Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

International audienceThe verification of floating-point mathematical libraries requires computing numerical bounds on approximation errors. Due to the tightness of these bounds and the peculiar structure of approximation errors, such a verification is out of the reach of generic tools such as computer algebra systems. In fact, the inherent difficulty of computing such bounds often mandates a formal proof of them. In this paper, we present a tactic for the Coq proof assistant that is designed to automatically and formally prove bounds on univariate expressions. It is based on a formalization of floating-point and interval arithmetic, associated with an on-the-fly computation of Taylor expansions. All the computations are performed inside Coq's logic, in a reflexive setting. This paper also compares our tactic with various existing tools on a large set of examples

Нейросекреторная активность супраоптического ядра переднего гипоталамуса кроликов под действием транскутанной электростимуляции зрительного анализатора

На 8 кролях породи Метелик вивчали вплив непрямої черезшкірної електростимуляції зорового аналізатора на нейросекреторну активність магноцелюлярних клітин супраоптичного ядра переднього гіпоталамусу. На мікропрепаратах інтактних тварин переважали нейрони II морфофункціонального типу, що перебувають у стадії синтезу нейросекрету. Показано, що за дії электростимуляції спостерігається перерозподіл головних морфо-функціональних типів нейронів. Відзначено збільшення змісту клітин I й III типів, відповідно у стадіях спокою після виведення секрету й накопичення, що вказує на активацію процесів звільнення нейросекрету і його акумуляції. Виразність реакції нервової тканини однакова при силі стимулюючого струму 100 мкА й 300 мкА.The influence of indirect through-skin electrostimulation (different doses) of the optical analyser on neurosecretory activity of anterior hypothalamus magnocellular nucleus was stading during chronic experiment. The stady was carried out on rabbits. Five morphological types of neurons was exposed in the supraoptical nucleus of control animal groop: I type- phase of rest after neurosecrets leading, II- phase of synthesis, III- phase of accumulation, IV - leading phase, V - phase of degerneration, but neurons of II types was prevalenced (51%). The indirect electrostimulation of the optical analyser provokes quantitative changes of keeping same neurons types. The number of I and III types neurons increases (on 20% and 7%) . The kind of changes is indicative of electrostimulation activation influense on neurosecrets leading and accumulation. Expression of nervous tissue reaction was identical under different doses (100 mkA and 300 mkA) of afferent electrostimulation

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

Formation and Growth of Oligomers: A Monte Carlo Study of an Amyloid Tau Fragment

Small oligomers formed early in the process of amyloid fibril formation may be the major toxic species in Alzheimer's disease. We investigate the early stages of amyloid aggregation for the tau fragment AcPHF6 (Ac-VQIVYK-NH2) using an implicit solvent all-atom model and extensive Monte Carlo simulations of 12, 24, and 36 chains. A variety of small metastable aggregates form and dissolve until an aggregate of a critical size and conformation arises. However, the stable oligomers, which are β-sheet-rich and feature many hydrophobic contacts, are not always growth-ready. The simulations indicate instead that these supercritical oligomers spend a lengthy period in equilibrium in which considerable reorganization takes place accompanied by exchange of chains with the solution. Growth competence of the stable oligomers correlates with the alignment of the strands in the β-sheets. The larger aggregates seen in our simulations are all composed of two twisted β-sheets, packed against each other with hydrophobic side chains at the sheet–sheet interface. These β-sandwiches show similarities with the proposed steric zipper structure for PHF6 fibrils but have a mixed parallel/antiparallel β-strand organization as opposed to the parallel organization found in experiments on fibrils. Interestingly, we find that the fraction of parallel β-sheet structure increases with aggregate size. We speculate that the reorganization of the β-sheets into parallel ones is an important rate-limiting step in the formation of PHF6 fibrils

Dimer Formation Enhances Structural Differences between Amyloid β-Protein (1–40) and (1–42): An Explicit-Solvent Molecular Dynamics Study

Amyloid -protein (A) is central to the pathology of Alzheimer's disease. A 5% difference in the primary structure of the two predominant alloforms, A and A, results in distinct assembly pathways and toxicity properties. Discrete molecular dynamics (DMD) studies of A and A assembly resulted in alloform-specific oligomer size distributions consistent with experimental findings. Here, a large ensemble of DMD–derived A and A monomers and dimers was subjected to fully atomistic molecular dynamics (MD) simulations using the OPLS-AA force field combined with two water models, SPCE and TIP3P. The resulting all-atom conformations were slightly larger, less compact, had similar turn and lower -strand propensities than those predicted by DMD. Fully atomistic A and A monomers populated qualitatively similar free energy landscapes. In contrast, the free energy landscape of A dimers indicated a larger conformational variability in comparison to that of A dimers. A dimers were characterized by an increased flexibility in the N-terminal region D1-R5 and a larger solvent exposure of charged amino acids relative to A dimers. Of the three positively charged amino acids, R5 was the most and K16 the least involved in salt bridge formation. This result was independent of the water model, alloform, and assembly state. Overall, salt bridge propensities increased upon dimer formation. An exception was the salt bridge propensity of K28, which decreased upon formation of A dimers and was significantly lower than in A dimers. The potential relevance of the three positively charged amino acids in mediating the A oligomer toxicity is discussed in the light of available experimental data

Combining Coq and Gappa for Certifying Floating-Point Programs

Formal verification of numerical programs is notoriously difficult. On the one hand, there exist automatic tools specialized in floating-point arithmetic, such as Gappa, but they target very restrictive logics. On the other hand, there are interactive theorem provers based on the LCF approach, such as Coq, that handle a general-purpose logic but that lack proof automation for floating-point properties. To alleviate these issues, we have implemented a mechanism for calling Gappa from a Coq interactive proof. This paper presents this combination and shows on several examples how this approach offers a significant speedup in the process of verifying floating-point programs

Strengths and weaknesses of data-driven docking in critical assessment of prediction of interactions

The recent CAPRI rounds have introduced new docking challenges in the form of protein-RNA complexes, multiple alternative interfaces, and an unprecedented number of targets for which homology modeling was required. We present here the performance of HADDOCK and its web server in the CAPRI experiment and discuss the strengths and weaknesses of data-driven docking. HADDOCK was successful for 6 out of 9 complexes (6 out of 11 targets) and accurately predicted the individual interfaces for two more complexes. The HADDOCK server, which is the first allowing the simultaneous docking of generic multi-body complexes, was successful in 4 out of 7 complexes for which it participated. In the scoring experiment, we predicted the highest number of targets of any group. The main weakness of data-driven docking revealed from these last CAPRI results is its vulnerability for incorrect experimental data related to the interface or the stoichiometry of the complex. At the same time, the use of experimental and/or predicted information is also the strength of our approach as evidenced for those targets for which accurate experimental information was available (e.g., the 10 three-stars predictions for T40!). Even when the models show a wrong orientation, the individual interfaces are generally well predicted with an average coverage of 60% 6 26% over all targets. This makes data-driven docking particularly valuable in a biological context to guide experimental studies like, for example, targeted mutagenesis