Robustness and fragility of Boolean models for genetic regulatory networks

An algebra-based method for inferring gene regulatory networks

BACKGROUND

The inference of gene regulatory networks (GRNs) from experimental observations is at the heart of systems biology. This includes the inference of both the network topology and its dynamics. While there are many algorithms available to infer the network topology from experimental data, less emphasis has been placed on methods that infer network dynamics. Furthermore, since the network inference problem is typically underdetermined, it is essential to have the option of incorporating into the inference process, prior knowledge about the network, along with an effective description of the search space of dynamic models. Finally, it is also important to have an understanding of how a given inference method is affected by experimental and other noise in the data used.

RESULTS

This paper contains a novel inference algorithm using the algebraic framework of Boolean polynomial dynamical systems (BPDS), meeting all these requirements. The algorithm takes as input time series data, including those from network perturbations, such as knock-out mutant strains and RNAi experiments. It allows for the incorporation of prior biological knowledge while being robust to significant levels of noise in the data used for inference. It uses an evolutionary algorithm for local optimization with an encoding of the mathematical models as BPDS. The BPDS framework allows an effective representation of the search space for algebraic dynamic models that improves computational performance. The algorithm is validated with both simulated and experimental microarray expression profile data. Robustness to noise is tested using a published mathematical model of the segment polarity gene network in Drosophila melanogaster. Benchmarking of the algorithm is done by comparison with a spectrum of state-of-the-art network inference methods on data from the synthetic IRMA network to demonstrate that our method has good precision and recall for the network reconstruction task, while also predicting several of the dynamic patterns present in the network.

CONCLUSIONS

Boolean polynomial dynamical systems provide a powerful modeling framework for the reverse engineering of gene regulatory networks, that enables a rich mathematical structure on the model search space. A C++ implementation of the method, distributed under LPGL license, is available, together with the source code, at http://www.paola-vera-licona.net/Software/EARevEng/REACT.html.

Logical modelling of Drosophila signalling pathways

A limited number of signalling pathways are involved in the specification of cell fate during the development of all animals. Several of these pathways were originally identified in Drosophila. To clarify their roles, and possible cross-talk, we have built a logical model for the nine key signalling pathways recurrently used in metazoan development. In each case, we considered the associated ligands, receptors, signal transducers, modulators, and transcription factors reported in the literature. Implemented using the logical modelling software GINsim, the resulting models qualitatively recapitulate the main characteristics of each pathway, in wild type as well as in various mutant situations (e.g. loss-of-function or gain-of-function). These models constitute pluggable modules that can be used to assemble comprehensive models of complex developmental processes. Moreover, these models of Drosophila pathways could serve as scaffolds for more complicated models of orthologous mammalian pathways. Comprehensive model annotations and GINsim files are provided for each of the nine considered pathways.

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.

Modeling approaches for qualitative and semi-quantitative analysis of cellular signaling networks

A central goal of systems biology is the construction of predictive models of bio-molecular networks. Cellular networks of moderate size have been modeled successfully in a quantitative way based on differential equations. However, in large-scale networks, knowledge of mechanistic details and kinetic parameters is often too limited to allow for the set-up of predictive quantitative models.Here, we review methodologies for qualitative and semi-quantitative modeling of cellular signal transduction networks. In particular, we focus on three different but related formalisms facilitating modeling of signaling processes with different levels of detail: interaction graphs, logical/Boolean networks, and logic-based ordinary differential equations (ODEs). Albeit the simplest models possible, interaction graphs allow the identification of important network properties such as signaling paths, feedback loops, or global interdependencies. Logical or Boolean models can be derived from interaction graphs by constraining the logical combination of edges. Logical models can be used to study the basic input-output behavior of the system under investigation and to analyze its qualitative dynamic properties by discrete simulations. They also provide a suitable framework to identify proper intervention strategies enforcing or repressing certain behaviors. Finally, as a third formalism, Boolean networks can be transformed into logic-based ODEs enabling studies on essential quantitative and dynamic features of a signaling network, where time and states are continuous.We describe and illustrate key methods and applications of the different modeling formalisms and discuss their relationships. In particular, as one important aspect for model reuse, we will show how these three modeling approaches can be combined to a modeling pipeline (or model hierarchy) allowing one to start with the simplest representation of a signaling network (interaction graph), which can later be refined to logical and eventually to logic-based ODE models. Importantly, systems and network properties determined in the rougher representation are conserved during these transformations.

An effective network reduction approach to find the dynamical repertoire of discrete dynamic networks

Discrete dynamic models are a powerful tool for the understanding and modeling of large biological networks. Although a lot of progress has been made in developing analysis tools for these models, there is still a need to find approaches that can directly relate the network structure to its dynamics. Of special interest is identifying the stable patterns of activity, i.e., the attractors of the system. This is a problem for large networks, because the state space of the system increases exponentially with network size. In this work, we present a novel network reduction approach that is based on finding network motifs that stabilize in a fixed state. Notably, we use a topological criterion to identify these motifs. Specifically, we find certain types of strongly connected components in a suitably expanded representation of the network. To test our method, we apply it to a dynamic network model for a type of cytotoxic T cell cancer and to an ensemble of random Boolean networks of size up to 200. Our results show that our method goes beyond reducing the network and in most cases can actually predict the dynamical repertoire of the nodes (fixed states or oscillations) in the attractors of the system.

Introduction to focus issue: quantitative approaches to genetic networks

All cells of living organisms contain similar genetic instructions encoded in the organism's DNA. In any particular cell, the control of the expression of each different gene is regulated, in part, by binding of molecular complexes to specific regions of the DNA. The molecular complexes are composed of protein molecules, called transcription factors, combined with various other molecules such as hormones and drugs. Since transcription factors are coded by genes, cellular function is partially determined by genetic networks. Recent research is making large strides to understand both the structure and the function of these networks. Further, the emerging discipline of synthetic biology is engineering novel gene circuits with specific dynamic properties to advance both basic science and potential practical applications. Although there is not yet a universally accepted mathematical framework for studying the properties of genetic networks, the strong analogies between the activation and inhibition of gene expression and electric circuits suggest frameworks based on logical switching circuits. This focus issue provides a selection of papers reflecting current research directions in the quantitative analysis of genetic networks. The work extends from molecular models for the binding of proteins, to realistic detailed models of cellular metabolism. Between these extremes are simplified models in which genetic dynamics are modeled using classical methods of systems engineering, Boolean switching networks, differential equations that are continuous analogues of Boolean switching networks, and differential equations in which control is based on power law functions. The mathematical techniques are applied to study: (i) naturally occurring gene networks in living organisms including: cyanobacteria, Mycoplasma genitalium, fruit flies, immune cells in mammals; (ii) synthetic gene circuits in Escherichia coli and yeast; and (iii) electronic circuits modeling genetic networks using field-programmable gate arrays. Mathematical analyses will be essential for understanding naturally occurring genetic networks in diverse organisms and for providing a foundation for the improved development of synthetic genetic networks.

Boolean approach to signalling pathway modelling in HGF-induced keratinocyte migration

Motivation: Cell migration is a complex process that is controlled through the time-sequential feedback regulation of protein signalling and gene regulation. Based on prior knowledge and own experimental data, we developed a large-scale dynamic network describing the onset and maintenance of hepatocyte growth factor-induced migration of primary human keratinocytes. We applied Boolean logic to capture the qualitative behaviour as well as short-and long-term dynamics of the complex signalling network involved in this process, comprising protein signalling, gene regulation and autocrine feedback.

Results: A Boolean model has been compiled from time-resolved transcriptome data and literature mining, incorporating the main pathways involved in migration from initial stimulation to phenotype progress. Steady-state analysis under different inhibition and stimulation conditions of known key molecules reproduces existing data and predicts novel interactions based on our own experiments. Model simulations highlight for the first time the necessity of a temporal sequence of initial, transient MET receptor (met proto-oncogene, hepatocyte growth factor receptor) and subsequent, continuous epidermal growth factor/integrin signalling to trigger and sustain migration by autocrine signalling that is integrated through the Focal adhesion kinase protein. We predicted in silico and verified in vitro that long-term cell migration is stopped if any of the two feedback loops are inhibited.

Availability: The network file for analysis with the R BoolNet library is available in the Supplementary Information.

Contact:melanie.boerries@frias.uni-freiburg.de or hauke.busch@frias.uni-freiburg.de

Supplementary information:Supplementary data are available at Bioinformatics online.

Logic-based models in systems biology: a predictive and parameter-free network analysis method

Highly complex molecular networks, which play fundamental roles in almost all cellular processes, are known to be dysregulated in a number of diseases, most notably in cancer. As a consequence, there is a critical need to develop practical methodologies for constructing and analysing molecular networks at a systems level. Mathematical models built with continuous differential equations are an ideal methodology because they can provide a detailed picture of a network's dynamics. To be predictive, however, differential equation models require that numerous parameters be known a priori and this information is almost never available. An alternative dynamical approach is the use of discrete logic-based models that can provide a good approximation of the qualitative behaviour of a biochemical system without the burden of a large parameter space. Despite their advantages, there remains significant resistance to the use of logic-based models in biology. Here, we address some common concerns and provide a brief tutorial on the use of logic-based models, which we motivate with biological examples.

Network class superposition analyses

Networks are often used to understand a whole system by modeling the interactions among its pieces. Examples include biomolecules in a cell interacting to provide some primary function, or species in an environment forming a stable community. However, these interactions are often unknown; instead, the pieces' dynamic states are known, and network structure must be inferred. Because observed function may be explained by many different networks (e.g., ≈ 10(30) for the yeast cell cycle process), considering dynamics beyond this primary function means picking a single network or suitable sample: measuring over all networks exhibiting the primary function is computationally infeasible. We circumvent that obstacle by calculating the network class ensemble. We represent the ensemble by a stochastic matrix T, which is a transition-by-transition superposition of the system dynamics for each member of the class. We present concrete results for T derived from boolean time series dynamics on networks obeying the Strong Inhibition rule, by applying T to several traditional questions about network dynamics. We show that the distribution of the number of point attractors can be accurately estimated with T. We show how to generate Derrida plots based on T. We show that T-based Shannon entropy outperforms other methods at selecting experiments to further narrow the network structure. We also outline an experimental test of predictions based on T. We motivate all of these results in terms of a popular molecular biology boolean network model for the yeast cell cycle, but the methods and analyses we introduce are general. We conclude with open questions for T, for example, application to other models, computational considerations when scaling up to larger systems, and other potential analyses.

Canalization and control in automata networks: body segmentation in Drosophila melanogaster

We present schema redescription as a methodology to characterize canalization in automata networks used to model biochemical regulation and signalling. In our formulation, canalization becomes synonymous with redundancy present in the logic of automata. This results in straightforward measures to quantify canalization in an automaton (micro-level), which is in turn integrated into a highly scalable framework to characterize the collective dynamics of large-scale automata networks (macro-level). This way, our approach provides a method to link micro- to macro-level dynamics--a crux of complexity. Several new results ensue from this methodology: uncovering of dynamical modularity (modules in the dynamics rather than in the structure of networks), identification of minimal conditions and critical nodes to control the convergence to attractors, simulation of dynamical behaviour from incomplete information about initial conditions, and measures of macro-level canalization and robustness to perturbations. We exemplify our methodology with a well-known model of the intra- and inter cellular genetic regulation of body segmentation in Drosophila melanogaster. We use this model to show that our analysis does not contradict any previous findings. But we also obtain new knowledge about its behaviour: a better understanding of the size of its wild-type attractor basin (larger than previously thought), the identification of novel minimal conditions and critical nodes that control wild-type behaviour, and the resilience of these to stochastic interventions. Our methodology is applicable to any complex network that can be modelled using automata, but we focus on biochemical regulation and signalling, towards a better understanding of the (decentralized) control that orchestrates cellular activity--with the ultimate goal of explaining how do cells and tissues 'compute'.

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

MOTIVATION

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

RESULTS

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

A REDUCTION METHOD FOR BOOLEAN NETWORK MODELS PROVEN TO CONSERVE ATTRACTORS

Boolean models, wherein each component is characterized with a binary (ON or OFF) variable, have been widely employed for dynamic modeling of biological regulatory networks. However, the exponential dependencse of the size of the state space of these models on the number of nodes in the network can be a daunting prospect for attractor analysis of large-scale systems. We have previously proposed a network reduction technique for Boolean models and demonstrated its applicability on two biological systems, namely, the abscisic acid signal transduction network as well as the T-LGL leukemia survival signaling network. In this paper, we provide a rigorous mathematical proof that this method not only conserves the fixed points of a Boolean network, but also conserves the complex attractors of general asynchronous Boolean models wherein at each time step a randomly selected node is updated. This method thus allows one to infer the long-term dynamic properties of a large-scale system from those of the corresponding reduced model.

Effects of community structure on the dynamics of random threshold networks

Random threshold networks (RTNs) have been widely used as models of neural or genetic regulatory networks. Network topology plays a central role in the dynamics of these networks. Recently it has been shown that many social and biological networks are scale-free and also exhibit community structure, in which autonomous modules are wired together to perform relatively independent functions. In this study we use both synchronous and asynchronous models of RTNs to systematically investigate how community structure affects the dynamics of RTNs with scale-free topology. Extensive simulation experiments show that RTNs with high modularity have more attractors than those RTNs with low modularity, and RTNs with smaller communities tend to have more attractors. Damage resulting from perturbation of initial conditions spreads less effectively in RTNs with higher modularity and RTNs with smaller communities. In addition, RTNs with high modularity can coordinate their internal dynamics better than RTNs with low modularity under the synchronous update scheme, and it is the other way around under the asynchronous update. This study shows that community structure has a strong effect on the dynamics of RTNs.

Boolean modeling of biological regulatory networks: a methodology tutorial

Given the complexity and interactive nature of biological systems, constructing informative and coherent network models of these systems and subsequently developing efficient approaches to analyze the assembled networks is of immense importance. The integration of network analysis and dynamic modeling enables one to investigate the behavior of the underlying system as a whole and to make experimentally testable predictions about less-understood aspects of the processes involved. In this paper, we present a tutorial on the fundamental steps of Boolean modeling of biological regulatory networks. We demonstrate how to infer a Boolean network model from the available experimental data, analyze the network using graph-theoretical measures, and convert it into a predictive dynamic model. For each step, the pitfalls one may encounter and possible ways to circumvent them are also discussed. We illustrate these steps on a toy network as well as in the context of the Drosophila melanogaster segment polarity gene network.

Dynamics of influenza virus and human host interactions during infection and replication cycle

The replication and life cycle of the influenza virus is governed by an intricate network of intracellular regulatory events during infection, including interactions with an even more complex system of biochemical interactions of the host cell. Computational modeling and systems biology have been successfully employed to further the understanding of various biological systems, however, computational studies of the complexity of intracellular interactions during influenza infection is lacking. In this work, we present the first large-scale dynamical model of the infection and replication cycle of influenza, as well as some of its interactions with the host's signaling machinery. Specifically, we focus on and visualize the dynamics of the internalization and endocytosis of the virus, replication and translation of its genomic components, as well as the assembly of progeny virions. Simulations and analyses of the models dynamics qualitatively reproduced numerous biological phenomena discovered in the laboratory. Finally, comparisons of the dynamics of existing and proposed drugs, our results suggest that a drug targeting PB1:PA would be more efficient than existing Amantadin/Rimantaine or Zanamivir/Oseltamivir.

Autonomous Boolean modelling of developmental gene regulatory networks

During early embryonic development, a network of regulatory interactions among genes dynamically determines a pattern of differentiated tissues. We show that important timing information associated with the interactions can be faithfully represented in autonomous Boolean models in which binary variables representing expression levels are updated in continuous time, and that such models can provide a direct insight into features that are difficult to extract from ordinary differential equation (ODE) models. As an application, we model the experimentally well-studied network controlling fly body segmentation. The Boolean model successfully generates the patterns formed in normal and genetically perturbed fly embryos, permits the derivation of constraints on the time delay parameters, clarifies the logic associated with different ODE parameter sets and provides a platform for studying connectivity and robustness in parameter space. By elucidating the role of regulatory time delays in pattern formation, the results suggest new types of experimental measurements in early embryonic development.

A checkpoints capturing timing-robust Boolean model of the budding yeast cell cycle regulatory network

Background

Cell cycle process of budding yeast (Saccharomyces cerevisiae) consists of four phases: G1, S, G2 and M. Initiated by stimulation of the G1 phase, cell cycle returns to the G1 stationary phase through a sequence of the S, G2 and M phases. During the cell cycle, a cell verifies whether necessary conditions are satisfied at the end of each phase (i.e., checkpoint) since damages of any phase can cause severe cell cycle defect. The cell cycle can proceed to the next phase properly only if checkpoint conditions are met. Over the last decade, there have been several studies to construct Boolean models that capture checkpoint conditions. However, they mostly focused on robustness to network perturbations, and the timing robustness has not been much addressed. Only recently, some studies suggested extension of such models towards timing-robust models, but they have not considered checkpoint conditions.

Results

To construct a timing-robust Boolean model that preserves checkpoint conditions of the budding yeast cell cycle, we used a model verification technique, ‘model checking’. By utilizing automatic and exhaustive verification of model checking, we found that previous models cannot properly capture essential checkpoint conditions in the presence of timing variations. In particular, such models violate the M phase checkpoint condition so that it allows a division of a budding yeast cell into two before the completion of its full DNA replication and synthesis. In this paper, we present a timing-robust model that preserves all the essential checkpoint conditions properly against timing variations. Our simulation results show that the proposed timing-robust model is more robust even against network perturbations and can better represent the nature of cell cycle than previous models.

Conclusions

To our knowledge this is the first work that rigorously examined the timing robustness of the cell cycle process of budding yeast with respect to checkpoint conditions using Boolean models. The proposed timing-robust model is the complete state-of-the-art model that guarantees no violation in terms of checkpoints known to date.

Continuous time Boolean modeling for biological signaling: application of Gillespie algorithm

Mathematical modeling is used as a Systems Biology tool to answer biological questions, and more precisely, to validate a network that describes biological observations and predict the effect of perturbations. This article presents an algorithm for modeling biological networks in a discrete framework with continuous time.

Background

There exist two major types of mathematical modeling approaches: (1) quantitative modeling, representing various chemical species concentrations by real numbers, mainly based on differential equations and chemical kinetics formalism; (2) and qualitative modeling, representing chemical species concentrations or activities by a finite set of discrete values. Both approaches answer particular (and often different) biological questions. Qualitative modeling approach permits a simple and less detailed description of the biological systems, efficiently describes stable state identification but remains inconvenient in describing the transient kinetics leading to these states. In this context, time is represented by discrete steps. Quantitative modeling, on the other hand, can describe more accurately the dynamical behavior of biological processes as it follows the evolution of concentration or activities of chemical species as a function of time, but requires an important amount of information on the parameters difficult to find in the literature.

Results

Here, we propose a modeling framework based on a qualitative approach that is intrinsically continuous in time. The algorithm presented in this article fills the gap between qualitative and quantitative modeling. It is based on continuous time Markov process applied on a Boolean state space. In order to describe the temporal evolution of the biological process we wish to model, we explicitly specify the transition rates for each node. For that purpose, we built a language that can be seen as a generalization of Boolean equations. Mathematically, this approach can be translated in a set of ordinary differential equations on probability distributions. We developed a C++ software, MaBoSS, that is able to simulate such a system by applying Kinetic Monte-Carlo (or Gillespie algorithm) on the Boolean state space. This software, parallelized and optimized, computes the temporal evolution of probability distributions and estimates stationary distributions.

Conclusions

Applications of the Boolean Kinetic Monte-Carlo are demonstrated for three qualitative models: a toy model, a published model of p53/Mdm2 interaction and a published model of the mammalian cell cycle. Our approach allows to describe kinetic phenomena which were difficult to handle in the original models. In particular, transient effects are represented by time dependent probability distributions, interpretable in terms of cell populations.

Emergence of robustness against noise: A structural phase transition in evolved models of gene regulatory networks

We investigate the evolution of Boolean networks subject to a selective pressure which favors robustness against noise, as a model of evolved genetic regulatory systems. By mapping the evolutionary process into a statistical ensemble and minimizing its associated free energy, we find the structural properties which emerge as the selective pressure is increased and identify a phase transition from a random topology to a "segregated-core" structure, where a smaller and more densely connected subset of the nodes is responsible for most of the regulation in the network. This segregated structure is very similar qualitatively to what is found in gene regulatory networks, where only a much smaller subset of genes--those responsible for transcription factors-is responsible for global regulation. We obtain the full phase diagram of the evolutionary process as a function of selective pressure and the average number of inputs per node. We compare the theoretical predictions with Monte Carlo simulations of evolved networks and with empirical data for Saccharomyces cerevisiae and Escherichia coli.

Convenience experimentation

Systems biology aims at explaining life processes by means of detailed models of molecular networks, mainly on the whole-cell scale. The whole cell perspective distinguishes the new field of systems biology from earlier approaches within molecular cell biology. The shift was made possible by the high throughput methods that were developed for gathering 'omic' (genomic, proteomic, etc.) data. These new techniques are made commercially available as semi-automatic analytic equipment, ready-made analytic kits and probe arrays. There is a whole industry of supplies for what may be called convenience experimentation. My paper inquires some epistemic consequences of strong reliance on convenience experimentation in systems biology. In times when experimentation was automated to a lesser degree, modeling and in part even experimentation could be understood fairly well as either being driven by hypotheses, and thus proceed by the testing of hypothesis, or as being performed in an exploratory mode, intended to sharpen concepts or initially vague phenomena. In systems biology, the situation is dramatically different. Data collection became so easy (though not cheap) that experimentation is, to a high degree, driven by convenience equipment, and model building is driven by the vast amount of data that is produced by convenience experimentation. This results in a shift in the mode of science. The paper shows that convenience driven science is not primarily hypothesis-testing, nor is it in an exploratory mode. It rather proceeds in a gathering mode. This shift demands another shift in the mode of evaluation, which now becomes an exploratory endeavor, in response to the superabundance of gathered data.

A Boolean network model of the FA/BRCA pathway

MOTIVATION

Fanconi anemia (FA) is a chromosomal instability syndrome originated by inherited mutations that impair the Fanconi Anemia/Breast Cancer (FA/BRCA) pathway, which is committed to the repair of DNA interstrand cross-links (ICLs). The disease displays increased spontaneous chromosomal aberrations and hypersensitivity to agents that create DNA interstrand cross-links. In spite of DNA damage, FA/BRCA-deficient cells are able to progress throughout the cell cycle, probably due to the activity of alternative DNA repair pathways, or due to defects in the checkpoints that monitor DNA integrity.

RESULTS

We propose a Boolean network model of the FA/BRCA pathway, Checkpoint proteins and some alternative DNA repair pathways. To our knowledge, this is the largest network model incorporating a DNA repair pathway. Our model is able to simulate the ICL repair process mediated by the FA/BRCA pathway, the activation of Checkpoint proteins observed by recurrent DNA damage, as well as the repair of DNA double-strand breaks and DNA adducts. We generated a series of simulations for mutants, some of which have never been reported and thus constitute predictions about the function of the FA/BRCA pathway. Finally, our model suggests alternative DNA repair pathways that become active whenever the FA/BRCA pathway is defective.

Regulatory network analysis acceleration with reconfigurable hardware

In medical research it is of great importance to be able to quickly obtain answers to inquiries about system response to different stimuli. Modeling the dynamics of biological regulatory networks is a promising approach to achieve this goal, but existing modeling approaches suffer from complexity issues and become inefficient with large networks. In order to improve the efficiency, we propose the implementation of models of regulatory networks in hardware, which allows for highly parallel simulation of these networks. We find that our FPGA implementation of an example model of peripheral naïve T cell differentiation provides five orders of magnitude speedup when compared to software simulation.

Network model of immune responses reveals key effectors to single and co-infection dynamics by a respiratory bacterium and a gastrointestinal helminth

Co-infections alter the host immune response but how the systemic and local processes at the site of infection interact is still unclear. The majority of studies on co-infections concentrate on one of the infecting species, an immune function or group of cells and often focus on the initial phase of the infection. Here, we used a combination of experiments and mathematical modelling to investigate the network of immune responses against single and co-infections with the respiratory bacterium Bordetella bronchiseptica and the gastrointestinal helminth Trichostrongylus retortaeformis. Our goal was to identify representative mediators and functions that could capture the essence of the host immune response as a whole, and to assess how their relative contribution dynamically changed over time and between single and co-infected individuals. Network-based discrete dynamic models of single infections were built using current knowledge of bacterial and helminth immunology; the two single infection models were combined into a co-infection model that was then verified by our empirical findings. Simulations showed that a T helper cell mediated antibody and neutrophil response led to phagocytosis and clearance of B. bronchiseptica from the lungs. This was consistent in single and co-infection with no significant delay induced by the helminth. In contrast, T. retortaeformis intensity decreased faster when co-infected with the bacterium. Simulations suggested that the robust recruitment of neutrophils in the co-infection, added to the activation of IgG and eosinophil driven reduction of larvae, which also played an important role in single infection, contributed to this fast clearance. Perturbation analysis of the models, through the knockout of individual nodes (immune cells), identified the cells critical to parasite persistence and clearance both in single and co-infections. Our integrated approach captured the within-host immuno-dynamics of bacteria-helminth infection and identified key components that can be crucial for explaining individual variability between single and co-infections in natural populations.

Infections with different infecting agents can alter the immune response against any one parasite and the relative abundance and persistence of the infections within the host. This is because the immune system is not compartmentalized but acts as a whole to allow the host to maintain control of the infections as well as repair damaged tissues and avoid immuno-pathology. There is no comprehensive understanding of the immune responses during co-infections and of how systemic and local mechanisms interact. Here we integrated experimental data with mathematical modelling to describe the network of immune responses of single and co-infection by a respiratory bacterium and a gastrointestinal helminth. We were able to identify key cells and functions responsible for clearing or reducing both parasites and showed that some mechanisms differed between type of infection as a result of different signal outputs and cells contributing to the immune processes. This study highlights the importance of understanding the immuno-dynamics of co-infection as a host response, how immune mechanisms differ from single infections and how they may alter parasite persistence, impact and abundance.

Discrete dynamic modeling of signal transduction networks

Newly available experimental data characterizing different processes involved in signaling pathways have provided the opportunity for network analysis and modeling of these interacting pathways. Current approaches in studying the dynamics of signaling networks fall into two major groups, namely, continuous and discrete models. The lack of kinetic information for biochemical interactions has limited the wide applicability of continuous models. To address this issue, discrete dynamic models, based on a qualitative description of a system's variables, have been applied for the analysis of biological systems with many unknown parameters. The purpose of this chapter is to give a detailed description of Boolean modeling, the simplest type of discrete dynamic modeling, and the ways in which it can be applied to analyze the dynamics of signaling networks. This is followed by practical examples of a Boolean dynamic framework applied to the modeling of the abscisic acid signal transduction network in plants as well as the T-cell survival signaling network in humans.

Logical modelling of gene regulatory networks with GINsim

Discrete mathematical formalisms are well adapted to model large biological networks, for which detailed kinetic data are scarce. This chapter introduces the reader to a well-established qualitative (logical) framework for the modelling of regulatory networks. Relying on GINsim, a software implementing this logical formalism, we guide the reader step by step towards the definition and the analysis of a simple model of the lysis-lysogeny decision in the bacteriophage λ.

Dynamical and structural analysis of a T cell survival network identifies novel candidate therapeutic targets for large granular lymphocyte leukemia

The blood cancer T cell large granular lymphocyte (T-LGL) leukemia is a chronic disease characterized by a clonal proliferation of cytotoxic T cells. As no curative therapy is yet known for this disease, identification of potential therapeutic targets is of immense importance. In this paper, we perform a comprehensive dynamical and structural analysis of a network model of this disease. By employing a network reduction technique, we identify the stationary states (fixed points) of the system, representing normal and diseased (T-LGL) behavior, and analyze their precursor states (basins of attraction) using an asynchronous Boolean dynamic framework. This analysis identifies the T-LGL states of 54 components of the network, out of which 36 (67%) are corroborated by previous experimental evidence and the rest are novel predictions. We further test and validate one of these newly identified states experimentally. Specifically, we verify the prediction that the node SMAD is over-active in leukemic T-LGL by demonstrating the predominant phosphorylation of the SMAD family members Smad2 and Smad3. Our systematic perturbation analysis using dynamical and structural methods leads to the identification of 19 potential therapeutic targets, 68% of which are corroborated by experimental evidence. The novel therapeutic targets provide valuable guidance for wet-bench experiments. In addition, we successfully identify two new candidates for engineering long-lived T cells necessary for the delivery of virus and cancer vaccines. Overall, this study provides a bird's-eye-view of the avenues available for identification of therapeutic targets for similar diseases through perturbation of the underlying signal transduction network.

T-LGL leukemia is a blood cancer characterized by an abnormal increase in the abundance of a type of white blood cell called T cell. Since there is no known curative therapy for this disease, identification of potential therapeutic targets is of utmost importance. Experimental identification of manipulations capable of reversing the disease condition is usually a long, arduous process. Mathematical modeling can aid this process by identifying potential therapeutic interventions. In this work, we carry out a systematic analysis of a network model of T cell survival in T-LGL leukemia to get a deeper insight into the unknown facets of the disease. We identify the T-LGL status of 54 components of the system, out of which 36 (67%) are corroborated by previous experimental evidence and the rest are novel predictions, one of which we validate by follow-up experiments. By deciphering the structure and dynamics of the underlying network, we identify component perturbations that lead to programmed cell death, thereby suggesting several novel candidate therapeutic targets for future experiments.

Exploring phospholipase C-coupled Ca(2+) signalling networks using Boolean modelling

In this study, the authors explored the utility of a descriptive and predictive bionetwork model for phospholipase C-coupled calcium signalling pathways, built with non-kinetic experimental information. Boolean models generated from these data yield oscillatory activity patterns for both the endoplasmic reticulum resident inositol-1,4,5-trisphosphate receptor (IP(3)R) and the plasma-membrane resident canonical transient receptor potential channel 3 (TRPC3). These results are specific as randomisation of the Boolean operators ablates oscillatory pattern formation. Furthermore, knock-out simulations of the IP(3)R, TRPC3 and multiple other proteins recapitulate experimentally derived results. The potential of this approach can be observed by its ability to predict previously undescribed cellular phenotypes using in vitro experimental data. Indeed, our cellular analysis of the developmental and calcium-regulatory protein, DANGER1a, confirms the counter-intuitive predictions from our Boolean models in two highly relevant cellular models. Based on these results, the authors theorise that with sufficient legacy knowledge and/or computational biology predictions, Boolean networks can provide a robust method for predictive modelling of any biological system. [Includes supplementary material].

Mathematical modeling of gene expression: a guide for the perplexed biologist

The detailed analysis of transcriptional networks holds a key for understanding central biological processes, and interest in this field has exploded due to new large-scale data acquisition techniques. Mathematical modeling can provide essential insights, but the diversity of modeling approaches can be a daunting prospect to investigators new to this area. For those interested in beginning a transcriptional mathematical modeling project, we provide here an overview of major types of models and their applications to transcriptional networks. In this discussion of recent literature on thermodynamic, Boolean, and differential equation models, we focus on considerations critical for choosing and validating a modeling approach that will be useful for quantitative understanding of biological systems.

A tutorial on analysis and simulation of boolean gene regulatory network models

Driven by the desire to understand genomic functions through the interactions among genes and gene products, the research in gene regulatory networks has become a heated area in genomic signal processing. Among the most studied mathematical models are Boolean networks and probabilistic Boolean networks, which are rule-based dynamic systems. This tutorial provides an introduction to the essential concepts of these two Boolean models, and presents the up-to-date analysis and simulation methods developed for them. In the Analysis section, we will show that Boolean models are Markov chains, based on which we present a Markovian steady-state analysis on attractors, and also reveal the relationship between probabilistic Boolean networks and dynamic Bayesian networks (another popular genetic network model), again via Markov analysis; we dedicate the last subsection to structural analysis, which opens a door to other topics such as network control. The Simulation section will start from the basic tasks of creating state transition diagrams and finding attractors, proceed to the simulation of network dynamics and obtaining the steady-state distributions, and finally come to an algorithm of generating artificial Boolean networks with prescribed attractors. The contents are arranged in a roughly logical order, such that the Markov chain analysis lays the basis for the most part of Analysis section, and also prepares the readers to the topics in Simulation section.

Recent advances in intervention in markovian regulatory networks

Markovian regulatory networks constitute a class of discrete state-space models used to study gene regulatory dynamics and discover methods that beneficially alter those dynamics. Thereby, this class of models provides a framework to discover effective drug targets and design potent therapeutic strategies. The salient translational goal is to design therapeutic strategies that desirably modify network dynamics via external signals that vary the expressions of a control gene. The objective of an intervention strategy is to reduce the likelihood of the pathological cellular function related to a disease. The task of finding an effective intervention strategy can be formulated as a sequential decision making problem for a pre-defined cost of intervention and a cost-per-stage function that discriminates the gene-activity profiles. An effective intervention strategy prescribes the actions associated with an external signal that result in the minimum expected cost. This strategy in turn can be used as a treatment that reduces the long-run likelihood of gene expressions favorable to the disease. In this tutorial, we briefly summarize the first method proposed to design such therapeutic interventions, and then move on to some of the recent refinements that have been proposed. Each of these recent intervention methods is motivated by practical or analytical considerations. The presentation of the key ideas is facilitated with the help of two case studies.

"Immunetworks", intersecting circuits and dynamics

This paper proposes a study of biological regulation networks based on a multi-level strategy. Given a network, the first structural level of this strategy consists in analysing the architecture of the network interactions in order to describe it. The second dynamical level consists in relating the patterns found in the architecture to the possible dynamical behaviours of the network. It is known that circuits are the patterns that play the most important part in the dynamics of a network in the sense that they are responsible for the diversity of its asymptotic behaviours. Here, we pursue further this idea and argue that beyond the influence of underlying circuits, intersections of circuits also impact significantly on the dynamics of a network and thus need to be payed special attention to. For some genetic regulation networks involved in the control of the immune system ("immunetworks"), we show that the small number of attractors can be explained by the presence, in the underlying structures of these networks, of intersecting circuits that "inter-lock".

Absence of Tec family kinases interleukin-2 inducible T cell kinase (Itk) and Bruton's tyrosine kinase (Btk) severely impairs Fc epsilonRI-dependent mast cell responses

Mast cells are critical effector cells in the pathophysiology of allergic asthma and other IgE-mediated diseases. The Tec family of tyrosine kinases Itk and Btk serve as critical signal amplifiers downstream of antigen receptors. Although both kinases are expressed and activated in mast cells following FcεRI stimulation, their individual contributions are not clear. To determine whether these kinases play unique and/or complementary roles in FcεRI signaling and mast cell function, we generated Itk and Btk double knock-out mice. Analyses of these mice show decreased mast cell granularity and impaired passive systemic anaphylaxis responses. This impaired response is accompanied by a significant elevation in serum IgE in Itk/Btk double knock-out mice. In vitro analyses of bone marrow-derived mast cells (BMMCs) indicated that Itk/Btk double knock-out BMMCs are defective in degranulation and cytokine secretion responses downstream to FcεRI activation. These responses were accompanied by a significant reduction in PLCγ2 phosphorylation and severely impaired calcium responses in these cells. This defect also results in altered NFAT1 nuclear localization in double knock-out BMMCs. Network analysis suggests that although they may share substrates, Itk plays both positive and negative roles, while Btk primarily plays a positive role in mast cell FcεRI-induced cytokine secretion.

Computational and experimental analysis reveals a requirement for eosinophil-derived IL-13 for the development of allergic airway responses in C57BL/6 mice

Eosinophils are found in the lungs of humans with allergic asthma, as well as in the lungs of animals in models of this disease. Increasing evidence suggests that these cells are integral to the development of allergic asthma in C57BL/6 mice. However, the specific function of eosinophils that is required for this event is not known. In this study, we experimentally validate a dynamic computational model and perform follow-up experimental observations to determine the mechanism of eosinophil modulation of T cell recruitment to the lung during development of allergic asthma. We find that eosinophils deficient in IL-13 were unable to rescue airway hyperresponsiveness, T cell recruitment to the lungs, and Th2 cytokine/chemokine production in ΔdblGATA eosinophil-deficient mice, even if Th2 cells were present. However, eosinophil-derived IL-13 alone was unable to rescue allergic asthma responses in the absence of competence of other IL-13-producing cells. We further computationally investigate the role of other cell types in the production of IL-13, which led to the various predictions including early and late pulses of IL-13 during airway hyperresponsiveness. These experiments suggest that eosinophils and T cells have an interdependent relationship, centered on IL-13, which regulates T cell recruitment to the lung and development of allergic asthma.

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.

Combining network modeling and gene expression microarray analysis to explore the dynamics of Th1 and Th2 cell regulation

Two T helper (Th) cell subsets, namely Th1 and Th2 cells, play an important role in inflammatory diseases. The two subsets are thought to counter-regulate each other, and alterations in their balance result in different diseases. This paradigm has been challenged by recent clinical and experimental data. Because of the large number of genes involved in regulating Th1 and Th2 cells, assessment of this paradigm by modeling or experiments is difficult. Novel algorithms based on formal methods now permit the analysis of large gene regulatory networks. By combining these algorithms with in silico knockouts and gene expression microarray data from human T cells, we examined if the results were compatible with a counter-regulatory role of Th1 and Th2 cells. We constructed a directed network model of genes regulating Th1 and Th2 cells through text mining and manual curation. We identified four attractors in the network, three of which included genes that corresponded to Th0, Th1 and Th2 cells. The fourth attractor contained a mixture of Th1 and Th2 genes. We found that neither in silico knockouts of the Th1 and Th2 attractor genes nor gene expression microarray data from patients with immunological disorders and healthy subjects supported a counter-regulatory role of Th1 and Th2 cells. By combining network modeling with transcriptomic data analysis and in silico knockouts, we have devised a practical way to help unravel complex regulatory network topology and to increase our understanding of how network actions may differ in health and disease.

Attractor analysis of asynchronous Boolean models of signal transduction networks

Prior work on the dynamics of Boolean networks, including analysis of the state space attractors and the basin of attraction of each attractor, has mainly focused on synchronous update of the nodes' states. Although the simplicity of synchronous updating makes it very attractive, it fails to take into account the variety of time scales associated with different types of biological processes. Several different asynchronous update methods have been proposed to overcome this limitation, but there have not been any systematic comparisons of the dynamic behaviors displayed by the same system under different update methods. Here we fill this gap by combining theoretical analysis such as solution of scalar equations and Markov chain techniques, as well as numerical simulations to carry out a thorough comparative study on the dynamic behavior of a previously proposed Boolean model of a signal transduction network in plants. Prior evidence suggests that this network admits oscillations, but it is not known whether these oscillations are sustained. We perform an attractor analysis of this system using synchronous and three different asynchronous updating schemes both in the case of the unperturbed (wild-type) and perturbed (node-disrupted) systems. This analysis reveals that while the wild-type system possesses an update-independent fixed point, any oscillations eventually disappear unless strict constraints regarding the timing of certain processes and the initial state of the system are satisfied. Interestingly, in the case of disruption of a particular node all models lead to an extended attractor. Overall, our work provides a roadmap on how Boolean network modeling can be used as a predictive tool to uncover the dynamic patterns of a biological system under various internal and environmental perturbations.

Stabilizing patterning in the Drosophila segment polarity network by selecting models in silico

The segmentation of Drosophila is a prime model to study spatial patterning during embryogenesis. The spatial expression of segment polarity genes results from a complex network of interacting proteins whose expression products are maintained after successful segmentation. This prompted us to investigate the stability and robustness of this process using a dynamical model for the segmentation network based on Boolean states. The model consists of intra-cellular as well as inter-cellular interactions between adjacent cells in one spatial dimension. We quantify the robustness of the dynamical segmentation process by a systematic analysis of mutations. Our starting point consists in a previous Boolean model for Drosophila segmentation. We define mathematically the notion of dynamical robustness and show that the proposed model exhibits limited robustness in gene expression under perturbations. We applied in silico evolution (mutation and selection) and discover two classes of modified gene networks that have a more robust spatial expression pattern. We verified that the enhanced robustness of the two new models is maintained in differential equations models. By comparing the predicted model with experiments on mutated flies, we then discuss the two types of enhanced models. Drosophila patterning can be explained by modelling the underlying network of interacting genes. Here we demonstrate that simple dynamical considerations and in silico evolution can enhance the model to robustly express the expected pattern, helping to elucidate the role of further interactions.

Distribution of essential interactions in model gene regulatory networks under mutation-selection balance

Gene regulatory networks typically have low in-degrees, whereby any given gene is regulated by few of the genes in the network. They also tend to have broad distributions for the out-degree. What mechanisms might be responsible for these degree distributions? Starting with an accepted framework of the binding of transcription factors to DNA, we consider a simple model of gene regulatory dynamics. There, we show that selection for a target expression pattern leads to the emergence of minimum connectivities compatible with the selective constraint. As a consequence, these gene networks have low in-degree and "functionality" is parsimonious, i.e., is concentrated on a sparse number of interactions as measured for instance by their essentiality. Furthermore, we find that mutations of the transcription factors drive the networks to have broad out-degrees. Finally, these classes of models are evolvable, i.e., significantly different genotypes can emerge gradually under mutation-selection balance.

Monotonicity, frustration, and ordered response: an analysis of the energy landscape of perturbed large-scale biological networks

BACKGROUND

For large-scale biological networks represented as signed graphs, the index of frustration measures how far a network is from a monotone system, i.e., how incoherently the system responds to perturbations.

RESULTS

In this paper we find that the frustration is systematically lower in transcriptional networks (modeled at functional level) than in signaling and metabolic networks (modeled at stoichiometric level). A possible interpretation of this result is in terms of energetic cost of an interaction: an erroneous or contradictory transcriptional action costs much more than a signaling/metabolic error, and therefore must be avoided as much as possible. Averaging over all possible perturbations, however, we also find that unlike for transcriptional networks, in the signaling/metabolic networks the probability of finding the system in its least frustrated configuration tends to be high also in correspondence of a moderate energetic regime, meaning that, in spite of the higher frustration, these networks can achieve a globally ordered response to perturbations even for moderate values of the strength of the interactions. Furthermore, an analysis of the energy landscape shows that signaling and metabolic networks lack energetic barriers around their global optima, a property also favouring global order.

CONCLUSION

In conclusion, transcriptional and signaling/metabolic networks appear to have systematic differences in both the index of frustration and the transition to global order. These differences are interpretable in terms of the different functions of the various classes of networks.

Mathematical modeling of translation initiation for the estimation of its efficiency to computationally design mRNA sequences with desired expression levels in prokaryotes

Background

Within the emerging field of synthetic biology, engineering paradigms have recently been used to design biological systems with novel functionalities. One of the essential challenges hampering the construction of such systems is the need to precisely optimize protein expression levels for robust operation. However, it is difficult to design mRNA sequences for expression at targeted protein levels, since even a few nucleotide modifications around the start codon may alter translational efficiency and dramatically (up to 250-fold) change protein expression. Previous studies have used ad hoc approaches (e.g., random mutagenesis) to obtain the desired translational efficiencies for mRNA sequences. Hence, the development of a mathematical methodology capable of estimating translational efficiency would greatly facilitate the future design of mRNA sequences aimed at yielding desired protein expression levels.

Results

We herein propose a mathematical model that focuses on translation initiation, which is the rate-limiting step in translation. The model uses mRNA-folding dynamics and ribosome-binding dynamics to estimate translational efficiencies solely from mRNA sequence information. We confirmed the feasibility of our model using previously reported expression data on the MS2 coat protein. For further confirmation, we used our model to design 22 luxR mRNA sequences predicted to have diverse translation efficiencies ranging from 10^-5 to 1. The expression levels of these sequences were measured in Escherichia coli and found to be highly correlated (R² = 0.87) with their estimated translational efficiencies. Moreover, we used our computational method to successfully transform a low-expressing DsRed2 mRNA sequence into a high-expressing mRNA sequence by maximizing its translational efficiency through the modification of only eight nucleotides upstream of the start codon.

Conclusions

We herein describe a mathematical model that uses mRNA sequence information to estimate translational efficiency. This model could be used to design best-fit mRNA sequences having a desired protein expression level, thereby facilitating protein over-production in biotechnology or the protein expression-level optimization necessary for the construction of robust networks in synthetic biology.

Modeling reveals bistability and low-pass filtering in the network module determining blood stem cell fate

Combinatorial regulation of gene expression is ubiquitous in eukaryotes with multiple inputs converging on regulatory control elements. The dynamic properties of these elements determine the functionality of genetic networks regulating differentiation and development. Here we propose a method to quantitatively characterize the regulatory output of distant enhancers with a biophysical approach that recursively determines free energies of protein-protein and protein-DNA interactions from experimental analysis of transcriptional reporter libraries. We apply this method to model the Scl-Gata2-Fli1 triad—a network module important for cell fate specification of hematopoietic stem cells. We show that this triad module is inherently bistable with irreversible transitions in response to physiologically relevant signals such as Notch, Bmp4 and Gata1 and we use the model to predict the sensitivity of the network to mutations. We also show that the triad acts as a low-pass filter by switching between steady states only in response to signals that persist for longer than a minimum duration threshold. We have found that the auto-regulation loops connecting the slow-degrading Scl to Gata2 and Fli1 are crucial for this low-pass filtering property. Taken together our analysis not only reveals new insights into hematopoietic stem cell regulatory network functionality but also provides a novel and widely applicable strategy to incorporate experimental measurements into dynamical network models.

Hematopoiesis—blood cell development—has long served as a model for study of cellular differentiation and its control by underlying gene regulatory networks. The Scl-Gata2-Fli1 triad is a network module essential for the development of hematopoietic stem cells but its mechanistic role is not well understood. The transcription factors Scl, Gata2 and Fli1 act in combination to upregulate transcription of each other via distal enhancer site binding. Similar network architectures are essential in other multipotent cell lines. We propose a method that uses experimental results to circumvent the difficulties of mathematically modeling the combinatorial regulation of this triad module. Using this dynamical model we show that the triad exhibits robust bistable behavior. Environmental signals can irreversibly switch the triad between stable states in a manner that reflects the unidirectional switching in the formation and subsequent differentiation of hematopoietic stem cells. We also show that the triad makes reliable decisions in noisy environments by only switching in response to transient signals that persist longer than the threshold duration. These results suggest that the Scl-Gata2-Fli1 module possibly functions as a control switch for hematopoietic stem cell development. The proposed method can be extended for quantitative characterization of other combinatorial gene regulatory modules.