A symmetric dual feedback system provides a robust and entrainable oscillator.

Abstract

Many organisms have evolved molecular clocks to anticipate daily changes in their environment. The molecular mechanisms by which the circadian clock network produces sustained cycles have extensively been studied and transcriptional-translational feedback loops are common structures to many organisms. Although a simple or single feedback loop is sufficient for sustained oscillations, circadian clocks implement multiple, complicated feedback loops. In general, different types of feedback loops are suggested to affect the robustness and entrainment of circadian rhythms. To reveal the mechanism by which such a complex feedback system evolves, we quantify the robustness and light entrainment of four competing models: the single, semi-dual, dual, and redundant feedback models. To extract the global properties of those models, all plausible kinetic parameter sets that generate circadian oscillations are searched to characterize their oscillatory features. To efficiently perform such analyses, we used the two-phase search (TPS) method as a fast and non-biased search method and quasi-multiparameter sensitivity (QMPS) as a fast and exact measure of robustness to uncertainty of all kinetic parameters.So far the redundant feedback model has been regarded as the most robust oscillator, but our extensive analysis corrects or overcomes this hypothesis. The dual feedback model, which is employed in biology, provides the most robust oscillator to multiple parameter perturbations within a cell and most readily entrains to a wide range of light-dark cycles. The kinetic symmetry between the dual loops and their coupling via a protein complex are found critically responsible for robust and entrainable oscillations. We first demonstrate how the dual feedback architecture with kinetic symmetry evolves out of many competing feedback systems.

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A Symmetric Dual Feedback System Provides a Robust
and Entrainable Oscillator
Kazuhiro Maeda, Hiroyuki Kurata*
Department of Bioscience and Bioinformatics, Kyushu Institute of Technology, Iizuka, Fukuoka, Japan
Abstract
Many organisms have evolved molecular clocks to anticipate daily changes in their environment. The molecular
mechanisms by which the circadian clock network produces sustained cycles have extensively been studied and
transcriptional-translational feedback loops are common structures to many organisms. Although a simple or single
feedback loop is sufficient for sustained oscillations, circadian clocks implement multiple, complicated feedback loops. In
general, different types of feedback loops are suggested to affect the robustness and entrainment of circadian
rhythms.To reveal the mechanism by which such a complex feedback system evolves, we quantify the robustness and
light entrainment of four competing models: the single, semi-dual, dual, and redundant feedback models. To extract the
global properties of those models, all plausible kinetic parameter sets that generate circadian oscillations are searched to
characterize their oscillatory features. To efficiently perform such analyses, we used the two-phase search (TPS) method as a
fast and non-biased search method and quasi-multiparameter sensitivity (QMPS) as a fast and exact measure of robustness
to uncertainty of all kinetic parameters.So far the redundant feedback model has been regarded as the most robust
oscillator, but our extensive analysis corrects or overcomes this hypothesis. The dual feedback model, which is employed in
biology, provides the most robust oscillator to multiple parameter perturbations within a cell and most readily entrains to a
wide range of light-dark cycles. The kinetic symmetry between the dual loops and their coupling via a protein complex are
found critically responsible for robust and entrainable oscillations. We first demonstrate how the dual feedback architecture
with kinetic symmetry evolves out of many competing feedback systems.
Citation: Maeda K, Kurata H (2012) A Symmetric Dual Feedback System Provides a Robust and Entrainable Oscillator. PLoS ONE 7(2): e30489. doi:10.1371/
journal.pone.0030489
Editor: Gennady Cymbalyuk, Georgia State University, United States of America
Received July 26, 2011; Accepted December 16, 2011; Published February 20, 2012
Copyright: ? 2012 Maeda, Kurata. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by KAKENHI (Grant-in-Aid for Scientific Research) on Scientific Research (B) (22300101) from the Ministry of Education,
Culture, Sports, Science and Technology of Japan. KM was supported by Research Fellowships from the Japan Society for the Promotion of Science for Young
Scientists. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: kurata@bio.kyutech.ac.jp
Introduction
Many organisms have evolved molecular clocks to anticipate
daily changes in the environment [1,2]. The molecular mecha-
nisms by which the circadian clock network produces sustained
cycles have extensively been studied and transcriptional-transla-
tional feedback loops are known as common structures to many
organisms [1]. Although a simple or single feedback loop is
sufficient for sustained oscillations [3], circadian clock systems
implement complicated feedback loops. A current problem is to
reveal the mechanism by which such a complex feedback system
evolves.
Mathematical models for circadian clocks have been proposed
and extensively studied [4–12]. In most studies, nominal or
convenient values are assigned to kinetic parameters, because
experimental data are lacking. The simulated results depend on a
particular choice of kinetic parameters, while not only network
topology but also parameter values alter the system’s features [13–
16]. To understand the global properties of circadian clocks, it is
necessary to search all plausible kinetic parameter sets that
generate circadian oscillations and to characterize the oscillatory
features over all of the parameter sets. A shortage of the parameter
search may lead to a wrong conclusion. To efficiently search
parameters, we developed the two-phase search (TPS) method,
which combines a random search with genetic algorithms to
achieve global search while reducing computational cost [17].
Robustness is the ability to resume reliable operation in the face
of different types of perturbations: parameter uncertainty,
environmental and genetic changes, and stochastic fluctuations
[18–20]. The importance for robustness is a functional criterion to
characterize the performance of biochemical networks [21], and it
can be used as a measure for determining plausibility among
different competing models, assuming that biological designs
enhance robustness. We proposed quasi-multiparameter sensitivity
(QMPS) as a numerical and fast measure of robustness to the
uncertainty of all kinetic parameters [22].
In general, feedback loops can be distinguished in terms of
topological features: loop length, loop redundancy, and coupling
types of multiple loops. It is known that a negative feedback with a
long reaction chain generates an oscillator more readily than one
with short chains [23]. By using TPS and QMPS, we demonstrated
that long-chain feedback loop has potential to present a robust
oscillator through the mechanism of distributed time delays [22]. In
engineering and biology, redundancy is the main pillar of system’s
robustness. Genetic redundancy enables reliable development
against fluctuatingenvironment andmutations [24–27].Redundant
metabolic pathway reduces the sensitivity to enzyme activity for the
flux and concentration of end products [22,28]. In circadian clocks,
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it is critically important to understand how multiple, complex
feedback loops are designed for robust oscillations. Stelling et al.
characterized the robustness of three types of feedback models: the
single, dual, and redundant feedback models by using multi-
parameter perturbation analysis, suggesting that the most robust
model is not the dual feedback model (employed by real biological
systems) but the redundant feedback model (a hypothetical model)
[15]. It may be a widely-recognized hypothesis, but a question
arises: how the dual feedback architecture survives against the
redundant feedback architecture despite less robustness of it.
On the other hand, the entrainment to a fluctuating environment
was numerically analyzed to validate mathematical models for
circadian clocks [5,6,8,10]. Gonze and Goldbeter investigated the
occurrence of various modes of dynamic behaviors as a function of
theforcingperiodandoftheamplitudewith respecttolight-induced
changes in kinetic parameters [29]. Kurosawa and Goldbeter
examined how the entrainment of these rhythms is affected by the
free-running period (period under constant darkness) and by the
amplitude of the external light-dark cycle [30]. However, their
analyses largely depended on a particular choice of kinetic
parameters. Considering that kinetic parameters constantly fluctu-
ate within a cell, the network structure should be the main source of
the capability of entrainment. Therefore, global, firm analysis is
required to reveal how a particular structure in circadian clocks is
related to light entrainment.
By using rigorous numerical methods of TPS and QMPS, we
reveal the mechanism of how particular feedback architecture is
related to robustness and to entrainment to light-dark cycles in
circadian clocks. Furthermore, we demonstrate how the dual
feedback architecture evolves out of many competing feedback
systems, correcting or overcoming the existing hypothesis.
Results and Discussion
Biochemical Models
In many biological models feedback loops are connected in
different manners. Architectures of feedback loops are featured by
loop redundancy and coupling of multiple loops. To analyze how
the loop coupling logic affects robustness and entrainment, the
single, semi-dual, dual, and redundant feedback models were
constructed by simplifying or refining the feedback models
described elsewhere [4,5,12,15]. The network maps are shown
in Figure 1. The mathematical equations and their associated
parameters are shown in Tables 1 and 2, respectively.
Figure 1. Biochemical network maps of the circadian clock models with different types of loop coupling logics. A: The single feedback
model, B: the semi-dual feedback model, C: the dual feedback model, D: the redundant feedback model. The notation of CADLIVE [46–48] is used for
simplifying the diagram. The dashed circle represents nucleus.
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The single feedback model (Figure 1A) is a simplified version of
Drosophila PER feedback model [4]. In this model, X gene expression
isnegativelycontrolledbyclockproteinX.Thesinglefeedbackmodel
is used as the reference or control. The clock protein in nucleus is
regarded as the output component from the circadian system as the
clock proteins control gene expressions in vivo. In the single feedback
model, X in nucleus (X(nuc)) is considered as the output component.
The semi-dual feedback model (Figure 1B) is a refined version of
the dCLK-CYC feedback loop in the interlocked feedback model
for Drosophila circadian clock [12]. In the semi-dual feedback model,
the synthesis of protein X is negatively regulated by the heterodimer
of X:Y, while the synthesis of protein Y occurs constitutively. In this
model, the X:Y complex in nucleus (X:Y(nuc)) is considered as the
output component. In Drosophila, X and Y correspond to dCLK and
CYC, respectively. CYC is reported to be abundant compared to
dCLK [31]. The question arises as to whether Y level in the semi-
dual feedback model contributes to the performance in robustness
and entrainment.
The dual feedback model (Figure 1C) is a simplified version of
the Drosophila PER-TIM feedback model [5]. In this model, the X
and Y feedback loops are coupled via the complex of X:Y and the
syntheses of both X and Y are negatively regulated by X:Y. In
Drosophila, X and Y correspond to PER and TIM. Unlike the semi-
dual feedback model, the Y loop in the dual feedback model is
closed to form a negative feedback control, where the Y
concentration oscillates with X. In this model, the X:Y complex
in nucleus (X:Y(nuc)) is considered as the output component. The
dual feedback model has the symmetric structure of X and Y loops.
By assigning the same values to the corresponding kinetic
parameters of both the X and Y loops (e.g., S1to S3, D1to D3),
Table 1. Dynamic models of competing circadian clock models.
Model name Equation
The single feedback model (m=3)
d½mRNA(X)?
dt
~S1
K1h
K1hz½X(nuc)?h{D1
½mRNA(X)?
L1z½mRNA(X)?{D4½mRNA(X)?
½X(nuc)?
U2z½X(nuc)?{D2
½X(nuc)?
U2z½X(nuc)?{D3
K1h
K1hz½X : Y(nuc)?h{D1
1
d½X?
dt
~S2½mRNA(X)?{T1
½X?
U1z½X?zT2
½X?
L2z½X?{D5½X?
2
d½X(nuc)?
dt
~T1
½X?
U1z½X?{T2
½X(nuc)?
L3z½X(nuc)?{D6½X(nuc)?
½mRNA(X)?
L1z½mRNA(X)?{D7½mRNA(X)?
½X?
L2z½X?{D8½X?
3
The semi-dual feedback model (m=6)
d½mRNA(X)?
dt
~S1
1
d½X?
dt
~S2½mRNA(X)?{B1½X?½Y?zB2½X : Y?{D2
2
d½mRNA(Y)?
dt
~S3{D3
½mRNA(Y)?
L3z½mRNA(Y)?{D9½mRNA(Y)?
3
d½Y?
dt
~S4½mRNA(Y)?{B1½X?½Y?zB2½X : Y?{D4
½Y?
L4z½Y?{D10½Y?
½X : Y(nuc)?
U2z½X : Y(nuc)?
4
d½X : Y?
dt
~B1½X?½Y?{B2½X : Y?{T1
½X : Y?
L5z½X : Y?{D11½X : Y?
d½X : Y(nuc)?
dt
{D12½X : Y(nuc)?
The equations except No. 3 are the same as those of the semi-dual feedback model.
½X : Y?
U1z½X : Y?zT2
{D5
5
~T1
½X : Y?
U1z½X : Y?{T2
½X : Y(nuc)?
U2z½X : Y(nuc)?{D6
½X : Y(nuc)?
L6z½X : Y(nuc)?
6
The dual feedback model (m=6)
d½mRNA(Y)?
dt
~S3
K2h
K2hz½X : Y(nuc)?h{D3
K1h
K1hz(½X(nuc)?z½Y(nuc)?)h{D1
{D7½mRNA(X)?
½X?
U1z½X?zT2
½X?
U1z½X?{T2
d½mRNA(Y)?
dt
{D10½mRNA(Y)?
d½Y?
dt
½mRNA(Y)?
L3z½mRNA(Y)?{D9½mRNA(Y)?
½mRNA(X)?
L1z½mRNA(X)?
3
The redundant feedback model (m=6)
d½mRNA(X)?
dt
~S1
1
d½X?
dt
~S2½mRNA(X)?{T1
½X(nuc)?
U2z½X(nuc)?{D2
½X?
L2z½X?{D8½X?
2
d½X(nuc)?
dt
~T1
½X(nuc)?
U2z½X(nuc)?{D3
K2h
½X(nuc)?
L3z½X(nuc)?{D9½X(nuc)?
½mRNA(Y)?
L4z½mRNA(Y)?
3
~S3
K2hz(½X(nuc)?z½Y(nuc)?)h{D4
4
~S4½mRNA(Y)?{T3
½Y?
U3z½Y?zT4
½Y(nuc)?
U4z½Y(nuc)?{D6
½Y(nuc)?
U4z½Y(nuc)?{D5
½Y?
L5z½Y?{D11½Y?
5
d½Y(nuc)?
dt
~T3
½Y?
U3z½Y?{T4
½Y(nuc)?
L6z½Y(nuc)?{D12½Y(nuc)?
6
[mRNA(X)] is mRNA for protein X, [X] protein X, and [X(nuc)] protein X in nucleus. [mRNA(Y)], [Y], and [Y(nuc)] are named in the same manner. [X:Y] is the binding complex
of X and Y, [X:Y(nuc)] the complex in nucleus. m is the number of equations for each model.
doi:10.1371/journal.pone.0030489.t001
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the dual feedback model shows a kinetic symmetry between the X
and Y loops.
The redundant feedback model (Figure 1D), which is not seen in
organisms, was presented as a competitive model to the dual
feedback model [15]. It has a symmetric structure between the X
and Y loops, where X and Y independently regulate the syntheses
of themselves. The total amount of X and Y in nucleus
(X(nuc)+Y(nuc)) is the output. The redundant feedback model was
mentioned to be the most robust oscillator for all the models.
Robustness Analysis
We investigated the mechanism by which different coupling
logics: the single, semi-dual, dual, and redundant feedback models
provide robustness to perturbation to an entire system. To extract
the global properties of those models, all plausible kinetic
parameter sets that generate circadian oscillations in constant
darkness are searched to characterize their oscillatory features. It is
important that the conclusion is independent of a particular choice
of kinetic parameter values. To this end, we used two-phase search
(TPS) method and quasi-multiparameter sensitivity (QMPS). For
details, see Materials and Methods.
The semi-dual feedback model
For the semi-dual feedback model, we simulated the QMPS of
the period and amplitude of the oscillatory behaviors of the output
component (X:Y(nuc)). Especially, we analyzed how the robustness
depends on the Y level. The kinetic parameter values were searched
by TPS so as to provide typical oscillatory behaviors: The period
and amplitude for the oscillation of the output component were set
to 23–25 h and to 2–6 nM, respectively. In the parameter search,
the total Y level was constrained within the specific range (less than
10 nM, between 10 and 200 nM, or more than 200 nM). The
search parameter values were varied by 0.1–10 fold with respect to
the reference values as shown in Table 2. For each range of the Y
level, a thousand of the solution parameter sets were obtained that
produce the given oscillatory features. For all the solutions, QMPSs
were calculated with respect to the period and amplitude of the
output component oscillations. The cumulative frequency distribu-
tions for QMPS are shown in Figure 2. The QMPSs for period and
amplitude decrease with an increase in the amount of Y, and their
QMPSdistributionsapproachtothoseinthe singlefeedbackmodel.
With respect to robustness in oscillations, the semi-dual feedback
model can be comparable to the single feedback model, but cannot
overwhelm it.
The semi-dual feedback model was derived from the dCLK-
CYC feedback loop in Drosophila [12], where X and Y correspond
to dCLK and CYC, respectively. Our results predict that a high
amount of CYC leads to robust oscillations. Indeed, the
concentration of CYC is much higher than that of dCLK in
Drosophila [31]. In mammals, BMAL1 and CLK appear to
constitute a semi-dual feedback architecture [6], where X and Y
correspond to BMAL1 and CLK, respectively. The level of CLK is
higher than that of BMAL1 [32,33]. These observations indicate
that living systems have evolved to increase in the level of protein Y
in order to cope with parameter uncertainty.
The dual feedback model
The dual feedback model has the symmetric structure of the X
and Y loops. Here parameter r ($0) is introduced, which adjusts
the symmetry between X and Y loops in terms of kinetics. In TPS,
the search space for the kinetic parameters associated with the Y
loop (Si+2, i=1,2; K2; Di+2, i=1,2,7,8; Li+2, i=1,2) is provided by r
and by the kinetic parameters associated with the X loop:
Si
1zrƒSiz2ƒSi(1zr)(i~1,2)
K1
1zrƒK2ƒK1(1zr)
Di
1zrƒDiz2ƒDi(1zr)(i~1,2,7,8)
Li
1zrƒLiz2ƒLi(1zr)(i~1,2)
ð1Þ
As the value of r decreases, the values of the kinetic parameters for
the Y loop approach to those for the X loop, increasing kinetic
symmetry. When the value of r is zero, the values of the kinetic
parameters associated with the X and Y loops are exactly the same
and the two loops are perfectly symmetric. When the value of r is
large enough ($99), the kinetic parameter values for both the X
and Y loops are independently assigned.
We simulated the QMPS of the period and amplitude of the
output component (X:Y(nuc)) of the dual feedback model with
different values of r. The kinetic parameter values were
searched by TPS so as to provide typical oscillatory behaviors,
where they were varied by 0.1–10 fold with respect to the
reference values as shown in Table 2. In addition, the search
space for the kinetic parameters associated with the Y loop is
further constrained by Eqs. (1). For each r value, a thousand of
the solution parameter sets were obtained that produce the
target oscillatory behaviors. For all the solutions QMPSs were
calculated with respect to the period and amplitude of the output
component oscillations. The cumulative frequency distributions
for QMPS are shown in Figure 3. A decrease in the r value
decreases the QMPS values with respect to both period and
amplitude, indicating that the kinetic symmetry of feedback
loops enhances the robustness of the oscillation to uncertainty of
multiple parameters. When r equals to zero, the dual feedback
model provides the most robust oscillator and greatly over-
whelms the single feedback model.
Here, we present the hypothesis that the dual feedback model
evolves toward the increased kinetic symmetry between the X and Y
loops. A dual feedbackarchitecture isfoundasthe PER-TIM system
in the Drosophila, and the symmetry between the two feedback loops
are frequently assumed [5,10,11]. Although it is unclear whether
these feedback loops are kinetically symmetric in vivo, many
experimental data suggested that the processes of PER and TIM
have the similar values of kinetic parameters. The E-box motif,
which is a target for transcription factors dCLK and CYC, is located
upstream of both the per and tim genes [34–37]. Therefore, the
dCLK:CYC complex seems to have almost the same affinities to the
per and tim promoters. The per and tim transcripts cycle in abundance
with similar amplitudes and phases [38]. The time courses of PER
and TIM are similar in shape and largely overlap [31,39]. These
experimental data support the hypothesis that the PER-TIM dual
feedback system is designed to hold the kinetic symmetry, providing
the robustness to uncertainty of kinetic parameters.
The redundant feedback model
The redundant feedback model has the symmetric structure of
the X and Y feedback loops. As shown in the section for the dual
feedback model, we use parameter r ($0). In TPS, the search
space for the kinetic parameters associated with the Y loop (Si+2,
i=1,2; K2; Ti+2, i=1,2; Ui+2, i=1,2; Di+3, i=1,2,3,7,8,9; Li+3,
i=1,2,3) is determined by r and by the kinetic parameters
associated with the X loop:
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Si
1zrƒSiz2ƒSi(1zr)(i~1,2)
K1
1zrƒK2ƒK1(1zr)
Ti
1zrƒTiz2ƒTi(1zr)(i~1,2)
Ui
1zrƒUiz2ƒUi(1zr)(i~1,2)
Di
1zrƒDiz3ƒDi(1zr)(i~1,2,3,7,8,9)
Li
1zrƒLiz3ƒLi(1zr)(i~1,2,3)
ð2Þ
The QMPS for the period and amplitude of the output
(X(nuc)+Y(nuc)) was analyzed. The kinetic parameters were
searched by TPS so as to provide typical oscillatory behaviors,
where they were varied by 0.1–10 fold with respect to the
reference values as shown in Table 2. In addition, the search space
for the kinetic parameters associated with the Y loop is further
constrained by Eqs. (2). For each r value, a thousand of the
solution parameter sets were obtained. For all the solutions
QMPSs were calculated with respect to the period and amplitude
of the output oscillations. The cumulative frequency distributions
for QMPS are shown in Figure 4. At a r value of more than 0.1,
the QMPS values increase with a decrease in the r value. At a r
value of smaller than 0.1, the QMPS values decrease with a
decrease in r.
To understand the complex behaviors shown in Figure 4, the
quantitative balance of the X and Y feedback loops c is defined by:
c~
½X(nuc)?mean
½X(nuc)?meanz½Y(nuc)?mean
,
ð3Þ
where [X(nuc)]mean indicates the mean concentration for X in
nucleus and [Y(nuc)]meanthat for Y in nucleus. When c is close to
Table 2. List of kinetic parameters for competing circadian clock models.
ParameterDefinitionValue
S2i{1(i~1,2)Maximum rates for transcriptions 1.0 [nM h21]
S2i(i~1,2)Translation rate constants 1.0 [h21]
Ki(i~1,2)Affinity constants for transcriptions1.0 [nM]
B1
Association constant 1.0 [nM21h21]
B2
Dissociation constant 1.0 [h21]
T2i{1(i~1,2) Maximum rates for transportations (cytoplasm to nucleus)1.0 [nM h21]
T2i(i~1,2) Maximum rates for transportations (nucleus to cytoplasm)1.0 [nM h21]
U2i{1(i~1,2)Affinity constants for transportations (cytoplasm to nucleus)1.0 [nM]
U2i(i~1,2)Affinity constants for transportations (nucleus to cytoplasm)1.0 [nM]
Di(i~1,2,:::,m) Maximum rates for degradations1.0 [nM h21]
Dizm(i~1,2,:::,m) Degradation rate constants0.01 [h21]
Li(i~1,2,:::,m)Affinity constants for degradations 1.0 [nM]
h Hill coefficient4.0 (fixed)
m is the number of equations for each model. Since the values of kinetic parameters are hardly measured in circadian oscillators, the followings are employed as the
reference values.
doi:10.1371/journal.pone.0030489.t002
Figure 2. Cumulative frequency distributions of QMPS for the oscillatory behaviors in the semi-dual feedback model. A: QMPS for
period, B: QMPS for amplitude. The level of protein Y was changed in the semi-dual feedback model: Y,10 nM (cross), 10 nM#Y#200 nM (circle),
Y.200 nM (square). The single feedback model (plus) is the control model.
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zero, the effect of the X loop on the oscillator is weak, while the Y
loop is dominant. When c is close to 0.5, the effects of both the X
and Y loops are comparable. When c is close to one, the X loop is
dominant. The c distributions for the redundant feedback model
with respect to r are shown in Figure 5. When r is set to a large
value, the values of c are biased towards zero or one, indicating that
kinetic symmetry is not generated: the level of X in nucleus oscillates
with a negligible level of Y in nucleus, and vice versa. A value of c
approaches to 0.5 with a decrease in r. A r value of less than 0.01,
strong confinement to kinetic symmetry, is required to generate the
oscillations with the comparable levels of X and Y in nucleus.
As shown in Figure 4, the distribution of QMPS for the
redundant feedback model is the same as that of the single
feedback model at a large value of r, where c is almost zero or one
(Figure 5). Such an asymmetric redundant feedback model can be
the same as the single feedback model. Under a weak constrain to
kinetic symmetry or a r value of more than 0.1, QMPS increases
with a decrease in r. It is probably because the oscillatory
behaviors of the X and Y loops would interfere with each other,
i.e., the intrinsic cycles of the two loops are not consistent. Under
the strong constraint to kinetic symmetry or a r value of smaller
than 0.1, the QMPS values decrease with a decrease in r. When r
is set to zero, the frequency distribution is biased toward a small
value of QMPS. The redundant feedback model with exact kinetic
symmetry became a more robust oscillator to uncertainty of all
kinetic parameters than the single feedback model. The redundant
feedback model provides robustness, when the kinetics of both the
feedback loops is symmetric or either of the two feedback loops is
negligible or dominant (a virtual single feedback model). Under
the other conditions, the robustness is readily decreased, probably
because the cycles by the two feedback loops interfere with each
other.
The redundant feedback model has the potential to produce
more robust oscillation than that of the single feedback model.
However, the redundant feedback model seems to have difficulty
in evolving toward the kinetic symmetry of the two feedback loops,
because a break in the symmetry readily destroys the robust
oscillations (e.g. r=0.1). This may be the reason that redundant
feedback oscillators are not seen in biology.
In the dual and redundant feedback models, the two feedback
loops should coordinate to oscillate. Otherwise, the cycles of both
the feedback loops interfere with each other. Kinetic symmetry is
thus required for robust oscillators. In the dual feedback model,
the two feedback loops are tightly coordinated by binding the two
proteins. In the redundant feedback model, the two feedback loops
are loosely connected at the transcription regulations. Different
types of loop coupling alter the robustness.
Comparison among the feedback models
In summary, we compared the QMPS distributions for the
single, semi-dual, dual, and redundant feedback models. The
QMPS distributions with respect to period and amplitude are
shown in Figure 6. The distributions for the single and semi-dual
feedback models are almost identical. When the r value is set to
zero, the QMPS values of amplitude for the dual and redundant
feedback models are comparable and lower than other models.
With respect to period, the QMPS values of the dual feedback
models are lower than any other model. The dual feedback model
can be the most robust oscillator.
In [15], the authors used the Monte Carlo simulations to
compare the robustness to uncertainty of all parameters between
the competing models: the PER single feedback model [4], the
PER-TIM dual feedback model [5], and the PER-TIM redundant
feedback model. They suggested that the redundant feedback
model is the most robust oscillator for these models and the dual
feedback model is less robust than the single feedback model.
However, our extensive analysis presents an alternative hypothesis
to the existing one: the dual feedback model has the potential to
provide the most robust oscillator to multiple parameter
perturbations. The parameter search in their work [15] was very
short, where they used only dozens of parameter sets with respect
to each model. In the Monte Carlo method, they run only 200
simulations for each parameter set. To reliably compare
alternative models, the efficient search for large parameter space
is required and at least ten thousand simulation runs are needed
for each reference parameter set [22].
Entrainment Probability Maps for the Feedback Models
Since the values of kinetic parameters constantly fluctuate
within a cell, the entrainment of circadian rhythms should not rely
on the fine-tuned values of them. To elucidate a mechanism by
which oscillations are entrained to light-dark (LD) cycles, we
performed the entrainment analysis [29] for all the kinetic
parameter sets used in the section of Robustness Analysis. In
Drosophila, light induces the degradation of clock protein TIM,
which allows circadian oscillations to entrain to a diurnal cycle
[40–42]. To consider the light-increased degradation rate for the
clock protein, the maximum rate for degradation of X was
Figure 3. Cumulative frequency distributions of QMPS for the oscillatory behaviors in the dual feedback models. A: QMPS for period,
B: QMPS for amplitude. The kinetic symmetry (r) was changed in the dual feedback model: r$99 (cross), r=1 (circle), r=0.1 (square), r=0.01
(diamond), r=0 (triangle). The single feedback model (plus) is the control model. A decrease in r increases the kinetic symmetry.
doi:10.1371/journal.pone.0030489.g003
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increased during light phase. The increase rate in the parameter
and the forcing period of LD cycles are denoted as d and f,
respectively. For each parameter set that produces typical
circadian rhythm in constant darkness, the region where
oscillations are successfully entrained is provided as a function of
f and d. The probability of entrainment at point f-d is determined
by counting the number of parameter sets that successfully entrain
to LD cycles with f and d. For details, see Materials and Methods.
For the single, semi-dual, dual, and redundant feedback models,
the entrainment probability map was drawn with respect to the
period of LD cycles (f) and the intensity of light (d). The
entrainment probability map for the single feedback model is
shown in Figure 7A. The single feedback model entrains 24 h LD
cycle, which is almost equal to the free-running period (the period
under constant darkness). When the period of LD cycles is far from
24 h, irregular oscillations (quasi-periodic and chaotic oscillations)
occur and cannot entrain to LD cycles. Entrainment occurs with
relatively weak light stimuli, while strong light causes irregular
oscillations. The maps for the semi-dual feedback model with
various concentrations of Y are shown in Figure S1. The change in
the level of Y does not dramatically affect ability of entrainment. At
a Y level of more than 200 nM (Figure 7B) the entrainment
probability map is similar to that of the single feedback model. In
the dual feedback model, at r of more than one, kinetic symmetry
increases the probability of entrainment (Figure S2). The
kinetically symmetric dual feedback model (r=0) (Figure 7C)
readily entrains to LD cycles in the widest space of the forcing
period and light intensity. The dual feedback model with kinetic
symmetry is the most reasonable choice for light entrainment. In
the redundant feedback model, kinetic symmetry increases the
probability of entrainment (Figure S3), but the perfect kinetic
symmetry (r=0) (Figure 7D) does not entrain to LD cycles more
than the single feedback model.
For all the models, forcing periods away from the free-running
period or strong light stimuli caused irregular oscillations. These
results are consistent with the previous works [29,30]. It is
important for successful entrainment to incorporate the forcing
period in intrinsic oscillations while avoiding irregular oscillations.
In the dual feedback model, the light-perturbed X loop and the
unperturbed Y loop are merged at the complex X:Y that regulates
the transcriptions of X and Y. This coupling mechanism is
suggested to effectively incorporate the forcing period into the
system, avoiding irregular oscillations. In the redundant feedback
model, since the X and Y loops separately regulate transcriptions,
the difference in the oscillations between the light-perturbed X
loop and the unperturbed Y loop is directly transmitted to the
transcription regulations, thereby causing irregular oscillations or
failure in the entrainment to LD cycles.
Key Devices for Robust and Entrainable Oscillators
The complex feedback system of the circadian clocks is one of
the most extensively studied systems, which generates the
robustness to the uncertainty of kinetic parameters and entrains
to periodic changes in the environment. However, it remains
unclear how such complex feedback loops are designed, while the
single or simple feedback can be a sufficiently robust and
entrainable oscillator. By performing global numerical analyses
by TPS, we demonstrated that the dual feedback model is the most
reasonable choice for creating a robust and entrainable oscillator
out of various types of loop couplings, which corrects or overcomes
Figure 4. Cumulative frequency distributions of QMPS for the oscillatory behaviors in the redundant feedback models. A: QMPS for
period, B: QMPS for amplitude. The kinetic symmetry (r) was changed in the redundant feedback model: r$99 (cross), r=1 (circle), r=0.1 (square),
r=0.01 (diamond), r=0 (triangle). The single feedback model (plus) is the control model. A decrease in r increases the kinetic symmetry.
doi:10.1371/journal.pone.0030489.g004
Figure 5. Frequency distributions of the c values for the
parameter sets that yield circadian oscillation. The frequency
distributions of the quantitative balance between X and Y loops (c) were
simulated, while changing the kinetic symmetry (r): r$99 (cross), r=1
(circle), r=0.1 (square), r=0.01 (diamond). A decrease in r increases the
kinetic symmetry. c is the quantitative balance of the X and Y feedback
½X(nuc)?mean
½X(nuc)?meanz½Y(nuc)?mean, where [X(nuc)]mean
indicates the mean concentration for X in nucleus and [Y(nuc)]meanthat
for Y in nucleus. When c is close to zero, the effect of the X loop on the
oscillator is weak, while the Y loop is dominant. When c is close to 0.5, the
effects of both the X and Y loops are comparable. When c is close to one,
the X loop is dominant. At r=0 (perfect kinetic symmetry), c is always
equal to 0.5. The distribution for r=0 is not shown.
doi:10.1371/journal.pone.0030489.g005
loops, which is defined by: c~
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the existing hypothesis [15]. The dual feedback model, employed
by a real biological system, provides the most robust oscillator and
readily entrains LD cycles. Furthermore kinetic symmetry in the
dual feedback model enhances the robustness with respect to
uncertainty of multiple parameters and the ability to entrain to a
LD cycle, providing an insight on the mechanism by which the
dual feedback model evolves toward kinetic symmetry for
enhanced robustness and entrainments. We discovered smart,
intelligent devises hidden in the real biological circadian
oscillators. The key devices are the coupling of two feedback
loops by forming a protein complex and kinetic symmetry between
them, which generate remarkable robustness to perturbations
within a cell and enable entrainment to a wide range of light
signals.
Materials and Methods
Mathematical Comparison between Competing Models
In order to understand the mechanism by which a specific
regulation generates a particular cellular function, it is important
to compare the performance criteria between the competing
models: the model containing the specific regulation and that
without it. While system’s properties are affected by not only the
network structure but also the values of the kinetic parameters, the
conclusion of the comparison should be independent of a change
in parameters [15,22,43]. In competing models, the kinetic
parameter values are searched within a certain range by the
two-phase search (TPS) method to find all possible solutions that
generate target behaviors. TPS is presented to systematically
analyze the dynamic behaviors in a large parameter space by
searching all plausible parameter values without any biases [17].
TPS consists of a random search and an evolutionary search
(genetic algorithms: GAs) to effectively explore all possible solution
sets of kinetic parameters satisfying a target or desired dynamics.
The algorithm of TPS is described in Text S1 and Figure S4.
Our mathematical comparison method is basically the same as
mathematically controlled comparison introduced by Alves and
Savageau [43], where parameter search is randomly performed to
guarantee equivalence among alternative models, and then the
models are statistically compared with respect to the property of
interest. First, we search the values of the kinetic parameters
providing circadian oscillations to hold behavioral equivalence
among all feedback models. Second, we statistically compare the
distributions of QMPS among the feedback models. Alves and
Savageau employed random search to obtain the values of kinetic
parameters yielding biologically plausible behaviors, allowing fare
comparison among alternative models. Instead of random search,
we use TPS to greatly reduce computational cost. The
distributions of solutions obtained by TPS were demonstrated to
be statistically the same as those by random search [17].
Quasi-multiparameter Sensitivity (QMPS)
Generally a dynamic model for biochemical networks is
formulated by ordinary differential equations:
:x~F(t,x,p),
ð4Þ
where t is time, x is the vector whose elements are the variables for
molecular concentrations, p=(p1,p2,…,pn) is the kinetic parameter
vector, and n is the number of kinetic parameters. Let q(p) be a
given target function. A single-parameter sensitivity of the target
function with respect to a change in the ith parameter is defined
as:
si(q,p)~Lq(p)
Lpi
pi
q(p)~Lln q(p)
Lln pi
:
ð5Þ
A single-parameter sensitivity identifies sensitive or insensitive
reactions to a target function in a biochemical network, while it
yields only linear approximations of the target function to single
parameter perturbation. Single-parameter sensitivity analysis does
not estimate the robustness to the uncertainty of all parameters.
Assuming that the relative change in the target function is linear to
a change in each parameter, multiparameter sensitivity (MPS)
[44,45] is defined by:
MPS(q,p)
jj2~
X
n
i~1
si(q,p)2:
ð6Þ
For complex models (including Table 1), it is generally hard to
analytically compute MPS. As a practical solution, quasi-
multiparameter sensitivity (QMPS) is proposed, where the single-
parameter sensitivities are numerically simulated by providing a
small perturbation to a kinetic parameter. When each kinetic
parameter pi is perturbed as given by p’i~pi(1zD), MPS is
defined as the square sum of single-parameter sensitivities:
Figure 6. Cumulative frequency distributions of QMPS for the oscillatory behaviors in the competing models. A: QMPS for period, B:
QMPS for amplitude. The single feedback model (plus), the semi-dual feedback model with Y.200 nM (cross), the dual feedback model with r=0
(circle), the redundant feedback model with r=0 (square). r=0 indicates perfect symmetry between two feedback loops.
doi:10.1371/journal.pone.0030489.g006
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lim
D?0
hand, QMPS is defined by the square sum oflnq(p’){lnq(p)
lnq(p’){lnq(p)
lnp’i{lnpi
, where D approaches to zero. On the other
lnp’i{lnpi
,
where D is not nearly zero but a small value, e.g. D=0.001. The
algorithm for computing QMPS is described in Text S2. In this
study the single-parameter sensitivities are calculated by setting the
value of D to 0.001. Since the calculated values are consistent at D
of less than 0.01, they can be used as the single-parameter
sensitivities.
QMPS is not only computationally efficient but also consistent
with the normalized variance for the target function obtained by
Monte Carlo method. In general the Monte Carlo method is used
to quantify robustness to multiple parameter perturbations, where
all kinetic parameters are simultaneously varied and the
normalized variance for the target function can be used as the
indicator for robustness. On the other hand, QMPS is theoret-
ically defined under the condition that a change in all parameter
values is infinitesimal, while QMPS is demonstrated to practically
be available under the condition that variations in all kinetic
parameters are less than 10%. By using many mathematical
models, we demonstrated that QMPS is consistent with the
normalized variance by the Monte Carlo method even if the rate
of variations in all kinetic parameters is set to as high as 10% [22].
QMPS is employed with TPS, thereby enabling the numerical
comparison of robustness among alternative or competing models.
First, we search the kinetic parameter values by TPS so as to
provide typical oscillatory behaviors: in constant darkness the
period and amplitude for the oscillation of the output component
are set to 23–25 h and to 2–6 nM, respectively. The fitness
function is described in Text S3. In general, oscillations with
approximately 24 h period and large amplitude are regarded as
typical circadian oscillation. The target period and amplitude are
provided based on experimental data and theoretical studies
[4,5,31], and their ranges are conveniently determined to obtain
many solutions. Narrow ranges provide only a small number of
solutions due to calculation complexity. Second, the QMPSs are
calculated for all the solution parameter sets. When comparing the
robustness of two alternative dynamic models, one model with a
lower value of QMPS is more robust than the other. In this
analysis the cumulative frequency is used as a function of the value
of QMPS squared in order to characterize the robustness among
alternative models with a variety of kinetic parameter values,
generated by TPS. The cumulative frequency (CF) of a QMPS
value x is given by:
CF(x)~f(Xƒx),
ð7Þ
where X is a random variable and f represents the frequency that X
takes on a value less than or equal to x. According to this criterion,
a higher cumulative frequency distribution indicates higher
robustness and a higher cumulative frequency at a lower QMPS
provides higher robustness. When two cumulative frequency
curves to be compared intersect, statistical analysis by median is
Figure 7. Entrainment probability maps for different types of feedback models. A: the single feedback model, B: the semi-dual feedback
model with Y.200 nM, C: the dual feedback model with r=0, D: the redundant feedback model with r=0. r=0 indicates perfect symmetry
between the two feedback loops. The color indicates the probability of entrainment given by Eq. (9), the ratio of the parameter sets that entrain to
light-dark cycles to all the parameter sets. Red color indicates a high probability, where the cycle readily entrains to the light-dark cycle; blue a low
one.
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used instead of an intuitive analysis. The median corresponds to
the value of QMPS that provides a cumulative frequency of 0.5. A
low value of median provides high robustness.
Entrainment Probability Map
To elucidate a mechanism of how oscillations are entrained to
light-dark (LD) cycles, the entrainment analysis [29] is performed
for all the kinetic parameter sets, searched by TPS, that generate
typical oscillations in constant darkness. In Drosophila, light induces
the degradation of clock protein TIM, which allows circadian
oscillations to entrain to a diurnal cycle [40–42]. To consider the
light-increased degradation rate for the clock protein, the
maximum rate for degradation of clock protein X is increased
during light phase as follows:
D2’~D2(1zd)withdw0,
ð8Þ
where d is the factor that increases D2. In addition to the
degradation of X, in the semi-dual and dual feedback models, the
degradation of the complex of X and Y (X:Y) is enhanced in the
same manner as Eq. (8). The forcing period of LD cycles is defined
as f. Light and dark phases in the LD cycle are set to the same in
length. Using all the parameter sets that produce typical circadian
rhythms in constant darkness, the region where oscillation is
successfully entrained is provided as a function of f and d. The
probability of entrainment at point f-d is determined by:
Pr(f,d)~1
N
X
N
i~1
En(pi,f,d),
ð9Þ
where piis the ith parameter set, N is the total number of the
parameter sets, and En is the function that gives one when
successful entrainment occurs, otherwise zero. By ‘‘successful
entrainment’’, we mean that period and amplitude of oscillations
are constant (less than 1% and 10% of deviations, respectively) and
the period of oscillations is the same as the forcing period (less than
1% of deviation).
Supporting Information
Figure S1
dual feedback model with various amounts of Y. A:
Y,10 nM, B: 10 nM#Y#200 nM, C: Y.200 nM. The color
indicates the probability of entrainment given by Eq. (9), the ratio
of the parameter sets that entrain to light-dark cycles to all the
Entrainment probability maps for the semi-
parameter sets. Red color indicates a high probability, where the
cycle readily entrains to the light-dark cycle; blue a low one.
(PNG)
Figure S2
feedback model with various r values. A: r$99, B: r=1,
C: r=0.1, D: r=0.01. E: r=0. A decrease in r increases the
kinetic symmetry. The color indicates the probability of entrain-
ment given by Eq. (9), the ratio of the parameter sets that entrain
to light-dark cycles to all the parameter sets. Red color indicates a
high probability, where the cycle readily entrains to the light-dark
cycle; blue a low one.
(PNG)
Entrainment probability maps for the dual
Figure S3
dant feedback model with various r values. A: r$99, B:
r=1, C: r=0.1, D: r=0.01. E: r=0. A decrease in r increases
the kinetic symmetry. The color indicates the probability of
entrainment given by Eq. (9), the ratio of the parameter sets that
entrain to light-dark cycles to all the parameter sets. Red color
indicates a high probability, where the cycle readily entrains to the
light-dark cycle; blue a low one.
(PNG)
Entrainment probability maps for the redun-
Figure S4
in the two-phase search (TPS) method. In this figure, the
dimension of parameter space is assumed to be two. L=(L1,L2)
and U=(U1,U2) are the lower and upper bound vectors,
respectively. RS stands for random search.
(PNG)
Initial population for genetic algorithm (GA)
Text S1
method.
(PDF)
The algorithm of the two-phase search (TPS)
Text S2
rameter sensitivity (QMPS).
(PDF)
The algorithm for computing quasi-multipa-
Text S3
(PDF)
The fitness function for circadian oscillations.
Acknowledgments
The super-computing resource was provided by Human Genome Center,
Institute of Medical Science, University of Tokyo.
Author Contributions
Conceived and designed the experiments: KM HK. Performed the
experiments: KM. Analyzed the data: KM HK. Contributed reagents/
materials/analysis tools: KM HK. Wrote the paper: KM HK.
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