in Figure 15) and (R,R)-1,2diaminocyclohexane (2 in Figure 15), used in the synthesis of CC3, a Tri 4 Di 6 imine molecule. 27,81 The relative energies per [2+3] unit were compared for the lowest energy conformations of the Tri 2 Di 3 , Tri 4 Di 6 , Tri 4 2 Di 6 , Tri 6 Di 9 and Tri 8 Di 12 topologies. The structures of the lowest energy open conformations are shown in Figure 15, with the relative energies in Figure 16 and Table  3. The Tri 2 Di 3 , Tri 4 Di 6 , and Tri 4 2 Di 6 topologies exhibit shape persistency, whereas the larger Tri 6 Di 9 and Tri 8 Di 12 topologies are more flexible and collapse under MD simulations, losing their internal cavities. Constrained MD simulations on the latter two topologies were able to find higher energy open metastable states, 4 and 2 kJ mol −1 per [2+3] unit above the collapsed conformations for Tri 6 Di 9 and Tri 8 Di 12 respectively. We compare the relative energies of the open conformations only, as these are likely the structures that initially form in the reaction solution, and therefore comparison between these structures is most pertinent for determining the likely reaction outcome.
The Tri 4 Di 6 molecule lies lowest in energy and is the synthetically realised topology, which can therefore be rationalised on the energy of the gas phase conformations alone. This is followed by the higher energy metastable inflated conformers of Tri 8 Di 12 (5 kJ mol −1 per [2+3] unit) and Tri 6 Di 9 (14 kJ mol −1 per [2+3] unit). An alternative topology is available with a [4+6] reaction, in which precursors are assembled into a doubly-bonded ring to give Tri 4 2 Di 6 , however this is considerably higher in energy (39 kJ mol −1 per [2+3] unit). The Tri 2 Di 3 , in which the precursors are assembled into a capsule, is the topology with the highest energy, 52 kJ mol −1 per [2+3] unit higher, unsurprisingly the geometry is characterised by a high degree of strain. To compare not only the topological prediction, but also the structure prediction, we compared our predicted structure of the Tri 4 Di 6 CC3 molecule to that of the single crystal X-ray diffraction structure. 27 We found a Root Mean Square Deviation (RMSD) value of 0.107 Å, showing the conformation is well reproduced, as shown in Figure S1-A.

The Tritopic + Ditopic family: CC5 and CC8
For the Tritopic + Ditopic family, we investigated a further case that has been reported as an example of emergent behaviour. When tris(4-formylphenyl)amine (3 in Figure 17 and (1S,2S)cyclopentanediamine (4 in Figure 17) are combined, they produce the porous Tri 4 Di 6 CC5 molecule, but changing the diamine to (R,R)-1,2-diaminocyclohexene (5 in Figure 17), adding only an additional CH 2 group, results in a Tri 8 Di 12 CC8 molecule, which is non-porous and lacks shape persistence. 33,82 The relative energies per [2+3] unit and structures for the lowest energy conformations of the Tri 2 Di 3 , Tri 4 Di 6 , and Tri 8 Di 12 topologies were compared. The structures are show in Figure 17 and the energies in Figure 18 and Table 4. The Tri 2 Di 3 topology is too small to contain a cavity, whereas the Tri 4 Di 6 molecules are shape persistent with cavities~7 Å in diameter. However, both Tri 8 Di 12 structures collapse, as is known experimentally for the For both cage families, the Tri 2 Di 3 molecule, with only two carbon atoms bridging the tritopic BBs, is strained and consequently high in energy compared to the other topological possibilities in all cases (by >40 kJ mol −1 ). It is not surprising therefore that this topology is not experimentally observed. For the CC8 molecule, the lowest energy conformation has the large Tri 8 Di 12 topology, with approximately octahedral symmetry. This corresponds to the experimental reports of this molecule forming with the diaminocyclohexene group. 33 As we previously reported, 72 the computed open conformation for CC8 is a good match to the single crystal X-ray diffraction structure of the solvate (with a RMSD value of 1.672 Å, see Figure S2-A), which im- proves slightly when minimising the single crystal structure (with the RMSD value moving to 0.629 Å, see Figure S2-B), suggesting some influence of the solvent molecules in opening the vertices slightly outwards.
For the CC5 molecule, there was a very different result to the CC8 molecule, now there is no clear preference between the Tri 4 Di 6 and Tri 8 Di 12 molecules, with a relative energy difference of only 1 kJ mol −1 per [2+3] unit. This molecule has in fact been synthetically realised only as a Tri 4 Di 6 topology, which had a BET SA of 1333 m 2 g −1 . 82 An overlay of the computed and single crystal X-ray diffraction structure of the solvate (see Figure S3) finds a good match, with an RMSD of 0.706 Å. Again, this match improves when the solvate conformation is geometry optimised (RMSD of 0.449 Å), which contracts the molecule, suggesting a scaffolding effect of the solvent that is lost in the calculations when the conformation is completely energy minimised after our artificial inflation procedure. In this case therefore, the calculations would not have been able to successfully distinguish between whether a Tri 4 Di 6 or Tri 8 Di 12 topology was formed, although it was clearly a different scenario to the CC8 topological energy landscape. We attribute this to our not being able to consider all solvent effects in our simulations (as evidenced by the overlays). Further, as discussed in the computational challenges section, if the entropic contribution to the free energies was able to be considered, this can be expected to destabilise the larger Tri 8 Di 12 topology relative to the Tri 4 Di 6 .

The Tritopic + Tritopic family
For the Tritopic + Tritopic family, we investigated the reaction of tris(2-aminoethyl)amine (6 in Figure 19) and tris(4formylphenyl)aldehyde (7 in Figure 19), used in the synthesis of CC11, a tetrapodal-shape Tri 4 Tri 4 molecule. 68 The relative energies per [1+1] unit were compared for the lowest energy conformations of the Tri 1 Tri 1 , Tri 2 Tri 2 and Tri 4 Tri 4 topologies. The structures of the lowest energy conformations are shown in Figure 19, with the relative energies shown in Figure 20 and Table 5. Both Tri 1 Tri 1 and Tri 2 Tri 2 present very strained geometries and therefore their energy is hundreds of kJ mol −1 higher than that of Tri 4 Tri 4 (376 and 123 kJ mol −1 per [1+1] unit, respectively). This is in agreement with the experimental result, where it is the Tri 4 Tri 4 topology with a tetrapodal shape that is formed. When we compared to the single crystal X-ray diffraction structure, we find an RMSD of 0.548 Å, with a good match of our predicted  structure to the experimentally reported one (see Figure S1-B).

The Tetratopic + Ditopic family
For the Tetratopic + Ditopic family, we investigated the reaction of the tetraformylcavitand (8 in Figure 21) and ethylene-1,2diamine (9 in Figure 21), reported by Warmuth and co-workers to produce a mixture of Tet 4 2 Di 8 , Tet 6 Di 12 and Tet 8 Di 16 topology molecules, dependent upon the solvent used. 36 Five different molecules were generated, Tet 2 Di 4 , Tet 3 Di 6 , Tet 4 2 Di 8 , Tet 6 Di 12 and Tet 8 Di 16 . In the MD simulations, Tet 2 Di 4 , Tet 3 3 Di 6 and Tet 4 2 Di 8 were found to be shape persistent, whereas the larger Tet 6 Di 12 and Tet 8 Di 16 topologies were not. Constrained MD simulations on the latter two were able to locate metastable, higher energy open conformations. The structures of the lowest energy open conformations are shown in Figure 21, with the relative energies shown in Figure 22 and reported in Table 6.
The cage with the lowest energy was a partially collapsed conformer of Tet 8 Di 16 . The cage was not found to be shape persistent after high temperature MD, and the inflating procedure provided us with a very large number of higher energy metastable conformers, some of which were partially open, like the lowest energy one reported, and others containing a larger pore size (as shown in Figure S4). As no crystal structure was available and it was not possible to be certain whether a single conformation represented the open structure, we plot the range of energies found for conformations with varying pore size in Figure 22   and show the conformation with the widest pore diameter (10.4 Å) in Figure 21. This more open conformation is located 38 kJ mol −1 above the less inflated lower energy conformer, which has a smaller pore diameter of 8.4 Å. The degree of strain of the structure increases with the degree of inflation.
Experimentally, the formation of Tet 8 Di 16 was obtained using CH 2 Cl 2 as a solvent. We carried out a structural analysis on the resulting cage, in order to understand whether it was possible to relate the solvent used during the synthesis to the final topology. Tet 8 Di 16 has 10 windows, with diameters that range from 4 -6 Å for the 8 triangular windows and from 6 -9 Å for the 2 square windows, depending on the degree of inflation (see Table S1). About 22 and 21 kJ mol −1 above the lowest energy conformer of Tet 8 Di 16 are Tet 6 Di 12 and Tet 4 2 Di 8 , that were respectively found experimentally in THF and chloroform. We analysed their windows diameters (an average of 5.2 Å for the 8 triangular windows of the octahedral Tet 6 Di 12 , and an average of 5.6 Å for the 4 biggest windows of Tet 4 2 Di 8 ), but we were not able to find any geometric correlation between the solvent used for the synthesis and the cage topologies. We leave this particular aspect to further studies.
The hypothetical topologies we generated, Tet 3 3 Di 6 and Tet 2 Di 4 , are more strained and thus higher in energy, explaining the fact that they have not been experimentally observed. They are located respectively 17 and 27 kJ mol −1 higher than Tet 4 2 Di 8 . We then investigated the effect of changing the ditopic BB's geometry on the topological landscape. The study carried out by Warmuth and co-workers suggested that it was not possible to synthesise a cage with the shape of a Tet 2 Di 4 capsule when using ethylene-1,2-diamine, which is rationalisable by the fact that it lies 27 kJ mol −1 higher in energy per [1+2] unit than the observed Tet 4 2 Di 8 topology. 36 However, they did find that the Tet 2 Di 4 capsule could be formed by using different ditopic BBs; benzenes amino-functionalised in the meta or para position. We therefore generated two capsules in a [2+4] ratio, Tet 2 Di 4 -B and Tet 2 Di 4 -C, using m-Phenylenediamine (10) and the chiral precursor p-Xylylenediamine (11) respectively. As these molecules contain different BBs, we can not directly compare their relative energies per formula unit, and therefore instead compare the relative formation energies of the three capsular Tet 2 Di 4 molecules (see Table 6). In agreement with experimental results, both cage B and C are lower in energy than A, in particular Tet 2 Di 4 -C is 6 kJ mol −1 per imine bond lower than Tet 2 Di 4 -B, and Tet 2 Di 4 -C is 8 kJ mol −1 lower than Tet 2 Di 4 -A. This is because kinked or twisted diamines allow the two cavitands to align in a more stable and less hindered conformation, than when using BB 9. The latter should adopt a very strained gauche conformation to permit the cavitands to align in the coplanar conformation required to reduce the strain.
shown in Figure 23. The relative formation energies of the two topologies were compared (reported in Table 7) and found to be very similar (within 1 kJ mol −1 ). Such a small difference in energy would explain why both topologies were experimentally observed in solution. Although the presence of the molecules in solutions was identified by Warmuth and coworkers with different techniques such as 1 H NMR, 13 C NMR and MALDI-TOF, the structures were never isolated from the solvent and neither solvated or desolvated crystal structures are currently available, therefore a direct geometric comparison between experimental and computed structures in not currently possible. Nevertheless, the calculations seem to suggest the lowest energy conformers of both the capsule and rhombic dodecahedron cage are shape persistent and porous, therefore of synthetic interest. The void dimensions of both molecules were calculated and reported in Table 7. Tet 2 Tri 4 shows a void diameter 9.9 Å, and Tet 6 Tri 8 of 17.8 Å, which is very close to the 20.6 Å void diameter of the boronate cube synthesised by Mastalerz and coworkers, which currently holds the record of the molecule with the biggest intrinsic pore synthesised. Considering the possible interest in a porous molecule of this cavity volume, the effect of the size of the triamine BB was investigated and two similar molecules with the Tet 6 Tri 8 topology were generated. Cavitand 8 was mixed with 1,3,5-triaminobenzene (13) to give Tet 6 Tri 8 -B and with 4,4',4-triaminotriphenylamine (14) to obtain Tet 6 Tri 8 -C. Both molecules were found to be shape persistent and therefore the formation energies per bond were calculated and compared to that of Tet 6 Tri 8 -A (see Table 7). Tet 6 Tri 8 -B was found to have a formation energy~1 kJ mol −1 lower than Tet 6 Tri 8 -A, thus is potentially equally likely to form. Tet 6 Tri 8 -C shows a less strained geometry, we attribute this to the central nitrogen atom on each tritopic BB allowing rotational freedom, bringing the formation energy 4 kJ mol −1 per imine bond lower than Tet 6 Tri 8 -A. This suggests that Tet 6 Tri 8 -C could be a particularly promising synthetic target. Both molecules are porous, showing respectively a shape persistent pore diameter of 12.2 Å and 15.5 Å. Fig. 23 Molecular structures of the lowest energy conformations of the Tetratopic + Tritopic topologies formed from the reaction of tetraformyl cavitand (8) and 1,3,5-tris(4-aminephenyl) benzene (12). Also shown with the Tet 6 Tri 8 topology are the lowest energy conformations with 1,3,5-triaminobenzene (13) and 4,4',4-triaminotriphenylamine (14) as the tritopic BBs. Tetratopic precursors are in yellow, tritopic precursors in blue, the imine bonds formed in the reaction are in grey.

Conclusions
We have enumerated the 20 possible topologies that can be used as underlying structures for the design and the synthesis of new porous organic molecules. We have suggested a naming convention that relies on the type, number and multiplicity of connectivity of the BB and propose that this will cause less confusion than naming based on relationships to Platonic or Archimedean solids. Whilst 12 of these topologies have already been synthetically realised, several interesting target candidates have also been identified.
We have tested to what extent calculations focusing on relative internal energies can assist in the prediction of topological outcome for a given DCC reaction that should reach the thermodynamic product. This approach was successful in the majority of the BB combinations we examined, including for CC3, CC5, CC11, the (Tetratopic + Ditopic) family from Warmuth and the (Tetratopic + Tritopic) family from Warmuth. However, for the case of emergent behaviour in the CC5 and CC8 systems, whilst we can correctly identify that the cyclohexane BB leads to a Tri 8 Di 12 topology, for the cyclopentane BB, we could not distinguish between the relative energies of Tri 4 Di 6 and Tri 8 Di 12 , thus not correctly identifying the experimental outcome of a Tri 4 Di 6 topology. We believe this result is likely down to a combination of the influence of the solvent in disturbing the energy landscape and an entropic contribution disfavouring the larger topology, which we were not able to directly include in oour simulations. For the Warmuth (Tetratopic + Ditopic) family, where the topology is known to be influenced by solvent choice, we were not able to find any obvious geometric correlation between the structures and the successful solvent, thus this remains the subject of future research.
We highlight the design opportunity behind using simulations to consider the thermodynamic viability of a given target, which, together with chemical intuition, can provide new developments for this emerging field. The natural evolution of this project would be to explore the topological outcome for a much wider library of precursors with different topicity, length, angles, and rigidity, with the goal to rationalise which criteria would favour the formation of promising porous organic cages. However, we have shown here computational challenges with modelling a relatively small set of organic precursors, suggesting that brute force screening of large libraries is not currently computationally feasible. Instead, we would suggest it is better to focus on answering specific questions with smaller library subsets, such as: How does variation in BB geometry influence topological outcome? Which BBs provide the most promising route to a desired topology? Which BB features are most critical to shape persistence in a desired topol-