SYNCHRONISATION OF BIOLOGICAL CLOCK SIGNALSCapturing Coupled Repressilators from a Control Systems Perspective

Thomas Hinze, Mathias Schumann and Stefan SchusterDepartment of Bioinformatics, Friedrich Schiller University Jena, E.-Abbe-Platz 1-4, D-07743 Jena, Germany

{thomas.hinze, mathias.schumann, stefan.schu}@uni-jena.de

Keywords: Chronobiology, Coupled repressilators, Reaction-diffusion kinetics, Internal/external clock synchronisation,Control system, Phase-locked loop.

Abstract: Exploration of chronobiological systems emerges as a growing research field within bioinformatics focusingon various applications in medicine, agriculture, and material sciences. From a systems biological perspective,the question arises whether biological control systems forregulation of oscillative signals and their technicalcounterparts utilise similar mechanisms. If so, modellingapproaches and parameterisation adopted from build-ing blocks can help to identify general components for clocksynchronisation. Phase-locked loops could be aninteresting candidate in this context. Both, biology and engineering, can benefit from a unified view. In a firstexperimental study, we analyse a model of coupled repressilators. We demonstrate its ability to synchroniseclock signals in a monofrequential manner. Several oscillators initially deviate in phase difference and fre-quency with respect to explicit reaction and diffusion rates. Accordingly, the duration of the synchronisationprocess depends on dedicated reaction and diffusion parameters whose settings still lack to be sufficientlycaptured by comprehensive tools like the Kuramoto approach.

1 INTRODUCTION

In both spheres, biological and technical systems, os-cillatory signals play a major role in order to triggerand control time-dependent processes. Core oscilla-tors are the simplest devices for generation of contin-uously running clock signals. To this end, signal pro-cessing units consisting of at least one feedback loopcan suffice (Russo and di Bernardo, 2009). So, it isno surprise that probably numerous evolutionary ori-gins led to oscillative reaction networks while inde-pendently technical attempts succeeded in construc-tion of single clocks or clock generators.

The situation becomes more complicated if sev-eral of those core oscillators start to interact. Re-sulting biological systems are commonly driven toachieve a synchronous behaviour towards an evolu-tionary advantage. Correspondingly, clock synchro-nisation in technical systems is frequently inspired bythe need to follow a global time. Interestingly, theformalisation of clock synchronisation processes isquite distant from each other. While in distributedcomputer systems, stepwise algorithmic approaches(like Berkeley or Christian’s method, (Tanenbaumand van Steen, 2001)) predominate, biological sys-tems adjust their clock signals more gradually. Its

formalisation is either based on reaction-diffusion ki-netics or employs the more abstract Kuramoto method(Kuramoto, 1984), an analytic signal coherence mea-sure restricted to sinusoidal signal shape to counteractphase shift between each pair of core oscillators.

We define different temporally oscillating signalsto be synchronous to each other if and only if theymeet three conditions: (1) The oscillatory signal mustrunundamped to avoid signal weakening. (2)Asymp-totical or total harmonisation of the oscillatory sig-nals meaning that after a finite amount of time calledtsync (time to synchronisation), both temporal sig-nal courses converge within an arbitrarily smallε-neighbourhood. (3) The resulting oscillatory sig-nal after tsync has to bemonofrequential to ensurechronoscopy (constant progression of time measure).

The central prerequisite of a core oscillator tobe capable of synchronisation to others is its abil-ity to vary its oscillation frequency within a speci-fied range (Granada and Herzel, 2009). This variationcan be achieved byforcing, by resetting, or by spe-cific selective perturbations affecting the oscillatingsignal. Without any external influences, core oscilla-tors resume their individual free-running oscillatorybehaviour, mostly by loosing their synchronicity.

Topologically, clock synchronisation can be ac-

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complished by two different strategies calledexter-nal and internal (Pikovsky et al., 2001). Externalstrategies comprise a central leading clock that propa-gates its time signal throughout the whole network ofdownstream core oscillators which adjust their indi-vidual signals by accelerating or slowing down theirfrequency for a certain amount of time. Here, we ob-serve an unidirectional coupling from the leading cen-tral clock to all others. In contrast, internal strategiesaim at a mutual clock exchange between the networkmembers. The coupling topology is mostly bidirec-tional, and each involved core oscillator is going toadjust its signal based on a weighted sum of the sig-nals released by its adjacent clocks.

Within a case study, we exemplify internal syn-chronisation by a biological system composed ofbidirectionally coupled repressilators. To this end,we model the entire gene regulatory networks usingreaction-diffusion kinetics. Afterwards, we conducttwo comprehensive simulation studies. The first onediscloses the time to synchronisation subject to ini-tial phase shift between the elementary repressilators.Its balanced diffusion rate acts as coupling strength.It appears that synchronisation of initially antipha-sic signals is most time-consuming for weak couplingwhile it has a negligible effect for strong coupling.A second simulation study investigates the synchro-nisation behaviour with respect to different initial fre-quencies of the single repressilators. The obtainednumerical results are envisioned to identify buildingblocks and their parameterisation towards composi-tion of a control system following the concept ofphase-locked loops.

2 INTERNALSYNCHRONISATION:COUPLED REPRESSILATORS

2.1 Reaction Network and Kinetics

We identified a network of bidirectionally coupled re-pressilators to be an appropriate candidate to exploreinternal synchronisation within a biological system.A repressilator is a gene regulatory network consist-ing of three focal proteins (LacI, TetR, cI) that mu-tually inhibit their expression from genes (lacI, tetR,cI) (Elowitz and Leibler, 2000). We employ a systemcomposed of two coupled repressilators located intwo adjacent cells inspired by Garcia-Ojalvo (Garcia-Ojalvo et al., 2004), see Fig. 1.

Let TetR be a protein able to migrate between thecells, it acts as coupling element. Its diffusion ratediffspecifies the variable bidirectional coupling strength.

Figure 1: Network topology of the TetR-coupled repressi-lator model with diffusion between both core oscillators.

The dynamical behaviour of the network can be spec-ified by reaction-diffusion kinetics based on corre-sponding ordinary differential equations (ODEs). Forspecies names in the ODEs, we abbreviate (LacI,TetR, cI)= (lp, tp,cp) for the proteins and (lacI, tetR,cI) = (lr, tr,cr) for the mRNA. The set of equationsfor each single repressilator reads:

d lpd t

= klr · lr− klp · lp

d tpd t

= ktr · tr− ktp · tp− diff · tp+ diff · tpexternal

d cpd t

= kcr · cr− kcp · cp

d lrd t

= α0+α · kn

m

knm + cp

− klr · lr− klr2 · lr

d trd t

= α0+α · kn

m

knm + lp

− ktr · tr− ktr2 · tr

d crd t

= α0+α · kn

m

knm + tp

− kcr · cr− kcr2 · cr

We utilise the parameter settingα0 = 0.03,α =29.97,km = 40,n = 3,k{lp,tp,cp} = 0.069,k{lr,tr,cr} =6.93,k{lr2,tr2,cr2} = 0.347 resulted from a parame-ter fitting based on the available experimental data(Garcia-Ojalvo et al., 2004). Additionally, the ini-tial species concentrations in case of no phase shiftare chosen at the limit cycle, e.g.lr = 0.819, tr =2.388,cr = 0.068, lp = 36.263, tp = 166.685,cp =64.26.

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The repressilator’s oscillation frequency mainlydepends on the degradation reaction rates. Diffusionof TetR proteins from one repressilator to its adja-cent counterpart causes the same effect. This allowsto control the frequency just by forcing using a sus-tained dissipation of diffusing TetR proteins. Fig. 2illustrates a typical synchronisation run.

time steps

Tet

R a

bund

ance

Figure 2: Typical synchronisation run of two coupled re-pressilators, coupling strengthdiff= 0.04, initial phase shift182◦ (arbitrarily chosen). Simulation carried out with Co-pasi using ODEs and parameter settings given in Sec. 2.1.

2.2 Synchronising Initial Phase Shifts

For the synchronisation study, we set up both repres-silator’s initial concentrations at the individual limitcycle in order to avoid effects occurring within thetransient phase (stabilisation phase). Afterwards, atwo-dimensional parameter scan was conducted vary-ing the initial phase shift of both repressilators be-tween 0◦ and 360◦ and simultaneously varying thecoupling strength within the relevant rangediff= 0.01to 0.13 (weak to strong coupling). The time to syn-chronisation was obtained assuming a signal conver-gence of one minute per day (ε-neighbourhood’s in-terval length = 1

1440 of oscillation period), see Fig. 3.The simulation study exhibits a correlation be-

tween coupling strength (diff) and time to synchro-nisation. Since a strong coupling (diff = 0.13) hasa more significant effect on the adjacent repressila-tor’s behaviour, synchronisation is achieved fast. Inthis case, even the influence of different initial phaseshifts can be widely neglected. The situation becomesdifferent when considering a weak coupling. Here,the initial phase shift predominantly determines thetime to synchronisation. Initial antiphase rhythmicity(phase shift 180◦) between both repressilators causes

0 50 100 150 200 250 300 350

2000

4000

6000

8000

1000

012

000

1400

0

diff = 0.01

diff = 0.04

diff = 0.07

diff = 0.1

diff = 0.13

180°

phase shift φ

timeto

synchronize

initial phase shifttim

e to

syn

chro

nisa

tion

diff = 0.01

diff = 0.04

diff = 0.07

diff = 0.10

diff = 0.13

Figure 3: Time to synchronisation subject to various initialphase shifts. Parameterdiff= 0.01, . . . ,0.13 denotes cou-pling strength from weak to strong coupling. Initial an-tiphase rhythmicity (phase shift 180◦) between both repres-silators causes the highest effort to synchronise both oscil-latory signals by mutual forcing.

the highest effort to synchronise both oscillatory sig-nals by mutual forcing. In this context, it is inter-esting to mention that the ability of the repressilatorcoupling to synchronise initial antiphase rhythmicityis a direct consequence of the (slight) asymmetric os-cillatory signal shape. While symmetric oscillationcurves (like sinusoidal signals) persist in antiphasewhen coupled, hence unable to synchronise, asym-metric curves (like spiking signals) entail a kind ofunbalanced response to forcing. There is no equi-librium between forcing effects shortening and thoseadvancing the oscillatory period. The remaining ef-fect is sufficient to initiate synchronisation. The slightasymmetry of the diagram in Fig. 3 also results fromthe asymmetric shape of the repressilator’s oscilla-tory signal. Interestingly, a medium coupling strength(diff = 0.07) generates a behaviour in which timeto synchronisation for increasing initial phase shiftcan be compensated within a range of approximately50◦ . . .100◦ and 260◦ . . .310◦, respectively.

2.3 Synchronisation of Different InitialFrequencies

We demonstrate the ability of the repressilator cou-pling to synchronise different initial frequencies in theelementary repressilators. To this end, individual pro-

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tein degradation ratesklp,ktp,kcp had been modified inconjunction with setting up all initial concentrationsat the individual limit cycle. From this, we conducteda parameter scan taking into account the ratios of ini-tial frequencies.

The purpose of this case study is to answer fourquestions: (1) Is there any synchronisation window, acontinuous range of parameter settings, that runs theentire system into synchronisation? In other words,can we detect a variant of a so-called Arnold tongue?(2) If a synchronisation window could be identified,which of the three conditions necessary for synchro-nised oscillations become violated by leaving the de-limiting parameter settings? (3) How is the timeto synchronisation distributed within the synchroni-sation window? (4) Which synchronous frequencydoes result from the initially different frequencies af-ter synchronisation?

While question (1) seems suitable to be answeredin part using the Kuramoto method (Kuramoto, 1984),an analytical ODE-based technique, a sufficient clar-ification of questions (2), (3), and (4) requires an ex-plorative simulation study. An essential part of thisstudy is the calculation of the frequencies governedby an oscillatory signal. To this end, we utilise thediscrete Fast Fourier Transformation (FFT) for long-term data accompanied by sampling and counting oflocal oscillatory signals maxima or minima for short-time data series. Time to synchronisation is againmeasured by the number of elapsed time steps up toconvergence of one minute per day (cf. Sec. 2.2).

If synchronisation is obtained, we can distinguishtwo qualitative scenarios characterised by the result-ing synchronous frequency in relation to either initialfrequencies.

Fig. 4 depicts a typical temporal course towardssynchronisation of twomarginally different initialfrequencies (solid lines). During the synchronisa-tion process, both frequencies converge to a commonvalue (dashed curves). This value deviates from bothinitial frequencies but arises in between. The synchro-nisation itself runs rather fast.

In contrast, a stronger – howeverslight – devianceof initial frequencies turns the synchronisation into afinal frequency asymptotically converging to the max-imum initial frequency, see Fig. 5 for an example.Here, the synchronisation process takes more time.

The latter case coincides with arrival at the limitsof the synchronisation window marking the maximaldeviance of initial frequencies leading to synchroni-sation. Inside the synchronisation window, the syn-chronous frequency becomes adjusted in between ofboth initial frequencies, and the more we approach to-wards the boundaries of the synchronisation window,

0.00

150.

0016

0.00

17

time steps

sync

hron

ous

freq

uenc

y

0 2000 4000 6000 8000

initial frequency cell 1 (0.001645)

initial frequency cell 2 (0.001578)

frequencies cell 1 towards synchronisation

frequencies cell 2 towards synchronisation

Figure 4: Typical temporal course towards synchronisationof two marginally different initial frequencies (solid lines)converging to a common value (dashed curves). Couplingstrength:diff = 0.01, ratio of initial frequencies:0.001645

0.001578≈1.042. Synchronous frequency: 0.001616.

0.00

150.

0016

0.00

17

time steps

sync

hron

ous

freq

uenc

y

0 2000 4000 6000 8000

initial frequency cell 1 (0.001691)

initial frequency cell 2 (0.001578)

frequencies cell 1 towards synchronisation

frequencies cell 2 towards synchronisation

Figure 5: Typical temporal course towards synchronisa-tion at the boundary of the synchronisation window. Syn-chronous frequency asymptotically reaches the maximumof either initial frequencies (dashed curves). Initial frequen-cies marked by solid lines. Coupling strength:diff = 0.01,ratio of initial frequencies:0.001691

0.001578≈ 1.072. Synchronousfrequency: 0.001690.

the synchronous frequency converges to the maxi-mum of both initial frequencies.

We obtain a synchronisation window delimited bypolyfrequential oscillations with respect to the ratiosof initial frequencies and loss of undamped oscillationwith respect to the coupling strength, see Fig. 6. Wechecked whether an oscillatory signal is undamped ornot by evaluating the eigenvalues of the Jacobian ma-trix derived from the ODEs specifying the reaction-diffusion kinetics.

Moreover, the simulation results indicate that amedium coupling strength (diff = 0.07) enables syn-chronisation within the largest ratio of initial frequen-cies ranging from 0.697 to 1.294. This means in termsof systems application for clock synchronisation thata clock signal can be temporarily slowed down (post-

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0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

0.00

100.

0020

0.00

300.

0040

diff = 0.01

diff = 0.04

diff = 0.07

diff = 0.1

diff = 0.13

min/max Quotient

Freq 1

Freq 2== 1

frequency parameter ratio

synchronizedfrequency

ratio of initial frequencies

sync

hron

ous

freq

uenc

y

Figure 6: Synchronisation window: ratios of initial fre-quencies subject to synchronous frequency considering avariety of relevant coupling strengthsdiff = 0.01, . . . ,0.13(variant of an Arnold tongue, a circle map disclosing de-pendencies of system parameters within a range of stableoscillation). Due to the bidirectionally balanced couplingstrength, an almost symmetric synchronisation window canbe obtained which is delimited by polyfrequential oscilla-tions with respect to the ratios of initial frequencies and lossof undamped oscillation with respect to coupling strength.

pone the clock) and speeded up (put the clock for-ward) with up to approximately 30% of its velocity.The knowledge about parameterisation, capabilitiesand limits of an oscillatory system envisioned to actas a biological clock is essential for subsequent inte-grative modelling, synthesis, and implementation of acorresponding frequency control system.

Bidirectionally coupled repressilators exhibit theability to synchronise their oscillatory signals by forc-ing. It has been observed that arbitrary initial phaseshifts become compensated while an adaption of theentire system to different initial frequencies of the sin-gle oscillators spans a synchronisation window.

3 EXTERNALSYNCHRONISATION:REPRESSILATOR AS COREOSCILLATOR

The repressilator can be seen as an advantageous toolto conduct external synchronisation when embeddedas core oscillator into a frequency control systembased on the concept of phase-locked loop (Stensby,1997), PLL for short. These systems adapt their oscil-latory output signal to an external stimulus acting asreference. In contrast to internal synchronisation, the

external stimulus is not affected. A biological exam-ple is given by circadian clocks that harmonise theiroscillatory behaviour with the daily light-dark rhyth-micity (Bell-Pedersen, 2005). Here, the light actsas external stimulus. Fig. 7 illustrates the generalscheme of PLL. One or several coupled core oscil-lators constitute its central part. The signal compara-tor as downstream module determines the differencebetween core oscillator output and external stimulus.The phase shift between either signals is an ideal can-didate to form an error signal able to adjust the coreoscillator. The error signal passes a global feedbackpath along with damping and delay by dedicated low-pass filters. Finally, the resulting smoothened signalinfluences the core oscillator(s) by increasing or de-creasing its frequency.

We expect to demonstrate that all functional mod-ules required for a PLL control system can be imple-mented as interacting reaction networks. Both mod-ules, signal comparator and global feedback path, ef-ficiently employ low-pass filters. Signal transduc-tion cascades found in cell signalling networks area common biological motif to cover the functional-ity of low-pass filters (Marhl et al., 2005). Here, afocal protein alters its chemical state according to atrigger signal. Here, a chemical state is specified byaddition or removal of phosphate groups to/from thefocal protein. In case of low-frequency triggers, thesubsequent modification of the chemical state can fol-low. Along with increasing frequency of the trigger,a threshold exists denoting that the reaction system isnow too slow to follow the trigger and ends up in asteady state by means of a chemical equilibrium.

Having a chemical low-pass filter at hand, thefunctionality of the global feedback path is com-pletely covered. The signal comparator benefits fromlow-pass filters to obtain the fundamental frequencyof both signals, core oscillator output and externalstimulus. Then, the phase shift between both signalsor the signal difference, respectively, can be extractedby performing arithmetic operations. Reaction net-works to this task are effectively feasible assumingthat substrate species concentrations encode operandswhile product species concentrations (in steady state)constitute the operational output (Hinze et al., 2009).For example, the set of two reactionsX1 + X2 → Yand degradationY → /0 in conjunction with mass-action kinetics conducts a multiplication of the formY = X1(0) · X2(0) with initial concentrationsX1(0)andX2(0) as multipliers. Addition, non-negative sub-traction, and division can be expressed in a similarway. Altogether, this allows construction of a PLLexplicitly composed of reaction-diffusion networks.

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coupledone or several

core oscillator(s)

local feedback(s)

global feedback path

damping and delay)(loop filter for

affectsfrequency

signaltuning

signal comparator

frequency deviation)(phase difference or

signaloutput (reference)

stimuliexternal

errorsignal

Figure 7: General scheme of a frequency control system basedon the concept of phase-locked loop (PLL). The system adaptsits oscillatory output signal to an external stimulus acting as reference for external synchronisation.

4 CONCLUSIONS

Bidirectionally coupled repressilators synchronisetheir oscillatory signals by forcing. Arbitrary initialphase shifts become compensated while adaption todifferent initial frequencies spans a synchronisationwindow. Coupled repressilators can be seen as a partof a biological control system based on the conceptof phase-locked loops. Further research has beendirected to finalise the entire frequency control sys-tem by integration of additional components for sig-nal comparison and damping, demonstrated by low-pass filters biologically implemented as specific sig-nal transduction cascades. The simulations describedin this paper were carried out using Copasi (Hoops,2006), statistical evaluation using[R].

ACKNOWLEDGEMENTS

We gratefully acknowledge funding from the GermanFederal Ministry of Education and Research (projectno. 0315260A, Research Initiative in Systems Biol-ogy).

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