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Given the reaction graph for some chemical system, the principles of transition-state theory (TST) can be applied by associating the relative energies, energy gradients, and curvatures for the component species with their respective graph vertices. This facilitates an encoding of the essential features of a PES; this is what is meant by the reaction graph describing the associated chemical space. Further, this reaction-graph -PES relationship is a feature of the Polytope Formalism where graphs of the same topology can describe the idiosyncrasies of PESs for different chemical systems.
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These two simple examples demonstrate additional important details. First, while some species within the formalism may represent well-defined locations (critical points) on its PES, others will require a qualitative rule to be invoked. Second, each graph edge corresponds to a different elementary reaction coordinate with this associated with (local) bond-stretching vibrational modes that describe each end of the desmotropic reaction coordinate. In many cases, two or more graph edges in sequence comprise a "full reaction coordinate" (e.g., Fig. ). The last point to note is that not all atoms in a chemical system need undergo atom-connection permutations (e.g., as in Fig. ). Further, the full scope of atom-connection permutations need not be applied. For example, the "active" H-atom in both examples in Fig. have the constraint of maintaining at least one bond.
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For any chemical system, the configuration space resulting from a constraint within the analysis (e.g., fewer atoms, restriction on connectivity, etc.) is a subspace of the unconstrained configuration space. These focused analyses are more tractable than looking at the larger system. Conversely, a larger space can be constructed by expanding the definition of the chemical system (e.g., with additional atoms, inclusion of stereoisomerism, inclusion of spin states, etc.).
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Given the superexponential scaling of these combinatorics-based analyses, the modular structure theorem can be employed to facilitate focused analyses where atom-connectivity permutations are considered for only a small subset of atoms in a larger system (e.g., as in Fig. ). Many approaches to this end are possible and it is worth noting that the Polytope Formalism of constitutional isomerism, when constrained by valence and limiting discussion to a single molecular entity, returns the same results as established methods for constitutional isomer enumeration. The following approach is broadly applicable to many chemical scenarios, especially where "bond walks" are key and range from simple cases of tautomerism to complex systems such as ion transport through transmembrane channels. Fig. illustrates how molecular systems can be partitioned where the subset of "active" atoms experiencing connectivity permutation can be further divided into two categories based upon their roles. We identify "mobile" atoms as "bonders" ℬ along with several bonding sites 𝒮 typically anchored within one or more larger molecular fragments.
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Different atom-connection configurations arise by only permutating connectivity between the bonders and the sites. All other parts of the molecular system are called "spectator" atoms and are, within this methodology, characterised by their atom connections remaining fixed throughout. With m bonding sites 𝒮 and n bonders ℬ, this partitioning approach is termed the 𝒮 𝑚 ℬ 𝑛 class.
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The dividing up of a chemical system into separate parts with each being analysed or considered using distinct approaches is widespread throughout Chemistry. Perhaps the best-known examples are ligand docking in drug discovery and the multilevel modelling of large chemical systems as developed by the 2013 Nobel Laureates Karplus, Levitt, and Warshel .
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The examples in Figs. 5b -5d all embody 𝒮 3 ℬ 2 partitioning. where three O-atoms (solid blue shading) are the three sites. For Fig. the example is NaHCO3 featuring an H-atom and Naatom as the two bonders (light-red shading), that are permitted to undergo connectivity permutation. The central C-atom (light blue shading) is a spectator of the site complex, and the three sites (O-atoms) are equivalent (exhibiting D3h site-symmetry). In Fig. the example has the same site complex as Fig. ("carbonate") but with a pair of bonders (two equivalent methylene C-atoms) joined through spectators (the central methylene). Partitioning for the description of tautomerism in glycolic acid is shown in Fig. . Here, the symmetry of the sites is broken with two equivalent O-atoms (the "carboxylate") and one distinct (the "alcohol").
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Under the 𝒮 𝑚 ℬ 𝑛 partitioning approach to the Polytope Formalism of constitutional isomerism, the same set of connectivity permutations apply to the three distinct examples in Figs. . Consequently, for each of these examples in Figs. , the resultant number of families and species are identical and equal to 6 and 49, respectively. The different symmetry combinations of the site and bonder complexes will, however, lead to these species being distributed into different numbers of genera. In the limit where there is no site symmetry, the number of genera equals the number of species with each genus containing a single species (a "monotypic" genus). This will be discussed and demonstrated later. The examples in Figs. 5e and 5f feature molecular systems where the bonders are located inside the site complex. The example in Fig. is set up to describe tautomerism in free-base subporphyrin (free-base triphyrin[1.1.1]) where the bonders (Hatoms) can connect to any of the three sites (N-atoms). These sitesand the entire site complexexhibits D3h symmetry. Fig. shows a B2OF2-porphyrin complex partitioned as the 𝒮 4 ℬ 2 class to investigate the strepsisomerization 1 bond-walk mechanism. The four sites (N-atoms) and the macrocycle ("porphyrindiyl") exhibit D4h symmetry. The two bonders are the B-atoms with the O and F-atoms as spectators of the bonder complex.
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By focusing on the high-symmetry system with R = H, (Fig. ) these questions are addressed by partitioning free-base subporphyrin monoanion as the 𝒮 3 ℬ 1 class (Fig. ). The "subporphyrindiyl" fragment 5 (Fig. ), the monoanion without the inner hydrogen atom, exhibits the D3h symmetry point group from an atom-connectivity configuration perspective. Under the condition of non-dissociation to maintain a single molecular entity, a full connectivity permutation between the three sites (N-atoms) and the single bonder (H-atom) gives the seven species as shown in Fig. .
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There are three degenerate species (6a, 6b, and 6c) that each feature a single connection between the bonder and a site. These species comprise the genus 6. Further, in this case genus 6 also comprises family F1. There are also three species (7a, 7b, and 7c) that each feature the bonder bridging two sites. These three species comprise genus 7 which, in turn, comprises family F2. The seventh species 8a features the bonder triply bridging all three sites. The species 8a is the sole member of genus 8 which comprises family F3. This pattern of a single genus comprising each family is not a general feature of the formalism. More complex worked examples below illustrate the broader utility of the taxonomic system.
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With the D3h site symmetry broken (R H), as in Fig. , the three species 6a -6c become chemically distinguishable. These distinct species should thus be placed into separate genera as lowered symmetry dictates. Similarly, the three species 7a -7c would also become chemically distinguishable and require distribution into separate genera as appropriate.
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As the formalism generates numerous species, there is a need for unambiguous and compact description/naming. Further, the abstract polytope representation of an atom-connectivity configuration is not particularly "human readable". The following compact descriptive system based upon the connectivity (using an adaptation of the kappa convention ) and PES character (local minimum LM, transition-state structure TS, etc.) of the species is suggested. For example, in Fig. species 6aa local minimum structure with the single bonder on site atom 16 -is LMκ16 and 7aa transition-state structure with the bonder bridging site atoms 16 and 17 -is TSκ 2 16,17. See Supplementary Information for more detailed descriptions.
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These symbols also indicate relational information. For example, LMκ16 and TSκ 2 have common elements (site atom-16) and one connection difference (site atom-17). We refer to such a single connectivity-configuration difference as "first order" with the R de c process connecting these two species described as a "first-order motion". Similarly, if the differences in atomconnectivity configuration involves two connections (one bonder's configuration changing by 2 connections, or two bonders changing by 1 connection each), this is referred to as "second order". These terms find application when referring to pathways on the corresponding PESs. Formal definition for these motions-order relationships in the Supplementary Information.
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We observe that a first-order relationship generally corresponds to a change of ±1 negative eigenvalues of the species' Hessian matrix where these values correspond to the associated bond stretches (e.g., LM ⇌ TS [6a ⇌ 7a], or TS ⇌ 2S [7a ⇌ 8a], see Fig. ). Similarly, a second-order relationship typically corresponds to ±2 negative bond-stretching Hessian eigenvalues (e.g., LM ⇌ 2S [6a ⇌ 8a], see Fig. ). This simple correspondence though is not followed for some pericyclic reactions, e.g., Diels-Alder reactions (Further details are given in the Discussion and Supplementary Information).
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The atom-connectivity configuration differences for a pair of species indicates whether they are "adjacent" on their reaction graph, that is, a graph edge exists between those species. For free-base subporphyrin monoanion, only first-order and secondorder relationships are possible for unimolecular (nondissociative) H-tautomerism. Fig. shows the combined fistorder and second-order motions graphs. This corresponds to the traditional chemical-structure depiction of the tautomerism shown in Fig. with a layout corresponding to Fig. . In Fig. is the corresponding DFT-calculated qualitative PES for the tautomeric H-atom using a real-space polar coordinate system. On this PES are shown the locations of the seven species discussed and the minimum-energy pathways (MEPs). Note that the graph from Fig. maps onto the realspace PES, species, and MEPs. The white dashed paths map to the first-order motions and the light-red dashed paths map to the second-order motionsthey are isomorphic.
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Inspection of the DFT-modelled species reveals that bondstretching vibrational modes (e.g., the single imaginary vibrational frequency for species 7a and two imaginary vibrational frequency for species 8a) distort their respective geometries along the MEPs and towards their reaction-graph adjacent species. A directed graph that encodes this information is shown in Fig. following the approach described for stereoisomerism. Self-loops are required to account for both displacement directions of a vibration and here describe the steep walls of the PES. By assigning energies to each reaction graph vertex and transformational information (energy gradients and curvatures, and vibrational modes) to the graph edges, the resulting mathematical object as interpreted through the lens of TST is a compact encoding of the PES (Fig. ). The two PESs shown in Fig. have been calculated following this approach using interpolation and are isomorphic with the real-space PES shown in Fig. . While the reaction graphs produced within the formalism are based on the discrete and abstract nature of atom connectivity and hence do not represent geometric coordinates, their isomorphism with real-space PESs demonstrates a direct relationship to the underlying geometric coordinates. This capacity to give structure to multidimensional TST becomes important and powerful as the complexity of the chemical system increases as is demonstrated in the following Sections.
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Outputs from this software and instructions are also provided. An additional chemically sensible restriction is to limit the maximum number of bonders on a single site atom 𝒮. In such cases we write this modification to the 𝒮 𝑚 ℬ 𝑛 class as 𝒮 𝑚(𝑘) ℬ 𝑛 where k is the maximum number of bonders at any single site atom.
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Starting with the intrinsic form of each atom-connectivity species (see Fig. and Fig. ), it is possible to generate a simple atomic representation of the sites and bonders. This shift from atom connectivityan abstract conceptto a plausible molecular geometry in 3D space means that steric considerations arise. Consequently, more than one geometry for each atomconnectivity species may be possible as now isomers are defined by both atom connectivity and local and global stereoisomeric atom connectivityan abstract conceptto a plausible molecular geometry in 3D space means that steric considerations arise. Consequently, more than one geometry for each atomconnectivity species may be possible as now isomers are defined by both atom connectivity and local and global stereoisomeric details. As each aspect presents a finite number of possible structural variations, the total number of conceivable chemical structures remains countable. All structures envisaged by this procedure may not be chemically feasible, however, owing to steric demands of linking the bonding sites (and possibly the bonders, see Fig. ). If a realistic complete chemical model can be constructed for a species, then computational schemes can be employed to convert this into a 3D chemical structure. As a first step, we recommend the use molecular mechanics exploiting its explicit definitions for atom connectivity. While this process is manually feasible for smaller systems, the real power will come through automation.
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In order to provide scope for discussing the more nuanced details of the formalism, we examine two chemical systems (neutral free-base subporphyrin and H2[MO3], where M can be many elements) that are equally described as 𝒮 3 ℬ 2 , and triphyrin[2.2.2] which is partitioned as 𝒮 3 ℬ 3 where the number of bonders that can connect to any site is limited to 2, which we write as 𝒮 3(2) ℬ 3 .
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𝓢 𝟑 𝓑 𝟐 systems Building on our earlier example of tautomerism in free-base subporphyrin monoanion, an additional bonder (H + ) gives neutral free-base subporphyrin partitioned as 𝒮 3 ℬ 2 as shown in Fig. . Here, there are two mobile bonders (both H) free to move inside the inner cavity of the macrocycle. The second chemical system, also shown in Fig. , corresponds to the general formula H2[MO3] where the two mobile bonders (both H) are free to move around the periphery. The central atom M could be one of many chemical elements, for example, carbon.
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While looking very different, these two examples demonstrate that the 𝒮 𝑚 ℬ 𝑛 partitioning approach can be broadly applied whether it be a significantly larger molecule (as in the subporphyrin) or relatively small (H2[MO3]), or pertain to atom-connectivity permutations inside a molecule (the subporphyrin) or on the outside (H2[MO3]). The files "genera_D3h_S3B2.txt" and "species_D3h_S3B2.txt" in the Supplementary Information provide the raw data to generate the generic and specific structures, respectively, shown in Fig. .
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Although there has been the addition of only a single bonder, the scope of atom-connectivity configurations has grown markedly. Inspection of Table , Table , and Table indicate that for the 𝒮 3 ℬ 2 class, there are 6 families with, due to D3h site symmetry, the 49 resultant species are collected into 9 genera. Distinctions between genera within a single family are based upon the combinations of bonder-site connection multiplicities. For example, in F1, genus 9 has the bonders connected to the same site whereas genus 10 has the bonders connected to different sites. In each case, both bonders are non-bridging,
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For each, all Ng = 9 possible genera are shown as 2D chemical structures. Genera are color-coded into Nf = 6 families F1 through to F6, depicting different permuted connection totals and bonder linking patterns as described in the legend. Compact labels are given for a representative species of each free-base subporphyrin genus. Stereochemistries are only demonstrative.
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Graphical outputs from the accompanying graphing software as applied to genera 9 -14 for the 𝒮 3 ℬ 2 class are shown in Fig. . For the purposes of demonstration and reduced complexity, genera 15 -17 which feature at least one bonder connected to three sites are omitted from the analysis
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The combined first-order and second-order motions graph for genera 9 -14 are shown in Fig. with the graph edges coloured as first order: blue; and second order: red. Note, that the topology of this combined graph is that of a 2-torus. Careful inspection of this graph shows that it is composed of six sets of rings ("cycles") that are each composed of six species. The firstorder motions and second-order motions graphs are shown in Fig. and Fig. , respectively. The latter two graphs have the topology of a 2-torus as, by the modular structure theorem, each are subgraphs of the combined graph in Fig. . This second-order motions graph (Fig. ) forms a disconnected graph which indicates that second-order motions cannot interconvert species between the separate subgraph components Fig. Toroidal desmotropic reaction graphs of genera 9 -14 for the 𝒮 3 ℬ 2 class. All graph vertices are labeled by the bonder connectivity (omitting the κ i symbols) and edges are coloured by motions type (blue: first order; red: second order). a) Combined first-and second-order motions graph laid out to show the toroidal form. First-order and second-order motions indicated by blue and red graph edges, respectively. b) First-order motions graph. c) Second-order motions graph. This is a disconnected graph. Fig. Interpolated approximation of the DFT-calculated potential-energy surface for tautomerism in free-base subporphyrin partitioned as 𝓢 𝟑 𝓑 𝟐 using genera 9 -14. Surface generated from calculated energies for representative structures for each genus. a) 2D-periodic rendering of the 2-toroidal PES. All species located on the surface are labeled by their species number (see Supplementary Information for details). The piecewise reaction coordinates of each bonder are periodic with replicate configuration genera shown in gray (top row and right edge). Desmotropic reactions are indicated as first order (solid black lines) and two second order networks (red dashed and red dotted lines). b) 3D rendering of the PES. c) The six genera with representative structures and their compact symbols indicating PES character. Stereochemistries are only demonstrative.
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The PES for the genera 9 -14 configuration space can be approximated analogously with free-base subporphyrin monoanion (Fig. ). Given the topologies of the graphs for genera 9 -14 are that of a 2-torus composed of 6 × 6 species, it follows that two periodic reaction coordinates (composed of first-order edges) span the configuration space. Using the graph information and only six DFT-calculated energies, the resulting PES is generated and given in Fig. . In Fig. is the PES given as a 6 × 6 2D periodic grid. The first order motions (solid black lines) correspond the to the graph in Fig. and form a regular rectilinear grid. The diagonal red lines are the secondorder motions and comprise two independent and interpenetrating networks (solid and dotted). These correspond to the two parts of the disconnected graph in Fig. using dashing as indicated in Fig. . A 3D rendering of the surface in Fig. is shown in Fig. . The Supplementary Information lists all species configurations and energy values corresponding to PES in Fig. . Note that the PES characters for genus 9 are that of an inflection point rather than local minimum due to stereoisomeric effects.
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Other examples of the D3h symmetric 𝒮 3 ℬ 2 class include systems such as free-base triphyrin[2.2.2] monoanion (e.g., 28 less one inner H-atom). The analogous approximated PES for this can be found in the Supplementary Information. This expanded macrocycle admits larger atoms in place of the inner H-atoms and modulation of the corresponding PES (see Discussion).
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Free-base triphyrin[2.2.2] which features three bonders and three sites corresponds to the 𝒮 3 ℬ 3 class. Inspection of Table , Table , and Table indicate that there are 10 families, 23 genera, and 343 species. This includes species where all three bonders are attached to a single site atom (N-atom). Based on valence and steric arguments, we exclude such species from the analysis.
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With the maximum number of bonders for a site limited to 2, the class is written as 𝒮 3(2) ℬ 3 . For the 𝒮 3(2) ℬ 3 class there are 6 families, 10 genera, and 174 species. The corresponding genera and families shown in Fig. . Following the pattern for freebase subporphyrin monoanion (see Fig. ) and free-base subporphyrin (Figs. and), the graphs for genera 27 -34 has the topology of a 3-torusthough with missing graph vertices arising from the criterion of excluding three bonders on a single site. Electronic files describing the 𝒮 3(2) ℬ 3 partitioned free-base triphyrin[2.2.2] system are provided in the Supplementary Information.
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A more pronounced difference arises when there are extensive differences between systems, for example, as shown Fig. where free-base subporphyrin and H2[MO3] are demonstrated to form analogous sets of constitutional isomeric entities when analysed from a 𝒮 3 ℬ 2 partitioned perspective. Though these chemical systems are dramatically different, those parts of chemical space describing each of their 𝒮 3 ℬ 2 -partitioned constitutional changes have identical topologies; the shape of their respective chemical spaces are the same despite the idiosyncratic differences in energetics.
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It is the Polytope Formalism's unbiased treatment atomconnectivity permuted species that allows it to comprehensively describe isomeric possibilities and interrelationships regardless of "stability"; differences in the chemical elements involved can radically change the energetic landscape yet the topology remains the same. With previous approaches for describing constitutional isomerism only focusing on the low energy regions or just the local minima, each chemical scenario becomes idiosyncratic. Given our abstracted approach delivers more general configuration-space shape and structure results, it provides a complete picture of possibilities.
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A simple example demonstrating this is shown in Fig. using free-base subporphyrin monoanion. Species 8a represents a 2S structure and has the highest energy on its PES. Previous methods looking at isomers would not consider such a species. Replacing the H-atom bonder with BF inverts the energetic nature of its PES such that species 37athe BF-analogue of 8a represent a LM structure. Despite the contrasting energetics of each PES, their graph topologies (white lines) are identical. Consequently, the atom-connection configurations of the formalism can be thought of as "structural motifs". 12 | J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xx With this structural-motif principle in mind, the application of 𝒮 4 ℬ 𝑛 partitioning to the more extensively known chemistry of porphyrins and related macrocycle systems yields new insights and will be presented in a subsequent paper. Systems of five or more 𝒮-sites such as sapphyrins and texaphyrins (𝒮 5 ℬ 3 ), smaragdyrins (𝒮 5 ℬ 4 ), rosarins (𝒮 6 ℬ 3 ) and higher homologs will also be dealt with elsewhere.
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Nested spaces. Perhaps the most important point involves the implications of the modular structure theorem. The topologies that emerge from the formalism are contain simpler topologies but are themselves nested within more complex topologies. The "shape of chemical space" that emerges is made up of common forms that map across and through its grand structure. Perhaps should not be a surprise as it offers an explanation for why, in chemical synthesis, a given set of reaction conditions acting upon vastly different substrates will generally lead to predictable products; why reactions can be classified into general types; why we see the same geometric forms again and again.
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"Entropy" of a configuration space and connection to thermodynamic entropy. Within Combinatorics, Graph Theory, and Information Theory, the general concept of entropy, analogous to that used in thermodynamics, finds wide applications including the characterisation of graphs and structured sets. Given the Polytope Formalism generates mathematical objects of this type, it is natural to consider the application of entropic measures to the formalism.
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The key to this relationship involves using a directed graph encoding of the PES, for examples, as shown in Fig. where each directed graph edge has a vibration associated with it (e.g., 6a transforms from the displacements of vibrational normal modes ν6 and ν13). Assuming an ensemble, appropriate statistics ("weights") can be assigned to each graph vertex (species) and their contribution to the thermodynamic entropy calculated. Given the reaction graph and its PES describe a whole system of interconvertible species, the corresponding formulation of the entropy contribution is also reflective of the full system and not just a single species. It is important to note that these expressions do not reflect the full entropy as other vibrational modes describing, for example, stereoisomeric configurations of each species, will contribute to a degree reflective of their relative energy regimes.
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For concerted reactions where (at least) two atom connections form/break simultaneously such is the case for cycloadditions and chelotropic reactions, the list of atom-connectivity configuration species generated by the Polytope Formalism of constitutional isomerism does not differentiate between the cyclized species and the TS of its formation. An illustrative example is given in Fig. which shows the Diels-Alder reaction of [butadiene + ethene] 38a, to give cyclohexene 39a, via the TS 39aʹ. The corresponding atom-connectivity picture of these three species are as shown in Fig. . Both 39aʹ (TSκ1:κ4) and 39 (LMκ1:κ4) have the same atom-connectivities. A consequence of this is that the formalism under-samples the PES from the perspective of defining the important species involved. This is not a fatal limitation as knowing of it allows one to look for those additional species of the same configuration but different PES character. For example, when generating initial representative structures, the bonds (atom connections) in question can be constrained by either bond order or an appropriate bond length.
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For a full listing of the atom-connectivity species examples arising from application of the formalism for this system (Fig. ), including some density-functional theory (DFT) geometries, see the Supplementary Information. Integrability of the formalism with cheminformatic tools. In chemical databases (e.g., the CAS database) that provide structure and substructure search features, the structures are stored and processed as bonding matrices, that is, they are digitally represented as atom-connectivity lists augmented by bond multiplicities and stereodescriptors. Also, systematic chemical names are automatically generated from the same digital representation. The atom-connectivity permuted structures generated by the Polytope Formalism of constitutional isomerism (or its focused 𝒮 𝑚 ℬ 𝑛 partitioned approach) provide descriptions of "structures" that are compatible with these databases and functionality. Consequently, many existing cheminformatics tools are expected to be readily adaptable to the implementation of the principles and concepts described here.
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Whereas prior approaches to constitutional isomerism focused on the number of isomers for a given set of atoms, we asked the broader question: how do the isomers interconvert? To this end, we adapted the principles of the Polytope Formalism (as applied to stereoisomerism) to constitutional isomerism generating mathematically comprehensive structured lists of atomconnectivity configuration species (Fig. ). As the approach explicitly includes subvalent and hypervalent configurations, it generates species corresponding to isomerization intermediates. The configuration lists are highly structured and form discrete configuration spaces that encode the inter-configuration relationships. The configuration spaces map onto PESs and hence the graph representation of the configuration spaces provide compact descriptions of the associated PESs. Requisite terminology and a taxonomic system reflecting this configuration space structure was implemented.
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We showed that while the general features of the Polytope Formalism of constitutional isomerism were simple to define, many resulting configurations would be physically impossible and the superexponential scaling would prove problematic. To address these issues, we introduced a focused approach called 𝒮 𝑚 ℬ 𝑛 partitioning that could be practically applied to many chemical problems of interest. Through several worked examples of H-tautomerism, we demonstrated the principles and power of the analysis and utility of the terminology. Graphs representing the interconfiguration relationships were utilized with these being equivalent to traditional chemical reaction networks. Further, these graphs also showed a one-to-one correspondence with the corresponding PESs (they are isomorphic).
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Looking more broadly, in demonstrating that the conceptual framework of the Polytope Formalism could be applied to the qualitatively distinct phenomena of stereoisomerism and constitutional isomerism, we have demonstrated a deeper connection between them (Fig. ). As such, this lends support to our assertion that all aspects of isomerism (e.g., nuclear spin isomerism) can be described using a single unifying framework. In addition to those aspects of chemical-entity classes going under the banner of isomerism, there are the various quantum mechanical states whose inherently discrete nature lend themselves to description given the permutational structure that is the engine of the Polytope Formalism. Examples of these include the electronic states and rovibrational states of molecules (Fig. ). The detailed treatment for these state-permutational molecular properties will be developed and reported elsewhere.
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Given this prospect of describing all salient features of molecular entities, the Polytope Formalism provides a single unifying descriptive framework for all chemical systems. As such, the formalism defines chemical space in a mathematically rigorous and meaningful way. It follows that this provides Chemistry with the exploration roadmap and an essential tool for machine-led chemical research.
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Examples include the large-scale wildfires in the Pacific Northwest of North America in 2017 and the Australian "black summer" fires in 2019 -2020. Smoke is also ubiquitous in the upper troposphere and is mixed into the lowermost stratosphere. Smoke from wildfires consists of mostly organic aerosol, called biomass-burning organic aerosol (BBOA). For example, smoke sampled from wildfires in the western U.S.A. comprised of > 90% BBOA by mass. Wildfires also emit large amounts of volatile organic compounds (VOCs). In the atmosphere, hydroxyl radicals (OH) and ozone (O3) oxidize VOCs, and some of the products of these reactions condense onto existing particles, adding to the BBOA mass. OH and O3 can also react directly with BBOA, forming a more oxidized aerosol. These combined processes result in aged BBOA (Fig. ). In the stratosphere, sulfuric acid (H2SO4) can also condense on to the aged BBOA to form aged BBOA-H2SO4 particles (Fig. ). The sources of H2SO4 in the stratosphere includes the oxidation of sulfur dioxide (SO2) from pyroCb events and volcanic eruptions and the oxidation of carbonyl sulfide (COS) from natural and industrial sources. The conversion of SO2 to H2SO4 occurs on the order of several weeks in the stratosphere, while the conversion of COS to H2SO4 occurs on the order of tens of years. BBOA can linger for many months in the stratosphere and contribute to the depletion of the UVblocking stratospheric ozone layer. Key reactions that may occur on BBOA in the stratosphere and contribute to stratospheric ozone depletion include ClONO2+HCl, ClONO2+H2O, and HOCl+HCl. These reactions on BBOA could delay ozone layer recovery for years to come, increasing ultraviolet radiation exposure at the Earth's surface and raising the risk of skin cancer. Additionally, BBOA introduced into the stratosphere by pyroCb events may influence the formation of polar stratospheric clouds (PSCs), potentially causing further depletion of the stratospheric ozone layer. Two important properties of BBOA are the viscosity and phase state (i.e., liquid, semisolid, or amorphous solid). These properties are closely related: liquids have a viscosity < 10 2 Pa s, semisolids range from 10 2 -10 12 Pa s, and amorphous ("glassy") solids exhibit a viscosity > 10 Pa s. Knowledge of the viscosity and phase state of BBOA particles is needed to predict the rates of multiphase reactions, including the multiphase reactions mentioned above that can contribute to ozone depletion. For example, the rate for a multiphase reaction often depends on the square root of the diffusion coefficient within the aerosol particle. Since diffusion is related to 1/viscosity, this implies that the rate often depends on the square root of 1/viscosity.
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Consequently, high viscosities and the glassy state could inhibit multiphase reactions within the particle bulk, restricting multiphase chemistry to the particle surface. Knowledge of the viscosity and phase state of BBOA is also needed to predict the formation pathways of PSCs (Fig. ). If BBOA particles are in a non-glassy state, they can take up H2O and HNO3 at low temperatures to form non-glassy PSCs (pathway a). In addition, Ansmann et al. speculated that BBOA particles in a glassy state could heterogeneously nucleate crystalline ice or crystalline nitric acid hydrate (NAT) (pathways c and d). We also speculate that glassy BBOA may take up H2O and HNO3 to form non-glassy PSCs due to the plasticizing effect of small molecules like H2O and HNO3 (pathway b). Researchers previously measured or estimated the viscosity and phase state of BBOA as a function of temperature and relative humidity (RH). They used these results to predict the behavior of BBOA in the troposphere (altitude ≲ 15 km). However, the temperature and RH in the stratosphere differ substantially from those in the troposphere, and these differences can strongly impact viscosity and phase state (as discussed below). Additionally, no studies have quantified the effect of H2SO4 uptake, or the uptake of any inorganic species, on BBOA viscosity and phase state. In a recent modelling study of stratospheric chemistry, researchers assumed that BBOA-H2SO4 particles in the stratosphere exist in a non-glassy state, allowing reactions to occur in the bulk of the particles. However, this assumption has not been verified with experiments or calculations. As a result, the viscosity and phase state of BBOA under stratospheric conditions remain highly uncertain. Several previous studies have quantified the viscosity and phase state of secondary organic aerosol (SOA) in the troposphere. These studies show that SOA, when free of inorganic species, can exist in a glassy state in the upper troposphere. However, these studies did not consider the temperatures and RH in the stratosphere, and most did not consider the uptake of inorganic species, such as H2SO4. In the following, we use laboratory data and a thermodynamic model to predict the viscosity and phase state of BBOA in the stratosphere from pyroCb events. We focus on altitudes from 15 -32 km in the stratosphere, since wildfire plumes from pyroCb events have been observed at these altitudes. We first estimate the viscosity and phase state for unaged BBOA at these heights. Next, we estimate the viscosity and phase state of aged BBOA particles mixed with H2SO4 (aged BBOA-H2SO4 particles) at these heights. The implications of the results for stratospheric chemistry and PSCs are discussed, as well as observational constraints of the viscosity and phase state of BBOA in the stratosphere. We also provide an outlook for future research on the viscosity and phase state of BBOA in the stratosphere.
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2. Methods. 2.1 Temperature in the stratosphere relevant for BBOA from pyroCb events. The viscosity of organic aerosols depend strongly on temperature. Simulated zonal mean monthly temperatures for January and July were obtained from the Whole Atmosphere Community Climate Model (WACCM) for the year 2000 (see Fig. ). WACCM has 72 vertical levels from surface to about 140 km above surface level and a horizontal resolution of 1.9° latitude by 2.5° longitude. For the altitude range of 15 -32 km , the stratospheric temperatures vary from 177 -250 K (see Fig. ). As a result, we will focus on this temperature range throughout the remainder of this article.
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(3) P*w was calculated using the following equation from Murphy and Koop: 34 To align with these observations, we focus on CH2O values of 15 ppmv in our analysis. 277.0 K c 0.1 g Lodgepole pine BBOA generated from flaming DeRieux et al. 266.0 K d 0.1 g a Glass transition temperature determined from volatility measurements.
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The glass transition temperature of dry BBOA (Tg,BBOA) and the volume-based hygroscopicity parameter of the BBOA (κ) have been reported for several types of unaged BBOA (Table ). The variability in Tg,BBOA and κ is likely due to differences in experimental conditions used to generate and sample the BBOA, such as the fuel type, combustion conditions, and dilution. Here we calculated the viscosity of unaged BBOA using each of the Tg,BBOA and κ pairs shown in Table to determine the possible range of viscosities and phase states of unaged BBOA in the stratosphere.
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The viscosity was calculated from Tg,BBOA and κ using the same approach used by Li et al. to calculate viscosity of SOA in the troposphere. First, we calculated the water content of the BBOA at a given RH using κ and the following equations: 26 , (5) and , (6) where wBBOA and wH2O represent the mass fraction of BBOA and H2O in a BBOA-water mixture, respectively. In Equation , represents the density of water, and represents the density of BBOA. For , we used 1 g/cm , and for , we used 1.3 g/cm , consistent with previous assumptions and measurements. Subsequently, the glass transition temperature of the BBOA and water mixture (Tg,mix) was calculated using the Gordon-Taylor equation and the glass transition temperature of dry BBOA (Tg,BBOA): 18 (7) where represents the Gordon-Taylor constant, suggested to be 2.5, and Tg,w is the glass transition temperature of water, set to 136 K. Viscosity at a given temperature (T) was then calculated using the Vogel-Fulcher-Tammann (VFT) equation: is the viscosity at infinite temperature (set to 10 -5 Pa s), D is the fragility parameter, assumed to be 10 as done previously, and T0 is the Vogel temperature calculated using the following equation:
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To calculate the viscosity of aged BBOA-H2SO4 particles at a given RH and temperature, we first determined the viscosity of aged BBOA particles and H2SO4 particles separately at the given RH and temperature, as outlined below. To determine the viscosity of aged BBOA particles at a given RH and temperature, we followed the same method described in Section 2.3. The viscosities determined in Section 2.3 correspond to unaged BBOA. We assume that these viscosities represent lower limits for aged BBOA due to the following factors: a) aging through OH and O3 reactions has been shown to increase BBOA viscosity at room temperature and on the timescale of days, b) UV exposure at room temperature also increases BBOA viscosity on the timescale of days, and c) dilution also increases BBOA viscosity. Nevertheless, additional measurements are needed to confirm this assumption for stratospheric temperatures, relative humidities, and residence times. We hope our work will motivate further studies on this topic. In addition, it is important to keep in mind that unaged BBOA in the stratosphere will often be in a glassy state (see below), and hence aging reactions with OH and O3 should be limited for initial conditions in the stratosphere. To determine the viscosity of H2SO4 particles at a given RH and temperature, we used the AIOMFAC-VISC thermodynamic model. This model applies a semi-empirical approach based on the Eyring's absolute rate theory to describe the viscosity of aqueous electrolyte solutions. The model reproduces viscosity data often within 10% for a wide range of aqueous electrolyte systems. After determining the viscosity of the aged BBOA particles and H2SO4 particles separately at a given RH and temperature, we used the Zdanovskii-Stokes-Robinson (ZSR) approach to calculate the viscosity of aged BBOA-H2SO4 particles at the given RH and temperature. The ZSR approach is expressed mathematically as the following: , (10) Where , , and represents the viscosity of the BBOA-H2SO4 particles, BBOA particles, and H2SO4 particles, respectively, all at the same RH and temperature.
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is the unit viscosity (1 Pa s); and f1 and f2 are the mass fractions of the two subsystems at the given RH and temperature. Studies by Song et al. and Klein et al. show that the ZSR approach predicts viscosities within 1 -2 orders of magnitude for mixtures of organic and inorganic material. To predict the viscosity of the aged BBOA-H2SO4 particles using Equation 10, the H2SO4-to-BBOA (H2SO4:BBOA) mass ratio is needed.An upper limit to the H2SO4:BBOA mass ratio for pyroCb smoke in the stratosphere can be estimated based on limited measurements. In 2017, the NASA Atmospheric Tomography (ATom) mission sampled pryoCb smoke in the northern hemisphere and lowest part of the stratosphere from the large-scale wildfires in the Pacific Northwest of North America. Single particle measurements from this mission indicate that the average H2SO4:BBOA mass ratio was ~0.37 and ~0.79 for 2 months and 9 months, respectively, after the initial pryoCb injection into the stratosphere (Section S1 and Fig. ). The H2SO4:BBOA mass ratio was even smaller for the largest particles (≳ 0.5 µm) (Fig. ). Based on the size distributions of the typical smoke in the stratosphere and smoke from pryoCb events, the largest particles were likely mostly from pyroCb events, whereas the smaller particles were likely aged smoke that is widespread in the upper troposphere and lower stratosphere. Based on these limited measurements, we suggest that H2SO4:BBOA mass ratios of 0.37 and 0.79 are likely an upper limit for pyroCb smoke in the lower stratosphere with an aging time of 2 months and 9 months, respectively. To be consistent with the ATom measurements, in the following, we consider H2SO4:BBOA mass ratios of 0.37, 0.5, and 0.79.
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3. Results and discussion: 3.1 RH in the stratosphere Before discussing the viscosity of BBOA in the stratosphere, we first considered the RH in the stratosphere since the RH strongly impacts particle viscosity. As the RH decreases, the water activity (and hence water content) of BBOA particles decreases to maintain equilibrium with the gas phase and the viscosity increases due to the plasticizing effect of water. We calculated the RH as a function of temperature in the stratosphere at altitudes of 15 and 32 km and a water vapor mixing ratio of 15 ppmv. An altitude range of 15 -32 km corresponds to the range of altitudes where wildfire plumes have been observed in the stratosphere. A water vapor mixing ratio of 15 ppmv is consistent with the water vapour mixing ratio expected in wildfire plumes that have reached the stratosphere via pryoCb events. The results of the calculations are shown in Figure . The horizontal bar illustrates the temperature range of 177 -250 K, which is most relevant for the BBOA in the stratosphere from pyroCb events at altitudes of 15 -32 km (see Methods). Our calculations indicate that the RH reaches extremely low levels (≲ 1% RH) under a wide range of conditions relevant to BBOA in the stratosphere. In contrast, in the troposphere, the RH is almost always greater than 10%. The low RH levels relevant for BBOA in the stratosphere, combined with the cold stratospheric temperatures, have significant implications for the viscosity of BBOA in this region as discussed below.
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Viscosities extended beyond 10 20 Pa s, however the y-axis was capped at this value. Shown in Figure are the calculated viscosities for the 10 unaged BBOA types listed in Table . Similar to Figure , we carried out calculations for the stratosphere at altitudes of 15 and 32 km and a water vapour mixing ratio of 15 ppmv. For the temperature range of 200 -250 K, all the BBOA types have viscosities of approximately 10 7 Pa s or greater (Fig. ). These extremely high viscosities are due to a combination of low RH values and low temperatures (Fig ). Furthermore, 8 out of the 10 BBOA types exhibit viscosities exceeding 10 12 Pa s for the entire temperature range of 200 -250 K, indicating that they are predicted to remain in a glassy solid state throughout this range. For the 2 proxies studied by Gregson et al. a glassy solid state was observed over a narrower range of temperatures (200 -230 K for the hydrophobic phase and 200 -240 K for hydrophilic phase), but the glassy state still dominated between 200 -250 K. Overall, these results suggest that the glass state dominates for unaged BBOA in the stratosphere for the 200 -250 K temperature range. At temperatures ≲ 200 K all of the BBOA types switch from a glassy solid state to a semisolid or liquid state (Fig. ) due to the heightened RH in this low-temperature regime (Figure ). At an altitude of 15 km, the transition from a glassy state to a semisolid/liquid state occurs at about 195 -200 K. At an altitude of 32 km, the transition occurs at about 180 -185 K.
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3.3 Viscosity of aged BBOA-H2SO4 particles Figure also includes calculated viscosities for aged BBOA-H2SO4 particles with with H2SO4:BBOA mass ratios of 0.37, 0.5, and 0.79. Measurements from the ATom mission suggest that H2SO4:BBOA mass ratios of 0.37 and 0.79 are likely upper limit for pyroCb smoke in the lower stratosphere with aging times of 2 months and 9 months, respectively (see Methods). The H2SO4:BBOA mass ratio of 0.5 is provided as an intermediate value between the ratios of 0.37 and 0.79. To calculate the viscosities of aged BBOA-H2SO4 particles, we used the viscosities of unaged BBOA as proxies for aged BBOA. Consequently, the calculated viscosities should be considered as lower limits to the viscosities of aged BBOA-H2SO4 particles for several reasons that are outlined in the Methods section. For H2SO4:BBOA = 0.37 (Fig. ) at temperatures between 200 -250 K, the viscosity of the aged BBOA-H2SO4 particles is approximately 3 orders of magnitude lower, or more, compared to the unaged case (compare Fig. and). This decrease is attributed to the low viscosity of the H2SO4 and its associated water compared to BBOA at the same RH and temperature. Despite the reduction in viscosity, for the temperature range of 200 -250 K, all the BBOA types have viscosities of approximately 10 4 Pa s or greater. In addition, 6 out of the 10 systems still exhibited viscosities greater than 10 12 Pa s for the full temperature range of 200 -250 K. For the remaining 4 systems, the particles still reached 10 12 Pa s, although over a narrower temperature range. Overall, these results suggest that aged BBOA with a H2SO4:BBOA mass ratio of 0.37 will have a viscosity of approximately 10 4 Pa or greater and will often be in a glassy state for the temperature range of 200 -250K. For H2SO4:BBOA = 0.37 (Fig. ) at temperatures ≲ 200 K, all of the aged BBOA-H2SO4 types transition from a glassy solid state to a semisolid or liquid state due to the increased RH in this low-temperature regime. At an altitude of 15 km, the transition from a glassy state to a semisolid/liquid state occurs at about 195 -200 K, while at an altitude of 32 km, the transition occurs at about 180 -185 K. For H2SO4:BBOA = 0.79 (Fig. ) at temperatures between 200 and 250 K, the viscosity decreases by 1 order of magnitude, or more, compared to H2SO4:BBOA = 0.37. Again, this reduction is attributed to the low viscosity of the H2SO4 and its associated RH compared to BBOA at the same RH and temperature. For the temperature range of 200 -250 K, all the BBOA types have viscosities of approximately 10 3 Pa s or greater. In addition, 3 out of the 10 systems exhibited viscosities greater than 10 12 Pa s for the full temperature range of 200 -250 K. For the remaining 7 systems, the particles still reach 10 12 Pa s, although over a more narrow temperature range Overall, these results suggest that aged BBOA with a H2SO4:BBOA mass ratio 0.79 will have viscosities of approximately 10 3 Pa s or greater for the temperature range of 200 to 250 K. In addition, the glass state will form in these aged BBOA for some stratospheric conditions.
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3.4 Sensitivity analysis In the analysis above, we used a CH2O value of 15 ppmv, consistent with measurements in plumes that recently entered the stratosphere via pyroCb events. Over time, CH2O levels decrease due to dilution, eventually reaching background values of ~5 ppmv. To evaluate how a reduction in CH2O affects viscosity and phase state, we recalculated viscosities using a CH2O value of 5 ppmv (Figure ). The lower CH2O level increases viscosity. For example, for the hydrophobic phase studied by Gregson et al., H2SO4:BBOA =0.79, CH2O = 15 ppmv, and an altitude of 15 km, the highest observed viscosity was ∼10 12 Pa s. In contrast, under the same conditions but with CH2O = 5 ppmv, the highest observed viscosity was ∼10 Pa s. To calculate the viscosities of aged BBOA-H₂SO₄ particles, we used the ZSR method. Previous studies have demonstrated that the ZSR method can estimate the viscosity of mixed organic and inorganic particles within 1 -2 orders of magnitude. Figure presents the predicted viscosities of aged BBOA-H₂SO₄ particles (CH2O = 15 ppmv and H₂SO₄:BBOA ratio = 0.37, 0.5, and 0.79) with all of the predicted viscosities reduced by two orders of magnitude. Even after this reduction, the aged BBOA with a H2SO4:BBOA mass ratio of 0.37 are still predicted to be in a glassy state often for the 200 -250 K temperature range. In addition, the aged BBOA with a H2SO4:BBOA mass ratio of 0.79 will still form the glass state for some stratospheric conditions and BBOA types. To calculate the viscosity of BBOA particles from their glass transition temperatures, we assumed a fragility parameter (D) of 10, consistent with values used in previous studies. For individual organic molecules, D values typically range from 5 to 20. To assess the impact of this range, we recalculated viscosities using D values of 5 and 20, with a water vapour mixing ratios of 15 ppmv and H₂SO₄:BBOA mass ratios of 0, 0.37, 0.5, and 0.79 (Figure and). The largest reductions in viscosity were observed for a D value of 20. Nevertheless, even for D = 20, the aged BBOA with a H2SO4:BBOA mass ratio of 0.37 are still predicted to be in a glassy state often for the 200 -250 K temperature range. In addition, the aged BBOA with a H2SO4:BBOA mass ratio of 0.79 are still predicted to form the glass state for some stratospheric conditions and BBOA types.
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Recent studies have suggested that reactions occur on or in BBOA in the stratosphere, leading to unexpected depletion of the UV-blocking stratospheric ozone layer. In modelling studies, researchers have assumed that the BBOA-H2SO4 particles in the stratosphere are in a non-glassy state and that reactions occur in the bulk of the BBOA-H2SO4 particles. These assumptions have generally aligned with model-measurement comparisons. Our results suggest that wildfire smoke particles will have viscosities ≳ 10 3 Pa s, when the H2SO4:BBOA mass ratio is ≲ 0.79 and the temperature is 200 -250 K. These high viscosities can influence the characteristic reaction diffusive length scale and reactive uptake rates. For instance, reactive uptake rates often scale with the square root of 1/viscosity, meaning a two-order-of-magnitude increase in viscosity can reduce the reactive uptake rate by one order of magnitude. Consequently, these elevated viscosities may need to be considered to accurately describe the stratospheric chemistry of wildfire smoke particles. Our results also suggest that wildfire smoke particles can be in a glassy state (viscosity ≥10 Pa s) under certain temperatures and RHs in the stratosphere when the H2SO4:BBOA mass ratio is ≲ 0.79. In such a state, bulk reactions will be inhibited, and multiphase chemistry may be restricted to the surface of the particles. Therefore, surface reactions may need to be considered for some conditions in the stratosphere. As mentioned above, BBOA may influence the formation mechanisms of PSCs via several pathways (Fig. ). In the current study, we address the possibility of pathway c in Figure . In Figure , we show the RH and temperature conditions where vapor becomes supersaturated with respect to ice, indicated by vertical black lines. The solid black line corresponds to conditions at an altitude of 15 km, while the dashed black line corresponds to an altitude of 32 km. If viscosity is ≥ 10 12 Pa s to the left of the vertical black lines (i.e., colder temperatures) at the respective altitudes, BBOA can exist in a glassy state while the vapor is supersaturated with respect to ice. This glassy state may provide a surface for heterogeneous ice nucleation, leading to the formation of ice-containing PSCs (pathway c, Fig. ). For H2SO4:BBOA ratios of 0, 0.37, 0.5, and 0.79 at altitudes of 15 km and 32 km (Fig. ), the viscosity is ≥ 10 12 Pa s to the left of the corresponding vertical black lines. This indicates that under certain stratospheric conditions, the particles are in a glassy state, the vapor is supersaturated with respect to ice, and the particles can act as potential surfaces for heterogeneous ice nucleation, possibly leading to the formation of ice-containing PSCs (pathway c, Fig. ).
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Lidar observations indicate that fresh smoke particles in the stratosphere exhibit a high depolarization ratio, suggesting the presence of non-spherical particles. This elevated depolarization could signal that BBOA is in a glassy or highly viscous state in the stratosphere. In newly emitted smoke plumes, BBOA particles can coagulate into fractal aggregates. If BBOA is in a glassy or highly viscous state, the material will not flow, maintaining its fractal structure after coagulation, which would contribute to a high depolarization ratio. Additionally, black carbon particles from combustion sources initially have a fractal geometry. When coated with a low-viscosity material, the fractal geometry can become more compact and spherical, with a low depolarization ratio. However, if black carbon is coated with BBOA in a glassy or highly viscous state, black carbon can maintain its fractal geometry, and the depolarization ratio can stay high. Nevertheless, interpretation of lidar depolarization is complicated because depolarization depends on particle size as well as the degree of nonsphericity. Lidar data also show that the depolarization ratio of smoke in the stratosphere decreases over a period of 3 -4 months, eventually approaching near-zero values (< 0.05). This trend may indicate that BBOA transition from a highly viscous state to a less viscous one. A decrease in viscosity would allow fractal aggregates of BBOA to transform into spherical particles, which typically exhibit a near-zero depolarization ratio. Similarly, a reduction in BBOA viscosity would enable fractal black carbon aggregates coated with BBOA to compact into a core-shell structure, also associated with a low depolarization ratio. However, it is important to note that glassy or highly viscous BBOA, such as tar balls, that have not coagulated into fractal aggregates in the atmosphere will possess a spherical geometry and exhibit a low depolarization ratio. Therefore, a low depolarization ratio does not definitively indicate that BBOA particles are in a non-glassy and non-viscous state.
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Baars et al. pointed out that the decrease in depolarization ratio of smoke in the stratosphere with time is consistent with the aging of smoke particles and the addition of a coating around the solid black carbon core aggregates, which would change the shape towards a spherical form. We add here that the coating would also need to be in a non-glassy and lower viscosity state, which may be accomplished by the uptake of H2SO4 present in the stratosphere, as shown above.
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We assumed that BBOA forms a single phase when mixed with H2SO4. Previous studies have shown that when secondary organic aerosol with an average O/C ratio ≳ 0.8 is mixed with ammonium sulfate aerosols, the resulting particles form a single phase. Studies are needed to confirm if a similar threshold applies for BBOA mixed with H2SO4. 3) To calculate the viscosity of BBOA particles from their glass transition temperatures, we assumed a fragility parameter (D) of 10, consistent with values used in previous studies. Measurements are needed to confirm that a fragility of 10 is appropriate for BBOA. 4) More measurements of the H2SO4 content in wildfire aerosols as a function of age in the stratosphere are needed. These measurements could be carried out by high-altitude aircraft or by using infrared spectra of wildfire smoke in the stratosphere from satellite measurements. 13 5) Observational constraints on the viscosity and phase state of BBOA in the stratosphere are essential. Lidar depolarization measurements of wildfire smoke in the stratosphere offer valuable insights into these properties, but additional measurements are needed. Lidar depolarization measurements indicate that wildfire smoke in the stratosphere is often nonspherical, but depolarization is an indirect measurement that does not uniquely determine phase and viscosity. 6) Measurements should also differentiate between ordinary smoke particles slowly lofted into the stratosphere and smoke rapidly injected into the stratosphere through large pyroCb events. PyroCb smoke is processed through a cloud during its ascent. This could result in a different mix of organic molecules than smoke that is not cloud-processed. 7) Our results suggest that aged BBOA and aged BBOA mixed with H2SO4 will be in a glassy state for some conditions in the stratosphere. Previous studies have shown that certain types of organic aerosols in a glassy state can nucleate ice heterogeneously. However, the ice nucleation efficiency of glassy organic aerosols is still an area of active research. Studies are needed to determine whether BBOA in a glassy state can nucleate crystalline PSCs. 8) We considered the uptake of water and H2SO4 by BBOA in the stratosphere. At the coldest temperatures, BBOA particles can also take up nitric acid (HNO3), similar to background H2SO4 particles in the stratosphere, which could further change the composition, and hence, the viscosity of the wildfire smoke particles. Smoke particles can also take up HCl and potentially other chlorine compounds. This uptake should be considered in future studies.
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A key challenge in small molecule drug discovery is to find novel chemical compounds with desirable properties. Computational methods have long been used to guide and accelerate the search through the huge chemical space of druglike molecules. In virtual screening, for instance, computational models can be utilized to rank virtual libraries of chemical structures regarding selected properties such as the predicted activity towards a target of interest . However, given the estimated vast amount of druglike molecules (10 23 -10 60 ) , a complete search through this space is computationally infeasible. An alternative approach is to computationally generate new molecules (de novo design) with optimized properties without the need for enumerating large virtual libraries. Heuristic methods such as genetic algorithms were used to optimize selected properties on-the-fly . However, due to the discrete nature of the chemical space, defining rules to transform one molecule into another (e.g. mutation and crossover rules for the genetic algorithms) largely depends on human expert knowledge. Moreover, defining a finite set of possible transformation rules limits the optimization possibilities and thereby promising molecules might be missed. With the recent rise of Deep Learning in the field of computational chemistry, new approaches for de novo drug design have emerged. Segler et al. trained a Recurrent Neural Network (RNN) to model a larger set of molecules represented by the Simplified Molecular Input Line Entry Specification (SMILES) notation . The resulting model was not only able to reproduce the molecules in the training set, but also to generate novel structures. By further training on a focused set of structures with a certain property distribution (e.g. the activity towards a biological target) the novel generated molecules could be enriched with structures following this desired property distribution. Another strategy for fine-tuning a generative model is Reinforcement Learning . Reinforcement Learning aims at learning the optimal set of actions to optimize a defined reward in a given environment. In the case of de novo design, the reward can e.g. be defined by the molecular properties to be optimized. this concept to alter the generative process of a pre-trained RNN, in order to generate more molecules with desirable properties . Besides RNNs trained on the SMILES representation, other groups also utilized Generative Adversarial Neural Networks or other molecular representations such as the molecular graph . While these method differ in how they generate molecules, they all apply Reinforcement Learning to enrich the generated molecules with structures that have desirable properties. The main drawback of such methods is the need to retrain the generative model every time the reward function changes. This becomes impractical in a typical drug discovery project as the optimization criteria usually change over time. A method that decouples the generation of molecules from the optimization problem was originally proposed by Gomez-Bombarelli et al. . In their work, a variational autoencoder was trained on the SMILES notation of a large set of molecules. As a result, a new continuous vector representation of chemical structures was obtained. Points in this continuous space correspond to molecules in the discrete chemical space (as represented by the SMILES notation) and vice versa. Novel structures can be generated by sampling arbitrary points in the continuous space and then transforming them back to the SMILES notation. A molecular transformation can be achieved by a simple shift in the vector representation. Thus, optimizing chemical structures with respect to selected properties can be directly performed by optimizing a reward function in the continuous space. In their work, Gomez-Bombarelli et al. utilized Bayesian Optimization to find points in the space that correspond to molecules with a high drug-likeness and synthetic accessibility. More recently, Jin et al. also used Bayesian Optimization to optimize molecules generated by a variational autoencoder based on molecular graphs . Bayesian Optimization is a powerful method that has proven useful in the optimization of functions that are computationally expensive to evaluate as it needs a comparably low amount of function evaluations . However, its computational complexity increases exponentially with the number of dimensions of the optimization space . In the case of molecular optimization, though, function evaluations are relatively cheap (prediction of molecular properties) and the dimensionality of the search space (continuous molecular representation) relatively high. In this work, we propose the use of a more light weight heuristic optimization method termed Particle Swarm Optimization (PSO). Hartenfeller et al. already proposed in 2008 to apply PSO on a discrete chemical space represented by a large library of molecular building blocks and chemical reactions . Here, we apply PSO in our continuous chemical space reported previously . In three different experiments we show how our proposed method can be utilized to optimize molecules with respect to a single objective, under constraints with regard to chemical substructures and with respect to a multi-objective value function.
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Our approach can be used with any continuous representation of the chemical space, such as those found in . In this study we build upon the continuous molecular representation framework reported earlier . This molecular representation was learned using a Deep Neural Network to translate from one molecular string representation (e.g. SMILES) to another. In this way, the model learns the common "essence" between both syntactically different string notations, i.e. the molecule which both notations are representing. By introducing a bottleneck in the architecture of the neural network, the molecule is encoded in a compressed embedding, that can be utilized as latent representation of the chemical space. As the model is trained on a huge dataset of approximately 75 million chemical structures stemming from various sources, the resulting latent space represents a wide range of the chemical space that can be explored. In this earlier work, we also showed that the learned molecular representation can be utilized as powerful molecular descriptors for quantitative structure activity relationships (QSAR) models. Moreover, transformations in the latent space result, if decoded back, in smooth transformations in the discrete chemical space in regard of structural as well as molecular properties. The interested reader is directed to the original publication for technical details of our framework .
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Particle Swarm Optimization (PSO) is a stochastic optimization technique that mimics swarm intelligence to find an optimal point in a search space as defined by an objective function. The particle swarm consists of individual agents (particles) that explore the search space, utilizing information gained during their search and "communicating" with other particles in the swarm . This concept can be defined by a few simple equations. The N particles in the swarm are defined by their position x and velocity v. The potential surface of the search space can be evaluated by the objective function f . The movement of the i-th particle at iteration step k is influenced by the best point it has yet discovered
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where c1 and c2 are constants that weight the contribution of the particles individual experience versus the swarm experience. r 1 and r 2 are random numbers drawn from independent uniform distributions between 0 and 1. The so called inertia weight w is a constant that controls the momentum of the particle from the previous iteration.
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The search of the Particle Swarm is guided by the objective function that is defined to be maximized. In drug discovery, the optimized objective can be both complex, conflicting, ill-defined or evolving over time. For example, at the early stages of lead discovery, a higher emphasis is put on increasing biological activity and gaining structure-activity-relationship knowledge. A set of targets that should not be hit by the compounds can also be introduced at that stage. Later on, when the overall activity landscape is well understood by the team, the focus of the optimization evolves more towards pharmacokinetics-related properties (ADME) such as improving solubility, metabolic stability or cell permeability, etc. These different objectives can contradict themselves, for example increasing the solubility of a compound might lead to permeability problems. This makes a multi-parameter optimization notoriously challenging.
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• rewards for defined substructures (e.g. fixing a scaffold) or similarity to a certain compound These functions either work directly on the continous representation (e.g. QSAR models, similarity) or on the decoded SMILES representation utilizing the chemoinformatics library RDKit . In this study, we build two biological activity models for prediction of Epidermal Growth Factor Receptor (EGFR) and asparyl protease b-site APP cleaving enzyme-1 (BACE1) activity. These targets were arbitrarily choosen. Compounds with reported IC50 values were extracted from ChEMBL and preprocessed as described in . We encoded the molecules in the continuous representation and trained Support Vector Machine (SVM) regression models (as implemented in the Python library scikit-learn ) on predicting the IC50 values of the compounds. Moreover, we trained SVMs on solubility, metabolic stability and membrane permeability endpoints utilizing in-house data. SVM hyperparameters were optimized in a 5-fold cross-validation.
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In order to filter for unwanted substructures, we extracted known toxic substructures from a published collection by SureChEMBL . Moreover, to filter for possible unstable structures we created a list with extended-connectivity fingerprints (ECFP4) of substructures that occur in more than 5 individual compounds in ChEMBL. During the optimization, generated structures are penalized if they contain such known toxic substructures or have uncommon ECFP4 bits.
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In order to combine the scores of the different functions in a multi-objective setting, we follow the approach reported in and scale each function between 0 and 1 reflecting values of low to high desirability (see Supporting Information for more details). The scaled scores of each function are combined as the weighted average, where the weights correspond to priorities in different tasks. The resulting desirability score (dscore) is subsequently used as the objective function for the PSO algorithm.
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The final optimization model combines the parts mentioned above, i.e. the continuous molecular representation, the optimization algorithm and the objective functions. Either a query molecule is encoded in the continuous space or a random point is sampled. The PSO algorithm is initialized by generating a fixed amount of particles (in the order of 100) at this position with randomly drawn initial velocities. After the first position update, the objective function is evaluated for each individual particle of the swarm. The search is continued until a certain number of iterations or a break condition (e.g. desired value) is met. Since the PSO algorithm is a stochastic optimization technique, multiple restarts are performed.
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By combining our encoder-decoder framework, computational models to predict properties and/or biological activities of compounds, and an optimization algorithm, we optimize a query molecule with respect to the objective function resulting in a set of compounds with more desirable (predicted) properties. In the first part we show the optimization of molecules with regard to a single objective. Then, we further restrict the optimization by adding substructure constraints. Finally, we demonstrate that our framework can be utilized to perform multi-objective optimization of compounds.
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As a first proof-of-principle, we run experiments on the optimization of molecules with respect to single molecular properties. Similar to related works , we perform individual optimizations on the drug-likeness, the octanol-water partition coefficient logP as well as the synthetic accessibility of a molecule. We utilize the RDKit implementation of the Quantitative Estimate of Druglikeness (QED) score to evaluate the drug-likeness and the penalized logP score that combines the partition coefficient and the synthetic accessibility (SA) score of a compound. Moreover, we optimize compounds with respect to their predicted biological activity on EGFR and BACE1.
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For each optimization task, we run the PSO algorithm 100 times for 100 steps each. Table shows the summarized results with the average, highest and lowest scores. For the drug-likeness and the penalized logP tasks, we compare our method to the best results of state-of-the-art optimization models as reported by You et al. . Our proposed method achieves the same performance on the drug-likeness task as the best state-of-the-art approach and outperforms all other approaches on the penalized logP task. Moreover, our method consistently generated molecules with very high predicted binding-affinities (IC50 < 1 nM) for both biological targets respectively. Figure depicts the average scores during the optimization process for the different tasks. To better understand the impact of the starting point for the optimization, we show Table : Results for the different single-objective optimization tasks. Our Method is compared to the results of three recently published optimizations methods as reported in .
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ORGAN JT-VAE GCPN results for a fixed starting point (benzene) and for variable starting points, randomly sampled from ChEMBL. In all tasks, the model consistently optimizes the respective property, reaching relatively high scores already after a few iterations. Although starting from different points, the variance between the scores at a given optimization step is similar to the variance obtained when repeatedly starting from benzene. As a matter of fact, the variance seems to be higher for the fixed starting point. Moreover, starting from a molecule picked from ChEMBL seems to result in faster and more successful optimizations, except for the optimization of the penalized logP score. This is probably due to the increased size and structural complexity of these molecules compared to a simple benzene ring. Moreover, initializing the particle swarm algorithm with more particles helps finding more optimal points in the search space in less iterations (at the expense of increased computational time). Using one GPU for passing the molecules in the swarm through the encoder-decoder model and one CPU core to perform the PSO algorithm and objective function evaluation, computational time is in the order of a few minutes for a 100-steps run. The run time on a 32-core CPU machine without GPU support is in the order of 10 minutes. This is in contrast to baseline methods in Table which have a reported wall clock running time of multiple hours to one day on a 32-core CPU machine .
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Figure shows a few example molecules randomly picked from the final iteration for each optimization task. It is evident that, by optimizing a single objective function, the model exploits its ability to freely navigate the chemical space and solely focuses on this very objective. While the optimization of drug-likeness obviously results in pleasant-looking molecules, the three other optimization tasks result into comparably "unusual" chemistry. Since long aliphatic chains are both maximizing the partition coefficient while being scored as easy to synthesize, they are the inevitable outcome when optimizing for penalized logP. Moreover, if the objective is solely maximizing the prediction of a QSAR model, the resulting molecules will aggregate the evidence the QSAR model found in the potent molecules of its training set. This, however, will most likely guide the optimizer into parts of the chemical space that are not well covered by training set molecules, leading to inaccurate and overoptimistic predictions. Thus, final molecules are likely to be artifacts of the underlying QSAR model. Summing up, we demonstrated that our method is able to very quickly improve upon a given starting molecule in terms of predicted drug-likeness or predicted biological activity. This confirms that our optimization method is able to navigate the chemical space defined by our pretrained embedding and perform single-parameter optimization in a timely fashion. However, these examples are far from actual drug design cases. At this stage, we do not apply any structural constraints for guiding the structure generation. This means that the new, optimized compounds might be structurally remote from the defined starting points or contain toxic or unstable moieties (e.g. 1,3-Pentadiyne substructure in Figure ). Usually, drug discovery projects are confined within chemical series and closely related analogues. Hence, we propose to add constraints on the chemical structure during optimization in the following section.
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In this experiment, we perform a single property optimization, while constraining the explorable chemical space using a defined molecular scaffold that needs to be present in the generated molecules. This task is more closely related to a real drug-development process, as it mimics a typical lead optimization task. We base our experiment on a lead optimization paper by Stamford et al. in which an iminopyrimidine lead series was optimized for BACE1 inhibition . In order to evaluate whether our method can optimize for biological activity while constraining the explorable chemical space, we start the optimization from the same initial compound as in (Figure ). We fix the iminopyrimidinone scaffold (Figure ) by penalizing generated compounds that do not contain this substructure in the objective function. For the prediction of BACE1 activity we retrain the BACE1 model from the previous section, excluding all compounds with an iminopyrimidinone scaffold.
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On the 17 iminopyrimidinone compounds reported by Stamford et al. the QSAR model achieves a Spearman Correlation Coefficient of ρ = 0.7 (p-value=0.004) compared to the reported IC50 values. This means that although we did not include compounds with an iminopyrimidinone scaffold in the training, the BACE1 activity prediction model shows a reasonable performance in the part of the chemical space we are interested in (i.e. compounds with iminopyrimidinone scaffold).
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We run the optimization model for 100 steps and restart the optimization 200 times. Figure -h depicts the compounds with the best scores found in the in silico optimization. In the course of the optimization the most active BACE1 inhibitor (compound c) from Stamford et al. was passed by in 2 out of the 200 runs but was not part of the final set of proposed molecules. This can be explained as the members of the final set of molecules all have higher predicted activities than compound c. As a proof of principle, we also performed an experiment where the Euclidean distance of a particle to compound c's embedding was used as objective function.
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In this experiment, we could consistently (200 out of 200 times) recover compound c as the optimal solution. Hence, we suppose that our approach does not contain compound c within the final set of molecules because this region of chemical space is not the most attractive for the applied BACE1 model. Our reported in silico solutions do contain the required scaffold and are predicted to have a higher potency, so they might possibly give useful new ideas to medicinal chemists working on BACE1 inhibitors with an iminopyrimidone scaffold. Step Objective Solubility Metabolic stability Permeability SA-Score QED Figure : Results for the best scoring (dscore) particle in the swarm over the course of optimization for 200 steps averaged over 100 runs for the three different optimization tasks. In addition to the objective (dscore) that is optimized, the average predicted potency (pIC50 in nM) on both BACE1 and EGFR as well as the average scaled predicted solubility, metabolic stability, permeability, SA score and QED of the best scoring (dscore) particle is shown.
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Additionally for all experiments, we maximize the predicted solubility, metabolic stability, cell permeability, drug-likeness as well as the predicted synthetic accessibility (SA) of the molecule and penalize for known toxic or uncommon substructures and molecular weight below 200 or above 600 Dalton. Each of the ten different objectives was scaled by a desirability function between 0 and 1, where low values correspond to undesirable ranges and high values to acceptable or good ranges of a molecular property (see Supporting Information for details on the scaling functions). The final optimization objective is the weighted average of the different scaled objective functions. In all experiments, we weighted the maximization of binding affinities with a factor of 5, minimization of binding affinity with a factor of 3 and all other properties with 1. Each of the three optimization tasks was run 100 times for 200 steps, starting from a randomly picked molecule from ChEMBL. The aggregated results for the different properties over the course of the optimization are depicted in Figure . Our proposed method is consistently able to optimize a query molecule with respect to the defined multi-objective value function. The weighted average of the different objective functions (dscore) increases on average from 0.64, 0.63 and 0.53 to 0.85, 0.9 and 0.82 for the three different experiments respectively. The method is able to find solutions that are predicted to meet the respective activity-profile while keeping desirable ADMET properties as well as QED and SA scores high. In experiment 3, however, the methods finds solutions that on average have a comparably lower SA score in order to meet the desired activity-profile. Table shows a few example molecules picked from the best final iteration for each of the three optimization tasks. Although the value function consists of many different and partially conflicting individual objectives our proposed method is consistently able to find molecules in the vast chemical space that meet the desirable ranges for all of the defined objectives.
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We propose the use of Particle Swarm Optimization (PSO) heuristic to optimize molecules in a continuous latent representation. We show that our model is able to consistently optimize molecules with respect to single objectives such as maximizing the predicted drug-likeliness, partition coefficient logP or target binding affinity as modeled by quantitative structure activity relationship (QSAR) model. Not only does our proposed method exhibit competitive or better performance in finding optimal solutions compared to baseline method, it also achieves significant reduction in computational time. In further experiments we show how the proposed method can be utilized to support medicinal chemists in a lead optimization process by proposing in silico solutions for relevant tasks. Finally, we perform multi-objective optimizations of compounds with respect to relevant properties such as specific target activity profiles and pharmacokineticsrelated properties. We demonstrate that our proposed method is consistently able to optimize the joined objective function and results in compounds that exhibit desirable ranges for all included computed properties. Although we show promising results for optimizing molecular properties in this work, it has to be noted that the optimization cycles are based on predicted values for the properties. This can be particularly problematic for QSAR models that are applied outside their domain of applicability. Hence, we advocate the use of our proposed method only in combination with reasonable constraints to parts of the chemical space that can be modeled by the applied functions with reasonable confidence. An even more suitable approach would be combining the in silico optimization with real world experiments in an active learning manner. By doing so, the QSAR model could be refitted while evolving into yet unexplored regions of the chemical space and hopefully remain reasonably accurate in its predictions. Table : Four example compounds as result of the in silico optimization for experiment 1,2 and 3 respectively, separated by horizontal lines. Each compound is shown with its desirability score (dscore) as well as its calculated and predicted molecular properties: Binding affinity to target Epidermal Growth Factor Receptor (EGFR), binding affinity to target asparyl protease b-site APP cleaving enzyme-1 (BACE1), metabolic stability (metstab), solubility from powder(sol), cell permeability (perm) drug-likeness (QED) and synthetic accessibility (SA)
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Monodisperse nanomaterials with precise morphologies are a vital component of many recent nanotechnologies. One promising synthetic route to prepare particles with defined nonspherical architectures is the crystallization-driven self-assembly (CDSA) of block copolymers. Living CDSA provides access to complex hierarchical nanostructures with impressive morphological and dimensional control; and this complexity enables wide-ranging applications in drug delivery, 10-13 colloid stabilization, catalysis, optoelectronics, and information storage. The 'livingness' of CDSA is achieved by initiating growth from uniform seed crystallites, which provide the nucleation that enables self-assembly to start from a common point (Fig. ). 6,8,10, Realizing the full potential of CDSA for nanostructure synthesis requires a precise understanding of how CDSA particles grow and how this self-assembly process can be controlled.
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Most previous reports monitoring the self-assembly of block copolymers derive kinetics from ensemble properties: For example, light scattering can be employed to report on the average size of CDSA micelles. However, these ensemble methods are insensitive to heterogeneity in the growth of individual nanoparticles and provide no direct information on particle morphology. Given the uniform complex hierarchical structures desired by CDSA, this ensemble averaging is a significant drawback in understanding and quantifying growth mechanics.
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Transmission electron microscopy (TEM) is arguably the most common method used to study the size evolution of individual polymer assemblies. 7, However, conventional TEM provides only 'snapshots' of the distribution of particle sizes during growth, lacking the capability to track the transformations of individual nanoparticles. Exciting recent advances in liquid cell TEM have opened a route to monitor the liquid phase reaction in-situ with millisecond to second temporal resolution, albeit at the expense of significant instru- mental complexity. Solution-phase atomic force microscopy (AFM) has also been applied to study the mechanism of interfacial seeded-growth of 1D micellar nanoparticles. However, tip-induced nanofiber fragmentation has been reported as a drawback; creating new active interfaces and affecting kinetics. Fluorescence microscopy offers a perhaps less invasive alternative, but at the expense of spatial resolution: For example, confocal laser scanning microscopy (CLSM) has been used to visualize 1D fiber growth at 100 ms temporal resolution with 120 nm lateral spatial resolution. Super-resolution fluorescence microscopy techniques can improve these limits with, for instance, 76 nm precision and 15 ms temporal resolution of the growth of 1D fiber. Although providing easy access to nanoparticle kinetics, fluorescence microscopy's significant drawback is the requirement for extrinsic labels; given the relative size of a fluorophore to component monomers, it is perhaps unsurprising that impacts on the CDSA kinetics are reported. Label-free techniques such as phase contrast or differential interference contrast microscopy can circumvent this issue but are typically limited in detection sensitivity. To address the need for an in-situ non-invasive method of characterizing living CDSA kinetics with high spatio-temporal single-particle resolution, here we apply interferometric scattering (iSCAT) microscopy to track CDSA in individual polymer particles. iSCAT is an intrinsically label-free technique capable of single-molecule resolution. To date, iSCAT has mainly been used to study biological systems, including the mass measurement of individual proteins, the formation of lipid membranes, and biological diffusion using single-particle tracking of metal nanoparticles. iSCAT has also been effectively utilized to monitor protein aggregation and actin polymerization processes. Given the similarities shared between actin polymerization and 1D CDSA fiber formation, we reasoned that iSCAT might also be productively applied to the study of CDSA.
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iSCAT relies on the interference between light scattered from an object of interest supported on a glass coverslip, and reference light reflected back from the same glass-sample interface (Fig. ). Compared to conventional interferometry, the common path agreement of iSCAT provides a simple and robust approach to measure changes in light scattering from small objects. For a diffraction-limited object, iSCAT signals appear as an Airy disc of concentric dark and light rings caused by the interference between the reflected and scattered signals. iSCAT offers lateral and axial resolutions of around 200 -300 nm and 10 -100 nm, respectively, and can further exploit optical super-resolution techniques to improve spatial precision. In addition, iSCAT can operate at high temporal resolutions (up to 1 µs) and monitor processes over long observation times without photobleaching.
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CDSA kinetics are highly dependent on core chemistry, and to date, the majority of studies examining CDSA-mediated growth have focused on 1D fibers, with typically slow (hours to days) formation kinetics. However, limited attention has been given to the study of 2D platelet kinetics, which are characterized by relatively rapid self-assembly processes, distinct morphology, and ease of modification. Here, we chose poly(ε-caprolactone) (PCL)based polymers as the core-forming block for the preparation of well-defined 1D fibers and 2D platelets. Compared with other commonly used core materials, the biocompatibility and biodegradability of PCL make it an attractive starting point for a range of materials for biomedical applications. 11, In addition, the crystallinity of the core-forming PCL block plays an essential role in determining the morphology and kinetics of CDSA particles: High crystallinity (i.e., high degree of polymerization) contributes to fast crystallization rates, commonly accompanied by self-nucleation and agglomeration; whereas core-forming blocks with lower crystallinity typically exhibit slower crystallization rates, resulting in irregular assembly shapes and a prolonged assembly process. Here, PCL 45 was selected, as it provides a good balance between fast kinetics and tight morphological control. Using iS-CAT, we mapped the kinetics and morphology of fiber and platelet growth from mixtures of PCL and poly(ε-caprolactone)-b-poly(N,N -dimethylacrylamide) block copolymer, PCLb-PDMA.
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To first summarize our overall method: Uniform seeds were prepared from polydisperse fibers of PCL 45 -b-PDMA 348 formed in ethanol (Fig. ), followed by sonication to produce short fibers with consistent size and length (mean length 24.6 ± 4.9 nm, Figs. &). Seeds were then attached to the coverslip via spin-coating, with a silicone spacer placed on top to form a reaction chamber. This sample was then mounted on a custom-built microscope, ready for imaging (Fig. ). Unimer solutions were then introduced into the reaction chamber, followed by immediate iSCAT imaging to monitor the CDSA process (Fig. ). An in-depth description of the experimental protocol and our instrumentation is provided in the Experimental Methods section and the Supplementary Information.
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We initially applied iSCAT microscopy to monitor seeded-growth of 1D fibers: 50 µL of 2.51 nM (0.1 µg/mL) PCL 45 -b-PDMA 348 seed solution was spin-coated onto the coverslip. PCL 73b-PDMA 204 unimer was added into methanol to reach a final concentration of 0.06 µM (1.67 µg/mL) before being introduced into the reaction chamber and imaged using iSCAT. Fig. depicts a representative time series of iSCAT images collected during the formation of 1D fibers (637 nm, 4 µW µm -2 , 400 µs exposure time, 3 s -1 per frame time-lapse). Upon unimer addition, uniform fiber elongation can be monitored as the reaction proceeds (the size of PCL 45 -b-PDMA 348 seeds is below the detection limit of our setup, thus they are not observable; Movie S1). Following image segmentation (see SI, Data Analysis), the length evolution of individual fibers can be determined as shown in Fig. . Applying a previously established kinetic model for 1D fiber growth, 30 rate constants were extracted by fitting to the trajectories of length vs. time (5.1 × 10 -3 s -1 , Fig. , Supplementary Methods section). Comparing these data to previous work on poly(ferrocenyldimethylsilane)-b-polydimethylsiloxane (PFS 63 -b-PDMS 513 ) 1D fibers (10 to 30 mg/mL unimer and 0.1 mg/mL seed), we observe faster kinetics in our PCL-based system (5.1×10 -3 s -1 compared to 1.8×10 -4 to 2.13×10 -4 s -1 ) despite lower unimer and seed concentration.
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Following our initial imaging of 1D fiber formation, we next focused on 2D platelet growth, where the characteristic morphology and fast controllable assembly of platelets might be used to assess the capabilities of iSCAT for CDSA measurement. Seeds were deposited on glass coverslips at a surface density of ∼0.16 µm -2 . Following the addition of 150 µL of 0.35 µM (3.33 µg/mL) PCL 45 :PCL 45 -b-PDMA 348 unimer mixtures, hexagonal platelets appear on the surface and uniformly grow in size as the reaction proceeds (Fig. , Movie S2; 637 nm, 4 µW µm -2 , 400 µs exposure time, 1.5 s -1 per frame time-lapse). We selected a PCL 45 :PCL 45 -b-PDMA 348 unimer mixture with the concentration of 0.35 µM (3.33 µg/mL) to ensure slow growth of well isolated platelets. Keeping the seed surface density fixed, increasing the unimer concentration results in faster kinetics. Further details are provided in the Experimental Methods section below. Control experiments without the presence of seeds show no characteristic fast growth (Fig. ). We also explored the effect of changing the core chemistry: Platelets prepared with a poly(η-octalactone) (POL)-based core-forming block exhibited less uniform morphology and size compared to the PCL-based system (Fig. ). Furthermore, we observed changes in platelet morphology with solvent (Fig. ), however, iSCAT was able to resolve CDSA assemblies under a wide range of solution conditions (Fig. ).
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Image sequences were then segmented (SI, Data Analysis) to extract parameters describing platelet morphology (area, long and short axis length, aspect ratio, and perimeter). Analysis of the time evolution of platelet area yields the kinetics of individual platelet growth (Fig. ). 210 platelets were recorded from 3 experimental repeats over a 25 min time-lapse.
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The individual growth trajectories of 40 representative platelets are shown in Fig. . All platelets show similar growth kinetics and consistent final platelet area (∼1.12 µm 2 ) at the end of the recording (Fig. ). The area distributions of platelets collected at the end of insitu recording (Fig. ) were independent of the position of the platelet in the sample (Fig. ). These data confirm the ability of living CDSA to produce uniform assemblies with controlled size and demonstrate that the in-situ iSCAT imaging captures growth kinetics representative of the entire population.
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During the early stages of CDSA, the platelet size is below the diffraction limit, and thus platelet area cannot be used to examine growth kinetics. However, the evolution of particle contrast can be used to report on this early stage (Figs. &). As platelets are detected, individual spots begin to appear on the surface with their absolute contrast values increasing over time (becomes more negative, i.e., darker). Eventually, these diffractionlimited spots grow sufficiently that the characteristic shape becomes discernible (Fig. ).
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2G & H). To demonstrate this capability, we examined the progression of the position of a platelet edge, with contrast profiles extracted from platelet cross-sections as depicted in Fig. . Data were collected with a temporal resolution of 333 µs (Fig. , Movie S4). A plot of the contrast profile evolution at 33.3 ms intervals with 100-frame averaging is shown in Fig. .
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We compared platelet parameters extracted from iSCAT imaging with those obtained from identical samples using AFM, TEM, and CLSM. Assembly of fluorescently-labeled platelets was initiated by adding 10 µL of 1 mM (10 mg/mL) aminochloromaleimide (ACM) -coupled homopolymer and block copolymer unimer mixtures (ACM-PCL 45 :ACM-PCL 45 -b-PDMA 348 , 1:1, w:w) into 1 mL 0.25 µM (10 µg/mL) PCL 45 -b-PDMA 348 seed solution in ethanol, followed by mixing via shaking. At predetermined time points, identical aliquots were taken, quenched by water addition, and analyzed using each technique. Further details for AFM TEM, and CLSM measurement are provided in the Experimental Methods section. Fig. shows that platelet morphology and area information extracted from AFM, TEM, CLSM, and iSCAT are comparable (Fig. ). Under these reaction conditions, platelet formation is rapid, with the assembly completed within two minutes of our first time point; essentially faster than the time resolution of AFM, TEM, and CLSM measurements.
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where N t is the number of unimers remaining in the reaction at time t; k is the overall growth rate constant; n a is the number of unimers in the exposed areas per unit perimeter; N 0 is the initial number of unimers; and B is the area contributed per unimer, calculated from the ratio of final platelet area to initial number of unimers (Eq. S6). The area per unimer (B) is considered as a constant, linking the experimentally determined platelet area to the consumption of unimers. Consequently, the evolution of platelet area (A t ) with time (t) can be expressed simply in terms of an effective overall rate constant, k ′ (see SI Methods):
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To quantify the kinetics of platelet growth, we then examined the dependence of platelet size and morphology on unimer concentration, seed concentration, and solvent conditions (Fig. ). Platelets were prepared in bulk under various reaction conditions, with sample aliquots removed at predefined time points to fully capture the reaction kinetics. These samples were then characterized by iSCAT (2 µW µm -2 , 637 nm, 900 µs exposure time; see further details in the Experimental Methods section). A minimum of 100 platelets were analyzed at each time point for each condition. As expected, a higher initial unimer concentration leads to faster platelet growth, producing platelets with larger final surface area (Fig. ). A linear dependence of the overall rate constant on unimer concentration was observed with an overall reaction order of 0.41 (Fig. ). This deviation from simple first-order kinetics has been reported previously and attributed to conformational effects on the block copolymer during assembly. We also examined the dependence of platelet formation on seed concentration (0.25 to 1.13 nM), keeping unimer concentration and solvent conditions fixed (Fig. ). As expected, a higher seed concentration produces platelets with smaller final sizes.
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The impact of tetrahydrofuran (THF), a good solvent for the crystallizable core-forming block (PCL), was investigated. Fig. illustrates the overall inhibitory effect of THF addition: Increasing the volume fraction of good solvent hinders the platelet growth rate, as crystallization is slowed due to enhanced polymer solubility. Generally, we observe that the final area of platelets generated from fixed unimer and seed concentrations remains consistent, regardless of solvent compositions (0 to 3% of THF). However, notably, this does not hold true for the highest THF concentration we explored (Fig. , purple, 5%).
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Similar results have been observed previously, attributing the increased final size to the solubilization of the seeds themselves at high THF content. This solubilization leads to a decrease in crystalline nuclei, thereby providing an additional source of unimers for remaining nuclei. Alongside measurements of growth kinetics based on platelet area, the impact of unimer, seed, and solvent variation on the platelet morphology was also examined. Figs. show the time evolution of the platelet aspect ratio (L 1 :L 2 ). Figs. show the underlying changes in long (L 1 ) and short (L 2 ) axis lengths with experimental conditions. Both unimer and seed concentrations had little impact on the platelet shape during assembly. Conversely, alterations in the THF content in the system (from 0 to 5%) significantly impacted platelet shape. While the platelet aspect ratio (L 1 :L 2 ) remained relatively constant during platelet formation for a particular THF concentration (Fig. ), an increased THF content led to a more pronounced preference for unimer addition along the longer axis (L 1 ), resulting in the formation of elongated platelets as illustrated in Figs. 4I & S14. However, the mechanism leading to this asymmetry remains unknown and warrants further study.
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Overall, for all conditions tested (unimer concentration, seed concentration, and solubility), our simple kinetic model (Eq. 2) for edge-directed growth provides a good description of our experimental observations. It can also be noted that the core chemistry significantly affects the kinetics. Compared with 1D fiber formation using a poly(ferrocenyldimethylsilane)-based core-forming blocks or 2D nanosheet formation from poly(cyclopentenylene vinylene)-based core-forming blocks, 32 our PCL-based system exhibits faster kinetics in both 1D and 2D assembly.
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| 18 |
For instance, changes in scattering properties have previously been utilized to study 2D lipid bilayer formation and phase transitions. Following CDSA, the platelet perimeter remains active for the templated edge-directed growth of additional platelet area. Therefore, multiannulus platelets can be prepared by sequential unimer addition. The growth occurs radially from the edges of the platelets, with each successive annulus extending the preceding one. By varying the unimer composition, the composition/thickness of each annulus can be adjusted, which allows us to utilize the contrast mechanism of iSCAT to investigate the kinetics and properties of individual annuli during platelet assembly. still identifiable (to clarify the boundaries, each annulus was color-coded as shown in Fig. ). The outermost boundary within each annulus appears discernibly darker (with a larger negative contrast) compared to the center. We speculate that this effect is potentially caused by a nonuniform distribution of PCL 45 and PCL 45 -b-PDMA 348 within each annulus due to differences in their crystallization rates. However, although providing some contrast variations, this effect was insufficient to render significant changes in growth kinetics. Fig. illustrates the progression of averaged platelet area (calculated from averaging three 3-annulus platelets displayed in Fig. ), with each successive unimer addition synchronized to the corresponding time of addition. The rate constants extracted for each annulus were essentially unchanged from those in Fig. . As expected, a higher unimer concentration resulted in a faster assembly rate. As a result, besides the dimensional and morphological information, iSCAT can also probe into the nanoscopic surface information of individual platelets based on their contrast variations.
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| 19 |
Taken together, our results demonstrate the potential of using iSCAT microscopy as a labelfree tool for the in-situ reporting the morphological and dimensional information of CDSA assemblies, ranging from 1D fiber formation to 3D thickness information of multi-annulus structures. With its excellent sensitivity and sub-millisecond temporal resolution, iSCAT successfully provides detailed insights into the CDSA process, including early-stage growth and high-resolution imaging of the growth dynamics.
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| 20 |
This method has enabled us to quantify the dependence of living CDSA kinetics on unimer concentration, seed concentration, and selective solubility. Living CDSA shares obvious similarities with living covalent polymerization, where unimer and exposed active sites at the platelet edges are analogous to the monomer and initiator in the reaction. Here, 2D platelet formation of PCL 45 :PCL 45 -b-PDMA 348 was well described by a kinetic model based on such assembly at the living edge of the forming platelet, with a sub-unitary overall order of reaction and dependence of growth kinetics on the nature of the block copolymer. This deviation from simple first-order kinetics has been previously attributed to conformational dispersity of the block copolymer. This is particularly intriguing when considering the variation in contrast within growth annuli in our examination of the growth of multi-annulus platelets (Fig. ). Secondly, observed variations in contrast across individual annuli are of note (Fig. ). One interpretation could be a change in composition, perhaps caused by a variation in the relative rates of deposition of PCL 45 and PCL 45 -b-PDMA 348 . These observations warrant further investigation, mapping how different building units might assemble differently as 2D platelet shape evolves.
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| 21 |
Beyond its capabilities for in-situ monitoring, iSCAT also holds much promise as a tool for ex-situ characterization of particle morphology due to its fast data collection, low cost, simple sample preparation, and high throughput. Additionally, this technique can be implemented using simple commercially available optical components compatible with fluorescence mi-croscopy. However, iSCAT is not without limitations: Microscopy limits the field of view in any single image (20 × 20 µm in our setup), requiring parallelization or surface scanning to monitor the growth of larger numbers of particles simultaneously. iSCAT also requires a reference light-field and thus is limited to studies at or near a surface. We note that here we saw no difference in kinetics between platelets grown in solution and those growing in-situ at the coverslip surface. However, the presence of this surface is inevitably different at some level from growth conditions in the bulk solution. Nevertheless, confocal variants of iSCAT hold the potential for 3D imaging much deeper into the sample solution. Here we have focused on 2D CDSA, but the interferometric nature of iSCAT also embeds axial information; enabling, for example, the measurement of nanoparticle axial position and size. Consequently, we envisage iSCAT might also be applied to profiling more complex CDSA processes, with three-dimensional shape at high spatio-temporal resolution. directly from a drying and degassing inert solvent tower system. Aminochloromaleimide (ACM) fluorescent dye was synthesized via the method reported previously. Polymer synthesis and characterization
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| 22 |
In a nitrogen-filled glove box (oxygen and water content lower than 0.1 ppm), solutions of diphenylphosphate (DPP, 17.0 mg, 1 eq) in dry toluene (2.5 mL) and dual-head CTA (17.0 mg, 1 eq) in dry toluene (1 mL) were added to ε-caprolactone dry toluene solution (543.3 mg, 70 eq in 1.5 mL). After stirring for 6 h at room temperature, the solution was removed from the glove box, precipitated three times into cold diethyl ether dropwise, and collected by centrifugation. PCL 45 (100.0 mg, 1 eq), DMA (736.3 mg, 400 eq), and AIBN (0.3 mg, 0.1 eq) were dissolved in 1,4-dioxane (1 mL) and placed in an ampule. The solution was then freeze-pump-thawed three times and heated for 2 h at 70 • C. The reaction was quenched by immersion of the ampule in liquid nitrogen and the polymer was precipitated in ice-cold diethyl ether three times before being dried under vacuum and analyzed. SEC (chloroform, PMMA standard):
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