Directed Supramolecular Surface Assembly of SNAP‐tag Fusion Proteins

Supramolecular assembly of proteins on surfaces and vesicles was investigated by site‐selective incorporation of a supramolecular guest element on proteins. Fluorescent proteins were site‐selectively labeled with bisadamantane by SNAP‐tag technology. The assembly of the bisadamantane functionalized SNAP‐fusion proteins on cyclodextrin‐coated surfaces yielded stable monolayers. The binding of the fusion proteins is specific and occurs with an affinity in the order of 10⁶ M ^?1 as determined by surface plasmon resonance. Reversible micropatterns of the fusion proteins on micropatterned cyclodextrin surfaces were visualized by using fluorescence microscopy. Furthermore, the guest‐functionalized proteins could be assembled out of solution specifically onto the surface of cyclodextrin vesicles. The SNAP‐tag labeling of proteins thus allows for assembly of modified proteins through a host–guest interaction on different surfaces. This provides a new strategy in fabricating protein patterns on surfaces and takes advantage of the high labeling efficiency of the SNAP‐tag with designed supramolecular elements.     

A Survey on Some Anisotropic Hardy-Type Function Spaces

Let $A$ be a general expansive matrix on $\mathbb{R}^n$. The aims of this article are twofold. The first one is to give a survey on the recent developments of anisotropic Hardy-type function spaces on $\mathbb{R}^n$, including anisotropic Hardy–Lorentz spaces, anisotropic variable Hardy spaces and anisotropic variable Hardy–Lorentz spaces as well as anisotropic Musielak–Orlicz Hardy spaces. The second one is to correct some errors and seal some gaps existing in the known articles. Some unsolved problems are also presented.     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators I

In this paper we study weighted function spaces of type B(?ⁿ, Q(x)) and F(?ⁿ, Q(x)), where Q(x) is a weight function of at most polynomial growth. Of special interest are the weight functions Q(x) = (1 + |x|²)^α/2 with α ? ?. The main result deals with estimates for the entropy numbers of compact embeddings between spaces of this type.     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators II

This paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type Ψ with χ χ ? ? and 0 ≤ γ ≤ 1 in the weighted function spaces B (?ⁿ, w(x)) and F (?ⁿ, w(x)) treated in [17]. Furthermore, we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b₂(x) b(x, D) b₁(x), where b₁(x) and b₂(x) are appropriate functions and b(x, D) ? Ψ. Finally, on the basis of the Birman-Schwinger principle, we deal with the “negative spectrum” (bound states) of related symmetric operators in L₂.     

Embeddings in Spaces of Lipschitz Type, Entropy and Approximation Numbers, and Applications

We consider the embeddings of certain Besov and Triebel–Lizorkin spaces in spaces of Lipschitz type. The prototype of such embeddings arises from the result of H. Brézis and S. Wainger (1980, Comm. Partial Differential Equations5, 773–789) about the “almost” Lipschitz continuity of elements of the Sobolev spaces H^1+n/p_p( ⁿ) when 1<p<∞. Two-sided estimates are obtained for the entropy and approximation numbers of a variety of related embeddings. The results are applied to give bounds for the eigenvalues of certain pseudo-differential operators and to provide information about the mapping properties of these operators.