Search for steady states of piecewise-linear differential equation models of genetic regulatory networks

Reverse engineering gene regulatory network based on complex-valued ordinary differential equation model

BACKGROUND

The growing researches of molecular biology reveal that complex life phenomena have the ability to demonstrating various types of interactions in the level of genomics. To establish the interactions between genes or proteins and understand the intrinsic mechanisms of biological systems have become an urgent need and study hotspot.

RESULTS

In order to forecast gene expression data and identify more accurate gene regulatory network, complex-valued version of ordinary differential equation (CVODE) is proposed in this paper. In order to optimize CVODE model, a complex-valued hybrid evolutionary method based on Grammar-guided genetic programming and complex-valued firefly algorithm is presented.

CONCLUSIONS

When tested on three real gene expression datasets from E. coli and Human Cell, the experiment results suggest that CVODE model could improve 20-50% prediction accuracy of gene expression data, which could also infer more true-positive regulatory relationships and less false-positive regulations than ordinary differential equation.

Qualitative Modeling, Analysis and Control of Synthetic Regulatory Circuits

Qualitative modeling approaches are promising and still underexploited tools for the analysis and design of synthetic circuits. They can make predictions of circuit behavior in the absence of precise, quantitative information. Moreover, they provide direct insight into the relation between the feedback structure and the dynamical properties of a network. We review qualitative modeling approaches by focusing on two specific formalisms, Boolean networks and piecewise-linear differential equations, and illustrate their application by means of three well-known synthetic circuits. We describe various methods for the analysis of state transition graphs, discrete representations of the network dynamics that are generated in both modeling frameworks. We also briefly present the problem of controlling synthetic circuits, an emerging topic that could profit from the capacity of qualitative modeling approaches to rapidly scan a space of design alternatives.

Gene Regulatory Network Construction Based on a Particle Swarm Optimization of a Long Short-term Memory Network

Background:

The Gene Regulatory Network (GRN) is a model for studying the function and behavior of genes by treating the genome as a whole, which can reveal the gene expression mechanism. However, due to the dynamics, nonlinearity, and complexity of gene expression data, it is a challenging task to construct a GRN precisely. And in the circulating cooling water system, the Slime-Forming Bacteria (SFB) is one of the bacteria that helps to form dirt. In order to explore the microbial fouling mechanism of SFB, constructing a GRN for the fouling-forming genes of SFB is significant.

Objective:

Propose an effective GRN construction method and construct a GRN for the foulingforming genes of SFB.

Methods:

In this paper, a combination method of Long Short-Term Memory Network (LSTM) and Mean Impact Value (MIV) was applied for GRN reconstruction. Firstly, LSTM was employed to establish a gene expression prediction model. To improve the performance of LSTM, a Particle Swarm Optimization (PSO) was introduced to optimize the weight and learning rate. Then, the MIV was used to infer the regulation among genes. In view of the fouling-forming problem of SFB, we have designed electromagnetic field experiments and transcriptome sequencing experiments to locate the fouling-forming genes and obtain gene expression data.

Results:

In order to test the proposed approach, the proposed method was applied to three datasets: a simulated dataset and two real biology datasets. By comparing with other methods, the experimental results indicate that the proposed method has higher modeling accuracy and it can be used to effectively construct a GRN. And at last, a GRN for fouling-forming genes of SFB was constructed using the proposed approach.

Conclusion:

The experiments indicated that the proposed approach can reconstruct a GRN precisely, and compared with other approaches, the proposed approach performs better in extracting the regulations among genes.

Dynamic modeling and network approaches for omics time course data: overview of computational approaches and applications

Inferring networks and dynamics of genes, proteins, cells and other biological entities from high-throughput biological omics data is a central and challenging issue in computational and systems biology. This is essential for understanding the complexity of human health, disease susceptibility and pathogenesis for Predictive, Preventive, Personalized and Participatory (P4) system and precision medicine. The delineation of the possible interactions of all genes/proteins in a genome/proteome is a task for which conventional experimental techniques are ill suited. Urgently needed are rapid and inexpensive computational and statistical methods that can identify interacting candidate disease genes or drug targets out of thousands that can be further investigated or validated by experimentations. Moreover, identifying biological dynamic systems, and simultaneously estimating the important kinetic structural and functional parameters, which may not be experimentally accessible could be important directions for drug-disease-gene network studies. In this article, we present an overview and comparison of recent developments of dynamic modeling and network approaches for time-course omics data, and their applications to various biological systems, health conditions and disease statuses. Moreover, various data reduction and analytical schemes ranging from mathematical to computational to statistical methods are compared including their merits, drawbacks and limitations. The most recent software, associated web resources and other potentials for the compared methods are also presented and discussed in detail.

Influence of parameter perturbations on the reachability of therapeutic target in systems with switchings

Background

Examination of physiological processes and the influences of the drugs on them can be efficiently supported by mathematical modeling. One of the biggest problems is related to the exact fitting of the parameters of a model. Conditions inside the organism change dynamically, so the rates of processes are very difficult to estimate. Perturbations in the model parameters influence the steady state so a desired therapeutic goal may not be reached. Here we investigate the effect of parameter deviation on the steady state in three simple models of the influence of a therapeutic drug on its target protein. Two types of changes in the model parameters are taken into account: small perturbations in the system parameter values, and changes in the switching time of a specific parameter. Additionally, we examine the systems response in case of a drug concentration decreasing with time.

Results

The models which we analyze are simplified, because we want to avoid influences of complex dynamics on the results. A system with a negative feedback loop is the most robust and the most rapid, so it requires the largest drug dose but the effects are observed very quickly. On the other hand a system with positive feedback is very sensitive to changes, so small drug doses are sufficient to reach a therapeutic target. In systems without feedback or with positive feedback, perturbations in the model parameters have a bigger influence on the reachability of the therapeutic target than in systems with negative feedback. Drug degradation or inactivation in biological systems enforces multiple drug applications to maintain the level of a drug’s target under the desired threshold. The frequency of drug application should be fitted to the system dynamics, because the response velocity is tightly related to the therapeutic effectiveness and the time for achieving the goal.

Conclusions

Systems with different types of regulation vary in their dynamics and characteristic features. Depending on the feedback loop, different types of therapy may be the most appropriate, and deviations in the model parameters have different influences on the reachability of the therapeutic target.

Computational dynamic approaches for temporal omics data with applications to systems medicine

Modeling and predicting biological dynamic systems and simultaneously estimating the kinetic structural and functional parameters are extremely important in systems and computational biology. This is key for understanding the complexity of the human health, drug response, disease susceptibility and pathogenesis for systems medicine. Temporal omics data used to measure the dynamic biological systems are essentials to discover complex biological interactions and clinical mechanism and causations. However, the delineation of the possible associations and causalities of genes, proteins, metabolites, cells and other biological entities from high throughput time course omics data is challenging for which conventional experimental techniques are not suited in the big omics era. In this paper, we present various recently developed dynamic trajectory and causal network approaches for temporal omics data, which are extremely useful for those researchers who want to start working in this challenging research area. Moreover, applications to various biological systems, health conditions and disease status, and examples that summarize the state-of-the art performances depending on different specific mining tasks are presented. We critically discuss the merits, drawbacks and limitations of the approaches, and the associated main challenges for the years ahead. The most recent computing tools and software to analyze specific problem type, associated platform resources, and other potentials for the dynamic trajectory and interaction methods are also presented and discussed in detail.

Recurrent neural network based hybrid model for reconstructing gene regulatory network

One of the exciting problems in systems biology research is to decipher how genome controls the development of complex biological system. The gene regulatory networks (GRNs) help in the identification of regulatory interactions between genes and offer fruitful information related to functional role of individual gene in a cellular system. Discovering GRNs lead to a wide range of applications, including identification of disease related pathways providing novel tentative drug targets, helps to predict disease response, and also assists in diagnosing various diseases including cancer. Reconstruction of GRNs from available biological data is still an open problem. This paper proposes a recurrent neural network (RNN) based model of GRN, hybridized with generalized extended Kalman filter for weight update in backpropagation through time training algorithm. The RNN is a complex neural network that gives a better settlement between biological closeness and mathematical flexibility to model GRN; and is also able to capture complex, non-linear and dynamic relationships among variables. Gene expression data are inherently noisy and Kalman filter performs well for estimation problem even in noisy data. Hence, we applied non-linear version of Kalman filter, known as generalized extended Kalman filter, for weight update during RNN training. The developed model has been tested on four benchmark networks such as DNA SOS repair network, IRMA network, and two synthetic networks from DREAM Challenge. We performed a comparison of our results with other state-of-the-art techniques which shows superiority of our proposed model. Further, 5% Gaussian noise has been induced in the dataset and result of the proposed model shows negligible effect of noise on results, demonstrating the noise tolerance capability of the model.

An Efficient Steady-State Analysis Method for Large Boolean Networks with High Maximum Node Connectivity

Boolean networks have been widely used to model biological processes lacking detailed kinetic information. Despite their simplicity, Boolean network dynamics can still capture some important features of biological systems such as stable cell phenotypes represented by steady states. For small models, steady states can be determined through exhaustive enumeration of all state transitions. As the number of nodes increases, however, the state space grows exponentially thus making it difficult to find steady states. Over the last several decades, many studies have addressed how to handle such a state space explosion. Recently, increasing attention has been paid to a satisfiability solving algorithm due to its potential scalability to handle large networks. Meanwhile, there still lies a problem in the case of large models with high maximum node connectivity where the satisfiability solving algorithm is known to be computationally intractable. To address the problem, this paper presents a new partitioning-based method that breaks down a given network into smaller subnetworks. Steady states of each subnetworks are identified by independently applying the satisfiability solving algorithm. Then, they are combined to construct the steady states of the overall network. To efficiently apply the satisfiability solving algorithm to each subnetwork, it is crucial to find the best partition of the network. In this paper, we propose a method that divides each subnetwork to be smallest in size and lowest in maximum node connectivity. This minimizes the total cost of finding all steady states in entire subnetworks. The proposed algorithm is compared with others for steady states identification through a number of simulations on both published small models and randomly generated large models with differing maximum node connectivities. The simulation results show that our method can scale up to several hundreds of nodes even for Boolean networks with high maximum node connectivity. The algorithm is implemented and available at http://cps.kaist.ac.kr/∼ckhong/tools/download/PAD.tar.gz.

ASP-G: an ASP-based method for finding attractors in genetic regulatory networks

MOTIVATION

Boolean network models are suitable to simulate GRNs in the absence of detailed kinetic information. However, reducing the biological reality implies making assumptions on how genes interact (interaction rules) and how their state is updated during the simulation (update scheme). The exact choice of the assumptions largely determines the outcome of the simulations. In most cases, however, the biologically correct assumptions are unknown. An ideal simulation thus implies testing different rules and schemes to determine those that best capture an observed biological phenomenon. This is not trivial because most current methods to simulate Boolean network models of GRNs and to compute their attractors impose specific assumptions that cannot be easily altered, as they are built into the system.

RESULTS

To allow for a more flexible simulation framework, we developed ASP-G. We show the correctness of ASP-G in simulating Boolean network models and obtaining attractors under different assumptions by successfully recapitulating the detection of attractors of previously published studies. We also provide an example of how performing simulation of network models under different settings help determine the assumptions under which a certain conclusion holds. The main added value of ASP-G is in its modularity and declarativity, making it more flexible and less error-prone than traditional approaches. The declarative nature of ASP-G comes at the expense of being slower than the more dedicated systems but still achieves a good efficiency with respect to computational time.

AVAILABILITY AND IMPLEMENTATION

The source code of ASP-G is available at http://bioinformatics.intec.ugent.be/kmarchal/Supplementary_Information_Musthofa_2014/asp-g.zip.

Composition and abstraction of logical regulatory modules: application to multicellular systems

MOTIVATION

Logical (Boolean or multi-valued) modelling is widely used to study regulatory or signalling networks. Even though these discrete models constitute a coarse, yet useful, abstraction of reality, the analysis of large networks faces a classical combinatorial problem. Here, we propose to take advantage of the intrinsic modularity of inter-cellular networks to set up a compositional procedure that enables a significant reduction of the dynamics, yet preserving the reachability of stable states. To that end, we rely on process algebras, a well-established computational technique for the specification and verification of interacting systems.

RESULTS

We develop a novel compositional approach to support the logical modelling of interconnected cellular networks. First, we formalize the concept of logical regulatory modules and their composition. Then, we make this framework operational by transposing the composition of logical modules into a process algebra framework. Importantly, the combination of incremental composition, abstraction and minimization using an appropriate equivalence relation (here the safety equivalence) yields huge reductions of the dynamics. We illustrate the potential of this approach with two case-studies: the Segment-Polarity and the Delta-Notch modules.

Extracting gene-gene interactions through curve fitting

This paper presents a simple and novel curve fitting approach for generating simple gene regulatory subnetworks from time series gene expression data. Microarray experiments simultaneously generate massive data sets and help immensely in the large-scale study of gene expression patterns. Initial biclustering reduces the search space in the high-dimensional microarray data. The least-squares error between fitting of gene pairs is minimized to extract a set of gene-gene interactions, involving transcriptional regulation of genes. The higher error values are eliminated to retain only the strong interacting gene pairs in the resultant gene regulatory subnetwork. Next the algorithm is extended to a generalized framework to enhance its capability. The methodology takes care of the higher-order dependencies involving multiple genes co-regulating a single gene, while eliminating the need for user-defined parameters. It has been applied to the time-series Yeast data, and the experimental results biologically validated using standard databases and literature.

Inference of biological S-system using the separable estimation method and the genetic algorithm

Reconstruction of a biological system from its experimental time series data is a challenging task in systems biology. The S-system which consists of a group of nonlinear ordinary differential equations (ODEs) is an effective model to characterize molecular biological systems and analyze the system dynamics. However, inference of S-systems without the knowledge of system structure is not a trivial task due to its nonlinearity and complexity. In this paper, a pruning separable parameter estimation algorithm (PSPEA) is proposed for inferring S-systems. This novel algorithm combines the separable parameter estimation method (SPEM) and a pruning strategy, which includes adding an l₁ regularization term to the objective function and pruning the solution with a threshold value. Then, this algorithm is combined with the continuous genetic algorithm (CGA) to form a hybrid algorithm that owns the properties of these two combined algorithms. The performance of the pruning strategy in the proposed algorithm is evaluated from two aspects: the parameter estimation error and structure identification accuracy. The results show that the proposed algorithm with the pruning strategy has much lower estimation error and much higher identification accuracy than the existing method.

An ensemble approach for inferring semi-quantitative regulatory dynamics for the differentiation of mouse embryonic stem cells using prior knowledge

The process of differentiation of embryonic stem cells (ESCs) is currently becoming the focus of many systems biologists not only due to mechanistic interest but also since it is expected to play an increasingly important role in regenerative medicine, in particular with the advert to induced pluripotent stem cells. These ESCs give rise to the formation of the three germ layers and therefore to the formation of all tissues and organs. Here, we present a computational method for inferring regulatory interactions between the genes involved in ESC differentiation based on time resolved microarray profiles. Fully quantitative methods are commonly unavailable on such large-scale data; on the other hand, purely qualitative methods may fail to capture some of the more detailed regulations. Our method combines the beneficial aspects of qualitative and quantitative (ODE-based) modeling approaches searching for quantitative interaction coefficients in a discrete and qualitative state space. We further optimize on an ensemble of networks to detect essential properties and compare networks with respect to robustness. Applied to a toy model our method is able to reconstruct the original network and outperforms an entire discrete boolean approach. In particular, we show that including prior knowledge leads to more accurate results. Applied to data from differentiating mouse ESCs reveals new regulatory interactions, in particular we confirm the activation of Foxh1 through Oct4, mediating Nodal signaling.

Genetic network analyzer: a tool for the qualitative modeling and simulation of bacterial regulatory networks

Genetic Network Analyzer (GNA) is a tool for the qualitative modeling and simulation of gene regulatory networks, based on so-called piecewise-linear differential equation models. We describe the use of this tool in the context of the modeling of bacterial regulatory networks, notably the network of global regulators controlling the adaptation of Escherichia coli to carbon starvation conditions. We show how the modeler, by means of GNA, can define a regulatory network, build a model of the network, determine the steady states of the system, perform a qualitative simulation of the network dynamics, and analyze the simulation results using model-checking tools. The example illustrates the interest of qualitative approaches for the analysis of the dynamics of bacterial regulatory networks.

Qualitative modelling and formal verification of the FLR1 gene mancozeb response in Saccharomyces cerevisiae

BACKGROUND

Qualitative models allow understanding the relation between the structure and the dynamics of gene regulatory networks. The dynamical properties of these models can be automatically analysed by means of formal verification methods, like model checking. This facilitates the model-validation process and the test of new hypotheses to reconcile model predictions with the experimental data.

RESULTS

The authors report in this study the qualitative modelling and simulation of the transcriptional regulatory network controlling the response of the model eukaryote Saccharomyces cerevisiae to the agricultural fungicide mancozeb. The model allowed the analysis of the regulation level and activity of the components of the gene mancozeb-induced network controlling the transcriptional activation of the FLR1 gene, which is proposed to confer multidrug resistance through its putative role as a drug eflux pump. Formal verification analysis of the network allowed us to confront model predictions with the experimental data and to assess the model robustness to parameter ordering and gene deletion.

CONCLUSIONS

This analysis enabled us to better understand the mechanisms regulating the FLR1 gene mancozeb response and confirmed the need of a new transcription factor for the full transcriptional activation of YAP1. The result is a computable model of the FLR1 gene response to mancozeb, permitting a quick and cost-effective test of hypotheses prior to experimental validation.

A SAT-based algorithm for finding attractors in synchronous Boolean networks

This paper addresses the problem of finding attractors in synchronous Boolean networks. The existing Boolean decision diagram-based algorithms have limited capacity due to the excessive memory requirements of decision diagrams. The simulation-based algorithms can be applied to larger networks, however, they are incomplete. We present an algorithm, which uses a SAT-based bounded model checking to find all attractors in a Boolean network. The efficiency of the presented algorithm is evaluated by analyzing seven networks models of real biological processes, as well as 150,000 randomly generated Boolean networks of sizes between 100 and 7,000. The results show that our approach has a potential to handle an order of magnitude larger models than currently possible.

SPABBATS: A pathway-discovery method based on Boolean satisfiability that facilitates the characterization of suppressor mutants

Background

Several computational methods exist to suggest rational genetic interventions that improve the productivity of industrial strains. Nonetheless, these methods are less effective to predict possible genetic responses of the strain after the intervention. This problem requires a better understanding of potential alternative metabolic and regulatory pathways able to counteract the targeted intervention.

Results

Here we present SPABBATS, an algorithm based on Boolean satisfiability (SAT) that computes alternative metabolic pathways between input and output species in a reconstructed network. The pathways can be constructed iteratively in order of increasing complexity. SPABBATS allows the accumulation of intermediates in the pathways, which permits discovering pathways missed by most traditional pathway analysis methods. In addition, we provide a proof of concept experiment for the validity of the algorithm. We deleted the genes for the glutamate dehydrogenases of the Gram-positive bacterium Bacillus subtilis and isolated suppressor mutant strains able to grow on glutamate as single carbon source. Our SAT approach proposed candidate alternative pathways which were decisive to pinpoint the exact mutation of the suppressor strain.

Conclusions

SPABBATS is the first application of SAT techniques to metabolic problems. It is particularly useful for the characterization of metabolic suppressor mutants and can be used in a synthetic biology setting to design new pathways with specific input-output requirements.

Model reduction using piecewise-linear approximations preserves dynamic properties of the carbon starvation response in Escherichia coli

The adaptation of the bacterium Escherichia coli to carbon starvation is controlled by a large network of biochemical reactions involving genes, mRNAs, proteins, and signalling molecules. The dynamics of these networks is difficult to analyze, notably due to a lack of quantitative information on parameter values. To overcome these limitations, model reduction approaches based on quasi-steady-state (QSS) and piecewise-linear (PL) approximations have been proposed, resulting in models that are easier to handle mathematically and computationally. These approximations are not supposed to affect the capability of the model to account for essential dynamical properties of the system, but the validity of this assumption has not been systematically tested. In this paper, we carry out such a study by evaluating a large and complex PL model of the carbon starvation response in E. coli using an ensemble approach. The results show that, in comparison with conventional nonlinear models, the PL approximations generally preserve the dynamics of the carbon starvation response network, although with some deviations concerning notably the quantitative precision of the model predictions. This encourages the application of PL models to the qualitative analysis of bacterial regulatory networks, in situations where the reference time scale is that of protein synthesis and degradation.

Genetic networks and soft computing

The analysis of gene regulatory networks provides enormous information on various fundamental cellular processes involving growth, development, hormone secretion, and cellular communication. Their extraction from available gene expression profiles is a challenging problem. Such reverse engineering of genetic networks offers insight into cellular activity toward prediction of adverse effects of new drugs or possible identification of new drug targets. Tasks such as classification, clustering, and feature selection enable efficient mining of knowledge about gene interactions in the form of networks. It is known that biological data is prone to different kinds of noise and ambiguity. Soft computing tools, such as fuzzy sets, evolutionary strategies, and neurocomputing, have been found to be helpful in providing low-cost, acceptable solutions in the presence of various types of uncertainties. In this paper, we survey the role of these soft methodologies and their hybridizations, for the purpose of generating genetic networks.

Applications of a formal approach to decipher discrete genetic networks

Background

A growing demand for tools to assist the building and analysis of biological networks exists in systems biology. We argue that the use of a formal approach is relevant and applicable to address questions raised by biologists about such networks. The behaviour of these systems being complex, it is essential to exploit efficiently every bit of experimental information. In our approach, both the evolution rules and the partial knowledge about the structure and the behaviour of the network are formalized using a common constraint-based language.

Results

In this article our formal and declarative approach is applied to three biological applications. The software environment that we developed allows to specifically address each application through a new class of biologically relevant queries. We show that we can describe easily and in a formal manner the partial knowledge about a genetic network. Moreover we show that this environment, based on a constraint algorithmic approach, offers a wide variety of functionalities, going beyond simple simulations, such as proof of consistency, model revision, prediction of properties, search for minimal models relatively to specified criteria.

Conclusions

The formal approach proposed here deeply changes the way to proceed in the exploration of genetic and biochemical networks, first by avoiding the usual trial-and-error procedure, and second by placing the emphasis on sets of solutions, rather than a single solution arbitrarily chosen among many others. Last, the constraint approach promotes an integration of model and experimental data in a single framework.