Critical intensities of Boolean models with different underlying convex shapes

We study the complexity of query satisfiability and entailment for the Boolean information retrieval models WP and AWV using techniques from propositional logic and computational complexity. WP and AWV can be used to represent and query textual information under the Boolean model using the concept of attribute with values of type text, the concept of word, and word proximity constraints. Variations of WP and AWP are in use in most deployed digital libraries using the Boolean model, text extenders for relational database systems (e.g., Oracle 10g), search engines, and P2P systems for information retrieval and filtering

BackgroundCellular decision-making is governed by molecular networks that are highly complex. An integrative understanding of these networks on a genome wide level is essential to understand cellular health and disease. In most cases however, such an understanding is beyond human comprehension and requires computational modeling. Mathematical modeling of biological networks at the level of biochemical details has hitherto relied on state transition models. These are typically based on enumeration of all relevant model states, and hence become very complex unless severely – and often arbitrarily – reduced. Furthermore, the parameters required for genome wide networks will remain underdetermined for the conceivable future. Alternatively, networks can be simulated by Boolean models, although these typically sacrifice molecular detail as well as distinction between different levels or modes of activity. However, the modeling community still lacks methods that can simulate genome scale networks on the level of biochemical reaction detail in a quantitative or semi quantitative manner.ResultsHere, we present a probabilistic bipartite Boolean modeling method that addresses these issues. The method is based on the reaction-contingency formalism, and enables fast simulation of large networks. We demonstrate its scalability by applying it to the yeast mitogen-activated protein kinase (MAPK) network consisting of 140 proteins and 608 nodes.ConclusionThe probabilistic Boolean model can be generated and parameterized automatically from a rxncon network description, using only two global parameters, and its qualitative behavior is robust against order of magnitude variation in these parameters. Our method can hence be used to simulate the outcome of large signal transduction network reconstruction, with little or no overhead in model creation or parameterization.

Minkowski tensor density formulas for Boolean models

In Systems Biology, Boolean models are gaining popularity in modeling and analysis of qualitative dynamics of gene regulatory mechanisms. With the development of advanced high-throughput technologies, the availability of experimental data on gene-gene, gene-protein interactions is ever increasing. Consequently, modern Boolean models are increasing in size and complexity. Software simulation of Boolean models does not scale for such state-of-the-art complex Boolean models. We propose Field Programmable Gate Arrays (FPGAs) for emulation of Boolean models, exploiting the highly parallel nature of FPGAs. Our emulation framework consists of converting a Boolean model to Verilog, connecting it to an execution core, which runs on an FPGA coherently attached to a POWER8 processor. We report an order of magnitude speed up over a multi-threaded software simulation tool running on the same processor.

IntroductionPseudomonas aeruginosa infections are one of the leading causes of death in immunocompromised patients with cystic fibrosis, diabetes, and lung diseases such as pneumonia and bronchiectasis. Furthermore, P. aeruginosa is one of the main multidrug-resistant bacteria responsible for nosocomial infections worldwide, including the multidrug-resistant CCBH4851 strain isolated in Brazil.MethodsOne way to analyze their dynamic cellular behavior is through computational modeling of the gene regulatory network, which represents interactions between regulatory genes and their targets. For this purpose, Boolean models are important predictive tools to analyze these interactions. They are one of the most commonly used methods for studying complex dynamic behavior in biological systems.Results and discussionTherefore, this research consists of building a Boolean model of the gene regulatory network of P. aeruginosa CCBH4851 using data from RNA-seq experiments. Next, the basins of attraction are estimated, as these regions and the transitions between them can help identify the attractors, representing long-term behavior in the Boolean model. The essential genes of the basins were associated with the phenotypes of the bacteria for two conditions: biofilm formation and polymyxin B treatment. Overall, the Boolean model and the analysis method proposed in this work can identify promising control actions and indicate potential therapeutic targets, which can help pinpoint new drugs and intervention strategies.

Consider a continuum percolation model in which, at each point of a d-dimensional Poisson process of rate λ, a ball of radius 1 is centred. We show that, for any d ≧ 3, there exists a phase where both the regions, occupied and vacant, contain unbounded components. The proof uses the concept of enhancement for the Boolean model, and along the way we prove that the critical intensity of a Boolean model defined on a slab is strictly larger than the critical intensity of a Boolean model defined on the whole space.

In the face of incomplete data on a system of interest, constraint-based Boolean modeling still allows for elucidating system characteristics by analyzing sets of models consistent with the available information. In this setting, methods not depending on consideration of every single model in the set are necessary for efficient analysis. Drawing from ideas developed in qualitative differential equation theory, we present an approach to analyze sets of monotonic Boolean models consistent with given signed interactions between systems components. We show that for each such model constraints on its behavior can be derived from a universally constructed state transition graph essentially capturing possible sign changes of the derivative. Reachability results of the modeled system, e.g., concerning trap or no-return sets, can then be derived without enumerating and analyzing all models in the set. The close correspondence of the graph to similar objects for differential equations furthermore opens up ways to relate Boolean and continuous models.

This article considers the problem whereby, given two metabolic networks N1 and N2, a set of source compounds, and a set of target compounds, we must find the minimum set of reactions whose removal (knockout) ensures that the target compounds are not producible in N1 but are producible in N2. Similar studies exist for the problem of finding the minimum knockout with the smallest side effect for a single network. However, if technologies of external perturbations are advanced in the near future, it may be important to develop methods of computing the minimum knockout for multiple networks (MKMN). Flux balance analysis (FBA) is efficient if a well-polished model is available. However, that is not always the case. Therefore, in this article, we study MKMN in Boolean models and an elementary mode (EM)-based model. Integer linear programming (ILP)-based methods are developed for these models, since MKMN is NP-complete for both the Boolean model and the EM-based model. Computer experiments are conducted with metabolic networks of clostridium perfringens SM101 and bifidobacterium longum DJO10A, respectively known as bad bacteria and good bacteria for the human intestine. The results show that larger networks are more likely to have MKMN solutions. However, solving for these larger networks takes a very long time, and often the computation cannot be completed. This is reasonable, because small networks do not have many alternative pathways, making it difficult to satisfy the MKMN condition, whereas in large networks the number of candidate solutions explodes. Our developed software minFvskO is available online.

Dynamic Boolean models

The purpose of the paper is to study the asymptotic geometry of a smooth-grained Boolean model (X[t])t≥0 restricted to a bounded domain as the intensity parameter t goes to ∞. Our approach is based on investigating the asymptotic properties as t → ∞ of the random sets X[t;β], β≥0, defined as the Gibbsian modifications of X[t] with the Hamiltonian given by βtμ(·), where μ is a certain normalized measure on the setting space. We show that our model exhibits a phase transition at a certain critical value of the inverse temperature β and we prove that at higher temperatures the behaviour of X[t;β] is qualitatively very similar to that of X[t] but it becomes essentially different in the low-temperature region. From these facts we derive information about the asymptotic properties of the original process X[t]. The results obtained include large- and moderate-deviation principles. We conclude the paper with an example application of our methods to analyse the asymptotic moderate-deviation properties of convex hulls of large uniform samples on a multidimensional ball. To translate the above problem to the Boolean model setting considered we use an appropriate representation of convex sets in terms of their support functions.

Prostate cancer is the second most occurring cancer in men worldwide. To better understand the mechanisms of tumorigenesis and possible treatment responses, we developed a mathematical model of prostate cancer which considers the major signalling pathways known to be deregulated. We personalised this Boolean model to molecular data to reflect the heterogeneity and specific response to perturbations of cancer patients. A total of 488 prostate samples were used to build patient-specific models and compared to available clinical data. Additionally, eight prostate cell line-specific models were built to validate our approach with dose-response data of several drugs. The effects of single and combined drugs were tested in these models under different growth conditions. We identified 15 actionable points of interventions in one cell line-specific model whose inactivation hinders tumorigenesis. To validate these results, we tested nine small molecule inhibitors of five of those putative targets and found a dose-dependent effect on four of them, notably those targeting HSP90 and PI3K. These results highlight the predictive power of our personalised Boolean models and illustrate how they can be used for precision oncology.

Consider a family of Boolean models, indexed by integers n≥1. The nth model features a Poisson point process in ℝn of intensity e{nρn}, and balls of independent and identically distributed radii distributed like X̅n√n. Assume that ρn→ρ as n→∞, and that X̅n satisfies a large deviations principle. We show that there then exist the three deterministic thresholds τd, the degree threshold, τp, the percolation probability threshold, and τv, the volume fraction threshold, such that, asymptotically as n tends to ∞, we have the following features. (i) For ρ<τd, almost every point is isolated, namely its ball intersects no other ball; (ii) for τd<ρ<τp, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless there is no percolation; (iii) for τp<ρ<τv, the volume fraction is 0 and nevertheless percolation occurs; (iv) for τd<ρ<τv, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless the volume fraction is 0; (v) for ρ>τv, the whole space is covered. The analysis of this asymptotic regime is motivated by problems in information theory, but it could be of independent interest in stochastic geometry. The relations between these three thresholds and the Shannon‒Poltyrev threshold are discussed.

We are interested in estimating the intensity parameter of a Boolean model of discs (the bombing model) from a single realization. To do so, we derive the conditional distribution of the points (germs) of the underlying Poisson process. We demonstrate how to apply coupling from the past to generate samples from this distribution, and use the samples thus obtained to approximate the maximum likelihood estimator of the intensity. We discuss and compare two methods: one based on a Monte Carlo approximation of the likelihood function, the other a stochastic version of the EM algorithm.

Retrievability is a measure of access that quantifies how easily documents can be found using a retrieval system. Such a measure is of particular interest within the patent domain, because if a retrieval system makes some patents hard to find, then patent searchers will have a difficult time retrieving these patents. This may mean that a patent searcher could miss important and relevant patents because of the retrieval system. In this paper, we describe measures of retrievability and how they can be applied to measure the overall access to a collection given a retrieval system. We then identify three features of best-match retrieval models that are hypothesized to lead to an improvement in access to all documents in the collection: sensitivity to term frequency, length normalization and convexity. Since patent searchers tend to favor Boolean models over best-match models, hybrid retrieval models are proposed that incorporate these features while preserving the desirable aspects of the traditional Boolean model. An empirical study conducted on four large patent corpora demonstrates that these hybrid models provide better access to the corpus of patents than the traditional Boolean model.

Let $Z$ be a Boolean model based on a stationary Poisson process $\\eta$ of compact, convex particles in Euclidean space $\\mathbb{R}^{d}$. Let $W$ denote a compact, convex observation window. For a large class of functionals $\\psi$, formulas for mean values of $\\psi(Z\\cap W)$ are available in the literature. The first aim of the present work is to study the asymptotic covariances of general geometric (additive, translation invariant and locally bounded) functionals of $Z\\cap W$ for increasing observation window $W$, including convergence rates. Our approach is based on the Fock space representation associated with $\\eta$. For the important special case of intrinsic volumes, the asymptotic covariance matrix is shown to be positive definite and can be explicitly expressed in terms of suitable moments of (local) curvature measures in the isotropic case. The second aim of the paper is to prove multivariate central limit theorems including Berry–Esseen bounds. These are based on a general normal approximation result obtained by the Malliavin–Stein method.