A cell complex in number theory

Let De_n be the simplicial complex of squarefree positive integers less than or equal to n ordered by divisibility. It is known that the asymptotic rate of growth of its Euler characteristic (the Mertens function) is closely related to deep properties of the prime number system. In this paper we study the asymptotic behavior of the individual Betti numbers and of their sum. We show that De_n has the homotopy type of a wedge of spheres, and that as n tends to infinity: \sum \be_k(\De_n) = \frac{2n}{\pi^2} + O(n^{\theta}),\;\; \mbox{for all} \theta > \frac{17}{54}. We also study a CW complex tDe_n that extends the previous simplicial complex. In tDe_n all numbers up to n correspond to cells and its Euler characteristic is the summatory Liouville function. This cell complex is shown to be homotopy equivalent to a wedge of spheres, and as n tends to infinity: \sum \be_k(\tDe_n) = \frac{n}{3} + O(n^{\theta}),\;\; \mbox{for all} \theta > \frac{22}{27}.Comment: 16 page

Effect of cell density on thrombin binding to a specific site on bovine vascular endothelial cells.

We studied thrombin binding to proliferating and confluent endothelial cells derived from bovine vascular endothelium. [125]thrombin was incubated with nonconfluent or confluent endothelial cells and both the total amount bound and the amount linked in a 77,000-dalton thrombin-cell complex were determined. Approximately 230,000 molecules of thrombin bound per cell in nonconfluent cultures compared to 12,800 molecules per cell in confluent cultures. Approximately 67,7000 thrombin molecules were bound in an apparently covalent complex, Mr = 77,000, with each cell in sparse cultures, whereas only 4,600 thrombin molecules per cell were bound in this complex with confluent cultures. Similar studies with [125I]thrombin and endothelial cells derived from bovine cornea revealed no difference either in the total amount of thrombin bound or in the amount bound in the 77,000-dalton complex using sparse or confluent cultures. When confluent vascular endothelial cultures were wounded, additional cellular binding sites for the 77,000-dalton complex with thrombin appeared within 24 h. A 237% increase in the amount of thrombin bound to these sites was induced by a wound which resulted in a 20% decrease in cell number in the monolayer. There was no significant increase in thrombin binding to other cellular sites at 24 h. These experiments provide evidence that the first change in thrombin binding after injury is an increase in the cellular sites involved in the 77,000-dalton complex, and suggest that thrombin binding to endothelial cells may be important in the vascular response to injury

Developmental motifs reveal complex structure in cell lineages

Many natural and technological systems are complex, with organisational structures that exhibit characteristic patterns, but defy concise description. One effective approach to analysing such systems is in terms of repeated topological motifs. Here, we extend the motif concept to characterise the dynamic behaviour of complex systems by introducing developmental motifs, which capture patterns of system growth. As a proof of concept, we use developmental motifs to analyse the developmental cell lineage of the nematode Caenorhabditis elegans, revealing a new perspective on its complex structure. We use a family of computational models to explore how biases arising from the dynamics of the developmental gene network, as well as spatial and temporal constraints acting on development, contribute to this complex organisation

Complex Systems Analysis of Cell Cycling Models in Carcinogenesis

A new approach to the modular, complex systems analysis of nonlinear dynamics in cell cycling network transformations involved in carcinogenesis is proposed. Carcinogenesis is a complex process that involves dynamically inter-connected biomolecules in the intercellular, membrane, cytosolic, nuclear and nucleolar compartments that form numerous inter-related pathways referred to as networks.
The variable biotopology of such dynamic networks is highly complex, and has a number of interesting properties that can be formally characterized at one level of organization by mathematical structures called 'biogroupoids'. 
One such family of pathways contains the cell cyclins. Cyclins are proteins that link several critical pro-apoptotic and other cell cycling/ division components, including the tumor suppressor gene TP53 and its product, the Thomsen-Friedenreich antigen (T antigen), Rb, mdm2, c-Myc, p21, p27, Bax, Bad and Bcl-2, which all play major roles in carcinogenesis of many cancers. A novel theoretical analysis is thus possible based on recently published studies of cyclin signaling, with special emphasis placed on the roles of cyclins D1 and E, suggests novel clinical trials and rational therapies of cancer through reestablishment of cell cycling inhibition in metastatic cancer cells

The Secreted Acid Phosphatase Domain-Containing GRA44 from Toxoplasma gondii Is Required for c-Myc Induction in Infected Cells.

During host cell invasion, the eukaryotic pathogen Toxoplasma gondii forms a parasitophorous vacuole to safely reside within the cell, while it is partitioned from host cell defense mechanisms. From within this safe niche, parasites sabotage multiple host cell systems, including gene expression, apoptosis, and intracellular immune recognition, by secreting a large arsenal of effector proteins. Many parasite proteins studied for active host cell manipulative interactions have been kinases. The translocation of effectors from the parasitophorous vacuole into the host cell is mediated by a putative translocon complex, which includes the proteins MYR1, MYR2, and MYR3. Whether other proteins are involved in the structure or regulation of this putative translocon is not known. We have discovered that the secreted protein GRA44, which contains a putative acid phosphatase domain, interacts with members of this complex and is required for host cell effects downstream of effector secretion. We have determined that GRA44 is processed in a region with homology to sequences targeted by protozoan proteases of the secretory pathway and that both major cleavage fragments are secreted into the parasitophorous vacuole. Immunoprecipitation experiments showed that GRA44 interacts with a large number of secreted proteins, including MYR1. Importantly, conditional knockdown of GRA44 resulted in a lack of host cell c-Myc upregulation, which mimics the phenotype seen when members of the translocon complex are genetically disrupted. Thus, the putative acid phosphatase GRA44 is crucial for host cell alterations during Toxoplasma infection and is associated with the translocon complex which Toxoplasma relies upon for success as an intracellular pathogen.IMPORTANCE Approximately one-third of humans are infected with the parasite Toxoplasma gondii Toxoplasma infections can lead to severe disease in those with a compromised or suppressed immune system. Additionally, infections during pregnancy present a significant health risk to the developing fetus. Drugs that target this parasite are limited, have significant side effects, and do not target all disease stages. Thus, a thorough understanding of how the parasite propagates within a host is critical in the discovery of novel therapeutic targets. Toxoplasma replication requires that it enter the cells of the infected organism. In order to survive the environment inside a cell, Toxoplasma secretes a large repertoire of proteins, which hijack a number of important cellular functions. How these Toxoplasma proteins move from the parasite into the host cell is not well understood. Our work shows that the putative phosphatase GRA44 is part of a protein complex responsible for this process

Tyrosine-Based Signals Regulate the Assembly of Daple⋅PARD3 Complex at Cell-Cell Junctions.

Polarized distribution of organelles and molecules inside a cell is vital for a range of cellular processes and its loss is frequently encountered in disease. Polarization during planar cell migration is a special condition in which cellular orientation is triggered by cell-cell contact. We demonstrate that the protein Daple (CCDC88C) is a component of cell junctions in epithelial cells which serves like a cellular "compass" for establishing and maintaining contact-triggered planar polarity. Furthermore, these processes may be mediated through interaction with the polarity regulator PARD3. This interaction, mediated by Daple's PDZ-binding motif (PBM) and the third PDZ domain of PARD3, is fine-tuned by tyrosine phosphorylation on Daple's PBM by receptor and non-receptor tyrosine kinases, such as Src. Hypophosphorylation strengthens the interaction, whereas hyperphosphorylation disrupts it, thereby revealing an unexpected role of Daple as a platform for signal integration and gradient sensing for tyrosine-based signals within the planar cell polarity pathway