A Bayesian framework for inference of the genotype-phenotype map for segregating populations

Complex genetic interactions lie at the foundation of many diseases. Understanding the nature of these interactions is critical to developing rational intervention strategies. In mammalian systems hypothesis testing in vivo is expensive, time consuming, and often restricted to a few physiological endpoints. Thus, computational methods that generate causal hypotheses can help to prioritize targets for experimental intervention. We propose a Bayesian statistical method to infer networks of causal relationships among genotypes and phenotypes using expression quantitative trait loci (eQTL) data from genetically randomized populations. Causal relationships between network variables are described with hierarchical regression models. Prior distributions on the network structure enforce graph sparsity and have the potential to encode prior biological knowledge about the network. An efficient Monte Carlo method is used to search across the model space and sample highly probable networks. The result is an ensemble of networks that provide a measure of confidence in the estimated network topology. These networks can be used to make predictions of system-wide response to perturbations. We applied our method to kidney gene expression data from an MRL/MpJ × SM/J intercross population and predicted a previously uncharacterized feedback loop in the local renin-angiotensin system.     

Protein fold recognition by total alignment probability

We present a protein fold-recognition method that uses a comprehensive statistical interpretation of structural Hidden Markov Models (HMMs). The structure/fold recognition is done by summing the probabilities of all sequence-to-structure alignments. The optimal alignment can be defined as the most probable, but suboptimal alignments may have comparable probabilities. These suboptimal alignments can be interpreted as optimal alignments to the "other" structures from the ensemble or optimal alignments under minor fluctuations in the scoring function. Summing probabilities for all alignments gives a complete estimate of sequence-model compatibility. In the case of HMMs that produce a sequence, this reflects the fact that due to our indifference to exactly how the HMM produced the sequence, we should sum over all possibilities. We have built a set of structural HMMs for 188 protein structures and have compared two methods for identifying the structure compatible with a sequence: by the optimal alignment probability and by the total probability. Fold recognition by total probability was 40% more accurate than fold recognition by the optimal alignment probability. Proteins 2000;40:451-462.     

A neural network approach to approximating MAP in belief networks

Bayesian belief networks (BBN) are a widely studied graphical model for representing uncertainty and probabilistic interdependence among variables. One of the factors that restricts the model's wide acceptance in practical applications is that the general inference with BBN is NP-hard. This is also true for the maximum a posteriori probability (MAP) problem, which is to find the most probable joint value assignment to all uninstantiated variables, given instantiation of some variables in a BBN. To circumvent the difficulty caused by MAP's computational complexity, we suggest in this paper a neural network approximation approach. With this approach, a BBN is treated as a neural network without any change or transformation of the network structure, and the node activation functions are derived based on an energy function defined over a given BBN. Three methods are developed. They are the hill-climbing style discrete method, the simulated annealing method, and the continuous method based on the mean field theory. All three methods are for BBN of general structures, with the restriction that nodes of BBN are binary variables. In addition, rules for applying these methods to noisy-or networks are also developed, which may lead to more efficient computation in some cases. These methods' convergence is analyzed, and their validity tested through a series of computer experiments with two BBN of moderate size and complexity. Although additional theoretical and empirical work is needed, the analysis and experiments suggest that this approach may lead to effective and accurate approximation for MAP problems.     

Causal inference in biomolecular pathways using a Bayesian network approach and an Implicit method

We introduce here the concept of Implicit networks which provide, like Bayesian networks, a graphical modelling framework that encodes the joint probability distribution for a set of random variables within a directed acyclic graph. We show that Implicit networks, when used in conjunction with appropriate statistical techniques, are very attractive for their ability to understand and analyze biological data. Particularly, we consider here the use of Implicit networks for causal inference in biomolecular pathways. In such pathways, an Implicit network encodes dependencies among variables (proteins, genes), can be trained to learn causal relationships (regulation, interaction) between them and then used to predict the biological response given the status of some key proteins or genes in the network. We show that Implicit networks offer efficient methodologies for learning from observations without prior knowledge and thus provide a good alternative to classical inference in Bayesian networks when priors are missing. We illustrate our approach by an application to simulated data for a simplified signal transduction pathway of the epidermal growth factor receptor (EGFR) protein.     

Predicting protein function from protein/protein interaction data: a probabilistic approach

MOTIVATION:The development of experimental methods for genome scale analysis of molecular interaction networks has made possible new approaches to inferring protein function. This paper describes a method of assigning functions based on a probabilistic analysis of graph neighborhoods in a protein-protein interaction network. The method exploits the fact that graph neighbors are more likely to share functions than nodes which are not neighbors. A binomial model of local neighbor function labeling probability is combined with a Markov random field propagation algorithm to assign function probabilities for proteins in the network. RESULTS: We applied the method to a protein-protein interaction dataset for the yeast Saccharomyces cerevisiae using the Gene Ontology (GO) terms as function labels. The method reconstructed known GO term assignments with high precision, and produced putative GO assignments to 320 proteins that currently lack GO annotation, which represents about 10% of the unlabeled proteins in S. cerevisiae.     

Using iterative ridge regression to explore associations between conditioned variables

We address a specific case of joint probability mapping, where the information presented is the probabilistic associations of random variables under a certain condition variable (conditioned associations). Bayesian and dependency networks graphically map the joint probabilities of random variables, though both networks may identify associations that are independent of the condition (background associations). Since the background associations have the same topological features as conditioned associations, it is difficult to discriminate between conditioned and non-conditioned associations, which results in a major increase in the search space. We introduce a modification of the dependency network method, which produces a directed graph, containing only condition-related associations. The graph nodes represent the random variables and the graph edges represent the associations that arise under the condition variable. This method is based on ridge-regression, where one can utilize a numerically robust and computationally efficient algorithm implementation. We illustrate the method's efficiency in the context of a medically relevant process, the emergence of drug-resistant variants of human immunodeficiency virus (HIV) in drug-treated, HIV-infected people. Our mapping was used to discover associations between variants that are conditioned by the initiation of a particular drug treatment regimen. We have demonstrated that our method can recover known associations of such treatment with selected resistance mutations as well as documented associations between different mutations. Moreover, our method revealed novel associations that are statistically significant and biologically plausible.     

An approximation method for solving the steady-state probability distribution of probabilistic Boolean networks

MOTIVATION: Probabilistic Boolean networks (PBNs) have been proposed to model genetic regulatory interactions. The steady-state probability distribution of a PBN gives important information about the captured genetic network. The computation of the steady-state probability distribution usually includes construction of the transition probability matrix and computation of the steady-state probability distribution. The size of the transition probability matrix is 2(n)-by-2(n) where n is the number of genes in the genetic network. Therefore, the computational costs of these two steps are very expensive and it is essential to develop a fast approximation method. RESULTS: In this article, we propose an approximation method for computing the steady-state probability distribution of a PBN based on neglecting some Boolean networks (BNs) with very small probabilities during the construction of the transition probability matrix. An error analysis of this approximation method is given and theoretical result on the distribution of BNs in a PBN with at most two Boolean functions for one gene is also presented. These give a foundation and support for the approximation method. Numerical experiments based on a genetic network are given to demonstrate the efficiency of the proposed method.     

Translated Chemical Reaction Networks

Many biochemical and industrial applications involve complicated networks of simultaneously occurring chemical reactions. Under the assumption of mass action kinetics, the dynamics of these chemical reaction networks are governed by systems of polynomial ordinary differential equations. The steady states of these mass action systems have been analyzed via a variety of techniques, including stoichiometric network analysis, deficiency theory, and algebraic techniques (e.g., Gröbner bases). In this paper, we present a novel method for characterizing the steady states of mass action systems. Our method explicitly links a network’s capacity to permit a particular class of steady states, called toric steady states, to topological properties of a generalized network called a translated chemical reaction network. These networks share their reaction vectors with their source network but are permitted to have different complex stoichiometries and different network topologies. We apply the results to examples drawn from the biochemical literature.     

Drug2ways: Reasoning over causal paths in biological networks for drug discovery

Elucidating the causal mechanisms responsible for disease can reveal potential therapeutic targets for pharmacological intervention and, accordingly, guide drug repositioning and discovery. In essence, the topology of a network can reveal the impact a drug candidate may have on a given biological state, leading the way for enhanced disease characterization and the design of advanced therapies. Network-based approaches, in particular, are highly suited for these purposes as they hold the capacity to identify the molecular mechanisms underlying disease. Here, we present drug2ways, a novel methodology that leverages multimodal causal networks for predicting drug candidates. Drug2ways implements an efficient algorithm which reasons over causal paths in large-scale biological networks to propose drug candidates for a given disease. We validate our approach using clinical trial information and demonstrate how drug2ways can be used for multiple applications to identify: i) single-target drug candidates, ii) candidates with polypharmacological properties that can optimize multiple targets, and iii) candidates for combination therapy. Finally, we make drug2ways available to the scientific community as a Python package that enables conducting these applications on multiple standard network formats.     

Probabilities of spurious connections in gene networks: application to expression time series

MOTIVATION: The reconstruction of gene networks from gene-expression microarrays is gaining popularity as methods improve and as more data become available. The reliability of such networks could be judged by the probability that a connection between genes is spurious, resulting from chance fluctuations rather than from a true biological relationship. RESULTS: Unlike the false discovery rate and positive false discovery rate, the decisive false discovery rate (dFDR) is exactly equal to a conditional probability without assuming independence or the randomness of hypothesis truth values. This property is useful not only in the common application to the detection of differential gene expression, but also in determining the probability of a spurious connection in a reconstructed gene network. Estimators of the dFDR can estimate each of three probabilities: (1) The probability that two genes that appear to be associated with each other lack such association. (2) The probability that a time ordering observed for two associated genes is misleading. (3) The probability that a time ordering observed for two genes is misleading, either because they are not associated or because they are associated without a lag in time. The first probability applies to both static and dynamic gene networks, and the other two only apply to dynamic gene networks.     

Biochemical networks with uncertain parameters

The modelling of biochemical networks becomes delicate if kinetic parameters are varying, uncertain or unknown. Facing this situation, we quantify uncertain knowledge or beliefs about parameters by probability distributions. We show how parameter distributions can be used to infer probabilistic statements about dynamic network properties, such as steady-state fluxes and concentrations, signal characteristics or control coefficients. The parameter distributions can also serve as priors in Bayesian statistical analysis. We propose a graphical scheme, the 'dependence graph', to bring out known dependencies between parameters, for instance, due to the equilibrium constants. If a parameter distribution is narrow, the resulting distribution of the variables can be computed by expanding them around a set of mean parameter values. We compute the distributions of concentrations, fluxes and probabilities for qualitative variables such as flux directions. The probabilistic framework allows the study of metabolic correlations, and it provides simple measures of variability and stochastic sensitivity. It also shows clearly how the variability of biological systems is related to the metabolic response coefficients.     

Some properties of transmission lines composed of random networks

This paper discusses some properties of a nerve-axon-like transmission line made up from random networks of threshold elements. A random network contains a large number of threshold elements of which the threshold values are Gaussian random variables, and it can act as a monostable or a bistable multivibrator. The signal wave propagation in a transmission line is analyzed by a statistical method. The results show that a signal wave can propagate along the line and that its waveform is shaped, during propagation, into a specific form peculiar to the line. A self-oscillatory system that consists of two random networks is also analyzed. Even in this simple system, various modes of oscillation can exist, with periods varying over a wide range according to the values of system parameters. Examples of self-oscillations obtained by a graphical method are presented, and also some results of computer experiments are shown.     

Intervention in a family of Boolean networks

MOTIVATION: Intervention in a gene regulatory network is used to avoid undesirable states, such as those associated with a disease. Several types of intervention have been studied in the framework of a probabilistic Boolean network (PBN), which is a collection of Boolean networks in which the gene state vector transitions according to the rules of one of the constituent networks and where network choice is governed by a selection distribution. The theory of automatic control has been applied to find optimal strategies for manipulating external control variables that affect the transition probabilities to desirably affect dynamic evolution over a finite time horizon. In this paper we treat a case in which we lack the governing probability structure for Boolean network selection, so we simply have a family of Boolean networks, but where these networks possess a common attractor structure. This corresponds to the situation in which network construction is treated as an ill-posed inverse problem in which there are many Boolean networks created from the data under the constraint that they all possess attractor structures matching the data states, which are assumed to arise from sampling the steady state of the real biological network. RESULTS: Given a family of Boolean networks possessing a common attractor structure composed of singleton attractors, a control algorithm is derived by minimizing a composite finite-horizon cost function that is a weighted average over all the individual networks, the idea being that we desire a control policy that on average suits the networks because these are viewed as equivalent relative to the data. The weighting for each network at any time point is taken to be proportional to the instantaneous estimated probability of that network being the underlying network governing the state transition. The results are applied to a family of Boolean networks derived from gene-expression data collected in a study of metastatic melanoma, the intent being to devise a control strategy that reduces the WNT5A gene's action in affecting biological regulation. AVAILABILITY: The software is available on request. SUPPLEMENTARY INFORMATION: The supplementary Information is available at http://ee.tamu.edu/~edward/tree     

A Bayesian network approach incorporating imputation of missing data enables exploratory analysis of complex causal biological relationships

Bayesian networks can be used to identify possible causal relationships between variables based on their conditional dependencies and independencies, which can be particularly useful in complex biological scenarios with many measured variables. Here we propose two improvements to an existing method for Bayesian network analysis, designed to increase the power to detect potential causal relationships between variables (including potentially a mixture of both discrete and continuous variables). Our first improvement relates to the treatment of missing data. When there is missing data, the standard approach is to remove every individual with any missing data before performing analysis. This can be wasteful and undesirable when there are many individuals with missing data, perhaps with only one or a few variables missing. This motivates the use of imputation. We present a new imputation method that uses a version of nearest neighbour imputation, whereby missing data from one individual is replaced with data from another individual, their nearest neighbour. For each individual with missing data, the subsets of variables to be used to select the nearest neighbour are chosen by sampling without replacement the complete data and estimating a best fit Bayesian network. We show that this approach leads to marked improvements in the recall and precision of directed edges in the final network identified, and we illustrate the approach through application to data from a recent study investigating the causal relationship between methylation and gene expression in early inflammatory arthritis patients. We also describe a second improvement in the form of a pseudo-Bayesian approach for upweighting certain network edges, which can be useful when there is prior evidence concerning their directions.     

Deviations from Michaelis-Menten kinetics. Computation of the probabilities of obtaining complex curves from simple kinetic schemes.

1. It is possible to calculate the intrinsic probability associated with any curve shape that is allowed for rational functions of given degree when the coefficients are independent or dependent random variables with known probability distributions. 2. Computations of such probabilities are described when the coefficients of the rational function are generated according to several probability distribution functions and in particular when rate constants are varied randomly for several simple model mechanisms. 3. It is concluded that each molecular mechanism is associated with a specific set of curve-shape probabilities, and this could be of value in discriminating between model mechanisms. 4. It is shown how a computer program can be used to estimate the probability of new complexities such as extra inflexions and turning points as the degree of rate equations increases. 5. The probability of 3 : 3 rate equations giving 2 : 2 curve shapes is discussed for unrestricted coefficients and also for the substrate-modifier mechanisms. 6. The probability associated with the numerical values of coefficients in rate equations is also calculated for this mechanism, and a possible method for determining the approximate magnitude of product-release steps is given. 7. The computer programs used in the computations have been deposited as Supplement SUP 50113 (21 pages) with the British Library Lending Division, Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies can be obtained on the terms indicated in Biochem, J. (1978) 169, 5.     

Predicting the probability of avian reproductive success and its components at a nesting site

Avian reproduction has four prime components: nesting, mating, hatching, and fledging. Predicting the probability of individual components helps in identifying the period of reproduction that would benefit from an increased conservation effort. Identification of biotic, abiotic, and sociological variables of the nesting sites is essential to calculate the component-wise success probabilities. There is no standard methodology to estimate these probability values separately. This study proposes a methodology to estimate the success probability of each component, identifies correlated environmental predictors, and provides a modelling framework to accurately predict the nesting success probabilities using Merops philippinus as a model species. Primary surveillance data and the proposed methodology indicate that the time window between the bird's nesting and mating is most vulnerable to environmental fluctuations. Both biotic and abiotic factors are crucial determinants of nesting success. Sociological factors also play a crucial role in determining the probabilities of these successes. Mating, hatching, and fledging success depend more on biotic factors than abiotic ones. Linear modelling frameworks are helpful in exploring which types of environments are better determinants of the success of a reproductive component. Artificial neural networks are useful in predicting mating, nesting, and overall reproductive success probabilities. Though the models in this study are developed using Merops philippinus data, the proposed methodology and modelling framework is also applicable for other bird species.     

Interdependent Multi-Layer Networks: Modeling and Survivability Analysis with Applications to Space-Based Networks

This article develops a novel approach and algorithmic tools for the modeling and survivability analysis of networks with heterogeneous nodes, and examines their application to space-based networks. Space-based networks (SBNs) allow the sharing of spacecraft on-orbit resources, such as data storage, processing, and downlink. Each spacecraft in the network can have different subsystem composition and functionality, thus resulting in node heterogeneity. Most traditional survivability analyses of networks assume node homogeneity and as a result, are not suited for the analysis of SBNs. This work proposes that heterogeneous networks can be modeled as interdependent multi-layer networks, which enables their survivability analysis. The multi-layer aspect captures the breakdown of the network according to common functionalities across the different nodes, and it allows the emergence of homogeneous sub-networks, while the interdependency aspect constrains the network to capture the physical characteristics of each node. Definitions of primitives of failure propagation are devised. Formal characterization of interdependent multi-layer networks, as well as algorithmic tools for the analysis of failure propagation across the network are developed and illustrated with space applications. The SBN applications considered consist of several networked spacecraft that can tap into each other''s Command and Data Handling subsystem, in case of failure of its own, including the Telemetry, Tracking and Command, the Control Processor, and the Data Handling sub-subsystems. Various design insights are derived and discussed, and the capability to perform trade-space analysis with the proposed approach for various network characteristics is indicated. The select results here shown quantify the incremental survivability gains (with respect to a particular class of threats) of the SBN over the traditional monolith spacecraft. Failure of the connectivity between nodes is also examined, and the results highlight the importance of the reliability of the wireless links between spacecraft (nodes) to enable any survivability improvements for space-based networks.     

Assessing intervention effects in a randomized trial within a social network

Studies of social networks provide unique opportunities to assess the causal effects of interventions that may impact more of the population than just those intervened on directly. Such effects are sometimes called peer or spillover effects, and may exist in the presence of interference, that is, when one individual's treatment affects another individual's outcome. Randomization-based inference (RI) methods provide a theoretical basis for causal inference in randomized studies, even in the presence of interference. In this article, we consider RI of the intervention effect in the eX-FLU trial, a randomized study designed to assess the effect of a social distancing intervention on influenza-like-illness transmission in a connected network of college students. The approach considered enables inference about the effect of the social distancing intervention on the per-contact probability of influenza-like-illness transmission in the observed network. The methods allow for interference between connected individuals and for heterogeneous treatment effects. The proposed methods are evaluated empirically via simulation studies, and then applied to data from the eX-FLU trial.     

Probabilities of transversions and transitions.

The values of the mean relative probabilities of transversions and transitions have been refined on the basis of the data collected by Jukes and found to be equal to 0.34 and 0.66, respectively. Evolutionary factors increase the probability of transversions to 0.44. The relative probabilities of individual substitutions have been determined, and a detailed classification of the nonsense mutations has been given. Such mutations are especially probable in the UGG (Trp) codon. The highest probability of AG, GA transitions correlates with the lowest mean change in the hydrophobic nature of the amino acids coded.