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Variational approximation methods are a way to approximate the posterior in Bayesian inference especially when the dataset has a large volume or high dimension. Factor covariance structure was introduced in previous work with three restrictions to handle the problem of computational infeasibility in Gaussian approximation. However, the three strong constraints on the covariance matrix could possibly break down during the process of the structure optimization, and the identification issue could still possibly exist within the final approximation. In this paper, we consider two types of manifold parameterization, Stiefel manifold and Grassmann manifold, to address the problems. Moreover, the Riemannian stochastic gradient descent method is applied to solve the resulting optimization problem while maintaining the orthogonal factors. Results from two experiments demonstrate that our model fixes the potential issue of the previous method with comparable accuracy and competitive converge speed even in high-dimensional problems.

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Prokaryotic Argonaute (pAgo) proteins, a class of DNA/RNA-guided programmable endonucleases, have been extensively utilized in nucleic acid biosensors. The specific binding and cleavage of nucleic acids by pAgo proteins, which are crucial processes for their applications, are dependent on the presence of Mn2+ bound in the pockets, as already verified through X-ray crystallography. However, a comprehensive understanding of how dissociated Mn2+ in the solvent affects the catalytic cycle, and its underlying regulatory role in this structure-function relationship, remains underdetermined. By combining experimental and computational methods, this study reveals that unbound Mn2+ significantly enhances the flexibility of diverse pAgo proteins, regardless of their classification as hyperthermophiles, thermophiles, or mesophiles. This increase in flexibility through decreasing the number of hydrogen bonds, induced by Mn2+, leads to higher affinity for substrates, thus facilitating cleavage. More importantly, Mn2+-induced structural flexibility increases the mismatch tolerance between guide-target pairs by increasing the conformational states, thereby enhancing the cleavage of mismatches. Further simulations indicated that the enhanced flexibility in linkers triggers conformational changes in the PAZ domain for recognizing various lengths of nucleic acids. Additionally, the Mn2+-induced dynamic alterations of the protein cause a conformational shift in the N domain and catalytic sites towards their functional form, resulting in a decreased energy penalty for target release and cleavage. These findings demonstrate that the dynamic conformations of pAgo proteins, resulting from the presence of the unbound Mn2+, significantly promote the catalytic cycle of endonucleases and the tolerance of cleavage to mismatches. This flexibility enhancement mechanism serves as a general strategy employed by Ago proteins from diverse prokaryotes to accomplish their catalytic functions and provide useful information for Ago-based precise molecular diagnostics.