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

Andrey Lazopulo, Sheyum Syed

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. 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. 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.