Dynamical modeling of syncytial mitotic cycles in Drosophila embryos

An autonomous mathematical model for the mammalian cell cycle

A mathematical model for the mammalian cell cycle is developed as a system of 13 coupled nonlinear ordinary differential equations. The variables and interactions included in the model are based on detailed consideration of available experimental data. A novel feature of the model is inclusion of cycle tasks such as origin licensing and initiation, nuclear envelope breakdown and kinetochore attachment, and their interactions with controllers (molecular complexes involved in cycle control). Other key features are that the model is autonomous, except for a dependence on external growth factors; the variables are continuous in time, without instantaneous resets at phase boundaries; mechanisms to prevent rereplication are included; and cycle progression is independent of cell size. Eight variables represent cell cycle controllers: the Cyclin D1-Cdk4/6 complex, APC^Cdh1, SCF^βTrCP, Cdc25A, MPF, NuMA, the securin-separase complex, and separase. Five variables represent task completion, with four for the status of origins and one for kinetochore attachment. The model predicts distinct behaviors corresponding to the main phases of the cell cycle, showing that the principal features of the mammalian cell cycle, including restriction point behavior, can be accounted for in a quantitative mechanistic way based on known interactions among cycle controllers and their coupling to tasks. The model is robust to parameter changes, in that cycling is maintained over at least a five-fold range of each parameter when varied individually. The model is suitable for exploring how extracellular factors affect cell cycle progression, including responses to metabolic conditions and to anti-cancer therapies.

Modeling the role for nuclear import dynamics in the early embryonic cell cycle

Nuclear composition determines nuclear function. The early embryos of many species begin life with large pools of maternally provided components that become rapidly imported into an increasing number of nuclei as the cells undergo repeated cleavage divisions. Because early cell cycles are too fast for nuclei to achieve steady-state nucleocytoplasmic partitioning, the composition of cleavage stage nuclei is likely dominated by nuclear import. The end of the rapid cleavage stage and onset of major zygotic transcription, known as the mid-blastula transition (MBT), is controlled by the ratio of nuclei/cytoplasm, indicating that changes in nuclear composition likely mediate MBT timing. Here, we explore how different nuclear import regimes can affect protein accumulation in the nucleus in the early Drosophila embryo. We find that nuclear import differs dramatically for a general nuclear cargo (NLS (nuclear localization signal)-mRFP) and a proposed MBT regulator (histone H3). We show that nuclear import rates of NLS-mRFP in a given nucleus remain relatively unchanged throughout the cleavage cycles, whereas those of H3 halve with each cycle. We model these two distinct modes of nuclear import as "nucleus-limited" and "import-limited" and examine how the two different modes can contribute to different protein accumulation dynamics. Finally, we incorporate these distinct modes of nuclear import into a model for cell-cycle regulation at the MBT and find that the import-limited H3 dynamics contribute to increased robustness and allow for stepwise cell-cycle slowing at the MBT.

Excess histone H3 is a competitive Chk1 inhibitor that controls cell-cycle remodeling in the early Drosophila embryo

The DNA damage checkpoint is crucial to protect genome integrity.^1,2 However, the early embryos of many metazoans sacrifice this safeguard to allow for rapid cleavage divisions that are required for speedy development. At the mid-blastula transition (MBT), embryos switch from rapid cleavage divisions to slower, patterned divisions with the addition of gap phases and acquisition of DNA damage checkpoints. The timing of the MBT is dependent on the nuclear-to-cytoplasmic (N/C ratio)^3-7 and the activation of the checkpoint kinase, Chk1.^8-17 How Chk1 activity is coupled to the N/C ratio has remained poorly understood. Here, we show that dynamic changes in histone H3 availability in response to the increasing N/C ratio control Chk1 activity and thus time the MBT in the Drosophila embryo. We show that excess H3 in the early cycles interferes with cell-cycle slowing independent of chromatin incorporation. We find that the N-terminal tail of H3 acts as a competitive inhibitor of Chk1 in vitro and reduces Chk1 activity in vivo. Using a H3-tail mutant that has reduced Chk1 inhibitor activity, we show that the amount of available Chk1 sites in the H3 pool controls the dynamics of cell-cycle progression. Mathematical modeling quantitatively supports a mechanism where titration of H3 during early cleavage cycles regulates Chk1-dependent cell-cycle slowing. This study defines Chk1 regulation by H3 as a key mechanism that coordinates cell-cycle remodeling with developmental progression.

Time to change your mind? Modelling transient properties of cortex formation highlights the importance of evolving cell division strategies

The successful development of the mammalian cerebral neocortex is linked to numerous cognitive functions such as language, voluntary movement, and episodic memory. Neocortex development occurs when neural progenitor cells divide and produce neurons. Critically, although the progenitor cells are able to self-renew they do not reproduce themselves endlessly. Hence, to fully understand the development of the neocortex we are faced with the challenge of understanding temporal changes in cell division strategy. Our approach to modelling neuronal production uses non-autonomous ordinary differential equations and allows us to use a ternary coordinate system in order to define a strategy space, through which we can visualise evolving cell division strategies. Using this strategy space, we fit the known data and use approximate Bayesian computation to predict the founding progenitor population sizes, currently unavailable in the experimental literature. Counter-intuitively, we show that humans can generate a larger number of neurons than a macaque's even when starting with a smaller number of progenitor cells. Accompanying the article is a self-contained piece of software, which provides the reader with immediate simulated results that will aid their intuition. The software can be found at www.dpag.ox.ac.uk/team/noemi-picco.

Tellurium notebooks-An environment for reproducible dynamical modeling in systems biology

The considerable difficulty encountered in reproducing the results of published dynamical models limits validation, exploration and reuse of this increasingly large biomedical research resource. To address this problem, we have developed Tellurium Notebook, a software system for model authoring, simulation, and teaching that facilitates building reproducible dynamical models and reusing models by 1) providing a notebook environment which allows models, Python code, and narrative to be intermixed, 2) supporting the COMBINE archive format during model development for capturing model information in an exchangeable format and 3) enabling users to easily simulate and edit public COMBINE-compliant models from public repositories to facilitate studying model dynamics, variants and test cases. Tellurium Notebook, a Python–based Jupyter–like environment, is designed to seamlessly inter-operate with these community standards by automating conversion between COMBINE standards formulations and corresponding in–line, human–readable representations. Thus, Tellurium brings to systems biology the strategy used by other literate notebook systems such as Mathematica. These capabilities allow users to edit every aspect of the standards–compliant models and simulations, run the simulations in–line, and re–export to standard formats. We provide several use cases illustrating the advantages of our approach and how it allows development and reuse of models without requiring technical knowledge of standards. Adoption of Tellurium should accelerate model development, reproducibility and reuse.

There is considerable value to systems and synthetic biology in creating reproducible models. An essential element of reproducibility is the use of community standards, an often challenging undertaking for modelers. This article describes Tellurium Notebook, a tool for developing dynamical models that provides an intuitive approach to building and reusing models built with community standards. Tellurium automates embedding human–readable representations of COMBINE archives in literate coding notebooks, bringing to systems biology this strategy central to other literate notebook systems such as Mathematica. We show that the ability to easily edit this human–readable representation enables users to test models under a variety of conditions, thereby providing a way to create, reuse, and modify standard–encoded models and simulations, regardless of the user’s level of technical knowledge of said standards.

A fully featured COMBINE archive of a simulation study on syncytial mitotic cycles in Drosophila embryos

COMBINE archives are standardised containers for data files related to a simulation study in computational biology. This manuscript describes a fully featured archive of a previously published simulation study, including (i) the original publication, (ii) the model, (iii) the analyses, and (iv) metadata describing the files and their origin. With the archived data at hand, it is possible to reproduce the results of the original work. The archive can be used for both, educational and research purposes. Anyone may reuse, extend and update the archive to make it a valuable resource for the scientific community.

Mathematical modeling of the female reproductive system: from oocyte to delivery

From ovulation to delivery, and through the menstrual cycle, the female reproductive system undergoes many dynamic changes to provide an optimal environment for the embryo to implant, and to develop successfully. It is difficult ethically and practically to observe the system over the timescales involved in growth and development (often hours to days). Even in carefully monitored conditions clinicians and biologists can only see snapshots of the development process. Mathematical models are emerging as a key means to supplement our knowledge of the reproductive process, and to tease apart complexity in the reproductive system. These models have been used successfully to test existing hypotheses regarding the mechanisms of female infertility and pathological fetal development, and also to provide new experimentally testable hypotheses regarding the process of development. This new knowledge has allowed for improvements in assisted reproductive technologies and is moving toward translation to clinical practice via multiscale assessments of the dynamics of ovulation, development in pregnancy, and the timing and mechanics of delivery. WIREs Syst Biol Med 2017, 9:e1353. doi: 10.1002/wsbm.1353 For further resources related to this article, please visit the WIREs website.

Qualitative Dynamical Modelling Can Formally Explain Mesoderm Specification and Predict Novel Developmental Phenotypes

Given the complexity of developmental networks, it is often difficult to predict the effect of genetic perturbations, even within coding genes. Regulatory factors generally have pleiotropic effects, exhibit partially redundant roles, and regulate highly interconnected pathways with ample cross-talk. Here, we delineate a logical model encompassing 48 components and 82 regulatory interactions involved in mesoderm specification during Drosophila development, thereby providing a formal integration of all available genetic information from the literature. The four main tissues derived from mesoderm correspond to alternative stable states. We demonstrate that the model can predict known mutant phenotypes and use it to systematically predict the effects of over 300 new, often non-intuitive, loss- and gain-of-function mutations, and combinations thereof. We further validated several novel predictions experimentally, thereby demonstrating the robustness of model. Logical modelling can thus contribute to formally explain and predict regulatory outcomes underlying cell fate decisions.

Role of Computational Modeling in Understanding Cell Cycle Oscillators

The periodic oscillations in the activity of the cell cycle regulatory program, drives the timely activation of key cell cycle events. Interesting dynamical systems, such as oscillators, have been investigated by various theoretical and computational modeling methods. Thanks to the insights achieved by these modeling efforts we have gained considerable insights about the underlying molecular regulatory networks that can drive cell cycle oscillations. Here we review the basic features and characteristics of biological oscillators, discussing from a computational modeling point of view their specific architectures and the current knowledge about the dynamics that the life evolution selected to drive cell cycle oscillations.

Quantitative imaging with Fucci and mathematics to uncover temporal dynamics of cell cycle progression

Cell cycle progression is strictly coordinated to ensure proper tissue growth, development, and regeneration of multicellular organisms. Spatiotemporal visualization of cell cycle phases directly helps us to obtain a deeper understanding of controlled, multicellular, cell cycle progression. The fluorescent ubiquitination-based cell cycle indicator (Fucci) system allows us to monitor, in living cells, the G1 and the S/G2/M phases of the cell cycle in red and green fluorescent colors, respectively. Since the discovery of Fucci technology, it has found numerous applications in the characterization of the timing of cell cycle phase transitions under diverse conditions and various biological processes. However, due to the complexity of cell cycle dynamics, understanding of specific patterns of cell cycle progression is still far from complete. In order to tackle this issue, quantitative approaches combined with mathematical modeling seem to be essential. Here, we review several studies that attempted to integrate Fucci technology and mathematical models to obtain quantitative information regarding cell cycle regulatory patterns. Focusing on the technological development of utilizing mathematics to retrieve meaningful information from the Fucci producing data, we discuss how the combined methods advance a quantitative understanding of cell cycle regulation.

From START to FINISH: computational analysis of cell cycle control in budding yeast

In the cell division cycle of budding yeast, START refers to a set of tightly linked events that prepare a cell for budding and DNA replication, and FINISH denotes the interrelated events by which the cell exits from mitosis and divides into mother and daughter cells. On the basis of recent progress made by molecular biologists in characterizing the genes and proteins that control START and FINISH, we crafted a new mathematical model of cell cycle progression in yeast. Our model exploits a natural separation of time scales in the cell cycle control network to construct a system of differential-algebraic equations for protein synthesis and degradation, post-translational modifications, and rapid formation and dissociation of multimeric complexes. The model provides a unified account of the observed phenotypes of 257 mutant yeast strains (98% of the 263 strains in the data set used to constrain the model). We then use the model to predict the phenotypes of 30 novel combinations of mutant alleles. Our comprehensive model of the molecular events controlling cell cycle progression in budding yeast has both explanatory and predictive power. Future experimental tests of the model’s predictions will be useful to refine the underlying molecular mechanism, to constrain the adjustable parameters of the model, and to provide new insights into how the cell division cycle is regulated in budding yeast.

Evidence for internuclear signaling in drosophila embryogenesis

BACKGROUND

Syncytial nuclei in Drosophila embryos undergo their first 13 divisions nearly synchronously. In the last several cell cycles, these division events travel across the anterior-posterior axis of the syncytial blastoderm in a wave. The phenomenon is well documented but the underlying mechanisms are not yet understood.

RESULTS

We study timing and positional data obtained from in vivo imaging of Drosophila embryos. We determine the statistical properties of the distribution of division times within and across generations with the null hypothesis that timing of division events is an independent random variable for each nucleus. We also compare timing data with a model of Drosophila cell cycle regulation that does not include internuclear signaling, and to a universal model of phase-dependent signaling to determine the probable form of internuclear signaling in the syncytial embryo.

CONCLUSIONS

The statistical variance of division times is lower than one would expect from uncoordinated activity. In fact, the variance decreases between the 10th and 11th divisions, which demonstrates a contribution of internuclear signaling to the observed synchrony and division waves. Our comparison with a coupled oscillator model leads us to conclude that internuclear signaling must be of Response/Signaling type with a positive impulse.

Validation of signalling pathways: Case study of the p16-mediated pathway

p16 is recognized as a tumor suppressor gene due to the prevalence of its genetic inactivation in all types of human cancers. Additionally, p16 gene plays a critical role in controlling aging, regulating cellular senescence, detection and maintenance of DNA damage. The molecular mechanism behind these events involves p16-mediated signaling pathway (or p16- Rb pathway), the focus of our study. Understanding functional dependence between dynamic behavior of biological components involved in the p16-mediated pathway and aforesaid molecular-level events might suggest possible implications in the diagnosis, prognosis and treatment of human cancer. In the present work, we employ reverse-engineering approach to construct the most detailed computational model of p16-mediated pathway in higher eukaryotes. We implement experimental data from the literature to validate the model, and under various assumptions predict the dynamic behavior of p16 and other biological components by interpreting the simulation results. The quantitative model of p16-mediated pathway is created in a systematic manner in terms of Petri net technologies.

A data-driven, mathematical model of mammalian cell cycle regulation

Few of >150 published cell cycle modeling efforts use significant levels of data for tuning and validation. This reflects the difficultly to generate correlated quantitative data, and it points out a critical uncertainty in modeling efforts. To develop a data-driven model of cell cycle regulation, we used contiguous, dynamic measurements over two time scales (minutes and hours) calculated from static multiparametric cytometry data. The approach provided expression profiles of cyclin A2, cyclin B1, and phospho-S10-histone H3. The model was built by integrating and modifying two previously published models such that the model outputs for cyclins A and B fit cyclin expression measurements and the activation of B cyclin/Cdk1 coincided with phosphorylation of histone H3. The model depends on Cdh1-regulated cyclin degradation during G1, regulation of B cyclin/Cdk1 activity by cyclin A/Cdk via Wee1, and transcriptional control of the mitotic cyclins that reflects some of the current literature. We introduced autocatalytic transcription of E2F, E2F regulated transcription of cyclin B, Cdc20/Cdh1 mediated E2F degradation, enhanced transcription of mitotic cyclins during late S/early G2 phase, and the sustained synthesis of cyclin B during mitosis. These features produced a model with good correlation between state variable output and real measurements. Since the method of data generation is extensible, this model can be continually modified based on new correlated, quantitative data.

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.

A mathematical model of mitotic exit in budding yeast: the role of Polo kinase

Cell cycle progression in eukaryotes is regulated by periodic activation and inactivation of a family of cyclin-dependent kinases (Cdk's). Entry into mitosis requires phosphorylation of many proteins targeted by mitotic Cdk, and exit from mitosis requires proteolysis of mitotic cyclins and dephosphorylation of their targeted proteins. Mitotic exit in budding yeast is known to involve the interplay of mitotic kinases (Cdk and Polo kinases) and phosphatases (Cdc55/PP2A and Cdc14), as well as the action of the anaphase promoting complex (APC) in degrading specific proteins in anaphase and telophase. To understand the intricacies of this mechanism, we propose a mathematical model for the molecular events during mitotic exit in budding yeast. The model captures the dynamics of this network in wild-type yeast cells and 110 mutant strains. The model clarifies the roles of Polo-like kinase (Cdc5) in the Cdc14 early anaphase release pathway and in the G-protein regulated mitotic exit network.

A dynamical model of oocyte maturation unveils precisely orchestrated meiotic decisions

Maturation of vertebrate oocytes into haploid gametes relies on two consecutive meioses without intervening DNA replication. The temporal sequence of cellular transitions driving eggs from G2 arrest to meiosis I (MI) and then to meiosis II (MII) is controlled by the interplay between cyclin-dependent and mitogen-activated protein kinases. In this paper, we propose a dynamical model of the molecular network that orchestrates maturation of Xenopus laevis oocytes. Our model reproduces the core features of maturation progression, including the characteristic non-monotonous time course of cyclin-Cdks, and unveils the network design principles underlying a precise sequence of meiotic decisions, as captured by bifurcation and sensitivity analyses. Firstly, a coherent and sharp meiotic resumption is triggered by the concerted action of positive feedback loops post-translationally activating cyclin-Cdks. Secondly, meiotic transition is driven by the dynamic antagonism between positive and negative feedback loops controlling cyclin turnover. Our findings reveal a highly modular network in which the coordination of distinct regulatory schemes ensures both reliable and flexible cell-cycle decisions.

In the life cycle of sexual organisms, a specialized cell division -meiosis- reduces the number of chromosomes in gametes or spores while fertilization or mating restores the original number. The essential feature that distinguishes meiosis from mitosis (the usual division) is the succession of two rounds of division following a single DNA replication, as well as the arrest at the second division in the case of oocyte maturation. The fact that meiosis and mitosis are similar but different raises several interesting questions: What is the meiosis-specific dynamics of cell-cycle regulators? Are there mechanisms which guarantee the occurence of two and only two rounds of division despite the presence of intrinsic and extrinsic noises ? The study of a model of the molecular network that underlies the meiotic maturation process in Xenopus oocytes provides unexpected answers to these questions. On the one hand, the modular organization of this network ensures separate controls of the first and second divisions. On the other hand, regulatory synergies ensure that these two stages are precisely and reliably sequenced during meiosis. We conclude that cells have evolved a sophisticated regulatory network to achieve a robust, albeit flexible, meiotic dynamics.

A hybrid model of mammalian cell cycle regulation

The timing of DNA synthesis, mitosis and cell division is regulated by a complex network of biochemical reactions that control the activities of a family of cyclin-dependent kinases. The temporal dynamics of this reaction network is typically modeled by nonlinear differential equations describing the rates of the component reactions. This approach provides exquisite details about molecular regulatory processes but is hampered by the need to estimate realistic values for the many kinetic constants that determine the reaction rates. It is difficult to estimate these kinetic constants from available experimental data. To avoid this problem, modelers often resort to ‘qualitative’ modeling strategies, such as Boolean switching networks, but these models describe only the coarsest features of cell cycle regulation. In this paper we describe a hybrid approach that combines the best features of continuous differential equations and discrete Boolean networks. Cyclin abundances are tracked by piecewise linear differential equations for cyclin synthesis and degradation. Cyclin synthesis is regulated by transcription factors whose activities are represented by discrete variables (0 or 1) and likewise for the activities of the ubiquitin-ligating enzyme complexes that govern cyclin degradation. The discrete variables change according to a predetermined sequence, with the times between transitions determined in part by cyclin accumulation and degradation and as well by exponentially distributed random variables. The model is evaluated in terms of flow cytometry measurements of cyclin proteins in asynchronous populations of human cell lines. The few kinetic constants in the model are easily estimated from the experimental data. Using this hybrid approach, modelers can quickly create quantitatively accurate, computational models of protein regulatory networks in cells.

The physiological behaviors of cells (growth and division, differentiation, movement, death, etc.) are controlled by complex networks of interacting genes and proteins, and a fundamental goal of computational cell biology is to develop dynamical models of these regulatory networks that are realistic, accurate and predictive. Historically, these models have divided along two basic lines: deterministic or stochastic, and continuous or discrete; with scattered efforts to develop hybrid approaches that bridge these divides. Using the cell cycle control system in eukaryotes as an example, we propose a hybrid approach that combines a continuous representation of slowly changing protein concentrations with a discrete representation of components that switch rapidly between ‘on’ and ‘off’ states, and that combines the deterministic causality of network interactions with the stochastic uncertainty of random events. The hybrid approach can be easily tailored to the available knowledge of control systems, and it provides both qualitative and quantitative results that can be compared to experimental data to test the accuracy and predictive power of the model.

How to turn a genetic circuit into a synthetic tunable oscillator, or a bistable switch

Systems and Synthetic Biology use computational models of biological pathways in order to study in silico the behaviour of biological pathways. Mathematical models allow to verify biological hypotheses and to predict new possible dynamical behaviours. Here we use the tools of non-linear analysis to understand how to change the dynamics of the genes composing a novel synthetic network recently constructed in the yeast Saccharomyces cerevisiae for In-vivo Reverse-engineering and Modelling Assessment (IRMA). Guided by previous theoretical results that make the dynamics of a biological network depend on its topological properties, through the use of simulation and continuation techniques, we found that the network can be easily turned into a robust and tunable synthetic oscillator or a bistable switch. Our results provide guidelines to properly re-engineering in vivo the network in order to tune its dynamics.

BioModels.net Web Services, a free and integrated toolkit for computational modelling software

Exchanging and sharing scientific results are essential for researchers in the field of computational modelling. BioModels.net defines agreed-upon standards for model curation. A fundamental one, MIRIAM (Minimum Information Requested in the Annotation of Models), standardises the annotation and curation process of quantitative models in biology. To support this standard, MIRIAM Resources maintains a set of standard data types for annotating models, and provides services for manipulating these annotations. Furthermore, BioModels.net creates controlled vocabularies, such as SBO (Systems Biology Ontology) which strictly indexes, defines and links terms used in Systems Biology. Finally, BioModels Database provides a free, centralised, publicly accessible database for storing, searching and retrieving curated and annotated computational models. Each resource provides a web interface to submit, search, retrieve and display its data. In addition, the BioModels.net team provides a set of Web Services which allows the community to programmatically access the resources. A user is then able to perform remote queries, such as retrieving a model and resolving all its MIRIAM Annotations, as well as getting the details about the associated SBO terms. These web services use established standards. Communications rely on SOAP (Simple Object Access Protocol) messages and the available queries are described in a WSDL (Web Services Description Language) file. Several libraries are provided in order to simplify the development of client software. BioModels.net Web Services make one step further for the researchers to simulate and understand the entirety of a biological system, by allowing them to retrieve biological models in their own tool, combine queries in workflows and efficiently analyse models.

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.

Temporal controls of the asymmetric cell division cycle in Caulobacter crescentus

The asymmetric cell division cycle of Caulobacter crescentus is orchestrated by an elaborate gene-protein regulatory network, centered on three major control proteins, DnaA, GcrA and CtrA. The regulatory network is cast into a quantitative computational model to investigate in a systematic fashion how these three proteins control the relevant genetic, biochemical and physiological properties of proliferating bacteria. Different controls for both swarmer and stalked cell cycles are represented in the mathematical scheme. The model is validated against observed phenotypes of wild-type cells and relevant mutants, and it predicts the phenotypes of novel mutants and of known mutants under novel experimental conditions. Because the cell cycle control proteins of Caulobacter are conserved across many species of alpha-proteobacteria, the model we are proposing here may be applicable to other genera of importance to agriculture and medicine (e.g., Rhizobium, Brucella).

Modular logical modelling of the budding yeast cell cycle

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

A general computational method for robustness analysis with applications to synthetic gene networks

Motivation: Robustness is the capacity of a system to maintain a function in the face of perturbations. It is essential for the correct functioning of natural and engineered biological systems. Robustness is generally defined in an ad hoc, problem-dependent manner, thus hampering the fruitful development of a theory of biological robustness, recently advocated by Kitano.

Results: In this article, we propose a general definition of robustness that applies to any biological function expressible in temporal logic LTL (linear temporal logic), and to broad model classes and perturbation types. Moreover, we propose a computational approach and an implementation in BIOCHAM 2.8 for the automated estimation of the robustness of a given behavior with respect to a given set of perturbations. The applicability and biological relevance of our approach is demonstrated by testing and improving the robustness of the timed behavior of a synthetic transcriptional cascade that could be used as a biological timer for synthetic biology applications.

Availability: Version 2.8 of BIOCHAM and the transcriptional cascade model are available at http://contraintes.inria.fr/BIOCHAM/

Contact:gregory.batt@inria.fr

Using in silico models to simulate dual perturbation experiments: procedure development and interpretation of outcomes

Background

A growing number of realistic in silico models of metabolic functions are being formulated and can serve as 'dry lab' platforms to prototype and simulate experiments before they are performed. For example, dual perturbation experiments that vary both genetic and environmental parameters can readily be simulated in silico. Genetic and environmental perturbations were applied to a cell-scale model of the human erythrocyte and subsequently investigated.

Results

The resulting steady state fluxes and concentrations, as well as dynamic responses to the perturbations were analyzed, yielding two important conclusions: 1) that transporters are informative about the internal states (fluxes and concentrations) of a cell and, 2) that genetic variations can disrupt the natural sequence of dynamic interactions between network components. The former arises from adjustments in energy and redox states, while the latter is a result of shifting time scales in aggregate pool formation of metabolites. These two concepts are illustrated for glucose-6 phosphate dehydrogenase (G6PD) and pyruvate kinase (PK) in the human red blood cell.

Conclusion

Dual perturbation experiments in silico are much more informative for the characterization of functional states than single perturbations. Predictions from an experimentally validated cellular model of metabolism indicate that the measurement of cofactor precursor transport rates can inform the internal state of the cell when the external demands are altered or a causal genetic variation is introduced. Finally, genetic mutations that alter the clinical phenotype may also disrupt the 'natural' time scale hierarchy of interactions in the network.

Computational systems biology of the cell cycle

One of the early success stories of computational systems biology was the work done on cell-cycle regulation. The earliest mathematical descriptions of cell-cycle control evolved into very complex, detailed computational models that describe the regulation of cell division in many different cell types. On the way these models predicted several dynamical properties and unknown components of the system that were later experimentally verified/identified. Still, research on this field is far from over. We need to understand how the core cell-cycle machinery is controlled by internal and external signals, also in yeast cells and in the more complex regulatory networks of higher eukaryotes. Furthermore, there are many computational challenges what we face as new types of data appear thanks to continuing advances in experimental techniques. We have to deal with cell-to-cell variations, revealed by single cell measurements, as well as the tremendous amount of data flowing from high throughput machines. We need new computational concepts and tools to handle these data and develop more detailed, more precise models of cell-cycle regulation in various organisms. Here we review past and present of computational modeling of cell-cycle regulation, and discuss possible future directions of the field.

Signaling gradients in cascades of two-state reaction-diffusion systems

Biological networks frequently use cascades, generally defined as chain-like arrangements of similar modules. Spatially lumped cascades can serve as noise filters, time-delay, or thresholding elements. The operation and functional capabilities of spatially distributed cascades are much less understood. Motivated by studies of pattern formation in the early Drosophila embryo, we analyze cascades of 2-state reaction-diffusion systems. At each stage within such as a cascade, a diffusible particle is reversibly bound by immobile traps and can be annihilated in both mobile and immobile states. When trapped, these particles drive the next stage by converting mobile particles of a different type from a passive to active form. The cascade initiated by injection of mobile particles into the first stage. We derive analytical expressions for the steady-state concentration profiles of mobile and immobile particles and analyze how the output of a cascade is controlled by properties of the constituent stages.

Spatial bistability generates hunchback expression sharpness in the Drosophila embryo

During embryonic development, the positional information provided by concentration gradients of maternal factors directs pattern formation by providing spatially dependent cues for gene expression. In the fruit fly, Drosophila melanogaster, a classic example of this is the sharp on-off activation of the hunchback (hb) gene at midembryo, in response to local concentrations of the smooth anterior-posterior Bicoid (Bcd) gradient. The regulatory region for hb contains multiple binding sites for the Bcd protein as well as multiple binding sites for the Hb protein. Some previous studies have suggested that Bcd is sufficient for properly sharpened Hb expression, yet other evidence suggests a need for additional regulation. We experimentally quantified the dynamics of hb gene expression in flies that were wild-type, were mutant for hb self-regulation or Bcd binding, or contained an artificial promoter construct consisting of six Bcd and two Hb sites. In addition to these experiments, we developed a reaction-diffusion model of hb transcription, with Bcd cooperative binding and hb self-regulation, and used Zero Eigenvalue Analysis to look for multiple stationary states in the reaction network. Our model reproduces the hb developmental dynamics and correctly predicts the mutant patterns. Analysis of our model indicates that the Hb sharpness can be produced by spatial bistability, in which hb self-regulation produces two stable levels of expression. In the absence of self-regulation, the bistable behavior vanishes and Hb sharpness is disrupted. Bcd cooperative binding affects the position where bistability occurs but is not itself sufficient for a sharp Hb pattern. Our results show that the control of Hb sharpness and positioning, by hb self-regulation and Bcd cooperativity, respectively, are separate processes that can be altered independently. Our model, which matches the changes in Hb position and sharpness observed in different experiments, provides a theoretical framework for understanding the data and in particular indicates that spatial bistability can play a central role in threshold-dependent reading mechanisms of positional information.

FERN - a Java framework for stochastic simulation and evaluation of reaction networks

Background

Stochastic simulation can be used to illustrate the development of biological systems over time and the stochastic nature of these processes. Currently available programs for stochastic simulation, however, are limited in that they either a) do not provide the most efficient simulation algorithms and are difficult to extend, b) cannot be easily integrated into other applications or c) do not allow to monitor and intervene during the simulation process in an easy and intuitive way. Thus, in order to use stochastic simulation in innovative high-level modeling and analysis approaches more flexible tools are necessary.

Results

In this article, we present FERN (Framework for Evaluation of Reaction Networks), a Java framework for the efficient simulation of chemical reaction networks. FERN is subdivided into three layers for network representation, simulation and visualization of the simulation results each of which can be easily extended. It provides efficient and accurate state-of-the-art stochastic simulation algorithms for well-mixed chemical systems and a powerful observer system, which makes it possible to track and control the simulation progress on every level. To illustrate how FERN can be easily integrated into other systems biology applications, plugins to Cytoscape and CellDesigner are included. These plugins make it possible to run simulations and to observe the simulation progress in a reaction network in real-time from within the Cytoscape or CellDesigner environment.

Conclusion

FERN addresses shortcomings of currently available stochastic simulation programs in several ways. First, it provides a broad range of efficient and accurate algorithms both for exact and approximate stochastic simulation and a simple interface for extending to new algorithms. FERN's implementations are considerably faster than the C implementations of gillespie2 or the Java implementations of ISBJava. Second, it can be used in a straightforward way both as a stand-alone program and within new systems biology applications. Finally, complex scenarios requiring intervention during the simulation progress can be modelled easily with FERN.

Biological switches and clocks

To introduce this special issue on biological switches and clocks, we review the historical development of mathematical models of bistability and oscillations in chemical reaction networks. In the 1960s and 1970s, these models were limited to well-studied biochemical examples, such as glycolytic oscillations and cyclic AMP signalling. After the molecular genetics revolution of the 1980s, the field of molecular cell biology was thrown wide open to mathematical modellers. We review recent advances in modelling the gene-protein interaction networks that control circadian rhythms, cell cycle progression, signal processing and the design of synthetic gene networks.