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Circadian time (/sɜːrˈkeɪdiən/), denoted as CT, is a form of subjective time derived from the free-running period of the endogenous circadian rhythm of an organism. A free-running period is the length of time it takes for an organism's endogenous rhythm to repeat (return to the same phase) in the absence of environmental time cues. Circadian time is distinct from zeitgeber time, denoted as ZT, which is an exogenous timing system driven by rhythms in the external environment, such as Earth’s 24-hour light/dark cycle. While it is known that circadian pacemaker cells are capable of tracking time, the exact molecular mechanisms that allow these cells to do so are still being investigated.

History
In 1729, Jean-Jacques d’Ortous de Mairan observed that the daily rhythms in leaf movement of Mimosa pudica plants persist in constant dark (DD) conditions. In 1832, Augustin Pyrame de Candolle observed that the mimosa plants had a free-running period close to 24 hours, indicating that the rhythm was circadian in nature. Though it is unknown exactly when the term "circadian time" was first used in the literature, Mairan and Candolle's experiments were some of the first instances in which organisms' endogenous timekeeping systems were observed to be distinct from standard time.

Circadian time versus zeitgeber time
There are several important differences between circadian time and zeitgeber time.

Mathematics of circadian time
Circadian time uses the same "hours:minutes:seconds" units that conventional time systems use. However, since circadian time is expressed using a 24-hour scale, a circadian hour could be longer or shorter than a standard hour if the intrinsic free-running period of the organism is not exactly 24 hours long. For example, if the intrinsic period of an organism is 23 hours long, then the organism will experience 24 circadian hours in the same duration as 23 standard hours, so one circadian hour will be equal to approximately 0.96 standard hours.

The length in standard hours of one hour in circadian time can be found using the following equation:

$$Length \ of \ 1 \ circadian \ hour \ (in \ standard \ hours) \ = \ Length \ of \ 1 \ intrinsic \ period \ (in \ standard \ hours) \ / \ 24 $$

Circadian time is typically rolled-over after CT23 back to CT0, though this is not necessary. For example, a time of 26 hours after CT0 could be denoted as either CT2 or CT26.

Entrainment and shifting of circadian time
An organism is considered to be free-running when it is in constant conditions, often constant dim light, and not under the influence of any zeitgebers. Entrainment occurs when a zeitgeber is used to synchronize the endogenous, circadian rhythms of the organism to the exogenous, environmental rhythms by manipulating the organism's phase and period.

Circadian time is often used synonymously with the term "phase" to describe reference points in daily endogenous rhythms. Phase response curves (PRC) are graphical representations of the effects of timed perturbations, often light pulses or transient temperature shifts, on the phase of an organism's rhythm. The phase effect of a perturbation on the organism's rhythm depends on the circadian time at which the perturbation occurs. Phase response curves can be used to determine how to effectively entrain an organism to any period within its range of entrainment through the use of phase shifts (advances and/or delays in the organism's phase). Perturbations during the early subjective night tend to result in a delay in the phase of an organism's circadian rhythms. Perturbations during the late subjective night tend to result in an advance in the phase of an organism's circadian rhythms.

Circadian time is a property of individual cells and/or sub-cellular structures, not a property of the organism as a whole. For example, it has been shown that liver cells prefer to entrain to food zeitgebers while suprachiasmatic nucleus (SCN) cells prefer to entrain to photic zeitgebers. As a result, if the rhythms of the food and photic zeitgebers are out of phase, it is possible for circadian molecular rhythms within SCN cells to be out of phase with the circadian molecular rhythms within the liver cells, demonstrating that the two types of cells would not be perceiving the same circadian time.

Example of circadian time entrainment
In this example, the organism has an intrinsic period of 24 hours and is being entrained to a 20-hour T-cycle (i.e. a light pulse occurs once every 20 hours). Since the intrinsic free-running period of the organism is 24 hours, the length of one circadian hour in this example is simply equal to one standard hour.

The phase response curve (PRC) shown below is used to represent the effects of timed perturbations on this organism's phase. On the x-axis, the CT at which the perturbation was administered is shown. On the y-axis, the resulting phase shift is shown, symbolized by Δф. Phase advances are positive and phase delays are negative.

In this example, the organism experiences the first light pulse at CT15 (other times can also be used as a starting point in these calculations). This perturbation causes a 4-hour phase delay, found using the PRC below, which instantaneously shifts the organism's circadian time. The organism now perceives its circadian time to be CT11. The next light pulse is administered after 20 standard hours, at CT7, causing no phase shift. The light pulses continue occurring once every 20 hours until the the light pulse falls at the same phase every cycle, indicating entrainment of the circadian rhythm. In this example, the light pulse occurs at CT22 every cycle, resulting in the same 4-hour phase advance each time, to entrain the organism.

This example illustrates how circadian time is subjective and can be manipulated using exogenous forces. In contrast, standard time is steady in its 24-hour cycle.