Structure of human Frizzled5 by fiducial-assisted cryo-EM supports a heterodimeric mechanism of canonical Wnt signaling

Frizzleds (Fzd) are the primary receptors for Wnt morphogens, which are essential regulators of stem cell biology, yet the structural basis of Wnt signaling through Fzd remains poorly understood. Here we report the structure of an unliganded human Fzd5 determined by single-particle cryo-EM at 3.7 Å resolution, with the aid of an antibody chaperone acting as a fiducial marker. We also analyzed the topology of low-resolution XWnt8/Fzd5 complex particles, which revealed extreme flexibility between the Wnt/Fzd-CRD and the Fzd-TM regions. Analysis of Wnt/β-catenin signaling in response to Wnt3a versus a ‘surrogate agonist’ that cross-links Fzd to LRP6, revealed identical structure-activity relationships. Thus, canonical Wnt/β-catenin signaling appears to be principally reliant on ligand-induced Fzd/LRP6 heterodimerization, versus the allosteric mechanisms seen in structurally analogous class A G protein-coupled receptors, and Smoothened. These findings deepen our mechanistic understanding of Wnt signal transduction, and have implications for harnessing Wnt agonism in regenerative medicine.

Dilations and Hahn Decompositions for Linear Maps

Suppose is a C*-algebra and H is a Hilbert space. Let denote the set of completely positive maps from into the set B(H) of (bounded linear) operators on H. This paper studies the vector space spanned by , i.e., the linear maps that are finite linear combinations of completely positive maps. From another viewpoint, a map ϕ is in precisely when it has a decomposition ϕ = (ϕ1 – ϕ2) + i(ϕ3 – ϕ4) with ϕ1, ϕ2, ϕ3, ϕ4 in CP ; this decomposition is analogous to the Hahn decomposition for measures [8, 111.4.10] (see also Theorem 20). The analogous class of maps with “completely positive” replaced by “positive” was studied by R. I. Loebl [11] and S.-K. Tsui [17], and when is commutative, this latter class coincides withi , since every positive linear map on a commutative C*-algebra is completely positive [16].

Minimally strong digraphs

Dirac (2) and Plummer (5) independently investigated the structure of minimally 2-connected graphs G, which are characterized by the property that for any line x of G, G–x is not 2-connected. In this paper we investigate an analogous class of strongly connected digraphs D such that for any arc x, D–x is not strong. Not surprisingly, these digraphs have much in common with the minimally 2-connected graphs, and a number of theorems similar to those in (2) and (5) are proved, notably our Theorems 9 and 12.