The 2+12+1 convex hull of a finite set

We study $\mathbb{R}^2\oplus\mathbb{R}$-separately convex hulls of finite sets of points in $\mathbb{R}^3$, as introduced in \cite{KirchheimMullerSverak2003}. When $\mathbb{R}^3$ is considered as a certain subset of $3\times 2$ matrices, this notion of convexity corresponds to rank-one convex convexity $K^{rc}$. If $\mathbb{R}^3$ is identified instead with a subset of $2\times 3$ matrices, it actually agrees with the quasiconvex hull, due to a recent result \cite{HarrisKirchheimLin18}. We introduce "$2+1$ complexes", which generalize $T_n$ constructions. For a finite set $K$, a "$2+1$ $K$-complex" is a $2+1$ complex whose extremal points belong to $K$. The "$2+1$-complex convex hull of $K$", $K^{cc}$, is the union of all $2+1$ $K$-complexes. We prove that $K^{cc}$ is contained in the $2+1$ convex hull $K^{rc}$. We also consider outer approximations to $2+1$ convexity based in the locality theorem \cite[4.7]{Kirchheim2003}. Starting with a crude outer approximation we iteratively chop off "$D$-prisms". For the examples in \cite{KirchheimMullerSverak2003}, and many others, this procedure reaches a "$2+1$ $K$-complex" in a finite number of steps, and thus computes the $2+1$ convex hull. We show examples of finite sets for which this procedure does not reach the $2+1$ convex hull in finite time, but we show that a sequence of outer approximations built with $D$-prisms converges to a $2+1$ $K$-complex. We conclude that $K^{rc}$ is always a "$2+1$ $K$-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