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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.  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
5.  Digital clocks: simple Boolean models can quantitatively describe circadian systems 
The gene networks that comprise the circadian clock modulate biological function across a range of scales, from gene expression to performance and adaptive behaviour. The clock functions by generating endogenous rhythms that can be entrained to the external 24-h day–night cycle, enabling organisms to optimally time biochemical processes relative to dawn and dusk. In recent years, computational models based on differential equations have become useful tools for dissecting and quantifying the complex regulatory relationships underlying the clock's oscillatory dynamics. However, optimizing the large parameter sets characteristic of these models places intense demands on both computational and experimental resources, limiting the scope of in silico studies. Here, we develop an approach based on Boolean logic that dramatically reduces the parametrization, making the state and parameter spaces finite and tractable. We introduce efficient methods for fitting Boolean models to molecular data, successfully demonstrating their application to synthetic time courses generated by a number of established clock models, as well as experimental expression levels measured using luciferase imaging. Our results indicate that despite their relative simplicity, logic models can (i) simulate circadian oscillations with the correct, experimentally observed phase relationships among genes and (ii) flexibly entrain to light stimuli, reproducing the complex responses to variations in daylength generated by more detailed differential equation formulations. Our work also demonstrates that logic models have sufficient predictive power to identify optimal regulatory structures from experimental data. By presenting the first Boolean models of circadian circuits together with general techniques for their optimization, we hope to establish a new framework for the systematic modelling of more complex clocks, as well as other circuits with different qualitative dynamics. In particular, we anticipate that the ability of logic models to provide a computationally efficient representation of system behaviour could greatly facilitate the reverse-engineering of large-scale biochemical networks.
doi:10.1098/rsif.2012.0080
PMCID: PMC3405750  PMID: 22499125
systems biology; circadian gene networks; Boolean logic; photoperiodism; Arabidopsis thaliana
6.  Modeling the Effects of Cell Cycle M-phase Transcriptional Inhibition on Circadian Oscillation 
PLoS Computational Biology  2008;4(3):e1000019.
Circadian clocks are endogenous time-keeping systems that temporally organize biological processes. Gating of cell cycle events by a circadian clock is a universal observation that is currently considered a mechanism serving to protect DNA from diurnal exposure to ultraviolet radiation or other mutagens. In this study, we put forward another possibility: that such gating helps to insulate the circadian clock from perturbations induced by transcriptional inhibition during the M phase of the cell cycle. We introduced a periodic pulse of transcriptional inhibition into a previously published mammalian circadian model and simulated the behavior of the modified model under both constant darkness and light–dark cycle conditions. The simulation results under constant darkness indicated that periodic transcriptional inhibition could entrain/lock the circadian clock just as a light–dark cycle does. At equilibrium states, a transcriptional inhibition pulse of certain periods was always locked close to certain circadian phases where inhibition on Per and Bmal1 mRNA synthesis was most balanced. In a light–dark cycle condition, inhibitions imposed at different parts of a circadian period induced different degrees of perturbation to the circadian clock. When imposed at the middle- or late-night phase, the transcriptional inhibition cycle induced the least perturbations to the circadian clock. The late-night time window of least perturbation overlapped with the experimentally observed time window, where mitosis is most frequent. This supports our hypothesis that the circadian clock gates the cell cycle M phase to certain circadian phases to minimize perturbations induced by the latter. This study reveals the hidden effects of the cell division cycle on the circadian clock and, together with the current picture of genome stability maintenance by circadian gating of cell cycle, provides a more comprehensive understanding of the phenomenon of circading gating of cell cycle.
Author Summary
Circadian clock and cell cycle are two important biological processes that are essential for nearly all eukaryotes. The circadian clock governs day and night 24 h periodic molecular processes and physiological behaviors, while cell cycle controls cell division process. It has been widely observed that cell division does not occur randomly across day and night, but instead is normally confined to specific times during day and night. These observations suggest that cell cycle events are gated by the circadian clock. Regarding the biological benefit and rationale for this intriguing gating phenomena, it has been postulated that circadian gating helps to maintain genome stability by confining radiation-sensitive cell cycle phases to night. Bearing in mind the facts that global transcriptional inhibition occurs at cell division and transcriptional inhibition shifts circadian phases and periods, we postulate that confining cell division to specific circadian times benefits the circadian clock by removing or minimizing the side effects of cell division on the circadian clock. Our results based on computational simulation in this study show that periodic transcriptional inhibition can perturb the circadian clock by altering circadian phases and periods, and the magnitude of the perturbation is clearly circadian phase dependent. Specifically, transcriptional inhibition initiated at certain circadian phases induced minimal perturbation to the circadian clock. These results provide support for our postulation. Our postulation and results point to the importance of the effect of cell division on the circadian clock in the interaction between circadian and cell cycle and suggest that it should be considered together with other factors in the exploitation of circadian cell cycle interaction, especially the phenomena of circadian gating of cell cycle.
doi:10.1371/journal.pcbi.1000019
PMCID: PMC2267494  PMID: 18369419
7.  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
8.  Circadian Rhythmicity by Autocatalysis 
PLoS Computational Biology  2006;2(7):e96.
The temperature compensated in vitro oscillation of cyanobacterial KaiC phosphorylation, the first example of a thermodynamically closed system showing circadian rhythmicity, only involves the three Kai proteins (KaiA, KaiB, and KaiC) and ATP. In this paper, we describe a model in which the KaiA- and KaiB-assisted autocatalytic phosphorylation and dephosphorylation of KaiC are the source for circadian rhythmicity. This model, based upon autocatalysis instead of transcription-translation negative feedback, shows temperature-compensated circadian limit-cycle oscillations with KaiC phosphorylation profiles and has period lengths and rate constant values that are consistent with experimental observations.
Synopsis
Circadian rhythms are a central feature of biological systems. In cyanobacteria, the clock involves three major proteins: KaiA, KaiB, and KaiC, with KaiC showing autophosphorylation in the presence of ATP. Remarkably, by incubating the purified Kai proteins with ATP, the clock can be reconstituted in vitro. The authors were intrigued by the simplicity of this oscillator and its connection to chemical oscillatory reactions, and saw the possibility for a realistic reaction kinetic representation. Their study represents a synthetic, predictive, and dynamic explanation for the in vitro KaiC circadian clock based on the self-amplifying response (“autocatalysis”) of autophosphorylating kinases. The presented model is based on existing biological and biochemical observations and recapitulates observed experimental features, including temperature compensation, which has been notably recalcitrant to explanation. The model provides several predictions including that period length and the ratio between phosphorylated and unphosphorylated KaiC in clock mutants are closely linked and related to the stability of the ternary complex formed between the three Kai proteins. The model also predicts the occurrence of bistability and that evolution can move simple bistable systems into oscillatory systems by changing only one rate constant.
doi:10.1371/journal.pcbi.0020096
PMCID: PMC1523307  PMID: 16863394
9.  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
10.  optPBN: An Optimisation Toolbox for Probabilistic Boolean Networks 
PLoS ONE  2014;9(7):e98001.
Background
There exist several computational tools which allow for the optimisation and inference of biological networks using a Boolean formalism. Nevertheless, the results from such tools yield only limited quantitative insights into the complexity of biological systems because of the inherited qualitative nature of Boolean networks.
Results
We introduce optPBN, a Matlab-based toolbox for the optimisation of probabilistic Boolean networks (PBN) which operates under the framework of the BN/PBN toolbox. optPBN offers an easy generation of probabilistic Boolean networks from rule-based Boolean model specification and it allows for flexible measurement data integration from multiple experiments. Subsequently, optPBN generates integrated optimisation problems which can be solved by various optimisers.
In term of functionalities, optPBN allows for the construction of a probabilistic Boolean network from a given set of potential constitutive Boolean networks by optimising the selection probabilities for these networks so that the resulting PBN fits experimental data. Furthermore, the optPBN pipeline can also be operated on large-scale computational platforms to solve complex optimisation problems. Apart from exemplary case studies which we correctly inferred the original network, we also successfully applied optPBN to study a large-scale Boolean model of apoptosis where it allows identifying the inverse correlation between UVB irradiation, NFκB and Caspase 3 activations, and apoptosis in primary hepatocytes quantitatively. Also, the results from optPBN help elucidating the relevancy of crosstalk interactions in the apoptotic network.
Summary
The optPBN toolbox provides a simple yet comprehensive pipeline for integrated optimisation problem generation in the PBN formalism that can readily be solved by various optimisers on local or grid-based computational platforms. optPBN can be further applied to various biological studies such as the inference of gene regulatory networks or the identification of the interaction's relevancy in signal transduction networks.
doi:10.1371/journal.pone.0098001
PMCID: PMC4077690  PMID: 24983623
11.  Network Features of the Mammalian Circadian Clock 
PLoS Biology  2009;7(3):e1000052.
The mammalian circadian clock is a cell-autonomous system that drives oscillations in behavior and physiology in anticipation of daily environmental change. To assess the robustness of a human molecular clock, we systematically depleted known clock components and observed that circadian oscillations are maintained over a wide range of disruptions. We developed a novel strategy termed Gene Dosage Network Analysis (GDNA) in which small interfering RNA (siRNA)-induced dose-dependent changes in gene expression were used to build gene association networks consistent with known biochemical constraints. The use of multiple doses powered the analysis to uncover several novel network features of the circadian clock, including proportional responses and signal propagation through interacting genetic modules. We also observed several examples where a gene is up-regulated following knockdown of its paralog, suggesting the clock network utilizes active compensatory mechanisms rather than simple redundancy to confer robustness and maintain function. We propose that these network features act in concert as a genetic buffering system to maintain clock function in the face of genetic and environmental perturbation.
Author Summary
The circadian clock is the biological clock found throughout the body that coordinates the timing of molecular and cellular processes on a 24-hour rhythm. It is composed of numerous transcription factors that feed back and control their own expression. To explore how the clock functions in the face of genetic perturbations, we disrupted its function by knocking down gene expression of known clock genes in a dose-dependent fashion. We measured the expression of clock genes following knockdown and constructed perturbation-based network models to describe, visualize, and mine the results. We reported several novel network features, such as signal propagation through interacting genetic modules and proportional responses whereby levels of expression are altered commensurately with changing levels of the gene. We also observed several examples where a gene is up-regulated following knockdown of its paralog, suggesting the clock network utilizes active compensatory mechanisms rather than simple redundancy to confer robustness and maintain function. We propose that the network features we observe act in concert as a genetic buffering system to maintain clock function in the face of genetic and environmental perturbation.
How does the circadian clock maintain function in the face of genetic perturbation? The authors construct gene dosage perturbation networks and uncover several underlying principles contributing to genetic buffering of the clock.
doi:10.1371/journal.pbio.1000052
PMCID: PMC2653556  PMID: 19278294
12.  Continuous time boolean modeling for biological signaling: application of Gillespie algorithm 
BMC Systems Biology  2012;6:116.
Mathematical modeling is used as a Systems Biology tool to answer biological questions, and more precisely, to validate a network that describes biological observations and predict the effect of perturbations. This article presents an algorithm for modeling biological networks in a discrete framework with continuous time.
Background
There exist two major types of mathematical modeling approaches: (1) quantitative modeling, representing various chemical species concentrations by real numbers, mainly based on differential equations and chemical kinetics formalism; (2) and qualitative modeling, representing chemical species concentrations or activities by a finite set of discrete values. Both approaches answer particular (and often different) biological questions. Qualitative modeling approach permits a simple and less detailed description of the biological systems, efficiently describes stable state identification but remains inconvenient in describing the transient kinetics leading to these states. In this context, time is represented by discrete steps. Quantitative modeling, on the other hand, can describe more accurately the dynamical behavior of biological processes as it follows the evolution of concentration or activities of chemical species as a function of time, but requires an important amount of information on the parameters difficult to find in the literature.
Results
Here, we propose a modeling framework based on a qualitative approach that is intrinsically continuous in time. The algorithm presented in this article fills the gap between qualitative and quantitative modeling. It is based on continuous time Markov process applied on a Boolean state space. In order to describe the temporal evolution of the biological process we wish to model, we explicitly specify the transition rates for each node. For that purpose, we built a language that can be seen as a generalization of Boolean equations. Mathematically, this approach can be translated in a set of ordinary differential equations on probability distributions. We developed a C++ software, MaBoSS, that is able to simulate such a system by applying Kinetic Monte-Carlo (or Gillespie algorithm) on the Boolean state space. This software, parallelized and optimized, computes the temporal evolution of probability distributions and estimates stationary distributions.
Conclusions
Applications of the Boolean Kinetic Monte-Carlo are demonstrated for three qualitative models: a toy model, a published model of p53/Mdm2 interaction and a published model of the mammalian cell cycle. Our approach allows to describe kinetic phenomena which were difficult to handle in the original models. In particular, transient effects are represented by time dependent probability distributions, interpretable in terms of cell populations.
doi:10.1186/1752-0509-6-116
PMCID: PMC3517402  PMID: 22932419
Boolean modeling; Continuous time; Markov process; Gillespie algorithm
13.  Quantification of Circadian Rhythms in Single Cells 
PLoS Computational Biology  2009;5(11):e1000580.
Bioluminescence techniques allow accurate monitoring of the circadian clock in single cells. We have analyzed bioluminescence data of Per gene expression in mouse SCN neurons and fibroblasts. From these data, we extracted parameters such as damping rate and noise intensity using two simple mathematical models, one describing a damped oscillator driven by noise, and one describing a self-sustained noisy oscillator. Both models describe the data well and enabled us to quantitatively characterize both wild-type cells and several mutants. It has been suggested that the circadian clock is self-sustained at the single cell level, but we conclude that present data are not sufficient to determine whether the circadian clock of single SCN neurons and fibroblasts is a damped or a self-sustained oscillator. We show how to settle this question, however, by testing the models' predictions of different phases and amplitudes in response to a periodic entrainment signal (zeitgeber).
Author Summary
Earth's 24-h-rotation around its axis is mirrored in the circadian clock that resides within each of our cells, controlling expression of ∼10% of all genes. The circadian clock is constructed as a negative feedback loop, in which clock proteins inhibit their own synthesis. During the last decade, a picture has emerged in which each cell is a self-sustained circadian oscillator that runs even without synchronizing cues. Here, we investigated state-of-the-art single-cell bioluminescence recordings of clock gene expression. It turns out that these time series are very well described by low-dimensional models, enabling us to extract descriptive parameters that characterize each cell. We find that different cell types do not differ much in their dynamics. However, different mutations in core clock genes yield different dynamic characteristics. Furthermore, we could not statistically reject the idea that the cells are in fact damped oscillators driven by noise. We thus declare the question of whether the circadian clock is a damped or self-sustained oscillator still unresolved. Further, we propose a way to resolve this question by examining the frequency-dependent response of single cells to periodic stimuli. We will then be in a better position to understand how cells coordinate and synchronize their circadian rhythms.
doi:10.1371/journal.pcbi.1000580
PMCID: PMC2776301  PMID: 19956762
14.  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
15.  Detection of attractors of large Boolean networks via exhaustive enumeration of appropriate subspaces of the state space 
BMC Bioinformatics  2013;14:361.
Background
Boolean models are increasingly used to study biological signaling networks. In a Boolean network, nodes represent biological entities such as genes, proteins or protein complexes, and edges indicate activating or inhibiting influences of one node towards another. Depending on the input of activators or inhibitors, Boolean networks categorize nodes as either active or inactive. The formalism is appealing because for many biological relationships, we lack quantitative information about binding constants or kinetic parameters and can only rely on a qualitative description of the type “A activates (or inhibits) B”. A central aim of Boolean network analysis is the determination of attractors (steady states and/or cycles). This problem is known to be computationally complex, its most important parameter being the number of network nodes. Various algorithms tackle it with considerable success. In this paper we present an algorithm, which extends the size of analyzable networks thanks to simple and intuitive arguments.
Results
We present lnet, a software package which, in fully asynchronous updating mode and without any network reduction, detects the fixed states of Boolean networks with up to 150 nodes and a good part of any present cycles for networks with up to half the above number of nodes. The algorithm goes through a complete enumeration of the states of appropriately selected subspaces of the entire network state space. The size of these relevant subspaces is small compared to the full network state space, allowing the analysis of large networks. The subspaces scanned for the analyses of cycles are larger, reducing the size of accessible networks. Importantly, inherent in cycle detection is a classification scheme based on the number of non-frozen nodes of the cycle member states, with cycles characterized by fewer non-frozen nodes being easier to detect. It is further argued that these detectable cycles are also the biologically more important ones. Furthermore, lnet also provides standard Boolean analysis features such as node loop detection.
Conclusions
lnet is a software package that facilitates the analysis of large Boolean networks. Its intuitive approach helps to better understand the network in question.
doi:10.1186/1471-2105-14-361
PMCID: PMC3882777  PMID: 24330355
Boolean network; Attractor; Fixed state; Cycle; Regulatory network; State space
16.  A model of the circadian clock in the cyanobacterium Cyanothece sp. ATCC 51142 
BMC Bioinformatics  2013;14(Suppl 2):S14.
Background
The over consumption of fossil fuels has led to growing concerns over climate change and global warming. Increasing research activities have been carried out towards alternative viable biofuel sources. Of several different biofuel platforms, cyanobacteria possess great potential, for their ability to accumulate biomass tens of times faster than traditional oilseed crops. The cyanobacterium Cyanothece sp. ATCC 51142 has recently attracted lots of research interest as a model organism for such research. Cyanothece can perform efficiently both photosynthesis and nitrogen fixation within the same cell, and has been recently shown to produce biohydrogen--a byproduct of nitrogen fixation--at very high rates of several folds higher than previously described hydrogen-producing photosynthetic microbes. Since the key enzyme for nitrogen fixation is very sensitive to oxygen produced by photosynthesis, Cyanothece employs a sophisticated temporal separation scheme, where nitrogen fixation occurs at night and photosynthesis at day. At the core of this temporal separation scheme is a robust clocking mechanism, which so far has not been thoroughly studied. Understanding how this circadian clock interacts with and harmonizes global transcription of key cellular processes is one of the keys to realize the inherent potential of this organism.
Results
In this paper, we employ several state of the art bioinformatics techniques for studying the core circadian clock in Cyanothece sp. ATCC 51142, and its interactions with other key cellular processes. We employ comparative genomics techniques to map the circadian clock genes and genetic interactions from another cyanobacterial species, namely Synechococcus elongatus PCC 7942, of which the circadian clock has been much more thoroughly investigated. Using time series gene expression data for Cyanothece, we employ gene regulatory network reconstruction techniques to learn this network de novo, and compare the reconstructed network against the interactions currently reported in the literature. Next, we build a computational model of the interactions between the core clock and other cellular processes, and show how this model can predict the behaviour of the system under changing environmental conditions. The constructed models significantly advance our understanding of the Cyanothece circadian clock functional mechanisms.
doi:10.1186/1471-2105-14-S2-S14
PMCID: PMC3549803  PMID: 23368635
17.  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
18.  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
19.  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
20.  Boolean network simulations for life scientists 
Modern life sciences research increasingly relies on computational solutions, from large scale data analyses to theoretical modeling. Within the theoretical models Boolean networks occupy an increasing role as they are eminently suited at mapping biological observations and hypotheses into a mathematical formalism. The conceptual underpinnings of Boolean modeling are very accessible even without a background in quantitative sciences, yet it allows life scientists to describe and explore a wide range of surprisingly complex phenomena. In this paper we provide a clear overview of the concepts used in Boolean simulations, present a software library that can perform these simulations based on simple text inputs and give three case studies. The large scale simulations in these case studies demonstrate the Boolean paradigms and their applicability as well as the advanced features and complex use cases that our software package allows. Our software is distributed via a liberal Open Source license and is freely accessible from
doi:10.1186/1751-0473-3-16
PMCID: PMC2603008  PMID: 19014577
21.  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
22.  Adult Circadian Behavior in Drosophila Requires Developmental Expression of cycle, But Not period 
PLoS Genetics  2011;7(7):e1002167.
Circadian clocks have evolved as internal time keeping mechanisms that allow anticipation of daily environmental changes and organization of a daily program of physiological and behavioral rhythms. To better examine the mechanisms underlying circadian clocks in animals and to ask whether clock gene expression and function during development affected subsequent daily time keeping in the adult, we used the genetic tools available in Drosophila to conditionally manipulate the function of the CYCLE component of the positive regulator CLOCK/CYCLE (CLK/CYC) or its negative feedback inhibitor PERIOD (PER). Differential manipulation of clock function during development and in adulthood indicated that there is no developmental requirement for either a running clock mechanism or expression of per. However, conditional suppression of CLK/CYC activity either via per over-expression or cyc depletion during metamorphosis resulted in persistent arrhythmic behavior in the adult. Two distinct mechanisms were identified that may contribute to this developmental function of CLK/CYC and both involve the ventral lateral clock neurons (LNvs) that are crucial to circadian control of locomotor behavior: (1) selective depletion of cyc expression in the LNvs resulted in abnormal peptidergic small-LNv dorsal projections, and (2) PER expression rhythms in the adult LNvs appeared to be affected by developmental inhibition of CLK/CYC activity. Given the conservation of clock genes and circuits among animals, this study provides a rationale for investigating a possible similar developmental role of the homologous mammalian CLOCK/BMAL1 complex.
Author Summary
The fruit fly Drosophila melanogaster is an excellent model system for studying the internal circadian clocks that animals use for daily time keeping. Since clocks exist and function in animals not only in adults, but also during prior development, the question arises if and how adult circadian rhythms depend on developmental clock circuits and components. To address this question we created transgenic flies in which the essential clock components CLOCK/CYCLE (CLK/CYC) and PERIOD (PER) can be manipulated via environmental temperature. Stopping the clock during development by depleting the negative regulator PER did not prevent restoration of circadian time keeping in the adult. However, a developmental arrest of the clock due to either depletion of the positive regulator CYC or overproduction of PER resulted in a persistent loss of clock-controlled behavior function in adults. Taken together, these observations indicate that adult clock function developmentally requires activity of the CLK/CYC transcription complex rather than a ticking clock. Based on the behavioral, molecular, and anatomical consequences of inhibiting CLK/CYC in circadian pacemaker neurons, we propose that the developmental requirement maps to these cells. It will be interesting to determine whether there is a comparable developmental requirement for the equivalent clock genes in humans.
doi:10.1371/journal.pgen.1002167
PMCID: PMC3131292  PMID: 21750685
23.  Stability and lability of circadian period of gene expression in the cyanobacterium Synechococcus elongatus 
Microbiology (Reading, England)  2009;155(Pt 2):635-641.
SUMMARY
Molecular aspects of the circadian clock in the cyanobacterium Synechococcus elongatus have been described in great detail. Three-dimensional structures have been determined for the three proteins, KaiA, KaiB, and KaiC, that comprise a central oscillator of the clock. Moreover, a temperature-compensated circadian rhythm of KaiC phosphorylation can be reconstituted in vitro with the addition of KaiA, KaiB, and ATP. These data suggest a relatively simple circadian system in which a single oscillator provides temporal information for all downstream processes. However, in vivo the situation is more complex, and additional components contribute to the maintenance of a normal period, the resetting of relative phases of circadian oscillations, and the control of rhythms of gene expression. We show here that two well-studied promoters in the S. elongatus genome report different circadian periods of expression under a given set of conditions in wild-type as well as mutant genetic backgrounds. Moreover, the period differs between these promoters with respect to modulation by light intensity, growth phase, and the presence or absence of a promoter-recognition subunit of RNA polymerase. These data contrast sharply with the current clock model in which a single Kai-based oscillator governs circadian period. Overall, these findings suggest that complex interactions between the circadian oscillator, perhaps other oscillators, and other cellular machinery result in a clock that is plastic and sensitive to the environment and to the physiological state of the cell.
doi:10.1099/mic.0.022343-0
PMCID: PMC2729554  PMID: 19202112
Circadian rhythms; cyanobacteria; circadian period; group 2 sigma factors
24.  Ras-Mediated Deregulation of the Circadian Clock in Cancer 
PLoS Genetics  2014;10(5):e1004338.
Circadian rhythms are essential to the temporal regulation of molecular processes in living systems and as such to life itself. Deregulation of these rhythms leads to failures in biological processes and eventually to the manifestation of pathological phenotypes including cancer. To address the questions as to what are the elicitors of a disrupted clock in cancer, we applied a systems biology approach to correlate experimental, bioinformatics and modelling data from several cell line models for colorectal and skin cancer. We found strong and weak circadian oscillators within the same type of cancer and identified a set of genes, which allows the discrimination between the two oscillator-types. Among those genes are IFNGR2, PITX2, RFWD2, PPARγ, LOXL2, Rab6 and SPARC, all involved in cancer-related pathways. Using a bioinformatics approach, we extended the core-clock network and present its interconnection to the discriminative set of genes. Interestingly, such gene signatures link the clock to oncogenic pathways like the RAS/MAPK pathway. To investigate the potential impact of the RAS/MAPK pathway - a major driver of colorectal carcinogenesis - on the circadian clock, we used a computational model which predicted that perturbation of BMAL1-mediated transcription can generate the circadian phenotypes similar to those observed in metastatic cell lines. Using an inducible RAS expression system, we show that overexpression of RAS disrupts the circadian clock and leads to an increase of the circadian period while RAS inhibition causes a shortening of period length, as predicted by our mathematical simulations. Together, our data demonstrate that perturbations induced by a single oncogene are sufficient to deregulate the mammalian circadian clock.
Author Summary
Living systems possess an endogenous time-generating system – the circadian clock - accountable for a 24 hours oscillation in the expression of about 10% of all genes. In mammals, disruption of oscillations is associated to several diseases including cancer. In this manuscript, we address the following question: what are the elicitors of a disrupted clock in cancer? We applied a systems biology approach to correlate experimental, bioinformatics and modelling data and could thereby identify key genes which discriminate strong and weak oscillators among cancer cell lines. Most of the discriminative genes play important roles in cell cycle regulation, DNA repair, immune system and metabolism and are involved in oncogenic pathways such as the RAS/MAPK. To investigate the potential impact of the Ras oncogene in the circadian clock we generated experimental models harbouring conditionally active Ras oncogenes. We put forward a direct correlation between the perturbation of Ras oncogene and an effect in the expression of clock genes, found by means of mathematical simulations and validated experimentally. Our study shows that perturbations of a single oncogene are sufficient to deregulate the mammalian circadian clock and opens new ways in which the circadian clock can influence disease and possibly play a role in therapy.
doi:10.1371/journal.pgen.1004338
PMCID: PMC4038477  PMID: 24875049
25.  Odefy -- From discrete to continuous models 
BMC Bioinformatics  2010;11:233.
Background
Phenomenological information about regulatory interactions is frequently available and can be readily converted to Boolean models. Fully quantitative models, on the other hand, provide detailed insights into the precise dynamics of the underlying system. In order to connect discrete and continuous modeling approaches, methods for the conversion of Boolean systems into systems of ordinary differential equations have been developed recently. As biological interaction networks have steadily grown in size and complexity, a fully automated framework for the conversion process is desirable.
Results
We present Odefy, a MATLAB- and Octave-compatible toolbox for the automated transformation of Boolean models into systems of ordinary differential equations. Models can be created from sets of Boolean equations or graph representations of Boolean networks. Alternatively, the user can import Boolean models from the CellNetAnalyzer toolbox, GINSim and the PBN toolbox. The Boolean models are transformed to systems of ordinary differential equations by multivariate polynomial interpolation and optional application of sigmoidal Hill functions. Our toolbox contains basic simulation and visualization functionalities for both, the Boolean as well as the continuous models. For further analyses, models can be exported to SQUAD, GNA, MATLAB script files, the SB toolbox, SBML and R script files. Odefy contains a user-friendly graphical user interface for convenient access to the simulation and exporting functionalities. We illustrate the validity of our transformation approach as well as the usage and benefit of the Odefy toolbox for two biological systems: a mutual inhibitory switch known from stem cell differentiation and a regulatory network giving rise to a specific spatial expression pattern at the mid-hindbrain boundary.
Conclusions
Odefy provides an easy-to-use toolbox for the automatic conversion of Boolean models to systems of ordinary differential equations. It can be efficiently connected to a variety of input and output formats for further analysis and investigations. The toolbox is open-source and can be downloaded at http://cmb.helmholtz-muenchen.de/odefy.
doi:10.1186/1471-2105-11-233
PMCID: PMC2873544  PMID: 20459647

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