Segmenting the fly embryo: a logical analysis of the pair-rule cross-regulatory module

Time and space in segmentation

Arthropod segmentation and vertebrate somitogenesis are leading fields in the experimental and theoretical interrogation of developmental patterning. However, despite the sophistication of current research, basic conceptual issues remain unresolved. These include: (i) the mechanistic origins of spatial organization within the segment addition zone (SAZ); (ii) the mechanistic origins of segment polarization; (iii) the mechanistic origins of axial variation; and (iv) the evolutionary origins of simultaneous patterning. Here, I explore these problems using coarse-grained models of cross-regulating dynamical processes. In the morphogenetic framework of a row of cells undergoing axial elongation, I simulate interactions between an 'oscillator', a 'switch' and up to three 'timers', successfully reproducing essential patterning behaviours of segmenting systems. By comparing the output of these largely cell-autonomous models to variants that incorporate positional information, I find that scaling relationships, wave patterns and patterning dynamics all depend on whether the SAZ is regulated by temporal or spatial information. I also identify three mechanisms for polarizing oscillator output, all of which functionally implicate the oscillator frequency profile. Finally, I demonstrate significant dynamical and regulatory continuity between sequential and simultaneous modes of segmentation. I discuss these results in the context of the experimental literature.

Dynamic Modeling of Transcriptional Gene Regulatory Networks

Diverse cellular phenotypes are determined by groups of transcription factors (TFs) and other regulators that influence each others' gene expression, forming transcriptional gene regulatory networks (GRNs). In many biological contexts, especially in development and associated diseases, the expression of the genes in GRNs is not static but evolves in time. Modeling the dynamics of GRN state is an important approach for understanding diverse cellular phenomena such as cell-fate specification, pluripotency and cell-fate reprogramming, oncogenesis, and tissue regeneration. In this protocol, we describe how to model GRNs using a data-driven dynamic modeling methodology, gene circuits. Gene circuits do not require knowledge of the GRN topology and connectivity but instead learn them from training data, making them very general and applicable to diverse biological contexts. We utilize the MATLAB-based gene circuit modeling software Fast Inference of Gene Regulation (FIGR) for training the model on quantitative gene expression data and simulating the GRN. We describe all the steps in the modeling life cycle, from formulating the model, training the model using FIGR, simulating the GRN, to analyzing and interpreting the model output. This protocol highlights these steps with the example of a dynamical model of the gap gene GRN involved in Drosophila segmentation and includes example MATLAB statements for each step.

Attractor - a new turning point in drug discovery

Drug discovery for complex diseases can be viewed as a challenging problem in which the influence of compounds on dynamic features of disease system should be considered, especially the strategies escaping from the disease attractors. Moreover, escaping from the disease-related attractors has been proved to be a cue for the treatment of the complex diseases. The drug discovery methodology based on the attractor theory indicates new solutions for target identification, drug discovery and drug combination design. The methodology is based on the holism level of the organism and the features of system dynamics, so it has advantages for the classification of complex diseases and drug discovery. Currently, research results of this method have increased, which expand the insight scope for drug discovery. This article introduces the major drug discovery methods in the history of pharmacy development and their characteristics, so as to illustrate the reasons and inevitability of the appearance of attractor method, its position in the history of pharmacy development, and its advantages for drug discovery and design, thereby to prove that the attractor method can indeed become the next major drug development method. In addition, it provides a comprehensive description about the concept of attractor, the pipeline of attractor analysis, the common methods of each process and its research progress, so as to provide a macroscopic framework and optional methods and tools for the follow-up researchers.

Algorithmic and Stochastic Representations of Gene Regulatory Networks and Protein-Protein Interactions

BACKGROUND

Latest studies reveal the importance of Protein-Protein interactions on physiologic functions and biological structures. Several stochastic and algorithmic methods have been published until now, for the modeling of the complex nature of the biological systems.

OBJECTIVE

Biological Networks computational modeling is still a challenging task. The formulation of the complex cellular interactions is a research field of great interest. In this review paper, several computational methods for the modeling of GRN and PPI are presented analytically.

METHODS

Several well-known GRN and PPI models are presented and discussed in this review study such as: Graphs representation, Boolean Networks, Generalized Logical Networks, Bayesian Networks, Relevance Networks, Graphical Gaussian models, Weight Matrices, Reverse Engineering Approach, Evolutionary Algorithms, Forward Modeling Approach, Deterministic models, Static models, Hybrid models, Stochastic models, Petri Nets, BioAmbients calculus and Differential Equations.

RESULTS

GRN and PPI methods have been already applied in various clinical processes with potential positive results, establishing promising diagnostic tools.

CONCLUSION

In literature many stochastic algorithms are focused in the simulation, analysis and visualization of the various biological networks and their dynamics interactions, which are referred and described in depth in this review paper.

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.

Logical Modeling and Analysis of Cellular Regulatory Networks With GINsim 3.0

The logical formalism is well adapted to model large cellular networks, in particular when detailed kinetic data are scarce. This tutorial focuses on this well-established qualitative framework. Relying on GINsim (release 3.0), a software implementing this formalism, we guide the reader step by step toward the definition, the analysis and the simulation of a four-node model of the mammalian p53-Mdm2 network.

Dynamic patterning by the Drosophila pair-rule network reconciles long-germ and short-germ segmentation

Drosophila segmentation is a well-established paradigm for developmental pattern formation. However, the later stages of segment patterning, regulated by the "pair-rule" genes, are still not well understood at the system level. Building on established genetic interactions, I construct a logical model of the Drosophila pair-rule system that takes into account the demonstrated stage-specific architecture of the pair-rule gene network. Simulation of this model can accurately recapitulate the observed spatiotemporal expression of the pair-rule genes, but only when the system is provided with dynamic "gap" inputs. This result suggests that dynamic shifts of pair-rule stripes are essential for segment patterning in the trunk and provides a functional role for observed posterior-to-anterior gap domain shifts that occur during cellularisation. The model also suggests revised patterning mechanisms for the parasegment boundaries and explains the aetiology of the even-skipped null mutant phenotype. Strikingly, a slightly modified version of the model is able to pattern segments in either simultaneous or sequential modes, depending only on initial conditions. This suggests that fundamentally similar mechanisms may underlie segmentation in short-germ and long-germ arthropods.

Ancient role of ten-m/odz in segmentation and the transition from sequential to syncytial segmentation

Background

Until recently, mechanisms of segmentation established for Drosophila served as a paradigm for arthropod segmentation. However, with the discovery of gene expression waves in vertebrate segmentation, another paradigm based on oscillations linked to axial growth was established. The Notch pathway and hairy delay oscillator are basic components of this mechanism, as is the wnt pathway. With the establishment of oscillations during segmentation of the beetle Tribolium, a common segmentation mechanism may have been present in the last common ancestor of vertebrates and arthropods. However, the Notch pathway is not involved in segmentation of the initial Drosophila embryo. In arthropods, the engrailed, wingless pair has a much more conserved function in segmentation than most of the hierarchy established for Drosophila.

Results

Here, we work backwards from this conserved pair by discussing possible mechanisms which could have taken over the role of the Notch pathway. We propose a pivotal role for the large transmembrane protein Ten-m/Odz. Ten-m/Odz may have had an ancient role in cell-cell communication, parallel to the Notch and wnt pathways. The Ten-m protein binds to the membrane with properties which resemble other membrane-based biochemical oscillators.

Conclusion

We propose that such a simple transition could have formed the initial scaffold, on top of which the hierarchy, observed in the syncytium of dipterans, could have evolved.

Odd-paired controls frequency doubling in Drosophila segmentation by altering the pair-rule gene regulatory network

The Drosophila embryo transiently exhibits a double-segment periodicity, defined by the expression of seven 'pair-rule' genes, each in a pattern of seven stripes. At gastrulation, interactions between the pair-rule genes lead to frequency doubling and the patterning of 14 parasegment boundaries. In contrast to earlier stages of Drosophila anteroposterior patterning, this transition is not well understood. By carefully analysing the spatiotemporal dynamics of pair-rule gene expression, we demonstrate that frequency-doubling is precipitated by multiple coordinated changes to the network of regulatory interactions between the pair-rule genes. We identify the broadly expressed but temporally patterned transcription factor, Odd-paired (Opa/Zic), as the cause of these changes, and show that the patterning of the even-numbered parasegment boundaries relies on Opa-dependent regulatory interactions. Our findings indicate that the pair-rule gene regulatory network has a temporally modulated topology, permitting the pair-rule genes to play stage-specific patterning roles.

Scaling of morphogenetic patterns in reaction-diffusion systems

Development of multicellular organisms is commonly associated with the response of individual cells to concentrations of chemical substances called morphogens. Concentration fields of morphogens form a basis for biological patterning and ensure its properties including ability to scale with the size of the organism. While mechanisms underlying the formation of morphogen gradients are reasonably well understood, little is known about processes responsible for their scaling. Here, we perform a formal analysis of scaling for chemical patterns forming in continuous systems. We introduce a quantity representing the sensitivity of systems to changes in their size and use it to analyse scaling properties of patterns forming in a few different systems. Particularly, we consider how scaling properties of morphogen gradients forming in diffusion-decay systems depend on boundary conditions and how the scaling can be improved by passive modulation of morphogens or active transport in the system. We also analyse scaling of morphogenetic signal caused by two opposing gradients and consider scaling properties of patterns forming in activator-inhibitor systems. We conclude with a few possible mechanisms which allow scaling of morphogenetic patterns.

Cooperative development of logical modelling standards and tools with CoLoMoTo

The identification of large regulatory and signalling networks involved in the control of crucial cellular processes calls for proper modelling approaches. Indeed, models can help elucidate properties of these networks, understand their behaviour and provide (testable) predictions by performing in silico experiments. In this context, qualitative, logical frameworks have emerged as relevant approaches, as demonstrated by a growing number of published models, along with new methodologies and software tools. This productive activity now requires a concerted effort to ensure model reusability and interoperability between tools. Following an outline of the logical modelling framework, we present the most important achievements of the Consortium for Logical Models and Tools, along with future objectives. Our aim is to advertise this open community, which welcomes contributions from all researchers interested in logical modelling or in related mathematical and computational developments.

Formal modelling of toll like receptor 4 and JAK/STAT signalling pathways: insight into the roles of SOCS-1, interferon-β and proinflammatory cytokines in sepsis

Sepsis is one of the major causes of human morbidity and results in a considerable number of deaths each year. Lipopolysaccharide-induced sepsis has been associated with TLR4 signalling pathway which in collaboration with the JAK/STAT signalling regulate endotoxemia and inflammation. However, during sepsis our immune system cannot maintain a balance of cytokine levels and results in multiple organ damage and eventual death. Different opinions have been made in previous studies about the expression patterns and the role of proinflammatory cytokines in sepsis that attracted our attention towards qualitative properties of TLR4 and JAK/STAT signalling pathways using computer-aided studies. René Thomas' formalism was used to model septic and non-septic dynamics of TLR4 and JAK/STAT signalling. Comparisons among dynamics were made by intervening or removing the specific interactions among entities. Among our predictions, recurrent induction of proinflammatory cytokines with subsequent downregulation was found as the basic characteristic of septic model. This characteristic was found in agreement with previous experimental studies, which implicate that inflammation is followed by immunomodulation in septic patients. Moreover, intervention in downregulation of proinflammatory cytokines by SOCS-1 was found desirable to boost the immune responses. On the other hand, interventions either in TLR4 or transcriptional elements such as NFκB and STAT were found effective in the downregulation of immune responses. Whereas, IFN-β and SOCS-1 mediated downregulation at different levels of signalling were found to be associated with variations in the levels of proinflammatory cytokines. However, these predictions need to be further validated using wet laboratory experimental studies to further explore the roles of inhibitors such as SOCS-1 and IFN-β, which may alter the levels of proinflammatory cytokines at different stages of sepsis.

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.

Mechanisms and constraints shaping the evolution of body plan segmentation

Segmentation of the major body axis into repeating units is arguably one of the major inventions in the evolution of animal body plan pattering. It is found in current day vertebrates, annelids and arthropods. Most segmented animals seem to use a clock-and-wavefront type mechanism in which oscillations emanating from a posterior growth zone become transformed into an anterior posterior sequence of segments. In contrast, few animals such as Drosophila use a complex gene regulatory hierarchy to simultaneously subdivide their entire body axis into segments. Here I discuss how in silico models simulating the evolution of developmental patterning can be used to investigate the forces and constraints that helped shape these two developmental modes. I perform an analysis of a series of previous simulation studies, exploiting the similarities and differences in their outcomes in relation to model characteristics to elucidate the circumstances and constraints likely to have been important for the evolution of sequential and simultaneous segmentation modes. The analysis suggests that constraints arising from the involved growth process and spatial patterning signal--posterior elongation producing a propagating wavefront versus a tissue wide morphogen gradient--and the evolutionary history--ancestral versus derived segmentation mode--strongly shaped both segmentation mechanisms. Furthermore, this implies that these patterning types are to be expected rather than random evolutionary outcomes and supports the likelihood of multiple parallel evolutionary origins.

Spatiotemporal network motif reveals the biological traits of developmental gene regulatory networks in Drosophila melanogaster

Background

Network motifs provided a “conceptual tool” for understanding the functional principles of biological networks, but such motifs have primarily been used to consider static network structures. Static networks, however, cannot be used to reveal time- and region-specific traits of biological systems. To overcome this limitation, we proposed the concept of a “spatiotemporal network motif,” a spatiotemporal sequence of network motifs of sub-networks which are active only at specific time points and body parts.

Results

On the basis of this concept, we analyzed the developmental gene regulatory network of the Drosophila melanogaster embryo. We identified spatiotemporal network motifs and investigated their distribution pattern in time and space. As a result, we found how key developmental processes are temporally and spatially regulated by the gene network. In particular, we found that nested feedback loops appeared frequently throughout the entire developmental process. From mathematical simulations, we found that mutual inhibition in the nested feedback loops contributes to the formation of spatial expression patterns.

Conclusions

Taken together, the proposed concept and the simulations can be used to unravel the design principle of developmental gene regulatory networks.

Robust non-linear differential equation models of gene expression evolution across Drosophila development

Background

This paper lies in the context of modeling the evolution of gene expression away from stationary states, for example in systems subject to external perturbations or during the development of an organism. We base our analysis on experimental data and proceed in a top-down approach, where we start from data on a system's transcriptome, and deduce rules and models from it without a priori knowledge. We focus here on a publicly available DNA microarray time series, representing the transcriptome of Drosophila across evolution from the embryonic to the adult stage.

Results

In the first step, genes were clustered on the basis of similarity of their expression profiles, measured by a translation-invariant and scale-invariant distance that proved appropriate for detecting transitions between development stages. Average profiles representing each cluster were computed and their time evolution was analyzed using coupled differential equations. A linear and several non-linear model structures involving a transcription and a degradation term were tested. The parameters were identified in three steps: determination of the strongest connections between genes, optimization of the parameters defining these connections, and elimination of the unnecessary parameters using various reduction schemes. Different solutions were compared on the basis of their abilities to reproduce the data, to keep realistic gene expression levels when extrapolated in time, to show the biologically expected robustness with respect to parameter variations, and to contain as few parameters as possible.

Conclusions

We showed that the linear model did very well in reproducing the data with few parameters, but was not sufficiently robust and yielded unrealistic values upon extrapolation in time. In contrast, the non-linear models all reached the latter two objectives, but some were unable to reproduce the data. A family of non-linear models, constructed from the exponential of linear combinations of expression levels, reached all the objectives. It defined networks with a mean number of connections equal to two, when restricted to the embryonic time series, and equal to five for the full time series. These networks were compared with experimental data about gene-transcription factor and protein-protein interactions. The non-uniqueness of the solutions was discussed in the context of plasticity and cluster versus single-gene networks.

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.

Evolution of networks for body plan patterning; interplay of modularity, robustness and evolvability

A major goal of evolutionary developmental biology (evo-devo) is to understand how multicellular body plans of increasing complexity have evolved, and how the corresponding developmental programs are genetically encoded. It has been repeatedly argued that key to the evolution of increased body plan complexity is the modularity of the underlying developmental gene regulatory networks (GRNs). This modularity is considered essential for network robustness and evolvability. In our opinion, these ideas, appealing as they may sound, have not been sufficiently tested. Here we use computer simulations to study the evolution of GRNs' underlying body plan patterning. We select for body plan segmentation and differentiation, as these are considered to be major innovations in metazoan evolution. To allow modular networks to evolve, we independently select for segmentation and differentiation. We study both the occurrence and relation of robustness, evolvability and modularity of evolved networks. Interestingly, we observed two distinct evolutionary strategies to evolve a segmented, differentiated body plan. In the first strategy, first segments and then differentiation domains evolve (SF strategy). In the second scenario segments and domains evolve simultaneously (SS strategy). We demonstrate that under indirect selection for robustness the SF strategy becomes dominant. In addition, as a byproduct of this larger robustness, the SF strategy is also more evolvable. Finally, using a combined functional and architectural approach, we determine network modularity. We find that while SS networks generate segments and domains in an integrated manner, SF networks use largely independent modules to produce segments and domains. Surprisingly, we find that widely used, purely architectural methods for determining network modularity completely fail to establish this higher modularity of SF networks. Finally, we observe that, as a free side effect of evolving segmentation and differentiation in combination, we obtained in-silico developmental mechanisms resembling mechanisms used in vertebrate development.

Robustness under functional constraint: the genetic network for temporal expression in Drosophila neurogenesis

Precise temporal coordination of gene expression is crucial for many developmental processes. One central question in developmental biology is how such coordinated expression patterns are robustly controlled. During embryonic development of the Drosophila central nervous system, neural stem cells called neuroblasts express a group of genes in a definite order, which leads to the diversity of cell types. We produced all possible regulatory networks of these genes and examined their expression dynamics numerically. From the analysis, we identified requisite regulations and predicted an unknown factor to reproduce known expression profiles caused by loss-of-function or overexpression of the genes in vivo, as well as in the wild type. Following this, we evaluated the stability of the actual Drosophila network for sequential expression. This network shows the highest robustness against parameter variations and gene expression fluctuations among the possible networks that reproduce the expression profiles. We propose a regulatory module composed of three types of regulations that is responsible for precise sequential expression. This study suggests that the Drosophila network for sequential expression has evolved to generate the robust temporal expression for neuronal specification.

Cell fate specification is of key importance in the development of multicellular organisms. To specify various cell fates correctly, genetic networks precisely coordinate spatial and temporal gene expression patterns during various developmental stages. One central question in developmental biology is to elucidate the relationship between the pattern formation and the network architecture. During embryonic development of the Drosophila central nervous system, the neural stem cells express a group of genes in a definite order, which is responsible for the diversity of neural cells. To elucidate the underlying mechanism of the process, we analyzed the structure and dynamics of the genetic network for the temporal changes occurring in the Drosophila neural stem cells. Searching all the possible regulatory networks of these genes using a computer program, we detected the requisite regulations that reproduce observed gene expression profiles. By comparing the stability of the dynamics among the functional networks, we uncovered the robust nature of the actual Drosophila network against environmental and intrinsic fluctuations. These results indicate that the genetic network for sequential expression has evolved to be robust under functional constraints. Our study proposes regulatory modules that are responsible for the precise sequential expressions, which might exist in genetic networks for other temporal patterning processes.

Spatial analysis of expression patterns predicts genetic interactions at the mid-hindbrain boundary

The isthmic organizer mediating differentiation of mid- and hindbrain during vertebrate development is characterized by a well-defined pattern of locally restricted gene expression domains around the mid-hindbrain boundary (MHB). This pattern is established and maintained by a regulatory network between several transcription and secreted factors that is not yet understood in full detail. In this contribution we show that a Boolean analysis of the characteristic spatial gene expression patterns at the murine MHB reveals key regulatory interactions in this network. Our analysis employs techniques from computational logic for the minimization of Boolean functions. This approach allows us to predict also the interplay of the various regulatory interactions. In particular, we predict a maintaining, rather than inducing, effect of Fgf8 on Wnt1 expression, an issue that remained unclear from published data. Using mouse anterior neural plate/tube explant cultures, we provide experimental evidence that Fgf8 in fact only maintains but does not induce ectopic Wnt1 expression in these explants. In combination with previously validated interactions, this finding allows for the construction of a regulatory network between key transcription and secreted factors at the MHB. Analyses of Boolean, differential equation and reaction-diffusion models of this network confirm that it is indeed able to explain the stable maintenance of the MHB as well as time-courses of expression patterns both under wild-type and various knock-out conditions. In conclusion, we demonstrate that similar to temporal also spatial expression patterns can be used to gain information about the structure of regulatory networks. We show, in particular, that the spatial gene expression patterns around the MHB help us to understand the maintenance of this boundary on a systems level.

Understanding brain formation during development is a tantalizing challenge. It is also essential for the fight against neurodegenerative diseases. In vertebrates, the central nervous system arises from a structure called the neural plate. This tissue is divided into four regions, which continue to develop into forebrain, midbrain, hindbrain and spinal cord. Interactions between locally expressed genes and signaling molecules are responsible for this patterning. Two key signaling molecules in this process are Fgf8 and Wnt1 proteins. They are secreted from a signaling center located at the boundary between prospective mid- and hindbrain (mid-hindbrain boundary, MHB) and mediate development of these two brain regions. Here, we logically analyze the spatial gene expression patterns at the MHB and predict interactions involved in the differentiation of mid- and hindbrain. In particular, our analysis indicates that Wnt1 depends on Fgf8 for stable maintenance. A time-course analysis of Wnt1 expression after implantation of Fgf8-coated beads in mouse neural plate/tube explants experimentally validates our prediction about the interactions between these two key patterning molecules. Subsequently, we demonstrate that available data allows construction of a mathematical model able to explain the maintenance of the signaling center at the MHB. We begin to understand this small aspect of brain formation on a systems level.

Modelling the Drosophila embryo

I provide a historical overview on the use of mathematical models to gain insight into pattern formation during early development of the fruit fly Drosophila melanogaster. It is my intention to illustrate how the aims and methodology of modelling have changed from the early beginnings of a theoretical developmental biology in the 1960s to modern-day systems biology. I show that even early modelling attempts addressed interesting and relevant questions, which were not tractable by experimental approaches. Unfortunately, their validation was severely hampered by a lack of specificity and appropriate experimental evidence. There is a simple lesson to be learned from this: we cannot deduce general rules for pattern formation from first principles or spurious reproduction of developmental phenomena. Instead, we must infer such rules (if any) from detailed and accurate studies of specific developmental systems. To achieve this, mathematical modelling must be closely integrated with experimental approaches. I report on progress that has been made in this direction in the past few years and illustrate the kind of novel insights that can be gained from such combined approaches. These insights demonstrate the great potential (and some pitfalls) of an integrative, systems-level investigation of pattern formation.

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.

Control and signal processing by transcriptional interference

A transcriptional activator can suppress gene expression by interfering with transcription initiated by another activator. Transcriptional interference has been increasingly recognized as a regulatory mechanism of gene expression. The signals received by the two antagonistically acting activators are combined by the polymerase trafficking along the DNA. We have designed a dual-control genetic system in yeast to explore this antagonism systematically. Antagonism by an upstream activator bears the hallmarks of competitive inhibition, whereas a downstream activator inhibits gene expression non-competitively. When gene expression is induced weakly, the antagonistic activator can have a positive effect and can even trigger paradoxical activation. Equilibrium and non-equilibrium models of transcription shed light on the mechanism by which interference converts signals, and reveals that self-antagonism of activators imitates the behavior of feed-forward loops. Indeed, a synthetic circuit generates a bell-shaped response, so that the induction of expression is limited to a narrow range of the input signal. The identification of conserved regulatory principles of interference will help to predict the transcriptional response of genes in their genomic context.

Discrete dynamic modeling with asynchronous update, or how to model complex systems in the absence of quantitative information

A major aim of systems biology is the study of the inter-relationships found within and between large biological data sets. Here we describe one systems biology method, in which the tools of network analysis and discrete dynamic (Boolean) modeling are used to develop predictive models of cellular signaling in cases where detailed temporal and kinetic information regarding the propagation of the signal through the system is lacking. This approach is also applicable to data sets derived from some other types of biological systems, such as transcription factor-mediated regulation of gene expression during the control of developmental fate, or host defense responses following pathogen attack, and is equally applicable to plant and non-plant systems. The method also allows prediction of how elimination of one or more individual signaling components will affect the ultimate outcome, thus allowing the researcher to model the effects of genetic knockout or pharmacological block. The method also serves as a starting point from which more quantitative models can be developed as additional information becomes available.

Boolean network simulations for life scientists

Modern life sciences research increasingly relies on computational solutions, from large scale data analyses to theoretical modeling. Within the theoretical models Boolean networks occupy an increasing role as they are eminently suited at mapping biological observations and hypotheses into a mathematical formalism. The conceptual underpinnings of Boolean modeling are very accessible even without a background in quantitative sciences, yet it allows life scientists to describe and explore a wide range of surprisingly complex phenomena. In this paper we provide a clear overview of the concepts used in Boolean simulations, present a software library that can perform these simulations based on simple text inputs and give three case studies. The large scale simulations in these case studies demonstrate the Boolean paradigms and their applicability as well as the advanced features and complex use cases that our software package allows. Our software is distributed via a liberal Open Source license and is freely accessible from

Implementing arithmetic and other analytic operations by transcriptional regulation

The transcriptional regulatory machinery of a gene can be viewed as a computational device, with transcription factor concentrations as inputs and expression level as the output. This view begs the question: what kinds of computations are possible? We show that different parameterizations of a simple chemical kinetic model of transcriptional regulation are able to approximate all four standard arithmetic operations: addition, subtraction, multiplication, and division, as well as various equality and inequality operations. This contrasts with other studies that emphasize logical or digital notions of computation in biological networks. We analyze the accuracy and precision of these approximations, showing that they depend on different sets of parameters, and are thus independently tunable. We demonstrate that networks of these "arithmetic" genes can be combined to accomplish yet more complicated computations by designing and simulating a network that detects statistically significant elevations in a time-varying signal. We also consider the much more general problem of approximating analytic functions, showing that this can be achieved by allowing multiple transcription factor binding sites on the promoter. These observations are important for the interpretation of naturally occurring networks and imply new possibilities for the design of synthetic networks.

Symbolic modeling of genetic regulatory networks

Understanding the functioning of genetic regulatory networks supposes a modeling of biological processes in order to simulate behaviors and to reason on the model. Unfortunately, the modeling task is confronted to incomplete knowledge about the system. To deal with this problem we propose a methodology that uses the qualitative approach developed by Thomas. A symbolic transition system can represent the set of all possible models in a concise and symbolic way. We introduce a new method based on model-checking techniques and symbolic execution to extract constraints on parameters leading to dynamics coherent with known behaviors. Our method allows us to efficiently respond to two kinds of questions: is there any model coherent with a certain hypothetic behavior? Are there behaviors common to all selected models? The first question is illustrated with the example of the mucus production in Pseudomonas aeruginosa while the second one is illustrated with the example of immunity control in bacteriophage lambda.

Dynamical modeling of syncytial mitotic cycles in Drosophila embryos

Immediately following fertilization, the fruit fly embryo undergoes 13 rapid, synchronous, syncytial nuclear division cycles driven by maternal genes and proteins. During these mitotic cycles, there are barely detectable oscillations in the total level of B-type cyclins. In this paper, we propose a dynamical model for the molecular events underlying these early nuclear division cycles in Drosophila. The model distinguishes nuclear and cytoplasmic compartments of the embryo and permits exploration of a variety of rules for protein transport between the compartments. Numerical simulations reproduce the main features of wild-type mitotic cycles: patterns of protein accumulation and degradation, lengthening of later cycles, and arrest in interphase 14. The model is consistent with mutations that introduce subtle changes in the number of mitotic cycles before interphase arrest. Bifurcation analysis of the differential equations reveals the dependence of mitotic oscillations on cycle number, and how this dependence is altered by mutations. The model can be used to predict the phenotypes of novel mutations and effective ranges of the unmeasured rate constants and transport coefficients in the proposed mechanism.

Three-dimensional morphology and gene expression in the Drosophila blastoderm at cellular resolution I: data acquisition pipeline

A suite of methods that provide the first quantitative three-dimensional description of gene expression and morphology with cellular resolution in whole Drosophila embryos is described.

Background

To model and thoroughly understand animal transcription networks, it is essential to derive accurate spatial and temporal descriptions of developing gene expression patterns with cellular resolution.

Results

Here we describe a suite of methods that provide the first quantitative three-dimensional description of gene expression and morphology at cellular resolution in whole embryos. A database containing information derived from 1,282 embryos is released that describes the mRNA expression of 22 genes at multiple time points in the Drosophila blastoderm. We demonstrate that our methods are sufficiently accurate to detect previously undescribed features of morphology and gene expression. The cellular blastoderm is shown to have an intricate morphology of nuclear density patterns and apical/basal displacements that correlate with later well-known morphological features. Pair rule gene expression stripes, generally considered to specify patterning only along the anterior/posterior body axis, are shown to have complex changes in stripe location, stripe curvature, and expression level along the dorsal/ventral axis. Pair rule genes are also found to not always maintain the same register to each other.

Conclusion

The application of these quantitative methods to other developmental systems will likely reveal many other previously unknown features and provide a more rigorous understanding of developmental regulatory networks.

Dynamical analysis of the regulatory network defining the dorsal-ventral boundary of the Drosophila wing imaginal disc

The larval development of the Drosophila melanogaster wings is organized by the protein Wingless, which is secreted by cells adjacent to the dorsal-ventral (DV) boundary. Two signaling processes acting between the second and early third instars and between the mid- and late third instar control the expression of Wingless in these boundary cells. Here, we integrate both signaling processes into a logical multivalued model encompassing four cells, i.e., a boundary and a flanking cell at each side of the boundary. Computer simulations of this model enable a qualitative reproduction of the main wild-type and mutant phenotypes described in the experimental literature. During the first signaling process, Notch becomes activated by the first signaling process in an Apterous-dependent manner. In silico perturbation experiments show that this early activation of Notch is unstable in the absence of Apterous. However, during the second signaling process, the Notch pattern becomes consolidated, and thus independent of Apterous, through activation of the paracrine positive feedback circuit of Wingless. Consequently, we propose that appropriate delays for Apterous inactivation and Wingless induction by Notch are crucial to maintain the wild-type expression at the dorsal-ventral boundary. Finally, another mutant simulation shows that cut expression might be shifted to late larval stages because of a potential interference with the early signaling process.

A method for the generation of standardized qualitative dynamical systems of regulatory networks

BACKGROUND

Modeling of molecular networks is necessary to understand their dynamical properties. While a wealth of information on molecular connectivity is available, there are still relatively few data regarding the precise stoichiometry and kinetics of the biochemical reactions underlying most molecular networks. This imbalance has limited the development of dynamical models of biological networks to a small number of well-characterized systems. To overcome this problem, we wanted to develop a methodology that would systematically create dynamical models of regulatory networks where the flow of information is known but the biochemical reactions are not. There are already diverse methodologies for modeling regulatory networks, but we aimed to create a method that could be completely standardized, i.e. independent of the network under study, so as to use it systematically.

RESULTS

We developed a set of equations that can be used to translate the graph of any regulatory network into a continuous dynamical system. Furthermore, it is also possible to locate its stable steady states. The method is based on the construction of two dynamical systems for a given network, one discrete and one continuous. The stable steady states of the discrete system can be found analytically, so they are used to locate the stable steady states of the continuous system numerically. To provide an example of the applicability of the method, we used it to model the regulatory network controlling T helper cell differentiation.

CONCLUSION

The proposed equations have a form that permit any regulatory network to be translated into a continuous dynamical system, and also find its steady stable states. We showed that by applying the method to the T helper regulatory network it is possible to find its known states of activation, which correspond the molecular profiles observed in the precursor and effector cell types.

A method for the generation of standardized qualitative dynamical systems of regulatory networks

Background

Modeling of molecular networks is necessary to understand their dynamical properties. While a wealth of information on molecular connectivity is available, there are still relatively few data regarding the precise stoichiometry and kinetics of the biochemical reactions underlying most molecular networks. This imbalance has limited the development of dynamical models of biological networks to a small number of well-characterized systems. To overcome this problem, we wanted to develop a methodology that would systematically create dynamical models of regulatory networks where the flow of information is known but the biochemical reactions are not. There are already diverse methodologies for modeling regulatory networks, but we aimed to create a method that could be completely standardized, i.e. independent of the network under study, so as to use it systematically.

Results

We developed a set of equations that can be used to translate the graph of any regulatory network into a continuous dynamical system. Furthermore, it is also possible to locate its stable steady states. The method is based on the construction of two dynamical systems for a given network, one discrete and one continuous. The stable steady states of the discrete system can be found analytically, so they are used to locate the stable steady states of the continuous system numerically. To provide an example of the applicability of the method, we used it to model the regulatory network controlling T helper cell differentiation.

Conclusion

The proposed equations have a form that permit any regulatory network to be translated into a continuous dynamical system, and also find its steady stable states. We showed that by applying the method to the T helper regulatory network it is possible to find its known states of activation, which correspond the molecular profiles observed in the precursor and effector cell types.

Regulatory network for cell shape changes during Drosophila ventral furrow formation

Rapid and sequential cell shape changes take place during the formation of the ventral furrow (VF) at the beginning of Drosophila gastrulation. At the cellular level, this morphogenetic event demands close coordination of the proteins involved in actin cytoskeletal reorganization. In order to construct a regulatory network that describes these cell shape changes, we have used published genetic and molecular data for 18 genes encoding transcriptional regulators and signaling pathway components. Based on the dynamic behavior of this network we explored the hypothesis that the combination of three recognizable phenotypes describing wild type or mutant cell types, during VF invagination, correspond to different activation states of a specific set of these gene products, which are point attractors of the regulatory network. From our results, we recognize missing components in the regulatory network and suggest alternative pathways in the regulation of cell shape changes during VF formation.

Minimum information requested in the annotation of biochemical models (MIRIAM)

Most of the published quantitative models in biology are lost for the community because they are either not made available or they are insufficiently characterized to allow them to be reused. The lack of a standard description format, lack of stringent reviewing and authors' carelessness are the main causes for incomplete model descriptions. With today's increased interest in detailed biochemical models, it is necessary to define a minimum quality standard for the encoding of those models. We propose a set of rules for curating quantitative models of biological systems. These rules define procedures for encoding and annotating models represented in machine-readable form. We believe their application will enable users to (i) have confidence that curated models are an accurate reflection of their associated reference descriptions, (ii) search collections of curated models with precision, (iii) quickly identify the biological phenomena that a given curated model or model constituent represents and (iv) facilitate model reuse and composition into large subcellular models.

GINsim: a software suite for the qualitative modelling, simulation and analysis of regulatory networks

This paper presents GINsim, a Java software suite devoted to the qualitative modelling, analysis and simulation of genetic regulatory networks. Formally, our approach leans on discrete mathematical and graph-theoretical concepts. GINsim encompasses an intuitive graph editor, enabling the definition and the parameterisation of a regulatory graph, as well as a simulation engine to compute the corresponding qualitative dynamical behaviour. Our computational approach is illustrated by a preliminary model analysis of the inter-cellular regulatory network activating Notch at the dorsal-ventral boundary in the wing imaginal disc of Drosophila. We focus on the cross-regulations between five genes (within and between two cells), which implements the dorsal-ventral border in the developing imaginal disc. Our simulations qualitatively reproduce the wild-type developmental pathway, as well as the outcome of various types of experimental perturbations, such as loss-of-function mutations or ectopically induced gene expression.

A network model for the control of the differentiation process in Th cells

T helper cells differentiate from a precursor type, Th0, to either the Th1 or Th2 phenotypes. While a number of molecules are known to participate in this process, it is not completely understood how they regulate each other to ensure differentiation. This article presents the core regulatory network controlling the differentiation of Th cells, reconstructed from published molecular data. This network encompasses 17 nodes, namely IFN-gamma, IL-4, IL-12, IL-18, IFN-beta, IFN-gammaR, IL-4R, IL-12R, IL-18R, IFN-betaR, STAT-1, STAT-6, STAT-4, IRAK, SOCS-1, GATA-3, and T-bet, as well as their cross-regulatory interactions. The reconstructed network was modeled as a discrete dynamical system, and analyzed in terms of its constituent feedback loops. The stable steady states of the Th network model are consistent with the stable molecular patterns of activation observed in wild type and mutant Th0, Th1 and Th2 cells.