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Theoretical and empirical comparisons have been made to assess the expressive power and performance of invariant and equivariant GNNs. However, there is currently no theoretical result comparing the expressive power of k-hop invariant GNNs and equivariant GNNs. Additionally, little is understood about whether the performance of equivariant GNNs, employing steerable features up to type-L, increases as L grows – especially when the feature dimension is held constant. In this study, we introduce a key lemma that allows us to analyze steerable features by examining their corresponding invariant features. The lemma facilitates us in understanding the limitations of k-hop invariant GNNs, which fail to capture the global geometric structure due to the loss of geometric information between local structures. Furthermore, we analyze the ability of steerable features to carry information by studying their corresponding invariant features. In particular, we establish that when the input spatial embedding has full rank, the informationcarrying ability of steerable features is characterized by their dimension and remains independent of the feature types. This suggests that when the feature dimension is constant, increasing L does not lead to essentially improved performance in equivariant GNNs employing steerable features up to type-L. We substantiate our theoretical insights with numerical evidence. 

Theoretical and empirical comparisons have been made to assess the expressive power and performance of invariant and equivariant GNNs. However, there is currently no theoretical result comparing the expressive power of k-hop invariant GNNs and equivariant GNNs. Additionally, little is understood about whether the performance of equivariant GNNs, employing steerable features up to type-L, increases as L grows – especially when the feature dimension is held constant. In this study, we introduce a key lemma that allows us to analyze steerable features by examining their corresponding invariant features. The lemma facilitates us in understanding the limitations of k-hop invariant GNNs, which fail to capture the global geometric structure due to the loss of geometric information between local structures. Furthermore, we analyze the ability of steerable features to carry information by studying their corresponding invariant features. In particular, we establish that when the input spatial embedding has full rank, the information carrying ability of steerable features is characterized by their dimension and remains independent of the feature types. This suggests that when the feature dimension is constant, increasing L does not lead to essentially improved performance in equivariant GNNs employing steerable features up to type-L. We substantiate our theoretical insights with numerical evidence. 

Abstract Drug nanoaggregates are particles that can deleteriously cause false positive results during drug screening efforts, but alternatively, they may be used to improve pharmacokinetics when developed for drug delivery purposes. The structural features of molecules that drive nanoaggregate formation remain elusive, however, and the prediction of intracellular aggregation and rational design of nanoaggregate-based carriers are still challenging. We investigate nanoaggregate self-assembly mechanisms using small molecule fragments to identify the critical molecular forces that contribute to self-assembly. We find that aromatic groups and hydrogen bond acceptors/donors are essential for nanoaggregate formation, suggesting that both π-π stacking and hydrogen bonding are drivers of nanoaggregation. We apply structure-assembly-relationship analysis to the drug sorafenib and discover that nanoaggregate formation can be predicted entirely using drug fragment substructures. We also find that drug nanoaggregates are stabilized in an amorphous core-shell structure. These findings demonstrate that rational design can address intracellular aggregation and pharmacologic/delivery challenges in conventional and fragment-based drug development processes.