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A mathematical model provides mechanistic links to temporal patterns in Drosophila daily activity.

Lazopulo A, Syed S - BMC Neurosci (2016)

Bottom Line: In the time domain, we find the timescales of the exponentials in our model to be ~1.5 h(-1) on average.Our results indicate that multiple spectral peaks from fly locomotion are simply harmonics of the circadian period rather than independent ultradian oscillators as previously reported.From timescales of the exponentials we hypothesize that model rates reflect activity of the neuropeptides that likely transduce signals of the circadian clock and the sleep-wake homeostat to shape behavioral outputs.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, University of Miami, 1320 Campo Sano Avenue, Coral Gables, FL, 33146, USA.

ABSTRACT

Background: Circadian clocks are endogenous biochemical oscillators that control daily behavioral rhythms in all living organisms. In fruit fly, the circadian rhythms are typically studied using power spectra of multiday behavioral recordings. Despite decades of study, a quantitative understanding of the temporal shape of Drosophila locomotor rhythms is missing. Locomotor recordings have been used mostly to extract the period of the circadian clock, leaving these data-rich time series largely underutilized. The power spectra of Drosophila and mouse locomotion often show multiple peaks in addition to the expected at T ~ 24 h. Several theoretical and experimental studies have previously used these data to examine interactions between the circadian and other endogenous rhythms, in some cases, attributing peaks in the T < 24 h regime to ultradian oscillators. However, the analysis of fly locomotion was typically performed without considering the shape of time series, while the shape of the signal plays important role in its power spectrum. To account for locomotion patterns in circadian studies we construct a mathematical model of fly activity. Our model allows careful analysis of the temporal shape of behavioral recordings and can provide important information about biochemical mechanisms that control fly activity.

Results: Here we propose a mathematical model with four exponential terms and a single period of oscillation that closely reproduces the shape of the locomotor data in both time and frequency domains. Using our model, we reexamine interactions between the circadian and other endogenous rhythms and show that the proposed single-period waveform is sufficient to explain the position and height of >88 % of spectral peaks in the locomotion of wild-type and circadian mutants of Drosophila. In the time domain, we find the timescales of the exponentials in our model to be ~1.5 h(-1) on average.

Conclusions: Our results indicate that multiple spectral peaks from fly locomotion are simply harmonics of the circadian period rather than independent ultradian oscillators as previously reported. From timescales of the exponentials we hypothesize that model rates reflect activity of the neuropeptides that likely transduce signals of the circadian clock and the sleep-wake homeostat to shape behavioral outputs.

No MeSH data available.


Model parameters are obtained by fitting the power spectrum. a Examples of data power spectrum (black line) and fit (red diamonds) for a wild type fly measured in LD (top) and DD (bottom). In the LD data, the 12 h peak is stronger than the 24 h peak and was therefore used as the reference peak in the fit. Comparison of model and data power spectra for low  shown in the inset. For low values of the period, the model predicts peak height  (inset, dashed line), a behavior not shown by the data. b Errors in peak height estimation from 10 flies (mean error ± standard deviation). On average, peak heights are estimated with error of 10 % (dashed lines) or better. c Parameters obtained from a are used to construct model of fly activity. Data (black line) shown in 20 min bins; day/night shown with white/black bars for LD and subjective day/night shown with white/grey bars for DD
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Fig4: Model parameters are obtained by fitting the power spectrum. a Examples of data power spectrum (black line) and fit (red diamonds) for a wild type fly measured in LD (top) and DD (bottom). In the LD data, the 12 h peak is stronger than the 24 h peak and was therefore used as the reference peak in the fit. Comparison of model and data power spectra for low shown in the inset. For low values of the period, the model predicts peak height (inset, dashed line), a behavior not shown by the data. b Errors in peak height estimation from 10 flies (mean error ± standard deviation). On average, peak heights are estimated with error of 10 % (dashed lines) or better. c Parameters obtained from a are used to construct model of fly activity. Data (black line) shown in 20 min bins; day/night shown with white/black bars for LD and subjective day/night shown with white/grey bars for DD

Mentions: We next determined the exponents  −  for wild-type flies in DD and LD. In our model, these exponents define the shape of the M and E peaks. The model parameters were obtained from the power spectra of activity data (Fig. 4a). Spectra were fitted with an analytical expression obtained by calculating the square of the Fourier transform of . The square of the Fourier transform yields peak heights at harmonics of the primary period [see Additional file 1: equations (5)–(8)]. In DD data, was determined from the peak at the circadian frequency. Since the fitting procedure is sensitive to the initial choice of parameters, as an initial guess we used parameters from a preliminary fitting of activity data with the model. The exponents obtained from the fits shown in Fig. 4 are , , , and for LD, and , , , and for DD. The analytical expression also produced good fits for the activity power spectra with the average peak height fit error of less than 10 % (Fig. 4b) (see “Methods” for details). Final values of the parameters determined from fitting the power spectrum were then used to construct a model for the activity recordings (Fig. 4c). For the measured wild-type flies, the constructed model shows good fit for locomotor activity with the average rate constant magnitude ~1.2 ± 2 h−1 (mean ± standard deviation). The parameters and obtained from fitting data can be either positive or negative, which means that the exponential terms with and in can be either concave or convex. Interestingly, we find that the parameters and are always greater than 0 and therefore corresponding terms are always concave. It should be noted here that our model does not impose any restrictions on the numerical values of the parameters.Fig. 4


A mathematical model provides mechanistic links to temporal patterns in Drosophila daily activity.

Lazopulo A, Syed S - BMC Neurosci (2016)

Model parameters are obtained by fitting the power spectrum. a Examples of data power spectrum (black line) and fit (red diamonds) for a wild type fly measured in LD (top) and DD (bottom). In the LD data, the 12 h peak is stronger than the 24 h peak and was therefore used as the reference peak in the fit. Comparison of model and data power spectra for low  shown in the inset. For low values of the period, the model predicts peak height  (inset, dashed line), a behavior not shown by the data. b Errors in peak height estimation from 10 flies (mean error ± standard deviation). On average, peak heights are estimated with error of 10 % (dashed lines) or better. c Parameters obtained from a are used to construct model of fly activity. Data (black line) shown in 20 min bins; day/night shown with white/black bars for LD and subjective day/night shown with white/grey bars for DD
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License 1 - License 2
Show All Figures
getmorefigures.php?uid=PMC4835852&req=5

Fig4: Model parameters are obtained by fitting the power spectrum. a Examples of data power spectrum (black line) and fit (red diamonds) for a wild type fly measured in LD (top) and DD (bottom). In the LD data, the 12 h peak is stronger than the 24 h peak and was therefore used as the reference peak in the fit. Comparison of model and data power spectra for low shown in the inset. For low values of the period, the model predicts peak height (inset, dashed line), a behavior not shown by the data. b Errors in peak height estimation from 10 flies (mean error ± standard deviation). On average, peak heights are estimated with error of 10 % (dashed lines) or better. c Parameters obtained from a are used to construct model of fly activity. Data (black line) shown in 20 min bins; day/night shown with white/black bars for LD and subjective day/night shown with white/grey bars for DD
Mentions: We next determined the exponents  −  for wild-type flies in DD and LD. In our model, these exponents define the shape of the M and E peaks. The model parameters were obtained from the power spectra of activity data (Fig. 4a). Spectra were fitted with an analytical expression obtained by calculating the square of the Fourier transform of . The square of the Fourier transform yields peak heights at harmonics of the primary period [see Additional file 1: equations (5)–(8)]. In DD data, was determined from the peak at the circadian frequency. Since the fitting procedure is sensitive to the initial choice of parameters, as an initial guess we used parameters from a preliminary fitting of activity data with the model. The exponents obtained from the fits shown in Fig. 4 are , , , and for LD, and , , , and for DD. The analytical expression also produced good fits for the activity power spectra with the average peak height fit error of less than 10 % (Fig. 4b) (see “Methods” for details). Final values of the parameters determined from fitting the power spectrum were then used to construct a model for the activity recordings (Fig. 4c). For the measured wild-type flies, the constructed model shows good fit for locomotor activity with the average rate constant magnitude ~1.2 ± 2 h−1 (mean ± standard deviation). The parameters and obtained from fitting data can be either positive or negative, which means that the exponential terms with and in can be either concave or convex. Interestingly, we find that the parameters and are always greater than 0 and therefore corresponding terms are always concave. It should be noted here that our model does not impose any restrictions on the numerical values of the parameters.Fig. 4

Bottom Line: In the time domain, we find the timescales of the exponentials in our model to be ~1.5 h(-1) on average.Our results indicate that multiple spectral peaks from fly locomotion are simply harmonics of the circadian period rather than independent ultradian oscillators as previously reported.From timescales of the exponentials we hypothesize that model rates reflect activity of the neuropeptides that likely transduce signals of the circadian clock and the sleep-wake homeostat to shape behavioral outputs.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, University of Miami, 1320 Campo Sano Avenue, Coral Gables, FL, 33146, USA.

ABSTRACT

Background: Circadian clocks are endogenous biochemical oscillators that control daily behavioral rhythms in all living organisms. In fruit fly, the circadian rhythms are typically studied using power spectra of multiday behavioral recordings. Despite decades of study, a quantitative understanding of the temporal shape of Drosophila locomotor rhythms is missing. Locomotor recordings have been used mostly to extract the period of the circadian clock, leaving these data-rich time series largely underutilized. The power spectra of Drosophila and mouse locomotion often show multiple peaks in addition to the expected at T ~ 24 h. Several theoretical and experimental studies have previously used these data to examine interactions between the circadian and other endogenous rhythms, in some cases, attributing peaks in the T < 24 h regime to ultradian oscillators. However, the analysis of fly locomotion was typically performed without considering the shape of time series, while the shape of the signal plays important role in its power spectrum. To account for locomotion patterns in circadian studies we construct a mathematical model of fly activity. Our model allows careful analysis of the temporal shape of behavioral recordings and can provide important information about biochemical mechanisms that control fly activity.

Results: Here we propose a mathematical model with four exponential terms and a single period of oscillation that closely reproduces the shape of the locomotor data in both time and frequency domains. Using our model, we reexamine interactions between the circadian and other endogenous rhythms and show that the proposed single-period waveform is sufficient to explain the position and height of >88 % of spectral peaks in the locomotion of wild-type and circadian mutants of Drosophila. In the time domain, we find the timescales of the exponentials in our model to be ~1.5 h(-1) on average.

Conclusions: Our results indicate that multiple spectral peaks from fly locomotion are simply harmonics of the circadian period rather than independent ultradian oscillators as previously reported. From timescales of the exponentials we hypothesize that model rates reflect activity of the neuropeptides that likely transduce signals of the circadian clock and the sleep-wake homeostat to shape behavioral outputs.

No MeSH data available.