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Random-scan Gibbs samplers possess a natural hierarchical structure. The structure connects Gibbs samplers targeting higher-dimensional distributions to those targeting lower-dimensional ones. This leads to a quasi-telescoping property of their spectral gaps. Based on this property, we derive three new bounds on the spectral gaps and convergence rates of Gibbs samplers on general domains. The three bounds relate a chain’s spectral gap to, respectively, the correlation structure of the target distribution, a class of random walk chains, and a collection of influence matrices. Notably, one of our results generalizes the technique of spectral independence, which has received considerable attention for its success on finite domains, to general state spaces. We illustrate our methods through a sampler targeting the uniform distribution on a corner of an n-cube. 

Site-specific proteolysis by the enzymatic cleavage of small linear sequence motifs is a key posttranslational modification involved in physiology and disease. The ability to robustly and rapidly predict protease–substrate specificity would also enable targeted proteolytic cleavage by designed proteases. Current methods for predicting protease specificity are limited to sequence pattern recognition in experimentally derived cleavage data obtained for libraries of potential substrates and generated separately for each protease variant. We reasoned that a more semantically rich and robust model of protease specificity could be developed by incorporating the energetics of molecular interactions between protease and substrates into machine learning workflows. We present Protein Graph Convolutional Network (PGCN), which develops a physically grounded, structure-based molecular interaction graph representation that describes molecular topology and interaction energetics to predict enzyme specificity. We show that PGCN accurately predicts the specificity landscapes of several variants of two model proteases. Node and edge ablation tests identified key graph elements for specificity prediction, some of which are consistent with known biochemical constraints for protease:substrate recognition. We used a pretrained PGCN model to guide the design of protease libraries for cleaving two noncanonical substrates, and found good agreement with experimental cleavage results. Importantly, the model can accurately assess designs featuring diversity at positions not present in the training data. The described methodology should enable the structure-based prediction of specificity landscapes of a wide variety of proteases and the construction of tailor-made protease editors for site-selectively and irreversibly modifying chosen target proteins. 

Couplings play a central role in the analysis of Markov chain Monte Carlo algorithms and appear increasingly often in the algorithms themselves, e.g. in convergence diagnostics, parallelization, and variance reduction techniques. Existing couplings of the Metropolis-Hastings algorithm handle the proposal and acceptance steps separately and fall short of the upper bound on one-step meeting probabilities given by the coupling inequality. This paper introduces maximal couplings which achieve this bound while retaining the practical advantages of current methods. We consider the properties of these couplings and examine their behavior on a selection of numerical examples.