Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations

We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main features of our work is an efficient implementation of the verification procedure which can prove non-trivial high-dimensional inequalities in several seconds. We developed the verification tool as a part of the Flyspeck project (a formal proof of the Kepler conjecture). The Flyspeck project includes about 1000 nonlinear inequalities. We successfully tested our method on more than 100 Flyspeck inequalities and estimated that the formal verification procedure is about 3000 times slower than an informal verification method implemented in C++. We also describe future work and prospective optimizations for our method.Comment: 15 page

Rigorous Polynomial Approximation using Taylor Models in Coq

International audienceOne of the most common and practical ways of representing a real function on machines is by using a polynomial approximation. It is then important to properly handle the error introduced by such an approximation. The purpose of this work is to offer guaranteed error bounds for a specific kind of rigorous polynomial approximation called Taylor model. We carry out this work in the Coq proof assistant, with a special focus on genericity and efficiency for our implementation. We give an abstract interface for rigorous polynomial approximations, parameter- ized by the type of coefficients and the implementation of polynomials, and we instantiate this interface to the case of Taylor models with inter- val coefficients, while providing all the machinery for computing them. We compare the performances of our implementation in Coq with those of the Sollya tool, which contains an implementation of Taylor models written in C. This is a milestone in our long-term goal of providing fully formally proved and efficient Taylor models

A formal proof of the Kepler conjecture

This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project

Mechanisms controlling anaemia in Trypanosoma congolense infected mice.

Trypanosoma congolense are extracellular protozoan parasites of the blood stream of artiodactyls and are one of the main constraints on cattle production in Africa. In cattle, anaemia is the key feature of disease and persists after parasitaemia has declined to low or undetectable levels, but treatment to clear the parasites usually resolves the anaemia. The progress of anaemia after Trypanosoma congolense infection was followed in three mouse strains. Anaemia developed rapidly in all three strains until the peak of the first wave of parasitaemia. This was followed by a second phase, characterized by slower progress to severe anaemia in C57BL/6, by slow recovery in surviving A/J and a rapid recovery in BALB/c. There was no association between parasitaemia and severity of anaemia. Furthermore, functional T lymphocytes are not required for the induction of anaemia, since suppression of T cell activity with Cyclosporin A had neither an effect on the course of infection nor on anaemia. Expression of genes involved in erythropoiesis and iron metabolism was followed in spleen, liver and kidney tissues in the three strains of mice using microarrays. There was no evidence for a response to erythropoietin, consistent with anaemia of chronic disease, which is erythropoietin insensitive. However, the expression of transcription factors and genes involved in erythropoiesis and haemolysis did correlate with the expression of the inflammatory cytokines Il6 and Ifng. The innate immune response appears to be the major contributor to the inflammation associated with anaemia since suppression of T cells with CsA had no observable effect. Several transcription factors regulating haematopoiesis, Tal1, Gata1, Zfpm1 and Klf1 were expressed at consistently lower levels in C57BL/6 mice suggesting that these mice have a lower haematopoietic capacity and therefore less ability to recover from haemolysis induced anaemia after infection

A High Throughput Screen Identifies Chemical Modulators of the Laminin-Induced Clustering of Dystroglycan and Aquaporin-4 in Primary Astrocytes

Background: Aquaporin-4 (AQP4) constitutes the principal water channel in the brain and is clusteredat the perivascular astrocyte endfeet. This specific distribution of AQP4 plays a major role in maintaining water homeostasis in the brain. A growing body of evidence points to a role ofthe dystroglycan complex and its interaction with perivascular laminin in the clusteringof AQP4 atperivascular astrocyte endfeet. Indeed, mice lacking components of this complex or in which laminindystroglycan interaction is disrupted show a delayed onset of brain edema due to a redistribution of AQP4 away from astrocyte endfeet. It is therefore important to identify inhibitory drugs of laminin-dependent AQP4 clustering which may prevent or reduce brain edema. Methodolgy/Principal Findings: In the present study we used primary rat astrocyte cultures toscreen a library of.3,500 chemicals and identified 6 drugs that inhibit the laminin-induced clustering of dystroglycan and AQP4. Detailed analysis of the inhibitory drug, chloranil, revealed that its inhibition of the clustering is due to the metalloproteinase-2-mediated ß-dystroglycan shedding and subsequent loss of laminin interaction with dystroglycan. Furthermore, chemical variants of chloranil induced a similar effect on ß-dystroglycan and this was prevented by the antioxidant N-acetylcysteine. Conclusion/Significance: These findings reveal the mechanism of action of chloranil in preventing the laminin-induced clustering of dystroglycan and AQP4 and validate the use of high-throughput screening as a tool to identify drugs tha

Modification of Collagen by 3-Deoxyglucosone Alters Wound Healing through Differential Regulation of p38 MAP Kinase

Background: Wound healing is a highly dynamic process that requires signaling from the extracellular matrix to the fibroblasts for migration and proliferation, and closure of the wound. This rate of wound closure is impaired in diabetes, which may be due to the increased levels of the precursor for advanced glycation end products, 3-deoxyglucosone (3DG). Previous studies suggest a differential role for p38 mitogen-activated kinase (MAPK) during wound healing; whereby, p38 MAPK acts as a growth kinase during normal wound healing, but acts as a stress kinase during diabetic wound repair. Therefore, we investigated the signaling cross-talk by which p38 MAPK mediates wound healing in fibroblasts cultured on native collagen and 3DG-collagen. Methodology/Principal Findings: Using human dermal fibroblasts cultured on 3DG-collagen as a model of diabetic wounds, we demonstrated that p38 MAPK can promote either cell growth or cell death, and this was dependent on the activation of AKT and ERK1/2. Wound closure on native collagen was dependent on p38 MAPK phosphorylation of AKT and ERK1/2. Furthermore, proliferation and collagen production in fibroblasts cultured on native collagen was dependent on p38 MAPK regulation of AKT and ERK1/2. In contrast, 3DG-collagen decreased fibroblast migration, proliferation, and collagen expression through ERK1/2 and AKT downregulation via p38 MAPK. Conclusions/Significance: Taken together, the present study shows that p38 MAPK is a key signaling molecule that plays

Targeting insulin-like growth factor pathways

Some cancer cells depend on the function of specific molecules for their growth, survival, and metastatic potential. Targeting of these critical molecules has arguably been the best therapy for cancer as demonstrated by the success of tamoxifen and trastuzumab in breast cancer. This review will evaluate the type I IGF receptor (IGF-IR) as a potential target for cancer therapy. As new drugs come forward targeting this receptor system, several issues will need to be addressed in the early clinical trials using these agents

Integrative Genomic Data Mining for Discovery of Potential Blood-Borne Biomarkers for Early Diagnosis of Cancer

Background: With the arrival of the postgenomic era, there is increasing interest in the discovery of biomarkers for the accurate diagnosis, prognosis, and early detection of cancer. Blood-borne cancer markers are favored by clinicians, because blood samples can be obtained and analyzed with relative ease. We have used a combined mining strategy based on an integrated cancer microarray platform, Oncomine, and the biomarker module of the Ingenuity Pathways Analysis (IPA) program to identify potential blood-based markers for six common human cancer types. Methodology/Principal Findings: In the Oncomine platform, the genes overexpressed in cancer tissues relative to their corresponding normal tissues were filtered by Gene Ontology keywords, with the extracellular environment stipulated and a corrected Q value (false discovery rate) cut-off implemented. The identified genes were imported to the IPA biomarker module to separate out those genes encoding putative secreted or cell-surface proteins as blood-borne (blood/serum/plasma) cancer markers. The filtered potential indicators were ranked and prioritized according to normalized absolute Student t values. The retrieval of numerous marker genes that are already clinically useful or under active investigation confirmed the effectiveness of our mining strategy. To identify the biomarkers that are unique for each cancer type, the upregulated marker genes that are in common between each two tumor types across the six human tumors were also analyzed by the IPA biomarker comparison function. Conclusion/Significance: The upregulated marker genes shared among the six cancer types may serve as a molecular tool to complement histopathologic examination, and the combination of the commonly upregulated and unique biomarkers may serve as differentiating markers for a specific cancer. This approach will be increasingly useful to discover diagnostic signatures as the mass of microarray data continues to grow in the ‘omics’ era