In this paper, we develop a new extension of the singular spectrum analysis (SSA) called functionalSSA to analyze functional time series. The new methodology is constructed by integrating ideasfrom functional data analysis and univariate SSA. Specifically, we i ntroduce a trajectory operatorin the functional world, which is equivalent to the trajectory matrix in the regular SSA. In theregular S SA, one needs to obtain the singular value decomposition (SVD) of the trajectorymatrix to decompose a given time series. Since there is no procedure to extract the functionalSVD (fSVD) of the trajectory operator, we introduce a computationally tr actable algorithm toobtain the fSVD components. The effectiveness of the proposed approach is illustrated by aninteresting example of remote sensing data. Also, we develop an efficient and user-friendly Rpackage and a shiny web application to allow interactive exploration of the results.

Copyright © 2020 American Chemical Society. The genetic material of viruses is protected by protein shells that are assembled from a large number of subunits in a process that is efficient and robust. Many of the mechanistic details underpinning efficient assembly of virus capsids are still unknown. The assembly mechanism of hepatitis B capsids has been intensively researched using a truncated core protein lacking the C-terminal domain responsible for binding genomic RNA. To resolve the assembly intermediates of hepatitis B virus (HBV), we studied the formation of nucleocapsids and empty capsids from full-length hepatitis B core proteins, using time-resolved small-angle X-ray scattering. We developed a detailed structural model of the HBV capsid assembly process using a combination of analysis with multivariate curve resolution, structural modeling, and Bayesian ensemble inference. The detailed structural analysis supports an assembly pathway that proceeds through the formation of two highly populated intermediates, a trimer of dimers and a partially closed shell consisting of around 40 dimers. These intermediates are on-path, transient and efficiently convert into fully formed capsids. In the presence of an RNA oligo that binds specifically to the C-terminal domain the assembly proceeds via a similar mechanism to that in the absence of nucleic acids. Comparisons between truncated and full-length HBV capsid proteins reveal that the unstructured C-terminal domain has a significant impact on the assembly process and is required to obtain a more complete mechanistic understanding of HBV capsid formation. These results also illustrate how combining scattering information from different time-points during time-resolved experiments can be utilized to derive a structural model of protein self-assembly pathways.

© 2016 American Statistical Association. This article develops a method for simultaneous estimation of density functions for a collection of populations of protein backbone angle pairs using a data-driven, shared basis that is constructed by bivariate spline functions defined on a triangulation of the bivariate domain. The circular nature of angular data is taken into account by imposing appropriate smoothness constraints across boundaries of the triangles. Maximum penalized likelihood is used to fit the model and an alternating blockwise Newton-type algorithm is developed for computation. A simulation study shows that the collective estimation approach is statistically more efficient than estimating the densities individually. The proposed method was used to estimate neighbor-dependent distributions of protein backbone dihedral angles (i.e., Ramachandran distributions). The estimated distributions were applied to protein loop modeling, one of the most challenging open problems in protein structure prediction, by feeding them into an angular-sampling-based loop structure prediction framework. Our estimated distributions compared favorably to the Ramachandran distributions estimated by fitting a hierarchical Dirichlet process model; and in particular, our distributions showed significant improvements on the hard cases where existing methods do not work well.