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Abstract A streamlined one-day protocol is described to produce isotopically methyl-labeled protein with high levels of deuterium for NMR studies. Using this protocol, the D₂O and²H-glucose content of the media and protonation level of ILV labeling precursors (ketobutyrate and ketovalerate) were varied. The relaxation rate of the multiple-quantum (MQ) state that is present during the HMQC-TROSY pulse sequence was measured for different labeling schemes and this rate was used to predict upper limits of molecular weights for various labeling schemes. The use of deuterated solvents (D₂O) or deuterated glucose is not required to obtain¹H–¹³C correlated NMR spectra of a 50 kDa homodimeric protein that are suitable for assignment by mutagenesis. High quality spectra of 100–150 kDa proteins, suitable for most applications, can be obtained without the use of deuterated glucose. The proton on the β-position of ketovalerate appears to undergo partial exchange with deuterium under the growth conditions used in this study. 

A Bayesian Network is a directed acyclic graph (DAG) on a set of n random variables (the vertices); a Bayesian Network Distribution (BND) is a probability distribution on the random variables that is Markovian on the graph. A finite k-mixture of such models is graphically represented by a larger graph which has an additional “hidden” (or “latent”) random variable U, ranging in {1,...,k}, and a directed edge from U to every other vertex. Models of this type are fundamental to causal inference, where U models an unobserved confounding effect of multiple populations, obscuring the causal relationships in the observable DAG. By solving the mixture problem and recovering the joint probability distribution with U, traditionally unidentifiable causal relationships become identifiable. Using a reduction to the more well-studied “product” case on empty graphs, we give the first algorithm to learn mixtures of non-empty DAGs. 

We present gOTzilla, a protocol for interactive zero-knowledge proofs for very large disjunctive statements of the following format: given publicly known circuit C, and set of values Y = {y1 , . . . , yn }, prove knowledge of a witness x such that C(x) = y1 ∨ C(x) = y2 ∨ · · · ∨ C(x) = yn . These type of statements are extremely important for the proof of assets (PoA) problem in cryptocurrencies where a prover wants to prove the knowledge of a secret key sk that associates with the hash of a public key H(pk) posted on the ledger. We note that the size of n in popular cryptocurrencies, such as Bitcoin, is estimated to 80 million. For the construction of gOTzilla, we start by observing that if we restructure the proof statement to an equivalent of proving knowledge of (x, y) such that (C(x) = y) ∧ (y = y1 ∨ · · · ∨ y = yn )), then we can reduce the disjunction of equalities to 1-out-of-N oblivious transfer (OT). Our overall protocol is based on the MPC in the head (MPCitH) paradigm. We additionally provide a concrete, efficient extension of our protocol for the case where C combines algebraic and non-algebraic statements (which is the case in the PoA application). We achieve an asymptotic communication cost of O(log n) plus the proof size of the underlying MPCitH protocol. While related work has similar asymptotic complexity, our approach results in concrete performance improvements. We implement our protocol and provide benchmarks. Concretely, for a set of size 1 million entries, the total run-time of our protocol is 14.89 seconds using 48 threads, with 6.18 MB total communication, which is about 4x faster compared to the state of the art when considering a disjunctive statement with algebraic and non-algebraic elements.