Circadian rhythms and protein turnover: the effect of temperature on the period lengths of clock mutants simulated by the Goodwin oscillator

Data-driven modelling captures dynamics of the circadian clock of Neurospora crassa

Eukaryotic circadian clocks are based on self-sustaining, cell-autonomous oscillatory feedback loops that can synchronize with the environment via recurrent stimuli (zeitgebers) such as light. The components of biological clocks and their network interactions are becoming increasingly known, calling for a quantitative understanding of their role for clock function. However, the development of data-driven mathematical clock models has remained limited by the lack of sufficiently accurate data. Here we present a comprehensive model of the circadian clock of Neurospora crassa that describe free-running oscillations in constant darkness and entrainment in light-dark cycles. To parameterize the model, we measured high-resolution time courses of luciferase reporters of morning and evening specific clock genes in WT and a mutant strain. Fitting the model to such comprehensive data allowed estimating parameters governing circadian phase, period length and amplitude, and the response of genes to light cues. Our model suggests that functional maturation of the core clock protein Frequency causes a delay in negative feedback that is critical for generating circadian rhythms.

Circadian rhythms are endogenous autonomous clocks that emancipate daily rhythms in physiology and behavior. Lately, a large body of research has contributed to our understanding of clocks’ genetic and mechanistic basis across kingdoms of life, i.e., mammals, fungi, plants, and bacteria. Several mathematical models have made key contributions to our current understanding of the design principles of the Neurospora crassa circadian clock and conditions for self-sustained oscillations. However, previous models uncovered and described the principle properties of the clock in generic manner due to a lack of experimental data. In this study, we developed a mathematical model based on systems of differential equations to describe the core clock components and estimated model parameters from luciferase data that capture experimental observations. We demonstrate the model predictive control simulation emphasizing the importance of functional maturation of the core clock protein Frequency in generating circadian rhythms.

The Goodwin Oscillator and its Legacy

In the 1960's Brian Goodwin published a couple of mathematical models showing how feedback inhibition can lead to oscillations and discussed possible implications of this behaviour for the physiology of the cell. He also presented key ideas about the rich dynamics that may result from the coupling between such biochemical oscillators. Goodwin's work motivated a series of theoretical investigations aiming at identifying minimal mechanisms to generate limit cycle oscillations and deciphering design principles of biological oscillators. The three-variable Goodwin model (adapted by Griffith) can be seen as a core model for a large class of biological systems, ranging from ultradian to circadian clocks. We summarize here main ideas and results brought by Goodwin and review a couple of modeling works directly or indirectly inspired by Goodwin's findings.

Crosstalk dynamics between the circadian clock and the mTORC1 pathway

Crosstalk between the circadian clock clockwork and cellular metabolic regulatory networks is crucial to ensure an adequate response of an organism to the day/night cycle. mTOR (mammalian/mechanistic target of rapamycin) is a master growth regulator and sensor of nutrient status, which is part of the mTOR complex 1 (mTORC1). While the circadian clock confers rhythmicity to the mTOR protein by regulating its degradation rate, mTORC1 activity diminishes period and augments amplitude of circadian oscillations at the cellular level by a currently unknown mechanism. Here, we develop a mathematical deterministic DAE (differential-algebraic equation) model, to explore the possible interactions that allow mTORC1 to display such regulation of the core circadian clock. Our results suggest that mTORC1 is capable of regulating amplitude by exerting translational control on core the clock protein BMAL1, and that period-tuning is achieved by controlling post-translational localization of BMAL1. Since, in our model, mTORC1 control of BMAL1 localization greatly diminishes the ability of the clock to oscillate, and regulation of BMAL1 translation reduces this effect, our results also suggest that both levels of regulation must be present to ensure the robustness of oscillations. Together, the above results emphasize the importance of the influence of mTORC1 on the 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.

Modeling-Based Investigation of the Effect of Noise in Cellular Systems

Noise is pervasive in cellular biology and inevitably affects the dynamics of cellular processes. Biological systems have developed regulatory mechanisms to ensure robustness with respect to noise or to take advantage of stochasticity. We review here, through a couple of selected examples, some insights on possible robustness factors and constructive roles of noise provided by computational modeling. In particular, we focus on (1) factors that likely contribute to the robustness of oscillatory processes such as the circadian clocks and the cell cycle, (2) how reliable coding/decoding of calcium-mediated signaling could be achieved in presence of noise and, in some cases, enhanced through stochastic resonance, and (3) how embryonic cell differentiation processes can exploit stochasticity to create heterogeneity in a population of identical cells.

The Neurospora Circadian System

The eukaryotic filamentous fungus Neurospora crassa has proven to be a durable and dependable model system for the analysis of the cellular and molecular bases of circadian rhythms. Pioneering genetic analyses identified clock genes, and beginning with the cloning of frequency ( frq), work over the past 2 decades has revealed the molecular basis of a core circadian clock feedback loop that has illuminated our understanding of circadian oscillators in microbes, plants, and animals. In this transcription/translation-based feedback loop, a heterodimer of the White Collar-1 (WC-1) and WC-2 proteins acts both as the circadian photoreceptor and, in the dark, as a transcription factor that promotes the expression of the frq gene. FRQ dimerizes and feeds back to block the activity of its activators (making a negative feedback loop), as well as feeding forward to promote the synthesis of its activator, WC-1. Phosphorylation of FRQ by several kinases leads to its ubiquitination and turnover, releasing the WC-1/WC-2 dimer to reactivate frq expression and restart the circadian cycle. Light resetting of the clock can be understood through the rapid light induction of frq expression and temperature resetting through the influence of elevated temperaturesin driving higher levels of FRQ. Several FRQ- and WC-independent, noncircadian FRQ-less oscillators (FLOs) have been described, each of which appears to regulate aspects of Neurospora growth or development. Overall, the FRQ/white collar complex feedback loop appears to coordinate the circadian system through its activity to regulate downstream-target clock-controlled genes, either directly or via regulation of driven FLOs.

Mathematical modeling and validation of glucose compensation of the neurospora circadian clock

Autonomous circadian oscillations arise from transcriptional-translational feedback loops of core clock components. The period of a circadian oscillator is relatively insensitive to changes in nutrients (e.g., glucose), which is referred to as "nutrient compensation". Recently, a transcription repressor, CSP-1, was identified as a component of the circadian system in Neurospora crassa. The transcription of csp-1 is under the circadian regulation. Intriguingly, CSP-1 represses the circadian transcription factor, WC-1, forming a negative feedback loop that can influence the core oscillator. This feedback mechanism is suggested to maintain the circadian period in a wide range of glucose concentrations. In this report, we constructed a mathematical model of the Neurospora circadian clock incorporating the above WC-1/CSP-1 feedback loop, and investigated molecular mechanisms of glucose compensation. Our model shows that glucose compensation exists within a narrow range of parameter space where the activation rates of csp-1 and wc-1 are balanced with each other, and simulates loss of glucose compensation in csp-1 mutants. More importantly, we experimentally validated rhythmic oscillations of the wc-1 gene expression and loss of glucose compensation in the wc-1(ov) mutant as predicted in the model. Furthermore, our stochastic simulations demonstrate that the CSP-1-dependent negative feedback loop functions in glucose compensation, but does not enhance the overall robustness of oscillations against molecular noise. Our work highlights predictive modeling of circadian clock machinery and experimental validations employing Neurospora and brings a deeper understanding of molecular mechanisms of glucose compensation.

Effect of Multiple Clock Gene Ablations on the Circadian Period Length and Temperature Compensation in Mammalian Cells

Most organisms have cell-autonomous circadian clocks to coordinate their activity and physiology according to 24-h environmental changes. Despite recent progress in circadian studies, it is not fully understood how the period length and the robustness of mammalian circadian rhythms are determined. In this study, we established a series of mouse embryonic stem cell (ESC) lines with single or multiplex clock gene ablations using the CRISPR/Cas9-based genome editing method. ESC-based in vitro circadian clock formation assay shows that the CRISPR-mediated clock gene disruption not only reproduces the intrinsic circadian molecular rhythms of previously reported mice tissues and cells lacking clock genes but also reveals that complexed mutations, such as CKIδ(m/m):CKIε(+/m):Cry2(m/m) mutants, exhibit an additively lengthened circadian period. By using these mutant cells, we also investigated the relation between period length alteration and temperature compensation. Although CKIδ-deficient cells slightly affected the temperature insensitivity of period length, we demonstrated that the temperature compensation property is largely maintained in all mutants. These results show that the ESC-based assay system could offer a more systematic and comprehensive approach to the genotype-chronotype analysis of the intracellular circadian clockwork in mammals.

Comprehensive modelling of the Neurospora circadian clock and its temperature compensation

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

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

Modeling circadian clocks: Roles, advantages, and limitations

Circadian rhythms are endogenous oscillations characterized by a period of about 24h. They constitute the biological rhythms with the longest period known to be generated at the molecular level. The abundance of genetic information and the complexity of the molecular circuitry make circadian clocks a system of choice for theoretical studies. Many mathematical models have been proposed to understand the molecular regulatory mechanisms that underly these circadian oscillations and to account for their dynamic properties (temperature compensation, entrainment by light dark cycles, phase shifts by light pulses, rhythm splitting, robustness to molecular noise, intercellular synchronization). The roles and advantages of modeling are discussed and illustrated using a variety of selected examples. This survey will lead to the proposal of an integrated view of the circadian system in which various aspects (interlocked feedback loops, inter-cellular coupling, and stochasticity) should be considered together to understand the design and the dynamics of circadian clocks. Some limitations of these models are commented and challenges for the future identified.

Modeling circadian clocks: From equations to oscillations

Circadian rhythms are generated at the cellular level by a small but tightly regulated genetic network. In higher eukaryotes, interlocked transcriptional-translational feedback loops form the core of this network, which ensures the activation of the right genes (proteins) at the right time of the day. Understanding how such a complex molecular network can generate robust, self-sustained oscillations and accurately responds to signals from the environment (such as light and temperature) is greatly helped by mathematical modeling. In the present paper we review some mathematical models for circadian clocks, ranging from abstract, phenomenological models to the most detailed molecular models. We explain how the equations are derived, highlighting the challenges for the modelers, and how the models are analyzed. We show how to compute bifurcation diagrams, entrainment, and phase response curves. In the subsequent paper, we discuss, through a selection of examples, how modeling efforts have contributed to a better understanding of the dynamics of the circadian regulatory network.

Lithium differentially affects clock gene expression in serum-shocked NIH-3T3 cells

Bipolar disorder has been associated with disturbances in circadian rhythms. Lithium is frequently used in the long-term treatment of bipolar disorder, and has been shown to prolong such rhythms in animals and humans. To examine whether lithium affects the expression of genes regulating the circadian clock, cultured NIH-3T3 cells were synchronized by serum-shocking, and the relative expression of the clock genes Period1 (Per1), Period2 (Per2), Period3 (Per3), Cryptochrome1 (Cry1), Cryptochrome2 (Cry2), Brain and muscle aryl hydrocarbon nuclear translocator-like 1 (Bmal1), Circadian locomotor output cycles kaput (Clock), Rev-Erb-α (Nr1d1), RAR-related orphan receptor α (Ror-α), Glycogen synthase kinase-3β (Gsk-3β), Casein kinase 1-ε (CK1-ε; Csnk1ε), E4 binding protein 4 (E4BP4; Nfil-3) and albumin D-binding protein (Dbp) was examined for three consecutive days in the presence of lithium (20 mM) or vehicle (20 mM NaCl). We found that lithium significantly increased the expression of Per2 and Cry1, whereas Per3, Cry2, Bmal1, E4BP4 and Rev-Erb-α expression was reduced. We also found that lithium prolonged the period of Per2. Taken together, these effects on clock gene expression may be relevant for the effects of lithium on biological rhythms and could also give new leads to further explore its mood-stabilizing actions in the treatment of bipolar disorder.

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.

Robust, tunable biological oscillations from interlinked positive and negative feedback loops

A simple negative feedback loop of interacting genes or proteins has the potential to generate sustained oscillations. However, many biological oscillators also have a positive feedback loop, raising the question of what advantages the extra loop imparts. Through computational studies, we show that it is generally difficult to adjust a negative feedback oscillator's frequency without compromising its amplitude, whereas with positive-plus-negative feedback, one can achieve a widely tunable frequency and near-constant amplitude. This tunability makes the latter design suitable for biological rhythms like heartbeats and cell cycles that need to provide a constant output over a range of frequencies. Positive-plus-negative oscillators also appear to be more robust and easier to evolve, rationalizing why they are found in contexts where an adjustable frequency is unimportant.

Simulating dark expressions and interactions of frq and wc-1 in the Neurospora circadian clock

Circadian rhythms are considered to play an essential part in the adaptation of organisms to their environments. The occurrence of circadian oscillations appears to be based on the presence of transcriptional-translational negative feedback loops. In Neurospora crassa, the protein FREQUENCY (FRQ) is part of such a negative feedback loop apparently by a direct interaction with its transcription factor WHITE COLLAR-1 (WC-1). Based on the observation that nuclear FRQ levels are significantly lower than nuclear WC-1 levels, it was suggested that FRQ would act more like a catalyst in inhibiting WC-1 rather than binding to WC-1 and making an inactive FRQ:WC-1 complex. Intrigued by this hypothesis, we constructed a model for the Neurospora circadian clock, which includes expression of the frq and the wc-1 genes and their possible interactions. The model suggests that even small amounts of nuclear FRQ-protein are capable of inhibiting frq transcription in a rhythmic manner by binding to WC-1 and promoting its degradation. Our model predicts the importance of a FRQ dependent degradation of WC-1 in closing the negative feedback loop. The model shows good agreement with experimental levels in nuclear and cytosolic FRQ and WC-1, their phase relationships, and several clock mutant phenotypes.

A model for the circadian rhythm of cyanobacteria that maintains oscillation without gene expression

An intriguing property of the cyanobacterial circadian clock is that endogenous rhythm persists when protein abundances are kept constant either in the presence of translation and transcription inhibitors or in the constant dark condition. Here we propose a regulatory mechanism of KaiC phosphorylation for the generation of circadian oscillations in cyanobacteria. In the model, clock proteins KaiA and KaiB are assumed to have multiple states, regulating the KaiC phosphorylation process. The model can explain 1), the sustained oscillation of gene expression and protein abundance when the expression of the kaiBC gene is regulated by KaiC protein, and 2), the sustained oscillation of phosphorylated KaiC when transcription and translation processes are inhibited and total protein abundance is fixed. Results of this work suggest that KaiA and KaiB strengthen the nonlinearity of KaiC phosphorylation, thereby promoting the circadian rhythm in cyanobacteria.

The relationship between FRQ-protein stability and temperature compensation in the Neurospora circadian clock

Temperature compensation is an important property of all biological clocks. In Neurospora crassa, negative-feedback regulation on the frequency (frq) gene's transcription by the FRQ protein plays a central role in the organism's circadian pacemaker. Earlier model calculations predicted that the stability of FRQ should determine the period length of Neurospora's circadian rhythm as well as the rhythm's temperature compensation. Here, we report experimental FRQ protein stabilities in frq mutants at 20 degrees C and 25 degrees C, and estimates of overall activation energies for mutant FRQ protein degradation. The results are consistent with earlier model predictions, i.e., temperature compensation of Neurospora's circadian rhythm is a highly regulated process where the stability of FRQ is an important factor in determining Neurospora's circadian period as well as the clock's temperature compensation. The partial loss of temperature compensation in frq7 and frq(S513I) mutants can be described by a simple negative-feedback model (the Goodwin oscillator) when the experimentally obtained activation energies of FRQ degradation for theses mutants are incorporated into the model.

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

We have analysed a first-order kinetic representation of a interlocking-feedback loop model for the Drosophila circadian clock. In this model, the transcription factor Drosophila CLOCK (dCLK) which activates the clock genes period (per) and timeless (tim) is subjected to positive and negative regulations by the proteins 'PAR Domain Protein 1' (PDP1) and VRILLE (VRI), whose transcription is activated by dCLK. The PER/TIM complex binds to dCLK and in this way reduces the activity of dCLK. The results of our simulations suggest that the positive and negative feedback loops of Pdp1 and vri are essential for the overall oscillations. Although self sustained oscillations can be obtained without per/tim, the model shows that the PER/TIM complex plays an important role in amplification and stabilization of the oscillations generated by the Pdp1/vri positive/negative feedback loops. We further show that in contrast to a single (per/tim) negative feedback loop oscillator, the interlocking-feedback loop model can readily account for the effect of gene dosages of per, vri, and Pdp1 on the period length. Calculations of phase resetting on a temperature compensated version of the model shows good agreement with experimental phase response curves for high and low temperature pulses. Also, the partial losses of temperature compensation in perS and perL mutants can be described, which are related to decreased stabilities of the PER/TIM complex in perS and the stronger/more stable inhibitory complex between dCLK and PER/TIM in perL, respectively. The model shows (somewhat surprisingly) poor entrainment properties, especially under extended light/dark (L/D) cycles, which suggests that parts of the L/D tracking or sensing system are not well represented.

Circadian rhythms in microorganisms: new complexities

Recent advances in understanding circadian (daily) rhythms in the genera Neurospora, Gonyaulax, and Synechococcus are reviewed and new complexities in their circadian systems are described. The previous model, consisting of a unidirectional flow of information from input to oscillator to output, has now expanded to include multiple input pathways, multiple oscillators, multiple outputs; and feedback from oscillator to input and output to oscillator. New posttranscriptional features of the frq/white-collar oscillator (FWC) of Neurospora are described, including protein phosphorylation and degradation, dimerization, and complex formation. Experimental evidence is presented for frq-less oscillator(s) (FLO) downstream of the FWC. Mathematical models of the Neurospora system are also discussed.

Temperature compensation in circadian clock models

Circadian clock of organisms has a free-running period that does not change much with ambient temperature. This property "temperature compensation" is studied when the rate of all reaction steps increase with temperature in the biochemical network generating the rhythm. The period becomes shorter when all the rate parameters are enhanced by the same factor. However, the period becomes longer as degradation rate of proteins and/or transcription rate of the clock gene increase (their elasticity is positive). This holds for a wide range of models, including N-variable model, and PER-TIM double oscillator model, provided that (1) branch reactions (e.g. degradation of proteins or mRNAs) are strongly saturated, and that (2) the cooperativity of transcription inhibition by nuclear proteins is not very large. A strong temperature sensitivity of degradation of PER proteins and/or temperature-sensitive alternative splicing of per gene, known for Drosophila, can be mechanisms for the temperature compensation of circadian clock.

A two variable delay model for the circadian rhythm of Neurospora crassa

A two variable model with delay in both the variables, is proposed for the circadian oscillations of protein concentrations in the fungal species Neurospora crassa. The dynamical variables chosen are the concentrations of FRQ and WC-1 proteins. Our model is a two variable simplification of the detailed model of Smolen et al. (J. Neurosci. 21 (2001) 6644) modeling circadian oscillations with interlocking positive and negative feedback loops, containing 23 variables. In our model, as in the case of Smolen's model, a sustained limit cycle oscillation takes place in both FRQ and WC-1 protein in continuous darkness, and WC-1 is anti-phase to FRQ protein, as observed in experiments. The model accounts for various characteristic features of circadian rhythms such as entrainment to light dark cycles, phase response curves and robustness to parameter variation and molecular fluctuations. Simulations are carried out to study the effect of periodic forcing of circadian oscillations by light-dark cycles. The periodic forcing resulted in a rich bifurcation diagram that includes quasiperiodicity and chaotic oscillations, depending on the magnitude of the periodic changes in the light controlled parameter. When positive feedback is eliminated, our model reduces to the generic one dimensional delay model of Lema et al. (J. Theor. Biol. 204 (2000) 565), delay model of the circadian pace maker with FRQ protein as the dynamical variable which represses its own production. This one-dimensional model also exhibits all characteristic features of circadian oscillations and gives rise to circadian oscillations which are reasonably robust to parameter variations and molecular noise.

Combining theoretical and experimental approaches to understand the circadian clock

This review is intended as a summary of our work carried out as part of the German Research Association (DFG) Center Program on Circadian Rhythms. Over the last six years, our approach to understanding circadian systems combined theoretical and experimental tools, and Gonyaulax and Neurospora have proven ideal for these efforts. Both of these model organisms demonstrate that even simple circadian systems can have multiple light input pathways and more than one rhythm generator. They have both been used to elaborate basic circadian features in conjunction with formal models. The models introduce the "zeitnehmer," i.e., a clock-regulated input pathway, to the conceptual framework of circadian systems, and proposes networks of individual feedbacks as the basis for circadian rhythmicity.

Cyanobacterial circadian clockwork: roles of KaiA, KaiB and the kaiBC promoter in regulating KaiC

Using model strains in which we ectopically express the cyanobacterial clock protein KaiC in cells from which the clock genes kaiA, kaiB and/or kaiC are deleted, we found that some features of circadian clocks in eukaryotic organisms are conserved in the clocks of prokaryotic cyanobacteria, but others are not. One unexpected difference is that the circadian autoregulatory feedback loop in cyanobacteria does not require specific clock gene promoters as it does in eukaryotes, because a heterologous promoter can functionally replace the kaiBC promoter. On the other hand, a similarity between eukaryotic clock proteins and the cyanobacterial KaiC protein is that KaiC is phosphorylated in vivo. The other essential clock proteins KaiA and KaiB modulate the status of KaiC phosphorylation; KaiA inhibits KaiC dephosphorylation and KaiB antagonizes this action of KaiA. Based upon an analysis of clock mutants, we conclude that the circadian period in cyanobacteria is determined by the phosphorylation status of KaiC and also by the degradation rate of KaiC. These observations are integrated into a model proposing rhythmic changes in chromosomal status.

Circadian period lengths of lipid synthesis mutants (cel, chol-1) of Neurospora show defective temperature, but intact pH-compensation

The influence of extracellular pH on the circadian sporulation rhythm of Neurospora crassa has been investigated for the mutants chol-1 and cel. Both mutants have a defect in the lipid synthesis pathway and require either choline or palmitate, respectively, as supplements for normal growth. The chol-1 and cel mutants also show an impaired temperature-compensation when growing on minimal medium. We investigated the possible correlation between loss of temperature- and pH-compensation in cel and chol-1 similar to the correlation found earlier for the frq7 mutant. Our results show that the cel and the chol-1 mutants, although defective in temperature-compensation have an intact pH-compensation of their circadian rhythms. At present, the products of the frq-locus are the only components of the clock that affect the sporulation rhythm of Neurospora both through pH- and temperature-compensation.

Genetic and molecular analysis of circadian rhythms in Neurospora

Over the course of the past 40 years Neurospora has become a well-known and uniquely tractable model system for the analysis of the molecular basis of eukaryotic circadian oscillatory systems. Molecular bases for the period length and sustainability of the rhythm, light, and temperature resetting of the circadian system and for gating of light input and light effects are becoming understood, and Neurospora promises to be a suitable system for examining the role of coupled feedback loops in the clock. Many of these insights have shown or foreshadow direct parallels in mammalian systems, including the mechanism of light entrainment, the involvement of PAS:PAS heterodimers as transcriptional activators in essential clock-associated feedback loops, and dual role of FRQ in the loop as an activator and a repressor; similarities extend to the primary sequence level in at least one case, that of WC-1 and BMAL1. Work on circadian output in Neurospora has identified more than a dozen regulated genes and has been at the forefront of studies aimed at understanding clock control of gene expression.

Biological timing and the clock metaphor: oscillatory and hourglass mechanisms

Living organisms have developed a multitude of timing mechanisms--"biological clocks." Their mechanisms are based on either oscillations (oscillatory clocks) or unidirectional processes (hourglass clocks). Oscillatory clocks comprise circatidal, circalunidian, circadian, circalunar, and circannual oscillations--which keep time with environmental periodicities--as well as ultradian oscillations, ovarian cycles, and oscillations in development and in the brain, which keep time with biological timescales. These clocks mainly determine time points at specific phases of their oscillations. Hourglass clocks are predominantly found in development and aging and also in the brain. They determine time intervals (duration). More complex timing systems combine oscillatory and hourglass mechanisms, such as the case for cell cycle, sleep initiation, or brain clocks, whereas others combine external and internal periodicities (photoperiodism, seasonal reproduction). A definition of a biological clock may be derived from its control of functions external to its own processes and its use in determining temporal order (sequences of events) or durations. Biological and chemical oscillators are characterized by positive and negative feedback (or feedforward) mechanisms. During evolution, living organisms made use of the many existing oscillations for signal transmission, movement, and pump mechanisms, as well as for clocks. Some clocks, such as the circadian clock, that time with environmental periodicities are usually compensated (stabilized) against temperature, whereas other clocks, such as the cell cycle, that keep time with an organismic timescale are not compensated. This difference may be related to the predominance of negative feedback in the first class of clocks and a predominance of positive feedback (autocatalytic amplification) in the second class. The present knowledge of a compensated clock (the circadian oscillator) and an uncompensated clock (the cell cycle), as well as relevant models, are briefly re viewed. Hourglass clocks are based on linear or exponential unidirectional processes that trigger events mainly in the course of development and aging. An important hourglass mechanism within the aging process is the limitation of cell division capacity by the length of telomeres. The mechanism of this clock is briefly reviewed. In all clock mechanisms, thresholds at which "dependent variables" are triggered play an important role.

The Goodwin model: simulating the effect of light pulses on the circadian sporulation rhythm of Neurospora crassa

The Goodwin oscillator is a minimal model that describes the oscillatory negative feedback regulation of a translated protein which inhibits its own transcription. Now, over 30 years later this scheme provides a basic description of the central components in the circadian oscillators of Neurospora, Drosophila, and mammals. We showed previously that Neurospora's resetting behavior by pulses of temperature, cycloheximide or heat shock can be simulated by this model, in which degradation processes play an important role for determining the clock's period and its temperature-compensation. Another important environmental factor for the synchronization is light. In this work, we show that on the basis of a light-induced transcription of the frequency (frq) gene phase response curves of light pulses as well as the influence of the light pulse length on phase shifts can be described by the Goodwin oscillator. A relaxation variant of the model predicts that directly after a light pulse inhibition in frq -transcription occurs, even when the inhibiting factor Z (FRQ) has not reached inhibitory concentrations. This has so far not been experimentally investigated for frq transcription, but it complies with a current model of light-induced transcription of other genes by a phosphorylated white-collar complex. During long light pulses, the relaxational model predicts that the sporulation rhythm is arrested in a steady state of high frq -mRNA levels. However, experimental results indicate the possibility of oscillations around this steady state and more in favor of the results by the original Goodwin model. In order to explain the resetting behavior by two light pulses, a biphasic first-order kinetics recovery period of the blue light receptor or of the light signal transduction pathway has to be assumed.

pH homeostasis of the circadian sporulation rhythm in clock mutants of Neurospora crassa

The influence of environmental (extracellular) pH on the sporulation rhythm in Neurospora crassa was investigated for wild-type (frq+) and the mutants chr, frq1, frq7, and frq8. In all mutants, including wild type, the growth rate was found to be influenced strongly by extracellular pH in the range 4-9. On the other hand, for the same pH range, the period length of the sporulation rhythm is little influenced in wild type, chr, and frq1. A loss of pH homeostasis of the period, however, was observed in the mutants frq7 and frq8, which also are known to have lost temperature compensation. Concerning the influence of extracellular pH on growth rates, a clear correspondence between growth rates and the concentration of available H2PO4- ion has been found, indicating that the uptake of H2PO4- may be a limiting factor for growth under our experimental conditions. The loss of pH compensation in the frq7 and frq8 mutants may be related to less easily degradable FRQ7,8 proteins when compared with wild-type FRQ. Results from recent model considerations and experimental results predict that, with increasing extra-and intracellular pH, the FRQ7 protein degradation increases and should lead to shorter period lengths.

Microbial circadian oscillatory systems in Neurospora and Synechococcus: models for cellular clocks

Common regulatory patterns have emerged among the feedback loops lying within circadian systems. Significant progress in dissecting the mechanism of clock resetting by temperature and the role of the WC proteins in the Neurospora light response has accompanied documentation of the importance of nuclear localization and phosphorylation-induced turnover of FRQ to this circadian cycle. The long-awaited molecular description of a transcription/translation loop in the Synechococcus circadian system represents a quantal step forward, followed by the identification of additional important proteins and interactions. Finally, the adaptive significance of rhythms in Synechococcus and by extension in all clocks nicely ties up an extraordinary year.

Theoretical models for circadian rhythms in Neurospora and Drosophila

We examine theoretical models proposed for the molecular mechanism of circadian rhythms in Drosophila. The models are based on the negative feedback exerted by a complex between the PER and TIM proteins on the expression of the per and tim genes. We show that a similar model can account for circadian oscillations in Neurospora, where the protein FRQ negatively regulates the expression of the frq gene. The effect of light on the circadian rhythms is included by considering that it elicits a rise in the rate of TIM degradation in Drosophila, whereas in Neurospora it enhances the rate of frq transcription. The models account for the occurrence of sustained circadian oscillations in continuous darkness in Drosophila and Neurospora. Numerical simulations further indicate that the periodic forcing of circadian oscillations by light-dark cycles can result either in the entrainment to the external periodicity or in aperiodic oscillations (i.e. chaos), depending on the magnitude of the periodic changes in the light-controlled parameter.

The Goodwin oscillator: on the importance of degradation reactions in the circadian clock

This article focuses on the Goodwin oscillator and related minimal models, which describe negative feedback schemes that are of relevance for the circadian rhythms in Neurospora, Drosophila, and probably also in mammals. The temperature behavior of clock mutants in Neurospora crassa and Drosophila melanogaster are well described by the Goodwin model, at least on a semi-quantitative level. A similar semi-quantitative description has been found for Neurospora crassa phase response curves with respect to moderate temperature pulses, heat shock pulses, and pulses of cycloheximide. A characteristic feature in the Goodwin and related models is that degradation of clock-mRNA and clock protein species plays an important role in the control of the oscillator's period. As predicted by this feature, recent experimental results from Neurospora crassa indicate that the clock (FRQ) protein of the long period mutant frq7 is degraded approximately twice as slow as the corresponding wild-type protein. Quantitative RT-PCR indicates that experimental frq7-mRNA concentrations are significantly higher than wild-type levels. The latter findings cannot be modeled by the Goodwin oscillator. Therefore, a threshold inhibition mechanism of transcription is proposed.

Eukaryotic circadian systems: cycles in common

Common regulatory patterns can now be discerned among eukaryotic circadian systems, extending from fungi through to mammals. Complexes of two distinct PAS domain-containing transcription factors play positive roles in clock-associated feedback loops by turning on classic clock proteins such as FRQ, PER and TIM. These in turn appear to act as negative elements, interfering with their own activation and thus giving rise to an oscillatory negative feedback loop. Post-transcriptional control governs the amount and type of FRQ and makes the clock responsive to temperature.

Molecular circadian oscillators: an alternative hypothesis

Results from experiments in different organisms have shown that elements of input pathways can themselves be under circadian control and that outputs might feed back into the oscillator. In addition, it has become clear that there might be redundancies in the generation of circadian rhythmicity, even within single cells. In view of these results, it is worth reevaluating our current working hypotheses about the pacemaker's molecular mechanisms and the involvement of single autoregulatory genes. On one hand, redundancies in the generation of circadian rhythmicity might make the approach of defining a discrete circadian oscillator with the help of single gene mutations extremely difficult. On the other hand, many examples show that components of signal transduction pathways can indeed be encoded by single genes. The authors have constructed a model placing an autoregulatory gene and its products on an input pathway feeding into a separate oscillator. The behavior of this model can explain the majority of results of molecular circadian biology published to date. In addition, it shows that different qualities of the circadian system might be associated with different cellular functions that can exist independently and, only if put together, will lead to the known circadian phenotype.

Circadian and ultradian clock-controlled rhythms in unicellular microorganisms

The time structure of a biological system is at least as intricate as its spatial structure. Whereas we have detailed information about the latter, our understanding of the former is still rudimentary. As techniques for monitoring intracellular processes continuously in single cells become more refined, it becomes increasingly evident that periodic behaviour abounds in all time domains. Circadian timekeeping dominates in natural environments. Here the free-running period is about 24 h. Circadian rhythms in eukaryotes and prokaryotes allow predictive matching of intracellular states with environmental changes during the daily cycles. Unicellular organisms provide excellent systems for the study of these phenomena, which pervade all higher life forms. Intracellular timekeeping is essential. The presence of a temperature-compensated oscillator provides such a timer. The coupled outputs (epigenetic oscillations) of this ultradian clock constitute a special class of ultradian rhythm. These are undamped and endogenously driven by a device which shows biochemical properties characteristic of transcriptional and translational elements. Energy-yielding processes, protein turnover, motility and the timing of the cell-division cycle processes are all controlled by the ultradian clock. Different periods characterize different species, and this indicates a genetic determinant. Periods range from 30 min to 4 h. Mechanisms of clock control are being elucidated; it is becoming evident that many different control circuits can provide these functions.

Modeling temperature compensation in chemical and biological oscillators

All physicochemical and biological oscillators maintain a balance between destabilizing reactions (as, for example, intrinsic autocatalytic or amplifying reactions) and stabilizing processes. These two groups of processes tend to influence the period in opposite directions and may lead to temperature compensation whenever their overall influence balances. This principle of "antagonistic balance" has been tested for several chemical and biological oscillators. The Goodwin negative feedback oscillator appears of particular interest for modeling the circadian clocks in Neurospora and Drosophila and their temperature compensation. Remarkably, the Goodwin oscillator not only gives qualitative, correct phase response curves for temperature steps and temperature pulses, but also simulates the temperature behavior of Neurospora frq and Drosophila per mutants almost quantitatively. The Goodwin oscillator predicts that circadian periods are strongly dependent on the turnover of the clock mRNA or clock protein. A more rapid turnover of clock mRNA or clock protein results, in short, a slower turnover in longer period lengths.

Temperature compensation of the circadian period length--a special case among general homeostatic mechanisms of gene expression?

In Neurospora crassa, as well as in other organisms, the expression of housekeeping genes is transiently suppressed after exposure to higher temperatures (30-45 degrees C); expression is then reactivated and adapts after a few-hours to values closer to the initial rates. Adaptive mechanisms apparently exist in the processes of transcription, RNA processing, and translation and render protein synthesis rates temperature compensated. Heat shock proteins (HSPs) play an important role within these mechanisms ("acquired thermotolerance of protein synthesis"), but their function is as yet not exactly known. Adaptive mechanisms seem also to involve intracellular ion changes after exposure to moderate temperature elevation. The expression of heat shock genes is transiently enhanced after exposure to higher temperatures and also adapts after a few hours. The adaptation mechanism includes inactivation of the heat shock transcription factor (HSF) by means of phosphorylation changes and possibly by binding of a gene product (HSP70)-a mechanism representing a negative feedback control. These examples demonstrate the existence of general adaptive mechanisms at different levels of gene expression that may also be at work in the temperature compensation of clock gene expression. Apart from such adaptation processes, antagonistic reactions within the processes of gene expression and protein modification might be equally enhanced or suppressed by temperature changes, leaving the equilibrium unaffected or balanced (antagonistic balance, see Ruoff et al., this issue of Chronobiology International). This principle is shown to apply to the effect of temperature elevation on total protein synthesis and degradation. It may also apply to other antagonistic processes such as phosphorylation-dephosphorylation or monomer-dimer formation. The circadian clock mechanism is assumed to consist of several processes that can either adapt or produce a balance. Single amino acid changes in a clock protein are assumed to partially upset this adaptation or balance.

Temporal organization and disorganization in organisms

The study of biological rhythms has always been a highly interdisciplinary field of research, one that is constantly breaking new ground. Developments during the past 10 years are transforming our ideas about the dynamics of physiological regulation in two ways: (i) homeostasis is really homeodynamics, with sophisticated but robust patterns of hierarchically nested rhythms covering several orders of magnitude in the frequency domain; (ii) there is much more irregularity than previously believed in processes as basic as the heart rate, suggesting either a chaotic generator, a complex pattern of interacting systems with different frequencies, or possibly both. This behavior is consistent with that expected of organisms as complex adaptive systems.