Understanding design principles of molecular interaction networks is an important goal of molecular systems biology. Some insights have been gained into features of their network topology through the discovery of graph theoretic patterns that constrain network dynamics. This paper contributes to the identification of patterns in the mechanisms that govern network dynamics. The control of nodes in gene regulatory, signaling, and metabolic networks is governed by a variety of biochemical mechanisms, with inputs from other network nodes that act additively or synergistically. This paper focuses on a certain type of logical rule that appears frequently as a regulatory pattern. Within the context of the multistate discrete model paradigm, a rule type is introduced that reduces to the concept of nested canalyzing function in the Boolean network case. It is shown that networks that employ this type of multivalued logic exhibit more robust dynamics than random networks, with few attractors and short limit cycles. It is also shown that the majority of regulatory functions in many published models of gene regulatory and signaling networks are nested canalyzing.
The set of regulatory interactions between genes, mediated by transcription factors, forms a species' transcriptional regulatory network (TRN). By comparing this network with measured gene expression data one can identify functional properties of the TRN and gain general insight into transcriptional control. We define the subnet of a node as the subgraph consisting of all nodes topologically downstream of the node, including itself. Using a large set of microarray expression data of the bacterium Escherichia coli, we find that the gene expression in different subnets exhibits a structured patt…
Genetic and environmental perturbation experiments have been used to study microbes in a bid to gain insight into transcriptional regulation, adaptive evolution, and other cellular dynamics. These studies have potential in enabling rational strain design. Unfortunately, experimentally determined intracellular flux distribution are often inconsistent or incomparable due to different experimental conditions and methodologies. Computational strain design relies on constraint-based reconstruction and analysis (COBRA) techniques to predict the effect of gene knockouts such as flux balance analysis …
How intracellular and extracellular signals are integrated by transcription factors is essential for understanding complex cellular patterns at the population level. In this Letter, by using a synthetic genetic oscillator coupled to a quorum-sensing apparatus, we propose an experimentally feasible cis-regulatory module (CRM) which performs four possible logic operations (ANDN, ORN, NOR and NAND) of input signals. We show both numerically and theoretically that these different CRMs drive fundamentally different dynamic patterns, such as synchronization, clustering and splay state.
Stochasticity (or noise) at cellular and molecular levels has been observed extensively as a universal feature for living systems. However, how living systems deal with noise while performing desirable biological functions remains a major mystery. Regulatory network configurations, such as their topology and timescale, are shown to be critical in attenuating noise, and noise is also found to facilitate cell fate decision. Here we review major recent findings on noise attenuation through regulatory control, the benefit of noise via noise-induced cellular plasticity during developmental patterni…
Interactions between genes and gene products give rise to complex circuits that enable cells to process information and respond to external signals. Theoretical studies often describe these interactions using continuous, stochastic, or logical approaches. We propose a new modeling framework for gene regulatory networks, that combines the intuitive appeal of a qualitative description of gene states with a high flexibility in incorporating stochasticity in the duration of cellular processes. We apply our methods to the regulatory network of the segment polarity genes, thus gaining novel insights…
Multistable gene regulatory systems sustain different levels of gene expression under identical external conditions. Such multistability is used to encode phenotypic states in processes including nutrient uptake and persistence in bacteria, fate selection in viral infection, cell cycle control, and development. Stochastic switching between different phenotypes can occur as the result of random fluctuations in molecular copy numbers of mRNA and proteins arising in transcription, translation, transport, and binding. However, which component of a pathway triggers such a transition is generally no…
Modeling stochasticity in gene regulatory networks is an important and complex problem in molecular systems biology. To elucidate intrinsic noise, several modeling strategies such as the Gillespie algorithm have been used successfully. This paper contributes an approach as an alternative to these classical settings. Within the discrete paradigm, where genes, proteins, and other molecular components of gene regulatory networks are modeled as discrete variables and are assigned as logical rules describing their regulation through interactions with other components. Stochasticity is modeled at th…
Expression of the Drosophila segment polarity genes is initiated by a prepattern of pair-rule gene products and maintained by a network of regulatory interactions throughout several stages of embryonic development. Analysis of a model of gene interactions based on differential equations showed that wild-type expression patterns of these genes can be obtained for a wide range of kinetic parameters, which suggests that the steady states are determined by the topology of the network and the type of regulatory interactions between components, not the detailed form of the rate laws. To investigate …
Genes are frequently regulated in complex manners, necessitating modelling approaches which go beyond simple (linear) gene-to-gene interactions and address the modularity of cis-regulatory regions and alternate transcription initiation sites. In particular, sharp expression patterns (peaks or stripes) indicate that gene regulation involves nonlinear transcription factor kinetics (beyond first-order). We propose a methodology for approaching this problem, using the example of the multiple cis-regulatory modules (CRMs) and two transcripts (P1 and P2) found in the Drosophila hunchback (hb) and (C…
A crucial step in the early development of multicellular organisms involves the establishment of spatial patterns of gene expression which later direct proliferating cells to take on different cell fates. These patterns enable the cells to infer their global position within a tissue or an organism by reading out local gene expression levels. The patterning system is thus said to encode positional information, a concept that was formalized recently in the framework of information theory. Here we introduce a toy model of patterning in one spatial dimension, which can be seen as an extension of W…
The relationship between the regulatory design and the functionality of molecular networks is a key issue in biology. Modules and motifs have been associated to various cellular processes, thereby providing anecdotal evidence for performance based localization on molecular networks. To quantify structure-function relationship we investigate similarities of proteins which are close in the regulatory network of the yeast Saccharomyces Cerevisiae. We find that the topology of the regulatory network show weak remnants of its history of network reorganizations, but strong features of co-regulated p…
The study of fluctuations in gene regulatory networks is extended to the case of Gaussian colored noise. Firstly, the solution of the corresponding Langevin equation with colored noise is expressed in terms of an Ito integral. Then, two important lemmas concerning the variance of an Ito integral and the covariance of two Ito integrals are shown. Based on the lemmas, we give the general formulae for the variances and covariance of molecular concentrations for a regulatory network near a stable equilibrium explicitly. Two examples, the gene auto-regulatory network and the toggle switch, are pres…
The structure of a stochastic nonlinear gene regulatory network is uncovered by studying its response to input signal generators. Four applications are studied in detail: a nonlinear connection of two linear systems, the design of a logic pulse, a molecular amplifier and the interference of three signal generators in E2F1 regulatory element. The gene interactions are presented using molecular diagrams that have a precise mathematical structure and retain the biological meaning of the processes.
Regulatory networks describe the interactions between molecular or cellular regulators, like transcription factors and genes in gene regulatory networks, kinases and their receptors in signalling networks, or neurons in neural networks. A long-standing aim of quantitative biology is to reconstruct such networks on the basis of large-scale data. Our aim is to leverage fluctuations around the non-equilibrium steady state for network inference. To this end, we use a stochastic model of gene regulation or neural dynamics and solve it approximately within a Gaussian mean-field theory. We develop a …
Gene regulatory networks (GRN) govern phenotypic adaptations and reflect the trade-offs between physiological responses and evolutionary adaptation that act at different time scales. To identify patterns of molecular function and genetic diversity in GRNs, we studied the drought response of the common sunflower, Helianthus annuus, and how the underlying GRN is related to its evolution. We examined the responses of 32,423 expressed sequences to drought and to abscisic acid and selected 145 co-expressed transcripts. We characterized their regulatory relationships in nine kinetic studies based on…
Gene regulatory networks with dynamics characterized by multiple stable states underlie cell fate-decisions. Quantitative models that can link molecular-level knowledge of gene regulation to a global understanding of network dynamics have the potential to guide cell-reprogramming strategies. Networks are often modeled by the stochastic Chemical Master Equation, but methods for systematic identification of key properties of the global dynamics are currently lacking. We present a method for analyzing global dynamics of gene networks using the Markov State Model (MSM) framework, which utilizes a …
We present a general methodology in order to build mathematical models of genetic regulatory networks. This approach is based on the mass action law and on the Jacob and Monod operon model. The mathematical models are built symbolically by the \emph{Mathematica} software package \emph{GeneticNetworks}. This package accepts as input the interaction graphs of the transcriptional activators and repressors and, as output, gives the mathematical model in the form of a system of ordinary differential equations. All the relevant biological parameters are chosen automatically by the software. Within t…
During organogenesis, developmental programs governed by Gene Regulatory Networks (GRN) define the functionality, size and shape of the different constituents of living organisms. Robustness, thus, is an essential characteristic that GRNs need to fulfill in order to maintain viability and reproducibility in a species. In the present work we analyze the robustness of the patterning for the ocellar complex formation in Drosophila melanogaster fly. We have systematically pruned the GRN that drives the development of this visual system to obtain the minimum pathway able to satisfy this pattern. We…
A widely used approach to describe the dynamics of gene regulatory networks is based on the chemical master equation, which considers probability distributions over all possible combinations of molecular counts. The analysis of such models is extremely challenging due to their large discrete state space. We therefore propose a hybrid approximation approach based on a system of partial differential equations, where we assume a continuous-deterministic evolution for the protein counts. We discuss efficient analysis methods for both modeling approaches and compare their performance. We show that …
We study the noise characteristics of stochastic oscillations in protein number dynamics of simple genetic oscillatory systems. Using the three-component negative feedback transcription regulatory system called the repressilator as a prototypical example, we quantify the degree of fluctuations in oscillation periods and amplitudes, as well as the noise propagation along the regulatory cascade in the stable oscillation regime via dynamic Monte Carlo simulations. For the single protein-species level, the fluctuation in the oscillation amplitudes is found to be larger than that of the oscillation…
Tumorigenic transformation of human epithelial cells in vitro has been described experimentally as the potential result of a process known as spontaneous immortalization. In this process a generic series of cell state transitions occur in which normal epithelial cells acquire a senescent state, later surpassed to attain first a mesenchymal state and then a final mesenchymal stem-like phenotype, with potential tumorigenic behavior. In this paper we integrate published data on the molecular components and interactions that have been described as key regulators of such cell states and transitions…
Gene regulatory networks arise in all living cells, allowing the control of gene expression patterns. The study of their topology has revealed that certain subgraphs of interactions or "motifs" appear at anomalously high frequencies. We ask here whether this phenomenon may emerge because of the functions carried out by these networks. Given a framework for describing regulatory interactions and dynamics, we consider in the space of all regulatory networks those that have a prescribed function. Monte Carlo sampling is then used to determine how these functional networks lead to specific motif s…
We show how one may analytically compute the stationary density of the distribution of molecular constituents in populations of cells in the presence of noise arising from either bursting transcription or translation, or noise in degradation rates arising from low numbers of molecules. We have compared our results with an analysis of the same model systems (either inducible or repressible operons) in the absence of any stochastic effects, and shown the correspondence between behaviour in the deterministic system and the stochastic analogs. We have identified key dimensionless parameters that c…
The current pandemic of SARS-CoV-2 has caused extensive damage to society. The characterization of SARS-CoV-2 profiles has been addressed by researchers globally with the aim of resolving this disruptive crisis. This investigation process is indispensable for an understanding of how SARS-CoV-2 behaves in human host cells. However, little is known about the systematic molecular mechanisms involved in the effect of SARS-CoV-2 infection on human host cells. Here, we have presented gene-to-gene regulatory networks in response to SARS-CoV-2 using a Bayesian network. We examined the dynamic changes …
Omics technologies enable unbiased investigation of biological systems through massively parallel sequence acquisition or molecular measurements, bringing the life sciences into the era of Big Data. A central challenge posed by such omics datasets is how to transform this data into biological knowledge. For example, how to use this data to answer questions such as: which functional pathways are involved in cell differentiation? Which genes should we target to stop cancer? Network analysis is a powerful and general approach to solve this problem consisting of two fundamental stages, network rec…
Background: Gene regulatory networks coordinate the expression of genes across physiological states and ensure a synchronized expression of genes in cellular subsystems, critical for the coherent functioning of cells. Here we address the questions whether it is possible to predict gene synchronization from network structure alone. We have recently shown that synchronized gene expression may be predicted from symmetries in the gene regulatory networks (GRN) and described by the concept of symmetry fibrations. We showed that symmetry fibrations partition the genes into groups called fibers based…
There are two phases of Wnt signalling in early vertebrate embryogenesis: very early, maternal Wnt signalling promotes dorsal development, and slightly later, zygotic Wnt signalling promotes ventral and lateral mesoderm induction. However, recent molecular biology analysis has revealed more complexity among the direct Wnt target genes, with at least five classes. Here in order to test the logic and the dynamics of a new Gene Regulatory Network model suggested by these discoveries we use mathematical modelling based on ordinary differential equations (ODEs). Our mathematical modelling of this G…
Genetic regulatory networks enable cells to respond to the changes in internal and external conditions by dynamically coordinating their gene expression profiles. Our ability to make quantitative measurements in these biochemical circuits has deepened our understanding of what kinds of computations genetic regulatory networks can perform and with what reliability. These advances have motivated researchers to look for connections between the architecture and function of genetic regulatory networks. Transmitting information between network's inputs and its outputs has been proposed as one such p…
Robustness to mutations and noise has been shown to evolve through stabilizing selection for optimal phenotypes in model gene regulatory networks. The ability to evolve robust mutants is known to depend on the network architecture. How do the dynamical properties and state-space structures of networks with high and low robustness differ? Does selection operate on the global dynamical behavior of the networks? What kind of state-space structures are favored by selection? We provide damage propagation analysis and an extensive statistical analysis of state spaces of these model networks to show …
Network motifs, the recurring regulatory structural patterns in networks, are able to self-organize to produce networks. Three major motifs, feedforward loop, single input modules and bi-fan are found in gene regulatory networks. The large ratio of genes to transcription factors (TFs) in genomes leads to a sharing of TFs by motifs and is sufficient to result in network self-organization. We find a common design principle of these motifs: short transcript's half-life (THL) TFs are significantly enriched in motifs and hubs. This enrichment becomes one of the driving forces for the emergence of t…
We present a highly efficient and accurate hybrid stochastic simulation algorithm (HSSA) for the purpose of simulating a subset of biochemical reactions of large gene regulatory networks (GRN). The algorithm relies on the separability of a GRN into two groups of reactions, A and B, such that the reactions in A can be simulated via a stochastic simulation algorithm (SSA), while those in group B can yield to a deterministic description via ordinary differential equations. First, we derive exact expressions needed to sample the next reaction time and reaction type, and then give two examples of h…
A Boolean network model of the cell-cycle regulatory network of fission yeast (Schizosaccharomyces Pombe) is constructed solely on the basis of the known biochemical interaction topology. Simulating the model in the computer, faithfully reproduces the known sequence of regulatory activity patterns along the cell cycle of the living cell. Contrary to existing differential equation models, no parameters enter the model except the structure of the regulatory circuitry. The dynamical properties of the model indicate that the biological dynamical sequence is robustly implemented in the regulatory n…
Interactions between individual members of the B-cell lymphoma 2 (Bcl-2) family of proteins form a regulatory network governing mitochondrial outer membrane permeabilization (MOMP). Bcl-2 family initiated MOMP causes release of the inter-membrane pro-apoptotic proteins to cytosol and creates a cytosolic environment suitable for the executionary phase of apoptosis. Using mathematical modeling and computational simulation, we have analyzed the response of this regulatory network caused by the up-/downregulation of each individual member of the Bcl-2 family. As a result, a non-linear stimulus-res…
Binding and unbinding of ligands to specific sites of a macromolecule are one of the most elementary molecular interactions inside the cell that embody the computational processes of biological regulations. The interaction between transcription factors and the operators of genes and that between ligands and binding sites of allosteric enzymes are typical examples of such molecular interactions. In order to obtain the general mathematical framework of biological regulations, we formulate these interactions as finite Markov processes and establish a computational theory of regulatory activities …
Statistical properties of environments experienced by biological signaling systems in the real world change, which necessitate adaptive responses to achieve high fidelity information transmission. One form of such adaptive response is gain control. Here we argue that a certain simple mechanism of gain control, understood well in the context of systems neuroscience, also works for molecular signaling. The mechanism allows to transmit more than one bit (on or off) of information about the signal independently of the signal variance. It does not require additional molecular circuitry beyond that …
Systemic properties of living cells are the result of molecular dynamics governed by so-called genetic regulatory networks (GRN). These networks capture all possible features of cells and are responsible for the immense levels of adaptation characteristic to living systems. At any point in time only small subsets of these networks are active. Any active subset of the GRN leads to the expression of particular sets of molecules (expression modes). The subsets of active networks change over time, leading to the observed complex dynamics of expression patterns. Understanding of this dynamics becom…
The "developmental hourglass" describes a pattern of increasing morphological divergence towards earlier and later embryonic development, separated by a period of significant conservation across distant species (the "phylotypic stage"). Recent studies have found evidence in support of the hourglass effect at the genomic level. For instance, the phylotypic stage expresses the oldest and most conserved transcriptomes. However, the regulatory mechanism that causes the hourglass pattern remains an open question. Here, we use an evolutionary model of regulatory gene interactions during development …
Robust systems, like the molecular networks of living cells are often resistant to single hits such as those caused by high-specificity drugs. Here we show that partial weakening of the Escherichia coli and Saccharomyces cerevisiae transcriptional regulatory networks at a small number (3-5) selected nodes can have a greater impact than the complete elimination of a single selected node. In both cases, the targeted nodes have the greatest possible impact; still the results suggest that in some cases broad specificity compounds or multitarget drug therapies may be more effective than individual …
Organisms are equipped with regulatory systems that display a variety of dynamical behaviours ranging from simple stable steady states, to switching and multistability, to oscillations. Earlier work has shown that oscillations in protein concentrations or gene expression levels are related to the presence of at least one negative feedback loop in the regulatory network. Here we study the dynamics of a very general class of negative feedback loops. Our main result is that in these systems the sequence of maxima and minima of the concentrations is uniquely determined by the topology of the loop …
Linearized catalytic reaction equations modeling e.g. the dynamics of genetic regulatory networks under the constraint that expression levels, i.e. molecular concentrations of nucleic material are positive, exhibit nontrivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems the inflation of the edge of chaos and multi-stability have been demonstrated to exist. The positivity constraint introduces a nonlinearity which makes chaotic dynamics possible. Despite the simplicity of such minimally nonlinear systems, their basic properties allow t…
The effect of signal integration through cis-regulatory modules (CRMs) on synchronization and clustering of populations of two-component genetic oscillators coupled by quorum sensing is in detail investigated. We find that the CRMs play an important role in achieving synchronization and clustering. For this, we investigate 6 possible cis-regulatory input functions (CRIFs) with AND, OR, ANDN, ORN, XOR, and EQU types of responses in two possible kinds of cell-to-cell communications: activator-regulated communication (i.e., the autoinducer regulates the activator) and repressor-regulated communic…
Phenotype transition takes place in many biological processes such as differentiation, and understanding how a cell reprograms its global gene expression profile is a problem of rate theories. A cell phenotype transition accompanies with switching of expression rates of clusters of genes, analogous to domain flipping in an Ising system. Here through analyzing single cell RNA sequencing data in the framework of transition path theory, we set to study how such a genome-wide expression program switching proceeds in three different cell transition processes. For each process after reconstructing a…
Regulatory networks consist of interacting molecules with a high degree of mutual chemical specificity. How can these molecules evolve when their function depends on maintenance of interactions with cognate partners and simultaneous avoidance of deleterious "crosstalk" with non-cognate molecules? Although physical models of molecular interactions provide a framework in which co-evolution of network components can be analyzed, most theoretical studies have focused on the evolution of individual alleles, neglecting the network. In contrast, we study the elementary step in the evolution of gene r…
The global dynamics of gene regulatory networks are known to show robustness to perturbations in the form of intrinsic and extrinsic noise, as well as mutations of individual genes. One molecular mechanism underlying this robustness has been identified as the action of so-called microRNAs that operate via feedforward loops. We present results of a computational study, using the modeling framework of stochastic Boolean networks, which explores the role that such network motifs play in stabilizing global dynamics. The paper introduces a new measure for the stability of stochastic networks. The r…
Profiling of whole transcriptomes has become a cornerstone of molecular biology and an invaluable tool for the characterization of clinical phenotypes and the identification of disease subtypes. Analyses of these data are becoming ever more sophisticated as we move beyond simple comparisons to consider networks of higher-order interactions and associations. Gene regulatory networks model the regulatory relationships of transcription factors and genes and have allowed the identification of differentially regulated processes in disease systems. In this perspective we discuss gene targeting score…
Molecular chaperones play a prominent role in signaling and transcriptional regulatory networks of the cell. Recent advances uncovered that chaperones act as genetic buffers stabilizing the phenotype of various cells and organisms and may serve as potential regulators of evolvability. Chaperones have weak links, connect hubs, are in the overlaps of network modules and may uncouple these modules during stress, which gives an additional protection for the cell at the network-level. Moreover, after stress chaperones are essential to re-build inter-modular contacts by their low affinity sampling o…
We show how to construct a reduced description of interacting genes in noisy, small regulatory networks using coupled binary "spin" variables. Treating both the protein number and gene expression state variables stochastically and on equal footing we propose a mapping which connects the molecular level description of networks to the binary representation. We construct a phase diagram indicating when genes can be considered to be independent and when the coupling between them cannot be neglected leading to synchrony or correlations. We find that an appropriately mapped boolean description repro…
Recent progress in experimental techniques has enabled us to quantitatively study stochastic and flexible behavior of biological systems. For example, gene regulatory networks perform stochastic information processing and their functionalities have been extensively studied. In gene regulatory networks, there are specific subgraphs called network motifs that occur at frequencies much higher than those found in randomized networks. Further understanding of the designing principle of such networks is highly desirable. In a different context, information thermodynamics has been developed as a theo…
The relationship between the design and functionality of molecular networks is now a key issue in biology. Comparison of regulatory networks performing similar tasks can give insights into how network architecture is constrained by the functions it directs. We here discuss methods of network comparison based on network architecture and signaling logic. Introducing local and global signaling scores for the difference between two networks we quantify similarities between evolutionary closely and distantly related bacteriophages. Despite the large evolutionary separation between phage $λ$ and 186…
Understanding the relationship between spontaneous stochastic fluctuations and the topology of the underlying gene regulatory network is of fundamental importance for the study of single-cell stochastic gene expression. Here by solving the analytical steady-state distribution of the protein copy number in a general kinetic model of stochastic gene expression with nonlinear feedback regulation, we reveal the relationship between stochastic fluctuations and feedback topology at the single-molecule level, which provides novel insights into how and to what extent a feedback loop can enhance or sup…
The cell cycle is a tightly controlled process, yet its underlying genetic network shows marked differences across species. Which of the associated structural features follow solely from the ability to impose the appropriate gene expression patterns? We tackle this question in silico by examining the ensemble of all regulatory networks which satisfy the constraint of producing a given sequence of gene expressions. We focus on three cell cycle profiles coming from baker's yeast, fission yeast and mammals. First, we show that the networks in each of the ensembles use just a few interactions that…
The promise of discovering a functional blueprint of a cellular system from large-scale and high-throughput sequence and experimental data is predicated on the belief that the same top-down investigative approach that proved successful in other biological problems (e.g. DNA sequencing) will be as effective when it comes to inferring more complex intracellular processes. The results in this paper address this fundamental issue in the specific context of transcription regulatory networks. Although simple recurring regulatory motifs have been identified in the past, due to the size and complexity…
