A new algorithm to accelerate harmonic analysis of time series

The Lomb periodogram has been traditionally a tool that allows us to elucidate if a frequency turns out to be important for explaining the behaviour of a given time series. Many linear and nonlinear reiterative harmonic processes that are used for studying the spectral content of a time series take into account this periodogram in order to avoid including spurious frequencies in their models due to the leakage problem of energy from one frequency to others. However, the estimation of the periodogram requires long computation time that makes the harmonic analysis slower when we deal with certain time series. Here we propose an algorithm that accelerates the extraction of the most remarkable frequencies from the periodogram, avoiding its whole estimation of the harmonic process at each iteration. This algorithm allows the user to perform a specific analysis of a given scalar time series. As a result, we obtain a functional model made of (1) a trend component, (2) a linear combination of Fourier terms, and (3) the so-called mixed secular terms by reducing the computation time of the estimation of the periodogram.

Detecting KaiC Phosphorylation Rhythms of the Cyanobacterial Circadian Oscillator In Vitro and In Vivo

The central oscillator of the cyanobacterial circadian clock is unique in the biochemical simplicity of its components and the robustness of the oscillation. The oscillator is composed of three cyanobacterial proteins: KaiA, KaiB, and KaiC. If very pure preparations of these three proteins are mixed in a test tube in the right proportions and with ATP and MgCl2, the phosphorylation states of KaiC will oscillate with a circadian period, and these states can be analyzed simply by SDS-PAGE. The purity of the proteins is critical for obtaining robust oscillation. Contaminating proteases will destroy oscillation by degradation of Kai proteins, and ATPases will attenuate robustness by consumption of ATP. Here, we provide a detailed protocol to obtain pure recombinant proteins from Escherichia coli to construct a robust cyanobacterial circadian oscillator in vitro. In addition, we present a protocol that facilitates analysis of phosphorylation states of KaiC and other phosphorylated proteins from in vivo samples.