Symplectic cohomology and q-intersection numbers

Given a symplectic cohomology class of degree 1, we define the notion of an equivariant Lagrangian submanifold. The Floer cohomology of equivariant Lagrangian submanifolds has a natural endomorphism, which induces a grading by generalized eigenspaces. Taking Euler characteristics with respect to the induced grading yields a deformation of the intersection number. Dehn twists act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz fibrations give fully computable examples. A key step in computations is to impose the "dilation" condition stipulating that the BV operator applied to the symplectic cohomology class gives the identity. Equivariant Lagrangians mirror equivariant objects of the derived category of coherent sheaves.Comment: 32 pages, 9 figures, expanded introduction, added details of example 7.5, added discussion of sign

Targeting of zyxin to sites of actin membrane interaction and to the nucleus.

The localization of proteins to particular intracellular compartments often regulates their functions. Zyxin is a LIM protein found prominently at sites of cell adhesion, faintly in leading lamellipodia, and transiently in cell nuclei. Here we have performed a domain analysis to identify regions in zyxin that are responsible for targeting it to different subcellular locations. The N-terminal proline-rich region of zyxin, which harbors binding sites for alpha-actinin and members of the Ena/VASP family, concentrates in lamellipodial extensions and weakly in focal adhesions. The LIM region of zyxin displays robust targeting to focal adhesions. When overexpressed in cells, the LIM region of zyxin causes displacement of endogenous zyxin from focal adhesions. Upon mislocalization of full-length zyxin, at least one member of the Ena/VASP family is also displaced, and the organization of the actin cytoskeleton is perturbed. Zyxin also has the capacity to shuttle between the nucleus and focal adhesion sites. When nuclear export is inhibited, zyxin accumulates in cell nuclei. The nuclear accumulation of zyxin occurs asynchronously with approximately half of the cells exhibiting nuclear localization of zyxin within 2.3 h of initiating leptomycin B treatment. Our results provide insight into the functions of different zyxin domains