Solving the Differential Biochemical Jacobian from Metabolomics Covariance Data

High-throughput molecular analysis has become an integral part in organismal systems biology. In contrast, due to a missing systematic linkage of the data with functional and predictive theoretical models of the underlying metabolic network the understanding of the resulting complex data sets is lacking far behind. Here, we present a biomathematical method addressing this problem by using metabolomics data for the inverse calculation of a biochemical Jacobian matrix, thereby linking computer-based genome-scale metabolic reconstruction and in vivo metabolic dynamics. The incongruity of metabolome coverage by typical metabolite profiling approaches and genome-scale metabolic reconstruction was solved by the design of superpathways to define a metabolic interaction matrix. A differential biochemical Jacobian was calculated using an approach which links this metabolic interaction matrix and the covariance of metabolomics data satisfying a Lyapunov equation. The predictions of the differential Jacobian from real metabolomic data were found to be correct by testing the corresponding enzymatic activities. Moreover it is demonstrated that the predictions of the biochemical Jacobian matrix allow for the design of parameter optimization strategies for ODE-based kinetic models of the system. The presented concept combines dynamic modelling strategies with large-scale steady state profiling approaches without the explicit knowledge of individual kinetic parameters. In summary, the presented strategy allows for the identification of regulatory key processes in the biochemical network directly from metabolomics data and is a fundamental achievement for the functional interpretation of metabolomics data.     

Metabolic network discovery through reverse engineering of metabolome data

Reverse engineering of high-throughput omics data to infer underlying biological networks is one of the challenges in systems biology. However, applications in the field of metabolomics are rather limited. We have focused on a systematic analysis of metabolic network inference from in silico metabolome data based on statistical similarity measures. Three different data types based on biological/environmental variability around steady state were analyzed to compare the relative information content of the data types for inferring the network. Comparing the inference power of different similarity scores indicated the clear superiority of conditioning or pruning based scores as they have the ability to eliminate indirect interactions. We also show that a mathematical measure based on the Fisher information matrix gives clues on the information quality of different data types to better represent the underlying metabolic network topology. Results on several datasets of increasing complexity consistently show that metabolic variations observed at steady state, the simplest experimental analysis, are already informative to reveal the connectivity of the underlying metabolic network with a low false-positive rate when proper similarity-score approaches are employed. For experimental situations this implies that a single organism under slightly varying conditions may already generate more than enough information to rightly infer networks. Detailed examination of the strengths of interactions of the underlying metabolic networks demonstrates that the edges that cannot be captured by similarity scores mainly belong to metabolites connected with weak interaction strength.     

Sets2Networks: network inference from repeated observations of sets

Background

The skeleton of complex systems can be represented as networks where vertices represent entities, and edges represent the relations between these entities. Often it is impossible, or expensive, to determine the network structure by experimental validation of the binary interactions between every vertex pair. It is usually more practical to infer the network from surrogate observations. Network inference is the process by which an underlying network of relations between entities is determined from indirect evidence. While many algorithms have been developed to infer networks from quantitative data, less attention has been paid to methods which infer networks from repeated co-occurrence of entities in related sets. This type of data is ubiquitous in the field of systems biology and in other areas of complex systems research. Hence, such methods would be of great utility and value.

Results

Here we present a general method for network inference from repeated observations of sets of related entities. Given experimental observations of such sets, we infer the underlying network connecting these entities by generating an ensemble of networks consistent with the data. The frequency of occurrence of a given link throughout this ensemble is interpreted as the probability that the link is present in the underlying real network conditioned on the data. Exponential random graphs are used to generate and sample the ensemble of consistent networks, and we take an algorithmic approach to numerically execute the inference method. The effectiveness of the method is demonstrated on synthetic data before employing this inference approach to problems in systems biology and systems pharmacology, as well as to construct a co-authorship collaboration network. We predict direct protein-protein interactions from high-throughput mass-spectrometry proteomics, integrate data from Chip-seq and loss-of-function/gain-of-function followed by expression data to infer a network of associations between pluripotency regulators, extract a network that connects 53 cancer drugs to each other and to 34 severe adverse events by mining the FDA’s Adverse Events Reporting Systems (AERS), and construct a co-authorship network that connects Mount Sinai School of Medicine investigators. The predicted networks and online software to create networks from entity-set libraries are provided online at http://www.maayanlab.net/S2N.

Conclusions

The network inference method presented here can be applied to resolve different types of networks in current systems biology and systems pharmacology as well as in other fields of research.     

A compound decision approach to covariance matrix estimation

Covariance matrix estimation is a fundamental statistical task in many applications, but the sample covariance matrix is suboptimal when the sample size is comparable to or less than the number of features. Such high-dimensional settings are common in modern genomics, where covariance matrix estimation is frequently employed as a method for inferring gene networks. To achieve estimation accuracy in these settings, existing methods typically either assume that the population covariance matrix has some particular structure, for example, sparsity, or apply shrinkage to better estimate the population eigenvalues. In this paper, we study a new approach to estimating high-dimensional covariance matrices. We first frame covariance matrix estimation as a compound decision problem. This motivates defining a class of decision rules and using a nonparametric empirical Bayes g-modeling approach to estimate the optimal rule in the class. Simulation results and gene network inference in an RNA-seq experiment in mouse show that our approach is comparable to or can outperform a number of state-of-the-art proposals.     

Formulas for intrinsic noise evaluation in oscillatory genetic networks

The linear noise approximation is a useful method for stochastic noise evaluations in genetic regulatory networks, where the covariance equation described as a Lyapunov equation plays a central role. We discuss the linear noise approximation method for evaluations of an intrinsic noise in autonomously oscillatory genetic networks; in such oscillatory networks, the covariance equation becomes a periodic differential equation that provides generally an unbounded covariance matrix, so that the standard method of noise evaluation based on the covariance matrix cannot be adopted directly. In this paper, we develop a new method of noise evaluation in oscillatory genetic networks; first, we investigate structural properties, e.g., orbital stability and periodicity, of the solutions to the covariance equation given as a periodic Lyapunov differential equation by using the Floquet-Lyapunov theory, and propose a global measure for evaluating stochastic amplitude fluctuations on the periodic trajectory; we also derive an evaluation formula for the period fluctuation. Finally, we apply our method to a model of circadian oscillations based on negative auto-regulation of gene expression, and show validity of our method by comparing the evaluation results with stochastic simulations.     

Identification of alterations in the Jacobian of biochemical reaction networks from steady state covariance data at two conditions

Model building of biochemical reaction networks typically involves experiments in which changes in the behavior due to natural or experimental perturbations are observed. Computational models of reaction networks are also used in a systems biology approach to study how transitions from a healthy to a diseased state result from changes in genetic or environmental conditions. In this paper we consider the nonlinear inverse problem of inferring information about the Jacobian of a Langevin type network model from covariance data of steady state concentrations associated to two different experimental conditions. Under idealized assumptions on the Langevin fluctuation matrices we prove that relative alterations in the network Jacobian can be uniquely identified when comparing the two data sets. Based on this result and the premise that alteration is locally confined to separable parts due to network modularity we suggest a computational approach using hybrid stochastic-deterministic optimization for the detection of perturbations in the network Jacobian using the sparsity promoting effect of $\ell _p$ -penalization. Our approach is illustrated by means of published metabolomic and signaling reaction networks.     

Inference of biochemical network models in S-system using multiobjective optimization approach

MOTIVATION: The inference of biochemical networks, such as gene regulatory networks, protein-protein interaction networks, and metabolic pathway networks, from time-course data is one of the main challenges in systems biology. The ultimate goal of inferred modeling is to obtain expressions that quantitatively understand every detail and principle of biological systems. To infer a realizable S-system structure, most articles have applied sums of magnitude of kinetic orders as a penalty term in the fitness evaluation. How to tune a penalty weight to yield a realizable model structure is the main issue for the inverse problem. No guideline has been published for tuning a suitable penalty weight to infer a suitable model structure of biochemical networks. RESULTS: We introduce an interactive inference algorithm to infer a realizable S-system structure for biochemical networks. The inference problem is formulated as a multiobjective optimization problem to minimize simultaneously the concentration error, slope error and interaction measure in order to find a suitable S-system model structure and its corresponding model parameters. The multiobjective optimization problem is solved by the epsilon-constraint method to minimize the interaction measure subject to the expectation constraints for the concentration and slope error criteria. The theorems serve to guarantee the minimum solution for the epsilon-constrained problem to achieve the minimum interaction network for the inference problem. The approach could avoid assigning a penalty weight for sums of magnitude of kinetic orders.     

Selective integration of multiple biological data for supervised network inference

MOTIVATION: Inferring networks of proteins from biological data is a central issue of computational biology. Most network inference methods, including Bayesian networks, take unsupervised approaches in which the network is totally unknown in the beginning, and all the edges have to be predicted. A more realistic supervised framework, proposed recently, assumes that a substantial part of the network is known. We propose a new kernel-based method for supervised graph inference based on multiple types of biological datasets such as gene expression, phylogenetic profiles and amino acid sequences. Notably, our method assigns a weight to each type of dataset and thereby selects informative ones. Data selection is useful for reducing data collection costs. For example, when a similar network inference problem must be solved for other organisms, the dataset excluded by our algorithm need not be collected. RESULTS: First, we formulate supervised network inference as a kernel matrix completion problem, where the inference of edges boils down to estimation of missing entries of a kernel matrix. Then, an expectation-maximization algorithm is proposed to simultaneously infer the missing entries of the kernel matrix and the weights of multiple datasets. By introducing the weights, we can integrate multiple datasets selectively and thereby exclude irrelevant and noisy datasets. Our approach is favorably tested in two biological networks: a metabolic network and a protein interaction network. AVAILABILITY: Software is available on request.     

Gradient directed regularization for sparse Gaussian concentration graphs, with applications to inference of genetic networks

Large-scale microarray gene expression data provide the possibility of constructing genetic networks or biological pathways. Gaussian graphical models have been suggested to provide an effective method for constructing such genetic networks. However, most of the available methods for constructing Gaussian graphs do not account for the sparsity of the networks and are computationally more demanding or infeasible, especially in the settings of high dimension and low sample size. We introduce a threshold gradient descent (TGD) regularization procedure for estimating the sparse precision matrix in the setting of Gaussian graphical models and demonstrate its application to identifying genetic networks. Such a procedure is computationally feasible and can easily incorporate prior biological knowledge about the network structure. Simulation results indicate that the proposed method yields a better estimate of the precision matrix than the procedures that fail to account for the sparsity of the graphs. We also present the results on inference of a gene network for isoprenoid biosynthesis in Arabidopsis thaliana. These results demonstrate that the proposed procedure can indeed identify biologically meaningful genetic networks based on microarray gene expression data.     

Sparse and Compositionally Robust Inference of Microbial Ecological Networks

16S ribosomal RNA (rRNA) gene and other environmental sequencing techniques provide snapshots of microbial communities, revealing phylogeny and the abundances of microbial populations across diverse ecosystems. While changes in microbial community structure are demonstrably associated with certain environmental conditions (from metabolic and immunological health in mammals to ecological stability in soils and oceans), identification of underlying mechanisms requires new statistical tools, as these datasets present several technical challenges. First, the abundances of microbial operational taxonomic units (OTUs) from amplicon-based datasets are compositional. Counts are normalized to the total number of counts in the sample. Thus, microbial abundances are not independent, and traditional statistical metrics (e.g., correlation) for the detection of OTU-OTU relationships can lead to spurious results. Secondly, microbial sequencing-based studies typically measure hundreds of OTUs on only tens to hundreds of samples; thus, inference of OTU-OTU association networks is severely under-powered, and additional information (or assumptions) are required for accurate inference. Here, we present SPIEC-EASI (SParse InversE Covariance Estimation for Ecological Association Inference), a statistical method for the inference of microbial ecological networks from amplicon sequencing datasets that addresses both of these issues. SPIEC-EASI combines data transformations developed for compositional data analysis with a graphical model inference framework that assumes the underlying ecological association network is sparse. To reconstruct the network, SPIEC-EASI relies on algorithms for sparse neighborhood and inverse covariance selection. To provide a synthetic benchmark in the absence of an experimentally validated gold-standard network, SPIEC-EASI is accompanied by a set of computational tools to generate OTU count data from a set of diverse underlying network topologies. SPIEC-EASI outperforms state-of-the-art methods to recover edges and network properties on synthetic data under a variety of scenarios. SPIEC-EASI also reproducibly predicts previously unknown microbial associations using data from the American Gut project.     

Inferring Gene Regulatory Networks by Singular Value Decomposition and Gravitation Field Algorithm

Reconstruction of gene regulatory networks (GRNs) is of utmost interest and has become a challenge computational problem in system biology. However, every existing inference algorithm from gene expression profiles has its own advantages and disadvantages. In particular, the effectiveness and efficiency of every previous algorithm is not high enough. In this work, we proposed a novel inference algorithm from gene expression data based on differential equation model. In this algorithm, two methods were included for inferring GRNs. Before reconstructing GRNs, singular value decomposition method was used to decompose gene expression data, determine the algorithm solution space, and get all candidate solutions of GRNs. In these generated family of candidate solutions, gravitation field algorithm was modified to infer GRNs, used to optimize the criteria of differential equation model, and search the best network structure result. The proposed algorithm is validated on both the simulated scale-free network and real benchmark gene regulatory network in networks database. Both the Bayesian method and the traditional differential equation model were also used to infer GRNs, and the results were used to compare with the proposed algorithm in our work. And genetic algorithm and simulated annealing were also used to evaluate gravitation field algorithm. The cross-validation results confirmed the effectiveness of our algorithm, which outperforms significantly other previous algorithms.     

Inferring Time-Varying Network Topologies from Gene Expression Data

Most current methods for gene regulatory network identification lead to the inference of steady-state networks, that is, networks prevalent over all times, a hypothesis which has been challenged. There has been a need to infer and represent networks in a dynamic, that is, time-varying fashion, in order to account for different cellular states affecting the interactions amongst genes. In this work, we present an approach, regime-SSM, to understand gene regulatory networks within such a dynamic setting. The approach uses a clustering method based on these underlying dynamics, followed by system identification using a state-space model for each learnt cluster—to infer a network adjacency matrix. We finally indicate our results on the mouse embryonic kidney dataset as well as the T-cell activation-based expression dataset and demonstrate conformity with reported experimental evidence.     

Granger causality in integrated GC–MS and LC–MS metabolomics data reveals the interface of primary and secondary metabolism

Metabolomics has emerged as a key technique of modern life sciences in recent years. Two major techniques for metabolomics in the last 10 years are gas chromatography coupled to mass spectrometry (GC–MS) and liquid chromatography coupled to mass spectrometry (LC–MS). Each platform has a specific performance detecting subsets of metabolites. GC–MS in combination with derivatisation has a preference for small polar metabolites covering primary metabolism. In contrast, reversed phase LC–MS covers large hydrophobic metabolites predominant in secondary metabolism. Here, we present an integrative metabolomics platform providing a mean to reveal the interaction of primary and secondary metabolism in plants and other organisms. The strategy combines GC–MS and LC–MS analysis of the same sample, a novel alignment tool MetMAX and a statistical toolbox COVAIN for data integration and linkage of Granger Causality with metabolic modelling. For metabolic modelling we have implemented the combined GC–LC–MS metabolomics data covariance matrix and a stoichiometric matrix of the underlying biochemical reaction network. The changes in biochemical regulation are expressed as differential Jacobian matrices. Applying the Granger causality, a subset of secondary metabolites was detected with significant correlations to primary metabolites such as sugars and amino acids. These metabolic subsets were compiled into a stoichiometric matrix N. Using N the inverse calculation of a differential Jacobian J from metabolomics data was possible. Key points of regulation at the interface of primary and secondary metabolism were identified.     

Estimation of Directed Effective Connectivity from fMRI Functional Connectivity Hints at Asymmetries of Cortical Connectome

The brain exhibits complex spatio-temporal patterns of activity. This phenomenon is governed by an interplay between the internal neural dynamics of cortical areas and their connectivity. Uncovering this complex relationship has raised much interest, both for theory and the interpretation of experimental data (e.g., fMRI recordings) using dynamical models. Here we focus on the so-called inverse problem: the inference of network parameters in a cortical model to reproduce empirically observed activity. Although it has received a lot of interest, recovering directed connectivity for large networks has been rather unsuccessful so far. The present study specifically addresses this point for a noise-diffusion network model. We develop a Lyapunov optimization that iteratively tunes the network connectivity in order to reproduce second-order moments of the node activity, or functional connectivity. We show theoretically and numerically that the use of covariances with both zero and non-zero time shifts is the key to infer directed connectivity. The first main theoretical finding is that an accurate estimation of the underlying network connectivity requires that the time shift for covariances is matched with the time constant of the dynamical system. In addition to the network connectivity, we also adjust the intrinsic noise received by each network node. The framework is applied to experimental fMRI data recorded for subjects at rest. Diffusion-weighted MRI data provide an estimate of anatomical connections, which is incorporated to constrain the cortical model. The empirical covariance structure is reproduced faithfully, especially its temporal component (i.e., time-shifted covariances) in addition to the spatial component that is usually the focus of studies. We find that the cortical interactions, referred to as effective connectivity, in the tuned model are not reciprocal. In particular, hubs are either receptors or feeders: they do not exhibit both strong incoming and outgoing connections. Our results sets a quantitative ground to explore the propagation of activity in the cortex.     

Metabolic network simulation using logical loop algorithm and Jacobian matrix

A novel method to accomplish efficient numerical simulation of metabolic networks for flux analysis was developed. The only inputs required are the set of stoichiometric balances and the atom mapping matrices of all components of the reaction network. The latter are used to automatically calculate isotopomer mapping matrices. Using the symbolic toolbox of MATLAB the analytical solution of the stoichiometric balance equation system, isotopomer balances and the analytical Jacobian matrix of the total set of stoichiometric and isotopomer balances are created automatically. The number of variables in the isotopomer distribution equation system is significantly reduced applying modified isotopomer mapping matrices. These allow lumping of several consecutive isotopomer reactions into a single one. The solution of the complete system of equations is improved by implementing an iterative logical loop algorithm and using the analytical Jacobian matrix. This new method provided quick and robust convergence to the root of such equation systems in all cases tested. The method was applied to a network of lysine producing Corynebacterium glutamicum. The resulting equation system with the dimension of 546 x 546 was directly derived from 12 isotopomer balance equations. The results obtained yielded identical labeling patterns for metabolites as compared to the relaxation method.     

Efficient reverse-engineering of a developmental gene regulatory network

Understanding the complex regulatory networks underlying development and evolution of multi-cellular organisms is a major problem in biology. Computational models can be used as tools to extract the regulatory structure and dynamics of such networks from gene expression data. This approach is called reverse engineering. It has been successfully applied to many gene networks in various biological systems. However, to reconstitute the structure and non-linear dynamics of a developmental gene network in its spatial context remains a considerable challenge. Here, we address this challenge using a case study: the gap gene network involved in segment determination during early development of Drosophila melanogaster. A major problem for reverse-engineering pattern-forming networks is the significant amount of time and effort required to acquire and quantify spatial gene expression data. We have developed a simplified data processing pipeline that considerably increases the throughput of the method, but results in data of reduced accuracy compared to those previously used for gap gene network inference. We demonstrate that we can infer the correct network structure using our reduced data set, and investigate minimal data requirements for successful reverse engineering. Our results show that timing and position of expression domain boundaries are the crucial features for determining regulatory network structure from data, while it is less important to precisely measure expression levels. Based on this, we define minimal data requirements for gap gene network inference. Our results demonstrate the feasibility of reverse-engineering with much reduced experimental effort. This enables more widespread use of the method in different developmental contexts and organisms. Such systematic application of data-driven models to real-world networks has enormous potential. Only the quantitative investigation of a large number of developmental gene regulatory networks will allow us to discover whether there are rules or regularities governing development and evolution of complex multi-cellular organisms.     

Validation of Inference Procedures for Gene Regulatory Networks

The availability of high-throughput genomic data has motivated the development of numerous algorithms to infer gene regulatory networks. The validity of an inference procedure must be evaluated relative to its ability to infer a model network close to the ground-truth network from which the data have been generated. The input to an inference algorithm is a sample set of data and its output is a network. Since input, output, and algorithm are mathematical structures, the validity of an inference algorithm is a mathematical issue. This paper formulates validation in terms of a semi-metric distance between two networks, or the distance between two structures of the same kind deduced from the networks, such as their steady-state distributions or regulatory graphs. The paper sets up the validation framework, provides examples of distance functions, and applies them to some discrete Markov network models. It also considers approximate validation methods based on data for which the generating network is not known, the kind of situation one faces when using real data.Key Words: Epistemology, gene network, inference, validation.     

Gene regulatory network reconstruction using Bayesian networks, the Dantzig Selector, the Lasso and their meta-analysis

Modern technologies and especially next generation sequencing facilities are giving a cheaper access to genotype and genomic data measured on the same sample at once. This creates an ideal situation for multifactorial experiments designed to infer gene regulatory networks. The fifth "Dialogue for Reverse Engineering Assessments and Methods" (DREAM5) challenges are aimed at assessing methods and associated algorithms devoted to the inference of biological networks. Challenge 3 on "Systems Genetics" proposed to infer causal gene regulatory networks from different genetical genomics data sets. We investigated a wide panel of methods ranging from Bayesian networks to penalised linear regressions to analyse such data, and proposed a simple yet very powerful meta-analysis, which combines these inference methods. We present results of the Challenge as well as more in-depth analysis of predicted networks in terms of structure and reliability. The developed meta-analysis was ranked first among the 16 teams participating in Challenge 3A. It paves the way for future extensions of our inference method and more accurate gene network estimates in the context of genetical genomics.