The transition from differential equations to Boolean networks: a case study in simplifying a regulatory network model

Leveraging quantum computing for dynamic analyses of logical networks in systems biology

The dynamics of cellular mechanisms can be investigated through the analysis of networks. One of the simplest but most popular modeling strategies involves logic-based models. However, these models still face exponential growth in simulation complexity compared with a linear increase in nodes. We transfer this modeling approach to quantum computing and use the upcoming technique in the field to simulate the resulting networks. Leveraging logic modeling in quantum computing has many benefits, including complexity reduction and quantum algorithms for systems biology tasks. To showcase the applicability of our approach to systems biology tasks, we implemented a model of mammalian cortical development. Here, we applied a quantum algorithm to estimate the tendency of the model to reach particular stable conditions and further revert dynamics. Results from two actual quantum processing units and a noisy simulator are presented, and current technical challenges are discussed.

Relative importance of composition structures and biologically meaningful logics in bipartite Boolean models of gene regulation

Boolean networks have been widely used to model gene networks. However, such models are coarse-grained to an extent that they abstract away molecular specificities of gene regulation. Alternatively, bipartite Boolean network models of gene regulation explicitly distinguish genes from transcription factors (TFs). In such bipartite models, multiple TFs may simultaneously contribute to gene regulation by forming heteromeric complexes, thus giving rise to composition structures. Since bipartite Boolean models are relatively recent, an empirical investigation of their biological plausibility is lacking. Here, we estimate the prevalence of composition structures arising through heteromeric complexes. Moreover, we present an additional mechanism where composition structures may arise as a result of multiple TFs binding to cis-regulatory regions and provide empirical support for this mechanism. Next, we compare the restriction in BFs imposed by composition structures and by biologically meaningful properties. We find that though composition structures can severely restrict the number of Boolean functions (BFs) driving a gene, the two types of minimally complex BFs, namely nested canalyzing functions (NCFs) and read-once functions (RoFs), are comparatively more restrictive. Finally, we find that composition structures are highly enriched in real networks, but this enrichment most likely comes from NCFs and RoFs.

Effective connectivity determines the critical dynamics of biochemical networks

Living systems comprise interacting biochemical components in very large networks. Given their high connectivity, biochemical dynamics are surprisingly not chaotic but quite robust to perturbations-a feature C.H. Waddington named canalization. Because organisms are also flexible enough to evolve, they arguably operate in a critical dynamical regime between order and chaos. The established theory of criticality is based on networks of interacting automata where Boolean truth values model presence/absence of biochemical molecules. The dynamical regime is predicted using network connectivity and node bias (to be on/off) as tuning parameters. Revising this to account for canalization leads to a significant improvement in dynamical regime prediction. The revision is based on effective connectivity, a measure of dynamical redundancy that buffers automata response to some inputs. In both random and experimentally validated systems biology networks, reducing effective connectivity makes living systems operate in stable or critical regimes even though the structure of their biochemical interaction networks predicts them to be chaotic. This suggests that dynamical redundancy may be naturally selected to maintain living systems near critical dynamics, providing both robustness and evolvability. By identifying how dynamics propagates preferably via effective pathways, our approach helps to identify precise ways to design and control network models of biochemical regulation and signalling.

Boolean modelling in plant biology

Signalling and genetic networks underlie most biological processes and are often complex, containing many highly connected components. Modelling these networks can provide insight into mechanisms but is challenging given that rate parameters are often not well defined. Boolean modelling, in which components can only take on a binary value with connections encoded by logic equations, is able to circumvent some of these challenges, and has emerged as a viable tool to probe these complex networks. In this review, we will give an overview of Boolean modelling, with a specific emphasis on its use in plant biology. We review how Boolean modelling can be used to describe biological networks and then discuss examples of its applications in plant genetics and plant signalling.

Boolean network analysis through the joint use of linear algebra and algebraic geometry

Among the various phenomena that can be modeled by Boolean networks, i.e., discrete-time dynamical systems with binary state variables, gene regulatory interactions are especially well known. Therefore, the analysis of Boolean networks is critical, e.g., to identify genetic pathways and to predict the effects of mutations on the cell functionality. Two methodologies (i.e., the semi-tensor product and the Gröbner bases over finite fields) have recently been proposed to tackle the problem of determining cycles and attractors (with the corresponding basin of attraction) for such systems. Here, it is shown that, by suitably coupling methodologies taken from these two fields (i.e., linear algebra and algebraic geometry), it is not only possible to determine cycles and attractors, but also to find closed-form solutions of the Boolean network. Such a goal is pursued by finding an immersion that recasts the Boolean dynamics in a linear form and by computing the closed-form solution of the latter system. The effectiveness of this technique is demonstrated by fully computing the solutions of the Boolean network modeling the differentiation of the Th-lymphocyte, a type of white blood cells involved in the human adaptive immune system.

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.

Modeling genotypes in their microenvironment to predict single- and multi-cellular behavior

A cell's phenotype is the set of observable characteristics resulting from the interaction of the genotype with the surrounding environment, determining cell behavior. Deciphering genotype-phenotype relationships has been crucial to understanding normal and disease biology. Analysis of molecular pathways has provided an invaluable tool to such understanding; however, typically it does not consider the physical microenvironment, which is a key determinant of phenotype. In this study, we present a novel modeling framework that enables the study of the link between genotype, signaling networks, and cell behavior in a three-dimensional microenvironment. To achieve this, we bring together Agent-Based Modeling, a powerful computational modeling technique, and gene networks. This combination allows biological hypotheses to be tested in a controlled stepwise fashion, and it lends itself naturally to model a heterogeneous population of cells acting and evolving in a dynamic microenvironment, which is needed to predict the evolution of complex multi-cellular dynamics. Importantly, this enables modeling co-occurring intrinsic perturbations, such as mutations, and extrinsic perturbations, such as nutrient availability, and their interactions. Using cancer as a model system, we illustrate how this framework delivers a unique opportunity to identify determinants of single-cell behavior, while uncovering emerging properties of multi-cellular growth. This framework is freely available at http://www.microc.org.

Ensembles, dynamics, and cell types: Revisiting the statistical mechanics perspective on cellular regulation

Genetic regulatory networks control ontogeny. For fifty years Boolean networks have served as models of such systems, ranging from ensembles of random Boolean networks as models for generic properties of gene regulation to working dynamical models of a growing number of sub-networks of real cells. At the same time, their statistical mechanics has been thoroughly studied. Here we recapitulate their original motivation in the context of current theoretical and empirical research. We discuss ensembles of random Boolean networks whose dynamical attractors model cell types. A sub-ensemble is the critical ensemble. There is now strong evidence that genetic regulatory networks are dynamically critical, and that evolution is exploring the critical sub-ensemble. The generic properties of this sub-ensemble predict essential features of cell differentiation. In particular, the number of attractors in such networks scales as the DNA content raised to the 0.63 power. Data on the number of cell types as a function of the DNA content per cell shows a scaling relationship of 0.88. Thus, the theory correctly predicts a power law relationship between the number of cell types and the DNA contents per cell, and a comparable slope. We discuss these new scaling values and show prospects for new research lines for Boolean networks as a base model for systems biology.

Criticality Distinguishes the Ensemble of Biological Regulatory Networks

The hypothesis that many living systems should exhibit near-critical behavior is well motivated theoretically, and an increasing number of cases have been demonstrated empirically. However, a systematic analysis across biological networks, which would enable identification of the network properties that drive criticality, has not yet been realized. Here, we provide a first comprehensive survey of criticality across a diverse sample of biological networks, leveraging a publicly available database of 67 Boolean models of regulatory circuits. We find all 67 networks to be near critical. By comparing to ensembles of random networks with similar topological and logical properties, we show that criticality in biological networks is not predictable solely from macroscale properties such as mean degree ⟨K⟩ and mean bias in the logic functions ⟨p⟩, as previously emphasized in theories of random Boolean networks. Instead, the ensemble of real biological circuits is jointly constrained by the local causal structure and logic of each node. In this way, biological regulatory networks are more distinguished from random networks by their criticality than by other macroscale network properties such as degree distribution, edge density, or fraction of activating conditions.

In Silico Prediction of Large-Scale Microbial Production Performance: Constraints for Getting Proper Data-Driven Models

Industrial bioreactors range from 10.000 to 700.000 L and characteristically show different zones of substrate availabilities, dissolved gas concentrations and pH values reflecting physical, technical and economic constraints of scale-up. Microbial producers are fluctuating inside the bioreactors thereby experiencing frequently changing micro-environmental conditions. The external stimuli induce responses on microbial metabolism and on transcriptional regulation programs. Both may deteriorate the expected microbial production performance in large scale compared to expectations deduced from ideal, well-mixed lab-scale conditions. Accordingly, predictive tools are needed to quantify large-scale impacts considering bioreactor heterogeneities. The review shows that the time is right to combine simulations of microbial kinetics with calculations of large-scale environmental conditions to predict the bioreactor performance. Accordingly, basic experimental procedures and computational tools are presented to derive proper microbial models and hydrodynamic conditions, and to link both for bioreactor modeling. Particular emphasis is laid on the identification of gene regulatory networks as the implementation of such models will surely gain momentum in future studies.

Dynamics of gene regulatory networks with stochastic propensities

Gene regulatory networks (GRNs) control the production of proteins in cells. It is well-known that this process is not deterministic. Numerous studies employed a non-deterministic transition structure to model these networks. However, it is not realistic to expect state-to-state transition probabilities to remain constant throughout an organism’s lifetime. In this work, we focus on modeling GRN state transition (edge) variability using an ever-changing set of propensities. We suspect that the source of this variation is due to internal noise at the molecular level and can be modeled by introducing additional stochasticity into GRN models. We employ a beta distribution, whose parameters are estimated to capture the pattern inherent in edge behavior with minimum error. Additionally, we develop a method for obtaining propensities from a pre-determined network.

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.

Ordinary differential equations and Boolean networks in application to modelling of 6-mercaptopurine metabolism

We consider two approaches to modelling the cell metabolism of 6-mercaptopurine, one of the important chemotherapy drugs used for treating acute lymphocytic leukaemia: kinetic ordinary differential equations, and Boolean networks supplied with one controlling node, which takes continual values. We analyse their interplay with respect to taking into account ATP concentration as a key parameter of switching between different pathways. It is shown that the Boolean networks, which allow avoiding the complexity of general kinetic modelling, preserve the possibility of reproducing the principal switching mechanism.

Estimation of delays in generalized asynchronous Boolean networks

A new generalized asynchronous Boolean network (GABN) model has been proposed in this paper. This continuous-time discrete-state model captures the biological reality of cellular dynamics without compromising the computational efficiency of the Boolean framework. The GABN synthesis procedure is based on the prior knowledge of the logical structure of the regulatory network, and the experimental transcriptional parameters. The novelty of the proposed methodology lies in considering different delays associated with the activation and deactivation of a particular protein (especially the transcription factors). A few illustrative examples of some well-studied network motifs have been provided to explore the scope of using the GABN model for larger networks. The GABN model of the p53-signaling pathway in response to γ-irradiation has also been simulated in the current paper to provide an indirect validation of the proposed schema.

Logical Modeling and Dynamical Analysis of Cellular Networks

The logical (or logic) formalism is increasingly used to model regulatory and signaling networks. Complementing these applications, several groups contributed various methods and tools to support the definition and analysis of logical models. After an introduction to the logical modeling framework and to several of its variants, we review here a number of recent methodological advances to ease the analysis of large and intricate networks. In particular, we survey approaches to determine model attractors and their reachability properties, to assess the dynamical impact of variations of external signals, and to consistently reduce large models. To illustrate these developments, we further consider several published logical models for two important biological processes, namely the differentiation of T helper cells and the control of mammalian cell cycle.

Inference of asynchronous Boolean network from biological pathways

Gene regulation is a complex process with multiple levels of interactions. In order to describe this complex dynamical system with tractable parameterization, the choice of the dynamical system model is of paramount importance. The right abstraction of the modeling scheme can reduce the complexity in the inference and intervention design, both computationally and experimentally. This article proposes an asynchronous Boolean network framework to capture the transcriptional regulation as well as the protein-protein interactions in a genetic regulatory system. The inference of asynchronous Boolean network from biological pathways information and experimental evidence are explained using an algorithm. The suitability of this paradigm for the variability of several reaction rates is also discussed. This methodology and model selection open up new research challenges in understanding gene-protein interactive system in a coherent way and can be beneficial for designing effective therapeutic intervention strategy.

Complementing ODE-Based System Analysis Using Boolean Networks Derived from an Euler-Like Transformation

In this paper, we present a systematic transition scheme for a large class of ordinary differential equations (ODEs) into Boolean networks. Our transition scheme can be applied to any system of ODEs whose right hand sides can be written as sums and products of monotone functions. It performs an Euler-like step which uses the signs of the right hand sides to obtain the Boolean update functions for every variable of the corresponding discrete model. The discrete model can, on one hand, be considered as another representation of the biological system or, alternatively, it can be used to further the analysis of the original ODE model. Since the generic transformation method does not guarantee any property conservation, a subsequent validation step is required. Depending on the purpose of the model this step can be based on experimental data or ODE simulations and characteristics. Analysis of the resulting Boolean model, both on its own and in comparison with the ODE model, then allows to investigate system properties not accessible in a purely continuous setting. The method is exemplarily applied to a previously published model of the bovine estrous cycle, which leads to new insights regarding the regulation among the components, and also indicates strongly that the system is tailored to generate stable oscillations.

Time-Delayed Models of Gene Regulatory Networks

We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternative modelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems.

From Boolean Network Model to Continuous Model Helps in Design of Functional Circuits

Computational circuit design with desired functions in a living cell is a challenging task in synthetic biology. To achieve this task, numerous methods that either focus on small scale networks or use evolutionary algorithms have been developed. Here, we propose a two-step approach to facilitate the design of functional circuits. In the first step, the search space of possible topologies for target functions is reduced by reverse engineering using a Boolean network model. In the second step, continuous simulation is applied to evaluate the performance of these topologies. We demonstrate the usefulness of this method by designing an example biological function: the SOS response of E. coli. Our numerical results show that the desired function can be faithfully reproduced by candidate networks with different parameters and initial conditions. Possible circuits are ranked according to their robustness against perturbations in parameter and gene expressions. The biological network is among the candidate networks, yet novel designs can be generated. Our method provides a scalable way to design robust circuits that can achieve complex functions, and makes it possible to uncover design principles of biological networks.

Boolean modeling: a logic-based dynamic approach for understanding signaling and regulatory networks and for making useful predictions

The biomolecules inside or near cells form a complex interacting system. Cellular phenotypes and behaviors arise from the totality of interactions among the components of this system. A fruitful way of modeling interacting biomolecular systems is by network-based dynamic models that characterize each component by a state variable, and describe the change in the state variables due to the interactions in the system. Dynamic models can capture the stable state patterns of this interacting system and can connect them to different cell fates or behaviors. A Boolean or logic model characterizes each biomolecule by a binary state variable that relates the abundance of that molecule to a threshold abundance necessary for downstream processes. The regulation of this state variable is described in a parameter free manner, making Boolean modeling a practical choice for systems whose kinetic parameters have not been determined. Boolean models integrate the body of knowledge regarding the components and interactions of biomolecular systems, and capture the system's dynamic repertoire, for example the existence of multiple cell fates. These models were used for a variety of systems and led to important insights and predictions. Boolean models serve as an efficient exploratory model, a guide for follow-up experiments, and as a foundation for more quantitative models.

A Boolean gene regulatory model of heterosis and speciation

BACKGROUND

Modelling genetic phenomena affecting biological traits is important for the development of agriculture as it allows breeders to predict the potential of breeding for certain traits. One such phenomenon is heterosis or hybrid vigor: crossing individuals from genetically distinct populations often results in improvements in quantitative traits, such as growth rate, biomass production and stress resistance. Heterosis has become a very useful tool in global agriculture, but its genetic basis remains controversial and its effects hard to predict. We have taken a computational approach to studying heterosis, developing a simulation of evolution, independent reassortment of alleles and hybridization of Gene Regulatory Networks (GRNs) in a Boolean framework. These artificial regulatory networks exhibit topological properties that reflect those observed in biology, and fitness is measured as the ability of a network to respond to external inputs in a pre-defined way.

RESULTS

Our model reproduced common experimental observations on heterosis using only biologically justified parameters, such as mutation rates. Hybrid vigor was observed and its extent was seen to increase as parental populations diverged, up until a point of sudden collapse of hybrid fitness. Thus, the model also describes a process akin to speciation due to genetic incompatibility of the separated populations. We also reproduce, for the first time in a model, the fact that hybrid vigor cannot easily be fixed by within a breeding line, currently an important limitation of the use of hybrid crops. The simulation allowed us to study the effects of three standard models for the genetic basis of heterosis: dominance, over-dominance, and epistasis.

CONCLUSION

This study describes the most detailed simulation of heterosis using gene regulatory networks to date and reproduces several phenomena associated with heterosis for the first time in a model. The level of detail in our model allows us to suggest possible warning signs of the impending collapse of hybrid vigor in breeding. In addition, the simulation provides a framework that can be extended to study other aspects of heterosis and alternative evolutionary scenarios.

Poised for survival: criticality, natural selection, and excitation-transcription coupling

Neurologically-complex species utilize two intricately coupled information-processing systems to adapt to their social and natural environments. Action potentials (APs) facilitate rapid responses to the nearly continuous fluctuations in the animal's surroundings. By contrast, genetic encodings comprise a molecular memory of ancient adaptive responses expressed as cognitive, emotional, or behavioral phenotypes. The linking of the two systems via intracellular Ca(2+) networks which address transcription factors - e.g., cAMP response element-binding protein (CREB) - is an appropriate focus for the biology of human behavior. Computational modeling utilizing Boolean networks (BNs) is a suitable qualitative method, requiring no kinetic information, for eliciting the systems' architectural properties. In particular, BNs can reveal critical intracellular regimes of possible evolutionary significance. As a platform for future research, we propose that those networks sufficiently robust to attenuate damaging intracellular noise and deleterious mutations, yet sufficiently close to chaos to permit or amplify adaptive noise and favorable mutations, would be favored by natural selection. Critical regimes of this type would be, literally, "poised for survival", and would stabilize and promote the survival of their correlated cultural phenotypes.

Piecewise linear and Boolean models of chemical reaction networks

Models of biochemical networks are frequently complex and high-dimensional. Reduction methods that preserve important dynamical properties are therefore essential for their study. Interactions in biochemical networks are frequently modeled using Hill functions ([Formula: see text]). Reduced ODEs and Boolean approximations of such model networks have been studied extensively when the exponent [Formula: see text] is large. However, while the case of small constant [Formula: see text] appears in practice, it is not well understood. We provide a mathematical analysis of this limit and show that a reduction to a set of piecewise linear ODEs and Boolean networks can be mathematically justified. The piecewise linear systems have closed-form solutions that closely track those of the fully nonlinear model. The simpler, Boolean network can be used to study the qualitative behavior of the original system. We justify the reduction using geometric singular perturbation theory and compact convergence, and illustrate the results in network models of a toggle switch and an oscillator.

Boolean networks with veto functions

Boolean networks are discrete dynamical systems for modeling regulation and signaling in living cells. We investigate a particular class of Boolean functions with inhibiting inputs exerting a veto (forced zero) on the output. We give analytical expressions for the sensitivity of these functions and provide evidence for their role in natural systems. In an intracellular signal transduction network [Helikar et al., Proc. Natl. Acad. Sci. USA 105, 1913 (2008)], the functions with veto are over-represented by a factor exceeding the over-representation of threshold functions and canalyzing functions in the same system. In Boolean networks for control of the yeast cell cycle [Li et al., Proc. Natl. Acad. Sci. USA 101, 4781 (2004); Davidich et al., PLoS ONE 3, e1672 (2008)], no or minimal changes to the wiring diagrams are necessary to formulate their dynamics in terms of the veto functions introduced here.

FOUNDATIONS FOR MODELING THE DYNAMICS OF GENE REGULATORY NETWORKS: A MULTILEVEL-PERSPECTIVE REVIEW

A promising alternative for unraveling the principles under which the dynamic interactions among genes lead to cellular phenotypes relies on mathematical and computational models at different levels of abstraction, from the molecular level of protein-DNA interactions to the system level of functional relationships among genes. This review article presents, under a bottom–up perspective, a hierarchy of approaches to modeling gene regulatory network dynamics, from microscopic descriptions at the single-molecule level in the spatial context of an individual cell to macroscopic models providing phenomenological descriptions at the population-average level. The reviewed modeling approaches include Molecular Dynamics, Particle-Based Brownian Dynamics, the Master Equation approach, Ordinary Differential Equations, and the Boolean logic abstraction. Each of these frameworks is motivated by a particular biological context and the nature of the insight being pursued. The setting of gene network dynamic models from such frameworks involves assumptions and mathematical artifacts often ignored by the non-specialist. This article aims at providing an entry point for biologists new to the field and computer scientists not acquainted with some recent biophysically-inspired models of gene regulation. The connections promoting intuition between different abstraction levels and the role that approximations play in the modeling process are highlighted throughout the paper.

The regulation of gene expression in eukaryotes: bistability and oscillations in repressilator models

To model the regulation of gene expression in eukaryotes by transcriptional activators and repressors, we introduce delays in conjugation with the mass action law. Delays are associated with the time gap between the mRNA transcription in the nucleoplasm and the protein synthesis in the cytoplasm. After re-parameterisation of the m-repressilator model with the Hill cooperative parameter n, for n=1, the m-repressilator is deducible from the mass action law and, in the limit n→∞, it is a Boolean type model. With this embedding and with delays, if m is odd and n>1, we show that there is always a choice of parameters for which the m-repressilator model has sustained oscillations (limit cycles), implying that the 1-repressilator is the simplest genetic mechanism leading to sustained oscillations in eukaryotes. If m is even and n>1, there is always a choice of parameters for which the m-repressilator model has bistability.

Boolean network model predicts knockout mutant phenotypes of fission yeast

Boolean networks (or: networks of switches) are extremely simple mathematical models of biochemical signaling networks. Under certain circumstances, Boolean networks, despite their simplicity, are capable of predicting dynamical activation patterns of gene regulatory networks in living cells. For example, the temporal sequence of cell cycle activation patterns in yeasts S. pombe and S. cerevisiae are faithfully reproduced by Boolean network models. An interesting question is whether this simple model class could also predict a more complex cellular phenomenology as, for example, the cell cycle dynamics under various knockout mutants instead of the wild type dynamics, only. Here we show that a Boolean network model for the cell cycle control network of yeast S. pombe correctly predicts viability of a large number of known mutants. So far this had been left to the more detailed differential equation models of the biochemical kinetics of the yeast cell cycle network and was commonly thought to be out of reach for models as simplistic as Boolean networks. The new results support our vision that Boolean networks may complement other mathematical models in systems biology to a larger extent than expected so far, and may fill a gap where simplicity of the model and a preference for an overall dynamical blueprint of cellular regulation, instead of biochemical details, are in the focus.

Understanding cancer mechanisms through network dynamics

Cancer is a complex, multifaceted disease. Cellular systems are perturbed both during the onset and development of cancer, and the behavioural change of tumour cells usually involves a broad range of dynamic variations. To an extent, the difficulty of monitoring the systemic change has been alleviated by recent developments in the high-throughput technologies. At both the genomic as well as proteomic levels, the technological advances in microarray and mass spectrometry, in conjunction with computational simulations and the construction of human interactome maps have facilitated the progress of identifying disease-associated genes. On a systems level, computational approaches developed for network analysis are becoming especially useful for providing insights into the mechanism behind tumour development and metastasis. This review emphasizes network approaches that have been developed to study cancer and provides an overview of our current knowledge of protein-protein interaction networks, and how their systemic perturbation can be analysed by two popular network simulation methods: Boolean network and ordinary differential equations.

On the relationship of steady states of continuous and discrete models arising from biology

For many biological systems that have been modeled using continuous and discrete models, it has been shown that such models have similar dynamical properties. In this paper, we prove that this happens in more general cases. We show that under some conditions there is a bijection between the steady states of continuous and discrete models arising from biological systems. Our results also provide a novel method to analyze certain classes of nonlinear models using discrete mathematics.

Stability of Boolean multilevel networks

The study of the interplay between the structure and dynamics of complex multilevel systems is a pressing challenge nowadays. In this paper, we use a semiannealed approximation to study the stability properties of random Boolean networks in multiplex (multilayered) graphs. Our main finding is that the multilevel structure provides a mechanism for the stabilization of the dynamics of the whole system even when individual layers work on the chaotic regime, therefore identifying new ways of feedback between the structure and the dynamics of these systems. Our results point out the need for a conceptual transition from the physics of single-layered networks to the physics of multiplex networks. Finally, the fact that the coupling modifies the phase diagram and the critical conditions of the isolated layers suggests that interdependency can be used as a control mechanism.

Binary threshold networks as a natural null model for biological networks

Spin models of neural networks and genetic networks are considered elegant as they are accessible to statistical mechanics tools for spin glasses and magnetic systems. However, the conventional choice of variables in spin systems may cause problems in some models when parameter choices are unrealistic from a biological perspective. Obviously, this may limit the role of a model as a template model for biological systems. Perhaps less obviously, also ensembles of random networks are affected and may exhibit different critical properties. We consider here a prototypical network model that is biologically plausible in its local mechanisms. We study a discrete dynamical network with two characteristic properties: Nodes with binary states 0 and 1, and a modified threshold function with Θ(0)(0)=0. We explore the critical properties of random networks of such nodes and find a critical connectivity K(c)=2.0 with activity vanishing at the critical point. Finally, we observe that the present model allows a more natural implementation of recent models of budding yeast and fission yeast cell-cycle control networks.

An information theoretic approach to constructing robust Boolean gene regulatory networks

We introduce a class of finite systems models of gene regulatory networks exhibiting behavior of the cell cycle. The model is an extension of a Boolean network model. The system spontaneously cycles through a finite set of internal states, tracking the increase of an external factor such as cell mass, and also exhibits checkpoints in which errors in gene expression levels due to cellular noise are automatically corrected. We present a 7-gene network based on Projective Geometry codes, which can correct, at every given time, one gene expression error. The topology of a network is highly symmetric and requires using only simple Boolean functions that can be synthesized using genes of various organisms. The attractor structure of the Boolean network contains a single cycle attractor. It is the smallest nontrivial network with such high robustness. The methodology allows construction of artificial cell cycle gene regulatory networks with the number of phases larger than in natural cell cycle.

Reverse engineering systems models of regulation: discovery, prediction and mechanisms

Biological systems can now be understood in comprehensive and quantitative detail using systems biology approaches. Putative genome-scale models can be built rapidly based upon biological inventories and strategic system-wide molecular measurements. Current models combine statistical associations, causative abstractions, and known molecular mechanisms to explain and predict quantitative and complex phenotypes. This top-down 'reverse engineering' approach generates useful organism-scale models despite noise and incompleteness in data and knowledge. Here we review and discuss the reverse engineering of biological systems using top-down data-driven approaches, in order to improve discovery, hypothesis generation, and the inference of biological properties.

Importance of metabolic coupling for the dynamics of gene expression following a diauxic shift in Escherichia coli

Gene regulatory networks consist of direct interactions, but also include indirect interactions mediated by metabolism. We investigate to which extent these indirect interactions arising from metabolic coupling influence the dynamics of the system. To this end, we build a qualitative model of the gene regulatory network controlling carbon assimilation in Escherichia coli, and use this model to study the changes in gene expression following a diauxic shift from glucose to acetate. In particular, we compare the relative variation in the steady-state concentrations of enzymes and transcription regulators during growth on glucose and acetate, as well as the dynamic response of gene expression to the exhaustion of glucose and the subsequent assimilation of acetate. We find significant differences between the dynamics of the system in the absence and presence of metabolic coupling. This shows that interactions arising from metabolic coupling cannot be ignored when studying the dynamics of gene regulatory networks.

Stability of Boolean and continuous dynamics

Regulatory dynamics in biology is often described by continuous rate equations for continuously varying chemical concentrations. Binary discretization of state space and time leads to Boolean dynamics. In the latter, the dynamics has been called unstable if flip perturbations lead to damage spreading. Here, we find that this stability classification strongly differs from the stability properties of the original continuous dynamics under small perturbations of the state vector. In particular, random networks of nodes with large sensitivity yield stable dynamics under small perturbations.

Decoding complex biological networks - tracing essential and modulatory parameters in complex and simplified models of the cell cycle

Background

One of the most well described cellular processes is the cell cycle, governing cell division. Mathematical models of this gene-protein network are therefore a good test case for assessing to what extent we can dissect the relationship between model parameters and system dynamics. Here we combine two strategies to enable an exploration of parameter space in relation to model output. A simplified, piecewise linear approximation of the original model is combined with a sensitivity analysis of the same system, to obtain and validate analytical expressions describing the dynamical role of different model parameters.

Results

We considered two different output responses to parameter perturbations. One was qualitative and described whether the system was still working, i.e. whether there were oscillations. We call parameters that correspond to such qualitative change in system response essential. The other response pattern was quantitative and measured changes in cell size, corresponding to perturbations of modulatory parameters. Analytical predictions from the simplified model concerning the impact of different parameters were compared to a sensitivity analysis of the original model, thus evaluating the predictions from the simplified model. The comparison showed that the predictions on essential and modulatory parameters were satisfactory for small perturbations, but more discrepancies were seen for larger perturbations. Furthermore, for this particular cell cycle model, we found that most parameters were either essential or modulatory. Essential parameters required large perturbations for identification, whereas modulatory parameters were more easily identified with small perturbations. Finally, we used the simplified model to make predictions on critical combinations of parameter perturbations.

Conclusions

The parameter characterizations of the simplified model are in large consistent with the original model and the simplified model can give predictions on critical combinations of parameter perturbations. We believe that the distinction between essential and modulatory perturbation responses will be of use for sensitivity analysis, and in discussions of robustness and during the model simplification process.

Boolean models can explain bistability in the lac operon

The lac operon in Escherichia coli has been studied extensively and is one of the earliest gene systems found to undergo both positive and negative control. The lac operon is known to exhibit bistability, in the sense that the operon is either induced or uninduced. Many dynamical models have been proposed to capture this phenomenon. While most are based on complex mathematical formulations, it has been suggested that for other gene systems network topology is sufficient to produce the desired dynamical behavior. We present a Boolean network as a discrete model for the lac operon. Our model includes the two main glucose control mechanisms of catabolite repression and inducer exclusion. We show that this Boolean model is capable of predicting the ON and OFF steady states and bistability. Further, we present a reduced model which shows that lac mRNA and lactose form the core of the lac operon, and that this reduced model exhibits the same dynamics. This work suggests that the key to model qualitative dynamics of gene systems is the topology of the network and Boolean models are well suited for this purpose.

Virtual mutagenesis of the yeast cyclins genetic network reveals complex dynamics of transcriptional control networks

Study of genetic networks has moved from qualitative description of interactions between regulators and regulated genes to the analysis of the interaction dynamics. This paper focuses on the analysis of dynamics of one particular network – the yeast cyclins network. Using a dedicated mathematical model of gene expression and a procedure for computation of the parameters of the model from experimental data, a complete numerical model of the dynamics of the cyclins genetic network was attained. The model allowed for performing virtual experiments on the network and observing their influence on the expression dynamics of the genes downstream in the regulatory cascade. Results show that when the network structure is more complicated, and the regulatory interactions are indirect, results of gene deletion are highly unpredictable. As a consequence of quantitative behavior of the genes and their connections within the network, causal relationship between a regulator and target gene may not be discovered by gene deletion. Without including the dynamics of the system into the network, its functional properties cannot be studied and interpreted correctly.

Redefining plant systems biology: from cell to ecosystem

Molecular biologists typically restrict systems biology to cellular levels. By contrast, ecologists define biological systems as communities of interacting individuals at different trophic levels that process energy, nutrient and information flows. Modern plant breeding needs to increase agricultural productivity while decreasing the ecological footprint. This requires a holistic systems biology approach that couples different aggregation levels while considering the variables that affect these biological systems from cell to community. The challenge is to generate accurate experimental data that can be used together with modelling concepts and techniques that allow experimentally verifying in silico predictions. The coupling of aggregation levels in plant sciences, termed Integral Quantification of Biological Organization (IQ(BiO)), might enhance our abilities to generate new desired plant phenotypes.

Reduced models of networks of coupled enzymatic reactions

The Michaelis-Menten equation has played a central role in our understanding of biochemical processes. It has long been understood how this equation approximates the dynamics of irreversible enzymatic reactions. However, a similar approximation in the case of networks, where the product of one reaction can act as an enzyme in another, has not been fully developed. Here we rigorously derive such an approximation in a class of coupled enzymatic networks where the individual interactions are of Michaelis-Menten type. We show that the sufficient conditions for the validity of the total quasi-steady state assumption (tQSSA), obtained in a single protein case by Borghans, de Boer and Segel can be extended to sufficient conditions for the validity of the tQSSA in a large class of enzymatic networks. Secondly, we derive reduced equations that approximate the network's dynamics and involve only protein concentrations. This significantly reduces the number of equations necessary to model such systems. We prove the validity of this approximation using geometric singular perturbation theory and results about matrix differentiation. The ideas used in deriving the approximating equations are quite general, and can be used to systematize other model reductions.

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.

Efficient parameter search for qualitative models of regulatory networks using symbolic model checking

Motivation: Investigating the relation between the structure and behavior of complex biological networks often involves posing the question if the hypothesized structure of a regulatory network is consistent with the observed behavior, or if a proposed structure can generate a desired behavior.

Results: The above questions can be cast into a parameter search problem for qualitative models of regulatory networks. We develop a method based on symbolic model checking that avoids enumerating all possible parametrizations, and show that this method performs well on real biological problems, using the IRMA synthetic network and benchmark datasets. We test the consistency between IRMA and time-series expression profiles, and search for parameter modifications that would make the external control of the system behavior more robust.

Availability: GNA and the IRMA model are available at http://ibis.inrialpes.fr/

Contact:gregory.batt@inria.fr

Supplementary information:Supplementary data are available at Bioinformatics online.

Evolution of form and function in a model of differentiated multicellular organisms with gene regulatory networks

The emergence of novelties, as a generator of diversity, in the form and function of the organisms have long puzzled biologists. The study of the developmental process and the anatomical properties of an organism provides scarce information into the means by which its morphology evolved. Some have argued that the very nature of novelty is believed to be linked to the evolution of gene regulation, rather than to the emergence of new structural genes. In order to gain further insight into the evolution of novelty and diversity, we describe a simple computational model of gene regulation that controls the development of locomotive multicellular organisms through a fixed set of simple structural genes. Organisms, modeled as two-dimensional spring networks, are simulated in a virtual environment to evaluate their steering skills for path-following. Proposed as a behavior-finding problem, this fitness function guides an evolutionary algorithm that produces structures whose function is well-adapted to the environment (i.e., good path-followers). We show that, despite the fixed simple set of structural genes, the evolution of gene regulation yields a rich variety of body plans, including symmetries, body segments, and modularity, resulting in a diversity of original behaviors to follow a simple path. These results suggest that the sole variation in the regulation of gene expression is a sufficient condition for the emergence of novelty and diversity.

Comparison of evolutionary algorithms in gene regulatory network model inference

BACKGROUND

The evolution of high throughput technologies that measure gene expression levels has created a data base for inferring GRNs (a process also known as reverse engineering of GRNs). However, the nature of these data has made this process very difficult. At the moment, several methods of discovering qualitative causal relationships between genes with high accuracy from microarray data exist, but large scale quantitative analysis on real biological datasets cannot be performed, to date, as existing approaches are not suitable for real microarray data which are noisy and insufficient.

RESULTS

This paper performs an analysis of several existing evolutionary algorithms for quantitative gene regulatory network modelling. The aim is to present the techniques used and offer a comprehensive comparison of approaches, under a common framework. Algorithms are applied to both synthetic and real gene expression data from DNA microarrays, and ability to reproduce biological behaviour, scalability and robustness to noise are assessed and compared.

CONCLUSIONS

Presented is a comparison framework for assessment of evolutionary algorithms, used to infer gene regulatory networks. Promising methods are identified and a platform for development of appropriate model formalisms is established.

Logical modelling of cell cycle control in eukaryotes: a comparative study

Dynamical modelling is at the core of the systems biology paradigm. However, the development of comprehensive quantitative models is complicated by the daunting complexity of regulatory networks controlling crucial biological processes such as cell division, the paucity of currently available quantitative data, as well as the limited reproducibility of large-scale experiments. In this context, qualitative modelling approaches offer a useful alternative or complementary framework to build and analyse simplified, but still rigorous dynamical models. This point is illustrated here by analysing recent logical models of the molecular network controlling mitosis in different organisms, from yeasts to mammals. After a short introduction covering cell cycle and logical modelling, we compare the assumptions and properties underlying these different models. Next, leaning on their transposition into a common logical framework, we compare their functional structure in terms of regulatory circuits. Finally, we discuss assets and prospects of qualitative approaches for the modelling of the cell cycle.

Dynamical approaches to modeling developmental gene regulatory networks

The network of interacting regulatory signals within a cell comprises one of the most complex and powerful computational systems in biology. Gene regulatory networks (GRNs) play a key role in transforming the information encoded in a genome into morphological form. To achieve this feat, GRNs must respond to and integrate environmental signals with their internal dynamics in a robust and coordinated fashion. The highly dynamic nature of this process lends itself to interpretation and analysis in the language of dynamical models. Modeling provides a means of systematically untangling the complicated structure of GRNs, a framework within which to simulate the behavior of reconstructed systems and, in some cases, suites of analytic tools for exploring that behavior and its implications. This review provides a general background to the idea of treating a regulatory network as a dynamical system, and describes a variety of different approaches that have been taken to the dynamical modeling of GRNs.

Modular logical modelling of the budding yeast cell cycle

Systems biologists are facing the difficult challenge of modelling and analysing regulatory networks encompassing numerous and diverse components and interactions. Furthermore, available data sets are often qualitative, which complicates the definition of truly quantitative models. In order to build comprehensive and predictive models, there is clearly a need for incremental strategies, enabling the progression from relatively small to large scale models. Leaning on former models, we have defined a logical model for three regulatory modules involved in the control of the mitotic cell cycle in budding yeast, namely the core cell cycle module, the morphogenetic checkpoint, and a module controlling the exit from mitosis. Consistency with available data has been assessed through a systematic analysis of model behaviours for various genetic backgrounds and other perturbations. Next, we take advantage of compositional facilities of the logical formalism to combine these three models in order to generate a single comprehensive model involving over thirty regulatory components. The resulting logical model preserves all relevant characteristics of the original modules, while enabling the simulation of more sophisticated experiments.