The brassinosteroid signal transduction pathway

Steroids function as signaling molecules in both animals and plants. While animal steroid hormones are perceived by nuclear receptor family of transcription factors, brassinosteroids （BR） in plants are perceived by a cell surface receptor kinase, BRI 1. Recent studies have demonstrated that BR binding to the extracellular domain of BRI 1 induces kinase activation and dimerization with another receptor kinase, BAKI. Activated BRI 1 or BAKI then regulate, possibly indirectly, the activities of BIN2 kinase and/or BSU 1 phosphatase, which directly regulate the phosphorylation status and nuclear accumulation of two homologous transcription factors, BZRI and BES 1. BZRI and BES 1 directly bind to promoters of BR responsive genes to regulate their expression. The BR signaling pathway has become a paradigm for both receptor kinase signaling in plants and steroid signaling by cell surface receptors in general.

Analysis of Deep Ritz Methods for Semilinear Elliptic Equations

In this paper,we propose a method for solving semilinear elliptical equa-tions using a ResNet with ReLU2 activations.Firstly,we present a comprehensive formulation based on the penalized variational form of the elliptical equations.We then apply the Deep Ritz Method,which works for a wide range of equations.We obtain an upper bound on the errors between the acquired solutions and the true solutions in terms of the depth D,width W of the ReLU2 ResNet,and the num-ber of training samples n.Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.

A Rate of Convergence of Physics Informed Neural Networks for the Linear Second Order Elliptic PDEs

In recent years,physical informed neural networks(PINNs)have been shown to be a powerful tool for solving PDEs empirically.However,numerical analysis of PINNs is still missing.In this paper,we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition,by establishing the upper bounds on the number of training samples,depth and width of the deep neural networks to achieve desired accuracy.The error of PINNs is decomposed into approximation error and statistical error,where the approximation error is given in C2 norm with ReLU^(3)networks(deep network with activation function max{0,x^(3)})and the statistical error is estimated by Rademacher complexity.We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with ReLU^(3)network,which is of immense independent interest.

Practical Absorbing Boundary Conditions forWave Propagation on Arbitrary Domain

This paper presents an absorbing boundary conditions(ABCs)for wave propagations on arbitrary computational domains.The purpose of ABCs is to eliminate the unwanted spurious reflection at the artificial boundaries and minimize the finite size effect.Traditional methods are usually complicate in theoretical derivation and implementation and work only for very limited types of boundary geometry.In contrast to other existing methods,our emphasis is placed on the ease of implementation.In particular,we propose a method for which the implementation can be done by fitting or learning from the simulation data in a larger domain,and it is insensitive to the geometry and space dimension of the computational domain.Furthermore,a stability criterion is imposed to ensure the stability of the proposed ABC.Numerical results are presented to demonstrate the effectiveness of our method.

Proteomic Study Identifies Proteins Involved in Brassinosteroid Regulation of Rice Growth

Brassinosteroids （BRs） are essential hormones for growth and development of plant. In rice, BRs regulate multiple developmental processes and affect many important traits such as height, leaf angle, fertility and seed filling. We identified brassinosteroid-regulated proteins in rice using proteomic approaches and performed functional analysis of some BR-regulated proteins by overexpression experiments. Using two-dimensional difference gel electrophoresis （2-D DIGE） followed by protein identification by mass spectrometry, we compared proteomic differences in the shoots and roots of the BR-insensitive mutant d61-4 and BR-deficient mutant brd1-3. We identified a large number of proteins differentially expressed in the mutants compared with wild type control. These include a glycine-rich RNA-binding protein （OsGRP1） and a DREPP2 protein, which showed reduced levels in the BR mutants. Overexpression of these two proteins partially suppressed the dwarf phenotype of the Arabidopsis BR-insensitive mutant bri1-5. In contrast to the reduced protein level, the RNA level of OsGRP1 was not significantly affected in the BR mutants or by BR treatment, suggesting BR regulation of OsGRP1 at the posttranslational level. This study identifies many BR-regulated proteins and demonstrates that OsGRP1 functions downstream in the BR signal transduction pathway to promote cell expansion.

The Discontinuous Galerkin Method by Divergence-Free Patch Reconstruction for Stokes Eigenvalue Problems

The discontinuous Galerkin method by divergence-free patch reconstruction is proposed for Stokes eigenvalue problems.It utilizes the mixed finite element framework.The patch reconstruction technique constructs two categories of approximation spaces.Namely,the local divergence-free space is employed to discretize the velocity space,and the pressure space is approximated by standard reconstruction space simultaneously.Benefit from the divergence-free constraint;the identical element patch serves two approximation spaces while using the element pair Pm+1/Pm.The optimal error estimate is derived under the inf-sup condition framework.Numerical examples are carried out to validate the inf-sup test and the theoretical results.

Hierarchical Absorbing Interface Conditions for Wave Propagation on Non-Uniform Meshes

In this paper,we propose hierarchical absorbing interface conditions to solve the problem of wave propagation in domains with a non-uniform space discretization or grid size inhomogeneity using Pad´e Via Lanczos(PVL)method.The proposed interface conditions add an auxiliary variable in the wave system to eliminate the spurious reflection at the interface between regions with different mesh sizes.The auxiliary variable with proper boundary condition can suppress the spurious reflection by cancelling the boundary source term produced by the space inhomogeneity in variational perspective.The new hierarchical interface conditions with the help of PVL implementation can effectively reduce the degree of freedom in solving the wave propagation problem.

Absorbing Interface Conditions for the Simulation of Wave Propagation on Non-Uniform Meshes

We proposed absorbing interface conditions for the simulation of linear wave propagation on non-uniform meshes.Based on the superposition principle of second-order linear wave equations,we decompose the interface condition problem into two subproblems around the interface:for the first one the conventional artificial absorbing boundary conditions is applied,while for the second one,the local analytic solutions can be derived.The proposed interface conditions permit a two-way transmission of low-frequency waves across mesh interfaces which can be supported by both coarse and fine meshes,and perform a one-way absorption of high-frequency waves which can only be supported by fine meshes when they travel from fine mesh regions to coarse ones.Numerical examples are presented to illustrate the efficiency of the proposed absorbing interface conditions.