What is Phase in Cellular Clocks?

Four inter-related measures of phase are described to study the phase synchronization of cellular oscillators, and computation of these measures is described and illustrated on single cell fluorescence data from the model filamentous fungus, Neurospora crassa. One of these four measures is the phase shift ϕ in a sinusoid of the form x(t) = A(cos(ωt + ϕ), where t is time. The other measures arise by creating a replica of the periodic process x(t) called the Hilbert transform x̃(t), which is 90 degrees out of phase with the original process x(t). The second phase measure is the phase angle FH(t) between the replica x̃(t) and x(t), taking values between -π and π. At extreme values the Hilbert Phase is discontinuous, and a continuous form FC(t) of the Hilbert Phase is used, measuring time on the nonnegative real axis (t). The continuous Hilbert Phase FC(t) is used to define the phase MC(t1,t0) for an experiment beginning at time t0 and ending at time t1. In that phase differences at time t0 are often of ancillary interest, the Hilbert Phase FC(t0) is subtracted from FC(t1). This difference is divided by 2π to obtain the phase MC(t1,t0) in cycles. Both the Hilbert Phase FC(t) and the phase MC(t1,t0) are functions of time and useful in studying when oscillators phase-synchronize in time in signal processing and circadian rhythms in particular. The phase of cellular clocks is fundamentally different from circadian clocks at the macroscopic scale because there is an hourly cycle superimposed on the circadian cycle.

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