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The Journal of Neuroscience, January 1, 1999, 19(1):40-47
A Mathematical Model for the Intracellular Circadian Rhythm
Generator
Tjeerd olde
Scheper1,
Don
Klinkenberg2,
Cyriel
Pennartz2, and
Jaap
van
Pelt2
1 Oxford Brookes University, School for Computing and
Math Science, Gipsy Lane Campus, OX3 0BP Headington Oxford, United
Kingdom, and 2 Graduate School Neurosciences Amsterdam,
Netherlands Institute for Brain Research, 1105 AZ Amsterdam, The
Netherlands
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ABSTRACT |
A mathematical model for the intracellular circadian rhythm
generator has been studied, based on a negative feedback of protein products on the transcription rate of their genes. The study is an
attempt at examining minimal but biologically realistic requirements for a negative molecular feedback loop involving considerably faster
reactions, to produce (slow) circadian oscillations. The model included
mRNA and protein production and degradation, along with a negative
feedback of the proteins upon mRNA production. The protein production
process was described solely by its total duration and a nonlinear
term, whereas also the feedback included nonlinear interactions among
protein molecules. This system was found to produce robust oscillations
in protein and mRNA levels over a wide range of parameter values.
Oscillations were slow, with periods much longer than the time
constants of any of the individual system parameters. Circadian
oscillations were obtained for realistic values of the parameters. The
system was readily entrainable to external periodic perturbations. Two
distinct classes of phase response curves were found, viz. with or
without a time domain within the circadian cycle in which external
perturbations fail to induce a phase shift ("dead zone"). The delay
and nonlinearity in the protein production and the cooperativity in the
negative feedback (Hill coefficient) were for this model found to be
necessary and sufficient to generate robust circadian oscillations. The similarities between model outcomes and empirical findings establish that circadian rhythmicity at the cellular level can plausibly emerge
from interactions among molecular systems which are not in themselves rhythmic.
Key words:
SCN; circadian rhythm; molecular clock; entrainment; phase-response curves; models
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INTRODUCTION |
Free-running behavioral rhythms with
a period close to 24 hr ("circadian") are found in many species and
constitute a fundamental mechanism for entrainment to environmental
day-night cycles. These rhythms are further characterized by typical
shifting of phase after external (light) stimulation, being dependent
on the phase point of application. Experimental evidence suggests an
intracellular origin for the generation of such rhythms, with a
critical involvement of negative feedback in the protein synthesis
system (Hardin et al., 1990 , 1992; Aronson et al., 1994; Sassone-Corsi,
1994).
The protein synthesis cascade involves much faster reactions than the
circadian period, making it not easily understood what are the critical
conditions for such highly stable and slow periodic protein
oscillations to occur. Mathematical models have been used by Goldbeter
(1995), Lewis et al. (1997), and L. F. Abbott (personal communication), to study circadian rhythm generation in
Drosophila, whereas Ruoff and Rensing (1996) concentrated on
temperature compensation using Goodwin's (1965) model. Different
mechanisms were proposed by these authors to obtain slow oscillations.
Goldbeter (1995) included protein phosphorylation (twofold) and nuclear
entry reactions of a bidirectional (equilibrium) type as well as a Hill
coefficient of n = 4 to account for cooperativity in
the negative feedback. Abbott (personal communication) did not
include a Hill coefficient but assumed up to 11 unidirectional
phosphorylation reactions to get a long enough delay in nuclear entry
to generate a 24 hr rhythm. Ruoff and Rensing (1996) included a single
after-processing step only, but needed to assume an unrealistically
high Hill coefficient of n = 9 to obtain circadian
oscillations. In a modified Goodwin model, Griffith (1968) found stable
limit cycles only with a Hill coefficient of eight or more. Lewis et
al. (1997) assumed a threshold type of reaction in the feedback loop
and included a delay of 8 hr for phosphorylation of period protein
(PER) and transport of the PER/TIM (timeless protein) complex into the
nucleus. Clearly, all these approaches were concerned primarily with
the question of how to generate slow oscillations from a feedback loop
consisting only of reactions with much faster kinetics. The models
differ among themselves in the selection of reaction steps in the
protein synthesis system that are modeled explicitly. When the reaction constants are used as optimization parameters, however, their outcomes
will necessarily strongly depend on the completeness and the precise
implementation of all reaction steps. Additionally, these approaches
require the availability of empirical data about the kinetics of each
of these reaction steps.
Our model avoids the strong constraint of specifying all the processes
involved in the production of the "effective protein" (mRNA
translation, protein postprocessing, transport, and nuclear entry) by
characterizing the full chain of reactions solely in terms of (1) the
total duration and (2) the nonlinear relationship between input and
(delayed) output of the reaction chain. Effective protein refers
to the molecular state directly capable of inhibiting mRNA production.
We show that these properties are sufficient for generating stable
and robust circadian oscillations for biologically realistic parameter
values. Additionally, the model shows realistic entrainment properties
and phase-response curves when subjected to periodic or single
external stimulation, respectively.
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MATERIALS AND METHODS |
Structure of the model. The essential components of
the reaction loop assumed to be involved in the generation of the
circadian rhythm are depicted in Figure
1A. The reaction loop
consists of a cascade for the production of the effective protein from
its mRNA and a negative feedback from the effective protein on the production of its mRNA. The protein production cascade involves the
translation and subsequent processing steps, such as phosphorylation (Edery et al., 1994b), dimerization, transport, and nuclear entry (Zerr
et al., 1990; Young, 1996; Saez and Young, 1996). Figure 1B depicts a functional abstraction of the reaction
loop emphasizing that the protein production cascade and the negative
feedback are assumed to be nonlinear processes, whereas the total
time involved in the protein production and subsequent processing (up to its negative effect on its own mRNA) is represented by a single delay term. It is assumed that these nonlinearities and the delay term
are the critical parameters in the feedback loop that determine the
free-running periodicity.
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Figure 1.
A, Schematic representation of the
biological elements of the protein synthesis cascade, assumed to be
elementary to the circadian rhythm generator. These include the auto
inhibition of the protein at translational or transcriptional level and
posttranslational processing such as phosphorylation, dimerization, and
transport. Protein* denotes the effective protein, being in the
molecular state capable of inhibiting mRNA production, as well as
expressing the circadian rhythm. B, Model interpretation
of A, emphasizing the delay ( ) and nonlinearity in
the protein production cascade, the nonlinear negative feedback, as
well as the mRNA and protein production and degradation. The mRNA and
protein production (rM,
rP) and degradation
(qM,
qP) rate constants, respectively, are
also used as targets for external stimulation (curly
arrows).
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To analyze the circadian behavior of the intracellular circadian
oscillator, the model is defined as follows:
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(1)
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(2)
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with M and P denoting relative
concentrations of mRNA and the effective protein, respectively,
rM the scaled mRNA production rate constant,
rP the protein production rate constant,
qM and qP the mRNA and
protein degradation rate constants, respectively, n the Hill
coefficient, the exponent m the nonlinearity in the protein
production cascade, the delay the total duration of protein
production from mRNA, and k a scaling constant. A derivation of these scaled equations is given in the . The formulation in
Equations 1 and 2 builds on the work of Goodwin (1965), who was the
first to study theoretically the negative feedback loop in protein synthesis.
The values for the seven parameters in the model have been chosen
according to the following considerations (Table
1). Based on data in Kornhauser et al.
(1990) and Zhang et al. (1996), a delay of 4 hr has been chosen. The
exponent m implements the nonlinearity in the protein
production term [i.e., a value of m different from one
indicates that the (delayed) protein production does not follow the
mRNA levels in a linear manner]. This occurs, for instance, when
multiple mRNA molecules are the substrate in the production of a
protein or when further processing steps introduce nonlinear interactions. The Hill coefficient n was given a value of 2 in view of the high probability of protein-protein interactions during the circadian oscillation and the putative role of dimerization in
negative feedback (Young, 1996; Zeng et al., 1996; King et al., 1997).
This value is lower than used by Goldbeter (1995) and Ruoff and Rensing
(1996). Degradation rate constants were chosen from biological
half-lives, for mRNA in the range of 0.07 and 1.39 hr 1 and for rapid degradation proteins from 0.35 to 4.16 hr1 (Alberts et al., 1989). We chose a
degradation rate constant of 0.21 hr1, and a
production rate constant of 1 hr1 for both mRNA
and protein. The scaling constant k was given the value of 1 throughout.
Method for solving the nonlinear delay equations. Delay
equations are notoriously difficult to solve analytically (Gumowski, 1981; Murray, 1989). The model was therefore analyzed by numerical integration of the equations, Equation 1 and Equation 2, with a
Runge-Kutta fourth-order differential integrator with step sizes of
0.01-1 hr. A step size of 0.1 hr proved to be the most efficient. Periods were calculated by determining the periodogram by means of
power spectrum estimation (Press et al., 1992).
Analysis of the unperturbed (free-running) system. Because
the system is defined by seven parameters {n,
m, , rM,
rP, qM, qP} it is not feasible to explore the behavior of
the system throughout the full seven-dimensional parameter space. We
have therefore chosen to investigate the behavior along single
dimensions around the set point (Table 1), by changing only one
parameter at a time while keeping the others at their set point value.
Additionally, the {n, m, } subspace was
investigated more extensively to assess the critical role of the
nonlinearity and delay parameters. It was especially important to find
those areas in parameter space in which the system has stable limit
cycles and those where the system has stable steady states. At the
boundaries between these areas, transitions in the qualitative behavior
of the system occur (bifurcations).
Analysis of the perturbed system. The effect of external
input was studied by briefly changing the production and the
degradation rate constants in the model. This choice was based on the
assumption that production and degradation rates are more sensitive to
external interference than the parameters n, m,
and , which reflect more structural than dynamic aspects of the
chemical reactions involved. Examples are light-induced TIM degradation
(Hunter-Ensor et al., 1996; Myers et al., 1996) whereas PER or TIM
induction shifts the phase of the clock (Edery et al., 1994a). To this
end, Equations 1 and 2 were extended with perturbation terms
SrM, SrP,
SqM, and SqP for
the parameters rM,
rP, qM, and
qP, respectively:
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(3)
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(4)
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The perturbation terms SrM,
SqM, SrP,
and SqP were kept at zero value in the
unperturbed model, but assumed either a positive or a negative value
during stimulation in the perturbed model.
"Entrainment" was studied by investigating the oscillatory behavior
of the system under conditions of periodic stimulation, which was
simulated by switching on at regular intervals the perturbation term
for 1 hr. "Entrainment windows", (i.e., range of stimulus periods
resulting in entrainment) were determined by changing the stimulus
period and monitoring for which range of stimulus periods the system
remained phase-locked to the stimulus. "Phase responses" have been
determined by applying a brief single stimulus of 1 hr to the system at
different points during the free-running cycle, and quantifying the
induced phase shift in the oscillation. This phase shift has been
determined from the time difference in peak values between the
free-running and the perturbed oscillation after the transient effect
of the stimulus had damped out. Both the mRNA and protein production
and degradation rates were subjected to a single pulse stimulation or
inhibition. "Phase-response curves" (PRCs) were obtained by
plotting the phase shift as a function of the time point of application
during the cycle.
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RESULTS |
Unperturbed (free-running) system
Dynamic states
Stable oscillations with a period of 24.6 hr were obtained for the
parameter values given in Table 1, with mRNA and protein concentrations
fluctuating over a wide range (Fig.
2A). The lag time
between the peak concentrations of mRNA and protein was 6 hr. The time
course of both oscillations deviated significantly from sinusoidal as
is also shown by the limit cycle contour (Fig. 2B)
with the mRNA concentration plotted versus the protein concentration. Nonsinusoidal behavior can be expected because the set of equations, Equation 1 and Equation 2, differs from that for a harmonic oscillator. The limit cycle contour is annotated with the time points in hours of
circadian model time (CMT), with the zero time point (0 hr CMT) set
arbitrarily at the maximum protein concentration. The mRNA
concentration fluctuated with a rising phase of  11.1 hr and a
falling phase of 13.5 hr, whereas the protein concentration oscillated with a rising phase of 7.8 hr and a falling phase of
16.8 hr. The dynamic behavior of the system (Eqs. 1 and 2) was
studied by an extensive search in parameter space for possible bifurcations. Apart from stable points and stable limit cycles, no
other behavior was encountered. Nevertheless, the possibility of
chaotic behavior cannot be excluded for remote areas of the parameter
space. Additionally, subharmonic oscillations might possibly be found
close to the bifurcation points.
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Figure 2.
Free-running oscillation for the parameter values
given in Table 1. Top panel, Time plot of the
oscillating protein concentration (continuous line) and
the oscillating mRNA concentration (dashed line).
Bottom panel, Corresponding limit cycle contour,
annotated with the time points within the 24.6 hr circadian cycle. The
zero time point (0 hr CMT) is taken at maximum protein
concentration.
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Boundaries in parameter space for limit cycle behavior, and the
resulting periods
Figure 3 shows how the period of a
free-running oscillation changes when one of the parameters is changed
with the other parameters remaining constant. The lowest value for a
given parameter in Figure 3 marks the bifurcation from a stable steady
state to a limit cycle, see Table 1. For all parameters, except
qM, the limit cycle behavior continued up
to the largest value indicated (10), and an even further increase (up
to 30) failed to change this pattern. For parameter
qM, on the other hand, a bifurcation from
limit cycle to stable steady state appeared at an upper value of 1.08. For all parameter values studied, the period of oscillation did not
become shorter than 13 hr. A more extensive exploration of the
{n, m, } subspace showed that at the
{n = 1, m = 1, = 4} point the
system always converged to a stable steady state for any choice of the
other parameters. For {n = 1, = 4} a
bifurcation was found at m 4, whereas the limit
cycles for m > 4 attain circadian periods at
m 6. For {m = 1, = 4} a
bifurcation was found at n 4, whereas the limit
cycles for n > 4 attain circadian periods at
n 5.5. Smaller values of the delay parameter required substantially larger values for n or m,
to obtain circadian oscillations. For instance, with = 3, such
oscillations were found at {n = 1, m = 9} or at {n = 14, m = 1}. For
larger values of , the parameter area for circadian oscillations
encounters the bifurcation boundary. For instance, with = 5, the
system is stable (for small values of n or m) or
oscillates with periods larger than 24 hr (for larger values of
n or m).
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Figure 3.
Period of oscillation plotted against the value of
(top panel) the mRNA production rate constant
rM, mRNA degradation rate constant
qM, and Hill coefficient
n, and (bottom panel) the protein
production rate constant rP, protein
degradation rate constant qP, and the
nonlinearity m and duration of the protein
production cascade. The intersections with the dotted
line indicate the set of parameter values for which the system
oscillates with a (free-running) period of 24.6 hr. Note, that each
parameter was varied while keeping the other parameters at their
original value.
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Sensitivity of periodicity to model parameters
The slope of each line in Figure 3 indicates how sensitively the
period depends on the parameter in question. Especially the nonlinearity m and the duration of the protein
production sequence strongly influence the period. Thus, changing the
protein production rate constant rP over almost
three orders of magnitude allowed the period increasing from 21 to
27 hr, whereas an increase of the delay term from 2 to 8 hr
caused the period to increase more than threefold. The curves for the
degradation rate constants qM and
qP were the only ones with negative slopes,
i.e., increased protein or mRNA degradation result in both cases in
shorter free-running periods.
Perturbed system
Entrainment to periodic external stimulation
Periodic external stimulation was studied by periodically
switching on for 1 hr one of the parameters
SrM, SqM,
SrP, SqP in Equations 3 and 4. For all four parameters entrainment of the oscillator was indeed found to occur to both shorter and longer cycles
than the free-running one (Fig. 4). At
the onset of periodic stimulation, the system went through a transient
episode lasting approximately one or two periods until it became
phase-locked. Entrainment appeared to depend on the stimulus strength:
small input strengths had little effect, whereas large input strengths made the model to become a slave oscillator of the external signal. Then, the stimulus cannot be considered anymore as a perturbation but
gets full control over the behavior of the system. The stimulus strength also determines the entrainment window, viz., the maximal deviation of the stimulus period from the free-running period for which
entrainment still occurs. An example is given in Figure 5 in which the entrainment window is
shown for different perturbations of the protein production rate
constant qP. For instance, with a perturbation
of SqP = 0.2 on the value of
qP of 0.21 hr1, the system
becomes phase-locked only between imposed cycles of 23.6 hr and
26.1 hr.
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Figure 4.
Entrainment of a free-running oscillator
(Tfr = 24.6 hr) to an external periodic
stimulation of the protein degradation rate
qP from 0.21 into 0.42 for 1 hr
(black bars) with stimulus period
Tstim = 24.0 (top
panel) or Tstim = 28.0 hr
(bottom panel). Note the onsets of entrainment of
the protein oscillation (continuous line). The
unperturbed free-running oscillations (24.6 hr) are included for
reference (dashed line).
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Figure 5.
Entrainment windows versus the strength of the
periodic 1 hr perturbation in the protein production rate
SqP. The entrainment window indicates the
range of stimulus periods for which entrainment occurs.
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Phase-response curves
Phase-response curves were obtained by perturbing for 1 hr the
mRNA production rate constant rM (Fig.
6A), the protein
production rate constant rP (Fig.
6B), the mRNA degradation rate constant qM (Fig. 6C), and the protein
degradation rate constant qP (Fig. 6D). The PRCs demonstrate that each type of
perturbation can result in a positive (phase advance) or in a negative
phase shift (phase delay). Additionally, stimulatory and inhibitory
perturbations had opposing effects on the phase shifts. For instance,
whereas mRNA production stimulation of 1 hr at 17 hr CMT resulted in a phase delay of 2.5 hr, mRNA production inhibition at 17 hr CMT results in a phase advance of 3 hr. Characteristic for the mRNA and
protein production PRCs (Fig.
6A,B), is that during some periods in the circadian cycle perturbations fail to produce any phase shift
(dead zone). For instance, perturbing the mRNA production rate constant
rM results in clear phase shifts when applied
between 5 and 23 hr CMT but had little or no effect when applied at the other time points during the circadian cycle. The dead zone for the
protein production perturbation occurs between ~6 and 12 hr CMT. The
degradation PRCs (Fig. 6C,D) show patterns quite
different from the production PRCs, in that they have no dead zones.
The PRCs for the four rate constants differ considerably in the phase point of maximal phase shift as well as in the phase advance and delay
time ranges. Investigating the PRCs at two other points in parameter
space {n = 1.5, m = 4, = 4} and
{n = 4, m = 1.5, = 4} yielded
similar results, with in the latter case only a shorter dead zone in
the rP PRC.
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Figure 6.
A, Phase-response curve for a
single pulse of 1 hr duration of mRNA production stimulation
(open squares) and inhibition (filled
squares). During stimulation, parameter
rM was changed from a value of 1 into a
value of 2, whereas during inhibition the value was set at zero.
B, Phase-response curve for a single pulse of 1 hr
duration of protein production stimulation (open
squares) and inhibition (filled squares).
During stimulation, parameter rP was changed
from a value of 1 into a value of 2, whereas during inhibition the
value was set at 0. C, Phase-response curve for a
single pulse of 1 hr duration of mRNA degradation stimulation
(open squares) and inhibition (filled
squares). During stimulation, parameter
qM was changed from a value of 0.21 into a
value of 0.42 (SqM = 0.21), whereas during
inhibition the value was set at 0 (SqM =  0.21). D, Phase-response curve for a single pulse of
1 hr duration of protein degradation stimulation (open
squares) and inhibition (filled squares).
During stimulation, parameter qP was changed
from a value of 0.21 into a value of 0.42 (SqP = 0.21), whereas during inhibition the
value was set at 0 (SqP = 0.21).
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DISCUSSION |
A minimal intracellular model for circadian rhythm generation is
shown in the present study to produce stable and robust circadian oscillations with realistic entrainment and phase-response properties. Characteristic for this model is that the complexity of the reaction cascade, involving protein synthesis, phosphorylation, transport, and
nuclear entry, is reduced to simply the overall duration of the
cascade along with a nonlinearity exponent m. By this
lumping, the strong constraint for a complete description of all the
reactions involved in the cascade and their respective kinetics is
avoided. The model thus distinguishes itself from other recent modeling studies, all of which included more detailed descriptions of the cascade of reactions which constitute the negative feedback loop of
proteins after the expression of their respective genes (Goldbeter, 1995; Lewis et al., 1997) (Abbott, personal communication).
Robustness of oscillation
At the chosen set point in parameter space, the system shows a
robust and stable free-running oscillation with a period of 24.6 hr.
Robust, because the precise parameter values are not critical for
oscillations to occur. Limit-cycle behavior was found for a large area
in parameter space (explored around the set point by changing only one
parameter at a time, and more extensively in the {n,
m, } subspace). Along the main axes this area is bounded by
lower limit values, all greater than one, where bifurcations are found,
and below which the system has only stable steady states. More
extensive investigation of the {n, m, }
subspace (by changing more than one parameter at the same time)
demonstrated that elimination of both nonlinearities, i.e.,
{n = 1, m = 1}, results in a system having only stable steady states. Elimination of only one of the nonlinearities requires substantially stronger nonlinearities for the
other to let the system oscillate. For instance, the condition m = 1 required a Hill coefficient of at least
n 6, and the condition n = 1 required a protein production nonlinearity of at least
m 6. Such large values for the nonlinearities seem
biologically unrealistic. Smaller values of the delay (e.g., = 3 hr) require even larger values for n or m,
whereas larger values (i.e., > 5 hr) made it impossible for the
system to oscillate with circadian periods (i.e., the system went to a
steady state or oscillated with larger periods). These findings
demonstrate that the duration , the nonlinearity m of the
protein synthesis cascade, and the protein-protein interaction in the
feedback loop (Hill coefficient n) are essential for the
emergence of oscillations in the system, with lower limit values along
the main axes of n = 1.27, = 2.21 hr, and
m = 1.81. The positive lower limits for the other
parameters further underscore that none of them can be excluded from
the description, making the present model sufficient as well as
necessary for the emergence of circadian rhythms in the molecular system.
Period of oscillation
For the area in parameter space where oscillations occur, the
period of oscillation was always longer than 13 hr. Clearly, this
period is much longer than the duration ( = 4 hr) of the protein
synthesis cascade and the protein and mRNA production and degradation
rates. The slow and robust oscillation must therefore be considered to
be an emergent property of the system, i.e., not arising simply from
the properties of any of its components. A rather strong dependence of
the period of oscillation was found for the delay parameter and the
nonlinearity parameter m of the protein synthesis cascade,
constituting the essential structure of this cascade. Much less
dependence was found for the protein production and degradation rate
constants. In other words, small changes in and m result
in large changes in the period. For survival an organism needs both a
robust circadian rhythm and the capacity to adapt and fine-tune its
intrinsic rhythm to environmental conditions. Because of the discrete
nature of reaction cascades, alterations in the core structure of the
feedback loop result in discrete and possibly large changes in the
period of oscillation. It is therefore plausible to assume that the
fine-tuning capacity is provided by graded adaptations in the
production and degradation rates, whereas the required robustness is
based on a stable core structure of the feedback loop (represented by
the delay and nonlinearity parameters). From an evolutionary point of
view, the structure of the feedback loop must have been a principle
target for natural selection as the period of circadian oscillation is
an "all-pervading characteristic of living organisms, conserved
throughout evolution and provided by highly stable and protected
regions of the clock genome" (Marques and Waterhouse, 1994). Because
of the sensitivity of the period for changes in the reaction cascade,
large and discrete differences in period are expected to occur by
mutations in the clock genes (Konopka and Benzer, 1971), suggesting
that this property has been an important factor in the natural
selection process.
Protein and mRNA oscillation
Although the duration chosen for the protein synthesis cascade
was only 4 hr, the peak in the simulated protein oscillation curve
occurred 6 hr later than the mRNA peak. This theoretical finding is in
excellent agreement with experimental data, which show either a lag of
6 hr between these two peak levels (Takahashi, 1991, 1995;
Aréchiga, 1993; Dunlap, 1996; King et al., 1997) or a lag of at
least 4 hr (Zeng et al., 1994). The time between the protein peak (and,
thus, the maximal negative feedback on mRNA production) and the
subsequent mRNA peak (~18 hr) is in good agreement with experimental
findings of Merrow et al. (1997) in which the elapsed time between
induced repression of frq transcript levels in
Neurospora by FRQ and full recovery was found to be 14-18 hr.
Entrainment to external periodic stimuli
The entrainment properties of the system have been studied by
periodically altering one of the production or degradation rate constants. These parameters were selected as targets for stimulation, assuming that light, too, interferes at one of these points in the
feedback loop. In all cases entrainment occurred to both shorter and
longer external periods but with a dependence on the intensity of the
stimulus and the difference between the stimulus and the free-running
period. For instance, sufficiently strong stimulation of the protein
degradation rate resulted in an entrainment window between 20 and 28 hr
(Fig. 5). This compares favorably with the empirical entrainment window
observed in the melatonin rhythm of rams under symmetrical light/dark
cycles, with a lower limit close to 20 hr cycles and an upper limit of
at least 28 hr (Ravault et al., 1989).
Phase-response curves
Phase shifting was studied by changing either the production or
the degradation rate constant of mRNA or protein during 1 hr. Both
facilitatory and inhibitory perturbations were given, either of which
could result in phase advance or phase delay, depending on the phase
point of application during the free-running cycle. A remarkable
observation was that almost all PRCs, obtained by perturbing the mRNA
or protein production rate, contained dead zones, i.e., ranges in the
free-running cycle in which perturbation failed to induce any phase
shift. The mRNA and protein degradation rate PRCs, on the contrary, did
not show dead zones, with phase shifts occurring at all points in the
free-running cycle except at zero crossings. The PRCs for the four rate
constants also turned out to differ considerably in the phase point of
maximal phase shift. Phase-response curves have been calculated for
three different points in parameter space, all yielding similar results
and underscoring the robustness of the PRCs for parameter changes and
the significance of their differences. Experimental PRCs are usually
derived from the observation of phase shifts in locomotor or other
activity brought about by light pulses. On the basis of our model
findings, the observation that the PRC for Drosophila (Myers
et al., 1996) lacks a dead zone suggests that light has targeted the
degradation rather than the production of mRNA or protein. This
conclusion is consistent with the finding that, in
Drosophila, light pulses indeed cause enhanced degradation
of TIM (one of the clock proteins) (Lee et al., 1996; Young, 1996; Zeng
et al., 1996; Dembinska et al., 1997).
Expression of circadian rhythms
The expression of molecular circadian rhythm generators varies
among species, and even tissues, taking the form of oscillating neuronal firing frequencies, hormonal levels, or behavioral expression, etc. Although one may assume that the period of oscillation is preserved in such expression, both the shape and the phase of the
rhythms at supracellular organizational levels can differ considerably
from the molecular "driver" rhythm. This consideration needs to be
taken into account when comparing the molecular simulations with
empirical findings.
Comparison with other modeling studies
The objective of the present study was to pinpoint those
components of intracellular feedback systems that are essential for producing circadian rhythmicity. The nonlinearities and the delay, identified as essential components in this model, find their
intracellular implementation in (complex) schemes of chemical
reactions, the precise nature of which is still unknown. One of the
mechanisms underlying the delay could be protein postprocessing and
transport, as studied by Goldbeter (1995), Abbott (personal
communication), and Lewis et al. (1997) by means of multiple PER
phosphorylation steps and nuclear entry, but also the time required for
protein synthesis itself contributes to the delay. With a value of 2 for the Hill coefficient, the present model needs only a moderate cooperativity (e.g., dimerization) in the feedback loop. In this sense,
it distinguishes itself from other theoretical studies, all of which
required stronger cooperativity (viz., higher values for the Hill
coefficient) in order for the studied systems to generate stable
circadian oscillations (Griffith, 1968; Goldbeter, 1995; Ruoff and
Rensing, 1996). The present model predicts a shorter period of
oscillation when the protein degradation rate increases (Fig. 3),
whereas Goldbeter's (1995) model shows a period increase under those
conditions. It is not clear whether this difference reflects
fundamental differences in the assumptions underlying the two models or
originates merely from different model implementations of the reaction schemes.
Conclusion
Nonlinearities and delay in the protein synthesis negative
feedback loop have been shown to be essential features in our model for
the generation of robust circadian oscillations. Although the model was
constructed on the basis of minimal requirements, it displays a rich
and realistic repertoire of circadian rhythm behavior with respect to
period, entrainment, and phase responses. Further outcomes of the
present study are: (1) the prediction of dead zones in the
phase-response curves for perturbations in the production rate but not
in those for degradation rate perturbations, and (2) quantitative
predictions of the dependence of the period of oscillation on the
parameters of the system. These findings not only contribute to a
better understanding of the putative molecular system underlying
circadian rhythm generation but also serve as a basis for interpreting
experimental findings as well as for formulating critical experiments
for validation.
| |
FOOTNOTES |
Received July 28, 1998; revised Oct. 14, 1998; accepted Oct. 16, 1998.
We thank Dr. Michael A. Corner for his critical comments and valuable
suggestions regarding this manuscript
Correspondence should be addressed to Dr. Jaap van Pelt, Netherlands
Institute for Brain Research, Meibergdreef 33, 1105 AZ Amsterdam, The Netherlands.
| |
APPENDIX |
The model of the intracellular circadian oscillator has been
defined as:
|
(A1)
|
|
(A2)
|
with M*(t) and P*(t)
denoting the concentrations of mRNA and the effective protein,
rM* the maximal mRNA production
rate, rP the protein production rate constant,
qM and qP the mRNA and
protein degradation rate constants, respectively, n the Hill
coefficient, the exponent m the nonlinearity in the protein
production cascade, the delay the total duration of protein
production from mRNA, and k* a scaling constant. Note, that
the maximal production rate r*M
carries the dimension [concentration]/[t], whereas the
production and degradation rate constants
rP, qM, and
qP carry the dimension 1/[t].
Introducing dimensionless quantities M =
M*/M0, p = P*/M0, and k
= k*/M0, and the
quantity rM =
rM*/M0
with dimension 1/[t], with M0
denoting the maximal mRNA concentration produced per unit of time, and
dividing both equations by M0 finally result
in Equations 1 and 2.
| |
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Abstract
Introduction
