Spin-state splittings, highest-occupied-molecular-orbital and lowest-unoccupied-molecular-orbital energies, and chemical hardness.

It is known that the exact density functional must give ground-state energies that are piecewise linear as a function of electron number. In this work we prove that this is also true for the lowest-energy excited states of different spin or spatial symmetry. This has three important consequences for chemical applications: the ground state of a molecule must correspond to the state with the maximum highest-occupied-molecular-orbital energy, minimum lowest-unoccupied-molecular-orbital energy, and maximum chemical hardness. The beryllium, carbon, and vanadium atoms, as well as the CH(2) and C(3)H(3) molecules are considered as illustrative examples. Our result also directly and rigorously connects the ionization potential and electron affinity to the stability of spin states.

Regulation of DLK-1 kinase activity by calcium-mediated dissociation from an inhibitory isoform.

MAPKKK dual leucine zipper-bearing kinases (DLKs) are regulators of synaptic development and axon regeneration. The mechanisms underlying their activation are not fully understood. Here, we show that C. elegans DLK-1 is activated by a Ca(2+)-dependent switch from inactive heteromeric to active homomeric protein complexes. We identify a DLK-1 isoform, DLK-1S, that shares identical kinase and leucine zipper domains with the previously described long isoform DLK-1L but acts to inhibit DLK-1 function by binding to DLK-1L. The switch between homo- or heteromeric DLK-1 complexes is influenced by Ca(2+) concentration. A conserved hexapeptide in the DLK-1L C terminus is essential for DLK-1 activity and is required for Ca(2+) regulation. The mammalian DLK-1 homolog MAP3K13 contains an identical C-terminal hexapeptide and can functionally complement dlk-1 mutants, suggesting that the DLK activation mechanism is conserved. The DLK activation mechanism is ideally suited for rapid and spatially controlled signal transduction in response to axonal injury and synaptic activity.