Physical Biology

Phys. Biol. 11 (2014) 045002 (14pp) doi:10.1088/1478-3975/11/4/045002

The Goodwin model revisited: Hopfbifurcation, limit-cycle, and periodicentrainment

Aurore Woller1,2, Didier Gonze3 and Thomas Erneux4

1 Unite Mixte de Recherche 1011, Universite Lille 2, INSERM, Institut Pasteur de Lille, Lille, France2 Laboratoire de Physique des Lasers, Atomes, Molecules, Unite Mixte de Recherche 8523, UniversiteLille 1, CNRS, Villeneuve d’Ascq, France3 Unite de Chronobiologie Theorique, CP 231, Universite Libre de Bruxelles, Bvd du Triomphe,B-1050 Bruxelles, Belgium4 Optique Nonlineaire Theorique, CP 231, Universite Libre de Bruxelles, Bvd du Triomphe,B-1050 Bruxelles, Belgium

E-mail: dgonze@ulb.ac.be and terneux@ulb.ac.be

Received 12 October 2013, revised 7 January 2014Accepted for publication 6 February 2014Published 30 July 2014

AbstractThe three-variable Goodwin oscillator is a minimal model demonstrating the emergence ofoscillations in simple biochemical feedback systems. As a prototypical oscillator, this modelwas extensively studied from a theoretical point of view and applied to various biologicalsystems, including circadian clocks. Here, we reexamine this model, derive analytically theamplitude equation near the Hopf bifurcation and investigate the effect of a periodicmodulation of the oscillator. In particular, we compare the entrainment performance when thefree oscillator displays either self-sustained or damped oscillations. We discuss the results inthe context of circadian oscillators.

Keywords: Goodwin model, Hopf bifurcation, limit-cycle oscillations, periodic forcing,circadian rhythms

1. Introduction

Temporal organization of cellular events is ensured bycomplex genetic and biochemical regulatory networks(Goodwin 1963, Alon 2006). Gene expression is dynamicallyregulated at the transcriptional level by transcriptionfactors. These transcription factors are themselves subjectto post-transcriptional modifications (phosphorylation, etc),translocation, and complex formations (with ligands,cofactors, or other proteins). These networks often encompassregulatory circuits, responsible for multistability (positivecircuits) and oscillations (negative circuits) (Thieffry 2007).Although large-scale networks can be simulated, a fullcomprehension of their dynamics starts with the understandingof the design principles of small regulatory motifs (Alon 2007,Novak and Tyson 2008).

Proposed in 1965, the Goodwin model is probably thefirst mathematical model demonstrating the occurrence of self-sustained oscillations in a minimal genetic system based solely

on a negative feedback loop (Goodwin 1965). This three-variable model was used to highlight the requirements andconditions to obtain self-sustained oscillations (Griffith 1968,Allwright 1977, Tyson 1979, Painter and Bliss 1981). As theBrusselator in chemistry, the Goodwin model is a prototypicalmodel in biology. Indeed, delayed negative feedback loopsreflect the underlying design of any biological oscillator (Tianaet al 2007).

The Goodwin model was specifically examined in thecontext of circadian rhythms (Ruoff et al 1999, 2001). Thismodel and its variants were used to investigate variousproperties of circadian oscillators, such as their response tolight pulses (Ruoff et al 2001), their robustness to noise (Gonzeand Goldbeter 2006), their entrainment and synchronizationproperties (Gonze et al 2005, Li et al 2006, Komin et al2011), their temperature compensation (Ruoff et al 1997,2005, Bodenstein et al 2012) or phase resetting upon lightpulses (Ruoff et al 2001, Pfeuty et al 2011). The Goodwin

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Phys. Biol. 11 (2014) 045002 A Woller et al

oscillator has also been used to model the segmentation clock(Zeiser et al 2007) and a Goodwin-like model has beenproposed for the NF-κB system (Krishna et al 2006, Jensen andKrishna 2012).

We revisit here the Goodwin model and examineits entrainment properties. We first derive analyticallythe amplitude equation for the limit-cycle near the Hopfbifurcation, thereby extending previous works on stabilityanalysis of this model. We then investigate the effect ofa periodic modulation of the oscillator. Although manytheoretical works have been devoted to periodically forcedoscillators (Tomita et al 1977, Tomita and Kai 1979, Aronsonet al 1986, Parlitz and Lauterborn 1987, Vance and Ross 1989,Glass and Sun 1994, Croisier et al 2009), to our knowledge,this is the first analytical study of the periodic forced Goodwinoscillator. We compare the entrainment performances whenthe free oscillator displays either self-sustained or dampedoscillations.

For a weakly forced Hopf bifurcation oscillator,Zhang and Golubistky (2011) have recently reviewed thepossible bifurcation diagrams. This analysis is based on thedetermination of the steady states of amplitude equations andthe study of their stability properties. We propose a similaranalysis for Goodwin oscillator. We determine the possiblebifurcation diagrams and investigate numerically the periodicsolutions of the amplitude equations. Of particular physicalinterest is the fact that pure entrainment is best achievedwhen the free oscillator exhibits damped oscillations. If thefree oscillator undergoes a Hopf bifurcation to sustainedoscillations, periodic regimes oscillating at the frequency ofthe external modulation are competing with quasiperiodicregimes in the bifurcation diagram. These quasiperiodicoscillations are low frequency modulation of fast oscillations.They are undesired for biological applications where goodsynchronization properties are needed. This is particularlyrelevant for circadian clocks, which may be damped underconstant condition.

The plan of the paper is as follows. In section 2, we presentthe Goodwin model and its biological rationale. In section 3,we investigate the Hopf bifurcation of Goodwin equations fortwo different control parameters. As we shall demonstrate,the bifurcation always leads to stable oscillations. Analyticalbifurcation diagrams are compared to diagrams obtained bynumerical simulations. In section 4, we consider the caseof a weakly forced Goodwin oscillator and determine thepossible bifurcation diagrams. We use perturbation techniquesfor solving the nonlinear problems and all details are relegatedto appendices. In section 5, analytical results are verified andextended through numerical simulation. Finally, in section 6,we discuss our results in the context of circadian oscillators.

2. The Goodwin model

The Goodwin model is schematized in figure 1. The variablesX , Y and Z can be interpreted as the concentration of agiven clock gene mRNA, the corresponding protein, and atranscriptional inhibitor, respectively. The latter could be thenuclear form or the activated (i.e. phosphorylated) form of

Figure 1. Scheme of the Goodwin model.

the protein. The feedback loop is achieved by the repressionexerted by the inhibitor to the mRNA synthesis.

The temporal evolution of this system is governed by thefollowing equations:

dX

dT= k1

Kni

Kni + Zn

− k2X (1)

dY

dT= k3X − k4Y (2)

dZ

dT= k5Y − k6Z (3)

All terms are linear except the repression term, whichis described by a Hill function. In the initial version of themodel (Goodwin 1963, 1965), the coefficient n was set ton = 1 (and actually did not appear explicitly in the equations).It was Griffith (1968) who introduced this exponent anddemonstrated that n must be larger than 8 to obtain limit-cycleoscillations in this model.

Hill functions are commonly used to model genetic andbiochemical systems. There are several ways to derive suchfunctions from detailed molecular mechanisms. In enzymekinetics, the Hill function may describe the cooperativebinding of multiple substrate or ligand molecules to an enzymeor a receptor (Segel 1975). At the transcriptional level, Hillfunction can be explained by the formation of repressor proteincomplexes or the cooperative binding of the repressor to thegene promoter (Keller 1995). All these processes rarely yieldHill coefficients higher than 3 or 4. Higher values can bereached by mechanisms such as multiphosphorylation of atranscription factor (Gonze and Abou-Jaoude 2013).

Due to the unusual high value of the Hill exponent, themodel was often criticized and variants have been proposed.These variants include Michaelian degradation or delay inthe feedback loop. Michaelian degradation of (at least) onecompound can be shown to be sufficient to allow a reductionof the Hill coefficient to 1 (Bliss et al 1982, Palsson andGroshans 1988, Tyson 2002). A Michaelis–Menten kineticscan result for example from post-transcriptional modificationsuch as phosphorylation, mediated by a saturable enzyme.Another factor that promotes the oscillations is the delay inthe repression. Extending the model by adding intermediary

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Phys. Biol. 11 (2014) 045002 A Woller et al

steps, even linear, allows a reduction of the Hill coefficient(Tyson and Othmer 1978, Invernizzi and Treu 1991, Zeiseret al 2007). Tyson and Othmer (1978) established a relationbetween the minimum value of the Hill coefficient to guaranteeoscillations and the number of intermediary steps. A similarconclusion is reached when the intermediary steps are replacedby an explicit delay (Tyson 2002).

3. Analytical results

In this section we study the Hopf bifurcation and deriveanalytical expression for the limit-cycle oscillations of theGoodwin oscillator.

In order to reduce the number of independent parameters,it is worthwhile to formulate Goodwin equations (1)–(3) indimensionless form. For mathematical simplicity, we considerthe case of equal degradation rates (k2 = k4 = k6). Introducingthe new variables

x = k3k5

k22Ki

X, y = k5

k2KiY, z = Z

Ki, and t = k2T

in equations (1)–(3). We obtain

x′ = α

1 + zn− x, (4)

y′ = x − y, (5)

z′ = y − z (6)

where

α ≡ k1k3k5

k32Ki

is the only control parameter for n fixed.The conditions for a Hopf bifurcation are documented in

Tyson (2002, p. 245) and are briefly rederived below. Little hasbeen done on the direction of the Hopf bifurcation. MacDonald(1977) and Invernizzi and Treu (1991) investigated thegeneralized Goodwin model with m variables (m � 3)

and derived conditions on the fixed parameters for a stableperiodic solution. They conjectured that Goodwin modelwith equal reaction rates always exhibits a bifurcation tostable oscillations. Here, we propose to construct the periodicsolution near its bifurcation point and compare analytical andnumerical bifurcation diagrams.

3.1. Hopf bifurcation

We consider α as our bifurcation parameter and propose toinvestigate the Hopf bifurcation of Goodwin equations (4)–(6). To this end, we simplify equations (4)–(6) by formulatinga third order equation for z only. Using (5) and then (6), wedetermine x as

x = y + y′ = z′ + z + z′′ + z′

= z′′ + 2z′ + z. (7)

Inserting (7) into equation (4) leads to the following third orderequation for z

z′′′ + 3z′′ + 3z′ + z = α

1 + zn. (8)

Figure 2. Stability diagram in terms of α and n. The line α = αH (n)is given by equation (15) and corresponds to a Hopf bifurcation.

The steady state solution z = zs(α) is given by (in implicitform)

α = zs(1 + zns ). (9)

From equation (8), we formulate the linearized equation forthe small deviation u = z − zs:

u′′′ + 3u′′ + 3u′ + u[1 + (

1 + zns

)−1nzn

s

] = 0. (10)

Inserting u = exp(λt) into equation (10), we obtain thefollowing characteristic equation for the growth rate λ

λ3 + 3λ2 + 3λ + [1 + (

1 + zns

)−1nzn

s

] = 0. (11)

This equation can be solved analytically and the three rootsare given by

λ1 = −1 − [(1 + zn

s

)−1nzn

s

]1/3< 0 (12)

λ2,3 = −1 + 12

[(1 + zn

s

)−1nzn

s

]1/3(1 ± i

√3). (13)

A Hopf bifurcation occurs if Re(λ2,3) = 0 and, using(13), we find that zs = zH is defined by

zH ≡(

8

n − 8

)1/n

(n > 8). (14)

The value of the bifurcation parameter α = αH is obtained byinserting (14) into (9). We find

αH =(

8

n − 8

)1/n n

n − 8(n > 8). (15)

Figure 2 represents the stability diagram. Note that αH → ∞as n → 8+. This particular feature of the Hopf bifurcationpoint will be important when we use n as the bifurcationparameter (see section 3.3). The steady state is stable (unstable)if α > αH (α < αH ). At the Hopf bifurcation point, thefrequency of the oscillations is given by

ωH = Im(λ2) =√

3. (16)

If n > 8, the steady state is always a stable or unstable focusfor all values of α. If n < 8, the steady state is always a stablefocus. This contrasts to other reference oscillators such as theBrusselator (Nicolis 1995) where the steady state is a focusonly in the vicinity of the Hopf bifurcation point.

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Phys. Biol. 11 (2014) 045002 A Woller et al

Figure 3. Amplitude R2 as a function of n (equation (21)). Thefunction emerges at n = 8, reaches a maximum near n = 14, anddecays to zero as n → ∞ following equation (22) (broken line).

3.2. Amplitude of the periodic solution

It is mathematically more convenient to consider zs as ourbifurcation parameter instead of α (α is related to zs byequation (9)). The steady state is stable (unstable) if zs < zH

(zs > zH ). Introducing the small parameter ε defined as

ε2 = zs − zH

c(17)

where c = ±1, we construct a periodic solution for z − zs

in power series of ε (Lindstedt–Poincare method (Drazin1992)). The perturbation analysis is long and tedious anddetails are relegated in appendix A. We find that the leadingapproximation of the solution is of the form

z � zH + 2εR cos(ωHt) (18)

y � z + ωHz′ = zH + 4εR cos(ωHt + π/3) (19)

x � z + 2ωHz′ + ω2Hz′′ = zH − 8εR cos(ωHt − π/3). (20)

R is obtained from equation (A.20). Specifically, R2 isgiven by

R2 = 2zH (n − 8)c[8863 (n − 17)2 − (n2 − 51n + 434)

] > 0. (21)

The amplitude R2 as a function of n is shown in figure 3 forn � 8 and c = 1. Setting c = 1 means that z is alwayslarger than zH (equation (17)). In other words, the branch ofperiodic solutions always overlaps the unstable steady state.The bifurcation is supercritical and according to Hopf theorem,the bifurcating limit-cycle solution is stable (Strogatz 2001).As n → ∞, R2 admits the limit

R2n→∞ = 126/(25n). (22)

We discuss this limiting behavior at the end of the section.In order to compare numerical and analytical bifurcation

diagrams, we need to relate x and α to z and zs, respectively.

Figure 4. Bifurcation diagram of the stable periodic solutions nearthe Hopf bifurcation point (n = 20). The full lines represent theextrema of x obtained numerically by integrating Goodwinequations (4)–(6) for long intervals of time (�t = 4000). Theparabolic broken line is Hopf bifurcation approximation given byequation (23).

We first note from (20) that the extrema of x are given by

xext = zH ± 8εR + O(ε2). (23)

Finally, we determine from (9) the relation

zs − zH = n − 8

9n(α − αH )(α − αH → 0). (24)

The analytical bifurcation diagram provided by combining(23) and (24) is compared to the numerical bifurcationdiagram of the extrema of x for n = 20 (see figure 4).The approximation of the minima agrees quantitatively withthe minima obtained numerically. For the maxima, theapproximation starts to deviate from the numerical data atα = 1.8. The period as a function of α remains close to its Hopfbifurcation value (2π/ωH ) and is not shown. A comparisonof the analytical and numerical limit-cycles for n = 20 andα = 1.8 is shown in figure 5.

An analysis of Hopf perturbation solution in the limitn large shows that its validity is restricted to the domainzs − zH � n−1. In Woller et al (2013), we investigated thiscase and showed that the solution profile is no more sinusoidalif zs − zH = O(n−1) or larger but is a combination of twoexponentials. Similarly, Hopf perturbation expansion becomessingular as n → 8 because zH → ∞ and αH → ∞. Weexamine this problem in the next section.

3.3. The limit α large

In the previous section, we considered α as our bifurcationparameter and kept n > 8 fixed. However, we are alsointerested to use n as our bifurcation parameter. We knowthat αH → ∞ if n − 8 → 0+. In order to explore the

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Phys. Biol. 11 (2014) 045002 A Woller et al

Figure 5. Comparison between numerical and analytical limit-cycle oscillations for n = 20 and α = 1.8. Left: limit-cycle in the y versus xphase plane. The outer and inner orbits correspond to the numerical and analytical limit-cycles, respectively. Right: the numerical andanalytical periodic solutions are in good quantitative agreement. Best fitting is obtained by replacing the mean value zH in equations (18)–(20) by the steady state value zs at α = 1.8.

bifurcation problem, we first need to examine the limit α → ∞of Goodwin equations (4)–(6). The singularity can be removedby introducing the new variables u, v, and w defined by

x = α1/(n+1)u, y = α1/(n+1)v, and z = α1/(n+1)w. (25)

Inserting (25) into equations (4)–(6) and taking the limitα → ∞, we obtain the following reduced problem

u′ = w−n − u, (26)

v′ = u − v, (27)

w′ = v − w. (28)

Equivalently, we may eliminate u, and v, and formulate anequation for w only. It is given by

w′′′ + 3w′′ + 3w′ + w = w−n. (29)

The steady state is ws = 1 and the linear stability analysisprovides the following expressions for the growth rates

λ1 = −1 − n1/3 < 0, (30)

λ2,3 = −1 + 12 n1/3(1 ± i

√3). (31)

A Hopf bifurcation appears at n = nH = 8 with frequencyωH = √

3. The bifurcation equation is derived in appendix Bby using the method of multiple time scales. The leadingapproximation of the stable periodic solution is

w − 1 = ε[A(τ ) exp(is) + c.c.] + O(ε2) (32)

where s ≡ ωHt, τ ≡ ε2t, and ε2 ≡ (n − nH )/nH . Theamplitude A satisfies equation (B.13) with c = 1 and is givenby

dA

dτ= 1

3A(1 + iωH )A + A2A∗(B + iC) (33)

where

B = − 277 and C = − 59

7 ωH . (34)

Inserting A = R exp(iφ) into equation (33) we obtaindR

dτ= 1

3R + R3B (35)

dφ

dτ= 1

3ωH + R2C. (36)

The stable steady state of equation (35) provides the limit-cycleamplitude

R =√

− 1

3B(37)

while equation (36) leads to the correction of the frequency

φ = ω2τ + φ(0) =(

1

3ωH − C

3B

)τ + φ(0). (38)

Consequently, the extrema of u are given by

uext = 1 ± 8εR = 1 ± 8

√− 1

3B

(n − nH )

nH(39)

and the leading expression of the period is of the form

P = 2π

ωH

(1 − n − nH

nHω2

). (40)

These approximations are compared to the numericalbifurcation diagrams of the extrema of u and of the periodP in figure 6. Excellent agreement is obtained for the periodwhile the approximation of the extrema start to deviate fromthe numerical values as n > 8.2.

4. Near-resonant periodic modulation

We next consider Goodwin problem with a periodicmodulation of parameter α. Instead of (8), the mathematicalproblem is now described by

z′′′ + 3z′′ + 3z′ + z = α(1 + ηh(t))

1 + zn(41)

where h(t) is a T -periodic function of t (T = 2π/σ ) and0 < η � 1. As in the previous section, we wish to considern as our bifurcation parameter and the limit α large now leadsto the following problem for w

w′′′ + 3w′′ + 3w′ + w = w−n(1 + ηh(t)). (42)

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Phys. Biol. 11 (2014) 045002 A Woller et al

Figure 6. Numerical bifurcation diagrams of equations (26)–(28)representing the extrema of u and the period P as function of n. Thebroken lines are provided by the approximations (39) and (40).

The analysis of the solutions of equation (42) is describedin appendix C for the case of a square-wave modulation. Theamplitude equations for R and φ are given by (C.11) and (C.12)and are of the form

dR

dτ= c

3R + BR3 − d cos(φ), (43)

dφ

dτ=

( c

3ωH − �

)+ CR2 + d

Rsin(φ) (44)

where

� = ε−2(σ − ωH ) and d = ε−3η/(12π) (45)

represent the frequency detuning and the amplitude of theexternal periodic modulation, respectively.

The steady states of equation (43)–(44) are the periodicsolutions of the original problem. They satisfy

c

3R + BR3 − d cos(φ) = 0, (46)

(c

3ωH − �

)+ CR2 + d

Rsin(φ) = 0. (47)

Eliminating the trigonometric function, we determine

d2 =(

c

3+ BR2

)2

R2 +( c

3ωH − � + CR2

)2R2. (48)

We are interested to determine the amplitude R as a functionof �. From equation (48), we have the solution (in implicitform)

� = c

3ωH + CR2 ±

√d2R−2 −

(c

3+ BR2

)2

(49)

(d2R−2 −

(c

3+ BR2

)2

� 0

). (50)

From the linear stability of these steady states, we determinethe following characteristic equation for the growth rate ζ

ζ 2 + f ζ + g = 0 (51)

where

f ≡ −(

2c

3+ 4BR2

), (52)

g ≡(

c

3+ 3BR2

)( c

3+ BR2

)

+[ c

3ωH − � + CR2

] ⎡⎣3CR2 + c

3ωH

−�

⎤⎦ . (53)

Stability means that f > 0 and g > 0. A Hopf bifurcation ofthe amplitude equations means a bifurcation to quasiperiodicoscillations for the original problem (torus bifurcation (TR)).The conditions for such a Hopf bifurcation are f = 0 andg > 0.

Two bifurcation diagrams of the amplitude equations (43)and (44) are shown in figure 7 for a case of strong modulation(d = 1, panels (a) and (b)) and for a case of weak modulation(d = 0.1, panels (c) and (d)). Recall that the steady andtime-periodic solutions of the amplitude equations correspondto time-periodic and quasiperiodic regimes of the originalGoodwin equations. Both left panels illustrate the subcriticalcase n < nH or c = −1. The two right panels illustrate thesupercritical case n > nH or c = 1. Hopf bifurcations ofthe amplitude equations (TR of Goodwin equations) are notpossible for the subcritical case (left panels) while they appearto play a major role in the supercritical case (right panels).Note that in figure 7(d), the left branch of periodic solutions ofthe amplitude equations terminates at an unstable steady state(homoclinic or infinite period orbit) located near � = −1.6.We also note a thin domain of coexistence between a stableperiodic solution and a stable steady state close to the Hopfbifurcation point located near � = 0.4.

An alternative to the bifurcation diagrams is to plotstability diagrams in the d versus � parameter space and toshow the stability boundaries given by the saddle-node (SN)bifurcation points of limit-cycles and the TR lines (figure 8).In the subcritical case (c = −1), the free oscillator exhibitsdamped oscillations if the steady state is perturbed. Drivenby an external modulation with a frequency close to the Hopfbifurcation frequency, entrainment is possible leading to anamplification of the response if the detuning is close to zero.Birhythmicity where two distinct forms of entrainment coexistfor the same range of values of the parameters is also possiblebut only if the modulation amplitude surpasses a critical

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Phys. Biol. 11 (2014) 045002 A Woller et al

(a) (b)

(c) (d)

Figure 7. Bifurcation diagrams of the steady states and stable time-periodic solutions of the amplitude equations (43) and (44). The figurerepresents the stable and unstable steady states by full and broken lines, respectively. The analytical expression of the steady states is givenin implicit form by equation (49). The red dots in panels (b) and (d) mark Hopf bifurcation points (of the amplitude equation) determinedfrom the linearized theory. The upper and lower blue points overlapping a branch of unstable steady states are the extrema of a stable time-periodic solution. Near the Hopf bifurcation, the phase is bounded but as soon as the minimum of the oscillations approaches zero, the phasebecomes unbounded. In panel (d), we note a sudden jump from a relatively large amplitude periodic solution to a stable steady state at� = 0.3867, as we progressively decrease � from � = 5. Since the Hopf bifurcation is located at �H = 0.3978, the numerical bifurcationdiagram suggests a small domain of coexistence between a stable steady state and a stable periodic solution. In this domain, the minimum ofthe oscillations is very close to zero. Parameter values: (a) c1 = −1 and d = 1; (b) c1 = 1 and d = 1; (c) c1 = −1 and d = 0.1; (d) c1 = 1and d = 0.1. SN and TR mean saddle-node bifurcation of limit-cycles and Torus bifurcation, respectively.

Figure 8. Stability diagram in the (�, d) parameter plane. The lines denoted by SN1 and SN2 (SN bifurcation point of limit-cycles) delimitthe domain of coexisting periodic solutions. The lines denoted by TR1 and TR2 for the case c = 1 are TR lines to quasiperiodic oscillations.They emerge from the SN lines from Bogdanov–Takens (or double zero) bifurcation points (dots in the figure). P: single periodic solution;PP: two stable periodic solutions (birhythmicity); QP: quasiperiodic solutions. Squares mark the values of the parameters used in oursimulations of the driven Goodwin equations (see figure 9). The simulations are in agreement with the predictions of the stability diagrams.

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Phys. Biol. 11 (2014) 045002 A Woller et al

0 10 20 300

0.5

1

1.5

2

2.5

3

(a)

x

time, t0 10 20 30

0

0.5

1

1.5

2

2.5

3

(b)

time, t

x

0 20 40 60 80 100 120 140 1600

1

2

3

4

(c)

time, t

x

Figure 9. Examples of times series obtained by numerical integration of the Goodwin oscillator forced by a periodic square-wave function.Parameter values: (a) α = 100, n = 7, η = 0.1, Tf = 3.6, (b) α = 100, n = 9, η = 0.1, Tf = 3.6, (c) α = 100, n = 9, η = 0.1, Tf = 4.These results have been generated with XPP-AUTO (Ermentrout 2002).

value. In the supercritical case (c = 1, right panel), the freeoscillator is oscillating and both periodic and quasiperiodicoscillations are possible for the driven oscillator. It is clear thatthe subcritical region is more interesting if a single entrainmentis desired for a large range of values of the detuning.

5. Numerical simulations

5.1. Verification of the analytical results

In order to verify the predictions of our analysis summarized infigure 8, we have simulated numerically the driven Goodwinequations for both the subcritical (n = 7) and the supercriticalcase (n = 9). For both cases, the definition of ε2 leads to thesame value

ε =√∣∣∣∣n − 8

8

∣∣∣∣ = 0.35. (54)

The normalized modulation amplitude d is defined in (45). Forall our simulations, we consider η = 0.1 which then implies

d = 0.06. (55)

We next concentrate on the normalized detuning � defined in(45). Note that ωH = √

3 and thus

� = ε−2(σ − ωH ) = ε−22π

(1

Tf− 1

TH

)(56)

where TH = 2π/ωH = 2π/√

3 � 3.63. If Tf = 3.6(figures 9(a) and (b)), we determine � = 0.11. If Tf = 4(figure 9(c)), we find � = −1.29. Figures 9(a) and (b)are cases of single entrainment and were predicted by thestability diagram of figure 8. Figure 9(c) illustrates a caseof quasiperiodic oscillations which is also in agreement withthe stability diagram (figure 8). The quasiperiodic oscillationsconsist of a low frequency modulation (ω1 ∼ ε2) of fastoscillations (ω2 ∼ ωH ).

It is worthwhile to determine the analytical prediction ofthe mean value of x given by xm = α1/(1+n). With α = 100for all figures, xm = α1/8 = 1.778 for figure 9(a) and xm =α1/10 = 1.58 for figures 9(b) and (c). From figures 9(a) and (b),we estimate the mean value as xm = xmin + (xmax − xin)/2. Wefind xm � 1.8 for both cases. In the case of figure 9(a) showinglow amplitude oscillations the agreement is quantitative. Forfigure 9(b) showing much larger amplitude oscillations, theagreement is semi-quantitative.

5.2. Extension of the analytical results

Our analysis of the forced Goodwin oscillator assumed α

large because it allowed us to consider n as a controlparameter. Simulations of Goodwin equations shown in

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Phys. Biol. 11 (2014) 045002 A Woller et al

1 2 3 4 5 6 70

1

2

3

Am

plitu

de

(a)

1 2 3 4 5 6 70

1

2

3

TR

SN

(b)

1 2 3 4 5 6 70

2

4

6

SN

SN

SN

Am

plitu

de

(c)

1 2 3 4 5 6 70

2

4

6

SN

SN

TR

SN

TR

(d)

1 2 3 4 5 6 70

5

10

15

SN

SN

SN

SN

Forcing period

Am

plitu

de

(e)

1 2 3 4 5 6 70

5

10

15

SN

SNSN

TRTR

Forcing period

( f )

Figure 10. Amplitude curve (xmax − xmin) as a function of the forcing period Tf , obtained for a forcing amplitude η = 0.1 (panels (a) and(b)), 1 (panels (c) and (d)), and 5 (panels (e) and ( f )), for n = 7 (panels (a),(c),(e)) and n = 10 (panels (b),(d),( f )). In all panels, α = 20.The SN and TR are indicated in red. The green points correspond to period doubling bifurcations. These curves have been obtainednumerically with XPP-AUTO (Ermentrout 2002).

figure 9 were performed for α = 100 and a weak modulationamplitude η. They are in good agreement with the analyticalpredictions (figure 8). Numerical simulations allow us toexplore the entrainment performance when α has a lowervalue and when the amplitude of the periodic forcing is notweak.

In figure 10, we computed numerically the bifurcationdiagrams of the unstable (n = 7, left panels) and stable (n =10, right panels) periodic solutions by using a continuationmethod (Ermentrout 2002). The periodic forcing takes theform of a square-wave function with amplitude η = 0.1 (panels(a) and (b)), η = 1 (panels (c) and (d)), and η = 5 (panels (e)

and ( f )). As in figure 7, we note the emergence of TR whenthe free oscillator exhibits sustained oscillations (right panels).When the forcing amplitude is weak, the damped oscillatorentrains very well, with a resonance 1:1, to the external forcingperiod (panel (a)), whereas the domain of entrainment of thelimit cycle oscillator is much reduced (panel (b)). We alsoobserve the appearance of other resonances. The bifurcationsnear Tf = 2 (panels (c)–( f )) and near Tf = 6 (panel (e))are period doubling bifurcations. In all cases the domain ofentrainment is larger when the free oscillator displays dampedoscillations.

6. Discussion

The three-variable Goodwin model is a minimal oscillatorbased solely on a negative feedback loop. Since suchmechanism is at the core of the circadian (and other) geneticnetwork, it is of interest to study the dynamical propertiesof this model. Many circadian models are extensions andvariants of the Goodwin model (Ruoff et al 1999, 2001,Gerard et al 2009).

Due to its relative simplicity, the Goodwin model allowsanalytical treatments. Early theoretical works on the Goodwinoscillator were mainly devoted to the existence of periodicsolutions (Griffith 1968, Tyson 2002). We revisited herethe stability analysis of the Goodwin model, and derivedanalytical equations for the limit-cycle close to the Hopfbifurcation. In a related study we also constructed the solutionof the system in the limit of strong feedback, i.e. for n large(Woller et al 2013).

In natural conditions, biological oscillators are not runningautonomously. They are affected by various dynamical factorsand must respond adequately to environmental changes (Glasset al 2001). For example, the circadian clock is not free-runningbut subject to the cyclical environmental conditions. There is

9

Phys. Biol. 11 (2014) 045002 A Woller et al

a priori no reason for living organisms to produce robust self-sustained oscillations in autonomous conditions. In contrast,the clock must have evolved to display a robust entrainmentunder the natural light dark cycle.

Whether circadian oscillations are damped or self-sustained in constant conditions is not yet clear. Manyexperiments suggest that rhythms in peripheral tissues maybe damped (Yamazaki et al 2000, Izumo et al 2003, Prolo et al2005). The suprachiasmatic nuclei (SCN), which constitutesthe circadian pacemaker, appears itself composed of anheterogeneous cell population with a large proportion of cellsexhibiting damped oscillations (Webb et al 2009). A recentquantitative analysis of experimental time series does notallow to confirm whether circadian clocks of single SCNneurons and fibroblasts are damped or self-sustained oscillator(Westermark et al 2009). We may thus wonder if dampedoscillations provide some adaptative advantage to the clock.

We studied here the entrainment by a periodic modulationof one control parameter, reflecting the daily alternance oflight and dark phases. Interestingly, analytical and numericalanalyses of the Goodwin model show very different patternsof behaviors, depending on whether the forced Goodwinoscillator exhibits damped or limit-cycle oscillations. Wefound that the entrainment region as a function of themodulating period is larger when the free oscillator is inthe damped regime. In this case, only a small region ofbirhythmicity is present (coexistence of two stable limit-cycles). In contrast, when the free oscillator is in the limit-cycleregion, it exhibits, upon entrainment, a much richer dynamics:quasi-periodicity, birhythmicity, or coexistence of limit-cycleand quasiperiodicity. These behaviors are likely undesired forbiological systems, in particular for circadian clocks whichneed to precisely set the phase of the oscillations with respectto the external light-dark cycles.

Our results provide insights on a possible advantage forcircadian clocks to be damped under constant conditions.Damped oscillators are more easily entrained over largedomains of periods and photoperiods (not shown) than limit-cycle oscillators. It should be noted, however, that theamplitude is reduced in the driven damped oscillator comparedto the entrained self-sustained oscillator. Circadian clocks mustnot only be well entrained, but also appropriately phase locked.In other words the production of clock protein must occur everyday at the same time and at a well defined time of the day. TheGoodwin model can be used to investigate the parameters thatdetermine the entrainment phase of the clock. It would be alsointeresting to study the resynchronization time (for ex. aftera jetlag) of damped versus self-sustained oscillations. Usingseveral models, including the Goodwin model, Granada andHerzel (2009) have shown that weak oscillators resynchronizefaster than strong oscillators. The ‘strength’ of the oscillator ishere related to the speed of the radial relaxation (i.e. the time tocome back to the limit-cycle). However they did not considerthe case of damped oscillators. Our results suggest that thelater may be even more flexible and more rapidly adaptable tochanges in the environment.

Coupled damped oscillators have also been shown toefficiently synchronize with each other (Gonze et al 2005,

Bernard et al 2007, Komin et al 2011). Thus, the largepopulation of cells exhibiting damped oscillations in theSCN may have evolved for their synchronization efficiency.Other studies have explored how sustained oscillations canarise in damped oscillators in presence of molecular noise(Thomas et al 2013, Mitarai et al 2013). These noise-inducedoscillations are most pronounced when the system is close to aHopf bifurcation. Interestingly, spectral analysis of single cellscircadian time series in fibroblast support the observation thatthese cells display noise-induced oscillations (Thomas et al2013). Damped oscillators thus present interesting dynamicalproperties, which may have been exploited by circadian clocks,as well as other biological oscillators.

Acknowledgments

The authors would like to thank Benjamin Pfeuty for fruitfuldiscussions. TE acknowledges the support of the FNRS(Belgium). AW is funded by Region Nord Pas-de-Calais,Centre Hospitalier Regional Universitaire de Lille, FrenchMinistry of Higher Education and Research and Labex CEMPI(ANR-11-LABX-0007).

Appendix A. α is the bifurcation parameter

A.1. Preliminaries

We consider the steady state value z = zs as our bifurcationparameter and wish to rewrite equation (8) in terms of thedeviation Z ≡ z − zs. The first algebraic difficulty is toreformulate the function

F ≡ z − α

1 + zn(A.1)

in terms of Z. We sequentially obtain

= zs + Z − α

1 + (zs + Z)n

F = zs + Z

−α

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(1 + zns )

−1 − (1 + zns )

−2nzn−1s Z

−[

−2(1 + zns )

−3n2z2n−2s

+(1 + zns )

−2n(n − 1)zn−2s

]Z2

2

−

⎡⎢⎢⎢⎣

6(1 + zns )

−4n3z3n−3s

−2(1 + zns )

−3n2(2n − 2)z2n−3s

−2(1 + zns )

−3n2(n − 1)z2n−3s

+(1 + zns )

−2n(n − 1)(n − 2)zn−3s

⎤⎥⎥⎥⎦ Z3

6 + · · ·

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(A.2)

Eliminating α by using the steady state equations (9),equation (A.2) simplifies as

F = Z −⎡⎢⎢⎢⎢⎢⎢⎣

−(1 + zns )

−1nzns Z

− [−2(1 + zns )

−2n2z2n−1s + (1 + zn

s )−1n(n − 1)zn−1

s

]Z2

2

−

⎡⎢⎢⎣

6(1 + zns )

−3n3z3n−2s

−2(1 + zns )

−2n2(2n − 2)z2n−2s

−2(1 + zns )

−2n2(n − 1)z2n−2s

+(1 + zns )

−1n(n − 1)(n − 2)zn−2s

⎤⎥⎥⎦ Z3

6 + · · ·

⎤⎥⎥⎥⎥⎥⎥⎦

.

(A.3)

10

Phys. Biol. 11 (2014) 045002 A Woller et al

Factorizing terms, we obtain

F = Z −⎡⎢⎢⎢⎢⎢⎢⎣

−(1 + zns )

−1nzns Z

−(1 + zns )

−2zn−1s

[−2n2zns + (1 + zn

s )n(n − 1)]

Z2

2

−(1 + zns )

−3zn−2s

⎡⎢⎢⎣

6n3z2ns

−2(1 + zns )n

2(2n − 2)zns

−2(1 + zns )n

2(n − 1)zns

+(1 + zns )

2n(n − 1)(n − 2)

⎤⎥⎥⎦ Z3

6 + · · ·

⎤⎥⎥⎥⎥⎥⎥⎦

,

or equivalently,

F = Z −⎡⎢⎢⎢⎢⎣

−(1 + zns )

−1nzns Z

−(1 + zns )

−2zn−1s n

[n − 1 − zn

s (n + 1)]

Z2

2

−(1 + zns )

−3zn−2s n

⎡⎣ 6n2z2n

s

−6(1 + zns )n(n − 1)zn

s+(1 + zn

s )2(n − 1)(n − 2)

⎤⎦ Z3

6 + · · ·

⎤⎥⎥⎥⎥⎦ .

(A.4)

A.2. The perturbation analysis

We introduce a small parameter ε2 defined by

zs = zH + ε2c (A.5)

where c = ±1 depending on the sign of zs − zH . Noting that

znH = 8

n − 8, 1 + zn

H = n

n − 8,

− (1 + zns )

−1nzns = −8 − 8z−1

H (n − 8)ε2c + O(ε4),

the expression (A.4) can be rewritten as

F = z − α

1 + zn

= Z −

⎡⎢⎣

[−8 − 8z−1H (n − 8)ε2c + O(ε4)

]Z

+[−8z−1H (n − 17) + O(ε2)

]Z2

2

+[−8z−2H (n2 − 51n + 434) + O(ε2)

]Z3

6 + · · ·

⎤⎥⎦ .

(A.6)

We apply the Lindstedt–Poincare method (Drazin 1992)and seek a 2π -periodic solution of the form

z = zs + εZ1(s) + ε2Z2(s) + · · · (A.7)

where

s ≡ ωt = (ωH + ε2ω2 + · · ·)t. (A.8)

We insert (A.7) and (A.8) into equation (8), use (A.6), andequate to zero the coefficients of each power of ε. We obtaina sequence of linear problems for the unknown functions Z1,

Z2, and Z3.

O(ε) problem: The leading problem for Z1 is

LZ1 ≡ ω3HZ1sss + 3ω2

HZ1ss + 3ωHZ1s + 9Z1 = 0 (A.9)

and admits the solution

Z1 = A exp(is) + c.c.

where A is an unknown amplitude.

O(ε2) problem: We examine the next problem for Z2 givenby

Lz2 = −8z−1H (n − 17)

Z21

2. (A.10)

Noting that

Z21 = A2 exp(2is) + c.c. + 2AA,

the solution of equation (A.10) is

Z2 = B exp(is) + c.c. + AAp0 + A2 exp(2is) + c.c. (A.11)

where B is a new unknown amplitude and

p0 = − 8

9z−1

H (n − 17),

p2 = 4

9z−1

H (n − 17)1

2i√

3 + 3= 4

9z−1

H (n − 17)3 − 2i

√3

21.

O(ε3) problem: The amplitude A is still unknown and weneed to examine the next problem for Z3 and apply a solvabilitycondition. The equation for Z3 is

Lz3 = −8z−1H (n − 17)Z1Z2 − 8z−2

H (n2 − 51n + 434)Z3

1

6− 8z−1

H (n − 8)cZ1 − ω2(3ω2HZ1sss + 6ωHZ1ss + 3Z1s).

(A.12)

Solvability requires that the coefficient of exp(is) in theexpansion of the right hand side is zero. This conditionis[−8z−1

H (n − 17)A2A [p2 + p0] − 4z−2H (n2 − 51n + 434)A2A

−8z−1H (n − 8)cA + 6iω2(1 − i

√3)

]= 0

or ⎡⎢⎣A2A

{−8z−2

H (n − 17)2[

49

3−2i√

321 − 8

9

]−4z−2

H (n2 − 51n + 434)

}

−8z−1H (n − 8)cA + 6iω2(1 − i

√3)

⎤⎥⎦ = 0. (A.13)

Multiplying by 1/2 (A.13) becomes⎡⎢⎣2A2Az−2

H

{−8(n − 17)2

[19

3−2i√

321 − 2

9

]−(n2 − 51n + 434)

}

−4z−1H (n − 8)cA + 3iω2(1 − i

√3)A

⎤⎥⎦ = 0. (A.14)

Dividing by 3(1 − i√

3), (A.14) reduces to

= 1 + i√

3

12

⎡⎢⎣2A2Az−2

H

{−8(n − 17)2

[19

3−2i√

321 − 2

9

]−(n2 − 51n + 434)

}

−4z−1H (n − 8)cA

⎤⎥⎦

+ iω2 = 0. (A.15)

From the real and imaginary parts, we finally obtain twoequations for R2 and ω2[

13 z−1

H (n − 8)c + BR2]R = 0, (A.16)

ω2 −√

33 z−1

H (n − 8)c − CR2 = 0 (A.17)

11

Phys. Biol. 11 (2014) 045002 A Woller et al

where

B ≡ − 16 z−2

H

[8863 (n − 17)2 − (n2 − 51n + 434)

], (A.18)

C ≡ − 16 z−2

H

√3[

8×419×21 (n − 17)2 − (n2 − 51n + 434)

]. (A.19)

From (A.16), we determine the amplitude R2 of the periodicsolution

R2 = −13 z−1

H (n − 8)c

B> 0 (A.20)

and from equation (A.17), we may extract the correction ofthe frequency ω2.

Appendix B. n is the bifurcation parameter

We analyze the Hopf bifurcation by the method of multipletime scales (Bender and Orszag 1978). We introduce the smallparameter ε defined as

n = nH (1 + cε2) (B.1)

where c = ±1 and seek a solution of the form

W = w − 1 = εW1(s, τ ) + ε2W2(s, τ ) + · · · (B.2)

where

s = ωHt and τ = ε2t (B.3)

are two distinct time scales. It implies the chain rules

W ′ = ωHWs + ε2Wτ (B.4)

W ′′ = ω2HWss + 2ε2ωHWsτ + O(ε4) (B.5)

W ′′′ = ω3HWsss + 3ε2ω2

HWssτ + O(ε4) (B.6)

where the subscripts mean partial derivatives with respect to sor τ. Inserting (B.1), (B.2), (B.4)–(B.6) into equation (29) andequating to zero the coefficients of each power of ε leads to asequence of linear problems for the functions W1, W2, and W3.

They are given by

LW1 ≡ ω3HW1sss + 3ω2

HW1ss + 3ωHW1s + 9W1 = 0 (B.7)

LW2 = (4 × 9)W 21 (B.8)

LW3 = (8x9)W1W2 − (4 × 3 × 10)W 31 − cnHW1

− 3ω2HW1ssτ − 6ωHW1sτ − 3W1τ . (B.9)

The solution of the first two problems are of the form

W1 = A(τ ) exp(is) + c.c. + e dt (B.10)

W2 = B(τ ) exp(is) + c.c. + e dt + AA∗ p0

+ A2 p2 exp(2is) + c.c. (B.11)

where A and B are two unknown functions of τ and e dt meansone exponential decaying function of s. The coefficients p0

and p2 are defined by

p0 = 8 and p2 = −4(3 − 2iωH )

21. (B.12)

Finally, the solvability condition for equation (B.9) leads tothe following amplitude equation for A(τ )

dA

dτ= c

3A(1 + iωH )A + A2A∗(B + iC) (B.13)

where

B = −27

7and C = −59

7ωH . (B.14)

Appendix C. Periodically modulated oscillator

We consider equation (42) and propose to determine smallamplitude periodic solutions. To this end, we are using thesame approach as in appendix B. The small parameter ε isdefined by (B.1) and is related to the small deviation of n fromthe Hopf bifurcation value nH = 8. We seek a solution of theform (B.2) where the fast time s ≡ σ t now is the time of theexternal modulation. The slow time τ ≡ ε2t is motivated bythe real part of the linear growth rate near the Hopf bifurcationpoint. Instead of (B.4)–(B.6), the chain rules are

W ′ = σWs + ε2Wτ , (C.1)

W ′′ = σ 2Wss + 2ε2σWsτ + O(ε4), (C.2)

W ′′′ = σ 3Wsss + 3ε2σ 2Wssτ + O(ε4). (C.3)

We are interested in the near-resonant case σ ∼ ωH and expandσ as

σ = ωH + ε2� + · · · . (C.4)

The success of our perturbation analysis also depend on thescaling

η = ε3a + · · · . (C.5)

The two first problems for W1 and W2 are exactly the sameas the problems documented in appendix B. The solutions areprovided by (B.10) and (B.11). The problem for W3 is howeverdifferent. It is given by

LW3 = (8x9)W1W2 − (4 × 3 × 10)W 31 − cnHW1 − 3ω2

HW1ssτ

− 6ωHW1sτ − 3W1τ − �(3ω2HW1sss + 6ωHW1ss + 3W1s)

+ ah(s). (C.6)

It solvability condition now leads to the following amplitudeequation for A

A′ =[ c

3(1 + iωH ) − i�

]A + (B + iC)A2A∗ − exp(iπ/3)d

(C.7)

where

d ≡ a

12I and I ≡ 1

2π

∫ 2π

0h(s) exp(−is) ds. (C.8)

If the modulation is a square-wave of the form

h(s) = 1(0 � s < π),

0(π � s < 2π),(C.9)

as in our numerical simulations of Goodwin original equations,we find I = −1/π = exp(iπ)/π . The phase factor exp(iπ)

as well as the factor exp(iπ/3) appearing in equation (C.7)do not contribute to the bifurcation diagram since they can beremoved by redefining A. Specifically, we introduce

A = R exp(4iπ/3 + iφ) (C.10)

into equation (C.7) and separate the real and imaginary parts.We obtain two coupled equations for R and φ given by

R′ = c

3R + BR3 − d cos(φ), (C.11)

φ′ =( c

3ωH − �

)+ CR2 + d

Rsin(φ) (C.12)

where

d = a

12π. (C.13)

12

Phys. Biol. 11 (2014) 045002 A Woller et al

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