From the nucleus, histone deacetylase 4 (HDAC4) regulates a variety of cellular processes, including growth, differentiation, and survival, by orchestrating transcriptional changes. Extracellular signals control its repressive influence mostly through regulating its nuclear-cytoplasmic shuttling. In particular, specific posttranslational modifications such as phosphorylation and caspase-mediated proteolytic processing operate on HDAC4 to promote its nuclear accumulation or export. To understand the signaling properties of this deacetylase, we investigated its cell death-promoting activity and the transcriptional repression potential of different mutants that accumulate in the nucleus. Here we show that, compared to that of other nuclear forms of HDAC4, a caspase-generated nuclear fragment exhibits a stronger cell death-promoting activity coupled with increased repressive effect on Runx2- or SRF-dependent transcription. However, this mutant displays reduced repressive action on MEF2C-driven transcription. Photobleaching experiments and quantitative analysis of the raw data, based on a two-binding-state compartmental model, demonstrate the existence of two nuclear pools of HDAC4 with different chromatin-binding properties. The caspase-generated fragment is weakly bound to chromatin, whereas an HDAC4 mutant defective in 14-3-3 binding or the wild-type HDAC5 protein forms a more stable complex. The tightly bound species show an impaired ability to induce cell death and repress Runx2- or SRF-dependent transcription less efficiently. We propose that, through specific posttranslation modifications, extracellular signals control two distinct nuclear pools of HDAC4 to differentially dictate cell death and differentiation. These two nuclear pools of HDAC4 are characterized by different repression potentials and divergent dynamics of chromatin interaction.

One of the most relevant tasks in network analysis is the detection of community structures, or clustering. Most popular techniques for community detection are based on the maximization of a quality function called modularity, which in turn is based upon particular quadratic forms associated to a real symmetric modularity matrix $M$, defined in terms of the adjacency matrix and a rank-one null model matrix. That matrix could be posed inside the set of relevant matrices involved in graph theory, alongside adjacency and Laplacian matrices. In this paper we analyze certain spectral properties of modularity matrices, which are related to the community detection problem. In particular, we propose a nodal domain theorem for the eigenvectors of $M$; we point out several relations occurring between the graph's communities and nonnegative eigenvalues of $M$; and we derive a Cheeger-type inequality for the graph modularity.

Let u be harmonic in the interior of a rectangle Ω and satisfy the third-kind boundary condition u_n + yu = Φ where Φ ⩾ 0, y ⩾ 0 with supports included in the bottom and in the top side of Ω, respectively. Recovering y from a knowledge of Φ and of the trace of u on the bottom Ω is a nonlinear inverse problem of interest in the field of nondestructive evaluation. A convergent Galerkin method for approximating y is proposed and tested in numerical experiments.