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1.  Quantitative analysis of regulatory flexibility under changing environmental conditions 
Day length changes with the seasons in temperate latitudes, affecting the many biological rhythms that entrain to the day/night cycle: we measure these effects on the expression of Arabidopsis clock genes, using RNA and reporter gene readouts, with a new method of phase analysis.Dusk sensitivity is proposed as a simple, natural and general mathematical measure to analyse and manipulate the changing phase of a clock output relative to the change in the day/night cycle.Dusk sensitivity shows how increasing the numbers of feedback loops in the Arabidopsis clock models allows more flexible regulation, consistent with a previously-proposed, general operating principle of biological networks.The Arabidopsis clock genes show flexibility of regulation that is characteristic of a three-loop clock model, validating aspects of the model and the operating principle, but some clock output genes show greater flexibility arising from direct light regulation.
The analysis of dynamic, non-linear regulation with the aid of mechanistic models is central to Systems Biology. This study compares the predictions of mechanistic, mathematical models of the circadian clock with molecular time-series data on rhythmic gene expression in the higher plant Arabidopsis thaliana. Analysis of the models helps us to understand (explain and predict) how the clock gene circuit balances regulation by external and endogenous factors to achieve particular behaviours. Such multi-factorial regulation is ubiquitous in, and characteristic of, living systems.
The Earth's rotation causes predictable changes in the environment, notably in the availability of sunlight for photosynthesis. Many biological processes are driven by the environmental input via sensory pathways, for example, from photoreceptors. Circadian clocks provide an alternative strategy. These endogenous, 24-h rhythms can drive biological processes that anticipate the regular environmental changes, rather than merely responding. Many rhythmic processes have both light and clock control. Indeed, the clock components themselves must balance internal timing with external inputs, because circadian clocks are reset daily through light regulation of one or more clock components. This process of entrainment is complicated by the change in day length. When the times of dawn and dusk move apart in summer, and closer together in winter, does the clock track dawn, track dusk or interpolate between them?
In plants, the clock controls leaf and petal movements, the opening and closing of stomatal pores, the discharge of floral fragrances, and many metabolic activities, especially those associated with photosynthesis. Centuries of physiological studies have shown that these rhythms can behave differently. Flowering in Ipomoea nil (Pharbitis nil, Japanese morning glory) is controlled by a rhythm that tracks the time of dusk, to give a classic example. We showed that two other rhythms associated with vegetative growth track dawn in this species (Figure 5A), so the clock system allows flexible regulation.
The relatively small number of components involved in the circadian clockwork makes it an ideal candidate for mathematical modelling. Molecular genetic studies in a variety of model eukaryotes have shown that the circadian rhythm is generated by a network of 6–20 genes. These genes form feedback loops generating a rhythm in mRNA production. A single negative feedback loop in which a gene encodes a protein that, after several hours, turns off transcription is capable of generating a circadian rhythm, in principle. A single light input can entrain the clock to ‘local time', synchronised with a light–dark cycle. However, real circadian clocks have proven to be more complicated than this, with multiple light inputs and interlocked feedback loops.
We have previously argued from mathematical analysis that multi-loop networks increase the flexibility of regulation (Rand et al, 2004) and have shown that appropriately deployed flexibility can confer functional robustness (Akman et al, 2010). Here we test whether that flexibility can be demonstrated in vivo, in the model plant, A. thaliana. The Arabidopsis clock mechanism comprises a feedback loop in which two partially redundant, myb transcription factors, LATE ELONGATED HYPOCOTYL (LHY) and CIRCADIAN CLOCK ASSOCIATED 1 (CCA1), repress the expression of their activator, TIMING OF CAB EXPRESSION 1 (TOC1). We previously modelled this single-loop circuit and showed that it was not capable of recreating important data (Locke et al, 2005a). An extended, two-loop model was developed to match observed behaviours, incorporating a hypothetical gene Y, for which the best identified candidate was the GIGANTEA gene (GI) (Locke et al, 2005b). Two further models incorporated the TOC1 homologues PSEUDO-RESPONSE REGULATOR (PRR) 9 and PRR7 (Locke et al, 2006; Zeilinger et al, 2006). In these circuits, a morning oscillator (LHY/CCA1–PRR9/7) is coupled to an evening oscillator (Y/GI–TOC1) via the original LHY/CCA1–TOC1 loop.
These clock models, like those for all other organisms, were developed using data from simple conditions of constant light, darkness or 12-h light–12-h dark cycles. We therefore tested how the clock genes in Arabidopsis responded to light–dark cycles with different photoperiods, from 3 h light to 18 h light per 24-h cycle (Edinburgh, 56° North latitude, has 17.5 h light in midsummer). The time-series assays of mRNA and in vivo reporter gene images showed a range of peak times for different genes, depending on the photoperiod (Figure 5C). A new data analysis method, mFourfit, was introduced to measure the peak times, in the Biological Rhythms Analysis Software Suite (BRASS v3.0). None of the genes showed the dusk-tracking behaviour characteristic of the Ipomoea flowering rhythm. The one-, two- and three-loop models were analysed to understand the observed patterns. A new mathematical measure, dusk sensitivity, was introduced to measure the change in timing of a model component versus a change in the time of dusk. The one- and two-loop models tracked dawn and dusk, respectively, under all conditions. Only the three-loop model (Figure 5B) had the flexibility required to match the photoperiod-dependent changes that we found in vivo, and in particular the unexpected, V-shaped pattern in the peak time of TOC1 expression. This pattern of regulation depends on the structure and light inputs to the model's evening oscillator, so the in vivo data supported this aspect of the model. LHY and CCA1 gene expression under short photoperiods showed greater dusk sensitivity, in the interval 2–6 h before dawn, than the three-loop model predicted, so these data will help to constrain future models.
The approach described here could act as a template for experimental biologists seeking to understand biological regulation using dynamic, experimental perturbations and time-series data. Simulation of mathematical models (despite known imperfections) can provide contrasting hypotheses that guide understanding. The system's detailed behaviour is complex, so a natural and general measure such as dusk sensitivity is helpful to focus on one property of the system. We used the measure to compare models, and to predict how this property could be manipulated. To enable additional analysis of this system, we provide the time-series data and experimental metadata online.
The circadian clock controls 24-h rhythms in many biological processes, allowing appropriate timing of biological rhythms relative to dawn and dusk. Known clock circuits include multiple, interlocked feedback loops. Theory suggested that multiple loops contribute the flexibility for molecular rhythms to track multiple phases of the external cycle. Clear dawn- and dusk-tracking rhythms illustrate the flexibility of timing in Ipomoea nil. Molecular clock components in Arabidopsis thaliana showed complex, photoperiod-dependent regulation, which was analysed by comparison with three contrasting models. A simple, quantitative measure, Dusk Sensitivity, was introduced to compare the behaviour of clock models with varying loop complexity. Evening-expressed clock genes showed photoperiod-dependent dusk sensitivity, as predicted by the three-loop model, whereas the one- and two-loop models tracked dawn and dusk, respectively. Output genes for starch degradation achieved dusk-tracking expression through light regulation, rather than a dusk-tracking rhythm. Model analysis predicted which biochemical processes could be manipulated to extend dusk tracking. Our results reveal how an operating principle of biological regulators applies specifically to the plant circadian clock.
doi:10.1038/msb.2010.81
PMCID: PMC3010117  PMID: 21045818
Arabidopsis thaliana; biological clocks; dynamical systems; gene regulatory networks; mathematical models; photoperiodism
2.  Data assimilation constrains new connections and components in a complex, eukaryotic circadian clock model 
Integrating molecular time-series data resulted in a more robust model of the plant clock, which predicts that a wave of inhibitory PRR proteins controls the morning genes LHY and CCA1.PRR5 is experimentally validated as a late-acting component of this wave.The family of sequentially expressed PRR proteins allows flexible entrainment of the clock, whereas a single protein could not, suggesting that the duplication of clock genes might confer this generic, functional advantage.The observed post-translational regulation of the evening protein TOC1 by interaction with ZTL and GI remains consistent with an indirect activation of TOC1 mRNA expression by GI, which was previously postulated from modelling.
Circadian rhythms are present in most eukaryotic organisms including plants. The core genes of the circadian clock are very important for plant physiology as they drive the rhythmic expression of around 30% of Arabidopsis genes (Edwards et al, 2006; Michael et al, 2008). The clock is normally entrained by daily environmental changes in light and temperature. Oscillations also persist under constant environmental conditions in a laboratory. The clock gene circuit in Arabidopsis is based on multiple interlocked feedback loops, which are typical of circadian genetic networks in other organisms (Dunlap and Loros, 2004; Bell-Pedersen et al, 2005). Mechanistic, mathematical models are increasingly useful in analysing and understanding how the observed molecular components give rise to the rhythmic behaviour of this dynamic, non-linear system.
Our previous model of Arabidopsis circadian clock (Locke et al, 2006) presented the core, three-loop structure of the clock, which comprised morning and evening oscillators and coupling between them (Figure 1). The morning loop included the dawn-expressed LATE ELONGATED HYPOCOTYL (LHY) and CIRCADIAN CLOCK ASSOCIATED 1 (CCA1) genes, which negatively regulate their expression through activation of the inhibitor proteins, PSEUDO-RESPONSE REGULATOR 9 (PRR9) and PRR7. These were described by a single, combined model component, PRR9/7. The evening loop included the dusk-expressed gene TIMING OF CAB EXPRESSION 1 (TOC1), which negatively regulates itself through inhibition of a hypothetical activator, gene Y. The evening-expressed gene GIGANTEA (GI) contributes to Y function. The morning and evening loops were connected through inhibition of the evening genes by LHY/CCA1 and activation of LHY/CCA1 expression by a hypothetical evening gene X. Here, we extend the previous model of circadian gene expression (Locke et al, 2006) based on recently published data (Figure 1). The new model retains the good match of our previous model to the large volume of molecular time-series data, and improves the behaviour of the model clock system under a range of light conditions and in a wider range of mutants.
The morning loop was extended by adding a hypothetical clock component, the night inhibitor (NI), which acts together with PRR9 and PRR7 to keep the expression of LHY and CCA1 at low levels over a broad interval spanning dusk. This regulation is important to set the phase of LHY/CCA1 expression at dawn. Data from the literature suggested that the PRR5 gene was a candidate for NI, leading us to predict that the sequentially expressed PRR9, PRR7 and PRR5 proteins together formed a wave of inhibitors of LHY and CCA1. This hypothesis was tested under discriminating light conditions, in which the light interval is replaced with the dawn and dusk pulses of light to form a ‘skeleton photoperiod'. Combining this protocol with mutation of the PRR7 and/or PRR5 genes, our new experimental results validated the model predictions and confirmed that PRR5 contributes to the function that we modelled as NI. During revision of this paper, that result received further experimental support (Nakamichi et al, 2010).
Model simulations revealed the functional importance of the inhibitor wave in entraining the clock to the light/dark cycle. Separating PRR9 from the other inhibitors in the model showed how the strong light activation observed for this gene contributes to more rapid entrainment. The observed, post-translation regulation of all three inhibitor proteins by light (Farre and Kay, 2007; Ito et al, 2007; Kiba et al, 2007) was also included in the model. Light-regulated degradation provides a molecular mechanism to explain the later phase of LHY and CCA1 expression under long photoperiods compared with short photoperiods, in line with experimental observations.
The connection between evening and morning loops was revised by including the inhibition of the morning gene PRR9 by the evening component TOC1, based on the data on TOC1-overexpressing plants (Makino et al, 2002; Ito et al, 2005). This inhibition causes a delay of PRR9 expression relative to LHY/CCA1, which allows LHY/CCA1 to reach a higher expression level at dawn. Our simulations showed that a partial mutant that lacks this inhibition of PRR9 by TOC1 is sufficient to cause the higher level of PRR9 and the short circadian period observed in toc1 mutant plants.
The evening loop was extended by introducing the observed, post-translational regulation of the TOC1 protein by the F-box protein ZEITLUPE (ZTL) and stabilization of ZTL by its interaction with GI in the presence of light (Kim et al, 2007). GI's function in the clock model has thus been revised according to the data: GI promotes an inhibition of TOC1 protein function through positive regulation of ZTL. This results, together with negative regulation of Y by TOC1, in indirect activation of TOC1 mRNA expression by GI, which agrees with our earlier experimental data (Locke et al, 2006). Simulations showed that the post-translational regulation of TOC1 by ZTL and GI results in the observed long period of the ztl mutant and fast dampening of rhythms in the lhy/cca1/gi triple mutant.
This is the first mathematical model that incorporates the observed post-translational regulation into the genetic network of the Arabidopsis clock. In addition to specific, mechanistic insights, the model shows a generic advantage from the duplication of clock genes and their expression at different phases. Such clock gene duplications are observed in eukaryotes with larger genomes, such as the mouse. Analogous, functional duplication can be achieved by differential regulation of a single clock gene in distinct cells, as in Drosophila.
Circadian clocks generate 24-h rhythms that are entrained by the day/night cycle. Clock circuits include several light inputs and interlocked feedback loops, with complex dynamics. Multiple biological components can contribute to each part of the circuit in higher organisms. Mechanistic models with morning, evening and central feedback loops have provided a heuristic framework for the clock in plants, but were based on transcriptional control. Here, we model observed, post-transcriptional and post-translational regulation and constrain many parameter values based on experimental data. The model's feedback circuit is revised and now includes PSEUDO-RESPONSE REGULATOR 7 (PRR7) and ZEITLUPE. The revised model matches data in varying environments and mutants, and gains robustness to parameter variation. Our results suggest that the activation of important morning-expressed genes follows their release from a night inhibitor (NI). Experiments inspired by the new model support the predicted NI function and show that the PRR5 gene contributes to the NI. The multiple PRR genes of Arabidopsis uncouple events in the late night from light-driven responses in the day, increasing the flexibility of rhythmic regulation.
doi:10.1038/msb.2010.69
PMCID: PMC2964123  PMID: 20865009
Arabidopsis thaliana; biological clocks; circadian rhythms; mathematical model; systems biology
3.  Extension of a genetic network model by iterative experimentation and mathematical analysis 
Molecular Systems Biology  2005;1:2005.0013.
We extend the current model of the plant circadian clock, in order to accommodate new and published data. Throughout our model development we use a global parameter search to ensure that any limitations we find are due to the network architecture and not to our selection of the parameter values, which have not been determined experimentally. Our final model includes two, interlocked loops of gene regulation and is reminiscent of the circuit structures previously identified by experiments on insect and fungal clocks. It is the first Arabidopsis clock model to show such good correspondence to experimental data.Our interlocked feedback loop model predicts the regulation of two unknown components. Experiments motivated by these predictions identify the GIGANTEA gene as a strong candidate for one component, with an unexpected pattern of light regulation.*
This study involves an iterative approach of mathematical modelling and experiment to develop an accurate mathematical model of the circadian clock in the higher plant Arabidopsis thaliana. Our approach is central to systems biology and should lead to a greater, quantitative understanding of the circadian clock, as well as being more widely relevant to research into genetic networks.
The day–night cycle caused by the Earth's rotation affects most organisms, and has resulted in the evolution of the circadian clock. The circadian clock controls 24-h rhythms in processes from metabolism to behaviour; in higher eukaryotes, the circadian clock controls the rhythmic expression of 5–10% of genes. In plants, the clock controls leaf and petal movements, the opening and closing of stomatal pores, the discharge of floral fragrances and many metabolic activities, especially those associated with photosynthesis.
The relatively small number of components involved in the central circadian network makes it an ideal candidate for mathematical modelling of complex biological regulation. Genetic studies in a variety of model organisms have shown that the circadian rhythm is generated by a central network of between 6 and 12 genes. These genes form feedback loops generating a rhythm in mRNA production. One negative feedback loop in which a gene encodes a protein that, after several hours, turns off transcription is, in principle, capable of creating a circadian rhythm. However, real circadian clocks have proven to be more complicated than this, with interlocked feedback loops. Networks of this complexity are more easily understood through mathematical modelling.
The clock mechanism in the model plant, A. thaliana, was first proposed to comprise a feedback loop in which two partially redundant genes, LATE ELONGATED HYPOCOTYL (LHY) and CIRCADIAN CLOCK ASSOCIATED 1 (CCA1), repress the expression of their activator, TIMING OF CAB EXPRESSION 1 (TOC1). We previously modelled this preliminary network and showed that it was not capable of recreating several important pieces of experimental data (Locke et al, 2005). Here, we extend the LHY/CCA1–TOC1 network in new mathematical models. To check the effects of each addition to the network, the outputs of the extended models are compared to published data and to new experiments.
As is the case for most biological networks, the parameter values in our model, such as the translation rate of TOC1 protein, are unknown. We employ here an optimisation method, which works well with noisy and varied data and allows a global search of parameter space. This should ensure that the limitations we find in our networks are due to the network structure, and not to our parameter choices.
Our final interlocked feedback loop model requires two hypothetical components, genes X and Y (Figure 4), but is the first Arabidopsis clock model to exhibit such a good correspondence with experimental data. The model simulates a residual short-period oscillation in the cca1;lhy mutant, as characterised by our experiments. No single-loop model is able to do this. Our model also matches experimental data under constant light (LL) conditions and correctly senses photoperiod. The model predicts an interlocked feedback loop structure similar to that seen in the circadian clock mechanisms of other organisms.
The interlocked feedback loop model predicts a distinctive pattern of Y mRNA accumulation in the wild type (WT) and in the cca1;lhy double mutant, with Y mRNA levels increasing transiently at dawn. We designed an experiment to identify Y based on this prediction. GIGANTEA (GI) mRNA levels fit very well to our predicted profile for Y (Figure 6), identifying GI as a strong candidate for Y.
The approach described here could act as a template for experimental biologists seeking to extend models of small genetic networks. Our results illustrate the usefulness of mathematical modelling in guiding experiments, even if the models are based on limited data. Our method provides a way of identifying suitable candidate networks and quantifying how these networks better describe a wide variety of experimental measurements. The characteristics of new putative genes are thereby obtained, facilitating the experimental search for new components. To facilitate future experimental design, we provide user-friendly software that is specifically designed for numerical simulation of circadian experiments using models for several species (Brown, 2004b).
*Footnote: Synopsis highlights were added on 5 July 2005.
Circadian clocks involve feedback loops that generate rhythmic expression of key genes. Molecular genetic studies in the higher plant Arabidopsis thaliana have revealed a complex clock network. The first part of the network to be identified, a transcriptional feedback loop comprising TIMING OF CAB EXPRESSION 1 (TOC1), LATE ELONGATED HYPOCOTYL (LHY) and CIRCADIAN CLOCK ASSOCIATED 1 (CCA1), fails to account for significant experimental data. We develop an extended model that is based upon a wider range of data and accurately predicts additional experimental results. The model comprises interlocking feedback loops comparable to those identified experimentally in other circadian systems. We propose that each loop receives input signals from light, and that each loop includes a hypothetical component that had not been explicitly identified. Analysis of the model predicted the properties of these components, including an acute light induction at dawn that is rapidly repressed by LHY and CCA1. We found this unexpected regulation in RNA levels of the evening-expressed gene GIGANTEA (GI), supporting our proposed network and making GI a strong candidate for this component.
doi:10.1038/msb4100018
PMCID: PMC1681447  PMID: 16729048
biological rhythms; gene network; mathematical modelling; parameter estimation
4.  A novel computational model of the circadian clock in Arabidopsis that incorporates PRR7 and PRR9 
We developed a mathematical model of the Arabidopsis circadian clock, including PRR7 and PRR9, which is able to predict several single, double and triple mutant phenotypes.Sensitivity Analysis was used to identify the properties and time sensing mechanisms of model structures.PRR7 and CCA1/LHY were identified as weak points of the mathematical model indicating where more experimental data is needed for further model development.Detailed dynamical studies showed that the timing of an evening light sensing element is essential for day length responsiveness
In recent years, molecular genetic techniques have revealed a complex network of components in the Arabidopsis circadian clock. Mathematical models allow for a detailed study of the dynamics and architecture of such complex gene networks leading to a better understanding of the genetic interactions. It is important to maintain a constant iteration with experimentation, to include novel components as they are discovered and use the updated model to design new experiments. This study develops a framework to introduce new components into the mathematical model of the Arabidopsis circadian clock accelerating the iterative model development process and gaining insight into the system's properties.
We used the interlocked feedback loop model published in Locke et al (2005) as the base model. In Arabidopsis, the first suggested regulatory loop involves the morning expressed transcription factors CIRCADIAN CLOCK-ASSOCIATED 1 (CCA1) and LATE ELONGATED HYPOCOTYL (LHY), and the evening expressed pseudo-response regulator TIMING OF CAB EXPRESSION (TOC1). The hypothetical component X had been introduced to realize a longer delay between gene expression of CCA1/LHY and TOC1. The introduction of Y was motivated by the need for a mechanism to reproduce the dampening short period rhythms of the cca1/lhy double mutant and to include an additional light input at the end of the day.
In this study, the new components pseudo-response regulators PRR7 and PRR9 were added in negative feedback loops based on the biological hypothesis that they are activated by LHY and in turn repress LHY transcription (Farré et al, 2005; Figure 1). We present three iterations steps of model development (Figure 1A–C).
A wide range of tools was used to establish and analyze new model structures. One of the challenges facing mathematical modeling of biological processes is parameter identification; they are notoriously difficult to determine experimentally. We established an optimization procedure based on an evolutionary strategy with a cost function mainly derived from wild-type characteristics. This ensured that the model was not restricted by a specific set of parameters and enabled us to use a large set of biological mutant information to assess the predictive capability of the model structure. Models were evaluated by means of an extended phenotype catalogue, allowing for an easy and fair comparison of the structures. We also carried out detailed simulation analysis of component interactions to identify weak points in the structure and suggest further modifications. Finally, we applied sensitivity analysis in a novel manner, using it to direct the model development. Sensitivity analysis provides quantitative measures of robustness; the two measures in this study were the traces of component concentrations over time (classical state sensitivities) and phase behavior (measured by the phase response curve). Three major results emerged from the model development process.
First, the iteration process helped us to learn about general characteristics of the system. We observed that the timing of Y expression is critical for evening light entrainment, which enables the system to respond to changes in day length. This is important for our understanding of the mechanism of light input to the clock and will add in the identification of biological candidates for this function. In addition, our results suggest that a detailed description of the mechanisms of genetic interactions is important for the systems behavior. We observed that the introduction of an experimentally based precise light regulation mechanism on PRR9 expression had a significant effect on the systems behavior.
Second, the final model structure (Figure 1C) was capable of predicting a wide range of mutant phenotypes, such as a reduction of TOC1 expression by RNAi (toc1RNAi), mutations in PRR7 and PRR9 and the novel mutant combinations prr9toc1RNAi and prr7prr9toc1RNAi. However, it was unable to predict the mutations in CCA1 and LHY.
Finally, sensitivity analysis identified the weak points of the system. The developed model structure was heavily based on the TOC1/Y feedback loop. This could explain the model's failure to predict the cca1lhy double mutant phenotype. More detailed information on the regulation of CCA1 and LHY expression will be important to achieve the right balance between the different regulatory loops in the mathematical model. This is in accordance with genetic studies that have identified several genes involved in the regulation of LHY and CCA1 expression. The identification of their mechanism of action will be necessary for the next model development.
In plants, as in animals, the core mechanism to retain rhythmic gene expression relies on the interaction of multiple feedback loops. In recent years, molecular genetic techniques have revealed a complex network of clock components in Arabidopsis. To gain insight into the dynamics of these interactions, new components need to be integrated into the mathematical model of the plant clock. Our approach accelerates the iterative process of model identification, to incorporate new components, and to systematically test different proposed structural hypotheses. Recent studies indicate that the pseudo-response regulators PRR7 and PRR9 play a key role in the core clock of Arabidopsis. We incorporate PRR7 and PRR9 into an existing model involving the transcription factors TIMING OF CAB (TOC1), LATE ELONGATED HYPOCOTYL (LHY) and CIRCADIAN CLOCK ASSOCIATED (CCA1). We propose candidate models based on experimental hypotheses and identify the computational models with the application of an optimization routine. Validation is accomplished through systematic analysis of various mutant phenotypes. We introduce and apply sensitivity analysis as a novel tool for analyzing and distinguishing the characteristics of proposed architectures, which also allows for further validation of the hypothesized structures.
doi:10.1038/msb4100101
PMCID: PMC1682023  PMID: 17102803
Arabidopsis; circadian rhythms; mathematical modeling; parameter optimization; sensitivity analysis
5.  Robustness of Circadian Clocks to Daylight Fluctuations: Hints from the Picoeucaryote Ostreococcus tauri 
PLoS Computational Biology  2010;6(11):e1000990.
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.
doi:10.1371/journal.pcbi.1000990
PMCID: PMC2978692  PMID: 21085637
6.  Coupling of a Core Post-Translational Pacemaker to a Slave Transcription/Translation Feedback Loop in a Circadian System 
PLoS Biology  2010;8(6):e1000394.
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.
doi:10.1371/journal.pbio.1000394
PMCID: PMC2885980  PMID: 20563306
7.  Entrainment of the Mammalian Cell Cycle by the Circadian Clock: Modeling Two Coupled Cellular Rhythms 
PLoS Computational Biology  2012;8(5):e1002516.
The cell division cycle and the circadian clock represent two major cellular rhythms. These two periodic processes are coupled in multiple ways, given that several molecular components of the cell cycle network are controlled in a circadian manner. For example, in the network of cyclin-dependent kinases (Cdks) that governs progression along the successive phases of the cell cycle, the synthesis of the kinase Wee1, which inhibits the G2/M transition, is enhanced by the complex CLOCK-BMAL1 that plays a central role in the circadian clock network. Another component of the latter network, REV-ERBα, inhibits the synthesis of the Cdk inhibitor p21. Moreover, the synthesis of the oncogene c-Myc, which promotes G1 cyclin synthesis, is repressed by CLOCK-BMAL1. Using detailed computational models for the two networks we investigate the conditions in which the mammalian cell cycle can be entrained by the circadian clock. We show that the cell cycle can be brought to oscillate at a period of 24 h or 48 h when its autonomous period prior to coupling is in an appropriate range. The model indicates that the combination of multiple modes of coupling does not necessarily facilitate entrainment of the cell cycle by the circadian clock. Entrainment can also occur as a result of circadian variations in the level of a growth factor controlling entry into G1. Outside the range of entrainment, the coupling to the circadian clock may lead to disconnected oscillations in the cell cycle and the circadian system, or to complex oscillatory dynamics of the cell cycle in the form of endoreplication, complex periodic oscillations or chaos. The model predicts that the transition from entrainment to 24 h or 48 h might occur when the strength of coupling to the circadian clock or the level of growth factor decrease below critical values.
Author Summary
The cell cycle and the circadian clock are two major cellular rhythms. These two periodic processes are tightly coupled through multiple regulatory interactions; several components of the cell cycle machinery are indeed controlled by the circadian network. By using detailed computational models for the cell cycle and circadian networks we investigate the conditions in which the mammalian cell cycle can be entrained by the circadian clock. We show that entrainment to a circadian period can occur when the period of the cell cycle prior to coupling is either smaller or larger than 24 h. Entrainment to 48 h can also be observed. The presence of multiple modes of coupling does not enlarge the domain of entrainment. Coupling to the circadian clock may also lead to complex oscillatory dynamics of the cell cycle in the form of endoreplication, complex periodic oscillations, or chaotic oscillations. The model predicts that entrainment of the cell cycle could also result from the circadian variation of a growth factor gating entry into G1, and that the transition from an entrained period of 24 h to 48 h might result from a decrease in coupling strength or in the level of growth factor.
doi:10.1371/journal.pcbi.1002516
PMCID: PMC3364934  PMID: 22693436
8.  Machine Learning Helps Identify CHRONO as a Circadian Clock Component 
PLoS Biology  2014;12(4):e1001840.
Two independent studies, one of them using a computational approach, identified CHRONO, a gene shown to modulate the activity of circadian transcription factors and alter circadian behavior in mice.
Over the last decades, researchers have characterized a set of “clock genes” that drive daily rhythms in physiology and behavior. This arduous work has yielded results with far-reaching consequences in metabolic, psychiatric, and neoplastic disorders. Recent attempts to expand our understanding of circadian regulation have moved beyond the mutagenesis screens that identified the first clock components, employing higher throughput genomic and proteomic techniques. In order to further accelerate clock gene discovery, we utilized a computer-assisted approach to identify and prioritize candidate clock components. We used a simple form of probabilistic machine learning to integrate biologically relevant, genome-scale data and ranked genes on their similarity to known clock components. We then used a secondary experimental screen to characterize the top candidates. We found that several physically interact with known clock components in a mammalian two-hybrid screen and modulate in vitro cellular rhythms in an immortalized mouse fibroblast line (NIH 3T3). One candidate, Gene Model 129, interacts with BMAL1 and functionally represses the key driver of molecular rhythms, the BMAL1/CLOCK transcriptional complex. Given these results, we have renamed the gene CHRONO (computationally highlighted repressor of the network oscillator). Bi-molecular fluorescence complementation and co-immunoprecipitation demonstrate that CHRONO represses by abrogating the binding of BMAL1 to its transcriptional co-activator CBP. Most importantly, CHRONO knockout mice display a prolonged free-running circadian period similar to, or more drastic than, six other clock components. We conclude that CHRONO is a functional clock component providing a new layer of control on circadian molecular dynamics.
Author Summary
Daily rhythms are ever-present in the living world, driving the sleep–wake cycle and many other physiological changes. In the last two decades, several labs have identified “clock genes” that interact to generate underlying molecular oscillations. However, many aspects of circadian molecular physiology remain unexplained. Here, we used a simple “machine learning” approach to identify new clock genes by searching the genome for candidate genes that share clock-like features such as cycling, broad-based tissue RNA expression, in vitro circadian activity, genetic interactions, and homology across species. Genes were ranked by their similarity to known clock components and the candidates were screened and validated for evidence of clock function in vitro. One candidate, which we renamed CHRONO (Gm129), interacted with the master regulator of the clock, BMAL1, disrupting its transcriptional activity. We found that Chrono knockout mice had prolonged locomotor activity rhythms, getting up progressively later each day. Our experiments demonstrated that CHRONO interferes with the ability of BMAL1 to recruit CBP, a bona fide histone acetylase and key transcriptional coactivator of the circadian clock.
doi:10.1371/journal.pbio.1001840
PMCID: PMC3988006  PMID: 24737000
9.  Adaptive Temperature Compensation in Circadian Oscillations 
PLoS Computational Biology  2012;8(7):e1002585.
A temperature independent period and temperature entrainment are two defining features of circadian oscillators. A default model of distributed temperature compensation satisfies these basic facts yet is not easily reconciled with other properties of circadian clocks, such as many mutants with altered but temperature compensated periods. The default model also suggests that the shape of the circadian limit cycle and the associated phase response curves (PRC) will vary since the average concentrations of clock proteins change with temperature. We propose an alternative class of models where the twin properties of a fixed period and entrainment are structural and arise from an underlying adaptive system that buffers temperature changes. These models are distinguished by a PRC whose shape is temperature independent and orbits whose extrema are temperature independent. They are readily evolved by local, hill climbing, optimization of gene networks for a common quality measure of biological clocks, phase anticipation. Interestingly a standard realization of the Goodwin model for temperature compensation displays properties of adaptive rather than distributed temperature compensation.
Author Summary
Circadian clocks are biological oscillators which evolved to couple the internal rhythm of animals, plants and even some bacteria to the alternation of light and day. Circadian oscillators are temperature compensated, i.e. they keep a 24-h period irrespective of the temperature of the organism. This is surprising, since many biochemical parameters, including average concentration of clock proteins, vary with temperature. From dynamical system theory, we therefore expect changes in both period and relative lengths of features in the phase response curve which are not seen. We couple mathematical modelling and computational evolution of gene networks to formulate a novel explanation for temperature compensation that accords better with experimental facts than alternatives. Our model has deep mathematical connections with the process of biochemical adaptation, by which cells respond to temporal gradients of signals rather than their absolute value.
doi:10.1371/journal.pcbi.1002585
PMCID: PMC3395600  PMID: 22807663
10.  Timing of Neuropeptide Coupling Determines Synchrony and Entrainment in the Mammalian Circadian Clock 
PLoS Computational Biology  2014;10(4):e1003565.
Robust synchronization is a critical feature of several systems including the mammalian circadian clock. The master circadian clock in mammals consists of about 20000 ‘sloppy’ neuronal oscillators within the hypothalamus that keep robust time by synchronization driven by inter-neuronal coupling. The complete understanding of this synchronization in the mammalian circadian clock and the mechanisms underlying it remain an open question. Experiments and computational studies have shown that coupling individual oscillators can achieve robust synchrony, despite heterogeneity and different network topologies. But, much less is known regarding the mechanisms and circuits involved in achieving this coupling, due to both system complexity and experimental limitations. Here, we computationally study the coupling mediated by the primary coupling neuropeptide, vasoactive intestinal peptide (VIP) and its canonical receptor, VPAC2R, using the transcriptional elements and generic mode of VIP-VPAC2R signaling. We find that synchrony is only possible if VIP (an inducer of Per expression) is released in-phase with activators of Per expression. Moreover, anti-phasic VIP release suppresses coherent rhythms by moving the network into a desynchronous state. Importantly, experimentally observed rhythms in VPAC2R have little effect on network synchronization, but can improve the amplitude of the SCN network rhythms while narrowing the network entrainment range. We further show that these findings are valid across several computational network models. Thus, we identified a general design principle to achieve robust synchronization: An activating coupling agent, such as VIP, must act in-phase with the activity of core-clock promoters. More generally, the phase of coupling is as critical as the strength of coupling from the viewpoint of synchrony and entrainment.
Author Summary
Synchronization among multiple oscillators is a common theme in many biological and engineered systems. Here, we look at its use by the mammalian biological clock to keep accurate time. Through biochemical interactions among a network of inaccurate neuron clocks, a strong precise clock is produced. Although we are gradually learning more about these biochemical interactions, the details still remain largely unclear. Studies, both computational and experimental, have shown that the strength of the rhythmic interaction critically decides if a system can synchronize, i.e., the interactions must be strong enough. In this work, we show that the rhythmic interaction between these neuronal clocks must be timed correctly (in the right phase) in addition to being strong enough to synchronize the network. Activating (repressing) interactions must coincide with other activators (repressors) in each neuron to achieve synchrony. Since this principle imposes certain properties on synchronizing interactions, these properties can be used to identify and understand novel interaction mechanisms. Further, these principles are applicable to interactions between cellular oscillators in other tissues and organisms.
doi:10.1371/journal.pcbi.1003565
PMCID: PMC3990482  PMID: 24743470
11.  Stochastic Boolean networks: An efficient approach to modeling gene regulatory networks 
BMC Systems Biology  2012;6:113.
Background
Various computational models have been of interest due to their use in the modelling of gene regulatory networks (GRNs). As a logical model, probabilistic Boolean networks (PBNs) consider molecular and genetic noise, so the study of PBNs provides significant insights into the understanding of the dynamics of GRNs. This will ultimately lead to advances in developing therapeutic methods that intervene in the process of disease development and progression. The applications of PBNs, however, are hindered by the complexities involved in the computation of the state transition matrix and the steady-state distribution of a PBN. For a PBN with n genes and N Boolean networks, the complexity to compute the state transition matrix is O(nN22n) or O(nN2n) for a sparse matrix.
Results
This paper presents a novel implementation of PBNs based on the notions of stochastic logic and stochastic computation. This stochastic implementation of a PBN is referred to as a stochastic Boolean network (SBN). An SBN provides an accurate and efficient simulation of a PBN without and with random gene perturbation. The state transition matrix is computed in an SBN with a complexity of O(nL2n), where L is a factor related to the stochastic sequence length. Since the minimum sequence length required for obtaining an evaluation accuracy approximately increases in a polynomial order with the number of genes, n, and the number of Boolean networks, N, usually increases exponentially with n, L is typically smaller than N, especially in a network with a large number of genes. Hence, the computational efficiency of an SBN is primarily limited by the number of genes, but not directly by the total possible number of Boolean networks. Furthermore, a time-frame expanded SBN enables an efficient analysis of the steady-state distribution of a PBN. These findings are supported by the simulation results of a simplified p53 network, several randomly generated networks and a network inferred from a T cell immune response dataset. An SBN can also implement the function of an asynchronous PBN and is potentially useful in a hybrid approach in combination with a continuous or single-molecule level stochastic model.
Conclusions
Stochastic Boolean networks (SBNs) are proposed as an efficient approach to modelling gene regulatory networks (GRNs). The SBN approach is able to recover biologically-proven regulatory behaviours, such as the oscillatory dynamics of the p53-Mdm2 network and the dynamic attractors in a T cell immune response network. The proposed approach can further predict the network dynamics when the genes are under perturbation, thus providing biologically meaningful insights for a better understanding of the dynamics of GRNs. The algorithms and methods described in this paper have been implemented in Matlab packages, which are attached as Additional files.
doi:10.1186/1752-0509-6-113
PMCID: PMC3532238  PMID: 22929591
12.  Modelling the widespread effects of TOC1 signalling on the plant circadian clock and its outputs 
BMC Systems Biology  2013;7:23.
Background
24-hour biological clocks are intimately connected to the cellular signalling network, which complicates the analysis of clock mechanisms. The transcriptional regulator TOC1 (TIMING OF CAB EXPRESSION 1) is a founding component of the gene circuit in the plant circadian clock. Recent results show that TOC1 suppresses transcription of multiple target genes within the clock circuit, far beyond its previously-described regulation of the morning transcription factors LHY (LATE ELONGATED HYPOCOTYL) and CCA1 (CIRCADIAN CLOCK ASSOCIATED 1). It is unclear how this pervasive effect of TOC1 affects the dynamics of the clock and its outputs. TOC1 also appears to function in a nested feedback loop that includes signalling by the plant hormone Abscisic Acid (ABA), which is upregulated by abiotic stresses, such as drought. ABA treatments both alter TOC1 levels and affect the clock’s timing behaviour. Conversely, the clock rhythmically modulates physiological processes induced by ABA, such as the closing of stomata in the leaf epidermis. In order to understand the dynamics of the clock and its outputs under changing environmental conditions, the reciprocal interactions between the clock and other signalling pathways must be integrated.
Results
We extended the mathematical model of the plant clock gene circuit by incorporating the repression of multiple clock genes by TOC1, observed experimentally. The revised model more accurately matches the data on the clock’s molecular profiles and timing behaviour, explaining the clock’s responses in TOC1 over-expression and toc1 mutant plants. A simplified representation of ABA signalling allowed us to investigate the interactions of ABA and circadian pathways. Increased ABA levels lengthen the free-running period of the clock, consistent with the experimental data. Adding stomatal closure to the model, as a key ABA- and clock-regulated downstream process allowed to describe TOC1 effects on the rhythmic gating of stomatal closure.
Conclusions
The integrated model of the circadian clock circuit and ABA-regulated environmental sensing allowed us to explain multiple experimental observations on the timing and stomatal responses to genetic and environmental perturbations. These results crystallise a new role of TOC1 as an environmental sensor, which both affects the pace of the central oscillator and modulates the kinetics of downstream processes.
doi:10.1186/1752-0509-7-23
PMCID: PMC3614443  PMID: 23506153
Circadian rhythms; Biological clocks; Gene regulatory networks; Mathematical model; Systems biology
13.  Effect of Network Architecture on Synchronization and Entrainment Properties of the Circadian Oscillations in the Suprachiasmatic Nucleus 
PLoS Computational Biology  2012;8(3):e1002419.
In mammals, the suprachiasmatic nucleus (SCN) of the hypothalamus constitutes the central circadian pacemaker. The SCN receives light signals from the retina and controls peripheral circadian clocks (located in the cortex, the pineal gland, the liver, the kidney, the heart, etc.). This hierarchical organization of the circadian system ensures the proper timing of physiological processes. In each SCN neuron, interconnected transcriptional and translational feedback loops enable the circadian expression of the clock genes. Although all the neurons have the same genotype, the oscillations of individual cells are highly heterogeneous in dispersed cell culture: many cells present damped oscillations and the period of the oscillations varies from cell to cell. In addition, the neurotransmitters that ensure the intercellular coupling, and thereby the synchronization of the cellular rhythms, differ between the two main regions of the SCN. In this work, a mathematical model that accounts for this heterogeneous organization of the SCN is presented and used to study the implication of the SCN network topology on synchronization and entrainment properties. The results show that oscillations with larger amplitude can be obtained with scale-free networks, in contrast to random and local connections. Networks with the small-world property such as the scale-free networks used in this work can adapt faster to a delay or advance in the light/dark cycle (jet lag). Interestingly a certain level of cellular heterogeneity is not detrimental to synchronization performances, but on the contrary helps resynchronization after jet lag. When coupling two networks with different topologies that mimic the two regions of the SCN, efficient filtering of pulse-like perturbations in the entrainment pattern is observed. These results suggest that the complex and heterogeneous architecture of the SCN decreases the sensitivity of the network to short entrainment perturbations while, at the same time, improving its adaptation abilities to long term changes.
Author Summary
In order to adapt to their cycling environment, virtually all living organisms have developed an internal timer, the circadian clock. In mammals, the circadian pacemaker is composed of about 20,000 neurons, called the suprachiasmatic nucleus (SCN) located in the hypothalamus. The SCN receives light signals from the retina and controls peripheral circadian clocks to ensure the proper timing of physiological processes. In each SCN neuron, a genetic regulatory network enables the circadian expression of the clock genes, but individual dynamics are highly heterogeneous in dispersed cell culture: many cells present damped oscillations and the period of the oscillations varies from cell to cell. In addition, the neurotransmitters that ensure the intercellular coupling, and thereby the synchronization of the cellular rhythms, differ between the two main regions of the SCN. We present here a mathematical model that accounts for this heterogeneous organization of the SCN and study the implication of the network topology on synchronization and entrainment properties. Our results show that cellular heterogeneity may help the resynchronization after jet lag and suggest that the complex architecture of the SCN decreases the sensitivity of the network to short entrainment perturbations while, at the same time, improving its adaptation abilities to long term changes.
doi:10.1371/journal.pcbi.1002419
PMCID: PMC3297560  PMID: 22423219
14.  Differentially Timed Extracellular Signals Synchronize Pacemaker Neuron Clocks 
PLoS Biology  2014;12(9):e1001959.
Circadian pacemaker neurons in Drosophila are regulated by two synchronizing signals that are released at opposite times of day, generating a rhythm in intracellular cyclic AMP.
Synchronized neuronal activity is vital for complex processes like behavior. Circadian pacemaker neurons offer an unusual opportunity to study synchrony as their molecular clocks oscillate in phase over an extended timeframe (24 h). To identify where, when, and how synchronizing signals are perceived, we first studied the minimal clock neural circuit in Drosophila larvae, manipulating either the four master pacemaker neurons (LNvs) or two dorsal clock neurons (DN1s). Unexpectedly, we found that the PDF Receptor (PdfR) is required in both LNvs and DN1s to maintain synchronized LNv clocks. We also found that glutamate is a second synchronizing signal that is released from DN1s and perceived in LNvs via the metabotropic glutamate receptor (mGluRA). Because simultaneously reducing Pdfr and mGluRA expression in LNvs severely dampened Timeless clock protein oscillations, we conclude that the master pacemaker LNvs require extracellular signals to function normally. These two synchronizing signals are released at opposite times of day and drive cAMP oscillations in LNvs. Finally we found that PdfR and mGluRA also help synchronize Timeless oscillations in adult s-LNvs. We propose that differentially timed signals that drive cAMP oscillations and synchronize pacemaker neurons in circadian neural circuits will be conserved across species.
Author Summary
Circadian molecular clocks are essential for daily cycles in animal behavior and we have a good understanding of how these clocks work in individual pacemaker neurons. However, the accuracy of these individual clocks is meaningless unless they are synchronized with one another. In this study we show that synchronizing the principal pacemaker LNv neurons in Drosophila larvae require two extracellular signals that are received at opposite times of day: namely, the neuropeptide PDF released from LNvs themselves at dawn and glutamate released from dorsal clock neurons at dusk. LNvs perceive both PDF and glutamate via G-protein coupled receptors that increase or decrease intracellular cAMP, respectively. The alternating phases of PDF and glutamate release generate oscillations in intracellular cyclic AMP. In addition to maintaining synchrony between LNvs, this rhythm is also required for molecular clock oscillations in individual larval LNvs. We show that disruption of PDF and glutamate signaling also reduces synchrony in adult LNvs. This impairs the oscillations of clock proteins and flies have delayed onset of sleep. Our data highlight the importance of intercellular signaling in ensuring synchrony between clock neurons within the circadian network. Our findings help extend the conservation of clock properties between Drosophila and mammals beyond clock genes to include clock circuitry.
doi:10.1371/journal.pbio.1001959
PMCID: PMC4181961  PMID: 25268747
15.  Minimum Criteria for DNA Damage-Induced Phase Advances in Circadian Rhythms 
PLoS Computational Biology  2009;5(5):e1000384.
Robust oscillatory behaviors are common features of circadian and cell cycle rhythms. These cyclic processes, however, behave distinctively in terms of their periods and phases in response to external influences such as light, temperature, nutrients, etc. Nevertheless, several links have been found between these two oscillators. Cell division cycles gated by the circadian clock have been observed since the late 1950s. On the other hand, ionizing radiation (IR) treatments cause cells to undergo a DNA damage response, which leads to phase shifts (mostly advances) in circadian rhythms. Circadian gating of the cell cycle can be attributed to the cell cycle inhibitor kinase Wee1 (which is regulated by the heterodimeric circadian clock transcription factor, BMAL1/CLK), and possibly in conjunction with other cell cycle components that are known to be regulated by the circadian clock (i.e., c-Myc and cyclin D1). It has also been shown that DNA damage-induced activation of the cell cycle regulator, Chk2, leads to phosphorylation and destruction of a circadian clock component (i.e., PER1 in Mus or FRQ in Neurospora crassa). However, the molecular mechanism underlying how DNA damage causes predominantly phase advances in the circadian clock remains unknown. In order to address this question, we employ mathematical modeling to simulate different phase response curves (PRCs) from either dexamethasone (Dex) or IR treatment experiments. Dex is known to synchronize circadian rhythms in cell culture and may generate both phase advances and delays. We observe unique phase responses with minimum delays of the circadian clock upon DNA damage when two criteria are met: (1) existence of an autocatalytic positive feedback mechanism in addition to the time-delayed negative feedback loop in the clock system and (2) Chk2-dependent phosphorylation and degradation of PERs that are not bound to BMAL1/CLK.
Author Summary
Molecular components and mechanisms that connect cell cycle and circadian rhythms are important for the well-being of an organism. Cell cycle machinery regulates the progress of cell growth and division while the circadian rhythm network generates an ∼24 h time-keeping mechanism that regulates the daily processes of an organism (i.e. metabolism, bowel movements, body temperature, etc.). It is observed that cell divisions usually occur during a certain time window of a day, which indicated that there are circadian-gated cell divisions. Moreover, it's been shown that mice are more prone to develop cancer when certain clock genes are mutated resulting in an arrhythmic clock. Recently, a cell cycle checkpoint regulator, Chk2, was identified as a component that influences a core clock component and creates mostly phase advances (i.e., jet lags due to traveling east) in circadian rhythms upon DNA damage. This phase response with minimum delays is an unexpected result, and the molecular mechanism behind this phenomenon remains unknown. Our computational analyses of a mathematical model reveal two molecular criteria that account for the experimentally observed phase responses of the circadian clock upon DNA damage. These results demonstrate how circadian clock regulation by cell cycle checkpoint controllers provides another layer of complexity for efficient DNA damage responses.
doi:10.1371/journal.pcbi.1000384
PMCID: PMC2677641  PMID: 19424508
16.  Synchronization-Induced Rhythmicity of Circadian Oscillators in the Suprachiasmatic Nucleus 
PLoS Computational Biology  2007;3(4):e68.
The suprachiasmatic nuclei (SCN) host a robust, self-sustained circadian pacemaker that coordinates physiological rhythms with the daily changes in the environment. Neuronal clocks within the SCN form a heterogeneous network that must synchronize to maintain timekeeping activity. Coherent circadian output of the SCN tissue is established by intercellular signaling factors, such as vasointestinal polypeptide. It was recently shown that besides coordinating cells, the synchronization factors play a crucial role in the sustenance of intrinsic cellular rhythmicity. Disruption of intercellular signaling abolishes sustained rhythmicity in a majority of neurons and desynchronizes the remaining rhythmic neurons. Based on these observations, the authors propose a model for the synchronization of circadian oscillators that combines intracellular and intercellular dynamics at the single-cell level. The model is a heterogeneous network of circadian neuronal oscillators where individual oscillators are damped rather than self-sustained. The authors simulated different experimental conditions and found that: (1) in normal, constant conditions, coupled circadian oscillators quickly synchronize and produce a coherent output; (2) in large populations, such oscillators either synchronize or gradually lose rhythmicity, but do not run out of phase, demonstrating that rhythmicity and synchrony are codependent; (3) the number of oscillators and connectivity are important for these synchronization properties; (4) slow oscillators have a higher impact on the period in mixed populations; and (5) coupled circadian oscillators can be efficiently entrained by light–dark cycles. Based on these results, it is predicted that: (1) a majority of SCN neurons needs periodic synchronization signal to be rhythmic; (2) a small number of neurons or a low connectivity results in desynchrony; and (3) amplitudes and phases of neurons are negatively correlated. The authors conclude that to understand the orchestration of timekeeping in the SCN, intracellular circadian clocks cannot be isolated from their intercellular communication components.
Author Summary
Circadian rhythms, characterized by a period close to 24 h, are observed in nearly all living organisms, from cyanobacteria to plants, insects, and mammals. In mammals, the central circadian clock is located in the suprachiasmatic nucleus (SCN) of the hypothalamus, where it receives light signals from the retina. In turn, the SCN controls circadian rhythms in peripheral tissues and behavioral activity. The SCN is composed of about 20,000 neurons characterized by a small size and a high density. Within each individual neuron, clock genes and proteins compose interlocked regulatory feedback loops that generate circadian oscillations on the molecular level. SCN neurons dispersed in cell cultures display cell-autonomous oscillations, with periods ranging from 20 h to 28 h. The ventrolateral part of the SCN receives light input from the retina, serving as a relay for the dorsomedial part. Coupling and synchronization among SCN neurons are ensured by neurotransmitters. A desire to understand how such a network of heterogeneous circadian oscillators achieves a synchronous and coherent output rhythm has motivated extensive experimental and theoretical work. In this paper, we present a molecular model combining intracellular and extracellular dynamics for the SCN circadian system, and propose a novel synchronization mechanism. Our results predict a dual role for the coupling factors within the SCN, both in maintaining the rhythmicity and in promoting the synchronization between the circadian oscillators.
doi:10.1371/journal.pcbi.0030068
PMCID: PMC1851983  PMID: 17432930
17.  Analysis of a Gene Regulatory Cascade Mediating Circadian Rhythm in Zebrafish 
PLoS Computational Biology  2013;9(2):e1002940.
In the study of circadian rhythms, it has been a puzzle how a limited number of circadian clock genes can control diverse aspects of physiology. Here we investigate circadian gene expression genome-wide using larval zebrafish as a model system. We made use of a spatial gene expression atlas to investigate the expression of circadian genes in various tissues and cell types. Comparison of genome-wide circadian gene expression data between zebrafish and mouse revealed a nearly anti-phase relationship and allowed us to detect novel evolutionarily conserved circadian genes in vertebrates. We identified three groups of zebrafish genes with distinct responses to light entrainment: fast light-induced genes, slow light-induced genes, and dark-induced genes. Our computational analysis of the circadian gene regulatory network revealed several transcription factors (TFs) involved in diverse aspects of circadian physiology through transcriptional cascade. Of these, microphthalmia-associated transcription factor a (mitfa), a dark-induced TF, mediates a circadian rhythm of melanin synthesis, which may be involved in zebrafish's adaptation to daily light cycling. Our study describes a systematic method to discover previously unidentified TFs involved in circadian physiology in complex organisms.
Author Summary
For most animals whose lives are dependent on the sun, circadian clocks govern their daily behaviors and physiology. In different animals, novel functions under the circadian clock's control can evolve as adaptations to their specific environment. A zebrafish demonstrates a remarkably high level of interplay between external light and its internal circadian clock due to its transparent nature. In a genome-wide study, we identified a large number of circadian oscillating genes as well as genes whose expression is highly sensitive to the light or dark in zebrafish. Our computational analysis of gene regulatory networks revealed a number of transcription factors (TFs) that mediate novel circadian functions. We investigated one example in depth, a key TF that relays the control of the circadian clock to the enzymes synthesizing melanin in a dark-induced pathway thus causing the daily change of pigmentation in zebrafish. This dark-induced circadian melanogenesis can lead to an anticipatory change in zebrafish skin color allowing zebrafish to adapt to its environment. This mechanism allows zebrafish to better evade predators and effectively adjust its daily light-sensitivity in the pigment cells. Our study provides an excellent example of how the circadian clock is adapted in a specific organism to control its behavior, thus enabling evolutionary adaptation to the organism's ecological niche.
doi:10.1371/journal.pcbi.1002940
PMCID: PMC3585402  PMID: 23468616
18.  Modeling the Drosophila melanogaster Circadian Oscillator via Phase Optimization 
Journal of biological rhythms  2008;23(6):525-537.
The circadian clock, which coordinates daily physiological behaviors of most organisms, maintains endogenous (approximately 24 h) cycles and simultaneously synchronizes to the 24-h environment due to its inherent robustness to environmental perturbations coupled with a sensitivity to specific environmental stimuli. In this study, the authors develop a detailed mathematical model that characterizes the Drosophila melanogaster circadian network. This model incorporates the transcriptional regulation of period, time-less, vrille, PAR-domain protein 1, and clock gene and protein counterparts. The interlocked positive and negative feedback loops that arise from these clock components are described primarily through mass-action kinetics (with the exception of regulated gene expression) and without the use of explicit time delays. System parameters are estimated via a genetic algorithm-based optimization of a cost function that relies specifically on circadian phase behavior since amplitude measurements are often noisy and do not account for the unique entrainment features that define circadian oscillations. Resulting simulations of this 29-state ordinary differential equation model comply with fitted wild-type experimental data, demonstrating accurate free-running (23.24-h periodic) and entrained (24-h periodic) circadian dynamics. This model also predicts unfitted mutant phenotype behavior by illustrating short and long periodicity, robust oscillations, and arrhythmicity. This mechanistic model also predicts light-induced circadian phase resetting (as described by the phase-response curve) that are in line with experimental observations.
doi:10.1177/0748730408325041
PMCID: PMC2675948  PMID: 19060261
circadian rhythms; phase; positive/negative feedback
19.  Coupling governs entrainment range of circadian clocks 
Circadian clock oscillator properties that are crucial for synchronization with the environment (entrainment) are studied in experiment and theory.The ratio between stimulus (zeitgeber) strength and oscillator amplitude, and the rigidity of the oscillatory system (relaxation rate upon perturbation) determine entrainment properties. Coupling among oscillators affects both qualities resulting in increased amplitude and rigidity.Uncoupled lung clocks entrain to extreme zeitgeber cycles, whereas the coupled oscillator system in the suprachiasmatic nucleus (SCN) does not; however, when coupling in the SCN is inhibited, larger ranges of entrainment can be achieved.
Daily rhythms in physiology, metabolism and behavior are controlled by an endogenous circadian timing system, which has evolved to synchronize an organism to periodically recurring environmental conditions, such as light–dark or temperature cycles. In mammals, the circadian system relies on cell-autonomous oscillators residing in almost every cell of the body. Cells of the SCN in the anterior hypothalamus are able to generate precise, long-lasting self-sustained circadian oscillations, which drive most rhythmic behavioral and physiological outputs, and which are believed to originate from the fact that the SCN tissue consists of tightly coupled cells (Aton and Herzog, 2005). In contrast, peripheral oscillators, such as lung tissue, exhibit seemingly damped and usually less precise oscillations, which are thought to be brought about by the lack of intercellular coupling.
Precise synchronization of these rhythms within the organism, but also with the environment (so-called entrainment), is an essential part of circadian organization. Entrainment is one of the cornerstones of circadian biology (Roenneberg et al, 2003). In evolution, the phase of a rhythmic variable is selective rather than its endogenous period. Thus, the synchronization of endogenous rhythms to zeitgeber cycles of the environment (resulting in a specific phase of entrainment) is fundamental for the adaptive value of circadian clocks. In this study, we systematically investigated the properties of circadian oscillators that are essential for entrainment behavior and describe coupling as a primary determinant.
As an experimental starting point of this study, we found that the circadian oscillators of lung tissue have a larger range of entrainment than SCN tissue—they readily entrained to extreme experimental temperature cycle of 20 or 28 h, whereas SCN tissue did not (Figure 4). For this purpose, we cultured SCN and lung slices derived from mice that express luciferase as fusion protein together with the clock protein PERIOD2 (Yoo et al, 2004). The detection of luciferase-driven bioluminescence allowed us to follow molecular clock gene activity in real-time over the course of several days.
In theoretical analyses, we show that both the ratio of amplitude and zeitgeber strength and, importantly, inter-oscillator coupling are major determinants for entrainment. The reason for coupling being critical is twofold: (i) Coupling makes an oscillatory system more rigid, i.e., it relaxes faster in response to a perturbation, and (ii) coupling increases the amplitude of the oscillatory system. Both of these consequences of coupling lead to a smaller entrainment range, because zeitgeber stimuli affect the oscillatory system less if the relaxation is fast and the amplitude is high (Figure 1). From these theoretical considerations, we conclude that the lung clock probably constitutes a weak oscillatory system, likely because a lack in coupling leads to a slow amplitude relaxation. (Circadian amplitude is not particularly low in lung (Figure 4).) In contrast, the SCN constitutes a rigid oscillator, whereby coupling and its described consequences probably are the primary causes for this rigidity. We then tested these theoretical predictions by experimentally perturbing coupling in the SCN (with MDL and TTX; O'Neill et al, 2008; Yamaguchi et al, 2003) and find that, indeed, reducing the coupling weakens the circadian oscillatory system in the SCN, which results in an enlargement of the entrainment range (Figure 6).
Why is the SCN designed to be a stronger circadian oscillator than peripheral organs? We speculate that the position of the SCN—as the tissue that conveys environmental timing information (i.e., light) to the rest of the body—makes it necessary to create a circadian clock that is robust against noisy environmental stimuli. The SCN oscillator needs to be robust enough to be protected from environmental noise, but flexible enough to fulfill its function as an entrainable clock even in extreme photoperiods (i.e., seasons). By the same token, peripheral clocks are more protected from the environmental zeitgebers due to intrinsic homeostatic mechanisms. Thus, they do not necessarily need to develop a strong oscillatory system (e.g., by strengthening the coupling), rather they need to stay flexible enough to respond to direct or indirect signals from the SCN, such as hormonal, neural, temperature or metabolic signals. Such a design ensures that only robust and persistent environmental signals trigger an SCN resetting response, while SCN signals can relatively easily be conveyed to the rest of the body. Thus, the robustness in the SCN clock likely serves as a filter for environmental noise.
In summary, using a combination of simulation studies, analytical calculations and experiments, we uncovered critical features for entrainment, such as zeitgeber-to-amplitude ratio and amplitude relaxation rate. Coupling is a primary factor that governs these features explaining important differences in the design of SCN and peripheral oscillators that ensure a robust, but also flexible circadian system.
Circadian clocks are endogenous oscillators driving daily rhythms in physiology and behavior. Synchronization of these timers to environmental light–dark cycles (‘entrainment') is crucial for an organism's fitness. Little is known about which oscillator qualities determine entrainment, i.e., entrainment range, phase and amplitude. In a systematic theoretical and experimental study, we uncovered these qualities for circadian oscillators in the suprachiasmatic nucleus (SCN—the master clock in mammals) and the lung (a peripheral clock): (i) the ratio between stimulus (zeitgeber) strength and oscillator amplitude and (ii) the rigidity of the oscillatory system (relaxation rate upon perturbation) determine entrainment properties. Coupling among oscillators affects both qualities resulting in increased amplitude and rigidity. These principles explain our experimental findings that lung clocks entrain to extreme zeitgeber cycles, whereas SCN clocks do not. We confirmed our theoretical predictions by showing that pharmacological inhibition of coupling in the SCN leads to larger ranges of entrainment. These differences between master and the peripheral clocks suggest that coupling-induced rigidity in the SCN filters environmental noise to create a robust circadian system.
doi:10.1038/msb.2010.92
PMCID: PMC3010105  PMID: 21119632
circadian clock; coupling; entrainment; mathematical modeling; oscillator
20.  Phase Coupling of a Circadian Neuropeptide With Rest/Activity Rhythms Detected Using a Membrane-Tethered Spider Toxin 
PLoS Biology  2008;6(11):e273.
Drosophila clock neurons are self-sustaining cellular oscillators that rely on negative transcriptional feedback to keep circadian time. Proper regulation of organismal rhythms of physiology and behavior requires coordination of the oscillations of individual clock neurons within the circadian control network. Over the last decade, it has become clear that a key mechanism for intercellular communication in the circadian network is signaling between a subset of clock neurons that secrete the neuropeptide pigment dispersing factor (PDF) and clock neurons that possess its G protein-coupled receptor (PDFR). Furthermore, the specific hypothesis has been proposed that PDF-secreting clock neurons entrain the phase of organismal rhythms, and the cellular oscillations of other clock neurons, via the temporal patterning of secreted PDF signals. In order to test this hypothesis, we have devised a novel technique for altering the phase relationship between circadian transcriptional feedback oscillation and PDF secretion by using an ion channel–directed spider toxin to modify voltage-gated Na+ channel inactivation in vivo. This technique relies on the previously reported “tethered-toxin” technology for cell-autonomous modulation of ionic conductances via heterologous expression of subtype-specific peptide ion channel toxins as chimeric fusion proteins tethered to the plasma membrane with a glycosylphosphatidylinositol (GPI) anchor. We demonstrate for the first time, to our knowledge, the utility of the tethered-toxin technology in a transgenic animal, validating four different tethered spider toxin ion channel modifiers for use in Drosophila. Focusing on one of these toxins, we show that GPI-tethered Australian funnel-web spider toxin δ-ACTX-Hv1a inhibits Drosophila para voltage-gated Na+ channel inactivation when coexpressed in Xenopus oocytes. Transgenic expression of membrane-tethered δ-ACTX-Hv1a in vivo in the PDF-secreting subset of clock neurons induces rhythmic action potential bursts and depolarized plateau potentials. These in vitro and in vivo electrophysiological effects of membrane-tethered δ-ACTX-Hv1a are consistent with the effects of soluble δ-ACTX-Hv1a purified from venom on Na+ channel physiological and biophysical properties in cockroach neurons. Membrane-tethered δ-ACTX-Hv1a expression in the PDF-secreting subset of clock neurons induces an approximately 4-h phase advance of the rhythm of PDF accumulation in their terminals relative to both the phase of the day:night cycle and the phase of the circadian transcriptional feedback loops. As a consequence, the morning anticipatory peak of locomotor activity preceding dawn, which has been shown to be driven by the clocks of the PDF-secreting subset of clock neurons, phase advances coordinately with the phase of the PDF rhythm of the PDF-secreting clock neurons, rather than maintaining its phase relationship with the day:night cycle and circadian transcriptional feedback loops. These results (1) validate the tethered-toxin technology for cell-autonomous modulation of ion channel biophysical properties in vivo in transgenic Drosophila, (2) demonstrate that the kinetics of para Na+ channel inactivation is a key parameter for determining the phase relationship between circadian transcriptional feedback oscillation and PDF secretion, and (3) provide experimental support for the hypothesis that PDF-secreting clock neurons entrain the phase of organismal rhythms via the temporal patterning of secreted PDF signals.
Author Summary
The regulation of the daily fluctuations that characterize an organism's physiology and behavior requires coordination of the cellular oscillations of individual “clock” neurons within the circadian control network. Clock neurons that secrete a neuropeptide called pigment dispersing factor (PDF) calibrate, or entrain, both the phase of organismal rhythms and the cellular oscillations of other clock neurons. In this study, we tested the hypothesis that phase of PDF secretion rhythms entrains phase of non-PDF neurons and locomotor rhythms using the tethered- toxin technique (which affixes toxins to the cell membrane) to express ion channel–specific peptide toxins in PDF neurons. A particular toxin inhibits inactivation of the Drosophila para sodium (Na+) channel. Inhibition of Na+ channel inactivation in PDF neurons of transgenic flies induces phase advance of PDF rhythm, and correlated phase advance of lights-on anticipatory locomotor activity, suggesting that phase of morning activity is determined by phase of PDF oscillation. Therefore, voltage-gated Na+ channels of Drosophila clock neurons play a key role in determining the phase relationship between circadian transcriptional feedback oscillation and PDF secretion, and PDF-secreting clock neurons entrain the phase of organismal rhythms via the temporal patterning of secreted PDF signals.
Cell-autonomous inhibition ofDrosophila para Na+ channel inactivation using a membrane-tethered spider toxin phase shifts circadian neuropeptide output from cellular oscillation, and the phase of morning anticipatory activity is determined by this phase-shifted neuropeptide output.
doi:10.1371/journal.pbio.0060273
PMCID: PMC2577701  PMID: 18986214
21.  Modeling an Evolutionary Conserved Circadian Cis-Element 
PLoS Computational Biology  2008;4(2):e38.
Circadian oscillator networks rely on a transcriptional activator called CLOCK/CYCLE (CLK/CYC) in insects and CLOCK/BMAL1 or NPAS2/BMAL1 in mammals. Identifying the targets of this heterodimeric basic-helix-loop-helix (bHLH) transcription factor poses challenges and it has been difficult to decipher its specific sequence affinity beyond a canonical E-box motif, except perhaps for some flanking bases contributing weakly to the binding energy. Thus, no good computational model presently exists for predicting CLK/CYC, CLOCK/BMAL1, or NPAS2/BMAL1 targets. Here, we use a comparative genomics approach and first study the conservation properties of the best-known circadian enhancer: a 69-bp element upstream of the Drosophila melanogaster period gene. This fragment shows a signal involving the presence of two closely spaced E-box–like motifs, a configuration that we can also detect in the other four prominent CLK/CYC target genes in flies: timeless, vrille, Pdp1, and cwo. This allows for the training of a probabilistic sequence model that we test using functional genomics datasets. We find that the predicted sequences are overrepresented in promoters of genes induced in a recent study by a glucocorticoid receptor-CLK fusion protein. We then scanned the mouse genome with the fly model and found that many known CLOCK/BMAL1 targets harbor sequences matching our consensus. Moreover, the phase of predicted cyclers in liver agreed with known CLOCK/BMAL1 regulation. Taken together, we built a predictive model for CLK/CYC or CLOCK/BMAL1-bound cis-enhancers through the integration of comparative and functional genomics data. Finally, a deeper phylogenetic analysis reveals that the link between the CLOCK/BMAL1 complex and the circadian cis-element dates back to before insects and vertebrates diverged.
Author Summary
Life on earth is subject to daily light/dark and temperature cycles that reflect the earth rotation about its own axis. Under such conditions, organisms ranging from bacteria to human have evolved molecularly geared circadian clocks that resonate with the environmental cycles. These clocks serve as internal timing devices to coordinate physiological and behavioral processes as diverse as detoxification, activity and rest cycles, or blood pressure. In insects and vertebrates, the clock circuitry uses interlocked negative feedback loops which are implemented by transcription factors, among which the heterodimeric activators CLOCK and CYCLE play a key role. The specific DNA elements recognized by this factor are known to involve E-box motifs, but the low information content of this sequence makes it a poor predictor of the targets of CLOCK/CYCLE on a genome-wide scale. Here, we use comparative genomics to build a more specific model for a CLOCK-controlled cis-element that extends the canonical E-boxes to a more complex dimeric element. We use functional data from Drosophila and mouse circadian experiments to test the validity and assess the performance of the model. Finally, we provide a phylogenetic analysis of the cis-elements across insect and vertebrates that emphasizes the ancient link between CLOCK/CYCLE and the modeled enhancer. These results indicate that comparative genomics provides powerful means to decipher the complexity of the circadian cis-regulatory code.
doi:10.1371/journal.pcbi.0040038
PMCID: PMC2242825  PMID: 18282089
22.  Robustness from flexibility in the fungal circadian clock 
BMC Systems Biology  2010;4:88.
Background
Robustness is a central property of living systems, enabling function to be maintained against environmental perturbations. A key challenge is to identify the structures in biological circuits that confer system-level properties such as robustness. Circadian clocks allow organisms to adapt to the predictable changes of the 24-hour day/night cycle by generating endogenous rhythms that can be entrained to the external cycle. In all organisms, the clock circuits typically comprise multiple interlocked feedback loops controlling the rhythmic expression of key genes. Previously, we showed that such architectures increase the flexibility of the clock's rhythmic behaviour. We now test the relationship between flexibility and robustness, using a mathematical model of the circuit controlling conidiation in the fungus Neurospora crassa.
Results
The circuit modelled in this work consists of a central negative feedback loop, in which the frequency (frq) gene inhibits its transcriptional activator white collar-1 (wc-1), interlocked with a positive feedback loop in which FRQ protein upregulates WC-1 production. Importantly, our model reproduces the observed entrainment of this circuit under light/dark cycles with varying photoperiod and cycle duration. Our simulations show that whilst the level of frq mRNA is driven directly by the light input, the falling phase of FRQ protein, a molecular correlate of conidiation, maintains a constant phase that is uncoupled from the times of dawn and dusk. The model predicts the behaviour of mutants that uncouple WC-1 production from FRQ's positive feedback, and shows that the positive loop enhances the buffering of conidiation phase against seasonal photoperiod changes. This property is quantified using Kitano's measure for the overall robustness of a regulated system output. Further analysis demonstrates that this functional robustness is a consequence of the greater evolutionary flexibility conferred on the circuit by the interlocking loop structure.
Conclusions
Our model shows that the behaviour of the fungal clock in light-dark cycles can be accounted for by a transcription-translation feedback model of the central FRQ-WC oscillator. More generally, we provide an example of a biological circuit in which greater flexibility yields improved robustness, while also introducing novel sensitivity analysis techniques applicable to a broader range of cellular oscillators.
doi:10.1186/1752-0509-4-88
PMCID: PMC2913929  PMID: 20576110
23.  Light and circadian regulation of clock components aids flexible responses to environmental signals 
The New Phytologist  2014;203(2):568-577.
The circadian clock measures time across a 24 h period, increasing fitness by phasing biological processes to the most appropriate time of day. The interlocking feedback loop mechanism of the clock is conserved across species; however, the number of loops varies. Mathematical and computational analyses have suggested that loop complexity affects the overall flexibility of the oscillator, including its responses to entrainment signals.We used a discriminating experimental assay, at the transition between different photoperiods, in order to test this proposal in a minimal circadian network (in Ostreococcus tauri) and a more complex network (in Arabidopsis thaliana).Transcriptional and translational reporters in O. tauri primarily tracked dawn or dusk, whereas in A. thaliana, a wider range of responses were observed, consistent with its more flexible clock. Model analysis supported the requirement for this diversity of responses among the components of the more complex network. However, these and earlier data showed that the O. tauri network retains surprising flexibility, despite its simple circuit.We found that models constructed from experimental data can show flexibility either from multiple loops and/or from multiple light inputs. Our results suggest that O. tauri has adopted the latter strategy, possibly as a consequence of genomic reduction.
doi:10.1111/nph.12853
PMCID: PMC4286021  PMID: 24842166
biological clocks; flexibility; marine algae; mathematical analysis; nonlinear dynamics; photoperiod; systems biology
24.  An Evaluation of Methods for Inferring Boolean Networks from Time-Series Data 
PLoS ONE  2013;8(6):e66031.
Regulatory networks play a central role in cellular behavior and decision making. Learning these regulatory networks is a major task in biology, and devising computational methods and mathematical models for this task is a major endeavor in bioinformatics. Boolean networks have been used extensively for modeling regulatory networks. In this model, the state of each gene can be either ‘on’ or ‘off’ and that next-state of a gene is updated, synchronously or asynchronously, according to a Boolean rule that is applied to the current-state of the entire system. Inferring a Boolean network from a set of experimental data entails two main steps: first, the experimental time-series data are discretized into Boolean trajectories, and then, a Boolean network is learned from these Boolean trajectories. In this paper, we consider three methods for data discretization, including a new one we propose, and three methods for learning Boolean networks, and study the performance of all possible nine combinations on four regulatory systems of varying dynamics complexities. We find that employing the right combination of methods for data discretization and network learning results in Boolean networks that capture the dynamics well and provide predictive power. Our findings are in contrast to a recent survey that placed Boolean networks on the low end of the “faithfulness to biological reality” and “ability to model dynamics” spectra. Further, contrary to the common argument in favor of Boolean networks, we find that a relatively large number of time points in the time-series data is required to learn good Boolean networks for certain data sets. Last but not least, while methods have been proposed for inferring Boolean networks, as discussed above, missing still are publicly available implementations thereof. Here, we make our implementation of the methods available publicly in open source at http://bioinfo.cs.rice.edu/.
doi:10.1371/journal.pone.0066031
PMCID: PMC3689729  PMID: 23805196
25.  A Multiscale Model to Investigate Circadian Rhythmicity of Pacemaker Neurons in the Suprachiasmatic Nucleus 
PLoS Computational Biology  2010;6(3):e1000706.
The suprachiasmatic nucleus (SCN) of the hypothalamus is a multicellular system that drives daily rhythms in mammalian behavior and physiology. Although the gene regulatory network that produces daily oscillations within individual neurons is well characterized, less is known about the electrophysiology of the SCN cells and how firing rate correlates with circadian gene expression. We developed a firing rate code model to incorporate known electrophysiological properties of SCN pacemaker cells, including circadian dependent changes in membrane voltage and ion conductances. Calcium dynamics were included in the model as the putative link between electrical firing and gene expression. Individual ion currents exhibited oscillatory patterns matching experimental data both in current levels and phase relationships. VIP and GABA neurotransmitters, which encode synaptic signals across the SCN, were found to play critical roles in daily oscillations of membrane excitability and gene expression. Blocking various mechanisms of intracellular calcium accumulation by simulated pharmacological agents (nimodipine, IP3- and ryanodine-blockers) reproduced experimentally observed trends in firing rate dynamics and core-clock gene transcription. The intracellular calcium concentration was shown to regulate diverse circadian processes such as firing frequency, gene expression and system periodicity. The model predicted a direct relationship between firing frequency and gene expression amplitudes, demonstrated the importance of intracellular pathways for single cell behavior and provided a novel multiscale framework which captured characteristics of the SCN at both the electrophysiological and gene regulatory levels.
Author Summary
Circadian rhythms are ∼24 hour cycles in biochemical, physiological and behavioral processes observed in a diverse range of organisms including Cyanobacteria, Neurospora, Drosophila, mice and humans. In mammals, the dominant circadian pacemaker that drives daily rhythms is located in the suprachiasmatic nucleus (SCN) of the hypothalamus. The SCN is composed of a highly connected network of ∼20,000 neurons. Within each individual SCN neuron core clock genes and proteins interact through intertwined regulatory loops to generate circadian oscillations on the molecular level. These neurons express daily rhythmicity in their firing frequency and other electrophysiological properties. The mechanisms by which the core clock produces synchronized rhythms in neural firing and gene expression are postulated to involve intracellular calcium, a second messenger that regulates many cellular processes. The interaction between the various clock components however remains unknown. In this paper, we present a single cell model that incorporates the circadian gene regulatory pathway, cellular electrophysiological properties, and cytosolic calcium dynamics. Our results suggest a possible system architecture that accounts for the robustness of the circadian clock at the single cell level. Our simulations predict a dual role for intracellular pathways instigated by intracellular calcium and VIP: maintaining the periodicity and amplitude of the core clock genes as well as the firing frequency oscillations.
doi:10.1371/journal.pcbi.1000706
PMCID: PMC2837390  PMID: 20300645

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