Modeling gene regulation networks using ordinary differential equations

Approaches for Benchmarking Single-Cell Gene Regulatory Network Methods

Gene regulatory networks are powerful tools for modeling genetic interactions that control the expression of genes driving cell differentiation, and single-cell sequencing offers a unique opportunity to build these networks with high-resolution genomic data. There are many proposed computational methods to build these networks using single-cell data, and different approaches are used to benchmark these methods. However, a comprehensive discussion specifically focusing on benchmarking approaches is missing. In this article, we lay the GRN terminology, present an overview of common gold-standard studies and data sets, and define the performance metrics for benchmarking network construction methodologies. We also point out the advantages and limitations of different benchmarking approaches, suggest alternative ground truth data sets that can be used for benchmarking, and specify additional considerations in this context.

TGPred: efficient methods for predicting target genes of a transcription factor by integrating statistics, machine learning and optimization

Four statistical selection methods for inferring transcription factor (TF)-target gene (TG) pairs were developed by coupling mean squared error (MSE) or Huber loss function, with elastic net (ENET) or least absolute shrinkage and selection operator (Lasso) penalty. Two methods were also developed for inferring pathway gene regulatory networks (GRNs) by combining Huber or MSE loss function with a network (Net)-based penalty. To solve these regressions, we ameliorated an accelerated proximal gradient descent (APGD) algorithm to optimize parameter selection processes, resulting in an equally effective but much faster algorithm than the commonly used convex optimization solver. The synthetic data generated in a general setting was used to test four TF-TG identification methods, ENET-based methods performed better than Lasso-based methods. Synthetic data generated from two network settings was used to test Huber-Net and MSE-Net, which outperformed all other methods. The TF-TG identification methods were also tested with SND1 and gl3 overexpression transcriptomic data, Huber-ENET and MSE-ENET outperformed all other methods when genome-wide predictions were performed. The TF-TG identification methods fill the gap of lacking a method for genome-wide TG prediction of a TF, and potential for validating ChIP/DAP-seq results, while the two Net-based methods are instrumental for predicting pathway GRNs.

Fuzzy and Rough Set Theory Based Computational Framework for Mining Genetic Interaction Triplets From Gene Expression Profiles for Lung Adenocarcinoma

Genetic interactions are very helpful in understanding different disease and discovering drugs for it. Compared to the gene pairs that represent the genetic interactions between two genes, the gene triplets are more informative and useful. However, existing works on genetic interactions among gene triplets have primarily focused on detecting gene triplets from time series gene expression profiles. Generating the time series gene expression profiles for humans is quite impracticable but the labeled gene expression profiles are available for different diseases in case of humans. In this paper, a computational framework has been proposed to detect gene triplets from labeled gene expression profiles. First, it employs Rough Set Theory for extracting the key genes and then designs a fuzzy inference system for generating possible gene triplets. Further, Root Mean Squared Error measure has been used to prune out the irrelevant gene triplets. In the present work, the proposed computational framework has been applied to labeled lung adenocarcinoma dataset and can be applied to any other labeled gene expression dataset. The extracted gene triplets and their functionalities have been verified with existing biological literature and benchmark databases and the results of verification signify that the proposed framework is promising in terms of finding useful genetic triplets. Further, the proposed framework has been found more efficient as compared to an existing mutual information-based technique in terms of detecting known genetic interactions.

Predictive landscapes hidden beneath biological cellular automata

To celebrate Hans Frauenfelder's achievements, we examine energy(-like) "landscapes" for complex living systems. Energy landscapes summarize all possible dynamics of some physical systems. Energy(-like) landscapes can explain some biomolecular processes, including gene expression and, as Frauenfelder showed, protein folding. But energy-like landscapes and existing frameworks like statistical mechanics seem impractical for describing many living systems. Difficulties stem from living systems being high dimensional, nonlinear, and governed by many, tightly coupled constituents that are noisy. The predominant modeling approach is devising differential equations that are tailored to each living system. This ad hoc approach faces the notorious "parameter problem": models have numerous nonlinear, mathematical functions with unknown parameter values, even for describing just a few intracellular processes. One cannot measure many intracellular parameters or can only measure them as snapshots in time. Another modeling approach uses cellular automata to represent living systems as discrete dynamical systems with binary variables. Quantitative (Hamiltonian-based) rules can dictate cellular automata (e.g., Cellular Potts Model). But numerous biological features, in current practice, are qualitatively described rather than quantitatively (e.g., gene is (highly) expressed or not (highly) expressed). Cellular automata governed by verbal rules are useful representations for living systems and can mitigate the parameter problem. However, they can yield complex dynamics that are difficult to understand because the automata-governing rules are not quantitative and much of the existing mathematical tools and theorems apply to continuous but not discrete dynamical systems. Recent studies found ways to overcome this challenge. These studies either discovered or suggest an existence of predictive "landscapes" whose shapes are described by Lyapunov functions and yield "equations of motion" for a "pseudo-particle." The pseudo-particle represents the entire cellular lattice and moves on the landscape, thereby giving a low-dimensional representation of the cellular automata dynamics. We outline this promising modeling strategy.

GENAVOS: A New Tool for Modelling and Analyzing Cancer Gene Regulatory Networks Using Delayed Nonlinear Variable Order Fractional System

Gene regulatory networks (GRN) are one of the etiologies associated with cancer. Their dysregulation can be associated with cancer formation and asymmetric cellular functions in cancer stem cells, leading to disease persistence and resistance to treatment. Systems that model the complex dynamics of these networks along with adapting to partially known real omics data are closer to reality and may be useful to understand the mechanisms underlying neoplastic phenomena. In this paper, for the first time, modelling of GRNs is performed using delayed nonlinear variable order fractional (VOF) systems in the state space by a new tool called GENAVOS. Although the tool uses gene expression time series data to identify and optimize system parameters, it also models possible epigenetic signals, and the results show that the nonlinear VOF systems have very good flexibility in adapting to real data. We found that GRNs in cancer cells actually have a larger delay parameter than in normal cells. It is also possible to create weak chaotic, periodic, and quasi-periodic oscillations by changing the parameters. Chaos can be associated with the onset of cancer. Our findings indicate a profound effect of time-varying orders on these networks, which may be related to a type of cellular epigenetic memory. By changing the delay parameter and the variable order functions (possible epigenetics signals) for a normal cell system, its behaviour becomes quite similar to the behaviour of a cancer cell. This work confirms the effective role of the miR-17-92 cluster as an epigenetic factor in the cancer cell cycle.

Pinning Control for the p53-Mdm2 Network Dynamics Regulated by p14ARF

p53 regulates the cellular response to genotoxic damage and prevents carcinogenic events. Theoretical and experimental studies state that the p53-Mdm2 network constitutes the core module of regulatory interactions activated by cellular stress induced by a variety of signaling pathways. In this paper, a strategy to control the p53-Mdm2 network regulated by p14ARF is developed, based on the pinning control technique, which consists into applying local feedback controllers to a small number of nodes (pinned ones) in the network. Pinned nodes are selected on the basis of their importance level in a topological hierarchy, their degree of connectivity within the network, and the biological role they perform. In this paper, two cases are considered. For the first case, the oscillatory pattern under gamma-radiation is recovered; afterward, as the second case, increased expression of p53 level is taken into account. For both cases, the control law is applied to p14ARF (pinned node based on a virtual leader methodology), and overexpressed Mdm2-mediated p53 degradation condition is considered as carcinogenic initial behavior. The approach in this paper uses a computational algorithm, which opens an alternative path to understand the cellular responses to stress, doing it possible to model and control the gene regulatory network dynamics in two different biological contexts. As the main result of the proposed control technique, the two mentioned desired behaviors are obtained.

Time Series Adjustment Enhancement of Hierarchical Modeling of Arabidopsis Thaliana Gene Interactions

Network models of gene interactions, using time course gene transcript abundance data, are computationally created using a genetic algorithm designed to incorporate hierarchical Bayesian methods with time series adjustments. The posterior probabilities of interaction between pairs of genes are based on likelihoods of directed acyclic graphs. This algorithm is applied to transcript abundance data collected from Arabidopsis thaliana genes. This study extends the underlying statistical and mathematical theory of the Norris-Patton likelihood by including time series adjustments.

Gene regulatory network inference resources: A practical overview

Transcriptional regulation is a fundamental molecular mechanism involved in almost every aspect of life, from homeostasis to development, from metabolism to behavior, from reaction to stimuli to disease progression. In recent years, the concept of Gene Regulatory Networks (GRNs) has grown popular as an effective applied biology approach for describing the complex and highly dynamic set of transcriptional interactions, due to its easy-to-interpret features. Since cataloguing, predicting and understanding every GRN connection in all species and cellular contexts remains a great challenge for biology, researchers have developed numerous tools and methods to infer regulatory processes. In this review, we catalogue these methods in six major areas, based on the dominant underlying information leveraged to infer GRNs: Coexpression, Sequence Motifs, Chromatin Immunoprecipitation (ChIP), Orthology, Literature and Protein-Protein Interaction (PPI) specifically focused on transcriptional complexes. The methods described here cover a wide range of user-friendliness: from web tools that require no prior computational expertise to command line programs and algorithms for large scale GRN inferences. Each method for GRN inference described herein effectively illustrates a type of transcriptional relationship, with many methods being complementary to others. While a truly holistic approach for inferring and displaying GRNs remains one of the greatest challenges in the field of systems biology, we believe that the integration of multiple methods described herein provides an effective means with which experimental and computational biologists alike may obtain the most complete pictures of transcriptional relationships. This article is part of a Special Issue entitled: Transcriptional Profiles and Regulatory Gene Networks edited by Dr. Federico Manuel Giorgi and Dr. Shaun Mahony.

Quantitative Kinetic Models from Intravital Microscopy: A Case Study Using Hepatic Transport

The liver performs critical physiological functions, including metabolizing and removing substances, such as toxins and drugs, from the bloodstream. Hepatotoxicity itself is intimately linked to abnormal hepatic transport, and hepatotoxicity remains the primary reason drugs in development fail and approved drugs are withdrawn from the market. For this reason, we propose to analyze, across liver compartments, the transport kinetics of fluorescein-a fluorescent marker used as a proxy for drug molecules-using intravital microscopy data. To resolve the transport kinetics quantitatively from fluorescence data, we account for the effect that different liver compartments (with different chemical properties) have on fluorescein's emission rate. To do so, we develop ordinary differential equation transport models from the data where the kinetics is related to the observable fluorescence levels by "measurement parameters" that vary across different liver compartments. On account of the steep non-linearities in the kinetics and stochasticity inherent to the model, we infer kinetic and measurement parameters by generalizing the method of parameter cascades. For this application, the method of parameter cascades ensures fast and precise parameter estimates from noisy time traces.

Ensemble Method of Feature Selection and Reverse Construction of Gene Logical Network Based on Information Entropy

In this paper, we propose a novel ensemble gene selection method to obtain a gene subset. Then we provide a reverse construction method of gene network derived from expression profile data of the gene subset. The uncertainty coefficient based on information entropy are used to define the existence of logical relations among these genes. If the uncertainty coefficient between some genes exceeds predefined thresholds, the gene nodes will be connected by directed edges. Thus, a gene network is generated, which we define as gene logical network. This method is applied to the breast cancer data including control group and experimental group, with comparisons of the 2nd-order logic type distribution, average degree as well as average path length of the networks. It is found that these structures with different networks are quite distinct. By the comparison of the degree difference between control group and experimental group, the key genes are picked up. By defining the dynamics evolution rules of state transition based on the logical regulation among the key genes in the network, the dynamic behaviors for normal breast cells and cells with cancer of different stages are simulated numerically. Some of them are highly related to the development of breast cancer through literature inquiry. The study may provide a useful revelation to the biological mechanism in the formation and development of cancer.

Inference of Large-scale Time-delayed Gene Regulatory Network with Parallel MapReduce Cloud Platform

Inference of gene regulatory network (GRN) is crucial to understand intracellular physiological activity and function of biology. The identification of large-scale GRN has been a difficult and hot topic of system biology in recent years. In order to reduce the computation load for large-scale GRN identification, a parallel algorithm based on restricted gene expression programming (RGEP), namely MPRGEP, is proposed to infer instantaneous and time-delayed regulatory relationships between transcription factors and target genes. In MPRGEP, the structure and parameters of time-delayed S-system (TDSS) model are encoded into one chromosome. An original hybrid optimization approach based on genetic algorithm (GA) and gene expression programming (GEP) is proposed to optimize TDSS model with MapReduce framework. Time-delayed GRNs (TDGRN) with hundreds of genes are utilized to test the performance of MPRGEP. The experiment results reveal that MPRGEP could infer more accurately gene regulatory network than other state-of-art methods, and obtain the convincing speedup.

fastBMA: scalable network inference and transitive reduction

Inferring genetic networks from genome-wide expression data is extremely demanding computationally. We have developed fastBMA, a distributed, parallel, and scalable implementation of Bayesian model averaging (BMA) for this purpose. fastBMA also includes a computationally efficient module for eliminating redundant indirect edges in the network by mapping the transitive reduction to an easily solved shortest-path problem. We evaluated the performance of fastBMA on synthetic data and experimental genome-wide time series yeast and human datasets. When using a single CPU core, fastBMA is up to 100 times faster than the next fastest method, LASSO, with increased accuracy. It is a memory-efficient, parallel, and distributed application that scales to human genome-wide expression data. A 10 000-gene regulation network can be obtained in a matter of hours using a 32-core cloud cluster (2 nodes of 16 cores). fastBMA is a significant improvement over its predecessor ScanBMA. It is more accurate and orders of magnitude faster than other fast network inference methods such as the 1 based on LASSO. The improved scalability allows it to calculate networks from genome scale data in a reasonable time frame. The transitive reduction method can improve accuracy in denser networks. fastBMA is available as code (M.I.T. license) from GitHub (https://github.com/lhhunghimself/fastBMA), as part of the updated networkBMA Bioconductor package (https://www.bioconductor.org/packages/release/bioc/html/networkBMA.html) and as ready-to-deploy Docker images (https://hub.docker.com/r/biodepot/fastbma/).

A method for estimating Hill function-based dynamic models of gene regulatory networks

Gene regulatory networks (GRNs) are quite large and complex. To better understand and analyse GRNs, mathematical models are being employed. Different types of models, such as logical, continuous and stochastic models, can be used to describe GRNs. In this paper, we present a new approach to identify continuous models, because they are more suitable for large number of genes and quantitative analysis. One of the most promising techniques for identifying continuous models of GRNs is based on Hill functions and the generalized profiling method (GPM). The advantage of this approach is low computational cost and insensitivity to initial conditions. In the GPM, a constrained nonlinear optimization problem has to be solved that is usually underdetermined. In this paper, we propose a new optimization approach in which we reformulate the optimization problem such that constraints are embedded implicitly in the cost function. Moreover, we propose to split the unknown parameter in two sets based on the structure of Hill functions. These two sets are estimated separately to resolve the issue of the underdetermined problem. As a case study, we apply the proposed technique on the SOS response in Escherichia coli and compare the results with the existing literature.

Network Analysis of Gene Expression

Studies have pointed out that the expression of genes are highly regulated, which result in a cascade of distinct patterns of coexpression forming a network. Identifying and understanding such patterns is crucial in deciphering molecular mechanisms that underlie the pathophysiology of diseases. With the advance of high throughput assay of messenger RNA (mRNA) and high performance computing, reconstructing such network from molecular data such as gene expression is now possible. This chapter discusses an overview of methods of constructing such networks, practical considerations, and an example.

Inference of time-delayed gene regulatory networks based on dynamic Bayesian network hybrid learning method

Gene regulatory networks (GRNs) research reveals complex life phenomena from the perspective of gene interaction, which is an important research field in systems biology. Traditional Bayesian networks have a high computational complexity, and the network structure scoring model has a single feature. Information-based approaches cannot identify the direction of regulation. In order to make up for the shortcomings of the above methods, this paper presents a novel hybrid learning method (DBNCS) based on dynamic Bayesian network (DBN) to construct the multiple time-delayed GRNs for the first time, combining the comprehensive score (CS) with the DBN model. DBNCS algorithm first uses CMI2NI (conditional mutual inclusive information-based network inference) algorithm for network structure profiles learning, namely the construction of search space. Then the redundant regulations are removed by using the recursive optimization algorithm (RO), thereby reduce the false positive rate. Secondly, the network structure profiles are decomposed into a set of cliques without loss, which can significantly reduce the computational complexity. Finally, DBN model is used to identify the direction of gene regulation within the cliques and search for the optimal network structure. The performance of DBNCS algorithm is evaluated by the benchmark GRN datasets from DREAM challenge as well as the SOS DNA repair network in Escherichia coli, and compared with other state-of-the-art methods. The experimental results show the rationality of the algorithm design and the outstanding performance of the GRNs.

In Silico Gene Regulatory Network of the Maurer's Cleft Pathway in Plasmodium falciparum

The Maurer's clefts (MCs) are very important for the survival of Plasmodium falciparum within an infected cell as they are induced by the parasite itself in the erythrocyte for protein trafficking. The MCs form an interesting part of the parasite's biology as they shed more light on how the parasite remodels the erythrocyte leading to host pathogenesis and death. Here, we predicted and analyzed the genetic regulatory network of genes identified to belong to the MCs using regularized graphical Gaussian model. Our network shows four major activators, their corresponding target genes, and predicted binding sites. One of these master activators is the serine repeat antigen 5 (SERA5), predominantly expressed among the SERA multigene family of P. falciparum, which is one of the blood-stage malaria vaccine candidates. Our results provide more details about functional interactions and the regulation of the genes in the MCs’ pathway of P. falciparum.

Natural biased coin encoded in the genome determines cell strategy

Decision making at a cellular level determines different fates for isogenic cells. However, it is not yet clear how rational decisions are encoded in the genome, how they are transmitted to their offspring, and whether they evolve and become optimized throughout generations. In this paper, we use a game theoretic approach to explain how rational decisions are made in the presence of cooperators and competitors. Our results suggest the existence of an internal switch that operates as a biased coin. The biased coin is, in fact, a biochemical bistable network of interacting genes that can flip to one of its stable states in response to different environmental stimuli. We present a framework to describe how the positions of attractors in such a gene regulatory network correspond to the behavior of a rational player in a competing environment. We evaluate our model by considering lysis/lysogeny decision making of bacteriophage lambda in E. coli.

Hierarchical Probabilistic Interaction Modeling for Multiple Gene Expression Replicates

Microarray technology allows for the collection of multiple replicates of gene expression time course data for hundreds of genes at a handful of time points. Developing hypotheses about a gene transcriptional network, based on time course gene expression data is an important and very challenging problem. In many situations there are similarities which suggest a hierarchical structure between the replicates. This paper develops posterior probabilities for network features based on multiple hierarchical replications. Through Bayesian inference, in conjunction with the Metropolis-Hastings algorithm and model averaging, a hierarchical multiple replicate algorithm is applied to seven sets of simulated data and to a set of Arabidopsis thaliana gene expression data. The models of the simulated data suggest high posterior probabilities for pairs of genes which have at least moderate signal partial correlation. For the Arabidopsis model, many of the highest posterior probability edges agree with the literature.

A review on the computational approaches for gene regulatory network construction

Many biological research areas such as drug design require gene regulatory networks to provide clear insight and understanding of the cellular process in living cells. This is because interactions among the genes and their products play an important role in many molecular processes. A gene regulatory network can act as a blueprint for the researchers to observe the relationships among genes. Due to its importance, several computational approaches have been proposed to infer gene regulatory networks from gene expression data. In this review, six inference approaches are discussed: Boolean network, probabilistic Boolean network, ordinary differential equation, neural network, Bayesian network, and dynamic Bayesian network. These approaches are discussed in terms of introduction, methodology and recent applications of these approaches in gene regulatory network construction. These approaches are also compared in the discussion section. Furthermore, the strengths and weaknesses of these computational approaches are described.

Tracking of time-varying genomic regulatory networks with a LASSO-Kalman smoother

It is widely accepted that cellular requirements and environmental conditions dictate the architecture of genetic regulatory networks. Nonetheless, the status quo in regulatory network modeling and analysis assumes an invariant network topology over time. In this paper, we refocus on a dynamic perspective of genetic networks, one that can uncover substantial topological changes in network structure during biological processes such as developmental growth. We propose a novel outlook on the inference of time-varying genetic networks, from a limited number of noisy observations, by formulating the network estimation as a target tracking problem. We overcome the limited number of observations (small n large p problem) by performing tracking in a compressed domain. Assuming linear dynamics, we derive the LASSO-Kalman smoother, which recursively computes the minimum mean-square sparse estimate of the network connectivity at each time point. The LASSO operator, motivated by the sparsity of the genetic regulatory networks, allows simultaneous signal recovery and compression, thereby reducing the amount of required observations. The smoothing improves the estimation by incorporating all observations. We track the time-varying networks during the life cycle of the Drosophila melanogaster. The recovered networks show that few genes are permanent, whereas most are transient, acting only during specific developmental phases of the organism.

Passing messages between biological networks to refine predicted interactions

Regulatory network reconstruction is a fundamental problem in computational biology. There are significant limitations to such reconstruction using individual datasets, and increasingly people attempt to construct networks using multiple, independent datasets obtained from complementary sources, but methods for this integration are lacking. We developed PANDA (Passing Attributes between Networks for Data Assimilation), a message-passing model using multiple sources of information to predict regulatory relationships, and used it to integrate protein-protein interaction, gene expression, and sequence motif data to reconstruct genome-wide, condition-specific regulatory networks in yeast as a model. The resulting networks were not only more accurate than those produced using individual data sets and other existing methods, but they also captured information regarding specific biological mechanisms and pathways that were missed using other methodologies. PANDA is scalable to higher eukaryotes, applicable to specific tissue or cell type data and conceptually generalizable to include a variety of regulatory, interaction, expression, and other genome-scale data. An implementation of the PANDA algorithm is available at www.sourceforge.net/projects/panda-net.

Modeling dynamic regulatory processes in stroke

The ability to examine the behavior of biological systems in silico has the potential to greatly accelerate the pace of discovery in diseases, such as stroke, where in vivo analysis is time intensive and costly. In this paper we describe an approach for in silico examination of responses of the blood transcriptome to neuroprotective agents and subsequent stroke through the development of dynamic models of the regulatory processes observed in the experimental gene expression data. First, we identified functional gene clusters from these data. Next, we derived ordinary differential equations (ODEs) from the data relating these functional clusters to each other in terms of their regulatory influence on one another. Dynamic models were developed by coupling these ODEs into a model that simulates the expression of regulated functional clusters. By changing the magnitude of gene expression in the initial input state it was possible to assess the behavior of the networks through time under varying conditions since the dynamic model only requires an initial starting state, and does not require measurement of regulatory influences at each time point in order to make accurate predictions. We discuss the implications of our models on neuroprotection in stroke, explore the limitations of the approach, and report that an optimized dynamic model can provide accurate predictions of overall system behavior under several different neuroprotective paradigms.

Computational modeling aims to use mathematical and algorithmic principles to link components of biological systems to predict system behavior. In the past such models have described a small set of carefully studied molecular interactions (proteins in signal transduction pathways) or larger abstract components (cell types or functional processes in the immune system). In this study we use data from global transcriptional analysis of the processes of neuroprotection in a mouse model of stroke to generate functional modules, groups of genes that coherently act to accomplish functions. We then derive equations relating the expression of these modules to one another, treating these individual equations as a closed system, and demonstrate that the model can be used to simulate the gene expression of the system over time. Our work is novel in describing the use of global transcriptomic data to develop dynamic models of expression in an animal model. We believe that the models developed will aid in understanding the complex dynamics of neuroprotection and provide ways to predict outcomes in terms of neuroprotection or injury. This approach will be broadly applicable to other problems and provides an approach to building dynamic models from the bottom up.