Potential landscape and flux framework ofnonequilibrium networks: Robustness, dissipation,and coherence of biochemical oscillationsJin Wang*†‡, Li Xu*, and Erkang Wang*‡

*State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin 130022,People’s Republic of China; and †Department of Chemistry, Physics, and Applied Mathematics, Stony Brook University, Stony Brook, NY 11790

Edited by Jose N. Onuchic, University of California at San Diego, La Jolla, CA, and approved June 3, 2008 (received for review January 22, 2008)

We established a theoretical framework for studying nonequilib-rium networks with two distinct natures essential for characteriz-ing the global probabilistic dynamics: the underlying potentiallandscape and the corresponding curl flux. We applied the idea toa biochemical oscillation network and found that the underlyingpotential landscape for the oscillation limit cycle has a distinctclosed ring valley (Mexican hat-like) shape when the fluctuationsare small. This global landscape structure leads to attractions of thesystem to the ring valley. On the ring, we found that the nonequi-librium flux is the driving force for oscillations. Therefore, bothstructured landscape and flux are needed to guarantee a robustoscillating network. The barrier height separating the oscillationring and other areas derived from the landscape topography isshown to be correlated with the escaping time from the limit cycleattractor and provides a quantitative measure of the robustnessfor the network. The landscape becomes shallower and the closedring valley shape structure becomes weaker (lower barrier height)with larger fluctuations. We observe that the period and theamplitude of the oscillations are more dispersed and oscillationsbecome less coherent when the fluctuations increase. We alsofound that the entropy production of the whole network, charac-terizing the dissipation costs from the combined effects of bothlandscapes and fluxes, decreases when the fluctuations decrease.Therefore, less dissipation leads to more robust networks. Ourapproach is quite general and applicable to other networks, dy-namical systems, and biological evolution. It can help in designingrobust networks.

entropy production � stability � attractor � landscape � nongradientforce � cellular network

Biological rhythms exist on many levels in living organisms. Thestudy of oscillation behavior in an integrated and coherent way

is crucial to understanding how rhythms function biologically (1, 2).Biological clock dynamics is often described by a network of

deterministic nonlinear chemical reactions of the correspondingaveraged protein concentrations in the bulk. In a cell, there is afinite number of molecules. Thus, intrinsic statistical fluctuationscan be significant for dynamics. However, external fluctuationsfrom highly dynamical and inhomogeneous cellular environmentscan also be important (3). It is therefore important to investigate theroles of statistical fluctuations in the robustness and stability ofoscillation. Instead of the averaged deterministic dynamics, we needto develop a probabilistic description to model the correspondingcellular process. This can be realized by constructing a masterequation for the intrinsic fluctuations or a diffusion equation forexternal fluctuations for probability evolution (4, 5). Even forintrinsic fluctuations, we can simplify the master equation into adiffusion equation in the weak noise or large number limit (5, 6).

By solving the diffusion equation, we can obtain the timeevolution and long-time steady state of the probability distributionin protein concentrations of the network. In analogy to equilibriumsystems, the generalized potential can be shown to be closelyassociated with the steady-state probability of the nonequilibrium

network, with a few applications (6–19). Once the network problemis formulated in terms of potential landscape, the issue of the globalstability or robustness is much easier to address (16–19). Althoughdeterministic dynamics might be nonlinear and chaotic, the corre-sponding probabilistic distribution obeying linear evolution equa-tions is usually ordered and can often be characterized globally. Aninteresting study is described in ref. 14 where the stability andemergence of competence cycles in Bacillus subtilis are controlledby the intrinsic statistical fluctuations and a nonconventional sto-chasticity from nonadiabatic conditions with finite binding/unbinding rates of the proteins to DNA. The resulting cycledynamics between stable vegetation and competence state is inco-herent (defined competent state duration but stochastic intervalsbetween) in low and coherent in high Com K expression. In thiswork, we will focus on coherent dynamics of limit cycles with certainperiodicity.

Landscape and Flux Framework for Nonequilibrium NetworksLandscape ideas were introduced for uncovering global principlesin biology for protein dynamics (20), protein folding, and interac-tions (21, 22). All of these ideas were based on a quasi-equilibriumassumption with known potentials. For a nonequilibrium opensystem constantly exchanging energies and information with out-side environments, the potential landscape is not known a priori andneeds to be uncovered. Even when the probability landscape couldbe discussed such as in population dynamics and developmentalbiology (23–25), it was not clear the relationship of landscape withdynamics. Furthermore, probability flux that is zero in equilibriumcase now becomes significant. It is the purpose of this article tostudy the global robustness and physical mechanism of nonequi-librium network through the introduction of the concepts andquantifications of the potential landscape and the nonequilibriumprobability flux, with an example of oscillations against fluctuationsin the cell.

The conventional way of describing the dynamics of a network isto write down the underlying chemical rate equations: dt

dx � F(x),where x is the concentration vector of N different protein species(x1, . . . , xN) and F(x) is a vector in concentration space representingthe chemical reaction diving force controlling the dynamics. Theabove network equations are the overdamped limit of the Newton‘ssecond law. In general, one cannot write F as a gradient of apotential: F(x) � ��x

�U (no potential). Yet, global physical propertiesof the network are hard to see without a potential. As we will see,

Author contributions: J.W. and E.W. designed research; J.W. and L.X. performed research;J.W. and E.W. contributed new reagents/analytic tools; J.W., L.X., and E.W. analyzed data;and J.W., L.X., and E.W. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

‡To whom correspondence may be addressed. E-mail: jin.wang.1@stonybrook.edu orekwang@ciac.jl.cn.

This article contains supporting information online at www.pnas.org/cgi/content/full/0800579105/DCSupplemental.

© 2008 by The National Academy of Sciences of the USA

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the dynamic driving force F can be decomposed into a gradient ofa potential and a rotational flow flux. Mathematically, this is similarto Helmholtz decomposition (7, 8, 12, 15, 16, 26, 27).

Here, we provide a general way to find potential and flux. Asmentioned, the cellular network is under intrinsic and externalfluctuations (3). The dynamics of the network system is thereforemore accurately described by the probabilistic approach: dt

dx � F(x)� �, where � is the noise force from the fluctuations. The statisticalnature of the noise can often be assumed as Gaussian (large numbertheorem) and white (no memory): ��(t)�(t�)� � 2D�(t � t�). D is thediffusion coefficient tensor (matrix) measuring the level of noisestrength.

Instead of deterministic trajectories, we will focus on the prob-abilistic evolution of diffusion equation covering the whole con-centration space: �t

�P � ��J(x, t) � 0. This represents a conservationlaw of probability (local change is due to net flux in or out) and theprobability flux vector J of the system for homogeneous (space xindependence) diffusion is defined as J(x, t) � FP � D��x

� P, whereJ(x, t) is the probability flux vector measuring the speed of the flowin concentration space x [see supporting information (SI) Text andFigs. S1 and S2].

If the steady state exists that is true for many network systems(see SI Text for conditions), �t

�P � 0, then ��J(x, t) � 0. It is obviousthat in the steady state the divergence of J must vanish. There aretwo possibilities; one is J � 0. This implies, from the definition ofthe flux, 0 � FPss � D��x

� Pss. Therefore, F � D��x� Pss/Pss �

�D��x� (�ln Pss) � �D��x

� U. So, the driving force F can be repre-sented as a gradient of a potential U that is linked with thesteady-state probability distribution Pss by U � �ln Pss. J � 0 is infact the detailed balance condition under which the system is inequilibrium. So we see, under equilibrium conditions, the famousBoltzmann relationship between equilibrium probability and un-derlying potential emerges. With detailed balance, the potentialdoes exist, and the gradient of which gives the driving force andcontrols the underlying dynamics.

For nonequilibrium systems in general, however, in steady state,��J(x, t) � 0 does not necessarily mean that J has to vanish; thereis no guarantee that the detailed balance condition J � 0 is satisfied.In general, the divergence-free nature implies flux J is a rotationalcurl or more precisely recurrent field: for example, in three dimen-sions, J � � A, with nonzero curl of vector A [for higherdimensions, see Hodge decomposition (28)]. This implies, from thedefinition of the flux, Jss � FPss � D��x

� Pss. Therefore, F �D��x

� Pss/Pss � Jss/Pss � �D��x� (�ln Pss) � Jss/Pss � �D��x

� U � Jss/Pss.In this way, we have decomposed F into a gradient of a generalizedpotential U linked with steady-state probability defined by U � �lnPss and steady-state divergent-free curl flux field Jss. Nonzero fluxreflects the lack of detailed balance in nonequilibrium systems.Cellular networks are open systems often with nonzero flux (con-stantly exchanging energies with the environments, for example,pumping in energies through ATP hydrolysis or phosphorylation,etc.), so the detailed balance conditions are not necessarily obeyed.

For nonequilibrium networks, the dynamics and global proper-ties are therefore determined not only by gradient of potentiallandscape but also by the divergent free curl flux field. This mayprovide the missing link between the probability landscape and theunderlying dynamics of networks (for example, in populationevolution dynamics). The dynamics of a nonequilibrium networkspirals (from flux) down the gradient (from potential) instead ofonly following the gradient as in the equilibrium case, just likeelectrons moving in both electric and magnetic field. As we shallsee, the best example of illustrating the interplay of both potentiallandscape and flux in action is the oscillatory network.

Landscape and Flux of Biochemical Oscillation NetworkTo explore the nature of the oscillation mechanism, we will studya simplified yet important example of biochemical network of cellcycle: a periodic accumulation and degradation of two types of

cyclins during the division cycle in budding yeast. The oscillationsconnected to dynamical interactions between CLN-type cyclins andCLB-type cyclins were found (29). CLN/CDC28s, which are CLN-type cyclins associated with Cdc28-kinase, activate their own syn-thesis (‘‘self-activation’’) and inhibit the degradation of CLB/CDC28s, which are CLB-type cyclins associated with CDC28kinase. As the concentration of CLB/CDC28 becomes larger, itinhibits the synthesis of CLN/CDC28. The mutual interplay ofCLN/CDC28 and CLB/CDC28 generates the periodic appearanceof their associated kinase activities, which drive bud emergence,DNA synthesis, mitosis, and cell division of the budding yeast cellcycle. Fig. 1 shows the mechanism of CLN/CDC28 and CLB/CDC28 oscillation. CLN/CDC28 and CLB/CDC28 subunits arelimited by cyclin availability, because kinase CDC28 is excessive (1).

For the protein network, based on the Michaelis–Menten en-zyme kinetic equations, one can derive a set of differential equa-tions that describe the variation rate of each component’s concen-tration in the network. We have two independent simplifiedequations:

dX1

dt� v1

�2 � X1/Km�2

1 � X1/Km�2

11 � X2/Kn

� k2X1 � F1X1, X2� [1]

dX2

dt� k3 �

k4X2

1 � X1/Kj�2 � F2X1, X2� [2]

In these equations, X1 and X2 are the average concentration ofCLN/CDC28 and CLB/CDC28, respectively. The k�’s are the rateconstants, K’s are the equilibrium binding constants, and the J’s arethe Michaelis constants. The first term describes the synthesis, andthe last term describes the decay. In terms of dimensionlessvariables (x1 � X1/Km, x2 � X2/Kn, t� � v1t/Km), these equationsbecome

dx1

dt��

�2 � x12

1 � x12

11 � x2

� ax1

�0dx2

dt�� b �

x2

1 � cx12

where a � k2Km/v1, b � k3/(k4Kn), c � (Km/Kj) 2, and �0 � v1/(k4Km).The corresponding diffusion equation for probability distribu-

tions of protein concentrations for x1 and x2 with noise due tofluctuations is

�Px1, x2, t��t

� ��

�x1�F1x1, x2�P �

�

�x2�F2x1, x2�P

� D��2P�x1

2� � D��2P�x2

2�.

Here, D is the diffusion coefficient tensor (or matrix) assumed tobe homogeneous and isotropic constant for simplicity (D11 � D22 �D and D12 � D21 � 0). The associated flux vector components in

A B

Fig. 1. Wiring diagram. (A) Cyclin fluctuations during the cell cycle inbudding yeast. (B) The gene network. Gene X (representing CLN/CDC28) isself-activating and inhibits gene Y (representing CLB/CDC28) degradation. Yinhibits the synthesis of X.

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two-dimensional protein concentration space are J1(x1, x2, t) �F1(x1, x2)P � D�x1

� P and J2(x1, x2, t) � F2(x1, x2)P � D�x2

� P.We fix all parameters except b and c. b represents the relative

effectiveness of production of x2, and c represents the relativeeffectiveness of inhibiting x2’s degradation. The other parametervalues are a � 0.1, � 0.1, and � � 5.0. Fig. 2 shows the phase planeof b and c from the analysis of the deterministic equations (Eqs. 1and 2). We can see that the system has three phase regions: anunstable limit cycle oscillation phase, a bistable phase, and amonostable phase. Large b and large c lead to effective inhibitionof x1 production and leave only with degradation of x1, and thereforeyield a monostable decay. Smaller b and c can provide the balancebetween the activation and degradation of x1. Therefore, when b isfixed to be small and c is large, oscillation emerges. In contrast,when c is fixed to be small and b is large, bistability emerges. When both c and b are small, there is neither effective production of x2 nor

effective inhibition of x2’s degradation. This leads to effectiveproduction of x1 and again monostability. We choose for conve-nience a set of specific parameters b � 0.1, c � 100, at which thefixed point is unstable and a limit cycle emerges.

To solve the diffusion equation, we impose the reflecting bound-ary condition, J � 0, in this work. We have also explored absorbingboundary conditions and obtained similar solutions. By givingcertain initial conditions (either homogeneous or inhomogeneous)and taking the long time limit, we obtained the same steady-statesolution using the finite difference method.

As we discussed before, from the steady-state probability distri-bution P, we can identify U(x) � �ln P(x, t3 �) � �ln Pss (when�P�t

� 0) from the solution of diffusion equation at long times, or inother words the steady-state solution, as the generalized potentialfunction of the nonequilibrium network system. In this way, we mapout the potential landscape. Fig. 3 shows the potential landscape U.We can see that when the fluctuations characterized by the diffu-sion coefficient are small, the underlying potential landscape has adistinct irregular and inhomogeneous closed ring valley or Mexicanhat-like shape as shown in Fig. 3A. The closed ring is around thedeterministic oscillation trajectories. This means that the potentialis lower (and probability is higher) along the oscillation path or onthe closed ring. Inside of the closed ring, the potential is higherforming a mountain or hat. Outside of the closed ring, the potentialis also higher. The system is therefore attracted to the closed ringrather than a particular stable point or basin. We can clearly see thatthe potential landscape is not uniformly distributed along the limitcycle path or the closed ring. This is because the time spent on eachstate of the averaged deterministic oscillation paths depends on therate at which the system passes through each state. The potential islower for lower passing rates and longer durations of stay in eachpoint (details in SI Text) (2). Because of the inhomogeneity of thepassing speed and the time spent, the potential landscape or thesteady-state probability along the closed ring is not uniform.

0

5

10 0

2

4

60

10

20

30

X2

A

X1

U

0

5

10 0

2

4

6

8

10

8

10

12

14

X2

B

X1

U

Fig. 3. Potential energy landscape with parameter b � 0.1, c � 100 withMexican hat like closed ring valley shape (diffusion coefficient D � 0.001) (A)and with shallow shape (D � 1.0) (B). The blue arrows represent the flux, andthe white arrows represent the force from negative gradient of the energylandscape.

X2

(F

lux)

Vector

C D

E F

0

1

2

3

4

5

6

Direction

X1X

2 (

Gra

dien

t for

ce)

0 2 4 6 8 100

1

2

3

4

5

6

X10 2 4 6 8 10

X2

(R

esid

ual f

orce

)

0

1

2

3

4

5

6

A B

Fig. 4. Flux and gradient potential force. Shown are the vector graphs of theflux (A), the residue force (C), the force from negative gradient of the energylandscape (E), and the direction of those forces with diffusion coefficient D �0.001 (B, D, and F).

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00

20

40

60

80

100

II

c

b

I

II

III

I: one unstable pointII: one stable pointIII: two stable point

Fig. 2. Phase diagram for the network.

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The potential landscape becomes flatter as the diffusion constantD grows larger, as indicated by the shallower energies along theclosed ring compared with both the inside and the outside of theclosed ring. The landscape transforms from a distinct irregularinhomogeneous closed ring valley into a flatter structure as shownin Fig. 3B. It implies that, when the system is under largerfluctuations, there is more freedom to go to other states andtherefore less attraction to the deterministic oscillation path. Lesstime is then spent on that path. The resulting underlying landscapedeparts from the clear closed ring valley structure of oscillations,reflecting the large fluctuations. When the diffusion coefficientincreases, the attraction to the limit cycle will be weaker; conversely,the weaker the fluctuation, the more robust the oscillation.

We can also obtain the steady-state probability flux, anotheressential quantity for the network system once we get the steady-state solution of the diffusion equation. At the steady state, thereis a circulating flow with nonzero curl as shown in Figs. 3 and 4. InFig. 3, the blue arrows represent the steady-state probability fluxand the white arrows represent the force from negative gradient ofthe potential landscape. Fig. 4A shows both the magnitude anddirection of the flux; Fig. 4B shows only the direction of the fluxflow. The magnitude of the flux is small inside and outside of theclosed ring but significant along the ring (Figs. 3 and 4). Thedirection of the flux near the ring is parallel to the oscillation path.The forces from negative gradient of the potential landscape areinsignificant along the closed ring and significant inside and outsideof the ring. So, inside and outside of the closed ring valley, thenetwork is attracted by the landscape toward the closed ring. Alongthe closed ring valley, the network is driven by the curl flux flow foroscillation.

The magnitude and direction of the residual curl flux forceF�(x) � F � D��U is also shown in Fig. 4C. Fig. 4D shows thedirection of the residual force F�(x). We see that the direction of curlflux J is parallel to that of the residual force F�(x). This is expectedfrom the force decomposition discussed earlier: F � D��U � Jss/Pss.The residual force is thus parallel to the flux J and is the drivingforce for the curl field of probability flux.

Without the landscape’s gradient-potential force (Figs. 3 and 4 Eand F), the system will not be attracted to the oscillation ring (majorstages such as G1, S, G2, and M of cell cycle). Without the curl fluxdriving the system (nutrition supply as the pump), the system willget stuck in low potential valleys on the ring without movingfurther(check points in cell cycle), and oscillation will not occur(Figs. 3 and 4 A–D). There is an important interplay between thedominant attractive force from the landscape inside as well asoutside the closed ring and the dominant driving force from the fluxalong the closed ring. So both landscape and flux are necessary tocharacterize this kind of nonequilibrium system, and this oscillation(of cell cycle) provides an excellent illustration of that necessity.

We also notice that, when the diffusion coefficient D is small,the curl flux Jss is almost parallel to real force F(x). This is becausethe gradient component of the force is proportional to D and theresidual force gives dominant contribution to total force when D issmall. In this case, dx/dt � Jss/Pss, so the period of oscillation can beapproximated through the loop integral of inverse flux along theoscillation path: T � �dl/(Jss

l /Pss). This provides a possibility throughthe observation of oscillation period and local speed to explore thenatures of the network flux.

Transition Time, Barrier Height, and RobustnessWe now study the stability and robustness of the network. Thestability is related to the escape time from the basins of attraction.

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

2

4

6

8

10

0.001 0.002 0.003 0.004 0.005

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10

0.0

2.0x104

4.0x104

6.0x104

8.0x104

1.0x105

1.2x105

1.4x105

Bar

rier

D

Barrier1 Barrier2

B

Bar

rie

r

D

A

τ

Barrier

Barrier1 Barrier2

Fig. 5. Barrier heights and escape time. (A) Barrier Heights (Ufix � Umin) and (Ufix � Umax) versus diffusion coefficient for b � 0.1, c � 100. (B) Escape time versusbarrier heights for different diffusion coefficients.

-1 0 1 2 3 4 5 6 7 8

0.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

0.0 0.2 0.4 0.6 0.8 1.00.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4A

EP

R

Barrier

Barrier1 Barrier2

B

EP

R

D

Fig. 6. Entropy production. (A) Entropy production rate versus diffusion coefficient for b � 0.1, c � 100. (B) Entropy production rate versus barrier heights.

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Because the system is characterized by the basins of attractor withlarge weights, the easier it is to escape, the less stable is the system.For the probabilistic description of the network above with diffu-sion equation, the mean first-passage time for escape �(x1, x2)starting from the point (x1, x2) obeys (30) F(x)��x� � D��x�x� � �1,and in our case of two dimensions:

F1

��

�x1� F2

��

�x2� D��2�

�x12 �

�2�

�x22� � �1.

It is essentially the average time it takes from a initial position toreach a given final position. The equation can be solved by anabsorbing boundary condition at the given site and reflectingboundary conditions for the rest.

For an equilibrium system, the barrier height on the potentiallandscape is intimately related to the escape time by Arrheniuslaw. The question is, Will there be still a direct relationshipbetween the escape time and barrier height for a nonequilibriumnetwork? If so, the landscape topography will then provide aquantitative measure of the hardness of the system to escapefrom the limit cycle attractor to outside and therefore of thestability and robustness.

We define the barrier heights as barrier1 � Ufix � Umin andbarrier2 � Ufix � Umax. Umax is the potential maximum along thelimit cycle attractor. Umin is the potential minimum along the limitcycle attractor. Ufix is the potential at the local maximum pointinside the limit cycle circle. In Fig. 5A, as the diffusion coefficientcharacterizing the fluctuations decreases, the barrier heights asso-ciated with escaping from the limit cycle attractor barrier1 andbarrier2 are higher. In Fig. 5B, we see a direct relationship betweenthe escape time and landscape barrier heights for nonequilibriumnetwork: as the barrier for escape becomes higher, the escape timebecomes longer. The resulting limit cycle attractor becomes morestable because it is harder to go from the ring to the outside.Therefore, small fluctuations and large barrier heights lead torobustness and stability in the oscillatory protein network.

Entropy Production, Barrier Height, and RobustnessThe nonequilibrium network is often an open system exchanginginformation and energies with its surroundings. Therefore, thenonequilibrium steady state dissipates energy and causes en-tropy, just as electric circuits dissipate heat due to the action of

both voltage (potential) and current (flux). Therefore, dissipa-tion can be determined globally with both the landscape and flux.In the steady state, the heat loss rate is equivalent to the entropyproduction rate. We will explore the dissipation cost via entropyproduction rate at the steady state (details in SI Text) (18, 19, 31,32). Fig. 6A shows the entropy production rate for differentdiffusion coefficients. We can see that the dissipation or theentropy production rate decreases when the diffusion coefficientdecreases. This implies that when the fluctuations of the systemsbecome smaller, the associated dissipation is smaller. Becausefewer fluctuations lead to more robust oscillations as shownabove in Fig. 5, less dissipation should be closely linked withfewer fluctuations and a more stable network. Indeed, we seethat less dissipation leads to higher barrier heights barrier1 andbarrier2, and therefore a more stable network (Fig. 6B). Becausethe entropy production is a global characterization of thenetwork, minimization of the dissipation cost might serve adesign principle for evolution of the network. It is intimatelyrelated to the robustness of the network.

Period, Amplitude, and Coherence of Oscillations AgainstFluctuationsTo address more of the robustness of the oscillations, we study thechemical reaction network equations under the fluctuating envi-ronments by simulating the stochastic dynamics for different valuesof D. That is, we follow the stochastic Brownian dynamics insteadof the deterministic average dynamics. Fig. 7 A and B shows thedistributions of the period of oscillations calculated for 1,400successive cycles. We can see that the distribution becomes morespread out with a mean period that is still close to the determin-istic period of oscillations ( � 343.3) when the fluctuationsincrease. The standard deviation � from the mean increases, andmore other possible values of the period of oscillations can appearwhen the fluctuations increase (Fig. 7C) (2). This implies that fewerfluctuations lead to more coherent oscillations with single period

0 20 40 60 80 1000.00

0.02

0.04

0.06

0.08

0.10

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

0 20 40 60 80 100

0

2

4

6

8

10

12

0.5 1.0 1.5 2.0 2.50

1

2

3

4

5

6

7

100 200 300 400 500 600 7000.0

0.1

0.2

0.3

Fre

quen

cy

Period

D: 0.01µ: 349.93σ: 4.56

100 200 300 400 500 600 7000.00

0.02

0.04

0.06

0.08

0.10D: 0.05µ: 338.88σ: 64.30

Fre

quen

cy

Period

EP

R

D

σ

DEPR

σ

Bar

rier

Barrier1 Barrier2

A

C D

Bar

rier

σ

B

Fig. 7. Period distribution against fluctuations. (A and B) Period distribu-tion (A) and variance (B) with different diffusion coefficients D for b � 0.1, c �100. (C) Diffusion D and entropy production rate EPR versus standard devia-tion �. (D) Barrier heights versus �.

0.00 0.05 0.10 0.15 0.200

1

2

3

0 2 4 6 8 100.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

σ

D

B

Fre

quen

cy(%

)

Amplitude(X2)

D=0.03 D=0.05 D=0.10 D=0.20

A

Fig. 8. Amplitude distribution against fluctuations. (A) Amplitude distribu-tion of different D for x2. (B) Standard deviation � of amplitude versusdiffusion D.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.4

0.8

0.0

2.0x10-4

4.0x10-4

6.0x10-4

0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8

10

0.997 0.998 0.999 1.0000

1

2

3

4

5

6

7

B

Barrier

EP

RD

Coherence

A D EPR

Coherence

Barrier1 Barrier2

Bar

rier

Coherence

Fig. 9. Coherence against fluctuations. (A) Diffusion coefficient D andentropy production rate versus phase coherence. (B) Barrier heights versusphase coherence.

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instead of multiple periods. We also see that the period distributionbecomes less dispersed when the entropy production rate is less.This shows that a less dissipated network can lead to a morecoherent oscillation with a unique period instead of a distributionof periods. We see also that higher barrier heights lead to lessdispersed period distribution (Fig. 7D). All of these show that morea robust network leads to more coherent oscillations focusing on asingle rather than multiple periods.

We also show the distributions of the amplitude for x2 as Dincreases. The distribution becomes more dispersed but keeps thesame mean value close to the deterministic amplitude of x2[Amplitude(x2) � 4.79], as the fluctuations increase in Fig. 8A. Thestandard deviation � increases when D goes up in Fig. 8B. This alsoshows that less fluctuation leads to a more robust and coherentoscillation.

The robustness of the oscillation can be quantified further bythe phase coherence � that measures the degree of periodicity of thetime evolution of a given variable (33) (details in SI Text). In thepresence of fluctuations, the more periodic the evolution, the largerthe value of �. In Fig. 9A, � decreases when the diffusion coefficientincreases. This means that larger fluctuations tend to destroy thecoherence of the oscillations and therefore the robustness. Incontrast, in Fig. 9B, � increases when barrier heights increase,showing that a more robust network leads to more coherentoscillations. � decreases when entropy production increases. Dissi-pation tends to destroy coherence.

ConclusionsWe have uncovered the underlying potential landscape and flux ofnonequilibrium networks, crucial for determining its dynamics andglobal robustness. The landscape of the oscillation network has aclosed ring valley shape attracting the system down, and the curl fluxalong the ring is the driving force for oscillation. The potentialbarrier height, shown here to be correlated with escape time,provides a measure of the likelihood of escaping from the limit cycleattractor, which determines the robustness of the network.

We observe that when the fluctuations increase, global dissipa-tion, period, and amplitude variations increase and barrier heightand coherence decrease. The period and amplitude variations canbe experimentally measured and compared with theoretical pred-ications (refs. 34–37; similar topology as here in refs. 35 and 37).Furthermore, minimizing the dissipation costs may lead to a generaldesign principle for robust networks. The framework and methodsin this article can be applied to more complicated and realisticnetworks and dynamical systems to explore the underlying globalpotential landscape and flux for probabilistic population dynamicsand biological evolution.

ACKNOWLEDGMENTS. We acknowledge the use of VCell software for part ofthe calculations in this work. J.W. thanks Profs. H. Qian, P. Ao, and P. Mitra fordiscussions. J.W. thanks the National Science Foundation for a Career Award. L.X.and E.W. are supported by National Natural Science Foundation of China Grants90713022 and 20735003.

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