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1. Chemical Biology Framework to Illuminate Proteostasis

   https://doi.org/10.1146/annurev-biochem-013118-111552  

   Sebastian, Rebecca M.; Shoulders, Matthew D. (June 2020, Annual Review of Biochemistry)

   null (Ed.)

   Protein folding in the cell is mediated by an extensive network of >1,000 chaperones, quality control factors, and trafficking mechanisms collectively termed the proteostasis network. While the components and organization of this network are generally well established, our understanding of how protein-folding problems are identified, how the network components integrate to successfully address challenges, and what types of biophysical issues each proteostasis network component is capable of addressing remains immature. We describe a chemical biology–informed framework for studying cellular proteostasis that relies on selection of interesting protein-folding problems and precise researcher control of proteostasis network composition and activities. By combining these methods with multifaceted strategies to monitor protein folding, degradation, trafficking, and aggregation in cells, researchers continue to rapidly generate new insights into cellular proteostasis. 

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2. Analysis of an hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

   https://doi.org/10.1051/m2an/2020015  

   Gong, Wei; Hu, Weiwei; Mateos, Mariano; Singler, John; Zhang, Yangwen (January 2020, ESAIM: Mathematical Modelling and Numerical Analysis)

   We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an $ L^2 $ penalty on the boundary control.  The contribution of this paper is twofold.  First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain.  The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary.  Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution.  For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary.  We present numerical experiments to demonstrate the performance of the HDG method. 

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3. A folding preprocess for the max k-cut problem

   https://doi.org/10.1007/s11590-025-02199-0  

   Fakhimi, Ramin; Validi, Hamidreza; Hicks, Illya V; Terlaky, Tamás; Zuluaga, Luis F (June 2025, Optimization Letters)

   Given graph G = (V, E) with vertex set V and edge set E, the max k-cut problem seeks to partition the vertex set V into at most k subsets that maximize the weight (number) of edges with endpoints in different parts. This paper proposes a graph folding procedure (i.e., a procedure that reduces the number of vertices and edges of graph G) for the weighted max k-cut problem that may help reduce the problem’s dimensionality. While our theoretical results hold for any k ≥ 2, our computational results show the effectiveness of the proposed preprocess only for k = 2 and on two sets of instances. Furthermore, we observe that the preprocess improves the performance of a MIP solver on a set of large-scale instances of the max cut problem. 

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4. Hidden convexity in the heat, linear transport, and Euler’s rigid body equations: A computational approach

   https://doi.org/10.1090/qam/1679  

   Kouskiya, Uditnarayan; Acharya, Amit (December 2024, Quarterly of Applied Mathematics)

   A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such ‘gauge invariance’ is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by noncausal dual problems. 

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5. Interpolation-based immersogeometric analysis methods for multi-material and multi-physics problems

   https://doi.org/10.1007/s00466-024-02506-z  

   Fromm, Jennifer E; Wunsch, Nils; Maute, Kurt; Evans, John A; Chen, Jiun-Shyan (June 2024, Computational Mechanics)

   Abstract Immersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted meshes, immersed boundary methods instead embed the computational domain in a structured background grid. Interpolation-based immersed boundary methods augment existing finite element software to non-invasively implement immersed boundary capabilities through extraction. Extraction interpolates the structured background basis as a linear combination of Lagrange polynomials defined on a foreground mesh, creating an interpolated basis that can be easily integrated by existing methods. This work extends the interpolation-based immersed isogeometric method to multi-material and multi-physics problems. Beginning from level-set descriptions of domain geometries, Heaviside enrichment is implemented to accommodate discontinuities in state variable fields across material interfaces. Adaptive refinement with truncated hierarchically refined B-splines (THB-splines) is used to both improve interface geometry representations and to resolve large solution gradients near interfaces. Multi-physics problems typically involve coupled fields where each field has unique discretization requirements. This work presents a novel discretization method for coupled problems through the application of extraction, using a single foreground mesh for all fields. Numerical examples illustrate optimal convergence rates for this method in both 2D and 3D, for partial differential equations representing heat conduction, linear elasticity, and a coupled thermo-mechanical problem. The utility of this method is demonstrated through image-based analysis of a composite sample, where in addition to circumventing typical meshing difficulties, this method reduces the required degrees of freedom when compared to classical boundary-fitted finite element methods. 

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