Biology by numbers: mathematical modelling in developmental biology

Making models match measurements: model optimization for morphogen patterning networks

Mathematical modeling of developmental signaling networks has played an increasingly important role in the identification of regulatory mechanisms by providing a sandbox for hypothesis testing and experiment design. Whether these models consist of an equation with a few parameters or dozens of equations with hundreds of parameters, a prerequisite to model-based discovery is to bring simulated behavior into agreement with observed data via parameter estimation. These parameters provide insight into the system (e.g., enzymatic rate constants describe enzyme properties). Depending on the nature of the model fit desired - from qualitative (relative spatial positions of phosphorylation) to quantitative (exact agreement of spatial position and concentration of gene products) - different measures of data-model mismatch are used to estimate different parameter values, which contain different levels of usable information and/or uncertainty. To facilitate the adoption of modeling as a tool for discovery alongside other tools such as genetics, immunostaining, and biochemistry, careful consideration needs to be given to how well a model fits the available data, what the optimized parameter values mean in a biological context, and how the uncertainty in model parameters and predictions plays into experiment design. The core discussion herein pertains to the quantification of model-to-data agreement, which constitutes the first measure of a model's performance and future utility to the problem at hand. Integration of this experimental data and the appropriate choice of objective measures of data-model agreement will continue to drive modeling forward as a tool that contributes to experimental discovery. The Drosophila melanogaster gap gene system, in which model parameters are optimized against in situ immunofluorescence intensities, demonstrates the importance of error quantification, which is applicable to a wide array of developmental modeling studies.

Do endothelial cells dream of eclectic shape?

Endothelial cells (ECs) exhibit dramatic plasticity of form at the single- and collective-cell level during new vessel growth, adult vascular homeostasis, and pathology. Understanding how, when, and why individual ECs coordinate decisions to change shape, in relation to the myriad of dynamic environmental signals, is key to understanding normal and pathological blood vessel behavior. However, this is a complex spatial and temporal problem. In this review we show that the multidisciplinary field of Adaptive Systems offers a refreshing perspective, common biological language, and straightforward toolkit that cell biologists can use to untangle the complexity of dynamic, morphogenetic systems.

Mathematical modeling of sub-cellular asymmetry of fat-dachsous heterodimer for generation of planar cell polarity

Planar Cell Polarity (PCP) is an evolutionarily conserved characteristic of animal tissues marked by coordinated polarization of cells or structures in the plane of a tissue. In insect wing epithelium, for instance, PCP is characterized by en masse orientation of hairs orthogonal to its apical-basal axis and pointing along the proximal-distal axis of the organ. Directional cue for PCP has been proposed to be generated by complex sets of interactions amongst three proteins - Fat (Ft), Dachsous (Ds) and Four-jointed (Fj). Ft and Ds are two atypical cadherins, which are phosphorylated by Fj, a Golgi kinase. Ft and Ds from adjacent cells bind heterophilically via their tandem cadherin repeats, and their binding affinities are regulated by Fj. Further, in the wing epithelium, sub-cellular levels of Ft-Ds heterodimers are seen to be elevated at the distal edges of individual cells, prefiguring their PCP. Mechanisms generating this sub-cellular asymmetry of Ft-Ds heterodimer in proximal and distal edges of cells, however, have not been resolved yet. Using a mathematical modeling approach, here we provide a framework for generation of this sub-cellular asymmetry of Ft-Ds heterodimer. First, we explain how the known interactions within Ft-Ds-Fj system translate into sub-cellular asymmetry of Ft-Ds heterodimer. Second, we show that this asymmetric localization of Ft-Ds heterodimer is lost when tissue-level gradient of Fj is flattened, or when phosphorylation of Ft by Fj is abolished, but not when tissue-level gradient of Ds is flattened or when phosphorylation of Ds is abrogated. Finally, we show that distal enrichment of Ds also amplifies Ft-Ds asymmetry. These observations reveal that gradient of Fj expression, phosphorylation of Ft by Fj and sub-cellular distal accumulation of Ds are three critical elements required for generating sub-cellular asymmetry of Ft-Ds heterodimer. Our model integrates the known experimental data and presents testable predictions for future studies.

Spatiotemporal control of gene expression using microfluidics

Accurate spatiotemporal regulation of genetic expression and cell microenvironment are both essential to epithelial morphogenesis during development, wound healing and cancer. In vivo, this is achieved through the interplay between intrinsic cellular properties and extrinsic signals. Amongst these, morphogen gradients induce specific concentration- and time-dependent gene expression changes that influence a target cell's fate. As systems biology attempts to understand the complex mechanisms underlying morphogenesis, the lack of experimental setup to recapitulate morphogen-induced patterning in vitro has become limiting. For this reason, we developed a versatile microfluidic-based platform to control the spatiotemporal delivery of chemical gradients to tissues grown in Petri dishes. Using this setup combined with a synthetic inducible gene expression system, we were able to restrict a target gene's expression within a confluent epithelium to bands of cells as narrow as four cell diameters with a one cell diameter accuracy. Applied to the targeted delivery of growth factor gradients to a confluent epithelium, this method further enabled the localized induction of epithelial-mesenchymal transitions and associated morphogenetic changes. Our approach paves the way for replicating in vitro the morphogen gradients observed in vivo to determine the relative contributions of known intrinsic and extrinsic factors in differential tissue patterning, during development and cancer. It could also be readily used to spatiotemporally control cell differentiation in ES/iPS cell cultures for re-engineering of complex tissues. Finally, the reversibility of the microfluidic chip assembly allows for pre- and post-treatment sample manipulations and extends the range of patternable samples to animal explants.

A computational model of the effect of gene misexpression on the development of cortical areas

Brain function depends on the specialisation of brain areas. In the murine cerebral cortex, the development of these areas depends on the coordinated expression of several genes in precise spatial patterns in the telencephalon during embryogenesis. Manipulating the expression of these genes during development alters the positions and sizes of cortical areas in the adult. Qualitative data also show that these genes regulate each other's expression during development so that they form a regulatory network with many feedback loops. However, it is currently unknown which regulatory interactions are critical to generating the correct expression patterns to lead to normal cortical development. Here, we formalise the relationships inferred from genetic manipulations into computational models. We simulate many different networks potentially consistent with the experimental data and show that a surprising diversity of networks produce similar results. This demonstrates that existing data cannot uniquely specify the network. We conclude by suggesting experiments necessary to constrain the model and help identify and understand the true structure of this regulatory network.

Mathematical modeling of regenerative processes

In many animals, regenerative processes can replace lost body parts. Organ and tissue regeneration consequently also hold great medical promise. The regulation of regenerative processes is achieved through concerted actions of multiple organizational levels of the organism, from diffusing molecules and cellular gene expression patterns up to tissue mechanics. Our intuition is usually not adapted well to this degree of complexity and the quantitative aspects of the regulation of regenerative processes remain poorly understood. One way out of this dilemma lies in the combination of experimentation and mathematical modeling within an iterative process of model development/refinement, model predictions for novel experimental conditions, quantitative experiments testing these predictions, and subsequent model refinement. This interdisciplinary approach has already provided key insights into smaller scale processes during embryonic development and a so-far limited number of more complex regeneration processes. This review discusses selected theoretical and interdisciplinary studies and is structured along the three phases of regeneration: (1) initiation of a regeneration response, (2) tissue patterning during regenerate growth, (3) arresting the regeneration response. Moreover, we highlight the opportunities provided by extensions of mathematical models from developmental processes toward the study of related regenerative processes.

Reconsidering the bio-detection of tolerance in renal transplantation

Operational tolerance in kidney transplantation tolerance is rare phenomenon. It concerns recipients who keep a good function of their graft without immunosuppressors for more than one year. A critical need in the field of transplantation tolerance is the identification of biomarkers able to detect precociously tolerance phenotype in stable recipient in order to adapt treatment and progressively stop immunosuppressive therapy. But many limitations in these studies slow the application in clinics of such tolerance signature. In this addendum article we talk about these limitations and potential new directions to improve our approach in the quest of tolerance biomarkers.

Stochastic modelling of biochemical systems of multi-step reactions using a simplified two-variable model

Background

A fundamental issue in systems biology is how to design simplified mathematical models for describing the dynamics of complex biochemical reaction systems. Among them, a key question is how to use simplified reactions to describe the chemical events of multi-step reactions that are ubiquitous in biochemistry and biophysics. To address this issue, a widely used approach in literature is to use one-step reaction to represent the multi-step chemical events. In recent years, a number of modelling methods have been designed to improve the accuracy of the one-step reaction method, including the use of reactions with time delay. However, our recent research results suggested that there are still deviations between the dynamics of delayed reactions and that of the multi-step reactions. Therefore, more sophisticated modelling methods are needed to accurately describe the complex biological systems in an efficient way.

Results

This work designs a two-variable model to simplify chemical events of multi-step reactions. In addition to the total molecule number of a species, we first introduce a new concept regarding the location of molecules in the multi-step reactions, which is the second variable to represent the system dynamics. Then we propose a simulation algorithm to compute the probability for the firing of the last step reaction in the multi-step events. This probability function is evaluated using a deterministic model of ordinary differential equations and a stochastic model in the framework of the stochastic simulation algorithm. The efficiency of the proposed two-variable model is demonstrated by the realization of mRNA degradation process based on the experimentally measured data.

Conclusions

Numerical results suggest that the proposed new two-variable model produces predictions that match the multi-step chemical reactions very well. The successful realization of the mRNA degradation dynamics indicates that the proposed method is a promising approach to reduce the complexity of biological systems.

Cellular resolution models for even skipped regulation in the entire Drosophila embryo

Transcriptional control ensures genes are expressed in the right amounts at the correct times and locations. Understanding quantitatively how regulatory systems convert input signals to appropriate outputs remains a challenge. For the first time, we successfully model even skipped (eve) stripes 2 and 3+7 across the entire fly embryo at cellular resolution. A straightforward statistical relationship explains how transcription factor (TF) concentrations define eve's complex spatial expression, without the need for pairwise interactions or cross-regulatory dynamics. Simulating thousands of TF combinations, we recover known regulators and suggest new candidates. Finally, we accurately predict the intricate effects of perturbations including TF mutations and misexpression. Our approach imposes minimal assumptions about regulatory function; instead we infer underlying mechanisms from models that best fit the data, like the lack of TF-specific thresholds and the positional value of homotypic interactions. Our study provides a general and quantitative method for elucidating the regulation of diverse biological systems. DOI:http://dx.doi.org/10.7554/eLife.00522.001.

Functional modelling of planar cell polarity: an approach for identifying molecular function

BACKGROUND

Cells in some tissues acquire a polarisation in the plane of the tissue in addition to apical-basal polarity. This polarisation is commonly known as planar cell polarity and has been found to be important in developmental processes, as planar polarity is required to define the in-plane tissue coordinate system at the cellular level.

RESULTS

We have built an in-silico functional model of cellular polarisation that includes cellular asymmetry, cell-cell signalling and a response to a global cue. The model has been validated and parameterised against domineering non-autonomous wing hair phenotypes in Drosophila.

CONCLUSIONS

We have carried out a systematic comparison of in-silico polarity phenotypes with patterns observed in vivo under different genetic manipulations in the wing. This has allowed us to classify the specific functional roles of proteins involved in generating cell polarity, providing new hypotheses about their specific functions, in particular for Pk and Dsh. The predictions from the model allow direct assignment of functional roles of genes from genetic mosaic analysis of Drosophila wings.

Multiscale computational models of complex biological systems

Integration of data across spatial, temporal, and functional scales is a primary focus of biomedical engineering efforts. The advent of powerful computing platforms, coupled with quantitative data from high-throughput experimental methodologies, has allowed multiscale modeling to expand as a means to more comprehensively investigate biological phenomena in experimentally relevant ways. This review aims to highlight recently published multiscale models of biological systems, using their successes to propose the best practices for future model development. We demonstrate that coupling continuous and discrete systems best captures biological information across spatial scales by selecting modeling techniques that are suited to the task. Further, we suggest how to leverage these multiscale models to gain insight into biological systems using quantitative biomedical engineering methods to analyze data in nonintuitive ways. These topics are discussed with a focus on the future of the field, current challenges encountered, and opportunities yet to be realized.

Bridging the genotype-phenotype gap: what does it take?

The genotype-phenotype map (GP map) concept applies to any time point in the ontogeny of a living system. It is the outcome of very complex dynamics that include environmental effects, and bridging the genotype-phenotype gap is synonymous with understanding these dynamics. The context for this understanding is physiology, and the disciplinary goals of physiology do indeed demand the physiological community to seek this understanding. We claim that this task is beyond reach without use of mathematical models that bind together genetic and phenotypic data in a causally cohesive way. We provide illustrations of such causally cohesive genotype-phenotype models where the phenotypes span from gene expression profiles to development of whole organs. Bridging the genotype-phenotype gap also demands that large-scale biological ('omics') data and associated bioinformatics resources be more effectively integrated with computational physiology than is currently the case. A third major element is the need for developing a phenomics technology way beyond current state of the art, and we advocate the establishment of a Human Phenome Programme solidly grounded on biophysically based mathematical descriptions of human physiology.

A multiscale road map of cancer spheroids – incorporating experimental and mathematical modelling to understand cancer progression

Computational models represent a highly suitable framework, not only for testing biological hypotheses and generating new ones but also for optimising experimental strategies. As one surveys the literature devoted to cancer modelling, it is obvious that immense progress has been made in applying simulation techniques to the study of cancer biology, although the full impact has yet to be realised. For example, there are excellent models to describe cancer incidence rates or factors for early disease detection, but these predictions are unable to explain the functional and molecular changes that are associated with tumour progression. In addition, it is crucial that interactions between mechanical effects, and intracellular and intercellular signalling are incorporated in order to understand cancer growth, its interaction with the extracellular microenvironment and invasion of secondary sites. There is a compelling need to tailor new, physiologically relevant in silico models that are specialised for particular types of cancer, such as ovarian cancer owing to its unique route of metastasis, which are capable of investigating anti-cancer therapies, and generating both qualitative and quantitative predictions. This Commentary will focus on how computational simulation approaches can advance our understanding of ovarian cancer progression and treatment, in particular, with the help of multicellular cancer spheroids, and thus, can inform biological hypothesis and experimental design.

The mechanistic basis of Myxococcus xanthus rippling behavior and its physiological role during predation

Myxococcus xanthus cells self-organize into periodic bands of traveling waves, termed ripples, during multicellular fruiting body development and predation on other bacteria. To investigate the mechanistic basis of rippling behavior and its physiological role during predation by this Gram-negative soil bacterium, we have used an approach that combines mathematical modeling with experimental observations. Specifically, we developed an agent-based model (ABM) to simulate rippling behavior that employs a new signaling mechanism to trigger cellular reversals. The ABM has demonstrated that three ingredients are sufficient to generate rippling behavior: (i) side-to-side signaling between two cells that causes one of the cells to reverse, (ii) a minimal refractory time period after each reversal during which cells cannot reverse again, and (iii) physical interactions that cause the cells to locally align. To explain why rippling behavior appears as a consequence of the presence of prey, we postulate that prey-associated macromolecules indirectly induce ripples by stimulating side-to-side contact-mediated signaling. In parallel to the simulations, M. xanthus predatory rippling behavior was experimentally observed and analyzed using time-lapse microscopy. A formalized relationship between the wavelength, reversal time, and cell velocity has been predicted by the simulations and confirmed by the experimental data. Furthermore, the results suggest that the physiological role of rippling behavior during M. xanthus predation is to increase the rate of spreading over prey cells due to increased side-to-side contact-mediated signaling and to allow predatory cells to remain on the prey longer as a result of more periodic cell motility.

Myxococcus xanthus cells collectively move on solid surfaces and reorganize their colonies in response to environmental cues. Under some conditions, cells exhibit an intriguing form of collective motility by self-organizing into bands of travelling alternating-density waves termed ripples. These waves are distinct from the waves originating from Turing instability in diffusion-reaction systems, as these counter-traveling waves do not annihilate but appear to pass through each other. Here we developed a new mathematical model of rippling behavior based on a recently observed contact signaling mechanism – cells that make side-to-side contacts can signal one another to reverse. We hypothesize that this signaling is enhanced by the presence of prey-associated macromolecules and compare modeling predictions with experimentally observed waves generated on E. coli prey cells. The model predicts a modified relationship between the wavelength and individual predatory cell motility parameters and provides a physiological role for rippling during predation. We show that ripples allow predatory cells to increase the rate of their spreading to quickly envelope the prey, and subsequently to decrease their random drift to remain in the prey region for longer. These and other predictions are confirmed by the experimental observations.

Computational approaches to developmental patterning

Computational approaches are breaking new ground in understanding how embryos form. Here, we discuss recent studies that couple precise measurements in the embryo with appropriately matched modeling and computational methods to investigate classic embryonic patterning strategies. We include signaling gradients, activator-inhibitor systems, and coupled oscillators, as well as emerging paradigms such as tissue deformation. Parallel progress in theory and experiment will play an increasingly central role in deciphering developmental patterning.

Direct comparison of small RNA and transcription factor signaling

Small RNAs (sRNAs) and proteins acting as transcription factors (TFs) are the principal components of gene networks. These two classes of signaling molecules have distinct mechanisms of action; sRNAs control mRNA translation, whereas TFs control mRNA transcription. Here, we directly compare the properties of sRNA and TF signaling using mathematical models and synthetic gene circuits in Escherichia coli. We show the abilities of sRNAs to act on existing target mRNAs (as opposed to TFs, which alter the production of future target mRNAs) and, without needing to be first translated, have surprisingly little impact on the dynamics. Instead, the dynamics are primarily determined by the clearance rates, steady-state concentrations and response curves of the sRNAs and TFs; these factors determine the time delay before a target gene’s expression can maximally respond to changes in sRNA and TF transcription. The findings are broadly applicable to the analysis of signaling in gene networks, and we demonstrate that they can be used to rationally reprogram the dynamics of synthetic circuits.

Understanding signaling dynamics through synthesis

Tissue-scale organization emerges from the action of sophisticated multiscale developmental programs. But the design rules for composing elementary signaling and information processing modules into such functional systems and for integrating them into the noisy and convoluted living context remain incompletely addressed. The construction of a synthetic gene circuit encoding contact-dependent signal propagation demonstrates one broadly applicable approach to this problem. The circuit comprises orthogonal signaling through the Delta ligand and the Notch receptor, multicellular positive feedback, and transcriptional signal amplification. Positive feedback and contact signaling proved sufficient for bistability and signal propagation across a population of mammalian cells, but only when combined with signal amplification. Thus, construction and characterization of synthetic gene circuits have made it possible to establish mechanistic sufficiency and the minimal requirements for the phenotype of interest.

On the evolution of morphogenetic models: mechano-chemical interactions and an integrated view of cell differentiation, growth, pattern formation and morphogenesis

In the 1950s, embryology was conceptualized as four relatively independent problems: cell differentiation, growth, pattern formation and morphogenesis. The mechanisms underlying the first three traditionally have been viewed as being chemical in nature, whereas those underlying morphogenesis have usually been discussed in terms of mechanics. Often, morphogenesis and its mechanical processes have been regarded as subordinate to chemical ones. However, a growing body of evidence indicates that the biomechanics of cells and tissues affect in striking ways those phenomena often thought of as mainly under the control of cell-cell signalling. This accumulation of data has led to a revival of the mechano-transduction concept in particular, and of complexity in general, causing us now to consider whether we should retain the traditional conceptualization of development. The researchers' semantic preferences for the terms 'patterning', 'pattern formation' or 'morphogenesis' can be used to describe three main 'schools of thought' which emerged in the late 1970s. In the 'molecular school', the term patterning is deeply tied to the positional information concept. In the 'chemical school', the term 'pattern formation' regularly implies reaction-diffusion models. In the 'mechanical school', the term 'morphogenesis' is more frequently used in relation to mechanical instabilities. Major differences among these three schools pertain to the concept of self-organization, and models can be classified as morphostatic or morphodynamic. Various examples illustrate the distorted picture that arises from the distinction among differentiation, growth, pattern formation and morphogenesis, based on the idea that the underlying mechanisms are respectively chemical or mechanical. Emerging quantitative approaches integrate the concepts and methods of complex sciences and emphasize the interplay between hierarchical levels of organization via mechano-chemical interactions. They draw upon recent improvements in mathematical and numerical morphogenetic models and upon considerable progress in collecting new quantitative data. This review highlights a variety of such models, which exhibit important advances, such as hybrid, stochastic and multiscale simulations.

Heterogeneity reduces sensitivity of cell death for TNF-stimuli

Background

Apoptosis is a form of programmed cell death essential for the maintenance of homeostasis and the removal of potentially damaged cells in multicellular organisms. By binding its cognate membrane receptor, TNF receptor type 1 (TNF-R1), the proinflammatory cytokine Tumor Necrosis Factor (TNF) activates pro-apoptotic signaling via caspase activation, but at the same time also stimulates nuclear factor κB (NF-κB)-mediated survival pathways. Differential dose-response relationships of these two major TNF signaling pathways have been described experimentally and using mathematical modeling. However, the quantitative analysis of the complex interplay between pro- and anti-apoptotic signaling pathways is an open question as it is challenging for several reasons: the overall signaling network is complex, various time scales are present, and cells respond quantitatively and qualitatively in a heterogeneous manner.

Results

This study analyzes the complex interplay of the crosstalk of TNF-R1 induced pro- and anti-apoptotic signaling pathways based on an experimentally validated mathematical model. The mathematical model describes the temporal responses on both the single cell level as well as the level of a heterogeneous cell population, as observed in the respective quantitative experiments using TNF-R1 stimuli of different strengths and durations. Global sensitivity of the heterogeneous population was quantified by measuring the average gradient of time of death versus each population parameter. This global sensitivity analysis uncovers the concentrations of Caspase-8 and Caspase-3, and their respective inhibitors BAR and XIAP, as key elements for deciding the cell's fate. A simulated knockout of the NF-κB-mediated anti-apoptotic signaling reveals the importance of this pathway for delaying the time of death, reducing the death rate in the case of pulse stimulation and significantly increasing cell-to-cell variability.

Conclusions

Cell ensemble modeling of a heterogeneous cell population including a global sensitivity analysis presented here allowed us to illuminate the role of the different elements and parameters on apoptotic signaling. The receptors serve to transmit the external stimulus; procaspases and their inhibitors control the switching from life to death, while NF-κB enhances the heterogeneity of the cell population. The global sensitivity analysis of the cell population model further revealed an unexpected impact of heterogeneity, i.e. the reduction of parametric sensitivity.

Statistical analysis of nonlinear dynamical systems using differential geometric sampling methods

Mechanistic models based on systems of nonlinear differential equations can help provide a quantitative understanding of complex physical or biological phenomena. The use of such models to describe nonlinear interactions in molecular biology has a long history; however, it is only recently that advances in computing have allowed these models to be set within a statistical framework, further increasing their usefulness and binding modelling and experimental approaches more tightly together. A probabilistic approach to modelling allows us to quantify uncertainty in both the model parameters and the model predictions, as well as in the model hypotheses themselves. In this paper, the Bayesian approach to statistical inference is adopted and we examine the significant challenges that arise when performing inference over nonlinear ordinary differential equation models describing cell signalling pathways and enzymatic circadian control; in particular, we address the difficulties arising owing to strong nonlinear correlation structures, high dimensionality and non-identifiability of parameters. We demonstrate how recently introduced differential geometric Markov chain Monte Carlo methodology alleviates many of these issues by making proposals based on local sensitivity information, which ultimately allows us to perform effective statistical analysis. Along the way, we highlight the deep link between the sensitivity analysis of such dynamic system models and the underlying Riemannian geometry of the induced posterior probability distributions.

[The discussion of the infiltrative model of mathematical knowledge to genetics teaching]

Genetics, the core course of biological field, is an importance major-basic course in curriculum of many majors related with biology. Due to strong theoretical and practical as well as abstract of genetics, it is too difficult to study on genetics for many students. At the same time, mathematics is one of the basic courses in curriculum of the major related natural science, which has close relationship with the establishment, development and modification of genetics. In this paper, to establish the intrinsic logistic relationship and construct the integral knowledge network and to help students improving the analytic, comprehensive and logistic abilities, we applied some mathematical infiltrative model genetic knowledge in genetics teaching, which could help students more deeply learn and understand genetic knowledge.

Exact and approximate distributions of protein and mRNA levels in the low-copy regime of gene expression

Gene expression at the single-cell level incorporates reaction mechanisms which are intrinsically stochastic as they involve molecular species present at low copy numbers. The dynamics of these mechanisms can be described quantitatively using stochastic master-equation modelling; in this paper we study a generic gene-expression model of this kind which explicitly includes the representations of the processes of transcription and translation. For this model we determine the generating function of the steady-state distribution of mRNA and protein counts and characterise the underlying probability law using a combination of analytic, asymptotic and numerical approaches, finding that the distribution may assume a number of qualitatively distinct forms. The results of the analysis are suitable for comparison with single-molecule resolution gene-expression data emerging from recent experimental studies.

Nutritional resources as positional information for morphogenesis in the stony coral Stylophora pistillata

We are interested in deciphering the mechanisms for morphogenesis in the Red Sea scleractinian coral Stylophora pistillata with the help of mathematical models. Previous mathematical models for coral morphogenesis assume that skeletal growth is proportional to the amount of locally available energetic resources like diffusible nutrients and photosynthetic products. We introduce a new model which includes factors like dissolved nutrients and photosynthates, but these resources do not serve as building blocks for growth but rather provide some kind of positional information for coral morphogenesis. Depending on this positional information side branches are generated, splittings of branches take place and branch growth direction is determined. The model results are supported by quantitative comparisons with experimental data obtained from young coral colonies.

Single-cell quantification of IL-2 response by effector and regulatory T cells reveals critical plasticity in immune response

The sensitivity of T cells to interleukin-2 (IL-2) can vary by three orders of magnitude and is determined by the surface densities of the IL-2 receptor α subunits.

Regulatory T cells inflict a double hit on effector T cells by lowering the bulk IL-2 concentration as well as the sensitivity of effector T cells to this crucial cytokine.

This double hit deprives weakly activated effector T cells of pSTAT5 survival signals while having only minimal effects on strongly activated effector cells that express increased levels of the IL-2 receptor.

Short-term signaling differences lead to a differential functional in terms of proliferation and cell division: regulatory T cell specifically suppress weakly activated effector T cells even at large numbers; small numbers of strongly activated effector T cells overcome the suppression.

Self-/non-self-discrimination in the adaptive immune system relies, to a large extent, on distinctions between self-antigens and foreign antigens as made by individual T cells. As such, single-cell decisions are prone to errors a reliable immune response can be expected to incorporate further proofreading schemes. One such scheme involves long time scale, population-level interactions between effector (T_eff) and regulatory (T_reg) T cells. T_reg cells are often described as immune suppressors; their role as immune regulators can be understood by mapping out the scenarios in which T_reg suppression is either significant or insignificant.

In this study, we have focused on one mechanism that allows T_reg cells to suppress T_eff survival, namely, interleukin-2 (IL-2) deprivation. Following antigen activation, T_eff cells secrete IL-2 and express the α subunit of the IL-2 receptor (IL-2r). The binding of extracellular IL-2 to the IL-2r is crucial for T_eff survival and proliferation and consequently for a full-blown immune response. T_reg cells deplete this IL-2 from the environment and deprive the T_eff cells of this important survival signal. In this tug-of-war for IL-2, we sought to quantitatively describe those scenarios in which IL-2 uptake by T_reg cells suffices to suppress T_eff cell activation and those where it does not.

The core of this competition for IL-2 lies in the fact that IL-2rα is expressed on both T_eff and T_reg cells. To understand how IL-2 binds to its receptor, we measured IL-2r subunit levels on single cells, together with STAT5 phosphorylation as evoked by varied IL-2. Contrary to previous descriptions that set the EC₅₀ of IL-2/IL-2r interaction at 10 pM, we found that the sensitivity of T cells to IL-2 varies over three orders of magnitude concentrations (Figure 1E, experiment). T_eff cells with higher levels of IL-2rα receptor subunit are more sensitive to IL-2, T_reg cells with higher levels of IL-2rα are more efficient in the scavenging of IL-2. IL-2rβ levels, on the other hand, determine response amplitudes. We describe a short time scale, two-step model to quantitatively describe IL-2 binding onto individual cells (Figure 1E, theory).

IL-2r expression levels are therefore a crucial parameter for determining the outcome of the competition for IL-2. We measured the regulation of IL-2r subunits on longer time scales in cultures of either T_eff or T_reg cells. For both cell types, IL-2r levels depend on the exposure to IL-2. For T_eff cells, there is a further dependence on the concentration of antigen by which they were activated.

We then measured IL-2r expression in cocultures of T_reg and T_eff cells. We show how IL-2 secreted by activated T_eff cells suffices in inducing IL-2rα upregulation in the T_reg population. We further show that the presence of T_reg cells decreased IL-2r upregulation in cocultured weakly activated T_eff cells. T_reg cells thus inflict a double hit on T_eff cells by reducing not only extracellular IL-2 concentrations but also the T_eff cells' ability to sense IL-2. T_eff cells activated by high-antigen concentrations exhibit sustained IL-2rα expression that is less prone to this effect. We compared IL-2r levels on T_reg cells and T_eff activated by varied antigen concentrations and found a critical crossover: at low-antigen concentrations T_reg cells have higher IL-2rα than T_eff cells, but this is reversed at high-antigen concentrations.

We constructed a long time scale computational model to quantify the significance of this crossover. The model describes IL-2/IL-2r binding and the regulation of IL-2 and IL-2r expression in populations of T_reg and T_eff cells. For a pure T_eff population, our model predicted a ‘quorum-sensing' threshold implying that sustained pSTAT5 signaling requires a minimal concentration of cells that increases with decreasing activation strength. The model further predicts that the addition of T_reg cells will greatly increase the quorum concentration for weakly activated T_eff cells but have no effect on strongly activated T_eff cells.

We validated the model's predictions in vitro. We show a quorum-sensing threshold for activated T_eff cells. We also show that the presence of a T_reg population suppressed pSTAT5 signaling in a large number of weakly but has little effect on even a few strongly activated T_eff cells (Figure 6C and D). On longer time scales, this translates to the suppression of cell division (Figure 6G and H) and proliferation (Figure 6I) in a manner that discriminates between strongly and weakly activated cells.

We then went to demonstrate that IL-2 deprivation by T_reg cells takes place in vivo. We used IL-2 injections to upregulate IL-2rα levels in T_reg cells. As predicted by our in vitro results, such treatment leads to a suppressive environment in which T_eff cells activated by subsequent antigen/LPS immunization proliferate to a lesser extent. We were able to reverse this suppressive effect by continuing IL-2 treatment post-immunization. This highlights IL-2 as a limiting factor for T_eff proliferation and renders its scavenging by T_reg cells an important mechanism of suppression in vivo.

In conclusion, we formulated a quantitative description of IL-2/IL-2r regulation in mixed population of T_reg and T_eff cells. Population feedback loops that depend on cell numbers, molecular cell surface densities, free molecular densities and timing critically affect the outcome of the competition for IL-2. Such a description allows us to precisely identify the scenarios in which IL-2 deprivation by T_reg cells has a major suppressive role in vitro and better understand the role of this mechanism in vivo.

Understanding how the immune system decides between tolerance and activation by antigens requires addressing cytokine regulation as a highly dynamic process. We quantified the dynamics of interleukin-2 (IL-2) signaling in a population of T cells during an immune response by combining in silico modeling and single-cell measurements in vitro. We demonstrate that IL-2 receptor expression levels vary widely among T cells creating a large variability in the ability of the individual cells to consume, produce and participate in IL-2 signaling within the population. Our model reveals that at the population level, these heterogeneous cells are engaged in a tug-of-war for IL-2 between regulatory (T_reg) and effector (T_eff) T cells, whereby access to IL-2 can either increase the survival of T_eff cells or the suppressive capacity of T_reg cells. This tug-of-war is the mechanism enforcing, at the systems level, a core function of T_reg cells, namely the specific suppression of survival signals for weakly activated T_eff cells but not for strongly activated cells. Our integrated model yields quantitative, experimentally validated predictions for the manipulation of T_reg suppression.

Modeling the control of planar cell polarity

A growing list of medically important developmental defects and disease mechanisms can be traced to disruption of the planar cell polarity (PCP) pathway. The PCP system polarizes cells in epithelial sheets along an axis orthogonal to their apical-basal axis. Studies in the fruitfly, Drosophila, have suggested that components of the PCP signaling system function in distinct modules, and that these modules and the effector systems with which they interact function together to produce emergent patterns. Experimental methods allow the manipulation of individual PCP signaling molecules in specified groups of cells; these interventions not only perturb the polarization of the targeted cells at a subcellular level, but also perturb patterns of polarity at the multicellular level, often affecting nearby cells in characteristic ways. These kinds of experiments should, in principle, allow one to infer the architecture of the PCP signaling system, but the relationships between molecular interactions and tissue-level pattern are sufficiently complex that they defy intuitive understanding. Mathematical modeling has been an important tool to address these problems. This article explores the emergence of a local signaling hypothesis, and describes how a local intercellular signal, coupled with a directional cue, can give rise to global pattern. We will discuss the critical role mathematical modeling has played in guiding and interpreting experimental results, and speculate about future roles for mathematical modeling of PCP. Mathematical models at varying levels of inhibition have and are expected to continue contributing in distinct ways to understanding the regulation of PCP signaling.

Disturbance analysis of nonlinear differential equation models of genetic SUM regulatory networks

Noise disturbances and time delays are frequently met in cellular genetic regulatory systems. This paper is concerned with the disturbance analysis of a class of genetic regulatory networks described by nonlinear differential equation models. The mechanisms of genetic regulatory networks to amplify (attenuate) external disturbance are explored, and a simple measure of the amplification (attenuation) level is developed from a nonlinear robust control point of view. It should be noted that the conditions used to measure the disturbance level are delay-independent or delay-dependent, and are expressed within the framework of linear matrix inequalities, which can be characterized as convex optimization, and computed by the interior-point algorithm easily. Finally, by the proposed method, a numerical example is provided to illustrate how to measure the attenuation of proteins in the presence of external disturbances.

Towards a unifying, systems biology understanding of large-scale cellular death and destruction caused by poorly liganded iron: Parkinson's, Huntington's, Alzheimer's, prions, bactericides, chemical toxicology and others as examples

Exposure to a variety of toxins and/or infectious agents leads to disease, degeneration and death, often characterised by circumstances in which cells or tissues do not merely die and cease to function but may be more or less entirely obliterated. It is then legitimate to ask the question as to whether, despite the many kinds of agent involved, there may be at least some unifying mechanisms of such cell death and destruction. I summarise the evidence that in a great many cases, one underlying mechanism, providing major stresses of this type, entails continuing and autocatalytic production (based on positive feedback mechanisms) of hydroxyl radicals via Fenton chemistry involving poorly liganded iron, leading to cell death via apoptosis (probably including via pathways induced by changes in the NF-κB system). While every pathway is in some sense connected to every other one, I highlight the literature evidence suggesting that the degenerative effects of many diseases and toxicological insults converge on iron dysregulation. This highlights specifically the role of iron metabolism, and the detailed speciation of iron, in chemical and other toxicology, and has significant implications for the use of iron chelating substances (probably in partnership with appropriate anti-oxidants) as nutritional or therapeutic agents in inhibiting both the progression of these mainly degenerative diseases and the sequelae of both chronic and acute toxin exposure. The complexity of biochemical networks, especially those involving autocatalytic behaviour and positive feedbacks, means that multiple interventions (e.g. of iron chelators plus antioxidants) are likely to prove most effective. A variety of systems biology approaches, that I summarise, can predict both the mechanisms involved in these cell death pathways and the optimal sites of action for nutritional or pharmacological interventions.

Switch and template pattern formation in a discrete reaction-diffusion system inspired by the Drosophila eye

We examine a spatially discrete reaction-diffusion model based on the interactions that create a periodic pattern in the Drosophila eye imaginal disc. This model is known to be capable of generating a regular hexagonal pattern of gene expression behind a moving front, as observed in the fly system. In order to better understand the novel "switch and template" mechanism behind this pattern formation, we present here a detailed study of the model's behavior in one dimension, using a combination of analytic methods and numerical searches of parameter space. We find that patterns are created robustly, provided that there is an appropriate separation of timescales and that self-activation is sufficiently strong, and we derive expressions in this limit for the front speed and the pattern wavelength. Moving fronts in pattern-forming systems near an initial linear instability generically select a unique pattern, but our model operates in a strongly nonlinear regime where the final pattern depends on the initial conditions as well as on parameter values. Our work highlights the important role that cellularization and cell-autonomous feedback can play in biological pattern formation.

A Boolean model of the gene regulatory network underlying Mammalian cortical area development

The cerebral cortex is divided into many functionally distinct areas. The emergence of these areas during neural development is dependent on the expression patterns of several genes. Along the anterior-posterior axis, gradients of Fgf8, Emx2, Pax6, Coup-tfi, and Sp8 play a particularly strong role in specifying areal identity. However, our understanding of the regulatory interactions between these genes that lead to their confinement to particular spatial patterns is currently qualitative and incomplete. We therefore used a computational model of the interactions between these five genes to determine which interactions, and combinations of interactions, occur in networks that reproduce the anterior-posterior expression patterns observed experimentally. The model treats expression levels as Boolean, reflecting the qualitative nature of the expression data currently available. We simulated gene expression patterns created by all possible networks containing the five genes of interest. We found that only of these networks were able to reproduce the experimentally observed expression patterns. These networks all lacked certain interactions and combinations of interactions including auto-regulation and inductive loops. Many higher order combinations of interactions also never appeared in networks that satisfied our criteria for good performance. While there was remarkable diversity in the structure of the networks that perform well, an analysis of the probability of each interaction gave an indication of which interactions are most likely to be present in the gene network regulating cortical area development. We found that in general, repressive interactions are much more likely than inductive ones, but that mutually repressive loops are not critical for correct network functioning. Overall, our model illuminates the design principles of the gene network regulating cortical area development, and makes novel predictions that can be tested experimentally.

Understanding the development of the brain is an important challenge. Progress on this problem will give insight into how the brain works and what can go wrong to cause developmental disorders like autism and learning disability. This paper examines the development of the outer part of the mammalian brain, the cerebral cortex. This part of the brain contains different areas with specialised functions. Over the past decade, several genes have been identified that play a major role in the development of cortical areas. During development, these genes are expressed in different patterns across the surface of the cortex. Experiments have shown that these genes interact with each other so that they each regulate how much other genes in the group are expressed. However, the experimental data are consistent with many different regulatory networks. In this study, we use a computational model to systematically screen many possible networks. This allows us to predict which regulatory interactions between these genes are important for the patterns of gene expression in the cortex to develop correctly.

Hybrid modeling of biological networks: mixing temporal and qualitative biological properties

Background

Modeling a dynamical biological system is often a difficult task since the a priori unknown parameters of such models are not always directly given by the experiments. Despite the lack of experimental quantitative knowledge, one can see a dynamical biological system as (i) the combined evolution tendencies (increase or decrease) of the biological compound concentrations, and: (ii) the temporal features, such as delays between two concentration peaks (i.e. the times when one of the components completes an increase (resp. decrease) phase and starts a decrease (resp. increase) phase).

Results

We propose herein a new hybrid modeling framework that follows such biological assumptions. This hybrid approach deals with both a qualitative structure of the system and a quantitative structure. From a theoretical viewpoint, temporal specifications are expressed as equality or inequality constraints between delay parameters, while the qualitative specifications are expressed as an ordered pattern of the concentrations peaks of the components. Using this new hybrid framework, the temporal specifications of a biological system can be obtained from incomplete experimental data. The model may be processed by a hybrid model-checker (e.g. Phaver) which is able to give some new constraints on the delay parameters (e.g. the delay for a given transition is exactly 5 hours after the later peak of a gene product concentration). Furthermore, by using a constraint solver on the previous results, it becomes possible to get the set of parameters settings which are consistent with given specifications. Such a modeling approach is particularly accurate for modeling oscillatory biological behaviors like those observed in the Drosophila circadian cycles. The achieved results concerning the parameters of this oscillatory system formally confirm the several previous studies made by numerical simulations. Moreover, our analysis makes it possible to propose an automatic investigation of the respective impact of per and tim on the circadian cycle.

Conclusions

A new hybrid technique for an automatic formal analysis of biological systems is developed with a special emphasis on their oscillatory behaviors. It allows the use of incomplete and empirical biological data.

Systems biology of embryogenesis

The development of a complete organism from a single cell involves extraordinarily complex orchestration of biological processes that vary intricately across space and time. Systems biology seeks to describe how all elements of a biological system interact in order to understand, model and ultimately predict aspects of emergent biological processes. Embryogenesis represents an extraordinary opportunity (and challenge) for the application of systems biology. Systems approaches have already been used successfully to study various aspects of development, from complex intracellular networks to four-dimensional models of organogenesis. Going forward, great advancements and discoveries can be expected from systems approaches applied to embryogenesis and developmental biology.

[Constructing the network of classic genetic knowledge and developing self-learning ability of students in genetic classroom]

With the quick increase of new knowledge in genetics, undergraduate teaching of genetics is becoming a challenge for many teachers. In this paper, the author suggested that it would be important to construct the knowledge network of genetics and to develop the self-learning ability of students. This could help students to read textbooks "from the thicker to the thinner in classroom" and "from the thinner to the thicker outside classroom", so that students would turn to be the talents with new ideas and have more competent ability in biology-related fields.

Ins and outs of systems biology vis-à-vis molecular biology: continuation or clear cut?

The comprehension of living organisms in all their complexity poses a major challenge to the biological sciences. Recently, systems biology has been proposed as a new candidate in the development of such a comprehension. The main objective of this paper is to address what systems biology is and how it is practised. To this end, the basic tools of a systems biological approach are explored and illustrated. In addition, it is questioned whether systems biology 'revolutionizes' molecular biology and 'transcends' its assumed reductionism. The strength of this claim appears to depend on how molecular and systems biology are characterised and on how reductionism is interpreted. Doing credit to molecular biology and to methodological reductionism, it is argued that the distinction between molecular and systems biology is gradual rather than sharp. As such, the classical challenge in biology to manage, interpret and integrate biological data into functional wholes is further intensified by systems biology's use of modelling and bioinformatics, and by its scale enlargement.

A dynamic model for stem cell homeostasis and patterning in Arabidopsis meristems

Plants maintain stem cells in their meristems as a source for new undifferentiated cells throughout their life. Meristems are small groups of cells that provide the microenvironment that allows stem cells to prosper. Homeostasis of a stem cell domain within a growing meristem is achieved by signalling between stem cells and surrounding cells. We have here simulated the origin and maintenance of a defined stem cell domain at the tip of Arabidopsis shoot meristems, based on the assumption that meristems are self-organizing systems. The model comprises two coupled feedback regulated genetic systems that control stem cell behaviour. Using a minimal set of spatial parameters, the mathematical model allows to predict the generation, shape and size of the stem cell domain, and the underlying organizing centre. We use the model to explore the parameter space that allows stem cell maintenance, and to simulate the consequences of mutations, gene misexpression and cell ablations.

Genetic algorithms and their application to in silico evolution of genetic regulatory networks

A genetic algorithm (GA) is a procedure that mimics processes occurring in Darwinian evolution to solve computational problems. A GA introduces variation through "mutation" and "recombination" in a "population" of possible solutions to a problem, encoded as strings of characters in "genomes," and allows this population to evolve, using selection procedures that favor the gradual enrichment of the gene pool with the genomes of the "fitter" individuals. GAs are particularly suitable for optimization problems in which an effective system design or set of parameter values is sought.In nature, genetic regulatory networks (GRNs) form the basic control layer in the regulation of gene expression levels. GRNs are composed of regulatory interactions between genes and their gene products, and are, inter alia, at the basis of the development of single fertilized cells into fully grown organisms. This paper describes how GAs may be applied to find functional regulatory schemes and parameter values for models that capture the fundamental GRN characteristics. The central ideas behind evolutionary computation and GRN modeling, and the considerations in GA design and use are discussed, and illustrated with an extended example. In this example, a GRN-like controller is sought for a developmental system based on Lewis Wolpert's French flag model for positional specification, in which cells in a growing embryo secrete and detect morphogens to attain a specific spatial pattern of cellular differentiation.

Algorithm of myogenic differentiation in higher-order organisms

Cell fate determination is governed by complex signaling molecules at appropriate concentrations that regulate the cell decision-making process. In vertebrates, however, concentration and kinetic parameters are practically unknown, and therefore the mechanism by which these molecules interact is obscure. In myogenesis, for example, multipotent cells differentiate into skeletal muscle as a result of appropriate interplay between several signaling molecules, which is not sufficiently characterized. Here we demonstrate that treatment of biochemical events with SAT (satisfiability) formalism, which has been primarily applied for solving decision-making problems, can provide a simple conceptual tool for describing the relationship between causes and effects in biological phenomena. Specifically, we applied the Łukasiewicz logic to a diffusible protein system that leads to myogenesis. The creation of an automaton that describes the myogenesis SAT problem has led to a comprehensive overview of this non-trivial phenomenon and also to a hypothesis that was subsequently verified experimentally. This example demonstrates the power of applying Łukasiewicz logic in describing and predicting any decision-making problem in general, and developmental processes in particular.

The calculus of integrating differentiation: timing control of T-bet

In this issue of Immunity, Schulz et al. (2009) use mathematical modeling to elucidate a signaling network controlling Ifng gene expression, thereby showing the importance of an Interleukin-12-dependent, Interferon-gamma-independent second phase of inducing the transcription factor T-bet.

Quantifying stochastic effects in biochemical reaction networks using partitioned leaping

"Leaping" methods show great promise for significantly accelerating stochastic simulations of complex biochemical reaction networks. However, few practical applications of leaping have appeared in the literature to date. Here, we address this issue using the "partitioned leaping algorithm" (PLA) [L. A. Harris and P. Clancy, J. Chem. Phys. 125, 144107 (2006)], a recently introduced multiscale leaping approach. We use the PLA to investigate stochastic effects in two model biochemical reaction networks. The networks that we consider are simple enough so as to be accessible to our intuition but sufficiently complex so as to be generally representative of real biological systems. We demonstrate how the PLA allows us to quantify subtle effects of stochasticity in these systems that would be difficult to ascertain otherwise as well as not-so-subtle behaviors that would strain commonly used "exact" stochastic methods. We also illustrate bottlenecks that can hinder the approach and exemplify and discuss possible strategies for overcoming them. Overall, our aim is to aid and motivate future applications of leaping by providing stark illustrations of the benefits of the method while at the same time elucidating obstacles that are often encountered in practice.

Predicting phenotypic diversity and the underlying quantitative molecular transitions

During development, signaling networks control the formation of multicellular patterns. To what extent quantitative fluctuations in these complex networks may affect multicellular phenotype remains unclear. Here, we describe a computational approach to predict and analyze the phenotypic diversity that is accessible to a developmental signaling network. Applying this framework to vulval development in C. elegans, we demonstrate that quantitative changes in the regulatory network can render ∼500 multicellular phenotypes. This phenotypic capacity is an order-of-magnitude below the theoretical upper limit for this system but yet is large enough to demonstrate that the system is not restricted to a select few outcomes. Using metrics to gauge the robustness of these phenotypes to parameter perturbations, we identify a select subset of novel phenotypes that are the most promising for experimental validation. In addition, our model calculations provide a layout of these phenotypes in network parameter space. Analyzing this landscape of multicellular phenotypes yielded two significant insights. First, we show that experimentally well-established mutant phenotypes may be rendered using non-canonical network perturbations. Second, we show that the predicted multicellular patterns include not only those observed in C. elegans, but also those occurring exclusively in other species of the Caenorhabditis genus. This result demonstrates that quantitative diversification of a common regulatory network is indeed demonstrably sufficient to generate the phenotypic differences observed across three major species within the Caenorhabditis genus. Using our computational framework, we systematically identify the quantitative changes that may have occurred in the regulatory network during the evolution of these species. Our model predictions show that significant phenotypic diversity may be sampled through quantitative variations in the regulatory network without overhauling the core network architecture. Furthermore, by comparing the predicted landscape of phenotypes to multicellular patterns that have been experimentally observed across multiple species, we systematically trace the quantitative regulatory changes that may have occurred during the evolution of the Caenorhabditis genus.

The diversity of metazoan life forms that we experience today arose as multicellular systems continually sampled new phenotypes that withstood ever changing selective pressures. This phenotypic diversification is driven by variations in the underlying regulatory network that instructs cells to form multicellular patterns and structures. Here, we computationally construct the phenotypic diversity that may be accessible through quantitative tuning of the regulatory network that drives multicellular patterning during C. elegans vulval development. We show that significant phenotypic diversity may be sampled through quantitative variations without overhauling the core regulatory network architecture. Furthermore, by comparing the predicted landscape of phenotypes to multicellular patterns that have been experimentally observed across multiple species, we systematically deduce the quantitative molecular changes that may have transpired during the evolution of the Caenorhabditis genus.

Bifurcation analysis informs Bayesian inference in the Hes1 feedback loop

Background

Ordinary differential equations (ODEs) are an important tool for describing the dynamics of biological systems. However, for ODE models to be useful, their parameters must first be calibrated. Parameter estimation, that is, finding parameter values given experimental data, is an inference problem that can be treated systematically through a Bayesian framework.

A Markov chain Monte Carlo approach can then be used to sample from the appropriate posterior probability distributions, provided that suitable prior distributions can be found for the unknown parameter values. Choosing these priors is therefore a vital first step in the inference process. We study here a negative feedback loop in gene regulation where an ODE incorporating a time delay has been proposed as a realistic model and where experimental data is available. Our aim is to show that a priori mathematical analysis can be exploited in the choice of priors.

Results

By focussing on the onset of oscillatory behaviour through a Hopf Bifurcation, we derive a range of analytical expressions and constraints that link the model parameters to the observed dynamics of the system. Computational tests on both simulated and experimental data emphasise the usefulness of this analysis.

Conclusion

Mathematical analysis not only gives insights into the possible dynamical behaviour of gene expression models, but can also be used to inform the choice of priors when parameters are inferred from experimental data in a Bayesian setting.

Biological systems from an engineer's point of view

The robust design of complex systems, such as cruise control, requires a careful balance of several objectives. As biological systems are no different, an engineering approach to these systems proves useful.