Riemannian optimization using three different metrics for Hermitian PSD fixed-rank constraints: an extended version

We consider smooth optimization problems with a Hermitian positive semi-definite fixed-rank constraint, where a quotient geometry with three Riemannian metrics $g^i(\cdot, \cdot)$ $(i=1,2,3)$ is used to represent this constraint. By taking the nonlinear conjugate gradient method (CG) as an example, we show that CG on the quotient geometry with metric $g^1$ is equivalent to CG on the factor-based optimization framework, which is often called the Burer--Monteiro approach. We also show that CG on the quotient geometry with metric $g^3$ is equivalent to CG on the commonly-used embedded geometry. We call two CG methods equivalent if they produce an identical sequence of iterates $\{X_k\}$. In addition, we show that if the limit point of the sequence $\{X_k\}$ generated by an algorithm has lower rank, that is $X_k\in \mathbb C^{n\times n}, k = 1, 2, \ldots$ has rank $p$ and the limit point $X_*$ has rank $r < p$, then the condition number of the Riemannian Hessian with metric $g^1$ can be unbounded, but those of the other two metrics stay bounded. Numerical experiments show that the Burer--Monteiro CG method has slower local convergence rate if the limit point has a reduced rank, compared to CG on the quotient geometry under the other two metrics. This slower convergence rate can thus be attributed to the large condition number of the Hessian near a minimizer

A Bubble Model for the Gating of Kv Channels

Voltage-gated Kv channels play fundamental roles in many biological processes, such as the generation of the action potential. The gating mechanism of Kv channels is characterized experimentally by single-channel recordings and ensemble properties of the channel currents. In this work, we propose a bubble model coupled with a Poisson-Nernst-Planck (PNP) system to capture the key characteristics, particularly the delay in the opening of channels. The coupled PNP system is solved numerically by a finite-difference method and the solution is compared with an analytical approximation. We hypothesize that the stochastic behaviour of the gating phenomenon is due to randomness of the bubble and channel sizes. The predicted ensemble average of the currents under various applied voltages across the channels is consistent with experimental observations, and the Cole-Moore delay is captured by varying the holding potential

Micro-Macro Modeling of Polymeric Fluids and Shear-Induced Microscopic Behaviors

This article delves into the micro-macro modeling of polymeric fluids, considering various microscopic potential energies, including the classical Hookean potential, as well as newly proposed modified Morse and Elastic-plastic potentials. These proposed potentials encompass microscopic-scale bond-breaking processes. The development of a thermodynamically consistent micro-macro model is revisited, employing the energy variational method. To validate the model's predictions, we conduct numerical simulations utilizing a deterministic particle-FEM method. Our numerical findings shed light on the distinct behaviors exhibited by polymer chains at the micro-scale in comparison to the macro-scale velocity and induced shear stresses of fluids under shear flow. Notably, we observe that polymer elongation, rotation, and bond breaking contribute to the zero polymer-induced stress in the micro-macro model when employing Morse and Elastic-plastic potentials. Furthermore, at high shear rates, polymer rotation is found to induce shear-thinning behavior in the model employing the classical Hookean potential

Mlsp : A bioinformatics tool for predicting molecular subtypes and prognosis in patients with breast cancer

The molecular landscape in breast cancer is characterized by large biological heterogeneity and variable clinical outcomes. Here, we performed an integrative multi-omics analysis of patients diagnosed with breast cancer. Using transcriptomic analysis, we identified three subtypes (cluster A, cluster B and cluster C) of breast cancer with distinct prognosis, clinical features, and genomic alterations: Cluster A was asso-ciated with higher genomic instability, immune suppression and worst prognosis outcome; cluster B was associated with high activation of immune-pathway, increased mutations and middle prognosis out-come; cluster C was linked to Luminal A subtype patients, moderate immune cell infiltration and best prognosis outcome. Combination of the three newly identified clusters with PAM50 subtypes, we pro-posed potential new precision strategies for 15 subtypes using L1000 database. Then, we developed a robust gene pair (RGP) score for prognosis outcome prediction of patients with breast cancer. The RGP score is based on a novel gene-pairing approach to eliminate batch effects caused by differences in heterogeneous patient cohorts and transcriptomic data distributions, and it was validated in ten cohorts of patients with breast cancer. Finally, we developed a user-friendly web-tool (https://sujiezhulab.shi-nyapps.io/BRCA/) to predict subtype, treatment strategies and prognosis states for patients with breast cancer.(c) 2022 The Authors. Published by Elsevier B.V. on behalf of Research Network of Computational and Structural Biotechnology. This is an open access article under the CC BY-NC-ND license (http://creative-commons.org/licenses/by-nc-nd/4.0/).Peer reviewe

Functional Specialization of Cellulose Synthase Isoforms in a Moss Shows Parallels with Seed Plants

The secondary cell walls of tracheary elements and fibers are rich in cellulose microfibrils that are helically oriented and laterally aggregated. Support cells within the leaf midribs of mosses deposit cellulose-rich secondary cell walls, but their biosynthesis and microfibril organization have not been examined. Although the Cellulose Synthase (CESA) gene families of mosses and seed plants diversified independently, CESA knockout analysis in the moss Physcomitrella patens revealed parallels with Arabidopsis (Arabidopsis thaliana) in CESA functional specialization, with roles for both subfunctionalization and neofunctionalization. The similarities include regulatory uncoupling of the CESAs that synthesize primary and secondary cell walls, a requirement for two or more functionally distinct CESA isoforms for secondary cell wall synthesis, interchangeability of some primary and secondary CESAs, and some CESA redundancy. The cellulose-deficient midribs of ppcesa3/8 knockouts provided negative controls for the structural characterization of stereid secondary cell walls in wild type P. patens. Sum frequency generation spectra collected from midribs were consistent with cellulose microfibril aggregation, and polarization microscopy revealed helical microfibril orientation only in wild type leaves. Thus, stereid secondary walls are structurally distinct from primary cell walls, and they share structural characteristics with the secondary walls of tracheary elements and fibers. We propose a mechanism for the convergent evolution of secondary walls in which the deposition of aggregated and helically oriented microfibrils is coupled to rapid and highly localized cellulose synthesis enabled by regulatory uncoupling from primary wall synthesis