A Simple Model.
To illustrate the resetting hypothesis, we use an overly simplified model (Appendix) of the circadian rhythm control system (10),
to focus on fundamental ideas rather than to be distracted by
mechanistic details. The fundamental idea behind the resetting
hypothesis is based on a generic bifurcation diagram and is in no way
dependent on special features of the illustrative model. In SI,
we present a more complex and realistic circadian rhythm model that has
the same sort of bifurcation diagram and, hence, the same general
properties as the simple model. Better models than these, as long as
they contain the right sort of interplay among positive and negative
feedback loops, will have the same potential for generating robust
oscillations by resetting.
Our simple model (Appendix) supplements the basic negative feedback loop (PER protein inhibits its own production by interfering with factors that promote
gene transcription) with a positive feedback loop (PER protein inhibits
its own degradation by forming homodimers that are less susceptible to
proteolysis). The interplay between these feedback loops creates the
potential for the control system to switch between a stable steady
state of low PER abundance and a limit-cycle oscillation during which
PER protein reaches very high abundance. To see this switching
potential, we plot in Fig. 1
A a one-parameter bifurcation diagram for the differential equations (Appendix) describing per mRNA and protein dynamics. As a function of translational efficiency, v
p, we plot [PER]ss, the steady state concentration of total PER protein (the S-shaped curve), and [PER]max and [PER]min during limit cycle oscillations. Limit cycles are found for vp values between 3.28 and 72. At vp = 72, limit cycles arise by a Hopf bifurcation (small amplitude, finite frequency); at vp = 3.28, they arise by a saddle-node on an invariant circle (SNIC) bifurcation (small frequency, finite amplitude). For a
small range of translational efficiencies, 2.98 < vp < 3.28, the control system has three steady-state solutions (one stable and two unstable).
The Resetting Hypothesis.
At this point, the usual approach would be to choose vp in the oscillatory region, say vp = 30, and model circadian rhythms as a limit cycle oscillation. The resetting hypothesis is more subtle: it posits (in this
case) that per mRNA translation rate is not constant but a regulated variable of the mechanism (more on this assumption later). That is,
vp is reinterpreted as a time-dependent variable rather than a rate constant. Suppose that vp(t) starts at a value <3.28 and increases exponentially, i.e., dvp/dt = μ·vp, for some constant value of μ. As long as vp < 3.28, the control system is attracted to the stable steady state with low [PER]. However, when vp passes through the SNIC bifurcation point, the stable steady state is lost and the control system begins an oscillation in
[PER]. We assume that, when [PER] drops below a threshold level, vp is reset by a factor σ < 1, which brings vp back below 3.28 (see the dash-dot curve in Fig. 1
A). In Fig. 2
A, we display endogenous oscillations of the resetting mechanism, plotting per mRNA, protein and vp as functions of time.
In the resetting model, the period of the oscillation is given exactly by T = μ−1·lnσ−1. Temperature compensation requires only that we balance the effects of μ and σ, the other rate constants in the mechanism may change considerably as a consequence of mutation without disturbing this balance.
For instance, the period of oscillation (T = 24.07 h) is unchanged by a 2-fold increase or decrease of any rate constant in the mechanism, except μ and σ (naturally)
and km. (If km is decreased below 0.17, then [PER] never drops below the threshold value, so vp is never reset; vp increases to some large value and the control system settles onto a stable steady state.) To illustrate this property of
the resetting model, we plot the bifurcation diagram and time courses of the system (in Figs. 1
B and 2
when all of the rate constants have been increased 2-fold (rough
simulation of a 10°C rise in temperature). The parameter σ is decreased
from 0.5 to 0.25 to compensate for the rise in μ. Notice that the
bifurcation points of the system move considerably, and the time
courses are much changed [e.g., vp(t) now increases and decreases 4-fold during an oscillation], but the period of the clock is still 24 h.
Is it reasonable to assume that PER translation rate might fluctuate over 24 h, as envisioned by the resetting hypothesis?
The efficiency with which PER protein is produced from mRNA may well be regulated by specific per mRNA-binding proteins (27) or microRNAs (28). Circadian oscillations of a translation-activating protein or a translation-silencing microRNA might carry vp back and forth across the bifurcation point, as envisioned in the resetting model in Figs. 1 and 2. However, there is no experimental evidence at present for such translational regulation of the circadian rhythm.
Robust Behavior of the Resetting Mechanism in the Face of Genetic Variability.
To claim that the resetting hypothesis gives a better account of properties i and ii than do limit cycle models, we must compare the behavior of the resetting model to some reasonable limit-cycle models of
circadian rhythms. We choose the Leloup–Goldbeter (LG) (18) and Tyson–Hong (TH) (10)
models in the limit-cycle regime. For a fair comparison, we must give
each limit cycle model the advantage of an arbitrary “compensation
relation” between any two of its most period-determining rate
constants, analogous to the relation σ = e
−24μ required of the resetting (RS) model. As explained in SI, we take these compensation relations to be vm = 0.43·(ks − 0.3)−0.16 for the LG model and Jp = 3.2 × 10−5·km−3.2
for the TH model. For each of the three models (RS, LG, and TH), we
perform two tests: (test A) variability of circadian period with
respect to simultaneous random perturbations of all of the kinetic
constants in the model (i.e., variability across individuals), and
(test B) ability to maintain temperature compensation in the face of
single mutations (which cause random changes in αi and Ei for some i).
each test, we generate a large sample of randomly perturbed
individuals. In test A, an individual is generated by multiplying each
basal parameter value (although not violating the compensation
relation) by a new random number drawn from N(1, σp), the normal distribution with mean 1.0 and standard deviation σp. In test B, a “mutant” organism is created by randomly selecting a rate constant ki = αie
−Ei/Rθ and altering both αi and Ei by random multiplicative factors drawn from N(1, σp). Then the mutant organism's period is computed for θ = 293, 294,…, 303 K and its ability to temperature compensate is measured
as ΔT = T
max − T
min. Both tests are run for many values of σp between 0.01 and 0.4, and the results (Fig. 3) plot the coefficient of variation of the period, CV = SD of period/mean of period (for test A) or the average value of ΔT (for test B) versus σp. For test A (Fig. 3
A), for CV to be ≈5% (as observed), we must constrain the rate constant perturbations to be <5% for LG and <12% for TH, but
there are no such constraints for RS. For test B (Fig. 3
we see that both LG and TH quickly lose the ability to temperature
compensate as mutations alter catalytic properties of circadian rhythm
components, but RS is robustly temperature compensated. These results
are a direct consequence of the fact that the limit cycle models spread
out control of the period to a large number of parameters. We can also
note that many perturbations for tests A and B cause a loss of
oscillation (or represent a transition into more complex rhythmic
behavior), giving us a separate measure of robustness (see Fig. 4).¶
General Requirements for Resetting.
The resetting mechanism does not depend on the specific assumptions we introduced to compute Fig. 1 or to make vp (the translational efficiency of per
mRNA) increase and decrease. It relies instead on having a regulatory
network of sufficient richness to generate a bifurcation that carries
the system from a stable steady state to a large amplitude oscillation,
and on having a resettable parameter that can carry the control system
back and forth across the bifurcation. Both SNIC bifurcations and
subcritical Hopf bifurcations (26, 29) are suitable for this purpose, and they are both commonly observed in regulatory networks with positive and negative feedback.
resetting to be consistent with a 24-h clock, the period of oscillation
close to the bifurcation point must be <24 h, because the control
system needs to spend some part of the 24-h cycle on the branch of
stable steady states and the rest of the cycle traversing (part of) the
limit cycle. This would seem to be a problem for a SNIC bifurcation
because the period of the limit cycle oscillation diverges to infinity
as the bifurcation parameter approaches the bifurcation point. However,
it is often the case that the period of oscillation decreases rapidly
as the bifurcation parameter moves away from a SNIC bifurcation, and so
it is possible to satisfy the timing requirement. In our case, for vp increasing beyond 3.28, the period drops precipitously to a value of about 15 h (Fig. 1
A Inset). Hence, the amount of time necessary for [PER] to increase to its maximum value and then drop again below the threshold,
when vp increases above 3.28, is ≈12 h. The control system spends about half the day in the stable steady state region and the other
half in the oscillatory region (Fig. 2
A). If the minimum period of oscillations in this region is larger than ≈20 h, then the resetting mechanism will not maintain
simple periodic repetitions of 24 h.
bifurcations are generic (their existence does not depend on delicate
mechanistic assumptions), and many different parameters in the
mechanism are candidates for the resetting role. Note that the
resetting hypothesis depends on the exponential change of some control
parameter, v, followed by proportional resetting of v
(multiplication by a factor σ). In the cell cycle context, these
requirements are quite natural, because cell size increases nearly
exponentially and is decreased by a factor of 0.5 at cell division. In
the context of circadian rhythms, proportional resetting of v is unlikely to be an
abrupt, stepwise change, but rather a rapid, continuous adjustment,
governed by some terms in a differential equation for dv/dt. When the resetting step is smoothed out in this fashion, the oscillation can now be thought of as a limit cycle for a system
of n + 1 differential equations
Nonetheless, the period of this limit cycle will be determined largely by the dynamics of v, i.e., by parameters μ and σ, and only very weakly by the rate constants k
2,…, etc., governing the dynamics of x. Circadian period will be robustly regulated ≈24 h if μ and σ satisfy a compensation relation like σ = e
It has been suggested (30)
that an increase in the complexity of the loop structure of a model
(i.e., the addition of more positive and negative feedback loops) leads
to an increased ability to meet several simultaneous evolutionary
constraints, such as temperature compensation and robustness to
parameter perturbations. The resetting paradigm described here achieves
some of the same goals with a simple mechanism comprising one positive
and one negative feedback loop.
A More Realistic Model.
we consider a model of PER dynamics in fruit flies, including
interactions with TIM, dCLK, and CYC proteins, nuclear transport, and
additional feedback loops. This model also displays a SNIC bifurcation (SI Fig. 6),
with the bifurcation parameter equal to the rate constant for nuclear
transport of PER/TIM complexes. In this context, resetting could
operate if the nuclear transport rate decreases exponentially during
the cycle, and is then reactivated when [PER] drops below some
threshold. We can imagine the following scenario: nuclear entry of PER
is progressively slowed down by posttranslational modification (e.g.,
phosphorylation) and/or by forming complexes with TIM, until a certain
phase of the cycle when PER's structure or phosphorylation state
changes to a form that enters the nucleus rapidly.
Recent evidence indicates that PER translocation between cytoplasm and nucleus is regulated during the circadian cycle. Meyer
et al. (31) showed that PER and TIM rapidly form complexes and accumulate in the cytoplasm, and after a delay of ≈6 h, they abruptly
dissociate and move into the nucleus. This perplexing behavior is consistent with the resetting picture in SI Fig. 6, where the rate of nuclear entry of PER decreases steadily during the circadian cycle and then increases abruptly. Furthermore,
PER translocation is intimately connected to circadian period and temperature compensation. The perL allele encodes a mutant protein PERL with a single amino acid substitution, resulting in long-period rhythms (≈28 h) (32) that are not temperature compensated (14). In mutant cells, nuclear translocation of PERL is delayed (31, 33). Meyer et al. (31) did not test for the timing of nuclear accumulation of PER as a function of temperature in wild-type PER vs. PERL expressing cells. We predict that the onset of PER nuclear translocation is further delayed with increasing temperature in
PERL expressing cells (hence longer periods at higher temperatures in perL cells; ref. 13), whereas the delay is invariant in wild-type PER-expressing cells. In light of the results of Meyer et al. and other evidence of regulated PER nuclear entry involving TIM, DBT, and various kinases (14, 34–36), we favor regulated nuclear import and/or export as a likely candidate for our resetting variable.
In contrast to delayed nuclear entry of PERL, TIMUL expressing cells exhibit advanced nuclear entry of PER compared with wild-type cells (37). (The timUL mutation, for which temperature compensation is intact, is a single amino acid substitution that causes prolonged accumulation
of PER/TIMUL in the nucleus, with an extended phase of repression of per and tim, which lengthens the period to 33 h.) It appears as though the timUL
mutation causes a longer period not by changing the onset of nuclear
accumulation of PER, but by altering the nuclear import and/or export
rate followed by delayed closure of the negative feedback loop. In the
context of our model, the timUL mutation might be causing changes in μ and/or σ to lengthen T, whereas maintaining the balance between μ and σ as a function of temperature variations.