Modeling the molecular regulatory mechanism of circadian rhythms in Drosophila

A stochastic vs deterministic perspective on the timing of cellular events

Cells are the fundamental units of life, and like all life forms, they change over time. Changes in cell state are driven by molecular processes; of these many are initiated when molecule numbers reach and exceed specific thresholds, a characteristic that can be described as "digital cellular logic". Here we show how molecular and cellular noise profoundly influence the time to cross a critical threshold-the first-passage time-and map out scenarios in which stochastic dynamics result in shorter or longer average first-passage times compared to noise-less dynamics. We illustrate the dependence of the mean first-passage time on noise for a set of exemplar models of gene expression, auto-regulatory feedback control, and enzyme-mediated catalysis. Our theory provides intuitive insight into the origin of these effects and underscores two important insights: (i) deterministic predictions for cellular event timing can be highly inaccurate when molecule numbers are within the range known for many cells; (ii) molecular noise can significantly shift mean first-passage times, particularly within auto-regulatory genetic feedback circuits.

Dynamic properties of a self-replicating peptide network with inhibition

In this paper, we report an open system consisting of three self-replicating peptides, in which peptide 1 inhibits the duplex template of peptide 2, peptide 2 inhibits duplex 3, and peptide 3 inhibits duplex 1 to complete the negative feedback loop. This interacting chemical network yields oscillations in the concentrations of all species over time and establishes a possible mechanism for pre-biotic chemical systems organization. The first focus of our analysis is the effect of altering rates of duplex formation and inhibition on oscillations. We then examine the autocatalytic rate constant in the symmetric and asymmetric cases.

Coordination of pancreatic islet rhythmic activity by delayed negative feedback

Secretion of insulin from the pancreas is pulsatile, driven by intrinsic oscillations within individual islets of Langerhans. The secretions are coordinated among the many islets distributed throughout the pancreas producing a synchronized rhythm in vivo that is essential for maintaining normal glucose levels. One hypothesized mechanism for the coordination of islet activity is negative feedback, whereby sequestration of glucose in response to elevated insulin leads to a reduction in the blood glucose level that is sensed by the islet population. This global signal of glucose then coordinates the individual islets. In this study, we tested how this coordination mechanism is affected by time delays in the negative feedback, using a microfluidic system to monitor Ca2+ levels in a small population of islets and implementing glucose control through a negative feedback system. We found that islet synchronization occurs even with time delays in the feedback of up to 7 min. We also found that a second, slower closed-loop oscillation period is produced during delayed feedback in which islet oscillations are clustered into episodes. The period of this second oscillatory mode increases with the time delay and appears to be a second stable behavior that coexists with the faster synchronized oscillation. The general conclusion is that islet coordination through negative feedback is a viable means of islet coordination that is robust to delays in the timing of the feedback, and could complement other potential coordination mechanisms such as entrainment by pancreatic ganglia.NEW & NOTEWORTHY Insulin secretion from islets of Langerhans is rhythmic, and these rhythms are coordinated to produce oscillatory plasma insulin levels. Using a combination of microfluidics and computational modeling, we demonstrate that coordination can occur through negative feedback of the type provided by the liver, even if that feedback is delayed by several minutes. We also demonstrate that a second, slower, mode of oscillations can occur when feedback is delayed where faster oscillations are grouped into episodes.

Existence and exponential stability of weighted pseudo-almost periodic solutions for genetic regulatory networks with time-varying delays

The importance of prediction for genetic regulatory network (GRNs) makes mathematical modeling a prominent tool. In this paper, we consider weighted pseudo-almost periodic solutions for a class of GRNs with time-varying delays. We establish the existence, uniqueness, and global exponential stability by employing the theory of dichotomy, the fixed point theorem, and differential inequality. A numerical example along with a graphical illustration are presented to support our main results. Our results extend existing GRNs models using almost periodic functions to support a wider range of regulatory processes.

A saturated reaction in repressor synthesis creates a daytime dead zone in circadian clocks

Negative feedback loops (NFLs) for circadian clocks include light-responsive reactions that allow the clocks to shift their phase depending on the timing of light signals. Phase response curves (PRCs) for light signals in various organisms include a time interval called a dead zone where light signals cause no phase shift during daytime. Although the importance of the dead zone for robust light entrainment is known, how the dead zone arises from the biochemical reactions in an NFL underlying circadian gene expression rhythms remains unclear. In addition, the observation that the light-responsive reactions in the NFL vary between organisms raises the question as to whether the mechanism for dead zone formation is common or distinct between different organisms. Here we reveal by mathematical modeling that the saturation of a biochemical reaction in repressor synthesis in an NFL is a common mechanism of daytime dead zone generation. If light signals increase the degradation of a repressor protein, as in Drosophila, the saturation of repressor mRNA transcription nullifies the effect of light signals, generating a dead zone. In contrast, if light signals induce the transcription of repressor mRNA, as in mammals, the saturation of repressor translation can generate a dead zone by cancelling the influence of excess amount of mRNA induced by light signals. Each of these saturated reactions is located next to the light-responsive reaction in the NFL, suggesting a design principle for daytime dead zone generation.

Light-entrainable circadian clocks form behavioral and physiological rhythms in organisms. The light-entrainment properties of these clocks have been studied by measuring phase shifts caused by light pulses administered at different times. The phase response curves of various organisms include a time window called the dead zone where the phase of the clock does not respond to light pulses. However, the mechanism underlying the dead zone generation remains unclear. We show that the saturation of biochemical reactions in feedback loops for circadian oscillations generates a dead zone. The proposed mechanism is generic, as it functions in different models of the circadian clocks and biochemical oscillators. Our analysis indicates that light-entrainment properties are determined by biochemical reactions at the single-cell level.

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.

An Unstable Singularity Underlies Stochastic Phasing of the Circadian Clock in Individual Cyanobacterial Cells

The endogenous circadian clock synchronizes with environmental time by appropriately resetting its phase in response to external cues. Of note, some resetting stimuli induce attenuated oscillations of clock output, which has been observed at the population-level in several organisms and in studies of individual humans. To investigate what is happening in individual cellular clocks, we studied the unicellular cyanobacterium S. elongatus. By measuring its phase-resetting responses to temperature changes, we found that population-level arrhythmicity occurs when certain perturbations cause stochastic phases of oscillations in individual cells. Combining modeling with experiments, we related stochastic phasing to the dynamical structure of the cyanobacterial clock as an oscillator and explored the physiological relevance of the oscillator structure for accurately timed rhythmicity in changing environmental conditions. Our findings and approach can be applied to other biological oscillators.

Computational modeling of the cell-autonomous mammalian circadian oscillator

This review summarizes various mathematical models of cell-autonomous mammalian circadian clock. We present the basics necessary for understanding of the cell-autonomous mammalian circadian oscillator, modern experimental data essential for its reconstruction and some special problems related to the validation of mathematical circadian oscillator models. This work compares existing mathematical models of circadian oscillator and the results of the computational studies of the oscillating systems. Finally, we discuss applications of the mathematical models of mammalian circadian oscillator for solving specific problems in circadian rhythm biology.

The synchronization of neuronal oscillators determined by the directed network structure of the suprachiasmatic nucleus under different photoperiods

The main function of the principal clock located in the suprachiasmatic nucleus (SCN) of mammals is synchronizing the body rhythms to the 24 h light-dark cycle. Additionally, the SCN is able to adapt to the photoperiod of the cycle which varies among seasons. Under the long photoperiod (LP), the synchronization degree of the SCN neurons is lower than that under the photoperiod (SP). In the present study, a potential explanation is given for this phenomenon. We propose that the asymmetrical coupling between the light-signal-sensitive part (the ventralateral part, abbreviation: VL) and the light-signal-insensitive part (the dorsalmedial part, abbreviation: DM) of the SCN plays a role in the synchronization degree, which is reflected by the ratio of the number of the directed links from the VL neurons to the DM neurons to the total links of both directions between the VL and the DM. The ratio is assumed to characterize the directed network structure under different photoperiods, which is larger under the SP and smaller under the LP. We found that with the larger ratio in the situation of the SP, the synchronization degree is higher. Our finding may shed new light on the asymmetrical coupling between the VL and the DM, and the network structure of the SCN.

Phase Sensitivity Analysis of Circadian Rhythm Entrainment

As a biological clock, circadian rhythms evolve to accomplish a stable (robust) entrainment to environmental cycles, of which light is the most obvious. The mechanism of photic entrainment is not known, but two models of entrainment have been proposed based on whether light has a continuous (parametric) or discrete (nonparametric) effect on the circadian pacemaker. A novel sensitivity analysis is developed to study the circadian entrainment in silico based on a limit cycle approach and applied to a model of Drosophila circadian rhythm. The comparative analyses of complete and skeleton photoperiods suggest a trade-off between the contribution of period modulation (parametric effect) and phase shift (nonparametric effect) in Drosophila circadian entrainment. The results also give suggestions for an experimental study to (in)validate the two models of entrainment.

Inverse problem studies of biochemical systems with structure identification of S-systems by embedding training functions in a genetic algorithm

An efficient inverse problem approach for parameter estimation, state and structure identification from dynamic data by embedding training functions in a genetic algorithm methodology (ETFGA) is proposed for nonlinear dynamical biosystems using S-system canonical models. Use of multiple shooting and decomposition approach as training functions has been shown for handling of noisy datasets and computational efficiency in studying the inverse problem. The advantages of the methodology are brought out systematically by studying it for three biochemical model systems of interest. By studying a small-scale gene regulatory system described by a S-system model, the first example demonstrates the use of ETFGA for the multifold aims of the inverse problem. The estimation of a large number of parameters with simultaneous state and network identification is shown by training a generalized S-system canonical model with noisy datasets. The results of this study bring out the superior performance of ETFGA on comparison with other metaheuristic approaches. The second example studies the regulation of cAMP oscillations in Dictyostelium cells now assuming limited availability of noisy data. Here, flexibility of the approach to incorporate partial system information in the identification process is shown and its effect on accuracy and predictive ability of the estimated model are studied. The third example studies the phenomenological toy model of the regulation of circadian oscillations in Drosophila that follows rate laws different from S-system power-law. For the limited noisy data, using a priori information about properties of the system, we could estimate an alternate S-system model that showed robust oscillatory behavior with predictive abilities.

The Neurospora photoreceptor VIVID exerts negative and positive control on light sensing to achieve adaptation

Light adaptation in Neurospora is mediated by the photoreceptor VIVID, which exerts both a negative and positive effect on light sensing. These apparently paradoxical roles of VIVID are explained by the dynamics of a network motif that utilizes futile cycling.

The fungus Neurospora detects relative changes in light intensity by adapting to the ambient light level and remaining responsive to increases in light intensity.

Both the downregulation of the acute light response and maintained responsiveness are mediated by the photoreceptor VIVID (VVD).

Data-based mathematical modeling shows that this paradoxical function of VVD can be realized by a futile-cycle network motif that turns feedback inhibition into sensory adaptation.

The light response in Neurospora is mediated by the photoreceptor and circadian transcription factor White Collar Complex (WCC). The expression rate of the WCC target genes adapts in daylight and remains refractory to moonlight, despite the extraordinary light sensitivity of the WCC. To explain this photoadaptation, feedback inhibition by the WCC interaction partner VIVID (VVD) has been invoked. Here we show through data-driven mathematical modeling that VVD allows Neurospora to detect relative changes in light intensity. To achieve this behavior, VVD acts as an inhibitor of WCC-driven gene expression and, at the same time, as a positive regulator that maintains the responsiveness of the photosystem. Our data indicate that this paradoxical function is realized by a futile cycle that involves the light-induced sequestration of active WCC by VVD and the replenishment of the activatable WCC pool through the decay of the photoactivated state. Our quantitative study uncovers a novel network motif for achieving sensory adaptation and defines a core input module of the circadian clock in Neurospora.

Deterministic versus stochastic models for circadian rhythms

Circadian rhythms which occur with a period close to 24 h in nearly all living organisms originate from the negative autoregulation of gene expression.Deterministic models based on genetic regulatory processes account for theoccurrence of circadian rhythms in constant environmental conditions (e.g.constant darkness), for entrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. At low numbers of protein and mRNA molecules, it becomes necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect of molecular noise by considering two stochastic versions of a core model for circadian rhythms. The deterministic version of this core modelwas previously proposed for circadian oscillations of the PER protein in Drosophila and of the FRQ protein in Neurospora. In the first, non-developed version of the stochastic model, we introduce molecular noise without decomposing the deterministic mechanism into detailed reaction steps while in the second, developed version we carry out such a detailed decomposition. Numerical simulations of the two stochastic versions of the model are performed by means of the Gillespie method. We compare the predictions of the deterministic approach with those of the two stochastic models, with respect both to sustained oscillations of the limit cycle type and to the influence of the proximity of a bifurcation point beyond which the system evolves to a stable steady state. The results indicate that robust circadian oscillations can occur even when the numbers of mRNA and nuclear protein involved in the oscillatory mechanism are reduced to a few tens orhundreds, respectively. The non-developed and developed versions of the stochastic model yield largely similar results and provide good agreement with the predictions of the deterministic model for circadian rhythms.

Modeling cortisol dynamics in the neuro-endocrine axis distinguishes normal, depression, and post-traumatic stress disorder (PTSD) in humans

Cortisol, secreted in the adrenal cortex in response to stress, is an informative biomarker that distinguishes anxiety disorders such as major depression and post-traumatic stress disorder (PTSD) from normal subjects. Yehuda et al. proposed a hypothesis that, in humans, the hypersensitive hypothalamus-pituitary-adrenal (HPA) axis is responsible for the occurrence of differing levels of cortisol in anxiety disorders. Specifically, PTSD subjects have lower cortisol levels during the late subjective night in comparison to normal subjects, and this was assumed to occur due to strong negative feedback loops in the HPA axis. In the present work, to address this hypothesis, we modeled the cortisol dynamics using nonlinear ordinary differential equations and estimated the kinetic parameters of the model to fit the experimental data of three categories, namely, normal, depressed, and PTSD human subjects. We concatenated the subjects (n = 3) in each category and created a model subject (n = 1) without considering the patient-to-patient variability in each case. The parameters of the model for the three categories were simultaneously obtained through global optimization. Bifurcation analysis carried out with the optimized parameters exhibited two supercritical Hopf points and, for the choice of parameters, the oscillations were found to be circadian in nature. The fitted kinetic parameters indicate that PTSD subjects have a strong negative feedback loop and, as a result, the predicted oscillating cortisol levels are extremely low at the nadir in contrast to normal subjects, albeit within the endocrinologic range. We also simulated the phenotypes for each of the categories and, as observed in the clinical data of PTSD patients, the simulated cortisol levels are consistently low at the nadir, and correspondingly the negative feedback was found to be extremely strong. These results from the model support the hypothesis that high stress intensity and strong negative feedback loop may cause hypersensitive neuro-endocrine axis that results in hypocortisolemia in PTSD.

PTSD is an anxiety disorder that occurs among persons exposed to a traumatic event involving life threat and injury. This is a co-morbid psychiatric disorder that occurs along with depression. Cortisol is an informative endocrine biomarker that can distinguish PTSD from other co-morbid disorders. In comparison to normal subjects, hypocortisolemia was observed during the night in PTSD, while hypercortisolemia was observed in depressed subjects. From analyzing the clinical data, Yehuda et al. hypothesized that hypocortisolemia in PTSD was due to the strong negative feedback loop operating in the neuroendocrine axis under severe stress. We complemented this hypothesis by constructing a mathematical model for cortisol dynamics in HPA axis and estimated the kinetic parameters that fitted the cortisol time series obtained from the clinical data of normal, depressed and PTSD patients. The parameters obtained from the simulated phenotypes also strongly support the hypothesis that, due to disruptive negative feedback loops, cortisol levels are different in normal, PTSD and depressed subjects during the night. Importantly, the model predicted the transitions from normal to various diseased states, and these transitions were shown to occur due to changes in the strength of the negative feedback loop and the stress intensity in the neuro-endocrine axis.

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.

MiR-206-mediated dynamic mechanism of the mammalian circadian clock

BACKGROUND

As a group of highly conserved small non-coding RNAs with a length of 21~23 nucleotides, microRNAs (miRNAs) regulate the gene expression post-transcriptionally by base pairing with the partial or full complementary sequences in target mRNAs, thus resulting in the repression of mRNA translation and the acceleration of mRNA degradation. Recent work has revealed that miRNAs are essential for the development and functioning of the skeletal muscles where they are. In particular, miR-206 has not only been identified as the only miRNA expressed in skeletal muscles, but also exhibited crucial roles in regulation of the muscle development. Although miRNAs are known to regulate various biological processes ranging from development to cancer, much less is known about their role in the dynamic regulation of the mammalian circadian clock.

RESULTS

A detailed dynamic model of miR-206-mediated mammalian circadian clock system was developed presently by using Hill-type terms, Michaelis-Menten type and mass action kinetics. Based on a system-theoretic approach, the model accurately predicts both the periodicity and the entrainment of the circadian clock. It also explores the dynamics properties of the oscillations mediated by miR-206 by means of sensitivity analysis and alterations of parameters. Our results show that miR-206 is an important regulator of the circadian clock in skeletal muscle, and thus by study of miR-206 the main features of its mediation on the clock may be captured. Simulations of these processes display that the amplitude and frequency of the oscillation can be significantly altered through the miR-206-mediated control.

CONCLUSIONS

MiR-206 has a profound effect on the dynamic mechanism of the mammalian circadian clock, both by control of the amplitude and control or alteration of the frequency to affect the level of the gene expression and to interfere with the temporal sequence of the gene production or delivery. This undoubtedly uncovers a new mechanism for regulation of the circadian clock at a post-transcriptional level and provides important insights into the normal development as well as the pathological conditions of skeletal muscles, such as the aging, chronic disease and cancer.

Practical limits for reverse engineering of dynamical systems: a statistical analysis of sensitivity and parameter inferability in systems biology models

The size and complexity of cellular systems make building predictive models an extremely difficult task. In principle dynamical time-course data can be used to elucidate the structure of the underlying molecular mechanisms, but a central and recurring problem is that many and very different models can be fitted to experimental data, especially when the latter are limited and subject to noise. Even given a model, estimating its parameters remains challenging in real-world systems. Here we present a comprehensive analysis of 180 systems biology models, which allows us to classify the parameters with respect to their contribution to the overall dynamical behaviour of the different systems. Our results reveal candidate elements of control in biochemical pathways that differentially contribute to dynamics. We introduce sensitivity profiles that concisely characterize parameter sensitivity and demonstrate how this can be connected to variability in data. Systematically linking data and model sloppiness allows us to extract features of dynamical systems that determine how well parameters can be estimated from time-course measurements, and associates the extent of data required for parameter inference with the model structure, and also with the global dynamical state of the system. The comprehensive analysis of so many systems biology models reaffirms the inability to estimate precisely most model or kinetic parameters as a generic feature of dynamical systems, and provides safe guidelines for performing better inferences and model predictions in the context of reverse engineering of mathematical models for biological systems.

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.

Circadian clocks and phosphorylation: Insights from computational modeling

Circadian clocks are based on a molecular mechanism regulated at the transcriptional, translational and post-translational levels. Recent experimental data unravel a complex role of the phosphorylations in these clocks. In mammals, several kinases play differential roles in the regulation of circadian rhythmicity. A dysfunction in the phosphorylation of one clock protein could lead to sleep disorders such as the Familial Advanced Sleep Phase Disorder, FASPS. Moreover, several drugs are targeting kinases of the circadian clocks and can be used in cancer chronotherapy or to treat mood disorders. In Drosophila, recent experimental observations also revealed a complex role of the phosphorylations. Because of its high degree of homology with mammals, the Drosophila system is of particular interest. In the circadian clock of cyanobacteria, an atypical regulatory mechanism is based only on three clock proteins (KaiA, KaiB, KaiC) and ATP and is sufficient to produce robust temperature-compensated circadian oscillations of KaiC phosphorylation. This review will show how computational modeling has become a powerful and useful tool in investigating the regulatory mechanism of circadian clocks, but also how models can give rise to testable predictions or reveal unexpected results.

Stochastic simulation of delay-induced circadian rhythms in Drosophila

Circadian rhythms are ubiquitous in all eukaryotes and some prokaryotes. Several computational models with or without time delays have been developed for circadian rhythms. Exact stochastic simulations have been carried out for several models without time delays, but no exact stochastic simulation has been done for models with delays. In this paper, we proposed a detailed and a reduced stochastic model with delays for circadian rhythms in Drosophila based on two deterministic models of Smolen et al. and employed exact stochastic simulation to simulate circadian oscillations. Our simulations showed that both models can produce sustained oscillations and that the oscillation is robust to noise in the sense that there is very little variability in oscillation period although there are significant random fluctuations in oscillation peaks. Moreover, although average time delays are essential to simulation of oscillation, random changes in time delays within certain range around fixed average time delay cause little variability in the oscillation period. Our simulation results also showed that both models are robust to parameter variations and that oscillation can be entrained by light/dark circles. Our simulations further demonstrated that within a reasonable range around the experimental result, the rates that dclock and per promoters switch back and forth between activated and repressed sites have little impact on oscillation period.

Age, but not experience, affects courtship gene expression in male Drosophila melanogaster

Mutation screens in model organisms have helped identify the foundation of many fundamental organismal phenotypes. An emerging question in evolutionary and behavioral biology is the extent to which these “developmental” genes contribute to the subtle individual variation that characterizes natural populations. A related question is whether individual differences arise from static differences in gene expression that arose during previous life stages, or whether they are due to dynamic regulation of expression during the life stage under investigation. Here, we address these questions using genes that have been discovered to control the development of normal courtship behavior in male Drosophila melanogaster. We examined whether these genes have static or dynamic expression in the heads of adult male flies of different ages and with different levels of social experience. We found that 16 genes of the 25 genes examined were statically expressed, and 9 genes were dynamically expressed with changes related to adult age. No genes exhibited rapid dynamic expression changes due to social experience or age*experience interaction. We therefore conclude that a majority of fly “courtship” genes are statically expressed, while a minority are regulated in adults with respect to age, but not with respect to relevant social experience. These results are consistent with those from a recent microarray analysis that found none of the canonical courtship genes changed expression in male flies after brief exposure to females.

On the functional diversity of dynamical behaviour in genetic and metabolic feedback systems

Background

Feedback regulation plays crucial roles in the robust control and maintenance of many cellular systems. Negative feedbacks are found to underline both stable and unstable, often oscillatory, behaviours. We explore the dynamical characteristics of systems with single as well as coupled negative feedback loops using a combined approach of analytical and numerical techniques. Particularly, we emphasise how the loop's characterising factors (strength and cooperativity levels) affect system dynamics and how individual loops interact in the coupled-loop systems.

Results

We develop an analytical bifurcation analysis based on the stability and the Routh- Hurwitz theorem for a common negative feedback system and a variety of its variants. We demonstrate that different combinations of the feedback strengths of individual loops give rise to different dynamical behaviours. Moreover, incorporating more negative feedback loops always tend to enhance system stability. We show that two mechanisms, in addition to the lengthening of pathway, can lower the Hill coefficient to a biologically plausible level required for sustained oscillations. These include loops coupling and end-product utilisation. We find that the degradation rates solely affect the threshold Hill coefficient for sustained oscillation, while the synthesis rates have more significant roles in determining the threshold feedback strength. Unbalancing the degradation rates between the system species is found as a way to improve stability.

Conclusion

The analytical methods and insights presented in this study demonstrate that reallocation of the feedback loop may or may not make the system more stable; the specific effect is determined by the degradation rates of the newly inhibited molecular species. As the loop moves closer to the end of the pathway, the minimum Hill coefficient for oscillation is reduced. Furthermore, under general (unequal) values of the degradation rates, system extension becomes more stable only when the added species degrades slower than it is being produced; otherwise the system is more prone to oscillation. The coupling of loops significantly increases the richness of dynamical bifurcation characteristics. The likelihood of having oscillatory behaviour is directly determined by the loops' strength: stronger loops always result in smaller oscillatory regions.

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.

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.

Mathematical identification of critical reactions in the interlocked feedback model

Dynamic simulations are necessary for understanding the mechanism of how biochemical networks generate robust properties to environmental stresses or genetic changes. Sensitivity analysis allows the linking of robustness to network structure. However, it yields only local properties regarding a particular choice of plausible parameter values, because it is hard to know the exact parameter values in vivo. Global and firm results are needed that do not depend on particular parameter values. We propose mathematical analysis for robustness (MAR) that consists of the novel evolutionary search that explores all possible solution vectors of kinetic parameters satisfying the target dynamics and robustness analysis. New criteria, parameter spectrum width and the variability of solution vectors for parameters, are introduced to determine whether the search is exhaustive. In robustness analysis, in addition to single parameter sensitivity analysis, robustness to multiple parameter perturbation is defined. Combining the sensitivity analysis and the robustness analysis to multiple parameter perturbation enables identifying critical reactions. Use of MAR clearly identified the critical reactions responsible for determining the circadian cycle in the Drosophila interlocked circadian clock model. In highly robust models, while the parameter vectors are greatly varied, the critical reactions with a high sensitivity are uniquely determined. Interestingly, not only the per-tim loop but also the dclk-cyc loop strongly affect the period of PER, although the dclk-cyc loop hardly changes its amplitude and it is not potentially influential. In conclusion, MAR is a powerful method to explore wide parameter space without human-biases and to link a robust property to network architectures without knowing the exact parameter values. MAR identifies the reactions critically responsible for determining the period and amplitude in the interlocked feedback model and suggests that the circadian clock intensively evolves or designs the kinetic parameters so that it creates a highly robust cycle.

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.

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.

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.

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.

Modeling the mammalian circadian clock: sensitivity analysis and multiplicity of oscillatory mechanisms

We extend the study of a computational model recently proposed for the mammalian circadian clock (Proc. Natl Acad. Sci. USA 100 (2003) 7051). The model, based on the intertwined positive and negative regulatory loops involving the Per, Cry, Bmal1, and Clock genes, can give rise to sustained circadian oscillations in conditions of continuous darkness. These limit cycle oscillations correspond to circadian rhythms autonomously generated by suprachiasmatic nuclei and by some peripheral tissues. By using different sets of parameter values producing circadian oscillations, we compare the effect of the various parameters and show that both the occurrence and the period of the oscillations are generally most sensitive to parameters related to synthesis or degradation of Bmal1 mRNA and BMAL1 protein. The mechanism of circadian oscillations relies on the formation of an inactive complex between PER and CRY and the activators CLOCK and BMAL1 that enhance Per and Cry expression. Bifurcation diagrams and computer simulations nevertheless indicate the possible existence of a second source of oscillatory behavior. Thus, sustained oscillations might arise from the sole negative autoregulation of Bmal1 expression. This second oscillatory mechanism may not be functional in physiological conditions, and its period need not necessarily be circadian. When incorporating the light-induced expression of the Per gene, the model accounts for entrainment of the oscillations by light-dark (LD) cycles. Long-term suppression of circadian oscillations by a single light pulse can occur in the model when a stable steady state coexists with a stable limit cycle. The phase of the oscillations upon entrainment in LD critically depends on the parameters that govern the level of CRY protein. Small changes in the parameters governing CRY levels can shift the peak in Per mRNA from the L to the D phase, or can prevent entrainment. The results are discussed in relation to physiological disorders of the sleep-wake cycle linked to perturbations of the human circadian clock, such as the familial advanced sleep phase syndrome or the non-24h sleep-wake syndrome.

Genetics and molecular biology of rhythms in Drosophila and other insects

Application of generic variants (Sections II-IV, VI, and IX) and molecular manipulations of rhythm-related genes (Sections V-X) have been used extensively to investigate features of insect chronobiology that might not have been experimentally accessible otherwise. Most such tests of mutants and molecular-genetic xperiments have been performed in Drosophila melanogaster. Results from applying visual-system variants have revealed that environmental inputs to the circadian clock in adult flies are mediated by external photoreceptive structures (Section II) and also by direct light reception chat occurs in certain brain neurons (Section IX). The relevant light-absorbing molecuLes are rhodopsins and "blue-receptive" cryptochrome (Sections II and IX). Variations in temperature are another clock input (Section IV), as has been analyzed in part by use of molecular techniques and transgenes involving factors functioning near the heart of the circadian clock (Section VIII). At that location within the fly's chronobiological system, approximately a half-dozen-perhaps up to as many as 10-clock genes encode functions that act and interact to form the circadian pacemaker (Sections III and V). This entity functions in part by transcriptional control of certain clock genes' expressions, which result in the production of key proteins that feed back negatively to regulate their own mRNA production. This occurs in part by interactions of such proteins with others that function as transcriptional activators (Section V). The implied feedback loop operates such that there are daily variations in the abundances of products put out by about one-half of the core clock genes. Thus, the normal expression of these genes defines circadian rhythms of their own, paralleling the effects of mutations at the corresponding genetic loci (Section III), which are to disrupt or apparently eliminate clock functioning. The fluctuations in the abundance of gene products are controlled transciptionally and posttranscriptionally. These clock mechanisms are being analyzed in ways that are increasingly complex and occasionally obscure; not all panels of this picture are comprehensive or clear, including problems revolving round the biological meaning or a given features of all this molecular cycling (Section V). Among the complexities and puzzles that have recently arisen, phenomena that stand out are posttranslational modifications of certain proteins that are circadianly regulated and regulating; these biochemical events form an ancillary component of the clock mechanism, as revealed in part by genetic identification of Factors (Section III) that turned out to encode protein kinases whose substrates include other pacemaking polypeptides (Section V). Outputs from insect circadian clocks have been long defined on formalistic and in some cases concrete criteria, related to revealed rhythms such as periodic eclosion and daily fluctuations of locomotion (Sections II and III). Based on the reasoning that if clock genes can regulate circadian cyclings of their own products, they can do the same for genes that function along output pathways; thus clock-regulated genes have been identified in part by virtue of their products' oscillations (Section X). Those studied most intensively have their expression influenced by circadian-pacemaker mutations. The clock-regulated genes discovered on molecular criteria have in some instances been analyzed further in their mutant forms and found to affect certain features of overt whole-organismal rhythmicity (Sections IV and X). Insect chronogenetics touches in part on naturally occurring gene variations that affect biological rhythmicity or (in some cases) have otherwise informed investigators about certain features of the organism's rhythm system (Section VII). Such animals include at least a dozen insect species other than D. melanogaster in which rhythm variants have been encountered (although usually not looked for systematically). The chronobiological "system" in the fruit fly might better be graced with a plural appellation because there is a myriad of temporally related phenomena that have come under the sway of one kind of putative rhythm variant or the other (Section IV). These phenotypes, which range well beyond the bedrock eclosion and locomotor circadian rhythms, unfortunately lead to the creation of a laundry list of underanalyzed or occult phenomena that may or may not be inherently real, whether or not they might be meaningfully defective under the influence of a given chronogenetic variant. However, such mutants seem to lend themselves to the interrogation of a wide variety of time-based attributes-those that fall within the experimental confines of conventionally appreciated circadian rhythms (Sections II, III, VI, and X); and others that consist of 24-hr or nondaily cycles defined by many kinds of biological, physiological, or biochemical parameters (Section IV).

Actin-based motility: from molecules to movement

Extensive progress has been made recently in understanding the mechanism by which cells move and extend protrusions using site-directed polymerization of actin in response to signalling. Insights into the molecular mechanism of production of force and movement by actin polymerization have been provided by a crosstalk between several disciplines, including biochemistry, biomimetic approaches and computational studies. This review focuses on the biochemical properties of the proteins involved in actin-based motility and shows how these properties are used to generate models of force production, how the predictions of different theoretical models are tested using a biochemically controlled reconstituted motility assay and how the changes in motility resulting from changes to the concentrations of components of the assay can help understand diverse aspects of the motile behavior of living cells.

Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell

The physiological responses of cells to external and internal stimuli are governed by genes and proteins interacting in complex networks whose dynamical properties are impossible to understand by intuitive reasoning alone. Recent advances by theoretical biologists have demonstrated that molecular regulatory networks can be accurately modeled in mathematical terms. These models shed light on the design principles of biological control systems and make predictions that have been verified experimentally.

Stochastic models for circadian rhythms: effect of molecular noise on periodic and chaotic behaviour

Circadian rhythms are endogenous oscillations that occur with a period close to 24 h in nearly all living organisms. These rhythms originate from the negative autoregulation of gene expression. Deterministic models based on such genetic regulatory processes account for the occurrence of circadian rhythms in constant environmental conditions (e.g., constant darkness), for entrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. When the numbers of protein and mRNA molecules involved in the oscillations are small, as may occur in cellular conditions, it becomes necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect of molecular noise by considering the stochastic version of a deterministic model previously proposed for circadian oscillations of the PER and TIM proteins and their mRNAs in Drosophila. The model is based on repression of the per and tim genes by a complex between the PER and TIM proteins. Numerical simulations of the stochastic version of the model are performed by means of the Gillespie method. The predictions of the stochastic approach compare well with those of the deterministic model with respect both to sustained oscillations of the limit cycle type and to the influence of the proximity from a bifurcation point beyond which the system evolves to stable steady state. Stochastic simulations indicate that robust circadian oscillations can emerge at the cellular level even when the maximum numbers of mRNA and protein molecules involved in the oscillations are of the order of only a few tens or hundreds. The stochastic model also reproduces the evolution to a strange attractor in conditions where the deterministic PER-TIM model admits chaotic behaviour. The difference between periodic oscillations of the limit cycle type and aperiodic oscillations (i.e. chaos) persists in the presence of molecular noise, as shown by means of Poincaré sections. The progressive obliteration of periodicity observed as the number of molecules decreases can thus be distinguished from the aperiodicity originating from chaotic dynamics. As long as the numbers of molecules involved in the oscillations remain sufficiently large (of the order of a few tens or hundreds, or more), stochastic models therefore provide good agreement with the predictions of the deterministic model for circadian rhythms.

Saturation of enzyme kinetics in circadian clock models

From the mathematical study of simple models for circadian rhythm, the authors identified a clear effect of saturation in the enzyme kinetics on the promotion or suppression of a sustained oscillation. In the models, a clock gene (per gene) is transcribed to produce mRNAs, which are translated to produce proteins in the cytosol which are then transported to the nucleus and suppress the transcription of the gene. The negative feedback loop with a long time delay creates sustained oscillation. All the enzymatic reactions (e.g., degradation, translation, and modification) are assumed to be of Michaelis-Menten type. The reaction rate increases with the amount of substrate but saturates when it is very large. The authors prove mathematically that the saturation in any of the reactions included in the feedback loop (in-loop reaction steps) suppresses the oscillation, whereas the saturation of both degradation steps and the back transport of the protein to cytosol (branch reaction steps) makes the oscillation more likely to occur. In the experimental measurements of enzyme kinetics and in published circadian clock simulators, in-loop reaction steps have a small saturation index whereas branch reaction steps have a large saturation index.

Engineered gene circuits

A central focus of postgenomic research will be to understand how cellular phenomena arise from the connectivity of genes and proteins. This connectivity generates molecular network diagrams that resemble complex electrical circuits, and a systematic understanding will require the development of a mathematical framework for describing the circuitry. From an engineering perspective, the natural path towards such a framework is the construction and analysis of the underlying submodules that constitute the network. Recent experimental advances in both sequencing and genetic engineering have made this approach feasible through the design and implementation of synthetic gene networks amenable to mathematical modelling and quantitative analysis. These developments have signalled the emergence of a gene circuit discipline, which provides a framework for predicting and evaluating the dynamics of cellular processes. Synthetic gene networks will also lead to new logical forms of cellular control, which could have important applications in functional genomics, nanotechnology, and gene and cell therapy.

Comparative study of circadian clock models, in search of processes promoting oscillation

We study simple models for circadian rhythm, and examine the condition in which the equilibrium is unstable, generating a sustained oscillation. In the models, a clock gene(s) is transcribed to produce mRNAs, which are translated to produce proteins that suppress the transcription of the clock gene(s). First, using a Lyapunov function, we prove under very general conditions that two-variable models cannot generate a stable oscillation, implying that additional structures are needed for the model to generate a sustainable rhythm. By comparing several models of different complexities using the Routh-Hurwitz criteria of stability, we show that a sustained oscillation is more likely to occur if the cell is compartmentalized and the proteins need to be transported from the cytosol to the nucleus, if the proteins have to be modified before entering the nucleus, if the kinetics of transcription inhibition or the transport to the nucleus have cooperativity with a nonlinear dependence on the substrate concentration, or if the products of two clock genes form a heterodimer that suppresses both of their own genes. We discuss the implications of these results.

A model for the enhancement of fitness in cyanobacteria based on resonance of a circadian oscillator with the external light-dark cycle

Fitness enhancement based on resonating circadian clocks has recently been demonstrated in cyanobacteria [Ouyang et al. (1998). Proc. Natl Acad. Sci. U.S.A.95, 8660-8664]. Thus, the competition between two cyanobacterial strains differing by the free-running period (FRP) of their circadian oscillations leads to the dominance of one or the other of the two strains, depending on the period of the external light-dark (LD) cycle. The successful strain is generally that which has an FRP closest to the period of the LD cycle. Of key importance for the resonance phenomenon are observations which indicate that the phase angle between the circadian oscillator and the LD cycle depends both on the latter cycle's length and on the FRP. We account for these experimental observations by means of a theoretical model which takes into account (i) cell growth, (ii) secretion of a putative cell growth inhibitor, and (iii) the existence of a cellular, light-sensitive circadian oscillator controlling growth as well as inhibitor secretion. Building on a previous analysis in which the phase angle was considered as a freely adjustable parameter [Roussel et al. (2000). J. theor. Biol.205, 321-340], we incorporate into the model a light-sensitive version of the van der Pol oscillator to represent explicitly the cellular circadian oscillator. In this way, the model automatically generates a phase angle between the circadian oscillator and the LD cycle which depends on the characteristic FRP of the strain and varies continuously with the period of the LD cycle. The model provides an explanation for the results of competition experiments between strains of different FRPs subjected to entrainment by LD cycles of different periods. The model further shows how the dominance of one strain over another in LD cycles can be reconciled with the observation that two strains characterized by different FRPs nevertheless display the same growth kinetics in continuous light or in LD cycles when present alone in the medium. Theoretical predictions are made as to how the outcome of competition depends on the initial proportions and on the FRPs of the different strains. We also determine the effect of the photoperiod and extend the analysis to the case of a competition between three cyanobacterial strains.

Robustness of circadian rhythms with respect to molecular noise

We use a core molecular model capable of generating circadian rhythms to assess the robustness of circadian oscillations with respect to molecular noise. The model is based on the negative feedback exerted by a regulatory protein on the expression of its gene. Such a negative regulatory mechanism underlies circadian oscillations of the PER protein in Drosophila and of the FRQ protein in Neurospora. The model incorporates gene transcription into mRNA, translation of mRNA into protein, reversible phosphorylation leading to degradation of the regulatory protein, transport of the latter into the nucleus, and repression of gene expression by the nuclear form of the protein. To assess the effect of molecular noise, we perform stochastic simulations after decomposing the deterministic model into elementary reaction steps. The oscillations predicted by the stochastic simulations agree with those obtained with the deterministic version of the model. We show that robust circadian oscillations can occur already with a limited number of mRNA and protein molecules, in the range of tens and hundreds, respectively. Entrainment by light/dark cycles and cooperativity in repression enhance the robustness of circadian oscillations with respect to molecular noise.

Time-patterned drug administration: insights from a modeling approach

The physiological effects of a drug depend not only on its molecular structure but also on the time-pattern of its administration. One of the main reasons for the importance of temporal patterns in drug action is biological rhythms--particularly those of circadian period. These rhythms affect most physiological functions as well as drug metabolism, clearance, and dynamic processes that may alter drug availability and target cell responsiveness with reference to biological time. We present an overview of the importance of time-patterned signals in physiology focused on the insights provided by a modeling approach. We first discuss examples of pulsatile intercellular communication by hormones such as gonadotropin-releasing hormone, and by cyclic adenosine monophosphate (cAMP) signals in Dictyostelium amoebae. Models based on reversible receptor desensitization account in both cases for the existence of optimal patterns of pulsatile signaling. Turning to circadian rhythms, we examine how models can be used to account for the response of 24h patterns to external stimuli such as light pulses or gene expression, and to predict how to restore the physiological characteristics of altered rhythms. Time-patterned treatments of cancer involve two distinct lines of research. The first, currently evaluated in clinical trials, relies on circadian chronomodulation of anticancer drugs, while the second, mostly based on theoretical studies, involves a resonance phenomenon with the cell-cycle length. We discuss the implications of modeling studies to improve the temporal patterning of drug administration.

Modeling circadian oscillations with interlocking positive and negative feedback loops

Both positive and negative feedback loops of transcriptional regulation have been proposed to be important for the generation of circadian rhythms. To test the sufficiency of the proposed mechanisms, two differential equation-based models were constructed to describe the Neurospora crassa and Drosophila melanogaster circadian oscillators. In the model of the Neurospora oscillator, FRQ suppresses frq transcription by binding to a complex of the transcriptional activators WC-1 and WC-2, thus yielding negative feedback. FRQ also activates synthesis of WC-1, which in turn activates frq transcription, yielding positive feedback. In the model of the Drosophila oscillator, PER and TIM are represented by a "lumped" variable, "PER." PER suppresses its own transcription by binding to the transcriptional regulator dCLOCK, thus yielding negative feedback. PER also binds to dCLOCK to de-repress dclock, and dCLOCK in turn activates per transcription, yielding positive feedback. Both models displayed circadian oscillations that were robust to parameter variations and to noise and that entrained to simulated light/dark cycles. Circadian oscillations were only obtained if time delays were included to represent processes not modeled in detail (e.g., transcription and translation). In both models, oscillations were preserved when positive feedback was removed.

Whole-cell simulation: a grand challenge of the 21st century

Study of the cell will never be complete unless its dynamic behavior is understood. The complex behavior of the cell cannot be determined or predicted unless a computer model of the cell is constructed and computer simulation is undertaken. Rapid accumulation of biological data from genome, proteome, transcriptome and metabolome projects can bring us to the point where it is no longer purely speculative to discuss how to construct virtual cells in silico. This article describes attempts to construct whole cell models. The E-CELL project has completed a couple of virtual cell models, and computer simulations have revealed some biological surprises.

Light regulation of circadian clock-controlled gene expression in rice

Using transgenic rice seedlings expressing a firefly luciferase (luc) gene under the control of a rice CAB (chlorophyll a/b binding protein) promoter, we demonstrated how light affects circadian clocks in the transcription of the CAB gene. Rhythmic luc expression was observed for more than 5 d under constant light and dark (DD) conditions after light/dark entrainment. After a light pulse was applied at different time points in DD various temporal patterns of CAB gene expression were individually observed. We first examined two distinct properties related to the entrainment mechanism of the circadian clock: fluence-rate dependence of free-running periods (FRPs) and phase resetting by a light pulse. Although fluence-rate dependent shortening of FRP was demonstrated, the FRP in DD was almost equal to that in constant light of a middle fluence-rate, indicating that this fluence-rate dependence may not fully describe the entrainment of the circadian clock in rice. Typical phase responses of the circadian clock by a single light pulse were also observed at the transcriptional level in rice seedlings. Thus, the phase resettings upon the light/dark transitions of daily cycles may be sufficient to explain the entrainment mechanisms of rice. We have further demonstrated that, in addition to having a gating effect to acute response, a light pulse can activate the circadian clock-controlled CAB1R gene expression at the first circadian peak in a phase-dependent manner. This suggests that light activates circadian clock activity in the diurnal CAB gene expression under daily light/dark cycles.

A molecular explanation for the long-term suppression of circadian rhythms by a single light pulse

With the use of a molecular model for circadian rhythms in Drosophila based on transcriptional regulation, we show how a single, critical pulse of light can permanently suppress circadian rhythmicity, whereas a second light pulse can restore the abolished rhythm. The phenomena occur via the pulsatile induction of either protein degradation or gene expression in conditions in which a stable steady state coexists with stable circadian oscillations of the limit cycle type. The model indicates that suppression by a light pulse can only be accounted for by assuming that the biochemical effects of such a pulse much outlast its actual duration. We determine the characteristics of critical pulses suppressing the oscillations as a function of the phase at which the rhythm is perturbed. The model predicts how the amplitude and duration of the biochemical changes induced by critical pulses vary with this phase. The results provide a molecular, dynamic explanation for the long-term suppression of circadian rhythms observed in a variety of organisms in response to a single light pulse and for the subsequent restoration of the rhythms by a second light pulse.