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Phase transition in thermodynamically consistent biochemical oscillatorsBasile Nguyen,1 Udo Seifert,1 and Andre C. Barato21)II. Institut fur Theoretische Physik, Universitat Stuttgart, 70550 Stuttgart,

Germany2)Max Planck Institute for the Physics of Complex Systems, Nothnizer Strasse 38, 01187 Dresden,

Germany

(Dated: 3 July 2018)

Biochemical oscillations are ubiquitous in living organisms. In an autonomous system, not influenced by anexternal signal, they can only occur out of equilibrium. We show that they emerge through a generic nonequi-librium phase transition, with a characteristic qualitative behavior at criticality. The control parameter is thethermodynamic force, which must be above a certain threshold for the onset of biochemical oscillations. Thiscritical behavior is characterized by the thermodynamic flux associated with the thermodynamic force, itsdiffusion coefficient, and the stationary distribution of the oscillating chemical species. We discuss metrics forthe precision of biochemical oscillations by comparing two observables, the Fano factor associated with thethermodynamic flux and the number of coherent oscillations. Since the Fano factor can be small even whenthere are no biochemical oscillations, we argue that the number of coherent oscillations is more appropriateto quantify the precision of biochemical oscillations. Our results are obtained with three thermodynamicallyconsistent versions of known models: the Brusselator, the activator-inhibitor model, and a model for KaiCoscillations.

I. INTRODUCTION

Living systems need biochemical oscillators1 for thetiming and control of several key processes such as cir-cadian rhythms2,3 and the cell cycle4. Oscillations canonly set in if the system is out of equilibrium. The con-trol parameter that has to be non-zero for the system tobe out of equilibrium is the thermodynamic force. For in-stance, for a system in contact with a bath that containsa fixed concentration of adenosine triphosphate (ATP),the thermodynamic force is the free energy liberated withthe hydrolysis of one ATP molecule.More specifically, in a recent study on the relation be-

tween energy dissipation and the precision of biochemicaloscillations, Cao et al.

5 have shown that the thermody-namic force must be above a certain threshold for theonset of biochemical oscillations. Therefore, a naturalquestion that arises is whether biochemical oscillatorsdisplay a phase transition. In other words, what kindof non-analytical behavior do physical observables dis-play at this critical thermodynamic force? It is worthnoting that the relation between biochemical oscillationsand thermodynamics has also been studied in6–10.In this paper, we show that a generic phase transition

takes place in biochemical oscillators. As in the well-known theory of nonequilibrium phase transitions11–15,this phase transition is associated with a Hopf bifur-cation, i.e., the onset of limit cycle, in the determinis-tic rate equations. An observable that characterizes thetransition is the steady state distribution of the chem-ical species that oscillates. This distribution becomesbimodal above the critical force. We analyze the crit-ical behavior of the fluctuating thermodynamic currentconjugate to the thermodynamic force. The average ofthis current is the rate at which the biochemical oscil-lator consumes ATP. The first derivative of this averagewith respect to the thermodynamic force is found to be

discontinuous at the critical point. We also investigatefluctuations of this thermodynamic current. In particu-lar, its diffusion coefficient is found to diverge there.

Since these biochemical oscillations can occur in sys-tems with a finite number of molecules that lead to rela-tively large fluctuations, it is natural to study the preci-sion of biochemical oscillations5,16,17. More broadly, therelation between precision and dissipation in biophysicshas been intensively investigated18–23. We analyze therelation between the Fano factor associated with the ther-modynamic current and the number of coherent oscilla-tions. The Fano factor has been analyzed for theoret-ical studies related to single molecule experiments24–28

and has been proposed as an observable that can quan-tify the precision of biochemical oscillators17,29. Interest-ingly, this Fano factor has a universal lower bound thatdepends solely on the thermodynamic force, which fol-lows from the thermodynamic uncertainty relation30–32.

The number of coherent oscillations, which is the num-ber of periods for which different stochastic realizationsremain coherent with each other, is a standard measure ofthe precision of biochemical oscillators5,16,33–38. It can beused to identify the onset of biochemical oscillations, i.e.,it is zero below the critical force and becomes non-zeroabove it5. We show that the Fano factor does not prop-erly quantify the precision of biochemical oscillations.This observable that diverges at the critical point, dueto the divergence of the diffusion coefficient, is shown tobe small below the critical point, indicating high preci-sion even if there are no biochemical oscillations.

We consider three different models for biochemicaloscillators: the Brusselator39, the activator-inhibitormodel5, and a model for the oscillations in the phospho-rylation level of KaiC40, which is a protein related to theregulation of the circadian rhythm of cyanobacteria41.

The paper is organized as follows. In Sec. II we in-troduce the three models and analyze their critical be-

2

havior. In Sec. III we discuss metrics for the precisionof biochemical oscillations, with the comparison betweenthe number of coherent oscillations and the Fano factor.We conclude in Sec. IV. Details of the activator-inhibitormodel and the KaiC model are provided in Appendix Aand Appendix B, respectively.

II. PHASE TRANSITION

A. BRUSSELATOR

1. Model definition

The Brusselator is a paradigmatic model for biochem-ical oscillations39,42–44. It consists of two intermediatespecies X and Y in a volume Ω. The external bathcontains two chemical species A and B at fixed concen-trations [A] and [B], respectively. The set of chemicalreactions is

Ak1−−−−k−1

X,

Bk2−−−−k−2

Y,

2X + Yk3−−k3

3X,

(1)

where k1, k−1, k2, k−2, k3 are transition rates. For con-venience we assume that the transition rates for the for-ward and backward direction in the third reaction arethe same. The system is driven out of equilibrium dueto a difference of chemical potential between A and B,which is written as∆µ ≡ µB−µA. For example, considerthe following cycle: a Y molecule is created with rate k2,then a Y molecule is transformed into anX molecule withrate k3 and, finally, an X molecule is degraded with ratek−1. This cycle leads to the consumption of substrate Band generation of product A. The thermodynamic forceassociated with this cycle is

∆µ ≡ lnk−1k2[B]

k−2k1[A], (2)

where the temperature T and Boltzmann’s constant kBare set to kBT = 1 throughout this paper. The above re-lation between the thermodynamic parameter ∆µ andthe transition rates is known as generalized detailedbalance45.

The state of the system is determined by two vari-ables, the total number of X molecules nX and the to-tal number of Y molecules nY . The time evolution ofP (nX , nY , t), the probability to find the system in state(nX , nY ) at time t, is governed by the chemical masterequation, which reads

∂tP (nX , nY , t) =Ωk1[A]

[ǫ−X − 1

]+Ωk2[B]

[ǫ−Y − 1

]

+ k−1

[(nX + 1)ǫ+X − nX

]+ k−2

[(nY + 1)ǫ+Y − nY

]

+k3Ω2

[(nX − 1)(nX − 2)(nY + 1)ǫ−Xǫ+Y

− nX(nX − 1)nY + (nX + 1)nX(nX − 1)ǫ+Xǫ−Y

− nX(nX − 1)(nX − 2)]P (nX , nY , t),

(3)where we define step operators as

ǫ±XP (nX , nY , t) ≡ P (nX ± 1, nY , t),

ǫ±Y P (nX , nY , t) ≡ P (nX , nY ± 1, t).(4)

The system reaches a nonequilibrium steady state witha steady-state distribution written as P (nX , nY ). Themarginal distribution of nX that we evaluate in numericalsimulations is defined as P (nX) ≡

∑nY

P (nX , nY )

From the master equation (3), we obtain the equationsfor the time evolution of the densities

x ≡∑

nX ,nY

nXP (nX , nY , t)/Ω,

y ≡∑

nX ,nY

nY P (nX , nY , t)/Ω,(5)

in the deterministic limit (Ω → ∞), which read

dx

dt= k1[A]− k−1x+ k3

(x2y − x3

),

dy

dt= k2[B]− k−2y − k3

(x2y − x3

).

(6)

We have performed continuous time Monte Carlo sim-ulations using the Gillespie algorithm46. We set the pa-rameters to k1 = k2 = 0.1, k−1 = k3 = 1, [A] = 1,[B] = 3. The rate k−2 is computed with ∆µ and the gen-eralized detailed balance relation in Eq. (2), where ∆µ isthe control parameter. In Fig. 1, we show stochastic tra-jectories of this model. For large enough∆µ, biochemicaloscillations set in.

2. Results

In the deterministic limit described by Eq. (6), for∆µ ≥ ∆µc ≃ 3.95 the stationary solution becomes nu-merically unstable and a limit cycle sets in. The ther-modynamic flux associated with this force is the rate ofgeneration of the product A per volume Ω, which mustbe equal the rate of consumption of B due to the conser-vation of the number of particles in the reservoir. In thedeterministic limit this flux takes the form

J = k−1x− k1[A], (7)

where x is the stationary solution of Eq. (6). The rateof entropy production is simply given by σ ≡ J∆µ in the

3

0 50 100 150 200

20,000

40,000

60,000

80,000

t

nX

∆µ = 3.7∆µ = 3.8∆µ = 3.9∆µ = 4.0∆µ = 4.1

FIG. 1. Trajectories of nX at different ∆µ in the Brusselatorwith volume Ω = 105, the remaining parameters are given inthe main text.

steady state. As shown in Fig. 2(a), there is a disconti-nuity in the first derivative ∂J/∂∆µ at the critical point∆µc in this deterministic limit.For a stochastic system, the average thermodynamic

flux reads

J = Ω−1k−1

∑

nX

nXP (nX)− k1[A]. (8)

In Fig. 2(a), we plot ∂J/∂∆µ as a function of ∆µ. Withincreasing system size, the curve tends to the determinis-tic result, with an increase of ∂J/∂∆µ close to the criticalpoint that gets steeper with increasing volume Ω.For a finite system, there is a crossover to oscillatory

behaviors, as illustrated in Fig. 1. The stationary proba-bility distribution P (nX) also changes at this crossover.Below some finite-size critical point, where biochemicaloscillations do not take place, this distribution is uni-modal, whereas above this critical point this distributionbecomes bimodal. This result is shown in Fig. 2(b). Wedefine the finite-size critical point ∆µc(Ω) as the mini-mal ∆µ for which the distribution displays a local min-imum. In Fig. 2(c), we show that ∆µc(Ω) converges to∆µc ≃ 3.95, where the difference∆µc(Ω)−∆µc decreasesas a power-law with the system size Ω.Fluctuations related to the thermodynamic flux can

be analyzed by considering a stochastic time-integratedcurrent Z, which is extensive in time. In a stochastictrajectory, this random variable increases by one if an Ais produced, which happens if the transition with rate k−1

takes place, and it decreases by one if an A is consumed,which happens if the transition with rate k1 takes place.The average flux in (8) can be defined as

J ≡ 〈Z〉/(TΩ), (9)

where T is the time interval and the brackets denote anaverage over stochastic trajectories. This time interval islarge enough compared to relaxation times so that thestationary regime is probed. The diffusion coefficient

3.6 3.8 4 4.2 4.4

0.1

0.15

0.2

0.25

∆µ

∂J∂∆µ

Ω = 103

Ω = 104

Ω = 105

Det.

(a)

0 200 400 600 800 1000 1200 14000

0.001

0.002

0.003

nX

P (nX)

∆µ = 4.25∆µ = 4.5∆µ = 4.75∆µ = 5.00∆µ = 1000.0

(b)

4 5 6 7 8 9 10

−3

−2

−1

0

1

lnΩ

ln(∆µc(Ω)−∆µc)

∝ Ω(−0.68±0.04)

(c)

FIG. 2. Phase transition in the Brusselator. (a) First deriva-tive of the thermodynamic flux J as a function of ∆µ. Thecritical point is ∆µc ≃ 3.95. (b) Stationary distribution ofchemical species X for Ω = 103 and different values of ∆µ.(c) Difference between the point at which the distribution in(b) becomes bimodal in a finite system ∆µc(Ω) and the crit-ical point ∆µc ≃ 3.95 obtained with the deterministic rateequations.

4

0 2 4 6 80

0.2

0.4

0.6

0.8

1

∆µ

D

Ω = 101

Ω = 102

Ω = 103

(a)

2 3 4 5 6

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

lnΩ

lnDc

Dc ∝ Ω(0.27±0.02)

(b)

FIG. 3. Diffusion coefficient for the Brusselator. (a) Diffusioncoefficient D as a function of ∆µ. (b) Maximum diffusioncoefficient Dc as a function of Ω.

(per volume) associated with Z is defined as

D ≡〈Z2〉 − 〈Z〉2

2TΩ. (10)

In Fig. 3(a) we show the diffusion coefficient D as a func-tion of ∆µ. It has a maximum close to the critical pointthat increases with the volume Ω. The maximum Dc asa function of the volume Ω follows a power law with ef-fective exponent 0.27± 0.02, as shown in Fig. 3(b). Thisfinite-size scaling indicates that D diverges at the criticalpoint.

M MpK

MR Mp

R X

∆µa1 d1 d2 a2

f−2

f2

f1

f−1

FIG. 4. Activator-inhibitor model. The transition rates forthe phosphorylation cycle of an enzyme M are representedwith black arrows. The completion of the counter clock-wisecycle leads to the hydrolysis of one ATP, which liberates afree energy ∆µ. Green arrows represent activation and thered lines represent inhibition.

B. ACTIVATOR-INHIBITOR MODEL

1. Model definition

The activator-inhibitor model5 is a more elaborate bio-chemical oscillator compared to the Brusselator. Themodel is depicted in Fig. 4. It consists of activators R, in-hibitors X , enzymes M and phosphatases K interactingin a volume Ω. The external bath contains fixed concen-trations of ATP, ADP, and Pi. The enzyme M can bein four different states (M,MR,MpK, Mp), where Mp

is the phosphorylated form of the enzyme. For a phos-phorylation reaction to take place, a R molecule mustbe bound to the enzyme M and for a dephosphorylationreaction to take place, a phosphatase K must be boundto the enzyme M . In the phosphorylation reaction, oneATP is transformed into an ADP, whereas in the de-phosphorylation reaction a Pi is released in the solution.In the anti-clockwise cycle shown in Fig. 4, one ATP istransformed into ADP + Pi, hence the thermodynamic

5

force associated with this cycle is

∆µ = µATP − µADP − µPi≡ ln

a1a2f1f2d1d2f−1f−2

, (11)

where the rates ai, di, f±i are given in Fig. 4.Activators R and inhibitors X are related to this phos-

phorylation cycle in a feedback loop. The system con-tains a fixed concentration of substrate S, which is con-sumed (produced) when a R or X molecule is produced(consumed). The phosphorylated form of the enzymeMp

catalyzes both the production of R with a rate k0 andthe production of X with a rate k3 (positive feedback).Inhibitors X degrade R with a rate k2 (negative feed-back) and can be spontaneously degraded with a rate k4.For thermodynamic consistency, we must include reverserates, which are given by δi, where i = 0, ..., 4. Thesereactions correspond to the lower part of Fig. 4 and canbe written as

Mp + Sk0−−δ0

Mp +R,

Sk1−−δ1

R,

X +R+ATPk2−−δ2

X + S +ADP + Pi,

Mp + Sk3−−δ3

Mp +X,

X +ATPk4−−δ4

S +ADP + Pi,

(12)

where generalized detailed balance45 requires

δ1 = k1δ0/k0,

δ2 = e−∆µk2k0/δ0,

δ4 = e−∆µk4k3/δ3.

(13)

In contrast to the model from5, we have added ATP con-sumption in the chemical reactions Eq. (12) for thermo-dynamic consistency.An additional feature of the activator-inhibitor model

in relation to the Brusselator is the competition for ascarce number of phosphatases nKtot

. In order for oscil-lations to set in, this number must be at some interme-diate optimal value. If nKtot

> nMtot, then the phospho-

rylation cycle shown in Fig. 4 for different enzymes doesnot synchronize, since there is always free phosphatase tobind to the enzyme, which is necessary for the dephos-phorylation reaction. If nKtot

is too small then only afew enzymes can complete their cycle in a synchronizedway. This competition for a scarce resource is a commonfeature in more realistic biochemical oscillators, such asthe model for KaiC oscillations in the next section.The chemical master equation and the respective de-

terministic rate equations for this model are shown inAppendix A. The state of the system is determinedby a vector of the numbers of molecules ni, with i =R,X,K,M,MR,Mp,MpK. This vector is subjected to

the constraints nMtot= nM + nMR + nMp

+ nMpK andnKtot

= nK + nMpK , where nMtotis the total num-

ber of enzymes and nKtotis the total number of phos-

phatases. The volume of the system is Ω and concen-trations are denoted by [i] ≡ ni/Ω. The concentra-tion of enzymes is set to [Mtot] = 10, the concentra-tion of phosphatases is [Ktot] = 0.8 and the concen-tration of substrates is [S] = 1. The rates are set tok0 = k2 = k3 = 0.02, k1 = 0.008, k4 = 0.01, δ0 = δ3 =0.001, f1 = f2 = d1 = d2 = 0.3, a1 = a2 = 2. The ratesδ1, δ2, δ4 are computed with the generalized detailed bal-ance relation in Eq. (13) and f−1 = f−2 = 2e−∆µ/2,where ∆µ is the control parameter.The chemical species that we observe in our numerical

simulations is X , which, depending on ∆µ, can displaybiochemical oscillations. The fluctuating thermodynamictime-integrated current Z in this model is the total num-ber of ATP consumed: if the reaction with rate f1 (f−1)in Fig. 4 takes place, the Z increases (decreases) by one.In addition, if the reactions with rate k2, k4 (δ2, δ4) inEq. (12) take place, Z also increases (decreases) by one.The average flux J and diffusion coefficient per volume

D are defined as in Eq. (7) and Eq. (10), respectively.

2. Results

As shown in Fig. 5, the critical behavior of theactivator-inhibitor model is qualitatively similar to theBrusselator. The first derivative of the flux ∂J/∂∆µhas a discontinuity at the critical point in the thermo-dynamic limit, as obtained from the deterministic rateequations shown in Appendix A. Furthermore, the sta-tionary distribution P (nX) becomes bimodal above thecritical point, which depends on the system size. Thediffusion coefficient D diverges at the critical point, asshown in Fig. 6. The effective exponent related to thefinite-size scaling of the maximum diffusion coefficient Dc

is 0.46± 0.02, which is different from the one found in theBrusselator.

C. KAIC MODEL

1. Model definition

There are several models for the oscillations of thephosphorylation level of KaiC proteins (see47 for a sum-mary). Here, we analyze a modified version of a modelintroduced in40. In particular, we make all transitionsreversible for thermodynamic consistency.The model contains KaiC molecules and KaiA

molecules in a volume Ω. Each KaiC molecule canbe in 14 different states denoted by Cj and Cj , wherej = 0, 1, . . . , 6. The variables j indicates the phosphory-lation level of the molecule, which has 6 phosphorylationsites. The state Cj indicates the molecule is active and

the state Cj indicates the molecule is inactive. The free

6

5 5.5 6 6.5 7

−0.08

−0.06

−0.04

−0.02

∆µ

∂J∂∆µ

Ω = 100Ω = 200Ω = 500Ω = 1000Det.

(a)

200 300 400 500 600 700 8000

0.001

0.002

0.003

0.004

nX

P (nX)

∆µ = 6.1∆µ = 6.2∆µ = 6.3∆µ = 6.4∆µ = 6.5

(b)

6.5 7 7.5 8

−2.5

−2

−1.5

lnΩ

ln(∆µc(Ω)−∆µc)

∝ Ω(−0.63±0.06)

(c)

FIG. 5. Phase transition for the activator-inhibitor model.(a) First derivative of the flux ∂J/∂∆µ as a function of ∆µ.The critical point is ∆µc ≃ 6.1. (b) Stationary probabilitydistribution of nX for different values of ∆µ and Ω = 100.The distribution becomes bimodal above the critical point,which is ∆µ ≃ 6.3 for this finite Ω. (c) Difference betweenthe point at which the distribution in (b) becomes bimodalin a finite system ∆µc(Ω) and the critical point ∆µc ≃ 6.1obtained with the deterministic rate equations.

4 5 6 7 80

2

4

6

8

10

∆µ

D

Ω = 50Ω = 100Ω = 200

(a)

4 4.5 5 5.5 6 6.5 7

2

2.5

3

lnΩ

lnDc

Dc ∝ Ω(0.46±0.02)

(b)

FIG. 6. Diffusion coefficient for the activator-inhibitor model.(a) Diffusion coefficient D as a function of ∆µ. (b) Maximumdiffusion coefficient Dc as a function of Ω.

energy of state Cj minus the free energy of state Cj isgiven by

∆Ej = −E + jE/3. (14)

If the molecule is active then a phosphorylation reactioncan happen and if the molecule is inactive a dephospho-rylation reaction can happen.An essential feature of the model is that a KaiA

molecule must bind to the KaiC molecule for a phospho-rylation reaction. The KaiA molecules play a role similarto the phosphatases in the activator-inhibitor model, i.e.,it is a scarce resource that synchronizes the phosphoryla-tion cycle of different KaiC molecules. The dissociationconstant for the binding of an A to a KaiC molecule instate Cj is Kj = K0a

j . The constant a > 1 makes thedissociation constant an increasing function of j, whichis necessary for the onset of biochemical oscillations40.The transition rates for the model are as follows.

KaiA(A) can bind with rate α to active states which arepartially phosphorylated (j = 0, ..., 5) to form a complexACj . The transition rate for the unbinding of A from a

7

CT0

CT1

CT5

...

C6

C0

C1

C5

...

C6

p0k

keE

p1k

ke2E/3

p5ke2E/3

k

keE

k

(1 − p0)γ [A]p1γK−10 eE/6−∆µ/2

(1 − p1)γ [A]p2γK−11 eE/6−∆µ/2

(1 − p4)γ [A]p5γK−14 eE/6−∆µ/2

(1 − p5)γ γK−15 eE/6−∆µ/2

γeE/6−∆µ/2 γ

γeE/6−∆µ/2 γ

γeE/6−∆µ/2 γ

γeE/6−∆µ/2 γ

FIG. 7. KaiC model with differential affinity. CTj = Cj+ACj

are the active KaiC states with j bound phosphates, whichcan be bound to a KaiA(A). We assume fast binding andunbinding of KaiA with Cj , which is characterized by a dis-sociation constant Kj = K0a

j and the fraction of unboundstates pj = Kj/(Kj + [A]). Note that the fully phosphory-

lated state C6 cannot bind to A. Cj are the inactive stateswith j bound phosphates. The transition rate from Cj to

Cj is keχE(j−3)/3 and the transition rate from Cj to Cj iskeχE(3−j)/3, where χ (χ) is an indicator function that is 0 (1)for j = 0, 1, 2, 3 and 1(0) for j = 4, 5, 6. See Appendix B forthe chemical master equation and rate equations.

KaiC in state ACj is αKj . The rate for a phosphoryla-tion reaction from ACj to Cj+1 is γ, while the rate for the

reverse reaction is rate γK−1j eE/6−∆µ/2. Dephosphoryla-

tion from Cj+1 to Cj occurs with rate γ , whereas the rate

for the reversed reaction is γeE/6−∆µ/2. The transition

rate from Cj to Cj is keχE(j−3)/3 and the transition rate

from Cj to Cj is ke(1−χ)E(3−j)/3, where χj is an indicator

function that is 0 for j = 0, 1, 2, 3 and 1 for j = 4, 5, 6.This transition rates are thermodynamically consistentsince the affinity of a phosporylation cycle is ∆µ. Thechemical master equation for this model is shown in Ap-pendix B.

11 11.2 11.4 11.6 11.8 12

0.008

0.01

0.012

0.014

0.016

0.018

0.02

∆µ

∂J∂∆µ

Ω = 10Ω = 100Det.

(a)

200 250 300 350 400 450 5000

0.002

0.004

0.006

nG

P (nG)

∆µ = 12.1∆µ = 12.3∆µ = 12.5∆µ = 12.7

(b)

3 4 5 6 7 8

−2

−1

0

lnΩ

ln(∆µc(Ω)−∆µc)

∝ Ω(−0.57±0.06)

(c)

FIG. 8. Phase transition for the KaiC model. (a) First deriva-tive of the flux ∂J/∂∆µ as a function of ∆µ. The criticalpoint is ∆µc ≃ 11.6. (b) Stationary probability distribu-tion of the phosphorylation level nG for different values of∆µ and Ω = 100. The distribution becomes bimodal abovethe critical point, which is ∆µ ≃ 12.3 for this finite Ω. (c)Difference between the point at which the distribution in (b)becomes bimodal in a finite system ∆µc(Ω) and the criticalpoint ∆µc ≃ 11.6 obtained with the deterministic rate equa-tions.

8

8 9 10 11 12 13 14

0

0.002

0.004

0.006

0.008

∆µ

D

Ω = 10Ω = 50Ω = 100

(a)

2.5 3 3.5 4

−5.05

−5

−4.95

−4.9

−4.85

−4.8

lnΩ

lnDc

Dc ∝ Ω0.11±0.01

(b)

FIG. 9. Diffusion coefficient for the KaiC model. (a) Diffusioncoefficient D as a function of ∆µ. (b) Maximum diffusioncoefficient Dc as a function of Ω.

We assume fast binding and unbinding kinetics of KaiAand introduce coarse-grained states CT

j = Cj + ACj forj = 0, .., 5. The transition rates for this coarse-grainedmodel are shown in Fig. 7. Within this assumption oftimescale separation, the entropy production associatedwith this coarse-grained model remains the same as theentropy production of the full model48. The fraction ofunbound states among CT

j is pj = Kj/(Kj + [A]), where[A] is the concentration of free KaiA. This concentrationfulfills the constraint

[Atot] = [A] +

6∑

j=0

[A][CT

j

]

Kj + [A], (15)

where [Atot] Ω is the total number of KaiA molecules.

The total number of KaiC is [Ctot]Ω, which leads to theconstraint

[Ctot] =6∑

j=0

([Cj ] + [Cj ]

). (16)

The chemical master equation and deterministic rateequations for this coarse-grained model are also shownin Appendix B.The concentration of KaiC is set to [Ctot] = 1, while

the concentration of KaiA is set to [Atot] = 0.04. Theparameters determining the transition rates are set toγ = 10, γ = 1, E = 20, k = e−E, a = 10,K0 = (1/3)10−7.The thermodynamic force∆µ is kept as a free parameter.In our simulations, we monitor the phosphorylation levelof the KaiC system, which is defined as

nG =6∑

j=1

j([Cj ] + [Cj ]

). (17)

Depending on ∆µ this quantity can exhibit oscillations.The probability distribution P (nG) is the steady stateprobability of this observable. The fluctuating thermody-namic time-integrated current Z in this model is the to-tal of number of ATP consumed. For the KaiC moleculein the active state, if a transition that increases j takesplace Z increases by one and if the reversed transitiontakes place Z decreases by one.

2. Results

The critical behavior of the present model is similar tothe critical behavior observed with the Brusselator andthe activator-inhibitor model, as shown in Fig. 8. Thefirst derivative of the flux ∂J/∂∆µ has a discontinuity atthe critical point in the thermodynamic limit and the sta-tionary distribution P (nG) becomes bimodal above thecritical point. For this model, concerning the quantity∂J/∂∆µ, we could not numerically access systems sizesthat are large enough for a better agreement between fi-nite systems and the result from deterministic rate equa-tions in Fig. 8(a). The diffusion coefficient D diverges atthe critical point, as shown in Fig. 9. The effective expo-nent related to the finite-size scaling of the maximum Dc

as a function of the the volume Ω is 0.11± 0.01, which issmaller then the effective exponents found for the othertwo models.

III. CRITERIA FOR THE PRECISION OF

BIOCHEMICAL OSCILLATIONS

Biochemical oscillations can occur in finite systemsthat display large fluctuations. In this section, we com-pare two quantities that can quantify the precision of bio-chemical oscillations, the number of coherent oscillationsand the Fano factor associated with the thermodynamic

9

0 20 40 60 80 100−0.5

0

0.5

1

t

CX

∆µ = 3.0∆µ = 3.5∆µ = 3.8∆µ = 3.9∆µ = 4.0

FIG. 10. Correlation function for species X (Eq. (18)) in theBrusselator for different ∆µ. The volume is Ω = 103.

0 2 4 6 80

2

4

6

8

∆µ

N

Ω = 102

Ω = 103

0 2 4 6 80

5

10

15

20

∆µ

F

2/∆µ

Ω = 101

Ω = 102

Ω = 103

(a)

0 5 10 15 200

2

4

6

8

∆µ

N

Ω = 50Ω = 100Ω = 200

0 5 10 15 200

50

100

150

200

∆µ

F

2/∆µΩ = 50Ω = 100Ω = 200

(b)

8 10 12 14 16 18 200

5

10

15

20

25

∆µ

N

Ω = 10Ω = 50Ω = 100

8 10 12 14 16 18 200

0.5

1

1.5

∆µ

F

2/∆µΩ = 10Ω = 50Ω = 100

(c)

FIG. 11. Comparison between the number of coherent oscil-lations N and the Fano factor F . (a) Brusselator (N couldnot be measured for Ω = 10). (b) Activator-inhibitor model.(c) Model for KaiC oscillations. The right panels also showthe lower bound (20) for F

flux. The first quantity is related to the correlation func-tion

CX(t) ≡⟨(nX(t)− 〈nX〉) (nX(0)− 〈nX〉)

⟩. (18)

where nX(t) is the number of X molecules at time t, theaverage number of molecules is 〈nX〉 ≡

∑nX

nXP (nX).Note that for the KaiC model the same expression isvalid with nG instead of nX . In our simulations of allthree models, the system starts at the initial time t = 0in the stationary state. In Fig. 10, we show the corre-lation function for species X in the Brusselator. Abovethe critical point, it displays oscillations with an expo-nentially decreasing amplitude . The number of coherentoscillations N is defined as the decay time divided bythe period. It gives the typical number of oscillations forwhich two different stochastic realizations would remaincoherent with each other. Therefore, a non-zero N is asignature of biochemical oscillations, since it is a result ofa non-zero imaginary part of the first excited eigenvalueof the stochastic matrix16,33.As shown in Fig. 11, for a finite system, N becomes

non-zero above some critical value of ∆µ. It is difficultto evaluate N numerically close to the critical point. Ifthe number of coherent oscillations is too small, it is notpossible to obtain the period of oscillation from directnumerical evaluation of the correlation function. Hence,with our numerical results, we cannot determine whetherN has a jump or approaches zero smoothly at the crit-ical point. However, for simple three-state models it ispossible to evaluate N by calculating the first non-zeroeigenvalue of the stochastic matrix16,33. In this analyticalcase, N approaches zero smoothly, without a discontinu-ity. Within our numerical simulations, the first ∆µ forwhich N becomes non-zero is smaller then the critical∆µ for which the probability distribution of the oscillat-ing species becomes bimodal, for all three models.For the Brusselator, the scaling of N with the volume

Ω has been analyzed for both ∆µ close to the criticalpoint35,36 and ∆µ above the critical point34. In the firstcase, the scaling N ∝ Ω1/2 has been observed, while inthe second N ∝ Ω1 has been obtained with the linearnoise approximation. We have observed that close to thecritical point our results for all three models, which areshown in Fig. 11, agree with the scalingN ∝ Ω1/2. Abovethe critical point, we find an exponent smaller than 1,which suggest that the linear noise approximation, whichis valid for large Ω, does not apply to the system sizeswe have considered. Furthermore, in36 Xiao et al showthat N becomes non-zero below the critical point of theBrusselator. We also observe a non-zero N below ∆µc

for the activator-inhibitor and KaiC models.The Fano factor F associated with the fluctuating cur-

rent Z is defined as

F ≡〈Z2〉 − 〈Z〉2

〈Z〉=

2D

J, (19)

which is a dimensionless parameter that quantifies thefluctuations of the thermodynamic time-integrated cur-

10

rent Z. A small Fano factor means that the current isprecise. In equilibrium, where there are no biochemi-cal oscillations, the Fano factor diverges, since the av-erage current is zero. Hence, this quantity is consistentwith the absence of biochemical oscillations in equilib-rium. Due to the divergence of the diffusion coefficientD, the Fano factor also diverges at the critical point inthe thermodynamic limit.The thermodynamic uncertainty relation30 implies the

bound

F ≥2

∆µ. (20)

The highest precision achievable increases with the ther-modynamic force ∆µ. Therefore, a large ∆µ is a neces-sary condition for a small Fano factor. This inequality isillustrated in Fig. 11. Close to the critical point, F is sub-stantially larger than the bound with a local maximumthat increases with the volume Ω.Two main motivations to consider the Fano factor as

a metric for the precision of biochemical oscillations areas follow. First, in light of the thermodynamic uncer-tainty relation, the Fano factor is a natural observable torelate precision with thermodynamics. Second, for a sim-ple unicyclic mode, the Fano factor can be related to thecycle completion time (i.e. the period of oscillations)17.Furthermore, the Fano factor associated with the ATPconsumption seems particularly appealing to quantifyingthe precision of oscillations in the KaiC model, for whichwe consider oscillations of the phosphorylation level.However, our results in Fig. 11 indicate that F does

not appropriately quantify the precision of biochemicaloscillations. In particular, for the activator-inhibitor inFig. 11(b), the Fano factor is minimal below the criticalpoint where there are no biochemical oscillations. In gen-eral, since the Fano factor diverges at the critical point inthe thermodynamic limit, a system that is large enoughshould display a smaller F below the critical point ascompared to the F is some region above the critical point.Note that it is more difficult to observe this effect in theKaiC model, due to the smaller effective exponent asso-ciated with the divergence of D. For the Brusselator andKaiC model, above the crossover to oscillatory behavior,an increase in∆µ leads to an increase inN and a decreasein F . However, for the activator-inhibitor model, thereis a region around ∆µ ≃ 8 where both N and F increasewith ∆µ. Hence, for this last model, even if we restrictto a region where biochemical oscillations exist, F doesnot appropriately correlate with the precision quantifiedby N .

IV. CONCLUSION

We have characterized a generic phase transition inbiochemical oscillators. A control parameter for thisphase transition is the thermodynamic force that drives

the system out of equilibrium. The stationary distribu-tion of the oscillating chemical species becomes bimodalabove the critical force, the first derivative of the thermo-dynamic flux with respect to the force is discontinuousat criticality, the diffusion coefficient (and Fano factor)associated with the thermodynamic flux diverges at crit-icality, and the number of coherent oscillations, which isa signature of biochemical oscillations, becomes nonzeroabove the critical point. We have estimated effective ex-ponents for the divergence of the diffusion coefficient witha limited range of volumes, a more reliable calculation ofthis exponent remains an open problem.Three different models that display a limit cycle in the

deterministic limit have been investigated. We expectthat models for biochemical oscillators with this featuredisplay qualitatively the same critical behavior. For mod-els with purely stochastic biochemical oscillations49, i.e.,models that do not have a limit cycle in the determin-istic limit, this phase transition remains unexplored sofar. The critical behavior of the entropy production hasalso been investigated in several different models withnonequilibrium phase transitions50–56. The present caseconstitutes an example for which the first derivative ofentropy production, which is the flux multiplied by theforce, displays a discontinuity at the phase transition.We expect that the diffusion coefficient associated withthe fluctuating entropy production diverges at the crit-ical point for other models with nonequilibrium phasetransitions.The precision of biochemical oscillations can be char-

acterized by the Fano factor F associated with the ther-modynamic flux (or related first passage variables17) andthe number of coherent oscillations N . We have shownthat the Fano factor F can indicate high precision evenif there are no biochemical oscillations. This result sug-gests that F does not quantify the precision of biochem-ical oscillations in a reliable way. We anticipate thatthis problem will also happen with probability currentsthat are different from the thermodynamic flux, whichcould be considered as more suitable for quantifying theprecision of biochemical oscillations, since their diffusioncoefficient will also diverge at criticality.It remains to be seen whether and how the phase tran-

sition that we have characterized here can be used tounderstand the operation and design principles of bio-chemical oscillators. We have restricted our study to au-tonomous biochemical oscillators. The theoretical inves-tigation of biochemical oscillators under the influence ofan external periodic signal is an appealing direction forfuture work.

Appendix A: Activator-inhibitor model

The time evolution of P (nR, nX , nK , nM, t), the prob-ability to find the system in state (nR, nX , nK , nM) attime t, is governed by the chemical master equation

11

∂tP (nR, nX , nK , nM, t) =k0[S]nMp

(ǫ−R − 1) + Ωk1[S](ǫ−

R − 1) + k3[S]nMp(ǫ−X − 1)

+ k2/Ω[nX(nR + 1)ǫ+R − nXnR

]+ k4[(nX + 1)ǫ+X − nX ]

+ δ0/Ω[nMp

(nR + 1)ǫ+R − nMpnR

]+ δ1

[(nR + 1)ǫ+R − nR

]

+ δ2[nX [S](ǫ−R − 1)

]+ δ3/Ω

[nMp

(nX + 1)ǫ+X − nMpnX

]+Ωδ4[S](ǫ

−

X − 1)

+ a1/Ω[(nM + 1)(nR − nMR + 1)ǫ+Mǫ−MR − (nR − nMR)nM

]

+ a2/Ω[(nK + 1)(nMp

+ 1)ǫ+Kǫ+Mpǫ−MpK

− nMpnK

]

+ d1[(nMR + 1)ǫ−Mǫ+MR − nMR

]+ d2

[(nMpK + 1)ǫ−Kǫ−Mp

ǫ+MpK− nMpK

]

+ f1

[(nMR + 1)ǫ+MRǫ

−

Mp− nMR

]+ f2

[(nMpK + 1)ǫ−Kǫ−M ǫ+MpK

− nMpK

]

+ f−1/Ω[(nMp

+ 1)(nR − nMR + 1)ǫ−MRǫ+Mp

− (nR − nMR)nMp

]

+ f−2/Ω[(nK + 1)(nM + 1)ǫ+Kǫ+Mǫ−MpK

− nKnM

]P (nR, nX , nK , nM, t),

(A1)

where we define step operators as ǫ±i P (nR, nX , nK , nM, t) ≡ P (..., ni ± 1, ..., t) for i = R,X,K,M,MR,Mp,MpK.

From the master equation (A1), we obtain the equations for the time evolution of the concentrations in the deter-ministic limit5, which reads

d[R]

dt= k0[Mp][S] + k1[S]− k2[X ][R]− δ0[Mp][R]− δ1[R] + δ2[X ][S]

d[X ]

dt= k3[Mp][S]− k4[X ]− δ3[Mp][X ] + δ4[S]

d[M ]

dt= f2[MpK] + d1[MR]− a1[M ]([R]− [MR])− f−2[M ][K]

d[MR]

dt= a1[M ]([R]− [MR]) + f−1[Mp]([R]− [MR])− (f1 + d1)[MR]

d[Mp]

dt= f1[MR] + d2[MpK]− a2[Mp][K]− f−1[Mp]([R]− [MR])

d[MpK]

dt= a2[Mp][K] + f−2[M ][K]− f2[MpK]− d2[MpK]

[Ktot] = [K] + [MpK].

(A2)

Note that [R] is the total concentration of activators and ([R] − [MR]) is the concentration of free activators whichcan bind an enzyme M.

Appendix B: KaiC model

For the KaiCmodel, the state of the system is determined by the number of free active KaiC nC = (nC0, nC1

, ..., nC6),

inactive KaiC nC= (nC0

, nC1, ..., nC6

), active KaiC bound with a KaiA nAC = (nAC0, nAC1

, ..., nAC5) and free KaiA

nA. The time evolution of P (nC, nC, nAC, nA, t), the probability to find the system in state (nC, nC

, nAC, nA, t) attime t, is governed by the chemical master equation

12

∂tP (nC, nC, nAC, nA, t) =

5∑

j=0

γ[(nACj

+ 1)ǫ+ACjǫ−Cj+1

− nACj

]+ γK−1

j eE/6−∆µ/2[(nCj+1

+ 1)ǫ+Cj+1ǫ−ACj

− nCj+1

]

+5∑

j=0

γeE/6−∆µ/2[(nCj

+ 1)ǫ+Cj

ǫ−Cj+1

− nCj

]+ γ

[(nCj+1

+ 1)ǫ+Cj+1

ǫ−Cj

− nCj+1

]

+

5∑

j=0

α/Ω[(nCj

+ 1)(nA + 1)ǫ+Cjǫ+Aǫ

−

ACj− nCj

nA

]+ αKj

[(nACj

+ 1)ǫ+ACjǫ−Cj

ǫ−A − nACj

]

+

6∑

j=0

keχE(3−j)/3[(nCj

+ 1)ǫ+Cjǫ−Cj

− nCj

]+ keχE(j−3)/3

[(nCj

+ 1)ǫ+Cj

ǫ−Cj− nCj

]

P (nC, nC

, nAC, nA, t),

(B1)where the step operators are defined as as ǫ±j P (nC, nC

, nAC, nA, t) ≡ P (..., nj ± 1, ..., t) for j =

C0, ..., C6, C0, ..., C6, AC0, ..., AC5, A.For the fast binding and unbinding kinetics of KaiA, we introduce coarse-grained active states nCT

j= nCj

+ nACj,

where j = 0, 5 and nCT6

= nC6(KaiA does not bind to C6). We define the unbound fraction of active states as

pj = Kj/(Kj + [A]). The concentration of free KaiA [A] is given implicitly by the constraint

[A] +

5∑

j=0

[A][CT

j

]

Kj + [A]= [Atot] . (B2)

The state of the coarse-grained system is determined by the total number of active KaiC nCT = (nCT1, ..., nCT

6) and

inactive KaiC nC

= (nC1, ..., nC6

). The time evolution of P (nCT , nC, t), the probability to find the system in state

(nCT , nC, t) at time t, is governed by the following chemical master equation

∂tP (nC, nC, t) =

5∑

j=0

(1− pj)γ

[(nCT

j+ 1)ǫ+n

CTj

ǫ−nCTj+1

− nCTj

]

+

4∑

j=0

[A]pjγK−1j eE/6−∆µ/2

[(nCT

j+1+ 1)ǫ+n

CTj+1

ǫ−nCTj

− nCTj+1

]

+ γK−15 eE/6−∆µ/2

[(nCT

6+ 1)ǫ+n

CT6

ǫ−nCT5

− nCT6

]

+

5∑

j=0

γeE/6−∆µ/2[(nCj

+ 1)ǫ+Cj

ǫ−Cj+1

− nCj

]+ γ

[(nCj+1

+ 1)ǫ+Cj+1

ǫ−Cj

− nCj+1

]

+

6∑

j=0

pjkeχE(3−j)/3

[(nCj

+ 1)ǫ+Cjǫ−Cj

− nCj

]+ keχE(j−3)/3

[(nCj

+ 1)ǫ+Cj

ǫ−Cj− nCj

]

P (nC, nC

, t).

(B3)

From the coarse-grained master equation (B3), we obtain the equations for the time evolution of the concentrationsin the deterministic limit

d[CT

j

]

dt= (1− pj−1)γ

[CT

j−1

]+ pj+1γK

−1j+1e

E/6−∆µ/2 [A][CT

j+1

]− (1 − pj)γ

[CT

j

](1− δj,6)

− pjγK−1j eE/6−∆µ/2 [A]

[CT

j

](1− δj,0) + keχE(3−j)/3

[Cj

]− pjke

χE(j−3)/3[CT

j

]

d[Cj

]

dt= γ

([Cj+1

]−[Cj

](1− δj,0)

)− γe∆µ/2

([Cj−1

]−[Cj

](1− δj,6)

)

+ pjkeχE(j−3)/3 [Cj ]− keχE(3−j)/3

[Cj

],

(B4)

13

where [A] is given by Eq. (B2).

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