Background: Recent developments have meant that network theory is making an important contribution to the topological study of biological networks, such as protein-protein interaction (PPI) networks. The identification of differentially expressed genes in DNA array experiments is a source of information regarding the molecular pathways involved in disease. Thus, considering PPI analysis and gene expression studies together may provide a better understanding of multifactorial neurodegenerative diseases such as Multiple Sclerosis (MS) and Alzheimer disease (AD). The aim of this study was to assess whether the parameters of degree and betweenness, two fundamental measures in network theory, are properties that differentiate between implicated (seed-proteins) and non-implicated nodes (neighbors) in MS and AD. We used experimentally validated PPI information to obtain the neighbors for each seed group and we studied these parameters in four networks: MS-blood network; MS-brain network; AD-blood network; and AD-brain network. Results: Specific features of seed-proteins were revealed, whereby they displayed a lower average degree in both diseases and tissues, and a higher betweenness in AD-brain and MS-blood networks. Additionally, the heterogeneity of the processes involved indicate that these findings are not pathway specific but rather that they are spread over different pathways. Conclusion: Our findings show differential centrality properties of proteins whose gene expression is impaired in neurodegenerative diseases.

We use the first Betti number of a complex to analyze the morphological structure of granular samples in mechanical equilibrium. We investigate two-dimensional granular packings after a tapping process by means of both simulations and experiments. States with equal packing fraction obtained with different tapping intensities are distinguished after the introduction of a filtration parameter which determines the particles (nodes in the network) that are joined by an edge. This is accomplished by just using the position of the particles obtained experimentally and no other information about the possible contacts, or magnitude of forces.

We study Pontryagin reflexivity in the class of precompact topological Abelian groups. We find reflexive groups among precompact not pseudocompact and among pseu- docompact not compact groups. Making use of Martin’s axiom we give an example of a reflexive countably compact not compact Abelian group. We also prove that every pseu- docompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.