A Simple Model of Circadian Rhythms Based on Dimerization and
Proteolysis of PER and TIM
John J. Tyson,* Christian I. Hong,* C. Dennis Thron,#
and Bela Novak§
*Department of Biology, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 USA; #
5 Barrymore Road,
Hanover, New Hampshire 03755 USA; and §
Department of Agricultural Chemical Technology, Technical University,
Budapest 1521, Hungary
ABSTRACT Many organisms display rhythms of physiology and behavior that are entrained to the 24-h cycle of light and
darkness prevailing on Earth. Under constant conditions of illumination and temperature, these internal biological rhythms
persist with a period close to 1 day (“circadian”), but it is usually not exactly 24 h. Recent discoveries have uncovered stunning
similarities among the molecular circuitries of circadian clocks in mice, fruit flies, and bread molds. A consensus picture is
coming into focus around two proteins (called PER and TIM in fruit flies), which dimerize and then inhibit transcription of their
own genes. Although this picture seems to confirm a venerable model of circadian rhythms based on time-delayed negative
feedback, we suggest that just as crucial to the circadian oscillator is a positive feedback loop based on stabilization of PER
upon dimerization. These ideas can be expressed in simple mathematical form (phase plane portraits), and the model
accounts naturally for several hallmarks of circadian rhythms, including temperature compensation and the perL
mutant
phenotype. In addition, the model suggests how an endogenous circadian oscillator could have evolved from a more
primitive, light-activated switch.
INTRODUCTION
Wild-type fruit flies, Drosophila melanogaster, exhibit endogenous
activity rhythms with a period of 24 h over a
broad temperature range (18–33°C). The first mutation to
interfere with this circadian rhythm was discovered by
Konopka and Benzer (1971), who called the gene period
(per, for short). Three mutant alleles of per have been
studied: perL
and perS
, with endogenous activity rhythms of
27 and 19 h, respectively (at 18°C), and per0
, a null allele
with no overt rhythm (Huang et al., 1995). Remarkably, the
perL
mutant has lost temperature compensation; the period
of its endogenous rhythm increases from 25 h at 15°C to
33 h at 30°C (Huang et al., 1995).
A second important gene, timeless or tim, encodes a
protein, TIM, that binds to PER (Gekakis et al., 1995;
Myers et al., 1995; Sehgal et al., 1994, 1995; Vosshall et al.,
1994; Zeng et al., 1996). Mutation of tim abolishes the
circadian rhythm (Sehgal et al., 1994). During endogenous
cycling in constant darkness, a brief light pulse causes a
phase shift of the circadian rhythm (Myers et al., 1996;
Pittendrigh, 1967). This phase shift has recently been attributed
to rapid degradation of TIM upon exposure to light
(Hunter-Ensor et al., 1996; Lee et al., 1996; Myers et al.,
1996; Zeng et al., 1996).
PER protein and per mRNA fluctuate with a 24-h period,
with protein lagging behind mRNA by 4–6 h (Hardin et al.,
1990; Zeng et al., 1994). When PER protein is overexpressed
from a constitutive promoter, expression of endogenous
per mRNA is repressed (Zeng et al., 1994), suggesting
that PER inhibits its own transcription (Hardin et al.,
1990). Binding to TIM seems to be necessary for translocation
of PER to the nucleus (Vosshall et al., 1994) to exert
its inhibitory effect. PER forms homo- and heterodimers
through its “PAS” domain (Gekakis et al., 1995; Huang et
al., 1995; Lee et al., 1996; Zeng et al., 1996), which it shares
with many transcription factors but not with TIM. The perL
mutation, which lies in the PAS domain, disrupts PER/PER
(Huang et al., 1995) and PER/TIM binding (Gekakis et al.,
1995). Expression of the per and tim genes is regulated by
a pair of transcription factors, dCLOCK (also called JRK)
and CYC, that appear to be inactivated by PER (Allada et
al., 1998; Darlington et al., 1998; Rutila et al., 1998). This
evidence for negative feedback of PER on transcription of
its own mRNA is the basis for most current theoretical
models of circadian rhythms (Goldbeter, 1995; Ruoff and
Rensing, 1996; Leloup and Goldbeter, 1998; Scheper et al.,
1999). However, we propose that a positive feedback loop,
based on stabilization of PER by dimerization with TIM,
may play an equally important role in generating oscillations.
This proposal is supported by recent discoveries on
PER phosphorylation and proteolysis.
PER is phosphorylated by a casein-like kinase called
DBT (encoded by the double-time gene), which is present at
roughly constant levels during the rhythm (Kloss et al.,
1998; Price et al., 1998). PER phosphorylation seems to be
a prelude to its degradation, as suggested by the phenotypes
of dbt mutants. In dbtP
, which codes for a nonfunctional
kinase and has no rhythm, PER accumulates in a hypophosphorylated
form. dbtS
codes for a more active kinase, accumulates
less PER than wild type, and has shorter cycles
(18 h in homozygote). dbtL
codes for a less active kinase,
accumulates more PER than wild type, and has longer
Received for publication 9 March 1999 and in final form 10 August 1999.
Address reprint requests to Dr. John J. Tyson, Department of Biology, MC
0406, Virginia Tech, Blacksburg, VA 24061. Tel.: 540-231-4662; Fax:
540-231-9307; E-mail: tyson@vt.edu.
© 1999 by the Biophysical Society
0006-3495/99/11/2411/07 $2.00
2411Biophysical Journal Volume 77 November 1999 2411–2417
cycles (26.8 h in homozygote). Experimental results suggest
that PER is stabilized on association with TIM (Kloss et al.,
1998; Price et al., 1998).
Other avenues of positive feedback are also possible in
Drosophila. For instance, Suri et al. (1999) present evidence
that PER/TIM dimers stabilize per mRNA, and the experiments
of Bae et al. (1998) suggest that PER and TIM are
transcriptional activators of dCLOCK, which in turn stimulates
transcription of the per and tim genes.
MECHANISM AND MATHEMATICAL MODEL
Following the lead of Kloss et al. (1998), we assume that
PER monomers are rapidly phosphorylated and degraded,
whereas PER/TIM dimers are less susceptible to proteolysis
(poorer substrates for either DBT or the proteolytic machinery).
Our model, summarized in Fig. 1, is similar in structure
to that of Leloup and Goldbeter (1998), but with a
crucial difference. In the Leloup-Goldbeter model, the role
of PER phosphorylation is to introduce a time delay into the
negative feedback loop. In our model, the role of PER
phosphorylation is to introduce positive feedback in PER
accumulation. As PER concentration increases, an ever
greater proportion of protein is dimerized and protected
from DBT. Therefore, as the total concentration of PER
(monomer ϩ dimer) increases, the rate of total PER degradation
does not increase proportionally. This nonlinearity is
a key factor in the following mathematical model of circadian
rhythms.
The mechanism in Fig. 1 could be translated into a set of
six differential equations, for per and tim mRNAs, PER and
TIM monomers, and PER/TIM dimers in the cytoplasm and
nucleus. Such a complicated set of equations would not
effectively illustrate the importance of positive feedback in
the reaction mechanism. Noticing that PER and TIM messages
and proteins follow roughly similar time courses in
vivo, we lump them together into a single pool of clock
proteins. In addition, we assume that the cytoplasmic and
nuclear pools of dimeric protein are in rapid equilibrium.
With these simplifying assumptions, our model reduces to
three differential equations for [mRNA] ϭ M, [monomer] ϭ
P1, and [dimer] ϭ P2:
dM
dt
ϭ
vm
1 ϩ ͑P2/Pcrit͒2 Ϫ kmM (1)
dP1
dt
ϭ vpM Ϫ
kЈp1P1
Jp ϩ P1 ϩ rP2
Ϫ kp3P1 Ϫ 2kaP1
2
ϩ 2kdP2 (2)
dP2
dt
ϭ kaP1
2
Ϫ kdP2 Ϫ
kp2P2
Jp ϩ P1 ϩ rP2
Ϫ kp3P2 (3)
In Eq. 1, we assume some cooperativity (represented by a
Hill coefficient of 2) for the inhibition of mRNA transcription
by PER/TIM dimers. This assumption is not essential:
the equations exhibit limit cycle oscillations even if the Hill
coefficient ϭ 1. However, we find it easier to fit some
properties of mutant fruit flies with a Hill coefficient of 2.
Of more importance is our assumption, in Eqs. 2 and 3, that
both monomers and dimers bind to DBT, but P1 is phosphorylated
more rapidly (kp1Ј ϾϾ kp2). It is essential for
oscillations that the DBT-catalyzed reaction shows saturable
kinetics (e.g., Michaelis-Menten) and that the dimer is a
competitive inhibitor of monomer phosphorylation. The extent
of competitive inhibition is determined by r, the ratio of
enzyme-substrate dissociation constants for the monomer
and dimer. Oscillations are observed for r as small as 0.2,
but not for r ϭ 0. These properties of the DBT-catalyzed
reaction have not yet been determined experimentally, so
they constitute testable predictions of our theory. Finally,
the terms involving kp3 in Eqs. 2 and 3, which represent
slow, first-order degradation of monomers and dimers, are
not essential for oscillations, but they allow a better fit to
some of the data.
If the dimerization reactions are fast (ka and kd large),
then P1 and P2 are always in equilibrium with each other.
Let Pt ϭ P1 ϩ 2P2 ϭ [total protein]. Then, from the
equilibrium condition, P2 ϭ KeqP1
2
, Keq ϭ ka/kd, we can
write P1 ϭ qPt and P2 ϭ 1⁄2(1 Ϫ q)Pt, where
q ϭ
2
1 ϩ ͱ1 ϩ 8KeqPt
(4)
Now our model reduces to a pair of nonlinear ordinary
differential equations:
dM
dt
ϭ
vm
1 ϩ ͑Pt͑1 Ϫ q͒/2Pcrit͒)2 Ϫ kmM (5)
FIGURE 1 A simple molecular mechanism for the circadian clock in
Drosophila, adapted from Price et al. (1998), Kloss et al. (1998), Wilsbacher
and Takahashi (1998), and Leloup and Goldbeter (1998). PER and
TIM proteins (rectangle and oval, respectively) are synthesized in the
cytoplasm, where they may be destroyed by proteolysis or they may
combine to form relatively stable heterodimers. Heteromeric complexes are
transported into the nucleus, where they inhibit transcription of per and tim
mRNA. We assume that PER monomers are rapidly phosphorylated by
DBT and then degraded. Dimers, we assume, are poorer substrates for
DBT.
2412 Biophysical Journal Volume 77 November 1999
dPt
dt
ϭ vpM Ϫ
kp1Ptq ϩ kp2Pt
Jp ϩ Pt
Ϫ kp3Pt (6)
supplemented by the algebraic equation (Eq. 4), which
specifies that q ϭ q(Pt). In Eq. 6, kp1 ϭ kp1Ј Ϫ kp2 Ϸ kp1Ј.
Also, in Eq. 6, we have assumed that r ϭ 2. If r 2, the
denominator of the Michaelis-Menten expression should be
written as Jp ϩ qPt ϩ (r/2)(1 Ϫ q)Pt. A typical solution of
Eqs. 4–6 is illustrated in Fig. 2.
PHASE PLANE PORTRAITS
To reach the pair of differential equations (Eqs. 5 and 6), we
have made a number of assumptions about interactions
between PER and TIM. In a later publication, we will relax
these assumptions and study the full set of kinetic equations
implied by Fig. 1. But for our present purpose of emphasizing
the role played by positive feedback in PER dynamics,
the two-equation model has a great advantage in being
amenable to the powerful tools of phase plane analysis
(Edelstein-Keshet, 1988). The phase plane for Eqs. 5 and 6
is the Cartesian coordinate system representing our two
variables, mRNA and total protein. In the phase plane (Fig.
3 A), we plot two nullclines: 1) where mRNA synthesis is
exactly balanced by mRNA degradation,
M ϭ
vm
kmͫ1 ϩ ͩPt͑1 Ϫ q͒
2Pcrit
ͪ2
ͬ (M-nullcline)
and 2) where protein synthesis and degradation are bal-
anced,
M ϭ
kp1Ptq ϩ kp2Pt
vp͑Jp ϩ Pt͒
ϩ
kp3Pt
vp
(P-nullcline)
The M-nullcline is sigmoidally shaped because we assume
cooperativity of the inhibition of transcription by PER/TIM
dimers. The P-nullcline is N-shaped because 1) dimer is more stable than monomer and 2) dimer competitively
inhibits binding of monomer to DBT.
A sketch of the nullclines in the phase plane is called a
“portrait” of the dynamical system, and it tells us much
about the system’s temporal behavior. For instance, the
portrait in Fig. 3 A shows the system oscillating around a
limit cycle in the phase plane. We obtain this portrait by
adjusting the parameters of the model so that the
M-nullcline intersects the P-nullcline on its intermediate
branch. Then the rate constants are scaled to give an oscillation
of (nearly) 24 h (see Table 1). If we were to change
some of the parameters (e.g., by mutation), the nullclines
would move across the phase plane, and the portrait would
change (Fig. 3 B). In particular, the period of oscillation
may change or the limit cycle may disappear altogether and
be replaced by a stable steady state.
In Fig. 4 we show how the temporal behavior of the
model depends on Keq and kp1. Within the U-shaped region,
the model exhibits stable limit cycle oscillations. Clearly,
oscillatory behavior requires that Keq be large enough (i.e.,
FIGURE 2 Numerical solution of model of Eqs. 5 and 6, given the
parameter values in Table 1.
FIGURE 3 Phase plane portraits. (A) M- and P-nullclines (see text) for
the parameter values in Table 1. The closed orbit is a stable limit cycle
oscillation. (B) Shift of the P nullcline as Keq is changed from 200 to 1.
When Keq ϭ 1, the system exhibits a stable steady state with abundant
mRNA but a low protein level.
Tyson et al. A Simple Model of Circadian Rhythms 2413
protein subunits tend to dimerize) and kp1 be large enough
(i.e., protein monomers are sufficiently unstable). For Keq Ͼ
100, the period of oscillation is close to 24 h and is quite
insensitive to changes in Keq or kp1, suggesting that the
rhythm of wild-type flies may be insensitive to temperatureinduced
changes in parameters. For Keq Ͻ 50, the period of
oscillation abruptly increases and becomes quite sensitive to
both Keq and kp1. It is known that the defect in perL
-encoded
protein reduces its tendency to form dimers (Gekakis et al.,
1995; Huang et al., 1995), which leads naturally to longer
periods of perL
mutants and the temperature sensitivity of
their rhythm (Fig. 4). Of course, variations of the other
parameters with temperature also affect the periods of wildtype
and mutant flies. By choosing appropriately (Table 1)
the activation energies for each parameter in the model
(Ruoff et al., 1997; Ruoff, 1992), we readily account for
temperature compensation of the wild-type rhythm and temperature
dependence of the oscillator in perL
flies (Table 2).
COMPARISON TO EXPERIMENTS
Our simple model can be tested against the phenotypes of
several other circadian rhythm mutants. We account for the
properties of dbtS
and dbtL
mutants (Table 2) by assuming
that kp1 is not much affected by these mutations (otherwise
the rhythm would likely be destroyed; see Fig. 4), but kp2 is
increased or decreased 10-fold. On the other hand, period is
not much affected by multiple doses of wild-type dbt (Table
2). The phenotypes of per0
(null mutant), perOP
(constitutive
promoter), and dbtp
(null mutant) are easily explained
(not shown). Regarding the dosage dependence of wild-type
per, the model predicts a slight increase in period with
increasing vm (the period at vm ϭ 1 is 0.6 h longer than the
period at vm ϭ 0.5), but genetic manipulations (Smith and
Konopka, 1982; Cote and Brody, 1986) show a slight decrease
in period (the period of perϩ
/perϩ
is 0.5 h shorter
than the period of perϩ
/per0
). In light of the simplicity of
the model, this discrepancy does not seem too serious.
Any model of circadian rhythms should also be tested for
its response to light pulses and its propensity to be entrained
by light-dark cycles. To simulate typical phase-response
curves (PRCs), we assume that the effect of light is to
reduce Keq. (The literature reports that light pulses increase
the degradation of TIM (Hunter-Ensor et al., 1996; Lee et
al., 1996; Myers et al., 1996; Zeng et al., 1996), suggesting
that we should model a light pulse by increasing kp3; however,
we found that this assumption does not produce correctly
shaped PRCs in our model, whereas decreasing Keq
does. Because monomeric protein is unstable in our model,
if light absorption were to destabilize dimers, then the clock
protein would rapidly be lost.) The pattern of delays and
advances (Fig. 5 A) is qualitatively similar to PRCs observed
experimentally (Myers et al., 1996; Pittendrigh,
1967), although close comparison will show many quantitative
discrepancies. The delay or advance shows up in the
first cycle after treatment, after which the oscillator is back
on its 24-h track.
In addition, the model is very rapidly entrained to an
external Zeitgeber with a period different from 24 h (Fig.
5 B).
TABLE 1 Parameter values suitable for circadian rhythm of wild-type fruit flies
Name Value Units* Ea/RT#
Description
vm 1 Cm hϪ1
6 Maximum rate of synthesis of mRNA
km 0.1 hϪ1
4 First-order rate constant for mRNA degradation
vp 0.5 Cp Cm
Ϫ1
hϪ1
6 Rate constant for translation of mRNA
kp1 10 Cp hϪ1
6 Vmax for monomer phosphorylation
kp2 0.03 Cp hϪ1
6 Vmax for dimer phosphorylation
kp3 0.1 hϪ1
6 First-order rate constant for proteolysis
Keq 200 Cp
Ϫ1
Ϫ12 Equilibrium constant for dimerization
Pcrit 0.1 Cp 6 Dimer concen at the half-maximum transcription rate
Jp 0.05 Cp Ϫ16 Michaelis constant for protein kinase (DBT)
*Cm and Cp represent characteristic concentrations for mRNA and protein, respectively.
#
Ea is the activation energy of each rate constant (necessarily positive) or the standard enthalpy change for each equilibrium binding constant (may be
positive or negative). These values are chosen to ensure temperature compensation of the wild-type oscillator.
FIGURE 4 Two-parameter bifurcation diagram for the model, calculated
with the software tool AUTO (Doedel, 1986). As Keq and kp1 are allowed
to vary, with all other parameter values fixed as in Table 1, we find a
U-shaped region of two-parameter space where limit cycle oscillations are
possible, bounded by a locus (H-H) of Hopf bifurcations. We foliate this
region with curves of constant period, from 24 to 40 h. Notice that the
period of oscillation varies little from 24 h in a large region of parameter
space, but then becomes quite sensitive to parameter values when Keq is
small.
2414 Biophysical Journal Volume 77 November 1999
DISCUSSION
The earliest attempts to model circadian rhythms were not
based on molecular mechanisms, because nothing of the sort
was known. Rather, they emphasized the generic properties
of limit cycle solutions to nonlinear dynamical systems
(Kalmus and Wigglesworth, 1960; Kronauer et al., 1982;
Pavlidis, 1967; Tyson et al., 1976; Wever, 1960; Winfree,
1970). To their credit, they revealed the sort of behavior that
can be expected of the circadian clock regardless of the
actual make-up of its springs, gears, and levers. As soon as
it became clear that repression of gene transcription plays an
important role in circadian timekeeping, models based on
Goodwin’s (1965) negative-feedback paradigm appeared
(Goldbeter, 1995; Ruoff and Rensing, 1996). Goodwin’s
equations, first used to model periodic enzyme synthesis in
bacteria, describe a “pure” negative feedback loop (no autocatalytic
terms):
dx1
dt
ϭ
1
1 ϩ xn
p Ϫ k1x1 ,
dxi
dt
ϭ xiϪ1 Ϫ kixi , i ϭ 2, . . . , n.
(7)
Although the equations are simple and elegant, their analysis
is complex and unintuitive. The conditions for sustained
oscillations can be severe: n Ն 3 (i.e., phase-plane
portraiture is impossible), p Ͼ 8 for n ϭ 3 (i.e., extreme
levels of cooperativity are often required), and ki Ϸ kj for all
i and j (i.e., all components of the loop must be comparably
unstable) (Griffith, 1968; Thron, 1991; Tyson and Othmer,
1978). Furthermore, in the more than 30 years since the
mechanism was first proposed, only one biochemical example
of a pure negative-feedback oscillator has been carefully
described (Bliss et al., 1982), even though end-product
repression is a common theme of biochemical pathways.
Instead of concentrating on the negative feedback loop
implied by PER’s inhibition of per transcription, we propose
that the crucial molecular interaction generating oscillatory
behavior of the PER network is the positive feedback
loop implied by the stabilization of PER protein upon dimer
formation. To emphasize the role that positive feedback
may play in the circadian system, we have drastically simplified
the molecular machinery (lumping together PER and
TIM) so that the model can be described by two differential
equations. Our simple two-component model has many
advantages over recent models based on pure negative feedback:
phase-plane analysis now gives useful insight into the
mechanism of oscillation, highly cooperative transitions are
no longer required, and reasonable rate constants can be
assigned to the elementary steps. Our model exhibits a
remarkable insensitivity of the oscillatory period to certain
crucial parameters (which preadapts the mechanism for
temperature-compensated rhythms), and it is consistent with
the phenotypes of many distinctive mutants at the per, tim,
and dbt loci. We can account for certain qualitative features
of phase resetting and synchronization in response to light
by assuming that Keq is light sensitive, but not by making
the more realistic assumption that light exposure increases
kp3. These strengths and weaknesses of the model suggest
that positive feedback plays a heretofore unrecognized role
in the dynamics of circadian rhythms, but that more comprehensive
molecular mechanisms and mathematical models
will be required for accurate representation of the detailed
properties of circadian rhythms in Drosophila.
The change in emphasis, from negative feedback to positive
feedback, provides a clue to the evolution of the
endogenous circadian clock. Consider a primitive mechanism
that lacks negative feedback on transcription. In this
case, dM/dt ϭ vm Ϫ kmM replaces Eq. 5, and the phaseplane
portrait of the system changes dramatically (Fig. 6 A).
An organism with this primitive mechanism would not
exhibit endogenous oscillations, but it could still be entrained
by external light/dark cycles. With positive feedback
in effect, the N-shaped P-nullcline creates a hysteresis loop
(Fig. 6 B) that can be traversed by changes in the equilibrium
constant for dimerization. When the lights are on, Keq
is small, and the protein level is small because it is mostly
monomeric and rapidly degraded. When the lights are off,
Keq is large, and the protein level is large because it is
mostly dimeric and more stable. Such an organism would
still know the time of day, as long as it was subjected to
external light/dark cycles. By adding negative feedback
later, the process of natural selection could convert a
“switch” into a “clock.” An unexpected benefit is that the
period of the endogenous oscillation is quite insensitive to
TABLE 2 Period of the endogenous rhythms of wild-type and mutant flies
Genotype Keq Temp* Period Genotype#
kp1 kp2 Period
Wild type 245 20 24.2 dbtϩ
(1ϫ) 10 0.03 24.2
200 25 24.2 (2ϫ) 15 0.06 24.4
164 30 24.2 (3ϫ) 20 0.09 25.7
perL
18.4 20 26.5 dbtS
10 0.3 17.6
15.0 25 28.7 dbtϩ
10 0.03 24.2
12.3 30 30.5 dbtL
10 0.003 25.2
*We assume that each parameter k varies with temperature according to k(T) ϭ k(298) exp{␧a(1 Ϫ 298/T)}, with values for k(298) and ␧a ϭ Ea/(0.592 kcal
molϪ1
) given in Table 1.
#
dbtϩ
(nϫ) means n copies of the wild-type allele.
Tyson et al. A Simple Model of Circadian Rhythms 2415
the mechanism’s crucial parameter values (Fig. 4), so it has
a built-in preadaptation for temperature compensation.
If by mutating the transcription factors, clock and cyc,
geneticists can isolate flies that synthesize PER and TIM
constitutively (i.e., no negative feedback); these mutant
flies, although they have no endogenous rhythm, would still
respond perfectly well to light, synthesizing PER at night
and degrading it during the day. Any model in which TIM
is rapidly degraded by exposure to light would predict this
effect, because TIM is necessary for PER accumulation.
What sets our model apart is the prediction that the switch
between synthesis and degradation should show hysteresis
as a function of intensity of illumination (Fig. 6 B). Furthermore,
by knocking out the phosphorylation sites on
PER, the hysteresis loop should be eliminated.
Our model of the circadian clock in Drosophila is surely
incomplete, because the molecular basis of circadian
rhythms is still vigorously studied and hotly debated. We
think of it as a skeletal or minimal model to be elaborated
and improved. We cannot expect such a simple model to
explain correctly all features of circadian rhythms, but it
does rest firmly on the current knowledge of the molecules
involved, and it gives new insight into the central roles
played by proteolysis, dimerization, and competitive inhibition
in generating bistability and oscillations. In addition,
FIGURE 5 Interaction of the endogenous oscillator with light. (A) Phase
response curve. At a chosen phase of the endogenous rhythm we perturb
the equations by setting Keq ϭ 15 for a definite period of time (10 or 30
min) and then return Keq to 200. We follow the perturbed rhythm for 100 h
and compare the time when Pt ϭ 1 to the time of this event in the
unperturbed rhythm. We plot the phase difference (advance or delay) as a
function of the phase at the onset of perturbation. Phase ϭ 0 (Pt ϭ 0.023,
M ϭ 0.82) corresponds roughly to subjective dusk (lights out). (B) Principal
entrainment band. Equation 4 is modified so that Keq ϭ 200 for
one-half of a Zeitgeber period (lights off), and Keq ϭ 200(1 Ϫ a) for the
other half (lights on). Presumably, a is proportional to the intensity of
illumination. As a function of Zeitgeber period (abscissa), we look for the
minimum value of a (ordinate) required for 1:1 entrainment of the circadian
oscillator to the Zeitgeber.
FIGURE 6 A circadian “switch” based on positive feedback alone. (A)
The P-nullcline is plotted for Keq ϭ 15 (day) and Keq ϭ 200 (night). The
M-nullcline is a horizontal line (M ϭ vm/km) because P does not inhibit
synthesis of M in this model. During the day, the system sits on a stable
steady state with Pt Х 0.008 (few dimers, rapid degradation of monomers).
At night the system sits on a stable steady state with Pt Х 10 (mostly
dimers, slow degradation). (B) Hysteresis loop. We plot the steady-state
level of Pt as a function of Keq. Assuming, as before, that Keq is inversely
proportional to the intensity of illumination, we predict that expression of
PER protein will show distinct thresholds: turning on at low intensity
(Keq ϭ 155), but not turning off until a much higher intensity is reached
(Keq ϭ 21.6).
2416 Biophysical Journal Volume 77 November 1999
our representation of the central control system by two
differential equations means that the intuitively pleasing,
geometric ideas of phase plane analysis can now be applied
to circadian rhythms.
We thank Arthur Winfree and Albert Goldbeter for helpful comments.
Kalimar Maia computed the bifurcation diagram in Fig. 4.
This work was supported by the Howard Hughes Medical Institute (75195-
512302) and the National Science Foundation (DMS-9525766).
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