Kernel Free Boundary Integral Method for 3D Stokes and Navier Equations on Irregular Domains

A second-order accurate kernel-free boundary integral method is presented for Stokes and Navier boundary value problems on three-dimensional irregular domains. It solves equations in the framework of boundary integral equations, whose corresponding discrete forms are well-conditioned and solved by the GMRES method. A notable feature of this approach is that the boundary or volume integrals encountered in BIEs are indirectly evaluated by a Cartesian grid-based method, which includes discretizing corresponding simple interface problems with a MAC scheme, correcting discrete linear systems to reduce large local truncation errors near the interface, solving the modified system by a CG method together with an FFT-based Poisson solver. No extra work or special quadratures are required to deal with singular or hyper-singular boundary integrals and the dependence on the analytical expressions of Green's functions for the integral kernels is completely eliminated. Numerical results are given to demonstrate the efficiency and accuracy of the Cartesian grid-based method

Kernel-free boundary integral method for two-phase Stokes equations with discontinuous viscosity on staggered grids

A discontinuous viscosity coefficient makes the jump conditions of the velocity and normal stress coupled together, which brings great challenges to some commonly used numerical methods to obtain accurate solutions. To overcome the difficulties, a kernel free boundary integral (KFBI) method combined with a modified marker-and-cell (MAC) scheme is developed to solve the two-phase Stokes problems with discontinuous viscosity. The main idea is to reformulate the two-phase Stokes problem into a single-fluid Stokes problem by using boundary integral equations and then evaluate the boundary integrals indirectly through a Cartesian grid-based method. Since the jump conditions of the single-fluid Stokes problems can be easily decoupled, the modified MAC scheme is adopted here and the existing fast solver can be applicable for the resulting linear saddle system. The computed numerical solutions are second order accurate in discrete $\ell^2$-norm for velocity and pressure as well as the gradient of velocity, and also second order accurate in maximum norm for both velocity and its gradient, even in the case of high contrast viscosity coefficient, which is demonstrated in numerical tests

Second order convergence of a modified MAC scheme for Stokes interface problems

Stokes flow equations have been implemented successfully in practice for simulating problems with moving interfaces. Though computational methods produce accurate solutions and numerical convergence can be demonstrated using a resolution study, the rigorous convergence proofs are usually limited to particular reformulations and boundary conditions. In this paper, a rigorous error analysis of the marker and cell (MAC) scheme for Stokes interface problems with constant viscosity in the framework of the finite difference method is presented. Without reformulating the problem into elliptic PDEs, the main idea is to use a discrete Ladyzenskaja-Babuska-Brezzi (LBB) condition and construct auxiliary functions, which satisfy discretized Stokes equations and possess at least second order accuracy in the neighborhood of the moving interface. In particular, the method, for the first time, enables one to prove second order convergence of the velocity gradient in the discrete $\ell^2$-norm, in addition to the velocity and pressure fields. Numerical experiments verify the desired properties of the methods and the expected order of accuracy for both two-dimensional and three-dimensional examples

Survival effect of PDGF-CC rescues neurons from apoptosis in both brain and retina by regulating GSK3β phosphorylation

Platelet-derived growth factor CC (PDGF-CC) is the third member of the PDGF family discovered after more than two decades of studies on the original members of the family, PDGF-AA and PDGF-BB. The biological function of PDGF-CC remains largely to be explored. We report a novel finding that PDGF-CC is a potent neuroprotective factor that acts by modulating glycogen synthase kinase 3β (GSK3β) activity. In several different animal models of neuronal injury, such as axotomy-induced neuronal death, neurotoxin-induced neuronal injury, 6-hydroxydopamine–induced Parkinson’s dopaminergic neuronal death, and ischemia-induced stroke, PDGF-CC protein or gene delivery protected different types of neurons from apoptosis in both the retina and brain. On the other hand, loss-of-function assays using PDGF-C null mice, neutralizing antibody, or short hairpin RNA showed that PDGF-CC deficiency/inhibition exacerbated neuronal death in different neuronal tissues in vivo. Mechanistically, we revealed that the neuroprotective effect of PDGF-CC was achieved by regulating GSK3β phosphorylation and expression. Our data demonstrate that PDGF-CC is critically required for neuronal survival and may potentially be used to treat neurodegenerative diseases. Inhibition of the PDGF-CC–PDGF receptor pathway for different clinical purposes should be conducted with caution to preserve normal neuronal functions

A conserved motif flags acyl carrier proteins for β-branching in polyketide synthesis

Type I PKSs often utilise programmed β-branching, via enzymes of an “HMG-CoA synthase (HCS) cassette”, to incorporate various side chains at the second carbon from the terminal carboxylic acid of growing polyketide backbones. We identified a strong sequence motif in Acyl Carrier Proteins (ACPs) where β-branching is known. Substituting ACPs confirmed a correlation of ACP type with β-branching specificity. While these ACPs often occur in tandem, NMR analysis of tandem β-branching ACPs indicated no ACP-ACP synergistic effects and revealed that the conserved sequence motif forms an internal core rather than an exposed patch. Modelling and mutagenesis identified ACP Helix III as a probable anchor point of the ACP-HCS complex whose position is determined by the core. Mutating the core affects ACP functionality while ACP-HCS interface substitutions modulate system specificity. Our method for predicting β-carbon branching expands the potential for engineering novel polyketides and lays a basis for determining specificity rules