Perturbation analysis analyzed—mathematical modeling of intact and perturbed gene regulatory circuits for animal development

Gene regulatory networks for animal development are the underlying mechanisms controlling cell fate specification and differentiation. The architecture of gene regulatory circuits determines their information processing properties and their developmental function. It is a major task to derive realistic network models from exceedingly advanced high throughput experimental data. Here we use mathematical modeling to study the dynamics of gene regulatory circuits to advance the ability to infer regulatory connections and logic function from experimental data. This study is guided by experimental methodologies that are commonly used to study gene regulatory networks that control cell fate specification. We study the effect of a perturbation of an input on the level of its downstream genes and compare between the cis-regulatory execution of OR and AND logics. Circuits that initiate gene activation and circuits that lock on the expression of genes are analyzed. The model improves our ability to analyze experimental data and construct from it the network topology. The model also illuminates information processing properties of gene regulatory circuits for animal development.     

Inference of gene regulatory networks and compound mode of action from time course gene expression profiles

MOTIVATION: Time series expression experiments are an increasingly popular method for studying a wide range of biological systems. Here we developed an algorithm that can infer the local network of gene-gene interactions surrounding a gene of interest. This is achieved by a perturbation of the gene of interest and subsequently measuring the gene expression profiles at multiple time points. We applied this algorithm to computer simulated data and to experimental data on a nine gene network in Escherichia coli. RESULTS: In this paper we show that it is possible to recover the gene regulatory network from a time series data of gene expression following a perturbation to the cell. We show this both on simulated data and on a nine gene subnetwork part of the DNA-damage response pathway (SOS pathway) in the bacteria E. coli. CONTACT: dibernardo@tigem.it SUPLEMENTARY INFORMATION: Supplementary data are available at http://dibernado.tigem.it     

Elementary network reconstruction: A framework for the analysis of regulatory networks in biological systems

Complexity of regulatory networks arises from the high degree of interaction between network components such as DNA, RNA, proteins, and metabolites. We have developed a modeling tool, elementary network reconstruction (ENR), to characterize these networks. ENR is a knowledge-driven, steady state, deterministic, quantitative modeling approach based on linear perturbation theory. In ENR we demonstrate a novel means of expressing control mechanisms by way of dimensionless steady state gains relating input and output variables, which are purely in terms of species abundances (extensive variables). As a result of systematic enumeration of network species in n×n matrix, the two properties of linear perturbation are manifested in graphical representations: transitive property is evident in a special L-shape structure, and additive property is evident in multiple L-shape structures arriving at the same matrix cell. Upon imposing mechanistic (lowest-level) gains, network self-assembly through transitive and additive properties results in elucidation of inherent topology and explicit cataloging of higher level gains, which in turn can be used to predict perturbation results. Application of ENR to the regulatory network behind carbon catabolite repression in Escherichia coli is presented. Through incorporation of known molecular mechanisms governing transient and permanent repressions, the ENR model correctly predicts several key features of this regulatory network, including a 50% downshift in intracellular cAMP level upon exposure to glucose. Since functional genomics studies are mainly concerned with redistribution of species abundances in perturbed systems, ENR could be exploited in the system-level analysis of biological systems.     

Modeling Genome-Wide Dynamic Regulatory Network in Mouse Lungs with Influenza Infection Using High-Dimensional Ordinary Differential Equations

The immune response to viral infection is regulated by an intricate network of many genes and their products. The reverse engineering of gene regulatory networks (GRNs) using mathematical models from time course gene expression data collected after influenza infection is key to our understanding of the mechanisms involved in controlling influenza infection within a host. A five-step pipeline: detection of temporally differentially expressed genes, clustering genes into co-expressed modules, identification of network structure, parameter estimate refinement, and functional enrichment analysis, is developed for reconstructing high-dimensional dynamic GRNs from genome-wide time course gene expression data. Applying the pipeline to the time course gene expression data from influenza-infected mouse lungs, we have identified 20 distinct temporal expression patterns in the differentially expressed genes and constructed a module-based dynamic network using a linear ODE model. Both intra-module and inter-module annotations and regulatory relationships of our inferred network show some interesting findings and are highly consistent with existing knowledge about the immune response in mice after influenza infection. The proposed method is a computationally efficient, data-driven pipeline bridging experimental data, mathematical modeling, and statistical analysis. The application to the influenza infection data elucidates the potentials of our pipeline in providing valuable insights into systematic modeling of complicated biological processes.     

Modeling of Hysteresis in Gene Regulatory Networks

Hysteresis, observed in many gene regulatory networks, has a pivotal impact on biological systems, which enhances the robustness of cell functions. In this paper, a general model is proposed to describe the hysteretic gene regulatory network by combining the hysteresis component and the transient dynamics. The Bouc-Wen hysteresis model is modified to describe the hysteresis component in the mammalian gene regulatory networks. Rigorous mathematical analysis on the dynamical properties of the model is presented to ensure the bounded-input-bounded-output (BIBO) stability and demonstrates that the original Bouc-Wen model can only generate a clockwise hysteresis loop while the modified model can describe both clockwise and counter clockwise hysteresis loops. Simulation studies have shown that the hysteresis loops from our model are consistent with the experimental observations in three mammalian gene regulatory networks and two E.coli gene regulatory networks, which demonstrate the ability and accuracy of the mathematical model to emulate natural gene expression behavior with hysteresis. A comparison study has also been conducted to show that this model fits the experiment data significantly better than previous ones in the literature. The successful modeling of the hysteresis in all the five hysteretic gene regulatory networks suggests that the new model has the potential to be a unified framework for modeling hysteresis in gene regulatory networks and provide better understanding of the general mechanism that drives the hysteretic function.     

Robustness and stability of the gene regulatory network involved in DV boundary formation in the Drosophila wing

Gene regulatory networks have been conserved during evolution. The Drosophila wing and the vertebrate hindbrain share the gene network involved in the establishment of the boundary between dorsal and ventral compartments in the wing and adjacent rhombomeres in the hindbrain. A positive feedback-loop between boundary and non-boundary cells and mediated by the activities of Notch and Wingless/Wnt-1 leads to the establishment of a Notch dependent organizer at the boundary. By means of a Systems Biology approach that combines mathematical modeling and both in silico and in vivo experiments in the Drosophila wing primordium, we modeled and tested this regulatory network and present evidence that a novel property, namely refractoriness to the Wingless signaling molecule, is required in boundary cells for the formation of a stable dorsal-ventral boundary. This new property has been validated in vivo, promotes mutually exclusive domains of Notch and Wingless activities and confers stability to the dorsal-ventral boundary. A robustness analysis of the regulatory network complements our results and ensures its biological plausibility.     

Integrating external biological knowledge in the construction of regulatory networks from time-series expression data

ABSTRACT: BACKGROUND: Inference about regulatory networks from high-throughput genomics data is of great interest in systems biology. We present a Bayesian approach to infer gene regulatory networks from time series expression data by integrating various types of biological knowledge. RESULTS: We formulate network construction as a series of variable selection problems and use linear regression to model the data. Our method summarizes additional data sources with an informative prior probability distribution over candidate regression models. We extend the Bayesian model averaging (BMA) variable selection method to select regulators in the regression framework. We summarize the external biological knowledge by an informative prior probability distribution over the candidate regression models. CONCLUSIONS: We demonstrate our method on simulated data and a set of time-series microarray experiments measuring the effect of a drug perturbation on gene expression levels, and show that it outperforms leading regression-based methods in the literature.     

Probabilistic representation of gene regulatory networks

MOTIVATION: Recent experiments have established unambiguously that biological systems can have significant cell-to-cell variations in gene expression levels even in isogenic populations. Computational approaches to studying gene expression in cellular systems should capture such biological variations for a more realistic representation. RESULTS: In this paper, we present a new fully probabilistic approach to the modeling of gene regulatory networks that allows for fluctuations in the gene expression levels. The new algorithm uses a very simple representation for the genes, and accounts for the repression or induction of the genes and for the biological variations among isogenic populations simultaneously. Because of its simplicity, introduced algorithm is a very promising approach to model large-scale gene regulatory networks. We have tested the new algorithm on the synthetic gene network library bioengineered recently. The good agreement between the computed and the experimental results for this library of networks, and additional tests, demonstrate that the new algorithm is robust and very successful in explaining the experimental data. AVAILABILITY: The simulation software is available upon request. SUPPLEMENTARY INFORMATION: Supplementary material will be made available on the OUP server.     

Design Principles for Riboswitch Function

Scientific and technological advances that enable the tuning of integrated regulatory components to match network and system requirements are critical to reliably control the function of biological systems. RNA provides a promising building block for the construction of tunable regulatory components based on its rich regulatory capacity and our current understanding of the sequence–function relationship. One prominent example of RNA-based regulatory components is riboswitches, genetic elements that mediate ligand control of gene expression through diverse regulatory mechanisms. While characterization of natural and synthetic riboswitches has revealed that riboswitch function can be modulated through sequence alteration, no quantitative frameworks exist to investigate or guide riboswitch tuning. Here, we combined mathematical modeling and experimental approaches to investigate the relationship between riboswitch function and performance. Model results demonstrated that the competition between reversible and irreversible rate constants dictates performance for different regulatory mechanisms. We also found that practical system restrictions, such as an upper limit on ligand concentration, can significantly alter the requirements for riboswitch performance, necessitating alternative tuning strategies. Previous experimental data for natural and synthetic riboswitches as well as experiments conducted in this work support model predictions. From our results, we developed a set of general design principles for synthetic riboswitches. Our results also provide a foundation from which to investigate how natural riboswitches are tuned to meet systems-level regulatory demands.     

Learning Gene Networks under SNP Perturbations Using eQTL Datasets

The standard approach for identifying gene networks is based on experimental perturbations of gene regulatory systems such as gene knock-out experiments, followed by a genome-wide profiling of differential gene expressions. However, this approach is significantly limited in that it is not possible to perturb more than one or two genes simultaneously to discover complex gene interactions or to distinguish between direct and indirect downstream regulations of the differentially-expressed genes. As an alternative, genetical genomics study has been proposed to treat naturally-occurring genetic variants as potential perturbants of gene regulatory system and to recover gene networks via analysis of population gene-expression and genotype data. Despite many advantages of genetical genomics data analysis, the computational challenge that the effects of multifactorial genetic perturbations should be decoded simultaneously from data has prevented a widespread application of genetical genomics analysis. In this article, we propose a statistical framework for learning gene networks that overcomes the limitations of experimental perturbation methods and addresses the challenges of genetical genomics analysis. We introduce a new statistical model, called a sparse conditional Gaussian graphical model, and describe an efficient learning algorithm that simultaneously decodes the perturbations of gene regulatory system by a large number of SNPs to identify a gene network along with expression quantitative trait loci (eQTLs) that perturb this network. While our statistical model captures direct genetic perturbations of gene network, by performing inference on the probabilistic graphical model, we obtain detailed characterizations of how the direct SNP perturbation effects propagate through the gene network to perturb other genes indirectly. We demonstrate our statistical method using HapMap-simulated and yeast eQTL datasets. In particular, the yeast gene network identified computationally by our method under SNP perturbations is well supported by the results from experimental perturbation studies related to DNA replication stress response.     

Amplitude Metrics for Cellular Circadian Bioluminescence Reporters

Bioluminescence rhythms from cellular reporters have become the most common method used to quantify oscillations in circadian gene expression. These experimental systems can reveal phase and amplitude change resulting from circadian disturbances, and can be used in conjunction with mathematical models to lend further insight into the mechanistic basis of clock amplitude regulation. However, bioluminescence experiments track the mean output from thousands of noisy, uncoupled oscillators, obscuring the direct effect of a given stimulus on the genetic regulatory network. In many cases, it is unclear whether changes in amplitude are due to individual changes in gene expression level or to a change in coherence of the population. Although such systems can be modeled using explicit stochastic simulations, these models are computationally cumbersome and limit analytical insight into the mechanisms of amplitude change. We therefore develop theoretical and computational tools to approximate the mean expression level in large populations of noninteracting oscillators, and further define computationally efficient amplitude response calculations to describe phase-dependent amplitude change. At the single-cell level, a mechanistic nonlinear ordinary differential equation model is used to calculate the transient response of each cell to a perturbation, whereas population-level dynamics are captured by coupling this detailed model to a phase density function. Our analysis reveals that amplitude changes mediated at either the individual-cell or the population level can be distinguished in tissue-level bioluminescence data without the need for single-cell measurements. We demonstrate the effectiveness of the method by modeling experimental bioluminescence profiles of light-sensitive fibroblasts, reconciling the conclusions of two seemingly contradictory studies. This modeling framework allows a direct comparison between in vitro bioluminescence experiments and in silico ordinary differential equation models, and will lead to a better quantitative understanding of the factors that affect clock amplitude.     

Biological networks 101: Computational modeling for molecular biologists

Computational modeling of biological networks permits the comprehensive analysis of cells and tissues to define molecular phenotypes and novel hypotheses. Although a large number of software tools have been developed, the versatility of these tools is limited by mathematical complexities that prevent their broad adoption and effective use by molecular biologists. This study clarifies the basic aspects of molecular modeling, how to convert data into useful input, as well as the number of time points and molecular parameters that should be considered for molecular regulatory models with both explanatory and predictive potential. We illustrate the necessary experimental preconditions for converting data into a computational model of network dynamics. This model requires neither a thorough background in mathematics nor precise data on intracellular concentrations, binding affinities or reaction kinetics. Finally, we show how an interactive model of crosstalk between signal transduction pathways in primary human articular chondrocytes allows insight into processes that regulate gene expression.     

An integer optimization algorithm for robust identification of non-linear gene regulatory networks

ABSTRACT: BACKGROUND: Reverse engineering gene networks and identifying regulatory interactions are integral to understanding cellular decision making processes. Advancement in high throughput experimental techniques has initiated innovative data driven analysis of gene regulatory networks. However, inherent noise associated with biological systems requires numerous experimental replicates for reliable conclusions. Furthermore, evidence of robust algorithms directly exploiting basic biological traits are few. Such algorithms are expected to be efficient in their performance and robust in their prediction. RESULTS: We have developed a network identification algorithm to accurately infer both the topology and strength of regulatory interactions from time series gene expression data in the presence of significant experimental noise and non-linear behavior. In this novel formulism, we have addressed data variability in biological systems by integrating network identification with the bootstrap resampling technique, hence predicting robust interactions from limited experimental replicates subjected to noise. Furthermore, we have incorporated non-linearity in gene dynamics using the S-system formulation. The basic network identification formulation exploits the trait of sparsity of biological interactions. Towards that, the identification algorithm is formulated as an integer-programming problem by introducing binary variables for each network component. The objective function is targeted to minimize the network connections subjected to the constraint of maximal agreement between the experimental and predicted gene dynamics. The developed algorithm is validated using both in-silico and experimental data-sets. These studies show that the algorithm can accurately predict the topology and connection strength of the in silico networks, as quantified by high precision and recall, and small discrepancy between the actual and predicted kinetic parameters. Furthermore, in both the in silico and experimental case studies, the predicted gene expression profiles are in very close agreement with the dynamics of the input data. CONCLUSIONS: Our integer programming algorithm effectively utilizes bootstrapping to identify robust gene regulatory networks from noisy, non-linear time-series gene expression data. With significant noise and non-linearities being inherent to biological systems, the present formulism, with the incorporation of network sparsity, is extremely relevant to gene regulatory networks, and while the formulation has been validated against in silico and E. Coli data, it can be applied to any biological system.     

Topology of gene expression networks as revealed by data mining and modeling

MOTIVATION: Interpretation of high-throughput gene expression profiling requires a knowledge of the design principles underlying the networks that sustain cellular machinery. Recently a novel approach based on the study of network topologies has been proposed. This methodology has proven to be useful for the analysis of a variety of biological systems, including metabolic networks, networks of protein-protein interactions, and gene networks that can be derived from gene expression data. In the present paper, we focus on several important issues related to the topology of gene expression networks that have not yet been fully studied. RESULTS: The networks derived from gene expression profiles for both time series experiments in yeast and perturbation experiments in cell lines are studied. We demonstrate that independent from the experimental organism (yeast versus cell lines) and the type of experiment (time courses versus perturbations) the extracted networks have similar topological characteristics suggesting together with the results of other common principles of the structural organization of biological networks. A novel computational model of network growth that reproduces the basic design principles of the observed networks is presented. Advantage of the model is that it provides a general mechanism to generate networks with different types of topology by a variation of a few parameters. We investigate the robustness of the network structure to random damages and to deliberate removal of the most important parts of the system and show a surprising tolerance of gene expression networks to both kinds of disturbance.     

Integration of Steady-State and Temporal Gene Expression Data for the Inference of Gene Regulatory Networks

We develop a new regression algorithm, cMIKANA, for inference of gene regulatory networks from combinations of steady-state and time-series gene expression data. Using simulated gene expression datasets to assess the accuracy of reconstructing gene regulatory networks, we show that steady-state and time-series data sets can successfully be combined to identify gene regulatory interactions using the new algorithm. Inferring gene networks from combined data sets was found to be advantageous when using noisy measurements collected with either lower sampling rates or a limited number of experimental replicates. We illustrate our method by applying it to a microarray gene expression dataset from human umbilical vein endothelial cells (HUVECs) which combines time series data from treatment with growth factor TNF and steady state data from siRNA knockdown treatments. Our results suggest that the combination of steady-state and time-series datasets may provide better prediction of RNA-to-RNA interactions, and may also reveal biological features that cannot be identified from dynamic or steady state information alone. Finally, we consider the experimental design of genomics experiments for gene regulatory network inference and show that network inference can be improved by incorporating steady-state measurements with time-series data.