On the dephasing of genetic oscillatorsDavit A. Potoyana,b and Peter G. Wolynesa,b,c,1

Departments of aChemistry and cPhysics and Astronomy, and bCenter for Theoretical Biological Physics, Rice University, Houston, TX 77005

Contributed by Peter G. Wolynes, December 18, 2013 (sent for review November 20, 2013)

The digital nature of genes combined with the associated low copynumbers of proteins regulating them is a significant source ofstochasticity, which affects the phase of biochemical oscillations. Weshow that unlike ordinary chemical oscillators, the dichotomicmolecular noise of gene state switching in gene oscillators affects thestochastic dephasing in a way that may not always be captured byphenomenological limit cycle-based models. Through simulations ofa realistic model of the NFκB/IκB network, we also illustrate thedephasing phenomena that are important for reconciling single-celland population-based experiments on gene oscillators.

gene networks | master equation | stochastic simulations

Cyclic rhythms are a common feature of many self-organizedsystems (1) manifesting themselves in myriad forms in bi-

ology, ranging from subcellular biochemical oscillations to celldivision and on to the familiar predator–prey cycles of ecology.An oscillatory response of a gene regulatory circuit, whethertransient or self-sustained, can provide several advantages overa temporally monotonic response (2, 3). The ability of copies ofa system to synchronize may lead to dramatic noise reductionand greater precision for timing in assemblies. On the subcellularlevel, rhythmic dynamics span time scales from a few seconds, asin the calcium oscillations, to days, as in the circadian rhythms, oryears for cicada cycles (1, 4–6). Ultradian genetic oscillations,which take place on an intermediate scale from minutes to a fewhours, are medically important. A singularly important case is theNuclear Factor Kappa B (NFκB) gene network, which organizesthe mammalian cell’s response to various types of external stressand plays a role in regulating inflammation levels in populations ofcells. The response of the NFκB circuit to continuous externalstimulation has been studied by Hoffmann et al. (7) who observeddamped oscillatory dynamics for NFκB, which has been linked tothe presence or absence of particular isoforms of the inhibitor IκB.On the other hand, experiments carried out on individual cellshave detected more sustainedNFκB/IκB oscillations that either arecompletely self-sustained (8, 9) or damp at a much slower rate (9)than found for the population, depending on the duration of ex-ternal stimulation. It follows that some type of averaging takesplace, but the physical mechanisms and the stochastic aspects ofthis population averaging are not fully understood. Many aspectsof the NFκB oscillatory dynamics may be rationalized using de-terministic mass action rate equations (10, 11), but how stochasticself-sustained oscillations average out at a cell population levelremains unclear.In this work, we provide a conceptual framework for un-

derstanding stochastic averaging as a result of “dephasing” of ge-netic oscillators. By dephasing, one essentially means the loss ofcommon phase or coherence of oscillations in populations ofmRNA or protein by-products of gene activation, caused by thestochastic molecular events in the course of an oscillator’s opera-tion that occur at different times in different cells. When thephases of genetic oscillators are not controlled by external cou-pling to other cells or by exogenous signals, the random characterof molecular events induces phase diffusion. In the absence ofsynchronizing forces, an ensemble of genetic oscillators in the longtime limit inevitably will be dephased completely as part of theentropic drive to randomize the phase distribution across the en-semble. In an extreme deterministic limit of ”newtonian” geneticoscillators, there would be no dephasing and the oscillators would

preserve the memories of their phases forever. The oscillators ofthe cell, on the other hand, are driven by inherently fluctuatingmolecular entities and will get out of phase and eventually losememory of their initial phase in a rather modest time (12, 13).We explore a particularly simple yet realistic model of theNFκB/

IκB circuit (Fig. 1) and demonstrate how self-sustained stochasticsingle-cell oscillatory dynamics yield damped oscillations at thepopulation level, as observed in the experiments of Hoffmann et al.(7). Another related but more fundamental question is how thesingle-molecule nature of the gene contributes to the stochasticityof gene oscillator networks. In our model, we explicitly account forthe highly non-Gaussian noise that comes from a single geneswitching on and off and investigate how the time scale of gene-state fluctuation affects the noisiness of the oscillations.

Discrete-State Markov Models of DephasingGenetic oscillators are chemically driven mesoscale systems oper-ating with a finite number of proteins and a handful of gene states.A satisfactory description for the dynamics of gene oscillators isgiven by a master equation, which captures the underlying dis-creteness and the probabilistic nature of molecular transactions. Insuch a description, the state of the system S is given by specifyingthe number of each type of protein together with the occupation ofstates of the genes. Once transition rates Wij between the pairs ofstates i, j ∈ S are assigned, the probability ket jPðtÞi can be de-scribed starting from a given initial condition jPðt= 0Þi. AssumingMarkovian dynamics for the transitions, we have

∂tjPðtÞi=WjPðtÞi: [1]

The elements of the stochastic rate matrix W, are the transitionrates from states j to i—Wij = hijWjji for i ≠ j—and the net escaperates from states i—Wii = hijWjii= −

Pjð≠iÞhjjWjii. The probability

of a microscopic state, z, then is given by pðz; tjz0; 0Þ= hzjPðtÞi.Owing to time-translation invariance, we can write the solutionusing the eigenvectors ðjV ðiÞiÞ and eigenvalues ðλiÞ of the ratematrix W. Assuming the eigenvalues are not degenerate, the gen-eral solution is jPðtÞi= eWtjPð0Þi=PN−1

i=0 eλi tjV ðiÞihV ðiÞjPð0Þi. The

Significance

Genetic oscillators are ubiquitous regulatory motifs in themolecular control circuits of living cells. Prominent examplesinclude the cell cycle and cellular signaling. There are two pri-mary sources of noise in these oscillators: the binary characterof gene states and the low copy numbers of proteins. Thismolecular noise induces dephasing in oscillators in analogy tothe way imperfections in clocks lead to disparities in time. Wecan study how these two distinct sources contribute todephasing and how well it can be approximated by addingnoise to a macroscopic model. Using the NFκB/IκB oscillator asan example, we find that gene noise leads to significant devi-ations from the often-used phenomenological models.

Author contributions: D.A.P. and P.G.W. designed research; D.A.P. performed research;D.A.P. analyzed data; and D.A.P. and P.G.W. wrote the paper.

The authors declare no conflict of interest.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. E-mail: pwolynes@rice.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1323433111/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1323433111 PNAS | February 11, 2014 | vol. 111 | no. 6 | 2391–2396

SYST

EMSBIOLO

GY

STATIST

ICS

Dow

nloa

ded

by g

uest

on

Mar

ch 1

3, 2

020

conditional probabilities for states z are then given by pðz; tjz0; 0Þ=PN−1i=0 hzjV ðiÞieλi thV ðiÞjPð0Þi. The Perron–Frobenius theorem re-

quires the existence of at least one purely real eigenvalue withreal part zero, whereas the rest of the eigenvalues must havestrictly negative real parts ðλ0 = 0;ReðλiÞ< 0; i= 1; . . .N − 1Þ.For systems with broken detailed balance, the rate matrix isnon-Hermitian (14), which may produce complex eigenvaluescorresponding to oscillatory motion along Nc states for whichratios of left and right fluxes do not balance W12W23...WNc1

W21W32...W1Nc≠ 1 (14,

15). Circular fluxes do not average out when system size is scaled,which leads to limit cycles on macroscopic scales (16). A conve-nient way to quantify temporal relaxation of an ensemble of oscil-lators to the steady state is via autocorrelation functions of theform CðτÞ= hzτz0i=

Rdzτ

Rdz0pðzτjz0Þpssðz0Þzτz0. The steady state

probability is obtained by simply taking the limit of very longtime: pssðzÞ= pðz;∞jz0; 0Þ= hzjV ð0ÞihV ð0ÞjPð0Þi= hzjV ð0Þi. The lastequality follows from the fact that the left eigenvector corre-sponding to the stationary state is unity, hV ð0Þj= 1. Plugging inthe expressions for the probabilities in the expression for CðτÞand integrating over variables zτ and z0 we obtain

CðτÞ= α20 +XNi= 1

eλi tαiβi; [2]

where the α’s and β’s are defined as αi =R hzjV ðiÞizdz, and

βi = hV ðiÞjPð0Þi · R hzjV ð0Þizdz. Because W has real elements, theeigenvalues and eigenvectors come in complex conjugate pairs, i.e.,αiβi = ðαi+ 1βi+ 1Þp and λi = λpi+1. Pairs with smaller γ=ReðλÞ corre-spond to the slow evolving modes, which for the oscillating system,also should have a large enough frequency component ω= ImðλÞso that the system goes through at least one cycle before decayingto the steady state. In other words, one expects the mode(s) withthe highest ω=γ= ImðλÞ=ReðλÞ and lowest jγj values to dominateoscillatory dynamics. If we assume there is a significant spectralgap between the first eigenvalue and the remaining ones (∀ i,ω1=γ1 � ωi=γi and γ1 � γi), we can subtract the constant station-ary fluctuation term, α20 = hzi2ss, to obtain a fairly simple expres-sion for the autocorrelation function:

CðτÞ≈R0e−jγjτcosðωτ+ϕ0Þ; [3]

where the R0 = jαβj and ϕ0 = argðαβÞ are constants set by theinitial condition. The slowest modes, according to our assumption,are linked to the motion along the cycle, whereas the faster modescome from fluctuations longitudinal to the cycle. The apparent

similarity of the form of correlation function 3 to that of morephenomenological approaches (see the next section, Eq. 11)provides a justification for using such models, which are basedon rather restrictive approximations. In the master equationformalism, one needs only invoke the much milder assumptionthat there is a sufficient spectral gap in the eigenvalues. The lastassumption can be tested readily on minimalist models of ge-netic oscillators. To illustrate the idea, we devised a simplemodel of a NFκB gene oscillator small enough to allow an exactnumerical solution of the master equation but complex enoughto include all the essential features of gene oscillators. Thedynamical variables describing the circuit are the gene state s,NFκB, number x, IκB, number y, and mRNA number z. Becausethere are no reactions that produce or consume NFκB, its totalamount is conserved (n = const), which allows full definition ofthe system through only four variables. A schematic representationof the circuit is depicted in Fig. 1. The rate coefficients of reactionsare chosen to ensure that at the macroscopic level, the correspond-ing mean-field ordinary differential equation (ODE)-based modelexhibits limit cycle behavior (see Table S1). Stochastic simulationsof a more complete model with variables accounting for cytoplas-mic and nuclear concentrations of NFκB/IκB and with experimen-tally determined rate coefficients are presented in the last section.The master equation of our simple model ð _Pðs; x; y; zÞ= WPÞ hasthe following explicit form:

W =Ω−1kcðx+ 1Þðy+ 1Þ+ gmðz+ 1Þ+ ktlz+Ωkts+ kdðN − x− s+ 1Þ+ koff ð1− sÞ+Ω−1konðx+ 1Þs−�Ω−1kcxy+ kdðN − x− sÞ+Ωkts+ gmz+ ktlz+Ω−1konxð1− sÞ+ koff s

�:

[4]

The k’s are rate coefficients and Ω is the dimensionless volumeor the size of the system. The terms inside the brackets corre-spond to the diagonal, and the terms outside the brackets ac-count for various off-diagonal elements of the rate matrix W.Numerically, the problem boils down to obtaining a spectral de-composition of a square non-Hermitian matrix with s × x × y × zrows (columns). Whereas the variables s and x have a priori clearupper limits (smax = 2 and xmax =N), no such limit exists for yand z (ymax =∞ and zmax =∞). Therefore, to obtain meaningfulsolutions, one must place boundary conditions and choose theupper values carefully so that transition probabilities beyond theboundaries will have a marginal impact on the qualitative fea-tures of the spectrum. We have carried out numerical computa-tions on grids with Ω = 1, n = 2 − 10, and ymax; zmax = 50− 150,obtaining consistent solutions in all cases. Physically, such anapproximation may be justified by realizing that having fewercopies of NFκB makes states with larger numbers of IκB andmRNA molecules less probable; hence, probabilities fall off suf-ficiently rapidly near the imposed upper boundaries.The computed spectrum (Fig. 2) reveals the existence of

a distinct rotational mode separated from the rest by a relativelylarge spectral gap. This mode also corresponds to the real ei-genvalue smallest in magnitude (or least negative) and thereforethe slowest component of the dynamics. All other rotationalmodes have higher decay rates and, as a result, contribute verylittle to the observed oscillations (Fig. S1). The spectral gapwould be expected to grow linearly proportional to the size of thesystem Ω based on the WKB expansion-like theories (17, 18) andstochastic simulations (18, 19) of other oscillating biochemicalnetworks. In the following sections, we show that in the case ofgene networks, besides macroscopic size, the spectral gap also isa function of the time scale for gene-state fluctuations. For suchgene networks, the precise relationship between the size of thesystem and the spectral gap has yet to be worked out; however,qualitatively one can demonstrate the trends using simplified toymodels. As a prototype for such cyclic processes, we considera cyclic graph composed of discrete states connected by one-way

Fig. 1. A minimalist model of an NFκB/IκB genetic oscillator. Bold arrows in-dicate binding (kon) and unbinding (koff) of NFκB to the gene. Once bound,mRNA is produced, which initiates the synthesis (ktl) of the IκB. The copies ofmRNA also are constantly degraded. The IκB inhibits the NFκB by binding to itand preventing the activation of the gene. The IKK drives the irreversibledegradation of IκB in the complex and prevents the full deactivation of NFκB.

2392 | www.pnas.org/cgi/doi/10.1073/pnas.1323433111 Potoyan and Wolynes

Dow

nloa

ded

by g

uest

on

Mar

ch 1

3, 2

020

transitions, e.g., 1 ���!k1 2 ���!k2 3 ���!k3 . . . ���! N ���!kN 1. Forthe cases of equal rates ðk1= k2 = . . . = kN = kÞ, the rate matrix ofsuch a directed cycle admits analytical solution, yielding theeigenvalues, λn = k

�eiπnN − 1

�. Going to the limit of large N, we have

γn ≈−kn2 π22N2 and ωn ≈ kn π

N. Therefore, as expected, the spectralgap in this model grows linearly proportional to the number ofstates in the system ω1

γ1− ω2

γ2∼N. The exact solution for the single-

gene master equation has been worked out for a few special cases(20). These studies show that slow operator-state fluctuationsbroaden and skew the product number distributions. To see howslow gene fluctuations accelerate the relaxation to steady state, onemust have a model with at least two time scales, one accounting forcyclic flux and another for the gene-state fluctuations coupled to it.As such, we consider aminimalistic toymodel of a directed cycle witha unit attached to an edge (Fig. 3).The role of the edge state is to mimic the binary nature of gene

states. This dichotomic variable moderates the cyclic dynamics toan extent depending on the switching time scale. By increasingthe rate G, the real part of the smallest nontrivial eigenvalueincreases, eventually saturating for high switching rates, implyingthat dephasing dynamics become G independent. The spectralsolution of Eq. 4 shows the same trend for the values of kon andkoff corresponding to the limit cycle solution in the macroscopiclimit. The small system sizes for which we can afford to solve themaster equation (Ω = 1) severely limit such explorations, be-cause the oscillators are quite noisy to begin with and dephasingtakes place during very short times (see Fig. S1). Fortunately, thestudy of dephasing in the large (Ω � 1) and biologically morerelevant limits may be carried out via kinetic Monte Carlo sim-ulations, which are elaborated in the last section.

Phenomenological Model of DephasingIn the near-deterministic limit, the complete master equationtreatment of gene networks becomes computationally intractablebecause of the steep rise of the dimensionality of the rate matrix.Fortunately, self-sustained oscillators in the deterministic limit maybe described with an autonomous system of equations with limitcycle attractors (21). This opens an avenue for including the noise ina top-down fashion (22, 23). For such attractors, trajectories rapidlyrelax toward a limit cycle, but once on that orbit, the phaseundergoes “free diffusion” driven by the underlying fluctuations.Let us first discuss the deterministic model of self-sustained oscil-lations that are born via a supercritical Andronov–Hopf (AH)

bifurcation (21). Classical examples (1, 4, 5) of chemical oscillationscreated via AH bifurcation include the Belousov–Zhabotinskyreaction, the Brusselator model, the Selkov model of glycolysis,and many others. The normal form of the AH bifurcation may bewritten (21)

_z= p1ðΛ−ΛcÞ z− p3jzj2 z; [5]

where z is the representation of the limit cycle in the complexplane in the vicinity of the bifurcation. In the context of genecircuits, z is the dynamical variable that describes the oscillatorycycle formed by any two components in the feedback loop. Sucha simple description is a consequence of planarity of the attractorin the phase space. The parameters p1 and p3 are complex ex-pansion coefficients pn = pn′+ ipn″, Λ is the bifurcation controlparameter, and Λc is its critical value. Transforming the variablesto polar form, one obtains the normal forms for amplitude_r= ðΛ−ΛcÞp1′ r− p3′ r3, and phase _ϕ= ðΛ−ΛcÞp1″ − p3″r2. Becauseof the attractor nature of the limit cycle, the amplitude equilibrateson a faster time scale, whereas the phase evolves freely. Therefore,

we may put r at its equilibrium value r0 =�ðΛ−ΛcÞp1′

p3′

�1=2so that the

long time behavior of phase will be

ϕ−ϕ0 = ðΛ−ΛcÞp1″−

p1′p3′p3″t=ωt: [6]

Deterministic limit cycle dynamics may be described as a combi-nation of an oscillation with a constant angular frequency _ϕ=ω0,and amplitude r = r0. The accuracy of the deterministic picturedeteriorates as we scale down the size of our system so that atsome point, we must account for the fluctuating molecular na-ture of the oscillator (24, 25). As a first-order approximation, wemay describe the dynamics of phase and amplitude as beingdriven by a Gaussian white noise ηðtÞ,

_ϕðtÞ=ω0 + v+ffiffiffiffiffiffiffiffiffi2Dϕ

p· ηðtÞ; [7]

_rðtÞ= αr− λr3 +ffiffiffiffiffiffiffiffi2Dr

p· ηðtÞ; [8]

where v= h _ϕi−ω0 is the noise-induced shift of the phase and Drand Dϕ are the diffusion coefficients for amplitude and phase.The λ and α are parameters that depend on the rate coefficients

Fig. 2. Spectral solution ofmaster equation for theNFκB/IκB gene oscillator. Theaxes are the decay rate γ=ReðλÞ, and the ratio of oscillation frequency ω= ImðλÞto decay rate γ. Different points correspond to different eigenvalues. Slowmodesare grouped around γ≈ 0. Dominant rotational modes have a high ω=γ ratio.

Fig. 3. A mockup model of a genetic oscillator: a six-state directed cyclewith an attached edge. The graph shows the dependence of the spectral gapon switching rate (G). (Inset) Dependence on G of the real part of the ei-genvalue smallest in magnitude.

Potoyan and Wolynes PNAS | February 11, 2014 | vol. 111 | no. 6 | 2393

SYST

EMSBIOLO

GY

STATIST

ICS

Dow

nloa

ded

by g

uest

on

Mar

ch 1

3, 2

020

in the gene network. For the limit cycle to be stable with stationaryamplitude r0 =

ffiffiffiffiffiffiffiffiα=λ

p, the parameters must be strictly positive α,

λ > 0. It is convenient to work with the corresponding Fokker–Plank equation ∂tPðr;ϕ; tÞ= LFPPðr;ϕ; tÞ. Writing ω=ω0 + v,the Fokker–Planck operator reads

LFP =∂∂r

�−α r+ λr3 +Dr

∂∂r

+�ω

∂∂ϕ

+Dϕ∂2

∂ϕ2

: [9]

Because the dynamics of the amplitude are decoupled fromphase diffusion, the P(r, ϕ, t) becomes simply a product of twoindividual distributions Pðr;ϕ; tÞ= pssðrÞpðϕ; tÞ. Because we alsoassume that amplitude dynamics take place on a time scale fasterthan the phase fluctuates, we may use the steady-state probabilitydistribution satisfying the equation ∂

∂r

�−αr+ λr3 +Dr

∂∂r

�pssðrÞ= 0.

The phase distribution is seen to be that of a simple Wienerprocess and is given by the standard form of Green’s functionfor the diffusion equation with ϕðtÞ∈R modulo 2π. Taking thetime evolution from a specified initial condition of phasePðr;ϕ; tÞ= eLFPtδðϕ−ϕ0Þ, one obtains the following form forthe probability distribution:

Pðr;ϕ; tÞ= 1NðtÞ e

1Dr ðα2 r2− λ

4 r4Þe−

ðϕ−ωt−ϕ0Þ24Dϕ t ; [10]

where the N−1(t) is a normalization factor. At this point, it becomesstraightforward to compute the correlation function by averagingthe time-lagged product of observables, CðτÞ= hrτcosϕτ · r0cosϕ0i.As phase diffusion is decoupled from amplitude fluctuations, wecan perform the averages separately and take the product at theend. The radial part yields hri2 = RR

rdrr0dr0pssðrÞpssðr0Þrr0. Numer-ical evaluation of the radial part, assuming some reasonable valuesfor the parameters, shows that hri2 ∼Dr , implying that noiseincreases the effective length of the oscillation amplitude. Forthe phase, it is more instructive to derive the correlation func-tion by first writing the phase, ϕðtÞ−ωt−ϕð0Þ= R t

0 ηðtÞdt. In thecomplex exponential

�eiðϕðtÞ−ϕð0ÞÞ

�representation, after taking

the average over realizations of noise ηðtÞ using the Gaussianwhite noise assumption, one finds hcosϕt · cosϕ0i= 1

2e−DϕtcosðωtÞ.

Combining the last two expressions for amplitude and phase, weobtain the final expression for the correlation function:

CðτÞ= 12hri2e−DϕτcosðωτÞ∼Dre−DϕτcosðωτÞ: [11]

The correlation function is a damped cosine oscillating at anaverage stochastic frequency ω with a constant average ampli-tude. In this mesoscopic description of dephasing, the dampingtime scale is set by the noise intensity Dϕ of the phase variable.It represents a “virtual” damping, because it results from dephas-ing of the trajectories due to stochastic fluctuations, whereas

Fig. 4. Normalized autocorrelation of steady-state fluctuations in an IκBpopulation as a function of gene binding/unbinding rate coefficients.

A B C

Fig. 5. Stationary probability distribution of noisylimit cycles of an NFκB/IκB oscillator as a function ofsystem size Ω. The panels correspond to the cases of(A) nonadiabatic: slow gene binding/unbinding; (B)adiabatic: fast gene binding/unbinding; and (C)nondichotomic: absence of dichotomic gene binding/unbinding noise.

2394 | www.pnas.org/cgi/doi/10.1073/pnas.1323433111 Potoyan and Wolynes

Dow

nloa

ded

by g

uest

on

Mar

ch 1

3, 2

020

individual trajectories themselves would appear to continue tooscillate. Finally, to connect the phenomenological picture withthe master equation-based description of the previous section,we express the solution of the phase diffusion equation via eigen-solutions of the equation LVn =ΛnVn, where L=ω ∂

∂ϕ+Dϕ∂2∂ϕ2 and

the eigenvalues and eigenfunctions are Λn = −Dϕn2 + inω andVn = 1ffiffiffiffi

2πp einϕ, respectively. From the expression of eigenvalues,

we find the spectral gap to be ∼ ωDϕ, where the dephasing coeffi-

cient is inversely proportional to the size of the system, Dϕ ∼Ω−1

(17), making the oscillations completely dominated by the mainrotational mode. The expression for the correlation function takesthe following form:

CðτÞ=X+∞−∞

Cneð−Dϕn2−inωÞτ; [12]

where Cn are constant parameters set by the initial conditions. Inthe near-macroscopic limit ðDϕ � 1Þ, one recovers Eq. 11.

Stochastic Simulations of Genetic Oscillator ModelsAt a simplified level, the core of the NFκB gene network involvesa negative feedback (Fig. 1) by the inhibitors IκB, which suppressthe binding of the NFκB to the gene that activates the synthesis ofthe IκB. In eukaryotic cells, the activation of NFκB is initiated byexternal cellular stimulation, which triggers a chain of reactionsleading to the synthesis of IKK proteins. The IKK phosphorylatesthe complex of NFκB with IκB, marking the IκB for degradationand thereby activating the NFκB. Once freed from IκB, the NFκBdiffuses to the cell nucleus and activates hundreds of genes,among which is also the IκB gene, which down-regulates the ac-tivity of NFκB to ensure the attainment of a homeostatic stateonce the external stimulus is over. The cyclic suppression of IκBsynthesis by IκB itself creates a closed time-delayed loop, result-ing in oscillatory behavior. Our model of the network is based onthe scheme proposed by Sneppen and coworkers (10, 11), whoexplored the oscillatory behavior of NFκB/IκB using a system ofdeterministic ODEs. A key difference between their model andthe present one is that we include the digital nature of gene ac-tivation (OFF + NFκB → ON; Fig. 1) as opposed to modelingtranscription using pseudo–first-order mass-action kinetics withHill coefficients. Although sharing the main qualitative features,the model we use for stochastic simulations is more sophisticatedthan the toy model of an NFκB oscillator used to solve the masterequation. Most importantly, this more sophisticated model accountsfor the cytoplasmic and nuclear concentration of all components and

uses rate coefficients extracted from the experiments (see SI Text andTable S2 for details). The stochastic simulations are set up to mimicthe experiment of Hoffmann et al. (7) in which the cells areunder constant exposure to external stimuli, which results insteady production of IKK and keeps the single-cell oscillationsundamped, whereas the cell population average decays overa finite period (9).We stochastically simulate the system of reactions via a kinetic

Monte Carlo algorithm (26) (see SI Text for details). Although bothdeterministic (see Fig. S2) and stochastic levels of treatment showsustained oscillations in the populations of NFκB/IκB, stochasticsimulations have gene noise-induced dephasing, which damps theoscillations at an ensemble level (Fig. 4 and Fig. S3). Averages aretaken over the independent stochastic trajectories initiated from thesame state. The latter is done to dissect the effects of gene noise onthe dephasing by eliminating the extrinsic noise. In contrast to themaster equation studies, here we use a much larger system size (Ω ∼103), which is a better approximation of the biochemical environ-ment in which NFκB IκB gene oscillators operate in mammaliancells. Gene noise is amplified by reducing binding/unbinding rates,which leads to faster dephasing in the ensemble of oscillators. Infact, theoretically one may achieve the same levels of dephasingobserved in the experiments of Hoffmann et al. (7) by simply de-creasing the binding (kon) and unbinding (koff) rates while main-taining the ratio equal to its known (27) in vitro valueðKd = koff=kon = 0:1Þ, as is done in our simulations. This provides analternative explanation for the damping of the oscillations from thatsuggested by the deterministic models, which would indicate thatthe experimentally observed damping must arise through the actionof the other members of the IκB family via independent reactionpathways.We see that the single-molecule nature of the gene in our model

turns out to be quite crucial for describing the dephasing of ge-netic oscillators, because the time scale of operator binding andrelease has a significant impact on the dephasing times τϕ. In thelanguage of the discrete Markov and phenomenological models,slowing of operator binding/release transitions boosts the dichot-omic gene noise, which makes the real part of the eigenvaluecorresponding to the oscillatory mode more negative and increa-ses the apparent phase diffusion hΔϕ2ðtÞi∼Dϕt (Fig. 4 andFigs. S1, S3, and S4), leading to faster damping of the oscillations.To better appreciate the nature of the single-gene dynamics

in the stochastic dephasing and to test the phenomenologicallimit cycle-based models, we compute 2D stationary probability

Fig. 6. Normalized autocorrelation of steady-state fluctuations in an IκBpopulation as a function of system size Ω.

Fig. 7. Dependence of the dephasing time τϕ on the rates of translation (ktl)and gene-state switching (koff).

Potoyan and Wolynes PNAS | February 11, 2014 | vol. 111 | no. 6 | 2395

SYST

EMSBIOLO

GY

STATIST

ICS

Dow

nloa

ded

by g

uest

on

Mar

ch 1

3, 2

020

distributions by varying the system size Ω (Fig. 5) for the casesof fast (adiabatic) and slow (nonadiabatic) gene binding/un-binding (28) and also in the absence of dichotomic gene noise(nondichotomic). The latter regime is achieved simply by scal-ing the gene number with Ω together with the rest of the proteinand mRNA populations. We then compare the 2D probabilitydistributions with the corresponding deterministic limit cycles(Eq. 5). First, from the comparison with the deterministic limitcycles, we see that in both adiabatic and nonadiabatic cases, thereare high-probability regions that lie outside the deterministicpath, showing that the deterministic description does fully captureall the features of stochastic gene oscillators. Second, in single-gene cases, there are relatively large fluctuations along the radialdirections, even in the near-deterministic limit Ω ∼ 103. Al-though the probability distribution is somewhat more localizedaround the deterministic cycle, in the adiabatic case, it still is notquite like the nondichotomic case, in which there is only a verynarrow distribution around the deterministic cycle at high Ω >102 values. It is interesting to note that contrary to the nondichotomiccase, both adiabatic and nonadiabatic oscillators never quite ap-proach the deterministic limit, even as Ω→∞. This indicates thatgene oscillators do not have a precise deterministic analog.By computing the autocorrelation function for various Ω values

(Fig. 6), we see that the dephasing time reaches a plateau instead ofdiverging, as is the case with the well-investigated case of chemicaloscillators with nondichotomic molecular noise. Thus, in the caseof genetic oscillators, there is residual noise associated with thegenetic degree of freedom, which is not completely eliminated byincreasing protein and mRNA numbers. In the master equationlanguage, this means that spectral gap is not a simple linear func-tion of the system size Ω.Besides making the oscillations more stochastic, the slowing of

gene-state fluctuations also delays the negative feedback exerted bythe IκB. The delay is reflected in the expansion of the limit cycle(Fig. S2 and Fig. 5) or lowering of the oscillation frequencies, as isseen from the systematic leftward shift of the power spectrum peak(29) (Fig. S4). Interestingly, the noise-induced expansion of the limitcycle is qualitatively captured by the phenomenological model (seethe expression for hri2), which predicts higher oscillation amplitudeswith more “noisy” oscillators. One expects the rate of dephasing tobe a function of the oscillation cycling rate or the period. In otherwords, oscillators that individually complete their cycles faster giventhe same levels of noise also will dephase faster in the ensemble.

We can tune the cycling rate without significantly perturbing thenoise levels by adjusting the irreversible rates that are part of thenegative feedback loop of the network while keeping the binding/unbinding rates fixed. By doing this, one finds that both faster cy-cling rates and slower binding/unbinding rates do indeed lead tofaster dephasing, and vice versa (Fig. 7). Lowering the gene bind-ing/unbinding rates, we observed a steady increase in the noise level(Fig. S3), whereas changing translation or transcription rates hadvirtually no impact on the noise level. This confirms our reasoningthat cycling rate modulates the dephasing time scale by propagatingnoise faster. Thus, the factors of phase noise and cycling rate areseen as the main contributors to dephasing, in which the formerquantifies the rate of decorrelation but only the latter is needed toset the time scale of the oscillations.It is expected that dephasing in vivo will be controlled by extrinsic

cellular machinery via coupling of different oscillators. This not onlyshould reduce the basal phase noise, but also may lead to a highdegree of temporal organization of the underlying biochemicalevents. The role of coupling strength in the dephasing rates will bepursued in future study. Quantitative understanding of the principlesof dephasing in gene oscillators should help in the interpretation andcomparison of single-cell and population-based experiments andalso may help in the design of synthetic gene oscillatory circuits.

ConclusionIn the present work, two mathematical approaches are elaborated,each having its own unique advantages for understanding thedephasing phenomena in gene oscillators. Discrete-Markov mod-els, although computationally demanding, provide a rigorous wayto treat dephasing as a nonequilibrium relaxation from a synchro-nized state to the state of uniform phase distribution. Phenome-nological models, although less rigorous, give valuable qualitativeinsights. However, care must be taken when using such modelsbecause the stochastic simulations on gene oscillators show thatdichotomic noise from gene-state switching leads to significantdeviations from the models that are rooted in the bifurcationspresented in the deterministic equations of motion.

ACKNOWLEDGMENTS. We thank Dr. Jin Wang, Dr. Aleksandra Walczak,Dr. Masaki Sasai, and Dr. Bin Zhang for critical reading of the manuscript.We gratefully acknowledge financial support by the D.R. Bullard-WelchChair at Rice University and PPG Grant P01 GM071862 from the NationalInstitute of General Medical Sciences.

1. Winfree A (2001) The Geometry of Biological Time (Springer, New York).2. Schultz D, Wolynes PG, Ben Jacob E, Onuchic JN (2009) Deciding fate in adverse times:

Sporulation and competence in Bacillus subtilis. Proc Natl Acad Sci USA 106(50):21027–21034.

3. Morelli LG, Jülicher F (2007) Precision of genetic oscillators and clocks. Phys Rev Lett98(22):228101.

4. Goldbeter A (1997) Biochemical Oscillations and Cellular Rhythms (Cambridge UnivPress, Cambridge, UK).

5. Novák B, Tyson JJ (2008) Design principles of biochemical oscillators. Nat Rev Mol CellBiol 9(12):981–991.

6. Wang J, Xu L, Wang E (2008) Robustness, dissipations and coherence of the oscillationof circadian clock: Potential landscape and flux perspectives. PMC Biophys 1(1):7.

7. Hoffmann A, Levchenko A, Scott ML, Baltimore D (2002) The IkappaB-NF-kappaB signalingmodule: Temporal control and selective gene activation. Science 298(5596):1241–1245.

8. Ashall L, et al. (2009) Pulsatile stimulation determines timing and specificity of NF-kappaB-dependent transcription. Science 324(5924):242–246.

9. Nelson DE, et al. (2004) Oscillations in NF-kappaB signaling control the dynamics ofgene expression. Science 306(5696):704–708.

10. Krishna S, Jensen MH, Sneppen K (2006) Minimal model of spiky oscillations in NF-kappaB signaling. Proc Natl Acad Sci USA 103(29):10840–10845.

11. Tiana G, Krishna S, Pigolotti S, Jensen MH, Sneppen K (2007) Oscillations and tem-poral signalling in cells. Phys Biol 4(2):R1–R17.

12. Yoda M, Ushikubo T, Inoue W, Sasai M (2007) Roles of noise in single and coupledmultiple genetic oscillators. J Chem Phys 126(11):115101.

13. Hensel Z, et al. (2012) Stochastic expression dynamics of a transcription factor re-vealed by single-molecule noise analysis. Nat Struct Mol Biol 19(8):797–802.

14. Qian H, Qian M (2000) Pumped biochemical reactions, nonequilibrium circulation,and stochastic resonance. Phys Rev Lett 84(10):2271–2274.

15. Schnakenberg J (1976) Network theory of microscopic and macroscopic behavior ofmaster equation systems. Rev Mod Phys 48:571.

16. Wang J, Xu L, Wang E (2008) Potential landscape and flux framework of non-equilibrium networks: Robustness, dissipation, and coherence of biochemical oscil-lations. Proc Natl Acad Sci USA 105(34):12271–12276.

17. Vance W, Ross J (1996) Fluctuations near limit cycles in chemical reaction systems.J Chem Phys 105:479.

18. Gonze D, Halloy J, Gaspard P (2002) Biochemical clocks and molecular noise: Theo-retical study of robustness factors. J Chem Phys 116:10997.

19. Gonze D, Goldbeter A (2006) Circadian rhythms and molecular noise. Chaos 16(2):026110.

20. Hornos JE, et al. (2005) Self-regulating gene: An exact solution. Phys Rev E Stat NonlinSoft Matter Phys 72(5 Pt 1):051907.

21. Guckenheimer J, Holmes P (1983) Nonlinear Oscillations, Dynamical Systems, and Bi-furcations of Vector Fields (Springer, New York).

22. Stratonovich R, Silverman R (1967) Topics in the Theory of Random Noise (Gordon andBreach, New York).

23. Baras F (1997) Stochastic Dynamics (Springer, New York), pp 167–178.24. Yuan R, Wang X, Ma Y, Yuan B, Ao P (2013) Exploring a noisy van der Pol type oscillator

with a stochastic approach. Phys Rev E Stat Nonlin Soft Matter Phys 87(6):062109.25. Baras F, Mansour MM, Van Den Broeck C (1982) Asymptotic properties of coupled

nonlinear langevin equations in the limit of weak noise. II: Transition to a limit cycle.J Stat Phys 28:577–587.

26. Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J PhysChem 81:2340–2361.

27. Bergqvist S, et al. (2009) Kinetic enhancement of NF-kappaBxDNA dissociation byIkappaBalpha. Proc Natl Acad Sci USA 106(46):19328–19333.

28. Feng H, Han B, Wang J (2012) Landscape and global stability of nonadiabatic andadiabatic oscillations in a gene network. Biophys J 102(5):1001–1010.

29. Li C, Wang E, Wang J (2011) Landscape, flux, correlation, resonance, coherence, stability,and key network wirings of stochastic circadian oscillation. Biophys J 101(6):1335–1344.

2396 | www.pnas.org/cgi/doi/10.1073/pnas.1323433111 Potoyan and Wolynes

Dow

nloa

ded

by g

uest

on

Mar

ch 1

3, 2

020