Research

We describe a significant update to the Macaulay2 package A1BrouwerDegrees. We extend several methods in the previous version of the package to the setting of finite étale algebras, allowing the computation of transfers along finite étale extensions. Additionally, we implement a number of new features for the computation of unstable $\mathbb{A}^{1}$-Brouwer degrees and manipulation of classes in the unstable Grothendieck-Witt group.

We give two geometric interpretations for the local type of a line that is highly tangent to a hypersurface in a single point. One interpretation is phrased in terms of the Wronski map, while the other interpretation relates to the fundamental forms of the hypersurface. These local types are the local contributions of an quadratic form-valued Euler number that depends on a choice of orientation.

We describe the Macaulay2 package "A1BrouwerDegrees" for computing local and global $\mathbb{A}^{1}$-Brouwer degrees and studying symmetric bilinear forms over the complex numbers, the real numbers, the rational numbers, and finite fields of characteristic not equal to 2.

In this paper we study circles tangent to conics. We show there are generically 184 complex circles tangent to three conics in the plane and we characterize the real discriminant of the corresponding polynomial system. We give an explicit example of 3 conics with 136 real circles tangent to them. We conjecture that 136 is the maximal number of real circles. Furthermore, we implement a hill-climbing algorithm to find instances of conics with many real circles, and we introduce a machine learning model that, given three real conics, predicts the number of circles tangent to these three conics.

The extreme surge of interest over the past decade surrounding the use of neural networks has inspired many groups to deploy them for predicting binding affinities of drug-like molecules to their receptors. A model that can accurately make such predictions has the potential to screen large chemical libraries and help streamline the drug discovery process. However, despite reports of models that accurately predict quantitative inhibition using protein kinase sequences and inhibitors’ SMILES strings, it is still unclear whether these models can generalize to previously unseen data. Here, we build a Convolutional Neural Network (CNN) analogous to those previously reported and evaluate the model over four datasets commonly used for inhibitor/kinase predictions. We find that the model performs comparably to those previously reported, provided that the individual data points are randomly split between the training set and the test set. However, model performance is dramatically deteriorated when all data for a given inhibitor is placed together in the same training/testing fold, implying that information leakage underlies the models’ performance. Through comparison to simple models in which the SMILES strings are tokenized, or in which test set predictions are simply copied from the closest training set data points, we demonstrate that there is essentially no generalization whatsoever in this model. In other words, the model has not learned anything about molecular interactions, and does not provide any benefit over much simpler and more transparent models. These observations strongly point to the need for richer structure-based encodings, to obtain useful prospective predictions of not-yet-synthesized candidate inhibitors.

In early stage drug discovery, the stage of hit-to-lead optimization (or "hit expansion") entails starting from a newly-identified active compound, and improving its potency or other properties. Traditionally this process relies on synthesizing and evaluating a series of analogs to build up structure-activity relationships. Here, we describe a computational strategy focused on kinase inhibitors, intended to expedite the process of identifying analogs with improved potency. Our protocol begins from an inhibitor of the target kinase, and generalizes the synthetic route used to access it. By searching for commercially-available replacements for the individual building blocks used to make the parent inhibitor, we compile an enumerated library of compounds that can be accessed using the same chemical transformations; these huge libraries can exceed many millions – or billions – of compounds. Because the resulting libraries are much too large for explicit virtual screening, we instead consider alternate approaches to identify the top-scoring compounds. We find that contributions from individual substituents are well-described by a pairwise additivity approximation, provided that the corresponding fragments position their shared core in precisely the same way relative to the binding site. This key insight allows us to determine which fragments are suitable for merging into a single new compounds, and which are not. Further, the use of the pairwise approximation allows interaction energies to be assigned to each compound in the library, without the need for any further structure-based modeling: interaction energies instead can be reliably estimated from the energies of the component fragments. We demonstrate this protocol using libraries built from five representative kinase inhibitors drawn from the literature, which target four different kinases: CDK9, CHK1, CDK2, and ACK1. In each example, the enumerated library includes additional analogs reported by the original study to have activity, and these analogs are successfully prioritized within the library. We envision that the insights from this work can facilitate the rapid assembly and screening of increasingly large libraries for focused hit-to-lead optimization. To encourage adoption of these methods and enable further analyses, we disseminate the computational tools needed to deploy this protocol.