Modeling the emergence of circadian rhythms in a clock neuron network

Circadian rhythms in pacemaker cells persist for weeks in constant darkness, while in other types of cells the molecular oscillations that underlie circadian rhythms damp rapidly under the same conditions. Although much progress has been made in understanding the biochemical and cellular basis of circadian rhythms, the mechanisms leading to damped or self-sustained oscillations remain largely unknown. There exist many mathematical models that reproduce the circadian rhythms in the case of a single cell of the Drosophila fly. However, not much is known about the mechanisms leading to coherent circadian oscillation in clock neuron networks. In this work we have implemented a model for a network of interacting clock neurons to describe the emergence (or damping) of circadian rhythms in Drosophila fly, in the absence of zeitgebers. Our model consists of an array of pacemakers that interact through the modulation of some parameters by a network feedback. The individual pacemakers are described by a well-known biochemical model for circadian oscillation, to which we have added degradation of PER protein by light and multiplicative noise. The network feedback is the PER protein level averaged over the whole network. In particular, we have investigated the effect of modulation of the parameters associated with (i) the control of net entrance of PER into the nucleus and (ii) the non-photic degradation of PER. Our results indicate that the modulation of PER entrance into the nucleus allows the synchronization of clock neurons, leading to coherent circadian oscillations under constant dark condition. On the other hand, the modulation of non-photic degradation cannot reset the phases of individual clocks subjected to intrinsic biochemical noise.     

Bifurcations in a mathematical model for circadian oscillations of clock genes

Circadian oscillations with a period of about 24h are observed in nearly all living organisms as conspicuous biological rhythms. In this paper, we investigate various kinds of bifurcation phenomena produced in a circadian oscillator model of Drosophila. In Drosophila, it is known that circadian oscillations in the levels of two proteins, PER and TIM, result from the negative feedback exerted by a PER-TIM complex on the expression of the per and tim genes that code for the two proteins. For studying circadian oscillations of proteins in Drosophila, a mathematical model has been proposed. The model cannot only account for regular circadian oscillations in environmental conditions such as constant darkness, but also give rise to more complex oscillatory phenomena including chaos and birhythmicity. By calculating bifurcations using Kawakami's method, we obtain detailed bifurcation diagrams related to stable and unstable invariant sets, and identify parameter regions in which the model generates complex oscillations as well as regular circadian oscillations. Moreover, we study bifurcations observed in the model incorporating the effect on a light-dark (LD) cycle and show that the waveform of the periodic variation in the light-induced parameter has a marked influence on the global bifurcation structure or the type of dynamic behavior resulting from the forcing term of the circadian oscillator by the LD cycles.     

Modeling the molecular regulatory mechanism of circadian rhythms in Drosophila

Thanks to genetic and biochemical advances on the molecular mechanism of circadian rhythms in Drosophila, theoretical models closely related to experimental observations can be considered for the regulatory mechanism of the circadian clock in this organism. Modeling is based on the autoregulatory negative feedback exerted by a complex between PER and TIM proteins on the expression of per and tim genes. The model predicts the occurrence of sustained circadian oscillations in continuous darkness. When incorporating light-induced TIM degradation, the model accounts for damping of oscillations in constant light, entrainment of the rhythm by light-dark cycles of varying period or photoperiod, and phase shifting by light pulses. The model further provides a molecular dynamical explanation for the permanent or transient suppression of circadian rhythmicity triggered in a variety of organisms by a critical pulse of light. Finally, the model shows that to produce a robust rhythm the various clock genes must be expressed at the appropriate levels since sustained oscillations only occur in a precise range of parameter values. BioEssays 22:84-93, 2000.     

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

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

Old flies have a robust central oscillator but weaker behavioral rhythms that can be improved by genetic and environmental manipulations

Sleep-wake cycles break down with age, but the causes of this degeneration are not clear. Using a Drosophila model, we addressed the contribution of circadian mechanisms to this age-induced deterioration. We found that in old flies, free-running circadian rhythms (behavioral rhythms assayed in constant darkness) have a longer period and an unstable phase before they eventually degenerate. Surprisingly, rhythms are weaker in light-dark cycles and the circadian-regulated morning peak of activity is diminished under these conditions. On a molecular level, aging results in reduced amplitude of circadian clock gene expression in peripheral tissues. However, oscillations of the clock protein PERIOD (PER) are robust and synchronized among different clock neurons, even in very old, arrhythmic flies. To improve rhythms in old flies, we manipulated environmental conditions, which can have direct effects on behavior, and also tested a role for molecules that act downstream of the clock. Coupling temperature cycles with a light-dark schedule or reducing expression of protein kinase A (PKA) improved behavioral rhythms and consolidated sleep. Our data demonstrate that a robust molecular timekeeping mechanism persists in the central pacemaker of aged flies, and reducing PKA can strengthen behavioral rhythms.     

Kinetics of doubletime kinase-dependent degradation of the Drosophila period protein

Robust circadian oscillations of the proteins PERIOD (PER) and TIMELESS (TIM) are hallmarks of a functional clock in the fruit fly Drosophila melanogaster. Early morning phosphorylation of PER by the kinase Doubletime (DBT) and subsequent PER turnover is an essential step in the functioning of the Drosophila circadian clock. Here using time-lapse fluorescence microscopy we study PER stability in the presence of DBT and its short, long, arrhythmic, and inactive mutants in S2 cells. We observe robust PER degradation in a DBT allele-specific manner. With the exception of doubletime-short (DBT(S)), all mutants produce differential PER degradation profiles that show direct correspondence with their respective Drosophila behavioral phenotypes. The kinetics of PER degradation with DBT(S) in cell culture resembles that with wild-type DBT and posits that, in flies DBT(S) likely does not modulate the clock by simply affecting PER degradation kinetics. For all the other tested DBT alleles, the study provides a simple model in which the changes in Drosophila behavioral rhythms can be explained solely by changes in the rate of PER degradation.     

Drosophila cryb mutation reveals two circadian clocks that drive locomotor rhythm and have different responsiveness to light

Cryptochrome (CRY) is a blue-light-absorbing protein involved in the photic entrainment of the circadian clock in Drosophila melanogaster. We have investigated the locomotor activity rhythms of flies carrying cryb mutant and revealed that they have two separate circadian oscillators with different responsiveness to light. When kept in constant light conditions, wild-type flies became arrhythmic, while cryb mutant flies exhibited free-running rhythms with two rhythmic components, one with a shorter and the other with a longer free-running period. The rhythm dissociation was dependent on the light intensities: the higher the light intensities, the greater the proportion of animals exhibiting the two oscillations. External photoreceptors including the compound eyes and the ocelli are the likely photoreceptors for the rhythm dissociation, since rhythm dissociation was prevented in so1;cryb and norpAP41;cryb double mutant flies. Immunohistochemical analysis demonstrated that the PERIOD expression rhythms in ventrally located lateral neurons (LNvs) occurred synchronously with the shorter period component, while those in the dorsally located per-expressing neurons showed PER expression most likely related to the longer period component, in addition to that synchronized to the LNvs. These results suggest that the Drosophila locomotor rhythms are driven by two separate per-dependent clocks, responding differentially to constant light.     

Effects of Different PER Translational Kinetics on the Dynamics of a Core Circadian Clock Model

Living beings display self-sustained daily rhythms in multiple biological processes, which persist in the absence of external cues since they are generated by endogenous circadian clocks. The period (per) gene is a central player within the core molecular mechanism for keeping circadian time in most animals. Recently, the modulation PER translation has been reported, both in mammals and flies, suggesting that translational regulation of clock components is important for the proper clock gene expression and molecular clock performance. Because translational regulation ultimately implies changes in the kinetics of translation and, therefore, in the circadian clock dynamics, we sought to study how and to what extent the molecular clock dynamics is affected by the kinetics of PER translation. With this objective, we used a minimal mathematical model of the molecular circadian clock to qualitatively characterize the dynamical changes derived from kinetically different PER translational mechanisms. We found that the emergence of self-sustained oscillations with characteristic period, amplitude, and phase lag (time delays) between per mRNA and protein expression depends on the kinetic parameters related to PER translation. Interestingly, under certain conditions, a PER translation mechanism with saturable kinetics introduces longer time delays than a mechanism ruled by a first-order kinetics. In addition, the kinetic laws of PER translation significantly changed the sensitivity of our model to parameters related to the synthesis and degradation of per mRNA and PER degradation. Lastly, we found a set of parameters, with realistic values, for which our model reproduces some experimental results reported recently for Drosophila melanogaster and we present some predictions derived from our analysis.     

Membrane electrical excitability is necessary for the free-running larval Drosophila circadian clock

Drosophila larvae and adult pacemaker neurons both express free-running oscillations of period (PER) and timeless (TIM) proteins that constitute the core of the cell-autonomous circadian molecular clock. Despite similarities between the adult and larval molecular oscillators, adults and larvae differ substantially in the complexity and organization of their pacemaker neural circuits, as well as in behavioral manifestations of circadian rhythmicity. We have shown previously that electrical silencing of adult Drosophila circadian pacemaker neurons through targeted expression of either an open rectifier or inward rectifier K(+) channel stops the free-running oscillations of the circadian molecular clock. This indicates that neuronal electrical activity in the pacemaker neurons is essential to the normal function of the adult intracellular clock. In the current study, we show that in constant darkness the free-running larval pacemaker clock-like that of the adult pacemaker neurons they give rise to-requires membrane electrical activity to oscillate. In contrast to the free-running clock, the molecular clock of electrically silenced larval pacemaker neurons continues to oscillate in diurnal (light-dark) conditions. This specific disruption of the free-running clock caused by targeted K(+) channel expression likely reflects a specific cell-autonomous clock-membrane feedback loop that is common to both larval and adult neurons, and is not due to blocking pacemaker synaptic outputs or disruption of pacemaker neuronal morphology.