Small Gene Regulatory Networks Research Articles

A data-driven optimization method for coarse-graining gene regulatory networks

One major challenge in systems biology is to understand how various genes in a gene regulatory network (GRN) collectively perform their functions and control network dynamics. This task becomes extremely hard to tackle in the case of large networks with hundreds of genes and edges, many of which have redundant regulatory roles and functions. The existing methods for model reduction usually require the detailed mathematical description of dynamical systems and their corresponding kinetic parameters, which are often not available. Here, we present a data-driven method for coarse-graining large GRNs, named SacoGraci, using ensemble-based mathematical modeling, dimensionality reduction, and gene circuit optimization by Markov Chain Monte Carlo methods. SacoGraci requires network topology as the only input and is robust against errors in GRNs. We benchmark and demonstrate its usage with synthetic, literature-based, and bioinformatics-derived GRNs. We hope SacoGraci will enhance our ability to model the gene regulation of complex biological systems.

Influence maximization in Boolean networks

The optimization problem aiming at the identification of minimal sets of nodes able to drive the dynamics of Boolean networks toward desired long-term behaviors is central for some applications, as for example the detection of key therapeutic targets to control pathways in models of biological signaling and regulatory networks. Here, we develop a method to solve such an optimization problem taking inspiration from the well-studied problem of influence maximization for spreading processes in social networks. We validate the method on small gene regulatory networks whose dynamical landscapes are known by means of brute-force analysis. We then systematically study a large collection of gene regulatory networks. We find that for about 65% of the analyzed networks, the minimal driver sets contain less than 20% of their nodes.

Modeling the Dynamics of a Family of Small Gene Regulatory Networks in which Single Edges Have Been Deleted Indicates Regulation by Hmo1, Msn2, and Cin5 Are Important for Controlling the Cold Shock Response in Saccharomyces cerevisiae

A gene regulatory network (GRN) is a set of transcription factors which regulate the level of expression of genes encoding other transcription factors. The dynamics of a GRN show how gene expression in the network changes over time. Microarray data were obtained from the Saccharomyces cerevisiae wild type strain and five transcription factor deletion strains (Δcin5, Δgln3, Δhap4, Δhmo1, Δzap1) before cold shock at 13°C and 15, 30, and 60 minutes after cold shock (NCBI GEO Series GSE83656). Genes that showed a significant change in expression were submitted to the YEASTRACT database to determine which transcription factors regulated them and to generate a candidate GRN of 15 nodes (transcription factors) and 28 edges (regulatory relationships). The edges of this intact network were then systematically deleted one‐at‐a‐time to create a family of 28 additional networks to determine the importance of each edge in the network. We used the open source software, GRNmap ( http://kdahlquist.github.io/GRNmap/), to model the dynamics of these GRNs. GRNmap models the change in expression for each gene as the production of mRNA minus its degradation using differential equations with a sigmoidal production function. Given published mRNA degradation rates (Neymotin et al. 2014; doi: 10.1261/rna.045104.114) and cold shock microarray data, GRNmap then estimated the production rates and expression thresholds for each gene, and regulatory weights for each edge, which denote the direction (activation or repression) and strength of the regulatory relationships. Edge weights were then visualized with the open source software GRNsight (Dahlquist et al. 2014; doi: 10.7717/peerj‐cs.85; https://dondi.github.io/GRNsight/). Figure 1 shows the intact GRN. Red edges indicate activation; blue edges indicate repression. Node stripes left to right indicate gene expression at 15, 30, and 60 minutes of cold shock, red is increased expression; blue is decreased. To evaluate the goodness of fit of the model, GRNmap reports the least squares error (LSE) between the model and the microarray data. The LSE was compared to the minimum theoretical least squares error (minLSE) achievable due to variation in the data, to determine the model performance between networks. In the 28 edge‐deletion networks, LSE:minLSE ratios indicated that five networks performed better than the intact network, ten networks performed about the same, and thirteen performed worse. The edge‐deletions involving the Hmo1, Msn2, and Cin5 transcription factors resulted in a poor fit of the model to data, indicating that those edges represent important regulatory relationships in the cold shock response. K‐means clustering was performed on the edge weight values from the intact network and edge‐deletion networks. An examination of the clusters showed that weight values deviated from those of the intact network for 6 of 9 edge‐deletions involving Msn2, 4 of 5 edge‐deletions involving Hmo1, and 3 of 6 edge‐deletions involving Cin5, further reinforcing their importance in the network controlling the cold shock response in yeast.Support or Funding InformationLoyola Marymount University Rains Research Assistant Program (A.K.F.)Figure 1

Reverse engineering directed gene regulatory networks from transcriptomics and proteomics data of biomining bacterial communities with approximate Bayesian computation and steady-state signalling simulations

BackgroundNetwork inference is an important aim of systems biology. It enables the transformation of OMICs datasets into biological knowledge. It consists of reverse engineering gene regulatory networks from OMICs data, such as RNAseq or mass spectrometry-based proteomics data, through computational methods. This approach allows to identify signalling pathways involved in specific biological functions. The ability to infer causality in gene regulatory networks, in addition to correlation, is crucial for several modelling approaches and allows targeted control in biotechnology applications.MethodsWe performed simulations according to the approximate Bayesian computation method, where the core model consisted of a steady-state simulation algorithm used to study gene regulatory networks in systems for which a limited level of details is available. The simulations outcome was compared to experimentally measured transcriptomics and proteomics data through approximate Bayesian computation.ResultsThe structure of small gene regulatory networks responsible for the regulation of biological functions involved in biomining were inferred from multi OMICs data of mixed bacterial cultures. Several causal inter- and intraspecies interactions were inferred between genes coding for proteins involved in the biomining process, such as heavy metal transport, DNA damage, replication and repair, and membrane biogenesis. The method also provided indications for the role of several uncharacterized proteins by the inferred connection in their network context.ConclusionsThe combination of fast algorithms with high-performance computing allowed the simulation of a multitude of gene regulatory networks and their comparison to experimentally measured OMICs data through approximate Bayesian computation, enabling the probabilistic inference of causality in gene regulatory networks of a multispecies bacterial system involved in biomining without need of single-cell or multiple perturbation experiments. This information can be used to influence biological functions and control specific processes in biotechnology applications.

Improvement of the memory function of a mutual repression network in a stochastic environment by negative autoregulation

BackgroundCellular memory is a ubiquitous function of biological systems. By generating a sustained response to a transient inductive stimulus, often due to bistability, memory is central to the robust control of many important biological processes. However, our understanding of the origins of cellular memory remains incomplete. Stochastic fluctuations that are inherent to most biological systems have been shown to hamper memory function. Yet, how stochasticity changes the behavior of genetic circuits is generally not clear from a deterministic analysis of the network alone. Here, we apply deterministic rate equations, stochastic simulations, and theoretical analyses of Fokker-Planck equations to investigate how intrinsic noise affects the memory function in a mutual repression network.ResultsWe find that the addition of negative autoregulation improves the persistence of memory in a small gene regulatory network by reducing stochastic fluctuations. Our theoretical analyses reveal that this improved memory function stems from an increased stability of the steady states of the system. Moreover, we show how the tuning of critical network parameters can further enhance memory.ConclusionsOur work illuminates the power of stochastic and theoretical approaches to understanding biological circuits, and the importance of considering stochasticity when designing synthetic circuits with memory function.

GRNsight: a web application and service for visualizing models of small- to medium-scale gene regulatory networks

GRNsight is a web application and service for visualizing models of gene regulatory networks (GRNs). A gene regulatory network (GRN) consists of genes, transcription factors, and the regulatory connections between them which govern the level of expression of mRNA and protein from genes. The original motivation came from our efforts to perform parameter estimation and forward simulation of the dynamics of a differential equations model of a small GRN with 21 nodes and 31 edges. We wanted a quick and easy way to visualize the weight parameters from the model which represent the direction and magnitude of the influence of a transcription factor on its target gene, so we created GRNsight. GRNsight automatically lays out either an unweighted or weighted network graph based on an Excel spreadsheet containing an adjacency matrix where regulators are named in the columns and target genes in the rows, a Simple Interaction Format (SIF) text file, or a GraphML XML file. When a user uploads an input file specifying an unweighted network, GRNsight automatically lays out the graph using black lines and pointed arrowheads. For a weighted network, GRNsight uses pointed and blunt arrowheads, and colors the edges and adjusts their thicknesses based on the sign (positive for activation or negative for repression) and magnitude of the weight parameter. GRNsight is written in JavaScript, with diagrams facilitated by D3.js, a data visualization library. Node.js and the Express framework handle server-side functions. GRNsight’s diagrams are based on D3.js’s force graph layout algorithm, which was then extensively customized to support the specific needs of GRNs. Nodes are rectangular and support gene labels of up to 12 characters. The edges are arcs, which become straight lines when the nodes are close together. Self-regulatory edges are indicated by a loop. When a user mouses over an edge, the numerical value of the weight parameter is displayed. Visualizations can be modified by sliders that adjust the force graph layout parameters and through manual node dragging. GRNsight is best-suited for visualizing networks of fewer than 35 nodes and 70 edges, although it accepts networks of up to 75 nodes or 150 edges. GRNsight has general applicability for displaying any small, unweighted or weighted network with directed edges for systems biology or other application domains. GRNsight serves as an example of following and teaching best practices for scientific computing and complying with FAIR principles, using an open and test-driven development model with rigorous documentation of requirements and issues on GitHub. An exhaustive unit testing framework using Mocha and the Chai assertion library consists of around 160 automated unit tests that examine nearly 530 test files to ensure that the program is running as expected. The GRNsight application (http://dondi.github.io/GRNsight/) and code (https://github.com/dondi/GRNsight) are available under the open source BSD license.

Cost and Precision in Small Gene Regulatory Networks

Genes in early fly development are expressed with astonishing precision, despite the molecular noise intrinsic to all biochemical reactions. In an attempt to understand the regulatory mechanisms in this system, I will discuss the nature of hunchback mRNA expression based on analysis of live imaging experiments. Inspired by fly development, I will then discuss trade-offs that gene regulatory circuits have to face in order to precisely respond to signals in the presence of molecular noise.

MCA: Multiresolution Correlation Analysis, a graphical tool for subpopulation identification in single-cell gene expression data.

BackgroundBiological data often originate from samples containing mixtures of subpopulations, corresponding e.g. to distinct cellular phenotypes. However, identification of distinct subpopulations may be difficult if biological measurements yield distributions that are not easily separable.ResultsWe present Multiresolution Correlation Analysis (MCA), a method for visually identifying subpopulations based on the local pairwise correlation between covariates, without needing to define an a priori interaction scale. We demonstrate that MCA facilitates the identification of differentially regulated subpopulations in simulated data from a small gene regulatory network, followed by application to previously published single-cell qPCR data from mouse embryonic stem cells. We show that MCA recovers previously identified subpopulations, provides additional insight into the underlying correlation structure, reveals potentially spurious compartmentalizations, and provides insight into novel subpopulations.ConclusionsMCA is a useful method for the identification of subpopulations in low-dimensional expression data, as emerging from qPCR or FACS measurements. With MCA it is possible to investigate the robustness of covariate correlations with respect subpopulations, graphically identify outliers, and identify factors contributing to differential regulation between pairs of covariates. MCA thus provides a framework for investigation of expression correlations for genes of interests and biological hypothesis generation.

An extended gene protein/products Boolean network model including post-transcriptional regulation.

BackgroundNetworks Biology allows the study of complex interactions between biological systems using formal, well structured, and computationally friendly models. Several different network models can be created, depending on the type of interactions that need to be investigated. Gene Regulatory Networks (GRN) are an effective model commonly used to study the complex regulatory mechanisms of a cell. Unfortunately, given their intrinsic complexity and non discrete nature, the computational study of realistic-sized complex GRNs requires some abstractions. Boolean Networks (BNs), for example, are a reliable model that can be used to represent networks where the possible state of a node is a boolean value (0 or 1). Despite this strong simplification, BNs have been used to study both structural and dynamic properties of real as well as randomly generated GRNs.ResultsIn this paper we show how it is possible to include the post-transcriptional regulation mechanism (a key process mediated by small non-coding RNA molecules like the miRNAs) into the BN model of a GRN. The enhanced BN model is implemented in a software toolkit (EBNT) that allows to analyze boolean GRNs from both a structural and a dynamic point of view. The open-source toolkit is compatible with available visualization tools like Cytoscape and allows to run detailed analysis of the network topology as well as of its attractors, trajectories, and state-space. In the paper, a small GRN built around the mTOR gene is used to demonstrate the main capabilities of the toolkit.ConclusionsThe extended model proposed in this paper opens new opportunities in the study of gene regulation. Several of the successful researches done with the support of BN to understand high-level characteristics of regulatory networks, can now be improved to better understand the role of post-transcriptional regulation for example as a network-wide noise-reduction or stabilization mechanisms.

Generation of oscillating gene regulatory network motifs

Using an improved version of an evolutionary algorithm originally proposed by François and Hakim [Proc. Natl. Acad. Sci. USA 101, 580 (2004)], we generated small gene regulatory networks in which the concentration of a target protein oscillates in time. These networks may serve as candidates for oscillatory modules to be found in larger regulatory networks and protein interaction networks. The algorithm was run for 10(5) times to produce a large set of oscillating modules, which were systematically classified and analyzed. The robustness of the oscillations against variations of the kinetic rates was also determined, to filter out the least robust cases. Furthermore, we show that the set of evolved networks can serve as a database of models whose behavior can be compared to experimentally observed oscillations. The algorithm found three smallest (core) oscillators in which nonlinearities and number of components are minimal. Two of those are two-gene modules: the mixed feedback loop, already discussed in the literature, and an autorepressed gene coupled with a heterodimer. The third one is a single gene module which is competitively regulated by a monomer and a dimer. The evolutionary algorithm also generated larger oscillating networks, which are in part extensions of the three core modules and in part genuinely new modules. The latter includes oscillators which do not rely on feedback induced by transcription factors, but are purely of post-transcriptional type. Analysis of post-transcriptional mechanisms of oscillation may provide useful information for circadian clock research, as recent experiments showed that circadian rhythms are maintained even in the absence of transcription.

Inference of gene regulatory networks using S-system: a unified approach

With the increased availability of DNA microarray time-series data, it is possible to discover dynamic gene regulatory networks (GRNs). S-system is a promising model to capture the rich dynamics of GRNs. However, owing to the complexity of the inference problem and limited number of available data comparing to the number of unknown kinetic parameters, S-system can only be applied to a very small GRN with few parameters. This significantly limits its applications. A unified approach to infer GRNs using the S-system model is proposed. In order to discover the structure of large-scale GRNs, a simplified S-system model is proposed that enables fast parameter estimation to determine the major gene interactions. If a detailed S-system model is desirable for a subset of genes, a two-step method is proposed where the range of the parameters will be determined first using genetic programming and recursive least square estimation. Then the mean values of the parameters will be estimated using a multi-dimensional optimisation algorithm. Both the downhill simplex algorithm and modified Powell algorithm are tested for multi-dimensional optimisation. A 50-dimensional synthetic model with 51 parameters for each gene is tested for the applicability of the simplified S-system model. In addition, real measurement data pertaining to yeast protein synthesis are used to demonstrate the effectiveness of the proposed two-step method to identify the detailed interactions among genes in small GRNs.

The capacity for multistability in small gene regulatory networks.

BackgroundRecent years have seen a dramatic increase in the use of mathematical modeling to gain insight into gene regulatory network behavior across many different organisms. In particular, there has been considerable interest in using mathematical tools to understand how multistable regulatory networks may contribute to developmental processes such as cell fate determination. Indeed, such a network may subserve the formation of unicellular leaf hairs (trichomes) in the model plant Arabidopsis thaliana.ResultsIn order to investigate the capacity of small gene regulatory networks to generate multiple equilibria, we present a chemical reaction network (CRN)-based modeling formalism and describe a number of methods for CRN analysis in a parameter-free context. These methods are compared and applied to a full set of one-component subnetworks, as well as a large random sample from 40,680 similarly constructed two-component subnetworks. We find that positive feedback and cooperativity mediated by transcription factor (TF) dimerization is a requirement for one-component subnetwork bistability. For subnetworks with two components, the presence of these processes increases the probability that a randomly sampled subnetwork will exhibit multiple equilibria, although we find several examples of bistable two-component subnetworks that do not involve cooperative TF-promoter binding. In the specific case of epidermal differentiation in Arabidopsis, dimerization of the GL3-GL1 complex and cooperative sequential binding of GL3-GL1 to the CPC promoter are each independently sufficient for bistability.ConclusionComputational methods utilizing CRN-specific theorems to rule out bistability in small gene regulatory networks are far superior to techniques generally applicable to deterministic ODE systems. Using these methods to conduct an unbiased survey of parameter-free deterministic models of small networks, and the Arabidopsis epidermal cell differentiation subnetwork in particular, we illustrate how future experimental research may be guided by network structure analysis.

Dynamical modeling of the cholesterol regulatory pathway with Boolean networks

BackgroundQualitative dynamics of small gene regulatory networks have been studied in quite some details both with synchronous and asynchronous analysis. However, both methods have their drawbacks: synchronous analysis leads to spurious attractors and asynchronous analysis lacks computational efficiency, which is a problem to simulate large networks. We addressed this question through the analysis of a major biosynthesis pathway. Indeed the cholesterol synthesis pathway plays a pivotal role in dislypidemia and, ultimately, in cancer through intermediates such as mevalonate, farnesyl pyrophosphate and geranyl geranyl pyrophosphate, but no dynamic model of this pathway has been proposed until now.ResultsWe set up a computational framework to dynamically analyze large biological networks. This framework associates a classical and computationally efficient synchronous Boolean analysis with a newly introduced method based on Markov chains, which identifies spurious cycles among the results of the synchronous simulation. Based on this method, we present here the results of the analysis of the cholesterol biosynthesis pathway and its physiological regulation by the Sterol Response Element Binding Proteins (SREBPs), as well as the modeling of the action of statins, inhibitor drugs, on this pathway. The in silico experiments show the blockade of the cholesterol endogenous synthesis by statins and its regulation by SREPBs, in full agreement with the known biochemical features of the pathway.ConclusionWe believe that the method described here to identify spurious cycles opens new routes to compute large and biologically relevant models, thanks to the computational efficiency of synchronous simulation.Furthermore, to the best of our knowledge, we present here the first dynamic systems biology model of the human cholesterol pathway and several of its key regulatory control elements, hoping it would provide a good basis to perform in silico experiments and confront the resulting properties with published and experimental data. The model of the cholesterol pathway and its regulation, along with Boolean formulae used for simulation are available on our web site . Graphical results of the simulation are also shown online. The SBML model is available in the BioModels database with submission ID: MODEL0568648427.

Effects of coupling strength and space on the dynamics of coupled toggle switches in stochastic gene networks with multiple-delayed reactions

Genetic toggle switches (TSs) are one of the best studied small gene regulatory networks (GRNs), due to their simplicity and relevant role. They have been interpreted as decision circuits in cell differentiation, a process long hypothesized to be bistable, or as cellular memory units. In these contexts, they must be reliable. Once a "decision" is made, the system must remain stable. One way to gain stability is by duplicating the genes of a TS and coupling the two TSs. Using a recent modeling strategy of GRNs, driven by a delayed stochastic simulation algorithm (delayed SSA) that allows modeling transcription and translation as multidelayed reactions, we analyze the stability of systems of coupled TSs. For this, we introduce the coupling strength (C), a parameter to characterize the GRN structure, against which we compare the GRN stability (S). We first show that time delays in transcription, associated to the promoter region release, ensure bistability of a TS, given no cooperative binding or self-activation reactions. Next, we couple two TSs and measure their toggling frequencies as C varies. Three dynamical regimes are observed: (i) for weak coupling, high frequency synchronized oscillations, (ii) for average coupling, low frequency synchronized oscillations, and (iii) for strong coupling the system becomes stable after a transient, in one of two steady states. The system stability, S, goes through a first order phase transition as C increases, in the average coupling regime. After, we study the effects of spatial separation in two compartments on the dynamics of two coupled TSs, where spatial separation is modeled as normally distributed random time delayed reactions. The phase transition of S, as C increases, occurs for lower values of C than when the two TSs are in the same compartment. Finally, we couple weakly and homogeneously several TSs within a single compartment and observe that as the number of coupled TSs increases, the system goes through the phase transition in S, from oscillatory to stable and for C values lower than in the two previous cases.