On Eigenvalues of Random Complexes

We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial-Meshulam model $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$ vertices. We show that for $p=\Omega(\log n/n)$, the eigenvalues of these matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of $(k-2)$-dimensional faces. Garland's result concerns the Laplacian; we develop an analogous result for the adjacency matrix. The same arguments apply to other models of random complexes which allow for dependencies between the choices of $k$-dimensional simplices. In the second part of the paper, we apply this to the question of possible higher-dimensional analogues of the discrete Cheeger inequality, which in the classical case of graphs relates the eigenvalues of a graph and its edge expansion. It is very natural to ask whether this generalizes to higher dimensions and, in particular, whether the higher-dimensional Laplacian spectra capture the notion of coboundary expansion - a generalization of edge expansion that arose in recent work of Linial and Meshulam and of Gromov. We show that this most straightforward version of a higher-dimensional discrete Cheeger inequality fails, in quite a strong way: For every $k\geq 2$ and $n\in \mathbb{N}$, there is a $k$-dimensional complex $Y^k_n$ on $n$ vertices that has strong spectral expansion properties (all nontrivial eigenvalues of the normalised $k$-dimensional Laplacian lie in the interval $[1-O(1/\sqrt{n}),1+O(1/\sqrt{n})]$) but whose coboundary expansion is bounded from above by $O(\log n/n)$ and so tends to zero as $n\rightarrow \infty$; moreover, $Y^k_n$ can be taken to have vanishing integer homology in dimension less than $k$.Comment: Extended full version of an extended abstract that appeared at SoCG 2012, to appear in Israel Journal of Mathematic

Dominant negative mutator mutations in the mutL gene of Escherichia coli.

The mutL gene product is part of the dam-directed mismatch repair system of Escherichia coli but has no known enzymatic function. It forms a complex on heteroduplex DNA with the mismatch recognition MutS protein and with MutH, which has latent endonuclease activity. An N-terminal hexahistidine-tagged MutL was constructed which was active in vivo. As a first stop to determine the functional domains of MutL, we have isolated 72 hydroxylamine-induced plasmid-borne mutations which impart a dominant-negative phenotype to the wild-type strain for increased spontaneous mutagenesis. None of the mutations complement a mutL deletion mutant, indicating that the mutant proteins by themselves are inactive. All the dominant mutations but one could be complemented by the wild-type mutL at about the same gene dosage. DNA sequencing indicated that the mutations affected 22 amino acid residues located between positions 16 and 549 of the 615 amino acid protein. In the N-terminal half of the protein, 12 out of 15 amino acid replacements occur at positions conserved in various eukaryotic MutL homologs. All but one of the sequence changes affecting the C-terminal end of the protein are nonsense mutations

The Escherichia coli MutL protein stimulates binding of Vsr and MutS to heteroduplex DNA.

Vsr DNA mismatch endonuclease is the key enzyme of very short patch (VSP) DNA mismatch repair and nicks the T-containing strand at the site of a T-G mismatch in a sequence-dependent manner. MutS is part of the mutHLS repair system and binds to diverse mismatches in DNA. The function of the mutL gene product is currently unclear but mutations in the gene abolish mutHLS -dependent repair. The absence of MutL severely reduces VSP repair but does not abolish it. Purified MutL appears to act catalytically to bind Vsr to its substrate; one-hundredth of an equivalent of MutL is sufficient to bring about a significant effect. MutL enhances binding of MutS to its substrate 6-fold but does so in a stoichiometric manner. Mutational studies indicate that the MutL interaction region lies within the N-terminal 330 amino acids and that the MutL multimerization region is at the C-terminal end. MutL mutant monomeric forms can stimulate MutS binding

Internalized FGF-2-Loaded Nanoparticles Increase Nuclear ERK1/2 Content and Result in Lung Cancer Cell Death

Innovative cancer treatments, which improve adjuvant therapy and reduce adverse events, are desperately needed. Nanoparticles provide controlled intracellular biomolecule delivery in the absence of activating external cell surface receptors. Prior reports suggest that intracrine signaling, following overexpression of basic fibroblast growth factor (FGF-2) after viral transduction, has a toxic effect on diseased cells. Herein, the research goals were to (1) encapsulate recombinant FGF-2 within stable, alginate-based nanoparticles (ABNs) for non-specific cellular uptake, and (2) determine the effects of ABN-mediated intracellular delivery of FGF-2 on cancer cell proliferation/survival. In culture, human alveolar adenocarcinoma basal epithelial cell line (A549s) and immortalized human bronchial epithelial cell line (HBE1s) internalized ABNs through non-selective endocytosis. Compared to A549s exposed to empty (i.e., blank) ABNs, the intracellular delivery of FGF-2 via ABNs significantly increased the levels of lactate dehydrogenase, indicating that FGF-2-ABN treatment decreased the transformed cell integrity. Noticeably, the nontransformed cells were not significantly affected by FGF-2-loaded ABN treatment. Furthermore, FGF-2-loaded ABNs significantly increased nuclear levels of activated-extracellular signal-regulated kinase ½ (ERK1/2) in A549s but had no significant effect on HBE1 nuclear ERK1/2 expression. Our novel intracellular delivery method of FGF-2 via nanoparticles resulted in increased cancer cell death via increased nuclear ERK1/2 activation

Random Chain Complexes

We study random, finite-dimensional, ungraded chain complexes over a finite field and show that for a uniformly distributed differential a complex has the smallest possible homology with the highest probability: either zero or one-dimensional homology depending on the parity of the dimension of the complex. We prove that as the order of the field goes to infinity the probability distribution concentrates in the smallest possible dimension of the homology. On the other hand, the limit probability distribution, as the dimension of the complex goes to infinity, is a super-exponentially decreasing, but strictly positive, function of the dimension of the homology

Phosphorylation by p38 MAPK as an alternative pathway for GSK3beta inactivation.

Glycogen synthase kinase 3beta (GSK3beta) is involved in metabolism, neurodegeneration, and cancer. Inhibition of GSK3beta activity is the primary mechanism that regulates this widely expressed active kinase. Although the protein kinase Akt inhibits GSK3beta by phosphorylation at the N terminus, preventing Akt-mediated phosphorylation does not affect the cell-survival pathway activated through the GSK3beta substrate beta-catenin. Here, we show that p38 mitogen-activated protein kinase (MAPK) also inactivates GSK3beta by direct phosphorylation at its C terminus, and this inactivation can lead to an accumulation of beta-catenin. p38 MAPK-mediated phosphorylation of GSK3beta occurs primarily in the brain and thymocytes. Activation of beta-catenin-mediated signaling through GSK3beta inhibition provides a potential mechanism for p38 MAPK-mediated survival in specific tissues