Stochastic simulation of the mammalian circadian clock

Weakly circadian cells improve resynchrony

The mammalian suprachiasmatic nuclei (SCN) contain thousands of neurons capable of generating near 24-h rhythms. When isolated from their network, SCN neurons exhibit a range of oscillatory phenotypes: sustained or damping oscillations, or arrhythmic patterns. The implications of this variability are unknown. Experimentally, we found that cells within SCN explants recover from pharmacologically-induced desynchrony by re-establishing rhythmicity and synchrony in waves, independent of their intrinsic circadian period We therefore hypothesized that a cell's location within the network may also critically determine its resynchronization. To test this, we employed a deterministic, mechanistic model of circadian oscillators where we could independently control cell-intrinsic and network-connectivity parameters. We found that small changes in key parameters produced the full range of oscillatory phenotypes seen in biological cells, including similar distributions of period, amplitude and ability to cycle. The model also predicted that weaker oscillators could adjust their phase more readily than stronger oscillators. Using these model cells we explored potential biological consequences of their number and placement within the network. We found that the population synchronized to a higher degree when weak oscillators were at highly connected nodes within the network. A mathematically independent phase-amplitude model reproduced these findings. Thus, small differences in cell-intrinsic parameters contribute to large changes in the oscillatory ability of a cell, but the location of weak oscillators within the network also critically shapes the degree of synchronization for the population.

Circadian rhythms are daily, near 24-h oscillations in biological processes that nearly all organisms on Earth experience. Single cells contain a molecular clock that drives circadian rhythms in physiology and, when many cells synchronize in a population, daily behaviors. We hypothesized that small differences in intrinsic cellular properties allow for a diversity of circadian periods and amplitudes across cells. We observed circadian cells and their synchrony before, during, and after limiting communication between cells and then compared their intrinsic properties to their resynchronization behavior. We found that arrhythmic, weakly oscillating, and self-sustained circadian cells rejoined the rhythmic population independent of their cell-intrinsic oscillations. Using a mechanistic computational model of circadian cells, we found that resynchronization could be enhanced by including more weak oscillators or by placing weak oscillators at more connected nodes in the network. We conclude that intrinsic properties (e.g. oscillator weakness and responsiveness) and network structure (e.g. positions of weak oscillators) can independently buffer tissue rhythms from perturbations. This reveals how cellular and network properties impose rules on systems of circadian cells that must achieve synchrony from a desynchronized state, for example during perinatal development or when forced to overcome societal constraints on sleep-wake behavior, such as working early or late shifts.

Systems biology of cellular rhythms

Rhythms abound in biological systems, particularly at the cellular level where they originate from the feedback loops present in regulatory networks. Cellular rhythms can be investigated both by experimental and modeling approaches, and thus represent a prototypic field of research for systems biology. They have also become a major topic in synthetic biology. We review advances in the study of cellular rhythms of biochemical rather than electrical origin by considering a variety of oscillatory processes such as Ca++ oscillations, circadian rhythms, the segmentation clock, oscillations in p53 and NF-κB, synthetic oscillators, and the oscillatory dynamics of cyclin-dependent kinases driving the cell cycle. Finally we discuss the coupling between cellular rhythms and their robustness with respect to molecular noise.

(Re)inventing the circadian feedback loop

For 20 years, researchers have thought that circadian clocks are defined by feedback loops of transcription and translation. The rediscovery of posttranslational circadian oscillators in diverse organisms forces us to rethink this paradigm. Meanwhile, the original "basic" feedback loops of canonical circadian clocks have swelled to include dozens of additional proteins acting in interlocked loops. We review several self-sustained clock mechanisms and propose that minimum requirements for diurnal timekeeping might be simpler than those of actual free-running circadian oscillators. Thus, complex mechanisms of circadian timekeeping might have evolved from random connections between unrelated feedback loops with independent but limited time-telling capability.

Entrainment of peripheral clock genes by cortisol

Circadian rhythmicity in mammals is primarily driven by the suprachiasmatic nucleus (SCN), often called the central pacemaker, which converts the photic information of light and dark cycles into neuronal and hormonal signals in the periphery of the body. Cells of peripheral tissues respond to these centrally mediated cues by adjusting their molecular function to optimize organism performance. Numerous systemic cues orchestrate peripheral rhythmicity, such as feeding, body temperature, the autonomic nervous system, and hormones. We propose a semimechanistic model for the entrainment of peripheral clock genes by cortisol as a representative entrainer of peripheral cells. This model demonstrates the importance of entrainer's characteristics in terms of the synchronization and entrainment of peripheral clock genes, and predicts the loss of intercellular synchrony when cortisol moves out of its homeostatic amplitude and frequency range, as has been observed clinically in chronic stress and cancer. The model also predicts a dynamic regime of entrainment, when cortisol has a slightly decreased amplitude rhythm, where individual clock genes remain relatively synchronized among themselves but are phase shifted in relation to the entrainer. The model illustrates how the loss of communication between the SCN and peripheral tissues could result in desynchronization of peripheral clocks.

MicroRNA-mediated regulation in the mammalian circadian rhythm

Mammalian circadian rhythms have been extensively studied for many years and many computational models have been presented. Most of the circadian rhythms are based on interlocked positive and negative feedback loops involving coding regions of some 'clock' genes. Recent works have implicated that microRNAs (miRNAs) may play crucial roles in modulating the circadian clock. Here we develop a computational model involving four genes, Per, Cry, Bmal1, and Clock, and two miRNAs, miRNA-219 and miRNA-132, to show their post-transcriptional roles in the modulation of the circadian rhythm. The model is based on experimental observations, by which the miRNAs are incorporated into a classic model including only coding genes. In agreement with experimental observations, the model predicts that miRNA-mediated regulation plays critical roles in modulating the circadian clock. In addition, parameter sensitivity analysis indicates that the period of circadian rhythm with miRNA-mediated regulation is more insensitive to perturbations, showing that the miRNA-mediated regulation can enhance the robustness of the circadian rhythms. This study may help us understand the microRNA-mediated regulation in the mammalian circadian rhythm more clearly.

Stochastic properties of the plant circadian clock

Circadian clocks are gene regulatory networks whose role is to help the organisms to cope with variations in environmental conditions such as the day/night cycle. In this work, we explored the effects of molecular noise in single cells on the behaviour of the circadian clock in the plant model species Arabidopsis thaliana. The computational modelling language Bio-PEPA enabled us to give a stochastic interpretation of an existing deterministic model of the clock, and to easily compare the results obtained via stochastic simulation and via numerical solution of the deterministic model. First, the introduction of stochasticity in the model allowed us to estimate the unknown size of the system. Moreover, stochasticity improved the description of the available experimental data in several light conditions: noise-induced fluctuations yield a faster entrainment of the plant clock under certain photoperiods and are able to explain the experimentally observed dampening of the oscillations in plants under constant light conditions. The model predicts that the desynchronization between noisy oscillations in single cells contributes to the observed damped oscillations at the level of the cell population. Analysis of the phase, period and amplitude distributions under various light conditions demonstrated robust entrainment of the plant clock to light/dark cycles which closely matched the available experimental data.

Persistent cell-autonomous circadian oscillations in fibroblasts revealed by six-week single-cell imaging of PER2::LUC bioluminescence

Biological oscillators naturally exhibit stochastic fluctuations in period and amplitude due to the random nature of molecular reactions. Accurately measuring the precision of noisy oscillators and the heterogeneity in period and strength of rhythmicity across a population of cells requires single-cell recordings of sufficient length to fully represent the variability of oscillations. We found persistent, independent circadian oscillations of clock gene expression in 6-week-long bioluminescence recordings of 80 primary fibroblast cells dissociated from PER2::LUC mice and kept in vitro for 6 months. Due to the stochastic nature of rhythmicity, the proportion of cells appearing rhythmic increases with the length of interval examined, with 100% of cells found to be rhythmic when using 3-week windows. Mean period and amplitude are remarkably stable throughout the 6-week recordings, with precision improving over time. For individual cells, precision of period and amplitude are correlated with cell size and rhythm amplitude, but not with period, and period exhibits much less cycle-to-cycle variability (CV 7.3%) than does amplitude (CV 37%). The time series are long enough to distinguish stochastic fluctuations within each cell from differences among cells, and we conclude that the cells do exhibit significant heterogeneity in period and strength of rhythmicity, which we measure using a novel statistical metric. Furthermore, stochastic modeling suggests that these single-cell clocks operate near a Hopf bifurcation, such that intrinsic noise enhances the oscillations by minimizing period variability and sustaining amplitude.

A NEW REGULATORY INTERACTION SUGGESTED BY SIMULATIONS FOR CIRCADIAN GENETIC CONTROL MECHANISM IN MAMMALS

Knowledge of molecular biological systems is increasing at an amazing pace. It is becoming harder to intuitively evaluate the significance of each interaction between the molecules of the complex biological systems. Hence, we need to develop an efficient computational method to explore the biological mechanisms. In this study, we employed a hybrid functional Petri net in order to analyze mammalian circadian genetic control mechanisms, which consists of feedback loops of clock genes and generates endogenous near 24 h rhythms in mammals. We constructed a computational model based on the available biological data, and by using Genomic Object Net, we performed computer simulations of the time courses of clock gene transcription and translation. Although the original model successfully reproduced most of the circadian genetic control mechanisms, two discrepancies remained despite a wide selection of the parameters. We found that addition of a hypothetical path into the original model result in successful simulation of time courses and phase relationships among clock genes. This also demonstrates the usefulness of the hybrid functional Petri net approach to biological systems.

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.

Molecular level dynamics of genetic oscillator--the effect of protein-protein interaction

Uncovering how interactions of a set of molecular components influence the system's dynamic behavior is important for understanding intracellular processes and elucidating design principles, but unfortunately, there are limited efforts for studying this issue. Here, we study the effect of distinct post-translational dynamics controlled by protein dimerization on oscillations in the repressilator. For this, we propose three biologically motivated model scenarios of the repressilator with monomer or dimer being the active form of repressor, and with protein-protein interactions. It is found that the dimer dissociation constant can tune oscillatory regions, frequency and amplitude. Introducing a modified linear noise approximation to evaluate fluctuations of amplitude and period in the oscillatory systems, we show that different dimerization leads to a different effect on period and amplitude in reducing noise. The manipulation of the circuit's biochemical properties provides a practical strategy for designing a robust and tunable oscillator.

Wavelet measurement suggests cause of period instability in mammalian circadian neurons

Cells in the suprachiasmatic nucleus (SCN) display remarkable precision, while either physically or chemically decoupling these cells from each other leads to a dramatic increase in period-to-period variability. Where previous studies have classified cells as either arrhythmic or circadian, our wavelet analysis reveals that individual cells, when removed from network interactions, intermittently express circadian and/or longer infradian periods. We reproduce the characteristic period distribution of uncoupled SCN cells with a stochastic model of the uncoupled SCN cell near a bifurcation in Bmal1 transcription repression. This suggests that the uncoupled cells may be switching between 2 oscillatory mechanisms: the indirect negative feedback of protein complex PER-CRY on the expression of Per and Cry genes, and the negative feedback of CLOCK-BMAL1 on the expression of the Bmal1 gene. The model is particularly sensitive near this bifurcation point, with only a small change in Bmal1 transcription repression needed to switch from the stable precision of coupled SCN cells to the unstable oscillations of decoupled individual cells, making this rate constant, an ideal target for cell signaling in the SCN.

Signal processing in cellular clocks

Many biochemical events within a cell need to be timed properly to occur at specific times of day, after other events have happened within the cell or in response to environmental signals. The cellular biochemical feedback loops that time these events have already received much recent attention in the experimental and modeling communities. Here, we show how ideas from signal processing can be applied to understand the function of these clocks. Consider two signals from the network s(t) and r(t), either two variables of a model or two experimentally measured time courses. We show how s(t) can be decomposed into two parts, the first being a function of r(t), and the second the derivative of a function of r(t). Geometric principles are then derived that can be used to understand when oscillations appear in biochemical feedback loops, the period of these oscillations, and their time course. Specific examples of this theory are provided that show how certain networks are prone or not prone to oscillate, how individual biochemical processes affect the period, and how oscillations in one chemical species can be deduced from oscillations in other parts of the network.

Computational modeling of mammalian signaling networks

One of the most exciting developments in signal transduction research has been the proliferation of studies in which a biological discovery was initiated by computational modeling. In this study, we review the major efforts that enable such studies. First, we describe the experimental technologies that are generally used to identify the molecular components and interactions in, and dynamic behavior exhibited by, a network of interest. Next, we review the mathematical approaches that are used to model signaling network behavior. Finally, we focus on three specific instances of 'model-driven discovery': cases in which computational modeling of a signaling network has led to new insights that have been verified experimentally.

Synchronization and entrainment of coupled circadian oscillators

Circadian rhythms in mammals are controlled by the neurons located in the suprachiasmatic nucleus of the hypothalamus. In physiological conditions, the system of neurons is very efficiently entrained by the 24 h light-dark cycle. Most of the studies carried out so far emphasize the crucial role of the periodicity imposed by the light-dark cycle in neuronal synchronization. Nevertheless, heterogeneity as a natural and permanent ingredient of these cellular interactions seemingly plays a major role in these biochemical processes. In this paper, we use a model that considers the neurons of the suprachiasmatic nucleus as chemically coupled modified Goodwin oscillators, and introduce non-negligible heterogeneity in the periods of all neurons in the form of quenched noise. The system response to the light-dark cycle periodicity is studied as a function of the interneuronal coupling strength, external forcing amplitude and neuronal heterogeneity. Our results indicate that the right amount of heterogeneity helps the extended system to respond globally in a more coherent way to the external forcing. Our proposed mechanism for neuronal synchronization under external periodic forcing is based on heterogeneity-induced oscillator death, damped oscillators being more entrainable by the external forcing than the self-oscillating neurons with different periods.

Multiscale complexity in the mammalian circadian clock

The field of systems biology studies how the interactions among individual components (e.g. genes and proteins) yield interesting and complex behavior. The circadian (daily) timekeeping system in mammals is an ideal system to study complexity because of its many biological scales (from genes to animal behavior). A wealth of data at each of these scales has recently been discovered. Within each scale, modeling can advance our understanding of challenging problems that arise in studying mammalian timekeeping. However, future work must focus on bridging the multiple spatial and temporal scales in the modeling of SCN network. Here we review recent advances, and then delve into a few areas that are promising research directions. We also discuss the flavor of modeling needed (simple or detailed) as well as new techniques that are needed to meet the challenges in modeling data across scales.

Distribution-based sensitivity metric for highly variable biochemical systems

Classical sensitivity analysis is routinely used to identify points of fragility or robustness in biochemical networks. However, intracellular systems often contain components that number in the thousands to tens or less and consequently motivate a stochastic treatment. Although methodologies exist to quantify sensitivities in stochastic models, they differ substantially from those used in deterministic regimes. Therefore it is not possible to tell whether observed differences in sensitivity measured in deterministic and stochastic elaborations of the same network are the result of methodology or model form. The authors introduce here a distribution-based methodology to measure sensitivity that is equally applicable in both regimes, and demonstrate its use and applicability on a sophisticated mathematical model of the mouse circadian clock that is available in both deterministic and stochastic variants. The authors use the method to produce sensitivity measurements on both variants. They note that the rank-order sensitivity of the clock to parametric perturbations is extremely well conserved across several orders of magnitude. The data show that the clock is fragile to perturbations in parameters common to the cellular machinery ('global' parameters) and robust to perturbations in parameters that are clock-specific ('local' parameters). The sensitivity measure can be used to reduce the model from its original 73 ordinary differential equations (ODEs) to 18 ODEs and to predict the degree to which parametric perturbation can distort the phase response curve of the clock. Finally, the method is employed to evaluate the effect of transcriptional and translational noise on clock function. [Includes supplementary material].

How molecular should your molecular model be? On the level of molecular detail required to simulate biological networks in systems and synthetic biology

The recent advance of genetic studies and the rapid accumulation of molecular data, together with the increasing performance of computers, led researchers to design more and more detailed mathematical models of biological systems. Many modeling approaches rely on ordinary differential equations (ODE) which are based on standard enzyme kinetics. Michaelis-Menten and Hill functions are indeed commonly used in dynamical models in systems and synthetic biology because they provide the necessary nonlinearity to make the dynamics nontrivial (i.e., limit-cycle oscillations or multistability). For most of the systems modeled, the actual molecular mechanism is unknown, and the enzyme equations should be regarded as phenomenological. In this chapter, we discuss the validity and accuracy of these approximations. In particular, we focus on the validity of the Michaelis-Menten function for open systems and on the use of Hill kinetics to describe transcription rates of regulated genes. Our discussion is illustrated by numerical simulations of prototype systems, including the Repressilator (a genetic oscillator) and the Toggle Switch model (a bistable system). We systematically compare the results obtained with the compact version (based on Michaelis-Menten and Hill functions) with its corresponding developed versions (based on "elementary" reaction steps and mass action laws). We also discuss the use of compact approaches to perform stochastic simulations (Gillespie algorithm). On the basis of these results, we argue that using compact models is suitable to model qualitatively biological systems.

Emergence of noise-induced oscillations in the central circadian pacemaker

Bmal1 is an essential transcriptional activator within the mammalian circadian clock. We report here that the suprachiasmatic nucleus (SCN) of Bmal1-null mutant mice, unexpectedly, generates stochastic oscillations with periods that overlap the circadian range. Dissociated SCN neurons expressed fluctuating levels of PER2 detected by bioluminescence imaging but could not generate circadian oscillations intrinsically. Inhibition of intercellular communication or cyclic-AMP signaling in SCN slices, which provide a positive feed-forward signal to drive the intracellular negative feedback loop, abolished the stochastic oscillations. Propagation of this feed-forward signal between SCN neurons then promotes quasi-circadian oscillations that arise as an emergent property of the SCN network. Experimental analysis and mathematical modeling argue that both intercellular coupling and molecular noise are required for the stochastic rhythms, providing a novel biological example of noise-induced oscillations. The emergence of stochastic circadian oscillations from the SCN network in the absence of cell-autonomous circadian oscillatory function highlights a previously unrecognized level of circadian organization.

Intrinsic noise and division cycle effects on an abstract biological oscillator

Oscillatory dynamics are common in biological pathways, emerging from the coupling of positive and negative feedback loops. Due to the small numbers of molecules typically contained in cellular volumes, stochastic effects may play an important role in system behavior. Thus, for moderate noise strengths, stochasticity has been shown to enhance signal-to-noise ratios or even induce oscillations in a class of phenomena referred to as "stochastic resonance" and "coherence resonance," respectively. Furthermore, the biological oscillators are subject to influences from the division cycle of the cell. In this paper we consider a biologically relevant oscillator and investigate the effect of intrinsic noise as well as division cycle which encompasses the processes of growth, DNA duplication, and cell division. We first construct a minimal reaction network which can oscillate in the presence of large or negligible timescale separation. We then derive corresponding deterministic and stochastic models and compare their dynamical behaviors with respect to (i) the extent of the parameter space where each model can exhibit oscillatory behavior and (ii) the oscillation characteristics, namely, the amplitude and the period. We further incorporate division cycle effects on both models and investigate the effect of growth rate on system behavior. Our results show that in the presence but not in the absence of large timescale separation, coherence resonance effects result in extending the oscillatory region and lowering the period for the stochastic model. When the division cycle is taken into account, the oscillatory region of the deterministic model is shown to extend or shrink for moderate or high growth rates, respectively. Further, under the influence of the division cycle, the stochastic model can oscillate for parameter sets for which the deterministic model does not. The division cycle is also found to be able to resonate with the oscillator, thereby enhancing oscillation robustness. The results of this study can give valuable insight into the complex interplay between oscillatory intracellular dynamics and various noise sources, stemming from gene expression, cell growth, and division.

Importance of cell variability for calcium signaling in rat airway myocytes

Calcium signaling controls several essential physiological functions in different cell types. Hence, it is not surprising that different aspects of Ca(2+) dynamics are in the focus of in-depth and extensive investigations. Efforts concentrate on the development of proper theoretical models that would provide a unified description of Ca(2+) signaling. Remarkably, experimentally recorded Ca(2+) signals exhibit a rather large diversity, which can be observed irrespective of the cell type, measuring techniques, or the nature of the signal. Our goal in the present study therefore is to present a theoretical explanation for the variability observed in experiments, whereby we focus on caffeine-induced Ca(2+) responses in isolated airway myocytes. By employing a stochastic model, we first test whether the observed variability can be attributed to intrinsic fluctuations that are a common feature of biochemical reactions that govern Ca(2+) signalization. We find that stochastic effects, within ranges that correspond to actual conditions in the cell, are far too modest to explain the large diversity observed in experimental data. Foremost, we reveal that only cell variability in theoretical modeling can appropriately describe the observed diversity in single-cell responses.

Light-dark cycle memory in the mammalian suprachiasmatic nucleus

The mammalian circadian oscillator, or suprachiasmatic nucleus (SCN), contains several thousand clock neurons in its ventrolateral division, many of which are spontaneous oscillators with period lengths that range from 22 to 28 h. In complete darkness, this network synchronizes through the exchange of action potentials that release vasoactive intestinal polypeptide, striking a compromise, free-running period close to 24 h long. We entrained Siberian hamsters to various light-dark cycles and then tracked their activity into constant darkness to show that they retain a memory of the previous light-dark cycle before returning to their own free-running period. Employing Leloup-Goldbeter mammalian clock neurons we model the ventrolateral SCN network and show that light acting weakly upon a strongly rhythmic vasoactive intestinal polypeptide oscillation can explain the observed light-dark cycle memory. In addition, light is known to initiate a mitogen-activated protein kinase signaling cascade that induces transcription of both per and mkp1 phosphatase. We show that the ensuing phosphatase-kinase interaction can account for the dead zone in the mammalian phase response curve and hypothesize that the SCN behaves like a lock-in amplifier to entrain to the light edges of the circadian day.

Weather and seasons together demand complex biological clocks

The 24-hour rhythms of the circadian clock [1] allow an organism to anticipate daily environmental cycles, giving it a competitive advantage [2, 3]. Although clock components show little protein sequence homology across phyla, multiple feedback loops and light inputs are universal features of clock networks [4, 5]. Why have circadian systems evolved such a complex structure? All biological clocks entrain a set of regulatory genes to the environmental cycle, in order to correctly time the expression of many downstream processes. Thus the question becomes: What aspects of the environment, and of the desired downstream regulation, are demanding the observed complexity? To answer this, we have evolved gene regulatory networks in silico, selecting for networks that correctly predict particular phases of the day under light/dark cycles. Gradually increasing the realism of the environmental cycles, we have tested the networks for the minimal characteristics of clocks observed in nature: oscillation under constant conditions, entrainment to light signals, and the presence of multiple feedback loops and light inputs. Realistic circadian gene networks are found to require a nontrivial combination of conditions, with seasonal differences in photoperiod as a necessary but not sufficient component.

Clustering predicted by an electrophysiological model of the suprachiasmatic nucleus

Despite the wealth of experimental data on the electrophysiology of individual neurons in the suprachiasmatic nuclei (SCN), the neural code of the SCN remains largely unknown. To predict the electrical activity of the SCN, the authors simulated networks of 10,000 GABAergic SCN neurons using a detailed model of the ionic currents within SCN neurons. Their goal was to understand how neuronal firing, which occurs on a time scale faster than a second, can encode a set phase of the circadian (24-h) cycle. The authors studied the effects of key network properties including: 1) the synaptic density within the SCN, 2) the magnitude of postsynaptic currents, 3) the heterogeneity of circadian phase in the neuronal population, 4) the degree of synaptic noise, and 5) the balance between excitation and inhibition. Their main result was that under a wide variety of conditions, the SCN network spontaneously organized into (typically 3) groups of synchronously firing neurons. They showed that this type of clustering can lead to the silencing of neurons whose intracellular clocks are out of circadian phase with the rest of the population. Their results provide clues to how the SCN may generate a coherent electrical output signal at the tissue level to time rhythms throughout the body.

Intrinsic, nondeterministic circadian rhythm generation in identified mammalian neurons

Circadian rhythms are modeled as reliable and self-sustained oscillations generated by single cells. The mammalian suprachiasmatic nucleus (SCN) keeps near 24-h time in vivo and in vitro, but the identity of the individual cellular pacemakers is unknown. We tested the hypothesis that circadian cycling is intrinsic to a unique class of SCN neurons by measuring firing rate or Period2 gene expression in single neurons. We found that fully isolated SCN neurons can sustain circadian cycling for at least 1 week. Plating SCN neurons at <100 cells/mm(2) eliminated synaptic inputs and revealed circadian neurons that contained arginine vasopressin (AVP) or vasoactive intestinal polypeptide (VIP) or neither. Surprisingly, arrhythmic neurons (nearly 80% of recorded neurons) also expressed these neuropeptides. Furthermore, neurons were observed to lose or gain circadian rhythmicity in these dispersed cell cultures, both spontaneously and in response to forskolin stimulation. In SCN explants treated with tetrodotoxin to block spike-dependent signaling, neurons gained or lost circadian cycling over many days. The rate of PERIOD2 protein accumulation on the previous cycle reliably predicted the spontaneous onset of arrhythmicity. We conclude that individual SCN neurons can generate circadian oscillations; however, there is no evidence for a specialized or anatomically localized class of cell-autonomous pacemakers. Instead, these results indicate that AVP, VIP, and other SCN neurons are intrinsic but unstable circadian oscillators that rely on network interactions to stabilize their otherwise noisy cycling.

Dynamical approaches to modeling developmental gene regulatory networks

The network of interacting regulatory signals within a cell comprises one of the most complex and powerful computational systems in biology. Gene regulatory networks (GRNs) play a key role in transforming the information encoded in a genome into morphological form. To achieve this feat, GRNs must respond to and integrate environmental signals with their internal dynamics in a robust and coordinated fashion. The highly dynamic nature of this process lends itself to interpretation and analysis in the language of dynamical models. Modeling provides a means of systematically untangling the complicated structure of GRNs, a framework within which to simulate the behavior of reconstructed systems and, in some cases, suites of analytic tools for exploring that behavior and its implications. This review provides a general background to the idea of treating a regulatory network as a dynamical system, and describes a variety of different approaches that have been taken to the dynamical modeling of GRNs.

Prevalence of stochasticity in experimentally observed responses of pancreatic acinar cells to acetylcholine

Calcium ions play an important role in both intra- and intercellular signaling. In pancreatic acinar cells intracellular Ca(2+) regulates exocytotic secretion and fluid secretion. In this paper we study the typical experimental traces of Ca(2+) responses in pancreatic acinar cells obtained in response to the physiological agonist acetylcholine. To determine whether they are stochastic or deterministic in nature, we analyze the traces with methods of nonlinear time series analysis. In particular, by performing surrogate data tests and employing a determinism test for short time series, we show that the responses of pancreatic acinar cells to acetylcholine are stochastic with only faintly expressed deterministic features. Presented results thus corroborate the notion that mathematical models should take stochasticity explicitly into account when describing intra- and intercellular processes, and that indeed further efforts should be directed toward this subject.

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.

Noise-induced coherence in multicellular circadian clocks

In higher organisms, circadian rhythms are generated by a multicellular genetic clock that is entrained very efficiently to the 24-h light-dark cycle. Most studies done so far of these circadian oscillators have considered a perfectly periodic driving by light, in the form of either a square wave or a sinusoidal modulation. However, in natural conditions, organisms are subject to nonnegligible fluctuations in the light level all through the daily cycle. In this article, we investigate how the interplay between light fluctuations and intercellular coupling affects the dynamics of the collective rhythm in a large ensemble of nonidentical, globally coupled cellular clocks modeled as Goodwin oscillators. On the basis of experimental considerations, we assume an inverse dependence of the cell-cell coupling strength on the light intensity, in such a way that the larger the light intensity, the weaker the coupling. Our results show a noise-induced rhythm generation for constant light intensities at which the clock is arrhythmic in the noise-free case. Importantly, the rhythm shows a resonancelike phenomenon as a function of the noise intensity. Such improved coherence can be only observed at the level of the overt rhythm and not at the level of the individual oscillators, thus suggesting a cooperative effect of noise, coupling, and the emerging synchronization between the oscillators.

A model of the cell-autonomous mammalian circadian clock

Circadian timekeeping by intracellular molecular clocks is evident widely in prokaryotes and eukaryotes. The clockworks are driven by autoregulatory feedback loops that lead to oscillating levels of components whose maxima are in fixed phase relationships with one another. These phase relationships are the key metric characterizing the operation of the clocks. In this study, we built a mathematical model from the regulatory structure of the intracellular circadian clock in mice and identified its parameters using an iterative evolutionary strategy, with minimum cost achieved through conformance to phase separations seen in cell-autonomous oscillators. The model was evaluated against the experimentally observed cell-autonomous circadian phenotypes of gene knockouts, particularly retention of rhythmicity and changes in expression level of molecular clock components. These tests reveal excellent de novo predictive ability of the model. Furthermore, sensitivity analysis shows that these knockout phenotypes are robust to parameter perturbation.

Guidance for data collection and computational modelling of regulatory networks

Many model regulatory networks are approaching the depth of characterisation of bacteriophage lambda, wherein the vast majority of individual components and interactions are identified, and research can focus on understanding whole network function and the role of interactions within that broader context. In recent years, the study of the system-wide behaviour of phage lambda's genetic regulatory network has been greatly assisted by the combination of quantitative measurements with theoretical and computational analyses. Such research has demonstrated the value of a number of general principles and guidelines for making use of the interplay between experiments and modelling. In this chapter we discuss these guidelines and provide illustration through reference to case studies from phage lambda biology.In our experience, computational modelling is best facilitated with a large and diverse set of quantitative, in vivo data, preferably obtained from standardised measurements and expressed as absolute units rather than relative units. Isolation of subsets of regulatory networks may render a system amenable to 'bottom-up' modelling, providing a valuable tool to the experimental molecular biologist. Decoupling key components and rendering their concentration or activity an independent experimental variable provide excellent information for model building, though conclusions drawn from isolated and/or decoupled systems should be checked against studies in the full physiological context; discrepancies are informative. The construction of a model makes possible in silico experiments, which are valuable tools for both the data analysis and the design of wet experiments.

Size-independent differences between the mean of discrete stochastic systems and the corresponding continuous deterministic systems

In this paper, it is shown that for a class of reaction networks, the discrete stochastic nature of the reacting species and reactions results in qualitative and quantitative differences between the mean of exact stochastic simulations and the prediction of the corresponding deterministic system. The differences are independent of the number of molecules of each species in the system under consideration. These reaction networks are open systems of chemical reactions with no zero-order reaction rates. They are characterized by at least two stationary points, one of which is a nonzero stable point, and one unstable trivial solution (stability based on a linear stability analysis of the deterministic system). Starting from a nonzero initial condition, the deterministic system never reaches the zero stationary point due to its unstable nature. In contrast, the result presented here proves that this zero-state is a stable stationary state for the discrete stochastic system, and other finite states have zero probability of existence at large times. This result generalizes previous theoretical studies and simulations of specific systems and provides a theoretical basis for analyzing a class of systems that exhibit such inconsistent behavior. This result has implications in the simulation of infection, apoptosis, and population kinetics, as it can be shown that for certain models the stochastic simulations will always yield different predictions for the mean behavior than the deterministic simulations.

Temporal switching and cell-to-cell variability in Ca2+ release activity in mammalian cells

Genetically identical cells in a uniform external environment can exhibit different phenotypes, which are often masked by conventional measurements that average over cell populations. Although most studies on this topic have used microorganisms, differentiated mammalian cells have rarely been explored. Here, we report that only approximately 40% of clonal human embryonic kidney 293 cells respond with an intracellular Ca²⁺ increase when ryanodine receptor Ca²⁺ release channels in the endoplasmic reticulum are maximally activated by caffeine. On the other hand, the expression levels of ryanodine receptor showed a unimodal distribution. We showed that the difference in the caffeine sensitivity depends on a critical balance between Ca²⁺ release and Ca²⁺ uptake activities, which is amplified by the regenerative nature of the Ca²⁺ release mechanism. Furthermore, individual cells switched between the caffeine-sensitive and caffeine-insensitive states with an average transition time of approximately 65 h, suggestive of temporal fluctuation in endogenous protein expression levels associated with caffeine response. These results suggest the significance of regenerative mechanisms that amplify protein expression noise and induce cell-to-cell phenotypic variation in mammalian cells.

Oscillations in the expression of a self-repressed gene induced by a slow transcriptional dynamics

We revisit the dynamics of a gene repressed by its own protein in the case where the transcription rate does not adapt instantaneously to protein concentration but is a dynamical variable. We derive analytical criteria for the appearance of sustained oscillations and find that they require degradation mechanisms much less nonlinear than for infinitely fast regulation. Deterministic predictions are confirmed by stochastic simulations of this minimal genetic oscillator.

Neurotransmitter-Mediated Collective Rhythms in Grouped Drosophila Circadian Clocks

Over the past decades, fly Drosophila melanogaster has being used as a premier model organism to study molecular and genetic bases of circadian rhythms. Here the authors propose a multicellular heterogeneous model for which the network of Drosophila circadian oscillators consists of two groups, the self-sustained lateral neurons (LNs) communicating to each other and the damped dorsal neurons (DNs) receiving neurotransmitters only from the LNs without interaction within this group. By simulating different experimental conditions, the authors find that the proposed model, except for being capable of reproducing some known experimental results well, also can predict some interesting phenomena: 1) The DNs need neuronal projections from the LNs to be rhythmic and to synchronize; 2) the effect of communication on mean amplitude and mean period of two oscillatory groups is different; 3) communication delay can facilitate the network synchronization of the LNs; and 4) only the LNs lose rhythmicity under constant light conditions. These results reveal the mechanism of an integrated pacemaker that would govern behavioral and physiological rhythmicity of the model organism.

Nature, nurture, or chance: stochastic gene expression and its consequences

Gene expression is a fundamentally stochastic process, with randomness in transcription and translation leading to cell-to-cell variations in mRNA and protein levels. This variation appears in organisms ranging from microbes to metazoans, and its characteristics depend both on the biophysical parameters governing gene expression and on gene network structure. Stochastic gene expression has important consequences for cellular function, being beneficial in some contexts and harmful in others. These situations include the stress response, metabolism, development, the cell cycle, circadian rhythms, and aging.

Synchrony and entrainment properties of robust circadian oscillators

Systems theoretic tools (i.e. mathematical modelling, control, and feedback design) advance the understanding of robust performance in complex biological networks. We highlight phase entrainment as a key performance measure used to investigate dynamics of a single deterministic circadian oscillator for the purpose of generating insight into the behaviour of a population of (synchronized) oscillators. More specifically, the analysis of phase characteristics may facilitate the identification of appropriate coupling mechanisms for the ensemble of noisy (stochastic) circadian clocks. Phase also serves as a critical control objective to correct mismatch between the biological clock and its environment. Thus, we introduce methods of investigating synchrony and entrainment in both stochastic and deterministic frameworks, and as a property of a single oscillator or population of coupled oscillators.

Rate constants rather than biochemical mechanism determine behaviour of genetic clocks

Many biological systems contain both positive and negative feedbacks. These are often classified as resonators or integrators. Resonators respond preferentially to oscillating signals of a particular frequency. Integrators, on the other hand, accumulate a response to signals. Computational neuroscientists often refer to neurons showing integrator properties as type I neurons and those showing resonator properties as type II neurons. Guantes & Poyatos have shown that type I or type II behaviour can be seen in genetic clocks. They argue that when negative feedback occurs through transcription regulation and post-translationally, genetic clocks act as integrators and resonators, respectively. Here we show that either behaviour can be seen with either design and in a wide range of genetic clocks. This highlights the importance of parameters rather than biochemical mechanism in determining the system behaviour.

Biological switches and clocks

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

Global parameter search reveals design principles of the mammalian circadian clock

BACKGROUND

Virtually all living organisms have evolved a circadian (~24 hour) clock that controls physiological and behavioural processes with exquisite precision throughout the day/night cycle. The suprachiasmatic nucleus (SCN), which generates these ~24 h rhythms in mammals, consists of several thousand neurons. Each neuron contains a gene-regulatory network generating molecular oscillations, and the individual neuron oscillations are synchronised by intercellular coupling, presumably via neurotransmitters. Although this basic mechanism is currently accepted and has been recapitulated in mathematical models, several fundamental questions about the design principles of the SCN remain little understood. For example, a remarkable property of the SCN is that the phase of the SCN rhythm resets rapidly after a 'jet lag' type experiment, i.e. when the light/dark (LD) cycle is abruptly advanced or delayed by several hours.

RESULTS

Here, we describe an extensive parameter optimization of a previously constructed simplified model of the SCN in order to further understand its design principles. By examining the top 50 solutions from the parameter optimization, we show that the neurotransmitters' role in generating the molecular circadian rhythms is extremely important. In addition, we show that when a neurotransmitter drives the rhythm of a system of coupled damped oscillators, it exhibits very robust synchronization and is much more easily entrained to light/dark cycles. We were also able to recreate in our simulations the fast rhythm resetting seen after a 'jet lag' type experiment.

CONCLUSION

Our work shows that a careful exploration of parameter space for even an extremely simplified model of the mammalian clock can reveal unexpected behaviours and non-trivial predictions. Our results suggest that the neurotransmitter feedback loop plays a crucial role in the robustness and phase resetting properties of the mammalian clock, even at the single neuron level.

Computational analysis of mammalian cell division gated by a circadian clock: quantized cell cycles and cell size control

Cell cycle and circadian rhythms are conserved from cyanobacteria to humans with robust cyclic features. Recently, molecular links between these two cyclic processes have been discovered. Core clock transcription factors, Bmal1 and Clock (Clk), directly regulate Wee1 kinase, which inhibits entry into the mitosis. We investigate the effect of this connection on the timing of mammalian cell cycle processes with computational modeling tools. We connect a minimal model of circadian rhythms, which consists of transcription-translation feedback loops, with a modified mammalian cell cycle model from Novak and Tyson (2004). As we vary the mass doubling time (MDT) of the cell cycle, stochastic simulations reveal quantized cell cycles when the activity of Wee1 is influenced by clock components. The quantized cell cycles disappear in the absence of coupling or when the strength of this link is reduced. More intriguingly, our simulations indicate that the circadian clock triggers critical size control in the mammalian cell cycle. A periodic brake on the cell cycle progress via Wee1 enforces size control when the MDT is quite different from the circadian period. No size control is observed in the absence of coupling. The issue of size control in the mammalian system is debatable, whereas it is well established in yeast. It is possible that the size control is more readily observed in cell lines that contain circadian rhythms, since not all cell types have a circadian clock. This would be analogous to an ultradian clock intertwined with quantized cell cycles (and possibly cell size control) in yeast. We present the first coupled model between the mammalian cell cycle and circadian rhythms that reveals quantized cell cycles and cell size control influenced by the clock.

Modeling the electrophysiology of suprachiasmatic nucleus neurons

Neurons in the SCN act as the central circadian (approximately 24-h) pacemaker in mammals. Using measurements of the ionic currents in SCN neurons, the authors fit a Hodgkin-Huxley-type model that accurately reproduces slow (approximately 28 Hz) neural firing as well as the contributions of ionic currents during an action potential. When inputs of other SCN neurons are considered, the model accurately predicts the fractal nature of firing rates and the appearance of random bursting. In agreement with experimental data, the molecular clock within these neurons modulates the firing rate through small changes in the concentration of internal calcium, calcium channels, or potassium channels. Predictions are made on how signals from other neurons can start, stop, speed up, or slow down firing. Only a slow sodium inactivation variable and voltage do not reach equilibrium during the interval between action potentials, and based on this finding, a reduced model is formulated.

Modelling reaction kinetics inside cells

In the past decade, advances in molecular biology such as the development of non-invasive single molecule imaging techniques have given us a window into the intricate biochemical activities that occur inside cells. In this chapter we review four distinct theoretical and simulation frameworks: (i) non-spatial and deterministic, (ii) spatial and deterministic, (iii) non-spatial and stochastic and (iv) spatial and stochastic. Each framework can be suited to modelling and interpreting intracellular reaction kinetics. By estimating the fundamental length scales, one can roughly determine which models are best suited for the particular reaction pathway under study. We discuss differences in prediction between the four modelling methodologies. In particular we show that taking into account noise and space does not simply add quantitative predictive accuracy but may also lead to qualitatively different physiological predictions, unaccounted for by classical deterministic models.

Establishing the stochastic nature of intracellular calcium oscillations from experimental data

Calcium has been established as a key messenger in both intra- and intercellular signaling. Experimentally observed intracellular calcium responses to different agonists show a variety of behaviors from simple spiking to complex oscillatory regimes. Here we study typical experimental traces of calcium oscillations in hepatocytes obtained in response to phenylephrine and ATP. The traces were analyzed with methods of nonlinear time series analysis in order to determine the stochastic/deterministic nature of the intracellular calcium oscillations. Despite the fact that the oscillations appear, visually, to be deterministic yet perturbed by noise, our analyses provide strong evidence that the measured calcium traces in hepatocytes are prevalently of stochastic nature. In particular, bursting calcium oscillations are temporally correlated Gaussian series distorted by a monotonic, instantaneous, time-independent function, whilst the spiking behavior appears to have a dynamical nonlinear component whereby the overall determinism level is still low. The biological importance of this finding is discussed in relation to the mechanisms incorporated in mathematical models as well as the role of stochasticity and determinism at cellular and tissue levels which resemble typical statistical and thermodynamic effects in physics.

Beyond intuitive modeling: combining biophysical models with innovative experiments to move the circadian clock field forward

Two major approaches have been used to model circadian clocks. Qualitative modeling, used prior to the recent wealth of detailed molecular knowledge, makes general predictions but cannot provide detailed mechanistic insights. The more recent biophysical approach, on the other hand, incorporates the biochemical events that drive the clock and can make detailed and testable molecular predictions. These predictions are being tested using new experimental techniques that measure reaction kinetics and the behavior of individual cells. A joint modeling and experimental approach has recently been used to understand how mutations affecting phosphorylation can lead to a short circadian period in tau mutant hamsters and in humans with familial advanced sleep phase syndrome (FASPS). Another recent study has revealed novel single-cell phenotypes of clock gene mutations, demanding revision of current biophysical models yet validating certain model predictions that were previously overlooked. A new paradigm for clock research is emerging in which modeling inspires new experimental efforts, experimental data inspire new modeling efforts, and joint modeling/experimental studies lead to a deeper understanding of mammalian circadian rhythms.