In summary, we
found that Ruoff's equation is not robust to mutation if it requires
delicate balancing of many rate constants in a limit cycle model for
the circadian rhythm mechanism. We propose that temperature
compensation and other indicators of the robustness of circadian period
to genetic variation are more likely the results of a molecular
mechanism for which only a few control parameters significantly affect
the period of oscillation, and we suggest a resetting hypothesis as a
candidate mechanism. Resetting works by moving an effective rate
constant back and forth across a SNIC bifurcation. SNIC bifurcations
are common features of regulatory networks with both positive and
negative feedback loops, of which the circadian machinery is richly
endowed. In general, many different rate constants in the mechanism can
serve the resetting role.
modeling presumes that circadian rhythm properties such as robust 24-h
period and temperature compensation are determined at the level of
single pacesetting cells. We have not considered any role for
intercellular communication in determining the period or the
temperature-independence of circadian rhythms. That circadian period
mutants commonly leave temperature compensation intact (Table 1)
may reflect a difference in levels of organization for these
properties. For example, oscillator period may be
temperature-compensated at the cellular level exactly as proposed by
Ruoff and colleagues, but the overt period of the rhythm in a
multicellular organism may be determined, in addition, by intercellular
couplings that are insensitive to temperature changes. If that were the
case, then mutations affecting intracellular interactions might change
the period without upsetting temperature compensation. Although this
alternative explanation may apply to Drosophila, it is unlikely for Neurospora.
oscillatory period of a cell-autonomous, limit-cycle model, based on
detailed biochemical interactions among circadian genes and proteins,
is a complicated function of all of the rate constants in the
mechanism. Because reaction rate constants increase rapidly with
temperature, the period-lengthening and period-shortening effects of
the parameters must be delicately balanced to achieve temperature
compensation. Consequently, temperature-independence of circadian
period (in this paradigm) should be fragile with respect to mutation.
By contrast, our resetting hypothesis concentrates all of the
period-determining effects on just two parameters (μ and σ), which
makes temperature compensation easier to achieve. Although the temporal
dynamics of the underlying reactions are still strongly temperature
dependent (within the resetting paradigm), the control system switches
back and forth between the domain of attraction of a stable steady
state and the domain of attraction of a stable limit cycle. As
temperature changes, any alterations in the relative timing of events
in the limit-cycle region are made up for by compensatory changes in
the time spent under attraction of the stable steady state. The 24-h
period is determined solely by the rules for switching between the two
the other hand, the resetting hypothesis may appear to be too robust:
only mutations that alter μ and σ impinge significantly on period and
temperature compensation. We are not proposing that the circadian
rhythm mechanism is such a simple process that only two parameters
dictate the period of the system. We suppose that, in reality, μ and σ
are functions of other molecular processes (phosphorylation,
ubiquitination, complex formation, etc.) and that mutations that
disrupt any of these processes may interfere with temperature
The resetting hypothesis makes a number of testable predictions.
many reasons independent of our theory, it seems reasonable to do a
thorough screen for genetic mutations that disrupt temperature
compensation. Are such mutations common and broadly distributed across
the circadian rhythm regulatory network, as the limit cycle hypothesis
would suggest, or are they rare and concentrated among a few components
of the network, as the resetting hypothesis would suggest?
Resetting requires a dynamic system with both positive and negative feedback loops that operates in a region of parameter
space exhibiting both multiple steady states (bistability) and limit cycle oscillations (see Fig. 1).
Hysteresis in our model relies on the autocatalytic increase of PER
based on its homodimerization and stabilization. One could test this
assumption directly by measuring the half-lives of monomeric PER (by
disrupting the PAS-binding domain) compared with PER–PER complexes. One
could also test for hysteresis directly, along the same lines that
proved successful in demonstrating bistability in the mitotic control
of frog eggs (38, 39), in a per-null mutant with the wild-type per
gene under the control of an inducible promoter (e.g., the Tet On/Off
system). When PER synthesis is ramped up from low rates, there should
be an abrupt increase in the PER expression level at a certain
threshold synthesis rate. Once the system is in the PER-high state, it
will stay there as the PER synthesis rate is ramped back down, until it
falls below a lower threshold synthesis rate for turning the bistable
bistability can be demonstrated in the circadian rhythm control system,
then it is likely on theoretical grounds that if the positive feedback
loop is genetically severed, then oscillations continue with shorter
period and smaller amplitude. This effect has been observed
experimentally in the analogous case for M-phase promoting factor in
frog egg extracts (40).
research was initiated at the Collegium Budapest, Hungary, with
financial support from the Santa Fe Institute, the Volkswagen Stiftung,
and the Defense Advanced Research Project Agency (AFRL 30602-02-0572).
- §To whom correspondence should be addressed. E-mail:
Author contributions: C.I.H. and E.D.C. contributed equally to this work; J.J.T. designed research; C.I.H. and E.D.C. performed
research; and C.I.H., E.D.C., and J.J.T. wrote the paper.
¶In test B, a small percentage of perturbations of some key parameters (Pcrit and Jp)
of the RS model lead to complex rhythmic oscillations that are not
exactly periodic but still “circadian,” i.e., the oscillations are
almost periodic, with a repeat interval of ≈24 h, and with peaks and
troughs varying up to ≈5%. These cases were considered to be
arrhythmic, to avoid any bias for the resetting model over limit cycle
models, neither of which is capable of such complex behavior.
The authors declare no conflict of interest.
This article is a PNAS direct submission.
This article contains supporting information online at www.pnas.org/cgi/content/full/0601378104/DC1.
The authors declare no conflict of interest.
- coefficient of variation.
Freely available online through the PNAS open access option.
- © 2007 by The National Academy of Sciences of the USA