Calculating activation energies for temperature compensation in circadian rhythms

Many biological species possess a circadian clock, which helps them anticipate daily variations in the environment. In the absence of external stimuli, the rhythm persists autonomously with a period of approximately 24 h. However, single pulses of light, nutrients, chemicals or temperature can shift the clock phase. In the case of light- and temperature-cycles, this allows entrainment of the clock to cycles of exactly 24 h. Circadian clocks have the remarkable property of temperature compensation, that is, the period of the circadian rhythm remains relatively constant within a physiological range of temperatures. For several organisms, temperature-regulated processes within the circadian clock have been identified in recent years. However, how these processes contribute to temperature compensation is not fully understood. Here, we theoretically investigate temperature compensation in general oscillatory systems. It is known that every oscillator can be locally temperature compensated around a reference temperature, if reactions are appropriately balanced. A balancing is always possible if the control coefficient with respect to the oscillation period of at least one reaction in the oscillator network is positive. However, for global temperature compensation, the whole physiological temperature range is relevant. Here, we use an approach which leads to an optimization problem subject to the local balancing principle. We use this approach to analyse different circadian clock models proposed in the literature and calculate activation energies that lead to temperature compensation.     

Temperature dependence and temperature compensation of kinetics of chemical oscillations; Belousov-Zhabotinskii reaction, glycolysis and circadian rhythms

Based on the distribution of activation energies around the experimental mean and averaging of rate constants we propose a theoretical scheme to examine the temperature dependence and temperature compensation of time periods of chemical oscillations. The critical finite width of the distribution is characteristic of endogeneous oscillations for compensating kinetics as observed in circadian oscillations, while the vanishing width corresponds to Arrhenius temperature dependent kinetics of non-endogeneous chemical oscillation in Belousov-Zhabotinskii reaction in a CSTR or glycolysis in cell-free yeast extracts. Our theoretical analysis is corroborated with experimental data.     

Temperature dependency and temperature compensation in a model of yeast glycolytic oscillations

Temperature sensitivities and conditions for temperature compensation have been investigated in a model for yeast glycolytic oscillations. The model can quantitatively simulate the experimental observation that the period length of glycolytic oscillations decreases with increasing temperature. Temperature compensation is studied by using control coefficients describing the effect of rate constants on oscillatory frequencies. Temperature compensation of the oscillatory period is observed when the positive contributions to the sum of products between control coefficients and activation energies balance the corresponding sum of the negative contributions. The calculations suggest that by changing the activation energies for one or several of the processes, i.e. by mutations, it could be possible to obtain temperature compensation in the yeast glycolytic oscillator.     

Loss of temperature compensation of circadian period length in the frq-9 mutant of Neurospora crassa

A new circadian clock mutant of Neurospora crassa has been isolated, whose most distinctive characteristic is the complete loss of temperature compensation of its period length. The Q10 of the period length was found to be equal to about 2 in the temperature range from 18 degrees to 30 degrees C. The period length was also found to be dependent on the composition of the medium, including the nature and concentration of both the carbon source and the nitrogen source. Although the rate of the clock and the growth rate were directly related when affected by varying the temperature, they were inversely related when altered by changing the composition of the medium. Therefore, the mutation has not simply coupled clock rate to growth rate in this strain. The mutation maps to the frq locus, where seven other clock mutations previously studied also map. Therefore, this mutant has been called frq-9. Since several of the other frq mutants show partial loss in temperature compensation, it is suggested that the frq gene or its product is closely related to the temperature compensation mechanism of the circadian clock of Neurospora.     

Adaptive significance of circadian clocks

Circadian clocks are ubiquitous and are found in organisms ranging from bacteria to mammals. This ubiquity of occurrence implies adaptive significance, but to date there has been no rigorous empirical evidence to support this. It is believed that an organism possessing circadian clocks gains fitness advantage in two ways: (i) by synchronizing its behavioral and physiological processes to cyclic environmental factors (extrinsic adaptive value); (ii) by coordinating its internal metabolic processes (intrinsic adaptive value). There is preliminary circumstantial evidence to support both. Several studies using organisms living in constant environments have shown that these organisms possess functional circadian clocks, suggesting that circadian clocks may have some intrinsic adaptive value. Studies to assess the adaptive value of circadian clocks in periodic environments suggest that organisms may have a fitness advantage in those periodic environments, which closely match their own intrinsic periodicity. Furthermore, evidence from organisms living in the wild, selection studies, and studies on latitudinal clines suggest that circadian clocks may have an extrinsic adaptive value as well. In this paper, I have presented several hypotheses for the emergence of circadian clocks and have reviewed some major empirical studies suggesting adaptive significance of circadian clocks.     

The amplitude of circadian oscillations: temperature dependence, latitudinal clines, and the photoperiodic time measurement.

This paper develops several propositions concerning the lability of the amplitude of Drosophila circadian pacemakers. The first is that the amplitude of the pacemaker's motion, unlike its period, is markedly temperature-dependent. The second is that latitudinal variation in pacemaker amplitude (higher in the north) is responsible for two very different sets of observations on Drosophila circadian systems at successively higher latitudes. One of these is a cline in D. auraria's phase-shifting response to light, which steadily weakens in a succession of more northerly strains. The other, concerning D. littoralis in the very far north, is a cline in the rate at which eclosion activity becomes arrhythmic (the circadian rhythm damps out) in constant darkness; damping is faster in the north. The third proposition concerns a plausible selection pressure for the cline in pacemaker amplitude that we propose underlies the two directly observed clines. Two points are emphasized: (1) The amplitude of the pacemaker's daily oscillation declines as the duration of the entraining light pulse (photoperiod) is increased; and (2) the duration of the daily photoperiods throughout the breeding season is steadily increased as one moves toward the poles. Selection for conservation of pacemaker amplitude (during the breeding season) would produce the latitudinal cline we propose. The fourth, and final proposition is that since the amplitude of the pacemaker's daily motion responds systematically to change in photoperiod, amplitude is clearly one way--and a temperature-dependent way--in which insect circadian systems may sense seasonal change. These propositions concerning the temperature and latitude dependence of pacemaker amplitude may be relevant to a wider array of circadian pacemakers than Drosophila.     

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.