Lactate is an important downstream product of glycolysis in living cells, and its level is highly related with diseases. On the basis of amorphous metal–organic frameworks (aMOFs), a multienzyme system consisting of lactate oxidase (LOx) and horseradish peroxidase (HRP) was established for intracellular lactate detection. By coencapsulation in aMOFs with proximity, LOx and HRP were delivered into cells, serving as artificially constructed organelles, exhibiting high activity and selectivity for the intracellular detection of the important metabolite lactate, which improved the signal to noise ratio by ∼650-fold. As demonstrated by both experimental and simulation results, the high efficiency was attributed to the short distance between the two types of enzymes coencapsulated in aMOFs. The concept of constructing multienzyme systems in this study shows promise for the detection of various intracellular metabolites.

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Structure-enforced matrix factorization (SeMF) represents a large class of mathematical models appearing in various forms of principal component analysis, sparse coding, dictionary learning and other machine learning techniques useful in many applications including neuroscience and signal processing. In this paper, we present a unified algorithm framework, based on the classic alternating direction method of multipliers (ADMM), for solving a wide range of SeMF problems whose constraint sets permit low-complexity projections. We propose a strategy to adaptively adjust the penalty parameters which is the key to achieving good performance for ADMM. We conduct extensive numerical experiments to compare the proposed algorithm with a number of state-of-the-art special-purpose algorithms on test problems including dictionary learning for sparse representation and sparse nonnegative matrix factorization. Results show that our unified SeMF algorithm can solve different types of factorization problems as reliably and as efficiently as special-purpose algorithms. In particular, our SeMF algorithm provides the ability to explicitly enforce various combinatorial sparsity patterns that, to our knowledge, has not been considered in existing approaches.

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