PER/TIM-mediated amplification, gene dosage effects and temperature compensation in an interlocking-feedback loop model of the Drosophila circadian clock

Hilbert transform-based time-series analysis of the circadian gene regulatory network

In this work, the authors propose the Hilbert transform (HT)‐based numerical method to analyse the time series of the circadian rhythms. They demonstrate the application of HT by taking both deterministic and stochastic time series that they get from the simulation of the fruit fly model Drosophila melanogaster and show how to extract the period, construct phase response curves, determine period sensitivity of the parameters to perturbations and build Arnold tongues to identify the regions of entrainment. They also derive a phase model that they numerically simulate to capture whether the circadian time series entrains to the forcing period completely (phase locking) or only partially (phase slips) or neither. They validate the phase model, and numerics with the experimental time series forced under different temperature cycles. Application of HT to the circadian time series appears to be a promising tool to extract the characteristic information about circadian rhythms.

Protein sequestration versus Hill-type repression in circadian clock models

Circadian (∼24 h) clocks are self-sustained endogenous oscillators with which organisms keep track of daily and seasonal time. Circadian clocks frequently rely on interlocked transcriptional-translational feedback loops to generate rhythms that are robust against intrinsic and extrinsic perturbations. To investigate the dynamics and mechanisms of the intracellular feedback loops in circadian clocks, a number of mathematical models have been developed. The majority of the models use Hill functions to describe transcriptional repression in a way that is similar to the Goodwin model. Recently, a new class of models with protein sequestration-based repression has been introduced. Here, the author discusses how this new class of models differs dramatically from those based on Hill-type repression in several fundamental aspects: conditions for rhythm generation, robust network designs and the periods of coupled oscillators. Consistently, these fundamental properties of circadian clocks also differ among Neurospora, Drosophila, and mammals depending on their key transcriptional repression mechanisms (Hill-type repression or protein sequestration). Based on both theoretical and experimental studies, this review highlights the importance of careful modelling of transcriptional repression mechanisms in molecular circadian clocks.

Hypothesis driven single cell dual oscillator mathematical model of circadian rhythms

Molecular mechanisms responsible for 24 h circadian oscillations, entrainment to external cues, encoding of day length and the time-of-day effects have been well studied experimentally. However, it is still debated from the molecular network point of view whether each cell in suprachiasmatic nuclei harbors two molecular oscillators, where one tracks dawn and the other tracks dusk activities. A single cell dual morning and evening oscillator was proposed by Daan et al., based on the molecular network that has two sets of similar non-redundant per1/cry1 and per2/cry2 circadian genes and each can independently maintain their endogenous oscillations. Understanding of dual oscillator dynamics in a single cell at molecular level may provide insight about the circadian mechanisms that encodes day length variations and its response to external zeitgebers. We present here a realistic dual oscillator model of circadian rhythms based on the series of hypotheses proposed by Daan et al., in which they conjectured that the circadian genes per1/cry1 track dawn while per2/cry2 tracks dusk and they together constitute the morning and evening oscillators (dual oscillator). Their hypothesis also provides explanations about the encoding of day length in terms of molecular mechanisms of per/cry expression. We frame a minimal mathematical model with the assumption that per1 acts a morning oscillator and per2 acts as an evening oscillator and to support and interpret this assumption we fit the model to the experimental data of per1/per2 circadian temporal dynamics, phase response curves (PRC's), and entrainment phenomena under various light-dark conditions. We also capture different patterns of splitting phenomena by coupling two single cell dual oscillators with neuropeptides vasoactive intestinal polypeptide (VIP) and arginine vasopressin (AVP) as the coupling agents and provide interpretation for the occurrence of splitting in terms of ME oscillators, though they are not required to explain the morning and evening oscillators. The proposed dual oscillator model based on Daan's hypothesis supports per1 and per2 playing the role of morning and evening oscillators respectively and this may be the first step towards the understanding of the core molecular mechanism responsible for encoding the day length.

Systems Biology-Derived Discoveries of Intrinsic Clocks

A systems approach to studying biology uses a variety of mathematical, computational, and engineering tools to holistically understand and model properties of cells, tissues, and organisms. Building from early biochemical, genetic, and physiological studies, systems biology became established through the development of genome-wide methods, high-throughput procedures, modern computational processing power, and bioinformatics. Here, we highlight a variety of systems approaches to the study of biological rhythms that occur with a 24-h period-circadian rhythms. We review how systems methods have helped to elucidate complex behaviors of the circadian clock including temperature compensation, rhythmicity, and robustness. Finally, we explain the contribution of systems biology to the transcription-translation feedback loop and posttranslational oscillator models of circadian rhythms and describe new technologies and "-omics" approaches to understand circadian timekeeping and neurophysiology.

Robust network topologies for generating oscillations with temperature-independent periods

Nearly all living systems feature a temperature-independent oscillation period in circadian clocks. This ubiquitous property occurs at the system level and is rooted in the network architecture of the clock machinery. To investigate the mechanism of this prominent property of the circadian clock and provide general guidance for generating robust genetic oscillators with temperature-compensated oscillations, we theoretically explored the design principle and core network topologies preferred by oscillations with a temperature-independent period. By enumerating all topologies of genetic regulatory circuits with three genes, we obtained four network motifs, namely, a delayed negative feedback oscillator, repressilator, activator-inhibitor oscillator and substrate-depletion oscillator; hybrids of these motifs constitute the vast majority of target network topologies. These motifs are biased in their capacities for achieving oscillations and the temperature sensitivity of the period. The delayed negative feedback oscillator and repressilator are more robust for oscillations, whereas the activator-inhibitor and substrate-depletion oscillators are superior for maintaining a temperature-independent oscillation period. These results suggest that thermally robust oscillation can be more plausibly achieved by hybridizing these two categories of network motifs. Antagonistic balance and temperature insulation mechanisms for achieving temperature compensation are typically found in these topologies with temperature robustness. In the temperature insulation approach, the oscillation period relies on very few parameters, and these parameters are influenced only slightly by temperature. This approach prevents the temperature from affecting the oscillation period and generates circadian rhythms that are robust against environmental perturbations.

The effects of time-varying temperature on delays in genetic networks

Delays in gene networks result from the sequential nature of protein assembly. However, it is unclear how models of gene networks that use delays should be modified when considering time-dependent changes in temperature. This is important, as delay is often used in models of genetic oscillators that can be entrained by periodic fluctuations in temperature. Here, we analytically derive the time dependence of delay distributions in response to time-varying temperature changes. We find that the resulting time-varying delay is nonlinearly dependent on parameters of the time-varying temperature such as amplitude and frequency, therefore, applying an Arrhenius scaling may result in erroneous conclusions. We use these results to examine a model of a synthetic gene oscillator with temperature compensation. We show that temperature entrainment follows from the same mechanism that results in temperature compensation. Under a common Arrhenius scaling alone, the frequency of the oscillator is sensitive to changes in the mean temperature but robust to changes in the frequency of a periodically time-varying temperature. When a mechanism for temperature compensation is included in the model, however, we show that the oscillator is entrained by periodically varying temperature even when maintaining insensitivity to the mean temperature.

Circadian rhythms. Decoupling circadian clock protein turnover from circadian period determination

The mechanistic basis of eukaryotic circadian oscillators in model systems as diverse as Neurospora, Drosophila, and mammalian cells is thought to be a transcription-and-translation-based negative feedback loop, wherein progressive and controlled phosphorylation of one or more negative elements ultimately elicits their own proteasome-mediated degradation, thereby releasing negative feedback and determining circadian period length. The Neurospora crassa circadian negative element FREQUENCY (FRQ) exemplifies such proteins; it is progressively phosphorylated at more than 100 sites, and strains bearing alleles of frq with anomalous phosphorylation display abnormal stability of FRQ that is well correlated with altered periods or apparent arrhythmicity. Unexpectedly, we unveiled normal circadian oscillations that reflect the allelic state of frq but that persist in the absence of typical degradation of FRQ. This manifest uncoupling of negative element turnover from circadian period length determination is not consistent with the consensus eukaryotic circadian model.

Velocity response curves demonstrate the complexity of modeling entrainable clocks

Circadian clocks are biological oscillators that regulate daily behaviors in organisms across the kingdoms of life. Their rhythms are generated by complex systems, generally involving interlocked regulatory feedback loops. These rhythms are entrained by the daily light/dark cycle, ensuring that the internal clock time is coordinated with the environment. Mathematical models play an important role in understanding how the components work together to function as a clock which can be entrained by light. For a clock to entrain, it must be possible for it to be sped up or slowed down at appropriate times. To understand how biophysical processes affect the speed of the clock, one can compute velocity response curves (VRCs). Here, in a case study involving the fruit fly clock, we demonstrate that VRC analysis provides insight into a clock׳s response to light. We also show that biochemical mechanisms and parameters together determine a model׳s ability to respond realistically to light. The implication is that, if one is developing a model and its current form has an unrealistic response to light, then one must reexamine one׳s model structure, because searching for better parameter values is unlikely to lead to a realistic response to light.

PDP1 regulates energy metabolism through the IIS-TOR pathway in the red flour beetle, Tribolium castaneum

The PAR-domain protein 1 (PDP1) is essential for locomotor activity of insects. However, its functions in insect growth and development have not been studied extensively, which prompted our hypothesis that PDP1 acts in energy metabolism. Here we report identification of TcPDP1 in the red flour beetle, Tribolium castaneum, and its functional analysis by RNAi. Treating larvae with dsTcPDP1 induced pupae developmental arrestment, accompanied by accelerated fat body degradation. dsTcPDP1 treatments in adults resulted in reduced female fecundity. Disruption of TcPDP1 expression affected the transcription of genes involved in insulin signaling transduction and mechanistic target of rapamycin (mTOR) pathway. These results support our hypothesis that TcPDP1 acts in energy metabolism in T. castaneum.

Calculating activation energies for temperature compensation in circadian rhythms

Many biological species possess a circadian clock, which helps them anticipate daily variations in the environment. In the absence of external stimuli, the rhythm persists autonomously with a period of approximately 24 h. However, single pulses of light, nutrients, chemicals or temperature can shift the clock phase. In the case of light- and temperature-cycles, this allows entrainment of the clock to cycles of exactly 24 h. Circadian clocks have the remarkable property of temperature compensation, that is, the period of the circadian rhythm remains relatively constant within a physiological range of temperatures. For several organisms, temperature-regulated processes within the circadian clock have been identified in recent years. However, how these processes contribute to temperature compensation is not fully understood. Here, we theoretically investigate temperature compensation in general oscillatory systems. It is known that every oscillator can be locally temperature compensated around a reference temperature, if reactions are appropriately balanced. A balancing is always possible if the control coefficient with respect to the oscillation period of at least one reaction in the oscillator network is positive. However, for global temperature compensation, the whole physiological temperature range is relevant. Here, we use an approach which leads to an optimization problem subject to the local balancing principle. We use this approach to analyse different circadian clock models proposed in the literature and calculate activation energies that lead to temperature compensation.

Robustness of circadian rhythms in the presence of molecular fluctuations: an investigation based on a mechanistic, statistical theory and a simulation algorithm

After a very brief introduction to a mechanistic and statistical theory of molecular fluctuations in chemical reactions developed by Joel Keizer, we explore the robustness of a circadian rhythm model by using the theory and the exact stochastic simulation (ESS). The comparative study shows that the theory reflects the effects of the dynamics of the model on the robustness more than ESS does. Even though the theory is a macroscopic one, the robustness of the model compares well with that computed from the ESS when the system size is larger than 50. The robustness increases nonlinearly with the system size and it reaches an asymptotic value at higher system sizes. As we can expect from the dynamics of the system, the robustness is minimum near the bifurcation point and as the most sensitive parameter increases away from the bifurcation point the robustness according to the theory as well as the ESS increases and then reaches to a steady value.

Circadian oscillators in eukaryotes

The biological clock, present in nearly all eukaryotes, has evolved such that organisms can adapt to our planet's rotation in order to anticipate the coming day or night as well as unfavorable seasons. As all modern high-precision chronometers, the biological clock uses oscillation as a timekeeping element. In this review, we describe briefly the discovery, historical development, and general properties of circadian oscillators. The issue of temperature compensation (TC) is discussed, and our present understanding of the underlying genetic and biochemical mechanisms in circadian oscillators are described with special emphasis on Neurospora crassa, mammals, and plants.

A biophysical interpretation of temperature-dependent body size in Drosophila aldrichi and D. buzzatii

The temperature-size rule, the observation that most ectotherms grow faster but reach smaller size at higher temperatures, has defied a general explanation. Here, the temperature-size rule in Drosophila aldrichi and Drosophila buzzatii is investigated, using data on development rate and adult dry weight at nine temperatures. In both species the linear regression of dry weight on temperature is negative. The data are used to infer the potential for a description of temperature dependent size by biophysical modelling. The biophysical Sharpe-Schoolfield model for biological rates and its derivative model for adult weight yield detailed patterns for the two species' development rate, growth rate, and adult weight. These detailed patterns do not confirm the existence of a simple temperature-size rule. The species differ significantly in the values of the parameters in the Sharpe-Schoolfield model, and as a consequence in different patterns of weight over temperatures. The different parameters of the Sharpe-Schoolfield model play distinct roles in the patterns of weight over temperatures. A temperature-size rule as a negative regression of weight on temperature might statistically follow from an upper temperature boundary for growth that is lower than the upper temperature boundary for development; as such a relation between the upper temperature boundaries for growth and development would lead to a decrease of weight at high temperature.

Modeling the Drosophila melanogaster circadian oscillator via phase optimization

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, timeless, 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.

Analytical approximations for the amplitude and period of a relaxation oscillator

BACKGROUND

Analysis and design of complex systems benefit from mathematically tractable models, which are often derived by approximating a nonlinear system with an effective equivalent linear system. Biological oscillators with coupled positive and negative feedback loops, termed hysteresis or relaxation oscillators, are an important class of nonlinear systems and have been the subject of comprehensive computational studies. Analytical approximations have identified criteria for sustained oscillations, but have not linked the observed period and phase to compact formulas involving underlying molecular parameters.

RESULTS

We present, to our knowledge, the first analytical expressions for the period and amplitude of a classic model for the animal circadian clock oscillator. These compact expressions are in good agreement with numerical solutions of corresponding continuous ODEs and for stochastic simulations executed at literature parameter values. The formulas are shown to be useful by permitting quick comparisons relative to a negative-feedback represillator oscillator for noise (10x less sensitive to protein decay rates), efficiency (2x more efficient), and dynamic range (30 to 60 decibel increase). The dynamic range is enhanced at its lower end by a new concentration scale defined by the crossing point of the activator and repressor, rather than from a steady-state expression level.

CONCLUSION

Analytical expressions for oscillator dynamics provide a physical understanding for the observations from numerical simulations and suggest additional properties not readily apparent or as yet unexplored. The methods described here may be applied to other nonlinear oscillator designs and biological circuits.

Biological switches and clocks

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

Complex dynamics in the Oregonator model with linear delayed feedback

The Belousov-Zhabotinsky (BZ) reaction can display a rich dynamics when a delayed feedback is applied. We used the Oregonator model of the oscillating BZ reaction to explore the dynamics brought about by a linear delayed feedback. The time-delayed feedback can generate a succession of complex dynamics: period-doubling bifurcation route to chaos; amplitude death; fat, wrinkled, fractal, and broken tori; and mixed-mode oscillations. We observed that this dynamics arises due to a delay-driven transition, or toggling of the system between large and small amplitude oscillations, through a canard bifurcation. We used a combination of numerical bifurcation continuation techniques and other numerical methods to explore the dynamics in the strength of feedback-delay space. We observed that the period-doubling and quasiperiodic route to chaos span a low-dimensional subspace, perhaps due to the trapping of the trajectories in the small amplitude regime near the canard; and the trapped chaotic trajectories get ejected from the small amplitude regime due to a crowding effect to generate chaotic-excitable spikes. We also qualitatively explained the observed dynamics by projecting a three-dimensional phase portrait of the delayed dynamics on the two-dimensional nullclines. This is the first instance in which it is shown that the interaction of delay and canard can bring about complex dynamics.

Internal noise-sustained circadian rhythms in a Drosophila model

Circadian rhythmic processes, mainly regulated by gene expression at the molecular level, have inherent stochasticity. Their robustness or resistance to internal noise has been extensively investigated by most of the previous studies. This work focuses on the constructive roles of internal noise in a reduced Drosophila model, which incorporates negative and positive feedback loops, each with a time delay. It is shown that internal noise sustains reliable oscillations with periods close to 24 h in a region of parameter space, where the deterministic kinetics would evolve to a stable steady state. The amplitudes of noise-sustained oscillations are significantly affected by the variation of internal noise level, and the best performance of the oscillations could be found at an optimal noise intensity, indicating the occurrence of intrinsic coherence resonance. In the oscillatory region of the deterministic model, the coherence of noisy circadian oscillations is suppressed by internal noise, while the period remains nearly constant over a large range of noise intensity, demonstrating robustness of the Drosophila model for circadian rhythms to intrinsic noise. In addition, the effects of time delay in the positive feedback on the oscillations are also investigated. It is found that the time delay could efficiently tune the performance of the noise-sustained oscillations. These results might aid understanding of the exploitation of intracellular noise in biochemical and genetic regulatory systems.

Effects of temperature on circadian rhythm in the Japanese honeybee, Apis cerana japonica

Temperature influences key aspects of insect circadian rhythms. The locomotor rhythm in foragers of the Japanese honeybee, Apis cerana japonica, was entrained to a skeleton temperature cycle. An initial warm temperature pulse was imposed at the beginning of subjective day and a second was applied at the end of the subjective day. A single warm pulse given every early subjective day in constant darkness (DD) entrained the locomotor rhythm without a second temperature pulse, but a single pulse given in late subjective day allowed a free-running rhythm. When honeybees were kept under a light-dark cycle, their body temperatures increased by 7-8 degrees C with locomotor activity. This temperature elevation remained during the photophase but followed the ambient environmental temperature at night. Body temperature oscillations continued to be circadian in DD, and temperature elevation occurred during the subjective day. In DD, the free-running period tau of locomotor activity increased when the ambient temperature increased from 27 to 37 degrees C, although these changes were within the range of temperature compensation for many organisms. Under continuous light conditions (LL), tau remained constant with more strict temperature compensation. Patterns of brain period mRNA levels of forager bees maintained at different temperatures in LL revealed that the free-running period of per mRNA rhythm was temperature compensated. In addition, temperature strongly influenced the amplitude of the circadian transcriptional rhythms during the free-run period in LL, which may confer temperature compensation. We also discuss the possibility that daily changes in forager body temperatures may act as an internal Zeitgeber by fluctuating hive temperature.

The PAS/LOV protein VIVID controls temperature compensation of circadian clock phase and development in Neurospora crassa

Circadian clocks are cellular timekeepers that regulate aspects of temporal organization on daily and seasonal time scales. To allow accurate time measurement, the period lengths of clocks are conserved in a range of temperatures--a phenomenon known as temperature compensation. Temperature compensation of circadian clock period aids in maintaining a stable "target time" or phase of clock-controlled events. Here we show that the Neurospora protein VIVID (VVD) buffers the circadian system against temperature fluctuations. In vvd-null mutants, the circadian period of clock-controlled events such as asexual sporulation (conidiation) is temperature compensated, but the phase of this clock time marker is not. Consistent with delayed conidiation at lower temperatures in vvd(KO) strains, the levels of vvd gene products in the wild type increase with decreasing temperatures. Moreover, vvd(C108A) mutants that lack the light function of VVD maintain a dark activity that transiently influences the phase of conidiation, indicating that VVD influences the time of conidiation downstream from the clock. FREQUENCY (FRQ) phosphorylation is altered in a vvd(KO) strain, suggesting a mechanism by which VVD can influence the timing of clock-controlled processes in the dark. Thus, temperature compensation of clock-controlled output is a key factor in maintaining temperature compensation of the entire circadian system.

A mathematical model of the Drosophila circadian clock with emphasis on posttranslational mechanisms

Experimental evidence points increasingly to the importance of posttranslational processes such as phosphorylation and translocation in the molecular circadian clocks of many organisms. We develop a mathematical model of the Drosophila circadian clock that incorporates the emerging details of the timing of nuclear translocation of the PERIOD and TIMELESS proteins. Most models assume that these proteins enter the nucleus as a complex, but recent experiments suggest that they in fact enter the nucleus separately. Our model reproduces observed patterns of intracellular localization of PERIOD and TIMELESS during light-dark cycles and in constant darkness, as well as phenotypes of several clock mutants. We also use the model to demonstrate how the Drosophila clock can exhibit robust oscillations with constant mRNA levels of period or timeless, and propose a possible mechanism for oscillations in double-rescue experiments of per(01)-tim(01) mutants. The model also explains (via posttranslational processes) the counter-intuitive observation that total dCLOCK levels are at their lowest at the circadian time when active nuclear dCLOCK must be peaking in order to activate transcription of other clock genes, implying that for dCLOCK a posttranslationally generated rhythm is more important than the transcriptionally generated rhythm. These results support the idea that posttranslational processes play key roles in generating as well as modulating robust circadian oscillations. While it appears that posttranslational mechanisms alone are not sufficient to generate rhythms in Drosophila, posttranslational mechanisms can greatly amplify a very weak transcriptional rhythm.

Temperature-compensated chemical reactions

Circadian rhythms are daily oscillations in behaviors that persist in constant light/dark conditions with periods close to 24 h. A striking feature of these rhythms is that their periods remain fairly constant over a wide range of physiological temperatures, a feature called temperature compensation. Although circadian rhythms have been associated with periodic oscillations in mRNA and protein levels, the question of how to construct a network of chemical reactions that is temperature compensated remains unanswered. We discuss a general framework for building such a network.

A proposal for robust temperature compensation of circadian rhythms

The internal circadian rhythms of cells and organisms coordinate their physiological properties to the prevailing 24-h cycle of light and dark on earth. The mechanisms generating circadian rhythms have four defining characteristics: they oscillate endogenously with period close to 24 h, entrain to external signals, suffer phase shifts by aberrant pulses of light or temperature, and compensate for changes in temperature over a range of 10 degrees C or more. Most theoretical descriptions of circadian rhythms propose that the underlying mechanism generates a stable limit cycle oscillation (in constant darkness or dim light), because limit cycles quite naturally possess the first three defining properties of circadian rhythms. On the other hand, the period of a limit cycle oscillator is typically very sensitive to kinetic rate constants, which increase markedly with temperature. Temperature compensation is therefore not a general property of limit cycle oscillations but must be imposed by some delicate balance of temperature dependent effects. However, "delicate balances" are unlikely to be robust to mutations. On the other hand, if circadian rhythms arise from a mechanism that concentrates sensitivity into a few rate constants, then the "balancing act" is likely to be more robust and evolvable. We propose a switch-like mechanism for circadian rhythms that concentrates period sensitivity in just two parameters, by forcing the system to alternate between a stable steady state and a stable limit cycle.

A temperature-compensated model for circadian rhythms that can be entrained by temperature cycles

From the viewpoint that reaction rates will change with temperature, we present a general method to build circadian clock models that generate circadian oscillations with almost constant period under different constant ambient temperature, and propose an algorithm estimating the parameter condition for compensated period against the change of temperature based on the PER single-feedback loop model of Goldbeter [1995. A model for circadian oscillations in the Drosophila period protein (PER). Proc. R. Soc. London Ser. B 261, 319-324] for Drosophila. We show that the model with derived parameters can realize the temperature compensation over a wide range of temperature, and simultaneously can realize the entrainment to temperature cycles.

Modelling of circadian rhythms in Drosophila incorporating the interlocked PER/TIM and VRI/PDP1 feedback loops

Circadian rhythms of gene activity, metabolism, physiology and behaviour are observed in all the eukaryotes and some prokaryotes. In this study, we present a model to represent the transcriptional regulatory network essential for the circadian rhythmicity in Drosophila. The model incorporates the transcriptional feedback loops revealed so far in the network of the circadian clock (PER/TIM and VRI/PDP1 loops). Conventional Hill functions are not assumed to describe the regulation of genes, instead of the explicit reactions of binding and unbinding processes of transcription factors to promoters are modelled. The model simulates sustained circadian oscillations in mRNA and protein concentrations in constant darkness in agreement with experimental observations. It also simulates entrainment by light-dark cycles, disappearance of the rhythmicity in constant light and the shape of phase response curves resembling that of the experimental results. The model is robust over a wide range of parameter variations. In addition, the simulated E-box mutation, per(S) and per(L) mutants are similar to that observed in the experiments. The deficiency between the simulated mRNA levels and experimental observations in per(01), tim(01) and clk(Jrk) mutants suggests some difference on the part of the model from reality.

Propensity of a circadian clock to adjust to the 24h day-night light cycle and its sensitivity to molecular noise

The circadian clock of Drosophila melanogaster and its tendency to adjust to the day-night light cycle is simulated by deterministic and stochastic methods. The robustness of the locking to the light-cycle with respect to molecular noise is studied. It is found that within the model studied, the molecular noise in the stochastic simulation erases the finer injection-locking structures, stronger injection signals are needed and the locking has the character of prolonged locked time intervals with cycle slips in between. The simulations are compared to a simple injection-locking model with noise that seems to describe the overall behavior well.