Stochastic simulation of the mammalian circadian clock

Cycling of CRYPTOCHROME proteins is not necessary for circadian-clock function in mammalian fibroblasts

BACKGROUND

An interlocked transcriptional-translational feedback loop (TTFL) is thought to generate the mammalian circadian clockwork in both the central pacemaker residing in the hypothalamic suprachiasmatic nuclei and in peripheral tissues. The core circadian genes, including Period1 and Period2 (Per1 and Per2), Cryptochrome1 and Cryptochrome2 (Cry1 and Cry2), Bmal1, and Clock are indispensable components of this biological clockwork. The cycling of the PER and CRY clock proteins has been thought to be necessary to keep the mammalian clock ticking.

RESULTS

We provide a novel cell-permeant protein approach for manipulating cryptochrome protein levels to evaluate the current transcription and translation feedback model of the circadian clockwork. Cell-permeant cryptochrome proteins appear to be functional on the basis of several criteria, including the abilities to (1) rescue circadian properties in Cry1(-/-)Cry2(-/-) mouse fibroblasts, (2) act as transcriptional repressors, and (3) phase shift the circadian oscillator in Rat-1 fibroblasts. By using cell-permeant cryptochrome proteins, we demonstrate that cycling of CRY1, CRY2, and BMAL1 is not necessary for circadian-clock function in fibroblasts.

CONCLUSIONS

These results are not supportive of the current version of the transcription and translation feedback-loop model of the mammalian clock mechanism, in which cycling of the essential clock proteins CRY1 and CRY2 is thought to be necessary.

Intercellular coupling confers robustness against mutations in the SCN circadian clock network

Molecular mechanisms of the mammalian circadian clock have been studied primarily by genetic perturbation and behavioral analysis. Here, we used bioluminescence imaging to monitor Per2 gene expression in tissues and cells from clock mutant mice. We discovered that Per1 and Cry1 are required for sustained rhythms in peripheral tissues and cells, and in neurons dissociated from the suprachiasmatic nuclei (SCN). Per2 is also required for sustained rhythms, whereas Cry2 and Per3 deficiencies cause only period length defects. However, oscillator network interactions in the SCN can compensate for Per1 or Cry1 deficiency, preserving sustained rhythmicity in mutant SCN slices and behavior. Thus, behavior does not necessarily reflect cell-autonomous clock phenotypes. Our studies reveal previously unappreciated requirements for Per1, Per2, and Cry1 in sustaining cellular circadian rhythmicity and demonstrate that SCN intercellular coupling is essential not only to synchronize component cellular oscillators but also for robustness against genetic perturbations.

Repressor dimerization in the zebrafish somitogenesis clock

The oscillations of the somitogenesis clock are linked to the fundamental process of vertebrate embryo segmentation, yet little is known about their generation. In zebrafish, it has been proposed that Her proteins repress the transcription of their own mRNA. However, in its simplest form, this model is incompatible with the fact that morpholino knockdown of Her proteins can impair expression of their mRNA. Simple self-repression models also do not account for the spatiotemporal pattern of gene expression, with waves of gene expression shrinking as they propagate. Here we study computationally the networks generated by the wealth of dimerization possibilities amongst transcriptional repressors in the zebrafish somitogenesis clock. These networks can reproduce knockdown phenotypes, and strongly suggest the existence of a Her1–Her7 heterodimer, so far untested experimentally. The networks are the first reported to reproduce the spatiotemporal pattern of the zebrafish somitogenesis clock; they shed new light on the role of Her13.2, the only known link between the somitogenesis clock and positional information in the paraxial mesoderm. The networks can also account for perturbations of the clock by manipulation of FGF signaling. Achieving an understanding of the interplay between clock oscillations and positional information is a crucial first step in the investigation of the segmentation mechanism.

Vertebrate embryos acquire a segmented structure along the anteroposterior axis. Segmentation is critical for patterning of other structures (such as nerves, vertebrae, muscles, and blood vessels) and occurs by the rhythmic separation of balls of cells, called somites, from the anterior end of their precursor tissue, called the presomitic mesoderm. These rhythmic events are associated with oscillatory gene expression in the presomitic mesoderm: waves of gene expression originate at the posterior end and spread anteriorly. When a wave reaches the anterior end, a pair of new somites detaches. The set of genes whose expression oscillates is termed the “somitogenesis clock.” Even though the zebrafish somitogenesis clock has been the subject of intensive study, it is not clear how its oscillations are generated. It has been proposed that the mechanism involves a simple negative feedback loop, with proteins of the Her family periodically repressing their own expression. However, this is incompatible with some experimental results and does not explain how the spatiotemporal pattern of gene expression is generated. Here I propose a model—based on physical interactions between Her proteins—that is compatible with experimental results, and that explains how positional information is used to generate the spatiotemporal pattern of gene expression.

Evolutionary mechanisms of circadian clocks

An intriguing question in biology is to know how circadian molecular networks could have evolved their particular topologies to adjust to a daily period. We analyze the mechanism of the evolution of such networks by using a computational design strategy that allows for the generation of synthetic networks with a targeted 24 hours oscillation. We have performed a systematic analysis of all possible two-gene network topologies based on a core activator-repressor frequently found in circadian mechanisms. We have considered transcriptional and post-translational regulations to implement this core. We have applied our analysis to both, eukaryotic and prokaryotic circadian machinery. Finally, we conjecture a possible mechanism for the evolution of circadian clocks.

Oscillation patterns in negative feedback loops

Organisms are equipped with regulatory systems that display a variety of dynamical behavior ranging from simple stable steady states, to switching and multistability, to oscillations. Earlier work has shown that oscillations in protein concentrations or gene expression levels are related to the presence of at least one negative feedback loop in the regulatory network. Here, we study the dynamics of a very general class of negative feedback loops. Our main result is that, when a single negative feedback loop dominates the dynamical behavior, the sequence of maxima and minima of the concentrations exhibit a pattern that uniquely identifies the interactions of the loop. This allows us to devise an algorithm to (i) test whether observed oscillating time series are consistent with a single underlying negative feedback loop, and if so, (ii) reconstruct the precise structure of the loop, i.e., the activating/repressing nature of each interaction. This method applies even when some variables are missing from the data set, or if the time series shows transients, like damped oscillations. We illustrate the relevance and the limits of validity of our method with three examples: p53-Mdm2 oscillations, circadian gene expression in cyanobacteria, and cyclic binding of cofactors at the estrogen-sensitive pS2 promoter.

Dynamical signatures of cellular fluctuations and oscillator stability in peripheral circadian clocks

Cell-autonomous and self-sustained molecular oscillators drive circadian behavior and physiology in mammals. From rhythms recorded in cultured fibroblasts we identified the dominant cause for amplitude reduction as desynchronization of self-sustained oscillators. Here, we propose a general framework for quantifying luminescence signals from biochemical oscillators, both in populations and individual cells. Our model combines three essential aspects of circadian clocks: the stability of the limit cycle, fluctuations, and intercellular coupling. From population recordings we can simultaneously estimate the stiffness of individual frequencies, the period dispersion, and the interaction strength. Consistent with previous work, coupling is found to be weak and insufficient to synchronize cells. Moreover, we find that frequency fluctuations remain correlated for longer than one clock cycle, which is confirmed from individual cell recordings. Using genetic models for circadian clocks, we show that this reflects the stability properties of the underlying circadian limit-cycle oscillators, and we identify biochemical parameters that influence oscillator stability in mammals. Our study thus points to stabilizing mechanisms that dampen fluctuations to maintain accurate timing in peripheral circadian oscillators.

A stochastic model for circadian rhythms from coupled ultradian oscillators

BACKGROUND

Circadian rhythms with varying components exist in organisms ranging from humans to cyanobacteria. A simple evolutionarily plausible mechanism for the origin of such a variety of circadian oscillators, proposed in earlier work, involves the non-disruptive coupling of pre-existing ultradian transcriptional-translational oscillators (TTOs), producing "beats," in individual cells. However, like other TTO models of circadian rhythms, it is important to establish that the inherent stochasticity of the protein binding and unbinding does not invalidate the finding of clear oscillations with circadian period.

RESULTS

The TTOs of our model are described in two versions: 1) a version in which the activation or inhibition of genes is regulated stochastically, where the 'unoccupied" (or "free") time of the site under consideration depends on the concentration of a protein complex produced by another site, and 2) a deterministic, "time-averaged" version in which the switching between the "free" and "occupied" states of the sites occurs so rapidly that the stochastic effects average out. The second case is proved to emerge from the first in a mathematically rigorous way. Numerical results for both scenarios are presented and compared.

CONCLUSION

Our model proves to be robust to the stochasticity of protein binding/unbinding at experimentally determined rates and even at rates several orders of magnitude slower. We have not only confirmed this by numerical simulation, but have shown in a mathematically rigorous way that the time-averaged deterministic system is indeed the fast-binding-rate limit of the full stochastic model.

Reversible protein phosphorylation regulates circadian rhythms

Protein phosphorylation regulates the period of the circadian clock within mammalian cells. Circadian rhythms are an approximately 24-hour cycle that regulates key biological processes. Daily fluctuations of wakefulness, stress hormones, lipid metabolism, immune function, and the cell division cycle are controlled by the molecular clocks that function throughout our bodies. Mutations in regulatory components of the clock can shorten or lengthen the timing of the rhythms and have significant physiological consequences. The clock is formed by a negative feedback loop of transcription, translation, and inhibition of transcription. The precision of clock timing is controlled by protein kinases and phosphatases. Casein kinase Iepsilon is a protein kinase that regulates the circadian clock by periodic phosphorylation of the proteins PER1 and PER2, controlling their stability and localization. The role of phosphorylation in regulating PER function in the clock has been explored in detail. Quantitative modeling has proven to be very useful in making important predictions about how changes in phosphorylation alter the clock's behavior. Quantitative data from biological studies can be used to refine the quantitative model and make additional testable predictions. A detailed understanding of how reversible protein phosphorylation regulates circadian rhythms and a detailed quantitative model that makes clear, testable, and accurate predictions about the clock and how we may manipulate it can have important benefits for human health. Pharmacological manipulation of rhythms could mitigate stress from jet lag, shift work, and perhaps even seasonal affective disorder.

Validation of an algorithm for delay stochastic simulation of transcription and translation in prokaryotic gene expression

The quantitative modeling of gene transcription and translation requires a treatment of two key features: stochastic fluctuations due to the limited copy numbers of key molecules (genes, RNA polymerases, ribosomes), and delayed output due to the time required for biopolymer synthesis. Recently proposed algorithms allow for efficient simulations of such systems. However, it is critical to know whether the results of delay stochastic simulations agree with those from more detailed models of the transcription and translation processes. We present a generalization of previous delay stochastic simulation algorithms which allows both for multiple delays and for distributions of delay times. We show that delay stochastic simulations closely approximate simulations of a detailed transcription model except when two-body effects (e.g. collisions between polymerases on a template strand) are important. Finally, we study a delay stochastic model of prokaryotic transcription and translation which reproduces observations from a recent experimental study in which a single gene was expressed under the control of a repressed lac promoter in E. coli cells. This demonstrates our ability to quantitatively model gene expression using these new methods.

Evolutionary design on a budget: robustness and optimality of bacteriophage T7

Exploring how biological systems have been 'designed' by evolution to achieve robust behaviours is now a subject of increasing research effort. Yet, it still remains unclear how environmental factors may contribute to this process. This issue is addressed by employing a detailed computer model for the intracellular growth of phage T7. More than 150 000 in silico T7 mutants were generated and the rates and efficiencies of their growth in two host environments, namely, a realistic environment that offered finite host resources for the synthesis of phage functions and a hypothetical environment where the phage was supplied infinite host resources, were evaluated. Results revealed two key properties of phage T7. First, T7 growth was overall robust with respect to perturbations in its parameters, but fragile with respect to changes in the ordering of its genetic elements. Secondly, the wild-type T7 had close to optimal fitness in the finite environment. Furthermore, a strong correlation was found between fitness and growth efficiency in the finite environment. The results underscore the potential importance of the environment in shaping robust design of a biological system. In particular, the strong correlation between fitness and growth efficiency suggests that T7 may have evolved to maximise its growth rate by minimising waste of finite resources.

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.

An opposite role for tau in circadian rhythms revealed by mathematical modeling

Biological clocks with a period of approximately 24 h (circadian) exist in most organisms and time a variety of functions, including sleep-wake cycles, hormone release, bioluminescence, and core body temperature fluctuations. Much of our understanding of the clock mechanism comes from the identification of specific mutations that affect circadian behavior. A widely studied mutation in casein kinase I (CKI), the CKIepsilon(tau) mutant, has been shown to cause a loss of kinase function in vitro, but it has been difficult to reconcile this loss of function with the current model of circadian clock function. Here we show that mathematical modeling predicts the opposite, that the kinase mutant CKIepsilon(tau) increases kinase activity, and we verify this prediction experimentally. CKIepsilon(tau) is a highly specific gain-of-function mutation that increases the in vivo phosphorylation and degradation of the circadian regulators PER1 and PER2. These findings experimentally validate a mathematical modeling approach to a complex biological function, clarify the role of CKI in the clock, and demonstrate that a specific mutation can be both a gain and a loss of function depending on the substrate.

Isochron-based phase response analysis of circadian rhythms

Circadian rhythms possess the ability to robustly entrain to the environmental cycles. This ability relies on the phase synchronization of circadian rhythm gene regulation to different environmental cues, of which light is the most obvious and important. The elucidation of the mechanism of circadian entrainment requires an understanding of circadian phase behavior. This article presents two phase analyses of oscillatory systems for infinitesimal and finite perturbations based on isochrons as a phase metric of a limit cycle. The phase response curve of circadian rhythm can be computed from the results of the analyses. The application to a mechanistic Drosophila circadian rhythm model gives experimentally testable hypotheses for the control mechanisms of circadian phase responses and evidence for the role of phase and period modulations in circadian photic entrainment.

Stochastic kinetics description of a simple transcription model

We study a stochastic model of transcription kinetics in order to characterize the distributions of transcriptional delay and of elongation rates. Transcriptional delay is the time which elapses between the binding of RNA polymerase to a promoter sequence and its dissociation from the DNA template strand with consequent release of the transcript. Transcription elongation is the process by which the RNA polymerase slides along the template strand. The model considers a DNA template strand with one promoter site and n nucleotide sites, and five types of reaction processes, which we think are key ones in transcription. The chemical master equation is a set of ordinary differential equations in 3(n) variables, where n is the number of bases in the template. This model is too huge to be handled if n is large. We manage to get a reduced Markov model which has only 2n independent variables and can well approximate the original dynamics. We obtain a number of analytical and numerical results for this model, including delay and transcript elongation rate distributions. Recent studies of single-RNA polymerase transcription by using optical-trapping techniques raise an issue of whether the elongation rates measured in a population are heterogeneous or not. Our model implies that in the cases studied, different RNA polymerase molecules move at different characteristic rates along the template strand. We also discuss the implications of this work for the mathematical modeling of genetic regulatory circuits.

Circadian rhythms and molecular noise

Circadian rhythms, characterized by a period of about 24 h, are the most widespread biological rhythms generated autonomously at the molecular level. The core molecular mechanism responsible for circadian oscillations relies on the negative regulation exerted by a protein on the expression of its own gene. Deterministic models account for the occurrence of autonomous circadian oscillations, for their entrainment by light-dark cycles, and for their phase shifting by light pulses. Stochastic versions of these models take into consideration the molecular fluctuations that arise when the number of molecules involved in the regulatory mechanism is low. Numerical simulations of the stochastic models show that robust circadian oscillations can already occur with a limited number of mRNA and protein molecules, in the range of a few tens and hundreds, respectively. Various factors affect the robustness of circadian oscillations with respect to molecular noise. Besides an increase in the number of molecules, entrainment by light-dark cycles, and cooperativity in repression enhance robustness, whereas the proximity of a bifurcation point leads to less robust oscillations. Another parameter that appears to be crucial for the coherence of circadian rhythms is the binding/unbinding rate of the inhibitory protein to the promoter of the clock gene. Intercellular coupling further increases the robustness of circadian oscillations.

Internal Noise Coherent Resonance for Mesoscopic Chemical Oscillations: A Fundamental Study

The effect of internal noise for a mesoscopic chemical oscillator is studied analytically in a parameter region outside, but close to, the supercritical Hopf bifurcation. By normal form calculation and a stochastic averaging procedure, we obtain stochastic differential equations for the oscillation amplitude r and phase theta that is solvable. Noise-induced oscillation and internal noise coherent resonance, which has been observed in many numerical experiments, are reproduced well by the theory.

Balance between DBT/CKIepsilon kinase and protein phosphatase activities regulate phosphorylation and stability of Drosophila CLOCK protein

The first circadian-relevant kinase to be identified was DOUBLE-TIME (DBT) in Drosophila, a homolog of vertebrate CKIepsilon, which regulates the progressive phosphorylation and stability of PERIOD (PER) proteins in animals. A negative feedback loop wherein PER directly inhibits the transcriptional activity of the CLOCK-CYCLE (CLK-CYC) heterodimer is central to the generation of molecular rhythms and normal progression of the clock in Drosophila. We show that DBT activity is required for the phase-specific hyperphosphorylation of CLK in vivo, an event that correlates with times of maximal repression in per RNA levels. The ability of DBT to hyperphosphorylate CLK, enhance its degradation, and evoke modest inhibition of CLK-dependent transactivation from circadian promoter elements was directly shown in cultured Drosophila cells. Intriguingly, DBT seems to function in close partnership with the PER-relevant protein phosphatase 2A, resulting in dynamic equilibrium between hypo- and hyperphosphorylated isoforms of CLK. This balancing mechanism might act to stabilize the limiting levels of CLK against stochastic fluctuations minimizing the propagation of "molecular noise" in the feedback circuitry. Also, the subcellular localization of CLK was altered from predominately nuclear to strong cytoplasmic staining in the presence of PER. These results suggest that, in contrast to mammalian clocks, circadian transcriptional inhibition in Drosophila involves displacement of the positive factors from chromatin. These results also demonstrate that DBT can target both negative and positive factors in circadian feedback loops and support a conserved role for dynamic regulation of reversible phosphorylation in directly modulating the activities of circadian transcription factors.

The sorting direct method for stochastic simulation of biochemical systems with varying reaction execution behavior

A key to advancing the understanding of molecular biology in the post-genomic age is the development of accurate predictive models for genetic regulation, protein interaction, metabolism, and other biochemical processes. To facilitate model development, simulation algorithms must provide an accurate representation of the system, while performing the simulation in a reasonable amount of time. Gillespie's stochastic simulation algorithm (SSA) accurately depicts spatially homogeneous models with small populations of chemical species and properly represents noise, but it is often abandoned when modeling larger systems because of its computational complexity. In this work, we examine the performance of different versions of the SSA when applied to several biochemical models. Through our analysis, we discover that transient changes in reaction execution frequencies, which are typical of biochemical models with gene induction and repression, can dramatically affect simulator performance. To account for these shifts, we propose a new algorithm called the sorting direct method that maintains a loosely sorted order of the reactions as the simulation executes. Our measurements show that the sorting direct method performs favorably when compared to other well-known exact stochastic simulation algorithms.

Dynamical principles of two-component genetic oscillators

Genetic oscillators based on the interaction of a small set of molecular components have been shown to be involved in the regulation of the cell cycle, the circadian rhythms, or the response of several signaling pathways. Uncovering the functional properties of such oscillators then becomes important for the understanding of these cellular processes and for the characterization of fundamental properties of more complex clocks. Here, we show how the dynamics of a minimal two-component oscillator is drastically affected by its genetic implementation. We consider a repressor and activator element combined in a simple logical motif. While activation is always exerted at the transcriptional level, repression is alternatively operating at the transcriptional (Design I) or post-translational (Design II) level. These designs display differences on basic oscillatory features and on their behavior with respect to molecular noise or entrainment by periodic signals. In particular, Design I induces oscillations with large activator amplitudes and arbitrarily small frequencies, and acts as an “integrator” of external stimuli, while Design II shows emergence of oscillations with finite, and less variable, frequencies and smaller amplitudes, and detects better frequency-encoded signals (“resonator”). Similar types of stimulus response are observed in neurons, and thus this work enables us to connect very different biological contexts. These dynamical principles are relevant for the characterization of the physiological roles of simple oscillator motifs, the understanding of core machineries of complex clocks, and the bio-engineering of synthetic oscillatory circuits.

Periodic variations in protein abundances are at the heart of important cellular processes, with common examples being the circadian rhythms or the cell cycle. What is the molecular basis of this behavior? Recent reports showed how simple architectures, based on the interaction of a few molecular components, are capable of inducing oscillatory dynamics. These structures are also of interest for the understanding of complex cellular oscillators, as they appear to be core constituents of them. The authors carefully analyze one of these architectures and uncover how different genetic implementations of such structures strikingly influence its dynamical behavior. They consider two genetic implementations of a repressor and activator element combined in a simple logical motif. While activation is transcriptionally implemented in both cases, repression may act either transcriptionally or post-translationally. These differences in design originate drastic changes in the way oscillations are produced, in the tolerance to molecular noise, or in the circuit response to external stimuli. Similar aspects have been discussed in relation to neural dynamics; therefore this work is able to connect two very different biological scenarios. Thus, simple genetic motifs exploit not only their connectivity pattern but their design to act as information processing units within living cells.

Optimal observability of sustained stochastic competitive inhibition oscillations at organellar volumes

When molecules are present in small numbers, such as is frequently the case in cells, the usual assumptions leading to differential rate equations are invalid and it is necessary to use a stochastic description which takes into account the randomness of reactive encounters in solution. We display a very simple biochemical model, ordinary competitive inhibition with substrate inflow, which is only capable of damped oscillations in the deterministic mass-action rate equation limit, but which displays sustained oscillations in stochastic simulations. We define an observability parameter, which is essentially just the ratio of the amplitude of the oscillations to the mean value of the concentration. A maximum in the observability is seen as the volume is varied, a phenomenon we name system-size observability resonance by analogy with other types of stochastic resonance. For the parameters of this study, the maximum in the observability occurs at volumes similar to those of bacterial cells or of eukaryotic organelles.

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

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

Stochasticity in gene expression: from theories to phenotypes

Genetically identical cells exposed to the same environmental conditions can show significant variation in molecular content and marked differences in phenotypic characteristics. This variability is linked to stochasticity in gene expression, which is generally viewed as having detrimental effects on cellular function with potential implications for disease. However, stochasticity in gene expression can also be advantageous. It can provide the flexibility needed by cells to adapt to fluctuating environments or respond to sudden stresses, and a mechanism by which population heterogeneity can be established during cellular differentiation and development.