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

Background

Phenomenological information about regulatory interactions is frequently available and can be readily converted to Boolean models. Fully quantitative models, on the other hand, provide detailed insights into the precise dynamics of the underlying system. In order to connect discrete and continuous modeling approaches, methods for the conversion of Boolean systems into systems of ordinary differential equations have been developed recently. As biological interaction networks have steadily grown in size and complexity, a fully automated framework for the conversion process is desirable.

Results

We present Odefy, a MATLAB- and Octave-compatible toolbox for the automated transformation of Boolean models into systems of ordinary differential equations. Models can be created from sets of Boolean equations or graph representations of Boolean networks. Alternatively, the user can import Boolean models from the CellNetAnalyzer toolbox, GINSim and the PBN toolbox. The Boolean models are transformed to systems of ordinary differential equations by multivariate polynomial interpolation and optional application of sigmoidal Hill functions. Our toolbox contains basic simulation and visualization functionalities for both, the Boolean as well as the continuous models. For further analyses, models can be exported to SQUAD, GNA, MATLAB script files, the SB toolbox, SBML and R script files. Odefy contains a user-friendly graphical user interface for convenient access to the simulation and exporting functionalities. We illustrate the validity of our transformation approach as well as the usage and benefit of the Odefy toolbox for two biological systems: a mutual inhibitory switch known from stem cell differentiation and a regulatory network giving rise to a specific spatial expression pattern at the mid-hindbrain boundary.

Conclusions

Odefy provides an easy-to-use toolbox for the automatic conversion of Boolean models to systems of ordinary differential equations. It can be efficiently connected to a variety of input and output formats for further analysis and investigations. The toolbox is open-source and can be downloaded at http://cmb.helmholtz-muenchen.de/odefy.

The gene networks that comprise the circadian clock modulate biological function across a range of scales, from gene expression to performance and adaptive behaviour. The clock functions by generating endogenous rhythms that can be entrained to the external 24-h day–night cycle, enabling organisms to optimally time biochemical processes relative to dawn and dusk. In recent years, computational models based on differential equations have become useful tools for dissecting and quantifying the complex regulatory relationships underlying the clock's oscillatory dynamics. However, optimizing the large parameter sets characteristic of these models places intense demands on both computational and experimental resources, limiting the scope of in silico studies. Here, we develop an approach based on Boolean logic that dramatically reduces the parametrization, making the state and parameter spaces finite and tractable. We introduce efficient methods for fitting Boolean models to molecular data, successfully demonstrating their application to synthetic time courses generated by a number of established clock models, as well as experimental expression levels measured using luciferase imaging. Our results indicate that despite their relative simplicity, logic models can (i) simulate circadian oscillations with the correct, experimentally observed phase relationships among genes and (ii) flexibly entrain to light stimuli, reproducing the complex responses to variations in daylength generated by more detailed differential equation formulations. Our work also demonstrates that logic models have sufficient predictive power to identify optimal regulatory structures from experimental data. By presenting the first Boolean models of circadian circuits together with general techniques for their optimization, we hope to establish a new framework for the systematic modelling of more complex clocks, as well as other circuits with different qualitative dynamics. In particular, we anticipate that the ability of logic models to provide a computationally efficient representation of system behaviour could greatly facilitate the reverse-engineering of large-scale biochemical networks.

The molecular bases of circadian clocks are complex and cannot be sufficiently explained by the relatively simple feedback loops, based on transcription and translation, of current models. The existence of additional oscillators has been demonstrated experimentally, but their mechanism(s) have so far resisted elucidation and any universally conserved clock components have yet to be identified. The fission yeast, Schizosaccharomyces pombe, as a simple and well-characterized eukaryote, is a useful model organism in the investigation of many aspects of cell regulation. In fast-growing cells of the yeast an ultradian clock operates, which can serve as a model system to analyse clock complexity. This clock shares strict period homeostasis and efficient entrainment with circadian clocks but, because of its short period of 30 min, mechanisms other than a transcription/translation-based feedback loop must be working. An initial systematic screen involving over 200 deletion mutants has shown that major cellular signalling pathways (calcium/phosphoinositide, mitogen-activated protein kinase and cAMP/protein kinase A) are crucial for the normal functioning of this ultradian clock. A comparative examination of the role of cellular signalling pathways in the S.pombe ultradian clock and in the circadian timekeeping of different eukaryotes may indicate common principles in biological timing processes that are universally conserved amongst eukaryotes.

Bioluminescence techniques allow accurate monitoring of the circadian clock in single cells. We have analyzed bioluminescence data of Per gene expression in mouse SCN neurons and fibroblasts. From these data, we extracted parameters such as damping rate and noise intensity using two simple mathematical models, one describing a damped oscillator driven by noise, and one describing a self-sustained noisy oscillator. Both models describe the data well and enabled us to quantitatively characterize both wild-type cells and several mutants. It has been suggested that the circadian clock is self-sustained at the single cell level, but we conclude that present data are not sufficient to determine whether the circadian clock of single SCN neurons and fibroblasts is a damped or a self-sustained oscillator. We show how to settle this question, however, by testing the models' predictions of different phases and amplitudes in response to a periodic entrainment signal (zeitgeber).

Author Summary

Earth's 24-h-rotation around its axis is mirrored in the circadian clock that resides within each of our cells, controlling expression of ∼10% of all genes. The circadian clock is constructed as a negative feedback loop, in which clock proteins inhibit their own synthesis. During the last decade, a picture has emerged in which each cell is a self-sustained circadian oscillator that runs even without synchronizing cues. Here, we investigated state-of-the-art single-cell bioluminescence recordings of clock gene expression. It turns out that these time series are very well described by low-dimensional models, enabling us to extract descriptive parameters that characterize each cell. We find that different cell types do not differ much in their dynamics. However, different mutations in core clock genes yield different dynamic characteristics. Furthermore, we could not statistically reject the idea that the cells are in fact damped oscillators driven by noise. We thus declare the question of whether the circadian clock is a damped or self-sustained oscillator still unresolved. Further, we propose a way to resolve this question by examining the frequency-dependent response of single cells to periodic stimuli. We will then be in a better position to understand how cells coordinate and synchronize their circadian rhythms.

The circadian clock is accountable for the regulation of internal rhythms in most living organisms. It allows the anticipation of environmental changes during the day and a better adaptation of physiological processes. In mammals the main clock is located in the suprachiasmatic nucleus (SCN) and synchronizes secondary clocks throughout the body. Its molecular constituents form an intracellular network which dictates circadian time and regulates clock-controlled genes. These clock-controlled genes are involved in crucial biological processes including metabolism and cell cycle regulation. Its malfunction can lead to disruption of biological rhythms and cause severe damage to the organism. The detailed mechanisms that govern the circadian system are not yet completely understood. Mathematical models can be of great help to exploit the mechanism of the circadian circuitry. We built a mathematical model for the core clock system using available data on phases and amplitudes of clock components obtained from an extensive literature search. This model was used to answer complex questions for example: how does the degradation rate of Per affect the period of the system and what is the role of the ROR/Bmal/REV-ERB (RBR) loop? Our findings indicate that an increase in the RNA degradation rate of the clock gene Period (Per) can contribute to increase or decrease of the period - a consequence of a non-monotonic effect of Per transcript stability on the circadian period identified by our model. Furthermore, we provide theoretical evidence for a potential role of the RBR loop as an independent oscillator. We carried out overexpression experiments on members of the RBR loop which lead to loss of oscillations consistent with our predictions. These findings challenge the role of the RBR loop as a merely auxiliary loop and might change our view of the clock molecular circuitry and of the function of the nuclear receptors (REV-ERB and ROR) as a putative driving force of molecular oscillations.

Author Summary

Most organisms have evolved an internal clock which allows them to anticipate and react to the light/dark daily rhythm and is able to generate oscillation with a circa 24 hour rhythm. A molecular network involving feedback loops is responsible for the rhythm generation. A large number of clock-controlled genes pass on time messages and control several biological processes. In spite of its medical importance (role in cancer, sleep disorders, diabetes and others) the mechanism of action of the circadian clock and the role of its constituent's feedback loops remains partially unknown. Using a mathematical model, we were able to bring insight in open circadian biology questions. Firstly, increasing the mRNA degradation rate of Per can contribute to increase or decrease of the period which might explain contradictory experimental findings. Secondly, our data points to a more relevant role of the ROR/Bmal/REV-ERB loop. In particular, that this loop can be an oscillator on its own. We provide experimental evidence that overexpression of members of the ROR/Bmal/REV-ERB lead to loss of Bmal reporter mRNA oscillations. The fact that REV-ERB and ROR are nuclear receptors and therefore important regulators in many cellular processes might have important implications for molecular biology and medicine.

The development of systemic approaches in biology has put emphasis on identifying genetic modules whose behavior can be modeled accurately so as to gain insight into their structure and function. However, most gene circuits in a cell are under control of external signals and thus, quantitative agreement between experimental data and a mathematical model is difficult. Circadian biology has been one notable exception: quantitative models of the internal clock that orchestrates biological processes over the 24-hour diurnal cycle have been constructed for a few organisms, from cyanobacteria to plants and mammals. In most cases, a complex architecture with interlocked feedback loops has been evidenced. Here we present the first modeling results for the circadian clock of the green unicellular alga Ostreococcus tauri. Two plant-like clock genes have been shown to play a central role in the Ostreococcus clock. We find that their expression time profiles can be accurately reproduced by a minimal model of a two-gene transcriptional feedback loop. Remarkably, best adjustment of data recorded under light/dark alternation is obtained when assuming that the oscillator is not coupled to the diurnal cycle. This suggests that coupling to light is confined to specific time intervals and has no dynamical effect when the oscillator is entrained by the diurnal cycle. This intringuing property may reflect a strategy to minimize the impact of fluctuations in daylight intensity on the core circadian oscillator, a type of perturbation that has been rarely considered when assessing the robustness of circadian clocks.

Author Summary

Circadian clocks keep time of day in many living organisms, allowing them to anticipate environmental changes induced by day/night alternation. They consist of networks of genes and proteins interacting so as to generate biochemical oscillations with a period close to 24 hours. Circadian clocks synchronize to the day/night cycle through the year principally by sensing ambient light. Depending on the weather, the perceived light intensity can display large fluctuations within the day and from day to day, potentially inducing unwanted resetting of the clock. Furthermore, marine organisms such as microalgae are subjected to dramatic changes in light intensities in the water column due to streams and wind. We showed, using mathematical modelling, that the green unicellular marine alga Ostreococcus tauri has evolved a simple but effective strategy to shield the circadian clock from daylight fluctuations by localizing coupling to the light during specific time intervals. In our model, as in experiments, coupling is invisible when the clock is in phase with the day/night cycle but resets the clock when it is out of phase. Such a clock architecture is immune to strong daylight fluctuations.

Molecular models have been described for the circadian clocks of representatives of several different taxa. Much of the work on the plant circadian system has been carried out using the thale cress, Arabidopsis thaliana, as a model. We discuss the roles of genes implicated in the plant circadian system, with special emphasis on Arabidopsis. Plants have an endogenous clock that regulates many aspects of circadian and photoperiodic behaviour. Despite the discovery of components that resemble those involved in the clocks of animals or fungi, no coherent model of the plant clock has yet been proposed. In this review, we aim to provide an overview of studies of the Arabidopsis circadian system. We shall compare these with results from different taxa and discuss them in the context of what is known about clocks in other organisms.

SUMMARY

Molecular aspects of the circadian clock in the cyanobacterium Synechococcus elongatus have been described in great detail. Three-dimensional structures have been determined for the three proteins, KaiA, KaiB, and KaiC, that comprise a central oscillator of the clock. Moreover, a temperature-compensated circadian rhythm of KaiC phosphorylation can be reconstituted in vitro with the addition of KaiA, KaiB, and ATP. These data suggest a relatively simple circadian system in which a single oscillator provides temporal information for all downstream processes. However, in vivo the situation is more complex, and additional components contribute to the maintenance of a normal period, the resetting of relative phases of circadian oscillations, and the control of rhythms of gene expression. We show here that two well-studied promoters in the S. elongatus genome report different circadian periods of expression under a given set of conditions in wild-type as well as mutant genetic backgrounds. Moreover, the period differs between these promoters with respect to modulation by light intensity, growth phase, and the presence or absence of a promoter-recognition subunit of RNA polymerase. These data contrast sharply with the current clock model in which a single Kai-based oscillator governs circadian period. Overall, these findings suggest that complex interactions between the circadian oscillator, perhaps other oscillators, and other cellular machinery result in a clock that is plastic and sensitive to the environment and to the physiological state of the cell.

Predictive understanding of cell signaling network operation based on general prior knowledge but consistent with empirical data in a specific environmental context is a current challenge in computational biology. Recent work has demonstrated that Boolean logic can be used to create context-specific network models by training proteomic pathway maps to dedicated biochemical data; however, the Boolean formalism is restricted to characterizing protein species as either fully active or inactive. To advance beyond this limitation, we propose a novel form of fuzzy logic sufficiently flexible to model quantitative data but also sufficiently simple to efficiently construct models by training pathway maps on dedicated experimental measurements. Our new approach, termed constrained fuzzy logic (cFL), converts a prior knowledge network (obtained from literature or interactome databases) into a computable model that describes graded values of protein activation across multiple pathways. We train a cFL-converted network to experimental data describing hepatocytic protein activation by inflammatory cytokines and demonstrate the application of the resultant trained models for three important purposes: (a) generating experimentally testable biological hypotheses concerning pathway crosstalk, (b) establishing capability for quantitative prediction of protein activity, and (c) prediction and understanding of the cytokine release phenotypic response. Our methodology systematically and quantitatively trains a protein pathway map summarizing curated literature to context-specific biochemical data. This process generates a computable model yielding successful prediction of new test data and offering biological insight into complex datasets that are difficult to fully analyze by intuition alone.

Author Summary

Over the past few years, many methods have been developed to construct large-scale networks from the literature or databases of genetic and physical interactions. With the advent of high-throughput biochemical methods, it is also possible to measure the states and activities of many proteins in these biochemical networks under different conditions of cellular stimulation and perturbation. Here we use constrained fuzzy logic to systematically compare interaction networks to experimental data. This systematic comparison elucidates interactions that were theoretically possible but not actually operating in the biological system of interest, as well as data that was not described by interactions in the prior knowledge network, pointing to a need to increase our knowledge in specific parts of the network. Furthermore, the result of this comparison is a trained, quantitative model that can be used to make a priori quantitative predictions about how the cellular protein network will respond in conditions not initially tested.

Background

Sensitivity and robustness are essential properties of circadian clock systems, enabling them to respond to the environment but resist noisy variations. These properties should be recapitulated in computational models of the circadian clock. Highly nonlinear kinetics and multiple loops are often incorporated into models to match experimental time-series data, but these also impact on model properties for clock models.

Methodology/Principal Findings

Here, we study the consequences of complicated structure and nonlinearity using simple Goodwin-type oscillators and the complex Arabidopsis circadian clock models. Sensitivity analysis of the simple oscillators implies that an interlocked multi-loop structure reinforces sensitivity/robustness properties, enhancing the response to external and internal variations. Furthermore, we found that reducing the degree of nonlinearity could sometimes enhance the robustness of models, implying that ad hoc incorporation of nonlinearity could be detrimental to a model's perceived credibility.

Conclusion

The correct multi-loop structure and degree of nonlinearity are therefore critical in contributing to the desired properties of a model as well as its capacity to match experimental data.

The negative feedback model for gene regulation of the circadian mechanism is described for the fruitfly, Drosophila melanogaster. The conservation of function of clock molecules is illustrated by comparison with the mammalian circadian system, and the apparent swapping of roles between various canonical clock gene components is highlighted. The role of clock gene duplications and divergence of function is introduced via the timeless gene. The impressive similarities in clock gene regulation between flies and mammals could suggest that variation between more closely related species within insects might be minimal. However, this is not borne out because the expression of clock molecules in the brain of the giant silk moth, Antheraea pernyi, is not easy to reconcile with the negative feedback roles of the period and timeless genes. Variation in clock gene sequences between and within fly species is examined and the role of co-evolution between and within clock molecules is described, particularly with reference to adaptive functions of the circadian phenotype.

An accurate mathematical model of the mammalian circadian clock provides novel insights into the mechanisms that generate 24-h rhythms. A double-negative feedback loop design is proposed for biological clocks whose period needs to be tightly regulated.

A 1–1 stoichiometric balance and tight binding between activators (PER–CRY) and repressors (BMAL1–CLOCK/NPAS2) is required for sustained rhythmicity.Stoichiometry is balanced by an additional negative feedback loop consisting of a stable activator.Our detailed model can explain more experimental data than previous models.Mathematical analysis of a simple model supports our claims.

Circadian (∼24 h) timekeeping is essential for the lives of many organisms. To understand the biochemical mechanisms of this timekeeping, we have developed a detailed mathematical model of the mammalian circadian clock. Our model can accurately predict diverse experimental data including the phenotypes of mutations or knockdown of clock genes as well as the time courses and relative expression of clock transcripts and proteins. Using this model, we show how a universal motif of circadian timekeeping, where repressors tightly bind activators rather than directly binding to DNA, can generate oscillations when activators and repressors are in stoichiometric balance. Furthermore, we find that an additional slow negative feedback loop preserves this stoichiometric balance and maintains timekeeping with a fixed period. The role of this mechanism in generating robust rhythms is validated by analysis of a simple and general model and a previous model of the Drosophila circadian clock. We propose a double-negative feedback loop design for biological clocks whose period needs to be tightly regulated even with large changes in gene dosage.

By regulating the metabolism of fatty acids, carbohydrates, and xenobiotic, the mammalian circadian clock plays a fundamental role on the liver physiology. At present, it is supposed that the circadian clock regulates metabolism mostly by regulating the expression of liver enzymes at the transcriptional level. However, recent evidences suggest that some signaling pathways synchronized by the circadian clock can also influence metabolism at a post-transcriptional level. In this context, we have recently shown that the circadian clock synchronizes the rhythmic activation of the IRE1α pathway in the endoplasmic reticulum. The absence of circadian clock perturbs this secondary clock, provokes deregulation of endoplasmic reticulum-localized enzymes, and leads to impaired lipid metabolism. We will describe here the additional pathways synchronized by the clock and discussed the influence of the circadian clock-controlled feeding rhythm on them.

Many organisms have evolved molecular clocks to anticipate daily changes in their environment. The molecular mechanisms by which the circadian clock network produces sustained cycles have extensively been studied and transcriptional-translational feedback loops are common structures to many organisms. Although a simple or single feedback loop is sufficient for sustained oscillations, circadian clocks implement multiple, complicated feedback loops. In general, different types of feedback loops are suggested to affect the robustness and entrainment of circadian rhythms.

To reveal the mechanism by which such a complex feedback system evolves, we quantify the robustness and light entrainment of four competing models: the single, semi-dual, dual, and redundant feedback models. To extract the global properties of those models, all plausible kinetic parameter sets that generate circadian oscillations are searched to characterize their oscillatory features. To efficiently perform such analyses, we used the two-phase search (TPS) method as a fast and non-biased search method and quasi-multiparameter sensitivity (QMPS) as a fast and exact measure of robustness to uncertainty of all kinetic parameters.

So far the redundant feedback model has been regarded as the most robust oscillator, but our extensive analysis corrects or overcomes this hypothesis. The dual feedback model, which is employed in biology, provides the most robust oscillator to multiple parameter perturbations within a cell and most readily entrains to a wide range of light-dark cycles. The kinetic symmetry between the dual loops and their coupling via a protein complex are found critically responsible for robust and entrainable oscillations. We first demonstrate how the dual feedback architecture with kinetic symmetry evolves out of many competing feedback systems.

Analysis of the cyanobacterial circadian biological clock reveals a complex interdependence between a transcription/translation feedback loop and a biochemical oscillator.

Cyanobacteria are the only model circadian clock system in which a circadian oscillator can be reconstituted in vitro. The underlying circadian mechanism appears to comprise two subcomponents: a post-translational oscillator (PTO) and a transcriptional/translational feedback loop (TTFL). The PTO and TTFL have been hypothesized to operate as dual oscillator systems in cyanobacteria. However, we find that they have a definite hierarchical interdependency—the PTO is the core pacemaker while the TTFL is a slave oscillator that quickly damps when the PTO stops. By analysis of overexpression experiments and mutant clock proteins, we find that the circadian system is dependent upon the PTO and that suppression of the PTO leads to damped TTFL-based oscillations whose temperature compensation is not stable under different metabolic conditions. Mathematical modeling indicates that the experimental data are compatible with a core PTO driving the TTFL; the combined PTO/TTFL system is resilient to noise. Moreover, the modeling indicates a mechanism by which the TTFL can feed into the PTO such that new synthesis of clock proteins can phase-shift or entrain the core PTO pacemaker. This prediction was experimentally tested and confirmed by entraining the in vivo circadian system with cycles of new clock protein synthesis that modulate the phosphorylation status of the clock proteins in the PTO. In cyanobacteria, the PTO is the self-sustained core pacemaker that can operate independently of the TTFL, but the TTFL damps when the phosphorylation status of the PTO is clamped. However, the TTFL can provide entraining input into the PTO. This study is the first to our knowledge to experimentally and theoretically investigate the dynamics of a circadian clock in which a PTO is coupled to a TTFL. These results have important implications for eukaryotic clock systems in that they can explain how a TTFL could appear to be a core circadian clockwork when in fact the true pacemaker is an embedded biochemical oscillator.

Author Summary

Many organisms from bacteria to humans have evolved circadian mechanisms for regulating biological processes on a daily time scale. In cyanobacteria, a minimal system for such cyclical regulation can be reconstituted in vitro from three proteins, called KaiA, KaiB, and KaiC. This three-protein oscillator is believed to regulate the cyclical activities in vivo through a post-translational mechanism that involves rhythmic phosphorylation of KaiC. Although this post-translational oscillator (PTO) is sufficient for generating rhythms in vitro, the cyanobacterial circadian system in vivo also includes a transcriptional/translational feedback loop (TTFL). The precise roles of the PTO and the TTFL and their interdependence in forming the complete clock system in vivo are unclear. By manipulating wild-type and mutant clock protein expression in vivo, we here show that the cyanobacterial circadian system is dependent upon the biochemical oscillator provided by the PTO and that suppression of the PTO leads to a residual damped (slave) oscillation that results from the TTFL. Mathematical modeling shows that the experimental data are compatible with a mechanism in which the PTO acts as a pacemaker to drive the activity of the TTFL. Moreover, our analyses suggest a mechanism by which the TTFL can feed back into the PTO such that new synthesis of the Kai proteins entrains the core PTO pacemaker. Therefore, the PTO and TTFL appear to have a definite hierarchical interdependency: the PTO is a self-sustained core pacemaker that can oscillate independently of the TTFL, but the TTFL is a slave oscillator that damps when the phosphorylation status of KaiC in the PTO is clamped. The core circadian pacemaker in eukaryotes is thought to be a TTFL, but our results with cyanobacteria have important implications for eukaryotic clock systems in that they can explain how a TTFL could appear to be the core clock when in fact the true pacemaker is an embedded biochemical oscillator.

Circadian rhythms in physiology and behavior are regulated by a master clock resident in the suprachiasmatic nucleus (SCN) of the hypothalamus, and dysfunctions in the circadian system can lead to serious health effects. This paper reviews the organization of the SCN as the brain clock, how it regulates gonadal hormone secretion, and how androgens modulate aspects of circadian behavior known to be regulated by the SCN. We show that androgen receptors are restricted to a core SCN region that receives photic input as well as afferents from arousal systems in the brain. We suggest that androgens modulate circadian behavior directly via actions on the SCN and that both androgens and estrogens modulate circadian rhythms through an indirect route, by affecting overall activity and arousal levels. Thus, this system has multiple levels of regulation; the SCN regulates circadian rhythms in gonadal hormone secretion, and hormones feed back to influence SCN functions.

The temperature compensated in vitro oscillation of cyanobacterial KaiC phosphorylation, the first example of a thermodynamically closed system showing circadian rhythmicity, only involves the three Kai proteins (KaiA, KaiB, and KaiC) and ATP. In this paper, we describe a model in which the KaiA- and KaiB-assisted autocatalytic phosphorylation and dephosphorylation of KaiC are the source for circadian rhythmicity. This model, based upon autocatalysis instead of transcription-translation negative feedback, shows temperature-compensated circadian limit-cycle oscillations with KaiC phosphorylation profiles and has period lengths and rate constant values that are consistent with experimental observations.

Synopsis

Circadian rhythms are a central feature of biological systems. In cyanobacteria, the clock involves three major proteins: KaiA, KaiB, and KaiC, with KaiC showing autophosphorylation in the presence of ATP. Remarkably, by incubating the purified Kai proteins with ATP, the clock can be reconstituted in vitro. The authors were intrigued by the simplicity of this oscillator and its connection to chemical oscillatory reactions, and saw the possibility for a realistic reaction kinetic representation. Their study represents a synthetic, predictive, and dynamic explanation for the in vitro KaiC circadian clock based on the self-amplifying response (“autocatalysis”) of autophosphorylating kinases. The presented model is based on existing biological and biochemical observations and recapitulates observed experimental features, including temperature compensation, which has been notably recalcitrant to explanation. The model provides several predictions including that period length and the ratio between phosphorylated and unphosphorylated KaiC in clock mutants are closely linked and related to the stability of the ternary complex formed between the three Kai proteins. The model also predicts the occurrence of bistability and that evolution can move simple bistable systems into oscillatory systems by changing only one rate constant.

A significant amount of attention has recently been focused on modeling of gene regulatory networks. Two frequently used large-scale modeling frameworks are Bayesian networks (BNs) and Boolean networks, the latter one being a special case of its recent stochastic extension, probabilistic Boolean networks (PBNs). PBN is a promising model class that generalizes the standard rule-based interactions of Boolean networks into the stochastic setting. Dynamic Bayesian networks (DBNs) is a general and versatile model class that is able to represent complex temporal stochastic processes and has also been proposed as a model for gene regulatory systems. In this paper, we concentrate on these two model classes and demonstrate that PBNs and a certain subclass of DBNs can represent the same joint probability distribution over their common variables. The major benefit of introducing the relationships between the models is that it opens up the possibility of applying the standard tools of DBNs to PBNs and vice versa. Hence, the standard learning tools of DBNs can be applied in the context of PBNs, and the inference methods give a natural way of handling the missing values in PBNs which are often present in gene expression measurements. Conversely, the tools for controlling the stationary behavior of the networks, tools for projecting networks onto sub-networks, and efficient learning schemes can be used for DBNs. In other words, the introduced relationships between the models extend the collection of analysis tools for both model classes.

Circadian clocks provide an internal measure of external time allowing organisms to anticipate and exploit predictable daily changes in the environment. Rhythms driven by circadian clocks have a temperature compensated periodicity of approximately 24 hours that persists in constant conditions and can be reset by environmental time cues. Computational modelling has aided our understanding of the molecular mechanisms of circadian clocks, nevertheless it remains a major challenge to integrate the large number of clock components and their interactions into a single, comprehensive model that is able to account for the full breadth of clock phenotypes. Here we present a comprehensive dynamic model of the Neurospora crassa circadian clock that incorporates its key components and their transcriptional and post-transcriptional regulation. The model accounts for a wide range of clock characteristics including: a periodicity of 21.6 hours, persistent oscillation in constant conditions, arrhythmicity in constant light, resetting by brief light pulses, and entrainment to full photoperiods. Crucial components influencing the period and amplitude of oscillations were identified by control analysis. Furthermore, simulations enabled us to propose a mechanism for temperature compensation, which is achieved by simultaneously increasing the translation of frq RNA and decreasing the nuclear import of FRQ protein.

Author Summary

Circadian clocks are internal timekeepers that integrate signals from the environment and orchestrate cellular events to occur at the most favourable time of day. Circadian clocks in animals, plants, fungi and bacteria have similar characteristic properties and molecular architecture. They have a periodicity of approximately 24 hours, persist in constant conditions and can be reset by environmental time cues such as light and temperature. Another essential property, whose molecular basis is poorly understood, is that the period is temperature compensated i.e. it remains the same over a range of temperatures. Computational modelling has become a valuable tool to predict and understand the underlying mechanisms of such complex molecular systems, but existing clock models are often restricted in the scope of molecular reactions they cover and in the breadth of conditions they are able to reproduce. We therefore built a comprehensive model of the circadian clock of the fungus Neurospora crassa, which encompasses existing knowledge of the biochemistry of the Neurospora clock. We validated this model against a wide range of experimental phenotypes and then used the model to investigate possible molecular explanations of temperature compensation. Our simulations suggest that temperature compensation of period is achieved by changing the abundance and cellular localisation of a key clock protein.

Circadian clocks organize behavior and physiology to adapt to daily environmental cycles. Genetic approaches in the fruit fly, Drosophila melanogaster, have revealed widely conserved molecular gears of these 24-h timers. Yet much less is known about how these cell-autonomous clocks confer temporal information to modulate cellular functions. Here we discuss our current knowledge of circadian clock function in Drosophila, providing an overview of the molecular underpinnings of circadian clocks. We then describe the neural network important for circadian rhythms of locomotor activity, including how these molecular clocks might influence neuronal function. Finally, we address a range of behaviors and physiological systems regulated by circadian clocks, including discussion of specific peripheral oscillators and key molecular effectors where they have been described. These studies reveal a remarkable complexity to circadian pathways in this “simple” model organism.

Background

A popular model for gene regulatory networks is the Boolean network model. In this paper, we propose an algorithm to perform an analysis of gene regulatory interactions using the Boolean network model and time-series data. Actually, the Boolean network is restricted in the sense that only a subset of all possible Boolean functions are considered. We explore some mathematical properties of the restricted Boolean networks in order to avoid the full search approach. The problem is modeled as a Constraint Satisfaction Problem (CSP) and CSP techniques are used to solve it.

Results

We applied the proposed algorithm in two data sets. First, we used an artificial dataset obtained from a model for the budding yeast cell cycle. The second data set is derived from experiments performed using HeLa cells. The results show that some interactions can be fully or, at least, partially determined under the Boolean model considered.

Conclusions

The algorithm proposed can be used as a first step for detection of gene/protein interactions. It is able to infer gene relationships from time-series data of gene expression, and this inference process can be aided by a priori knowledge available.

Complex systems are often modeled as Boolean networks in attempts to capture their logical structure and reveal its dynamical consequences. Approximating the dynamics of continuous variables by discrete values and Boolean logic gates may, however, introduce dynamical possibilities that are not accessible to the original system. We show that large random networks of variables coupled through continuous transfer functions often fail to exhibit the complex dynamics of corresponding Boolean models in the disordered (chaotic) regime, even when each individual function appears to be a good candidate for Boolean idealization. A suitably modified Boolean theory explains the behavior of systems in which information does not propagate faithfully down certain chains of nodes. Model networks incorporating calculated or directly measured transfer functions reported in the literature on transcriptional regulation of genes are described by the modified theory.

Motivation: Combinatorial effects, in which several variables jointly influence an output or response, play an important role in biological systems. In many settings, Boolean functions provide a natural way to describe such influences. However, biochemical data using which we may wish to characterize such influences are usually subject to much variability. Furthermore, in high-throughput biological settings Boolean relationships of interest are very often sparse, in the sense of being embedded in an overall dataset of higher dimensionality. This motivates a need for statistical methods capable of making inferences regarding Boolean functions under conditions of noise and sparsity.

Results: We put forward a statistical model for sparse, noisy Boolean functions and methods for inference under the model. We focus on the case in which the form of the underlying Boolean function, as well as the number and identity of its inputs are all unknown. We present results on synthetic data and on a study of signalling proteins in cancer biology.

Availability: go.warwick.ac.uk/sachmukherjee/sci

Contact: s.n.mukherjee@warwick.ac.uk

Background

Depression has been associated with several circadian rhythm perturbations, suggesting a disruption of the circadian clock system in affective disorders. The interaction of several circadian clock genes generates these daily circadian rhythms.

Methods

This cross-sectional study evaluated whether circadian gene expression differed between individuals with a history of depression and participants without a similar history. The participants were 60 healthy older adults. Half of the participants had a history of depression. Real-time quantitative polymerase chain reaction was used to measure the circadian gene Clock, BMAL1, Period1, and Period2 messenger RNA levels in peripheral blood leukocytes.

Results

Individuals with a history of depression had higher Clock, Period1, and Bmal1 mRNA levels, compared to non-depressed participants.

Limitations

Although circadian gene expression fluctuates throughout the day, clock genes mRNA levels were evaluated only in the morning.

Conclusions

These results suggest that disruptions of the molecular mechanisms underlying the circadian clock system may be associated with depression.