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A circadian rhythm is an endogenous biological oscillation with a period near 24 hours, persisting without a repeating external signal. A circadian clock times biological activities and anticipates daily changes without continual external instructions. A free-running circadian rhythm reveals its intrinsic period; entrainment of circadian rhythm coordinates it with an external cycle.^[ref]

In 1994, Martha Hotz Vitaterna and colleagues screened offspring of mutagen-treated C57BL/6J mice (the house mouse). Each mouse lived in a cage with a running wheel, inside a ventilated, light-tight chamber with timer-controlled lighting. Wheel revolutions were recorded electronically or on an event recorder. The initial screen used 14:10 LD for the first 41 animals and 12:12 LD thereafter; LD denotes hours of light and darkness. Transfer to constant darkness, DD, omitted the next scheduled lights-on; recording continued for at least three weeks. The first 41 animals also received brief light pulses, so those records were not uninterrupted DD.^[ref]

The subsequent-generation comparison in Figure 2 used 12:12 LD before DD. Animals without the Clock mutation and with mutant genotypes supplied the comparison: the B6 examples in panels A–C had initial DD periods of 23.6, 24.8 and 27.1 hours, respectively; the homozygote later lost its sustained circadian rhythm. Total activity remained comparable across genotypes in the interstrain comparison. The mutation therefore altered timing, beyond merely reducing running. Wheel activity is a behavioral readout; it does not directly measure every circadian clock or establish a human period.^[ref]

Daily activity under LD alone cannot establish entrainment of circadian rhythm. Light can directly change activity without resetting the circadian clock: masking of circadian rhythm. Removing the light cycle helps distinguish these effects; they can coexist.^[ref] In an experiment, other repeating inputs must also be controlled: a nominally dark room with daily handling is not automatically an environment without timing cues.

Entrainment of circadian rhythm synchronizes a circadian clock to a repeating cue, a zeitgeber. One-to-one entrainment of circadian rhythm gives one cycle per environmental cycle with a stable relative circadian phase.^[ref] The stable relation need not put an activity onset exactly at lights-off: a constant offset also expresses phase locking.

For an original local model, let $x_n$ be the activity-onset offset from lights-off on environmental cycle $n$, with a negative value meaning an earlier onset. Choose an intrinsic period $\tau =23.5\mathrm{h}$ and environmental period $T=24\mathrm{h}$. Let $D(x)$ be a stipulated daily delay, measured in hours; positive $D$ moves the next onset later. The recurrence relation isA fixed phase relationship $x_{\ast }$ requires $D(x_{\ast })=T-\tau$: here a half-hour delay balances the half-hour intrinsic advance each cycle. This equality alone describes balance; restoration after a small disturbance also requires fixed point stability for an iteration.

Choose the illustrative local law $D(x)=0.5\mathrm{h}-0.4x$. Thenfor the chosen starting offset $x_0=-3\mathrm{h}$. The error shrinks by a factor of $0.6$ each cycle, proving fixed point stability for an iteration in this model. After five cycles $x_5=-0.23328\mathrm{h}$, about fourteen minutes early. At the fixed point, the imposed delay is $0.5\mathrm{h}$ and the corrected onset interval is $\tau +D(x_{\ast })=24\mathrm{h}$, although the uncoupled period remains $23.5\mathrm{h}$.

The uncorrected comparison has $D=0$ and $x_n=-3\mathrm{h}+n(\tau -T)$, so after five cycles it is $5.5$ hours early. The two traces separate ongoing timekeeping from adjustment to a zeitgeber. The chosen linear delay law is only a local teaching construction: it is not a measured response to light, does not account for wrapping of circadian phase, and does not predict an organism's entrainment range. Nor does it include masking of circadian rhythm. In a real study, tracking several cycles after removal of the cue or after a timed perturbation supplies information that a single activity value under illumination cannot provide.

In masking of circadian rhythm, an external condition changes the observed output without necessarily changing the underlying circadian clock.

Plants use an entrained circadian clock to change their sensitivity to stimuli, while also responding directly to the current light and temperature environment. The same external stimulus can therefore have different effects at different internal phases. This circadian gating must be distinguished from a daily rhythm caused solely by daily variation in the stimulus itself.

The plant circadian clock contains interlocking regulation of gene expression, protein accumulation and protein activity. Morning-associated regulators such as CCA1 and LHY interact with later-acting components, including TOC1, to generate a sequence of phases rather than a single daily on–off switch. Light information, including inputs involving phytochromes and cryptochromes, and temperature cues adjust the clock through entrainment of circadian rhythm. Under suitably constant conditions, a free-running circadian rhythm persists. Anticipation of dawn or dusk is therefore possible before the corresponding environmental transition occurs.

Clock outputs can modify several points in a stimulus-response pathway. A receptor or transport protein may accumulate at a particular phase; signaling inhibitors may be removed at another. The clock can regulate hormone production, hormone inactivation, transport and sensitivity of downstream transcription factors. Post-translational regulation allows a phase-specific transcript to produce an output only under a second permissive condition. Consequently, a rhythmic hormone concentration and rhythmic responsiveness to that hormone are distinct explanations and may coexist.

Auxin provides direct experimental evidence for sensitivity gating. Timed auxin treatments and transcriptional reporters in Arabidopsis showed that an externally supplied hormone stimulus produces phase-dependent responses; growth responses were also gated. Because the stimulus was supplied experimentally, the result cannot be explained merely by assuming that the plant normally makes more auxin at one time of day. Reporter dynamics and pretreatment baselines still require appropriate controls, and a reporter is not a direct measurement of free hormone.^[ref]

An illustrative local description iswhere $S$ is a small standardized stimulus, $\varphi$ is circadian phase and $\Delta R$ is the response above the matched untreated baseline. A phase-dependent gain $g$ represents gating. In contrast, if $g$ is constant but the baseline response oscillates, observing different absolute values after treatment does not establish gating. Real pathways can saturate or change sign, so this equation illustrates the experimental distinction rather than defining a universal linear law.

Photoperiodism shows how internal time can be combined with an external cue to produce a developmental decision. In Arabidopsis, timed expression of CONSTANS interacts with light-dependent regulation of the protein. In long days, a permissive expression phase overlaps light sufficiently to support FLOWERING LOCUS T activation. The external coincidence model therefore responds to when light occurs relative to the clock, rather than merely adding up photons. Photoreceptor-dependent protein regulation provides a molecular connection between timing and the external environment.^[ref] FT protein can then communicate from leaves to the shoot apex. Phloem-specific expression and grafting experiments support FT protein movement as part of this long-distance floral signal.^[ref]

A convincing demonstration of circadian gating compares matched stimuli at several internal phases under constant external conditions, with untreated controls at every phase. Plants entrained to opposite schedules can help distinguish internal phase from laboratory clock time. Clock perturbations and restoration of clock function strengthen causal interpretation, provided effects on general health and growth are considered. A phase-shifted response in constant conditions supports clock control; disappearance immediately when the environmental cycle is removed suggests direct environmental control, although weak rhythms and altered physiology can complicate that inference. Masking of circadian rhythm, in which an environmental cue directly changes a measured output, can coexist with clock control.

Timing is useful because demands and risks recur. Coordinating stomatal conductance, photosynthetic preparation and growth with expected light availability can improve resource use. Yet the optimum phase depends on temperature, water availability and day length, and a strong acute stress can override a normal daily preference. Plants combine an internal estimate of time with current conditions, changing pathway sensitivity as well as the abundance of signals and response machinery.