476 research outputs found
The convex hull of a finite set
We study -separately convex hulls of finite
sets of points in , as introduced in
\cite{KirchheimMullerSverak2003}. When is considered as a
certain subset of matrices, this notion of convexity corresponds
to rank-one convex convexity . If is identified instead
with a subset of matrices, it actually agrees with the quasiconvex
hull, due to a recent result \cite{HarrisKirchheimLin18}.
We introduce " complexes", which generalize constructions. For a
finite set , a " -complex" is a complex whose extremal points
belong to . The "-complex convex hull of ", , is the union
of all -complexes. We prove that is contained in the
convex hull .
We also consider outer approximations to convexity based in the
locality theorem \cite[4.7]{Kirchheim2003}. Starting with a crude outer
approximation we iteratively chop off "-prisms". For the examples in
\cite{KirchheimMullerSverak2003}, and many others, this procedure reaches a
" -complex" in a finite number of steps, and thus computes the
convex hull.
We show examples of finite sets for which this procedure does not reach the
convex hull in finite time, but we show that a sequence of outer
approximations built with -prisms converges to a -complex. We
conclude that is always a " -complex", which has interesting
consequences
Divergent traits and ligand-binding properties of the cytomegalovirus CD48 gene family
The genesis of gene families by the capture of host genes and their subsequent duplication is a crucial process in the evolution of large DNA viruses. CD48 is a cell surface molecule that interacts via its N-terminal immunoglobulin (Ig) domain with the cell surface receptor 2B4 (CD244), regulating leukocyte cytotoxicity.We previously reported the presence of five CD48 homologs (vCD48s) in two related cytomegaloviruses, and demonstrated that one of them, A43, binds 2B4 and acts as a soluble CD48 decoy receptor impairing NK cell function. Here, we have characterized the rest of these vCD48s. We show that they are highly glycosylated proteins that display remarkably distinct features: divergent biochemical properties, cellular locations, and temporal expression kinetics. In contrast to A43, none of them interacts with 2B4. Consistent with this, molecular modeling of the N-terminal Ig domains of these vCD48s evidences notable changes as compared to CD48, suggesting that they interact with alternative targets. Accordingly, we demonstrate that one of them, S30, tightly binds CD2, a crucial T- and NK-cell adhesion and costimulatory molecule. Thus, our findings show how a key host immune receptor gene captured by a virus can be subsequently remodeled to evolve new immunoevasins with altered binding properties
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