The Solution Space of the Unitary Matrix Model String Equation and the Sato Grassmannian

The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. It is shown that the string equation is equivalent to simple conditions on points $V_1$ and $V_2$ in the big cell \Gr of the Sato Grassmannian $Gr$. This is a consequence of a well-defined continuum limit in which the string equation has the simple form \lb \cp ,\cq_- \rb =\hbox{\rm 1}, with \cp and \cq_- $2\times 2$ matrices of differential operators. These conditions on $V_1$ and $V_2$ yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation. This geometric formulation leads directly to the Virasoro constraints \L_n\,(n\geq 0), where \L_n annihilate the two modified-KdV \t-functions whose product gives the partition function of the Unitary Matrix Model.Comment: 21 page

A small molecule interacts with VDAC2 to block mouse BAK-driven apoptosis

Activating the intrinsic apoptosis pathway with small molecules is now a clinically validated approach to cancer therapy. In contrast, blocking apoptosis to prevent the death of healthy cells in disease settings has not been achieved. Caspases have been favored, but they act too late in apoptosis to provide long-term protection. The critical step in committing a cell to death is activation of BAK or BAX, pro-death BCL-2 proteins mediating mitochondrial damage. Apoptosis cannot proceed in their absence. Here we show that WEHI-9625, a novel tricyclic sulfone small molecule, binds to VDAC2 and promotes its ability to inhibit apoptosis driven by mouse BAK. In contrast to caspase inhibitors, WEHI-9625 blocks apoptosis before mitochondrial damage, preserving cellular function and long-term clonogenic potential. Our findings expand on the key role of VDAC2 in regulating apoptosis and demonstrate that blocking apoptosis at an early stage is both advantageous and pharmacologically tractable.Mark F. van Delft, Stephane Chappaz, Yelena Khakham, Chinh T. Bui, Marlyse A. Debrincat, Kym N. Lowes ... Benjamin T. Kile ... et al