Stochastic bifurcation in discrete models of biological oscillators

Max Veit

School of Physics and Astronomy, University of Minnesota,

Minneapolis, Minnesota 55455, USA

John Wentworth

Harvey Mudd College, Claremont, California 91711,

and Minnesota Supercomputing Institute,

University of Minnesota, Minneapolis, Minnesota 55455, USA

Mathieu Gaudreault

Department of Physics, McGill University,

3500 University St., Montreal, Quebec, H3A 2T8, Canada

Jorge Vinals

School of Physics and Astronomy, and Minnesota Supercomputing Institute,

University of Minnesota, Minneapolis, Minnesota 55455, USA

(Dated: June 26, 2014)

Abstract

We study the Hopf bifurcation threshold of two simple motifs of biological oscillation induced

by delayed feedback under noisy conditions: delayed protein degradation and dimer negative auto

regulation. We focus on fluctuations of intrinsic origin and study their effect on the bifurcation. We

contrast our numerical results based on chemical one step stochastic processes with two approximate

schemes that rely on the relative amplitude of fluctuations to averages: a fully macroscopic limit

described by stardard chemical kinetic equations with delay, and the weak noise (Langevin) limit,

also with delay. We find that fluctuations stabilize a stationary fixed point against oscillation,

in disagreement with macroscopic and weak noise predictions. Delayed conditional averages are

also studied in order to examine the validity of analytic decoupling schemes for one step process

equations with delay. We find that the decoupling fails close to the Hopf bifurcation.

1

I. INTRODUCTION

There is considerable interest in elucidating the design principles behind robust biological

oscillators as they underpin a wide variety of classes of biological control and response [1]. In

addition, significant effort is being devoted to the design, modeling, and in vivo realization

of synthetic oscillators [2–5], not only for their potential applications but also in search of

controlled models in which to study the key factors that contribute to oscillator performance:

robustness, tunability, and stability against fluctuations.

A large body of research has highlighted the importance of stochasticity, not only in

biological oscillators, but in gene regulation more generally. There is now direct evidence

of intrinsic stochastic behavior as the cause of differing gene expression levels across cell

populations in prokaryotes [6–8] as well as in eukaryotes [9, 10]. In fact, single cell trascrip-

tion experiments in E-coli have shown random fluctuations in mRNA levels down to single

molecule resolution [11]. Recent research in the specific case of biological oscillators has been

reviewed in Ref. 5. Modeling and in vivo studies range from the simplest single gene re-

pressor motif [12–14], through more complex repressilators [2, 15], to other multi gene based

oscillators [16–19]. In vivo experiments and modeling and simulation appear to be in rea-

sonable agreement regarding macroscopic (or physiological) observations such as parametric

ranges of oscillation and the period of the oscillation. However, it is also frequently the case

that stochastic oscillation is observed in ranges of parameters which differ from predictions

by chemical kinetics models. Such discrepancies have been documented, for example, for the

Smolen oscillator [14], for the repressilator [15], and for amplified, negative feedback oscilla-

tors [17]. We present here results on two different models that exhibit a Hopf bifurcation to

oscillation, and contrast the results of a stochastic model with those that would follow from

either macroscopic (chemical kinetics) or small noise (Langevin) descriptions.

We build on earlier work by Kepler and Elston [20] on stochasticity in transcriptional

switches (a direct instability rather than the oscillatory instability analyzed here). They fo-

cused on the effect of operator fluctuations in transcription, and presented a detailed analysis

of a stochastic discrete model that explicitly included the discrete states of the operators.

They also analyzed the limiting cases of a diffusion process in the weak noise limit, and the

macroscopic deterministic limit. Their analysis reveals a wealth of qualitative changes in the

bifurcation diagrams associated with the various limits. For example, stochastic bistability

2

can be present when the macroscopic limit reveals only a single stable state and, conversely,

deterministic bistability can be washed away by fluctuations. Also, new bifurcations appear

that can be traced back to the fluctuation rates of the operators. As expected, marked

differences between a discrete stochastic model and its continuum approximation appear

when the approximation fails: small transcript numbers, and fast but finite operator fluctu-

ations between its discrete states. We extend this analysis to the specific class of oscillators

that owe their oscillatory response to delayed feedback reactions, and in particular reexam-

ine some of the results of Ref. 13 who studied the effects arising from the combination of

delayed reactions in a gene regulation network and stochasticity of intrinsic origin.

Novak and Tyson [1] list three general principles underpinning biological oscillation. First,

nonlinearity is necessary for the existence of more than one stable attractor, and hence for the

exchange of stability as the model parameters are varied. Second, negative feedback loops

are generally responsible for oscillation in biological systems [21]. Third, delay is important

in producing robust oscillation in biological networks, especially when a delayed negative

feedback loop is engineered into the core of the oscillator [14]. Delay facilitates asymptotic

time dependence in cases in which competing but instantaneous forces would result in steady

states instead. For example, the importance of delay to oscillation in circadian clocks has

been documented both experimentally [22, 23] and theoretically [24, 25]. Prior studies of the

interaction between oscillation induced delayed feedback and stochasticity are given in [26–

30], but only in the diffusion limit. We extend these analyses below to two model systems

that undergo an oscillatory instability and in which fluctuations are not assumed small.

A macroscopic description of a biochemical system conventionally starts from mass action

kinetics for a set of chemical species present in numbers n = ni. If delayed reactions with

a single delay time τ are allowed, reaction kinetics is described by

niΩ

=∑j

gijAj(n(t)

Ω,n(t− τ)

Ω) (1)

where gij specifies the stoichiometric coefficient of species i in reaction j, and Aj is the

rate law for reaction j. Ω is the volume of the system. For generalized mass action rate

laws, Aj must be a polynomial with no additional dependence on Ω. The set of equations

(1) constitutes a system of delay differential equations [31]. It is worth noting here that

perturbative solutions to (1) in small τ may fail. For example, for the simple feedback

equation x(t) = ax(t) + bx(t− τ), a perturbation expansion in the delay τ is valid only up

3

to first order [32], and such an expansion misses its Hopf bifurcation entirely [26].

The macroscopic evolution laws, Eq. (1), are often extended to a one step (or discrete)

stochastic process in which the probability distribution function P (n, t) satisfies,

P (n, t) =∑m,j

(∏i

E−giji − 1)Aj(

n

Ω,m

Ω)P2(n, t;m, t− τ) (2)

Here Ei denotes the raising operator for species i, EiP (n, t) = P (n+ei, t), and P2(n, t;m, t−

τ) is the joint probability distribution function at times t and t − τ . Because of the finite

delay τ , the stochastic process is non Markovian [33].This implies that a description in terms

of a single probability distribution function P (n, t) is fundamentally incomplete. However,

in the case of a single delay time, Eq. (2) for P (n, t), though not closed, only depends on

the joint probability P2. This fact allows approximate closure approximations to obtain P ,

as will be discussed below. Similarly to delay differential equations, non Markovian terms

can be expanded in Taylor series under the assumption that time delay is small compared to

other characteristic scales of the system, leading to an effectively Markovian problem. Such

an expansion is also questionable in general.

Of course, Eq. (2) can be used to find the expectation values of n, leading to

〈ni(t)〉 =∑j

gijΩ〈Aj(n(t)

Ω,n(t− τ)

Ω)〉 (3)

If the Aj are affine in their arguments, then 〈Aj(n(t)Ω, n(t−τ)

Ω)〉 = Aj(

〈n(t)〉Ω, 〈n(t−τ)〉

Ω) and Eq.

(3) reduces to the macroscopic limit Eq. (1). This is not the case for higher order statistical

moments, however. More generally, Eq. (3) approaches Eq. (1) for any generalized mass

action system in the macroscopic limit of Ω → ∞ [34] . Stochastic fluctuations around

the average become negligible for systems with large numbers of each species. We wish to

explore below the opposite limit of Ω small as appropriate for networks in which at least

some of the variables need to remain discrete; either because of low copy number, or because

thay represent discrete states.

We study below two specific models in which delayed feedback leads to oscillation. Both

for large and small values of Ω characteristic temporal correlation times increase near the

onset of oscillation, and we find that the two point distribution function P2(n, t;m, t−τ) is a

convenient diagnostic of the bifurcation. Generally we find that the transition to oscillation

depends on Ω and does not coincide with the predictions of the macroscopic nor weak

4

noise limits. A key observation is that the boundary condition that captures the positivity

requirement of the discrete model (ni ≥ 0) provides for oscillation saturation above onset

of the Hopf bifurcation. On the other hand, in the first model studied of a single auto

repressor with degradation, its linearity both in the macroscopic and small noise limits leads

to unbounded oscillation above onset. Hence neither limit is defined above onset. We do not

believe that this is a shortcoming of the chemical model, nor that it argues against delayed

feedback providing for oscillation [35, 36], rather it is a difficulty confined to the macroscopic

and weak noise approximations.

II. SINGLE AUTO-REPRESSOR SYSTEM WITH DELAYED DEGRADATION

A stochastic description of a delayed feedback system, and the limiting behavior for

small fluctuations (large volume) have been extensively studied for the case of a single gene

autorepressor system. In its simplest form, we consider a single species with fixed production

rate and two paths for degradation, one instantaneous, and the other delayed by a fixed time

τ , [13]

∅ A−→ X XB−→ ∅ X

C⇒ ∅ (4)

As mentioned in Ref. 13, one example of such a system is found in [24], in which the

FRQ protein in the Neurospora crassa circadian clock has both delayed and non-delayed

degradation pathways with comparable rates. More generally, this model corresponds to

a linear approximation of a single variable, single delay chemical system undergoing an

oscillatory instability, and hence represents prototypical behavior.

The equation for the single point probability distribution function corresponding to (4)

for an extensive system of volume Ω is

P (n, t) = AΩ(E−1 − 1)P (n, t) +B(E − 1)P (n, t) +

+C∞∑m=0

m(E − 1)H(n)P2(n, t;m, t− τ) (5)

where the last term in the right hand side can be rewritten as,

∞∑m=0

m(E − 1)H(n)P2(n, t;m, t− τ) = (E − 1) [〈m, t− τ |n, t〉P (n, t)]

where n is the number of molecules of X at time t, m is the number of molecules at time

t−τ , and P the associated probability. E is the raising operator EP (n, t) = P (n+1, t), and

5

H(n) is the step function that enforces the constraint n(t) ≥ 0. The equation for P (n, t) is

not closed; rather it depends on the joint probability P2(n, t;m, t− τ).

One method that can be used to analyze Eq. (5) is to simulate stochastic trajectories

with Gillespie’s stochastic simulation algorithm (SSA), extended to accommodate delayed

reactions. One such extension is given in Ref. 13 which implements delayed reactions by

scheduling them to occur in the future, according to the selected reaction’s delay. Here

we employ a different method that more closely follows Eq. (5): When the propensities

of all reactions are computed at t in preparation for choosing the next reaction and time

step in the SSA, the propensities for the delayed reactions are computed using the system’s

past state at t − τ . The reaction constants AΩ, B, and C in Eq. (5) have units of inverse

time. Therefore, the unit system in which these constants are expressed as parameters of

the simulation sets the time scale. The delay time τ is expressed in the same units.

Typical trajectories that correspond to Eq. (5) either decay and fluctuate around a fixed

point, or show oscillatory behavior depending on the values of B, C and the delay time τ .

We show two such sample trajectories in Fig. 1. At fixed B, and for low enough values

of C, an initial state decays to a stationary state characterized by a unimodal distribution

centered at n ≈ A/(B+C). Alternatively, sustained oscillations are observed which saturate

to finite amplitude (n ≈ A/B), with a period of oscillation approximately equal to τ . It is

important to note that saturation in the oscillation amplitude in this model is due to the

constraint n ≥ 0.

A. Bifurcation analysis

The bifurcation between a steady state of n and sustained oscillation can be studied

analytically in the limit of large Ω. Expand n(t) = Ωx(t) + Ω1/2ξ(t) in Eq. (5) [28]. At

leading order, O(Ω1/2), x(t) obeys the deterministic equation,

x(t) = A−Bx(t)− Cx(t− τ). (6)

Equation (6) has a single fixed point x∗ = A/(B + C) or n∗ = AΩB+C

. Its stability can be

determined by expanding x(t) = x∗ + δe(µ+iω)t. Direct substitution yields,

µ = −B − Ce−µτ cos(ωτ) ω = Ce−µτ sin(ωτ). (7)

6

0 50 100 150 200 250 300

t

0

50

100

150

200

250

300

n(t)

0 50 100 150 200 250 300

t

0

50

100

150

200

250

300

350

n(t)

FIG. 1. Sample trajectories n(t) with C = 3 (below threshold, left) and C = 5 (above threshold,

right). Other parameters values were A = 1000, B = 4, τ = 10. The initial condition was n = 0

for t ∈ [−τ, 0].

The fixed point becomes unstable via a Hopf bifurcation when µ = 0, which defines the

critical line −B/C = cos(τ√C2 −B2

). The Hopf frequency is given by ω2 = C2 − B2.

Above the bifurcation, solutions of Eq. (6) grow exponentially without limit. This divergence

(and the associated negative and unphysical values of x that result) has been used to argue

against feedback loops such as the one in Eq. (4) as being the origin of oscillation in biological

systems (see, e.g. [35, 36]). It appears that this difficulty is confined to the approximation

in Eq. (6) (large n), but not to the original model (Eq. (4) or (5)). In the latter, oscillations

above threshold saturate, although the saturation is due to the constraint n ≥ 0, a condition

that has not been imposed on Eq. (6).

At the next order (O(1)), the following equation results for P (n, t) written as Π(ξ, t) (at

constant n, or on the line 0 = Ωdx+ Ω1/2dξ [37]),

∂Π(ξ, t)

∂t= − ∂

∂ξ

[(−Bξ − C

∫dξτξτΠ2(ξτ , t− τ ; ξ, t)

)Π(ξ, t)

]+

1

2(A+Bx(t) + Cx(t− τ))

∂2Π(ξ, t)

∂ξ2. (8)

Of course, this equation is only well defined in the non oscillatory range in which fluctuations

around x∗ are small. In this case, and at long times, one can approximate x(t) = x(t− τ) ≈

x∗, and the following simpler equation results

∂Π(ξ, t)

∂t= − ∂

∂ξ[(−Bξ − C〈ξτ , t− τ |ξ, t〉) Π(ξ, t)] + A

∂2Π(ξ, t)

∂ξ2. (9)

7

This equation has a stationary solution,

Π(ξ, t) ∝ exp

(−Bξ

2

2A− C

A

∫ ξ

dξ′〈ξτ , t− τ |ξ′, t〉). (10)

As shown in Ref. 28, Eq. (9) can be written as a Langevin equation with delay,

ξ = −Bξ(t)− Cξ(t− τ) + ζ(t) (11)

where ζ(t) is a random, Gaussian source of zero average and 〈ζ(t)ζ(t′)〉 = 2Aδ(t− t′).

The conditional average in Eq. (9) can be calculated near but below threshold of Eq.

(11). We expect that near threshold ξ(t) can be decomposed into an oscillatory part with an

amplitude that is slowly varying in time. In fact, we assume the amplitude to be a constant

here ξ(t) = a cos(ωt) + b sin(ωt). Then, (ξ−τ = ξ(t+ τ)),

〈ξ−τ |ξ〉 = 〈a cosω(t+ τ) + b sinω(t+ τ)|ξ〉

= 〈cos(ωτ)ξ +sin(ωτ)

ωξ|ξ〉 (12)

using the assumption for ξ(t) and trigonometric identities. Direct substitution of Eq. (11)

leads to

〈ξ−τ |ξ〉 = cos(ωτ)ξ +sin(ωτ)

ω(−Bξ − C〈ξτ |ξ〉) . (13)

If the process is stationary, then 〈ξ−τ |ξ〉 = 〈ξτ |ξ〉. This leads to,

〈ξτ |ξ〉 =

[ω cos(ωτ)−B sin(ωτ)

ω + C sin(ωτ)

]ξ. (14)

We then find for the stationary distribution function,

Π(ξ, t) ∝ exp

[−Bξ

2

2A−(Cξ2

2A

)ω cos(ωτ)−B sin(ωτ)

ω + C sin(ωτ)

]. (15)

This probability is defined as long as the coefficient of the exponential is positive. It becomes

zero when B+C cos(ωτ) = 0 which is identical to the macroscopic condition for instability,

Eq. (7).

The power spectrum associated with Eq. (11) below threshold has been given in Ref.

28. It describes exponentially decaying correlations of ξ(t) in time, with a correlation time

that is inversely proportional to B + C cos(ωτ). Hence the correlation time diverges at the

instability point, as expected (this also justifies the sinusoidal approximation for ξ(t) in the

estimation of 〈ξτ |ξ〉).

8

B. Bifurcation at finite Ω. Numerical study.

In order to make this simple problem analytically tractable when Ω is not large, a

mean field or independence assumption has been introduced [13]: P2(n, t;m, t − τ) ≈

P (n, t)P (m, t − τ). The reason given is that the time scale of cellular fluctuations is much

shorter than characteristic delay times in biological processes, and hence a similar decoupling

should hold in general. Introducing such a decoupling into Eq. (5) leads to

∂P (n, t)

∂t= AΩ(E−1 − 1)P (n, t) +B(E − 1)P (n, t) + C〈n(t− τ)〉(E − 1)H(n)P (n, t) (16)

so that the only dependence on delayed variables for any given trajectory is through the

delayed average. This approximation leads to an equation for 〈n(t)〉 for any Ω which is

identical to the macroscopic limit of Eq. (6). Therefore, a mean field approximation leads

to an instability point of the stationary solution for the first moment which is the same as

that given by both macroscopic and Langevin limits.

The mean field decoupling introduced is suspect as delayed reactions necessarily introduce

correlations at least as long as the delay time itself. In order to further investigate this issue,

we use a full numerical solution. We first validate our modified algorithm that incorporates

delayed reactions by calculating the stationary probability distribution function below onset.

Figure 2 shows our numerical results and compares them with the Langevin limit of Eq. (15).

Each distribution was obtained by evolving 300 SSA trajectories independently for 5 units

of simulation time, then computing histograms of the states of those trajectories at the end

of that period. The initial condition for each trajectory was a constant in [−τ, 0] randomly

sampled from the uniform distribution in [0, 180] in order to speed convergence. This process

was repeated 40 independent times using the same parameter set. The distributions shown

in Fig. 2 are averages over the 40 resulting distributions; the error bars on each bin are the

standard errors of those averages. The distribution function of fluctuations is a Gaussian far

from the bifurcation point, and agrees with the prediction of the Langevin model. As the

bifurcation point is approached, however, the distribution broadens and is not well described

by the linear model, Eq. (11).

In order to determine the bifurcation line of the stochastic model to oscillation we have

chosen to directly estimate the stationary joint probability P2(n,m) by calculating the his-

togram of occurrences of n(t) = n, n(t− τ) = m in simulated trajectories for each pair n,m.

Transient effects are eliminated by removing early points from the simulation as needed.

9

− 60 − 40 − 20 0 20 40 60

n-n *

0 .00

0 .01

0 .02

0 .03

0 .04

0 .05

ΠAna lyt ica l

SSA

− 100 − 50 0 50 100

n-n *

0 .000

0 .005

0 .010

0 .015

0 .020

Π

Ana lyt ica l

SSA

FIG. 2. Stationary probability distribution functions Π(ξ) for the delayed degradation model with

AΩ = 800, B = 4.5, and C = 4.75 (n∗ ≈ 86.5). Left: τ = 0.1, and far away from the bifurcation

(Cc ≈ 18.7). Right: τ = 1, closer to the bifurcation (Cc ≈ 5.20). The solid line in each plot is the

prediction of Eq. (15) using ω =√C2 −B2. The histogram bins are uniformly distributed over

the range n ∈ [0, 180] with a width of 3 (left plot) and 6 (right plot).

Figures 3–5 show our results for a typical set of model parameters; AΩ, B, and τ are held

constant while C is varied through the bifurcation. Figure 6 shows the distribution near the

bifurcation for a larger value of Ω. Below the bifurcation threshold Fig. 3 shows only one

peak centered at n ' m. This single peak corresponds (for large enough Ω) to the fixed

point n∗ = AΩB+C

, with stochastic fluctuations manifested by the width of the distribution.

As C is increased, the distribution broadens (Fig. 4) and eventually becomes bimodal (Fig.

5) indicating oscillation between the two states depicted in Fig. 1. We have chosen to de-

fine the bifurcation threshold as the set of parameters at which the distribution goes from

unimodal to bimodal.

Our results for the bifurcation line Cc = Cc(B) are shown in Fig. 7. The figure compares

the numerically determined threshold for Ω = 1, 10 (A = 100 so that n ∼ 10, 100 respec-

tively) and the macroscopic result. The bifurcation line shifts to a larger value of the delay

amplitude C for small Ω, but -as expected- agrees with the macroscopic limit for large Ω.

Therefore, for small Ω there is a fluctuation induced correction to the bifurcation line, unlike

both the limit of Ω large or the prediction from the mean field approximation. For small Ω, a

feedback or larger amplitude is needed to destabilize the fixed point in the stochastic system

10

0 5 10 15 20 25 30 35 40

n

0

5

10

15

20

25

30

35

40

m

0 .0

0 .8

1 .6

2 .4

3 .2

4 .0

4 .8

5 .6

6 .4

7 .2

1 e − 3

FIG. 3. Stationary joint probability matrix P2(n(t) = n, n(t − τ) = m) and conditional average

〈m, t − τ |n, t〉 (green line) with parameters AΩ = 100, B = 4.5, C = 1, and τ = 1. The magenta

line is the average value of m over the entire trajectory. The distribution is estimated from a single

trajectory run for 811 time units with the first 10 time units discarded.

relative to the deterministic approximation. In this particular model, noise acts to stabilize

the fixed point at the expense of the feedback, the latter being the cause for oscillation.

Figures 3–5 (overlaid) also show where applicable our numerical estimates of 〈m | n〉,

the prediction in the Langevin limit near threshold, Eq. (14), and the mean field prediction

〈m, t − τ |n, t〉 = 〈m〉. Away from threshold and in the unimodal range, the conditional

average is approximately independent of n, as suggested by the mean field approximation

(Fig. 3). As threshold is approached (and for sufficiently large Ω, Fig. 6), we find good

agreement with the Langevin result instead, except for small and large values of n (when the

oscillation saturates). The agreement deteriorates as Ω becomes smaller. In either case, the

conditional average is quite different from the mean field prediction. Finally, deep within the

oscillatory regime, the mean field approximation becomes accurate for large n. In summary,

not only does the conditional average deviate strongly from mean field behavior near the

bifurcation, but our results also suggest that the breakdown of the Langevin approximation

is related to the saturation in the amplitude of oscillation, an effect that becomes more

11

0 5 10 15 20 25 30 35 40

n

0

5

10

15

20

25

30

35

40

m

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0

3 .5

4 .0

4 .51 e − 3

FIG. 4. Stationary joint probability matrix and conditional average (green line) with parameters

AΩ = 100, B = 4.5, C = 5.25, and τ = 1, close to the stochastic bifurcation; the deterministic

bifurcation occurs at Cc = 5.2047. The blue line is the conditional average predicted by Eq. (14).

0 5 10 15 20 25 30 35 40

n

0

5

10

15

20

25

30

35

40

m

0 .00

0 .15

0 .30

0 .45

0 .60

0 .75

0 .90

1 .05

1 .20

1 .35

1e − 2

FIG. 5. Stationary joint probability matrix and conditional average (green line) with parameters

AΩ = 100, B = 4.5, C = 14, and τ = 1. The light blue line is the average value of m over only

those portions of the trajectory where m ≤ n.

12

0 20 40 60 80 100 120

n

0

20

40

60

80

100

120

m

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0

3 .5

4 .01 e − 4

FIG. 6. Stationary joint probability matrix and conditional average (green line) with parameters

AΩ = 400, B = 4.5, C = 5.25, and τ = 1. The blue line is the conditional average predicted by

Eq. (14).

pronounced as Ω is reduced. Concurrently, the bifurcation point as defined by P2 also

deviates from the Langevin threshold. We conclude that the physical constraint n ≥ 0

in the original model, not captured in the two asymptotic limits considered, modifies the

statistical properties of the process, and in particular the bifurcation threshold. This effect

disappears as Ω becomes large, and the constraint n ≥ 0 less relevant to the individual

trajectories.

III. DIMER NEGATIVE AUTOREGULATION SYSTEM

The auto repressor model studied above can be studied in cosiderable detail analytically,

but the dependence on Ω is relatively trivial. We have therefore extended our analysis to

a more complex case in which the system size dependence cannot be simply eliminated by

rescaling the model parameters. We consider, following Ref. 13, a model of a protein which

dimerizes to down regulate monomer production. Production is switched on or off by the

state of a single operator site. If the site is unoccupied, represented by D0 = 1, D1 = 0,

then delayed production occurs at rate A. If the site is occupied by a dimer X2, then D0 =

13

0 5 10

B

0

5

10Cc

Ω = 1

Ω = 10

FIG. 7. Critical values Cc at which the two point probability P2 becomes bimodal in the delayed

degradation model for various values of B. The solid line shown is the macroscopic bifurcation line

defined by Eq. (7). We show the case A = 100 and τ = 10.

0, D1 = 1 and no production occurs. The time required for transcription and translation is

summarized by a single delay in the production reaction, with a delay time τ . The reactions

defining the model are

D0A⇒ D0 +X,X

B−→ ∅

D0 +X2k1−→ D1,D1

k−1−−→ D0 +X2 (17)

X +Xk2−→ X2,X2

k−2−−→ X +X

This system possesses a single fixed point, and we study the Hopf bifurcation by changing

the delayed dimerization rate A.

We define the bifurcation threshold from the joint probability matrix P2(n,m) as the line

in parameter space separating regions of unimodal versus bimodal distributions. The bifur-

cation becomes quite distinct and the bifurcation line can be determined quite accurately.

Figure 8 shows our numerical results just below and above the stochastic bifurcation.

The bifurcation diagram for the macroscopic limit of Ω→∞ has to be computed numer-

ically. Given the fixed point (x∗, x∗2, d∗0) of Eq. (17), linearization of the governing equation

14

0 10 20 30 40 50n

0

10

20

30

40

50m

0.0000

0.0015

0.0030

0.0045

0.0060

0.0075

0.0090

0.0105

0.0120

0.0135

0 10 20 30 40 50n

0

10

20

30

40

50

m

0.0000

0.0008

0.0016

0.0024

0.0032

0.0040

0.0048

0.0056

0.0064

FIG. 8. Joint probability distribution P2(n,m) for the dimer feedback system with A = 70 (uni-

modal distribution) and A = 80 (bimodal distribution). Other parameters of the model were

k1 = 100, k−1 = 1000, k2 = 200, k−2 = 1000, B = 4,Ω = 1.

yields the following equation∣∣∣∣∣∣∣∣∣

λ 0 0

0 λ 0

Ae−λτ 0 λ

−−B − 2k2x

∗ 2k2x∗ 0

k−2 −k1d∗0 − k−2 k1d

∗0

0 −k−1 − k1x∗2 −k−1 − k1x

∗2

∣∣∣∣∣∣∣∣∣ = 0 (18)

for the critical eigenvalue λ. τ is the delay time in production. For any given set of model

parameters, we find the fixed point numerically, and find the bifurcation line Ac(B) by

requiring that the real part of the eigenvalue λ be zero. This is done numerically by iteration

of Newton’s method. Our results are shown in Fig. 9 (solid line).

Our numerical results for the bifurcation line are summarized in Fig. 9 for two values of

Ω. As was the case for the model of delayed protein degradation, the stochastic bifurcation

15

0 50 100 150 200

τΒ

0

500

1000

1500

2000

2500

3000

τA

c

Ω = 1

Ω = 10

FIG. 9. Critical value Ac for the Hopf bifurcation of the dimer system. Circles (blue online) and

crosses (red) show points obtained numerically for system sizes Ω = 1 and Ω = 10, respectively. The

solid line is the macroscopic bifurcation line. Other parameters were k1 = 100, k−1 = 1000, k2 =

200, k−2 = 1000, τ = 20.

line for the larger system size Ω agrees better with the predictions of the macroscopic limit,

but differs significantly from it when Ω is small. Again, random fluctuations act to stabilize

the fixed point of the system.

To conclude, we have shown results from both analytical approximations and numerical

simulations of two model systems that involve delayed feedback and stochasticity. In both

cases, delayed feedback leads to a Hopf bifurcation from a unimodal distribution (a single

steady state) to a bimodal distribution corresponding to oscillation. The bifurcation was

located by identifying the splitting of the unimodal peak in the joint probability distribu-

tion P2(n,m). As expected, the macroscopic and Langevin approximations were in good

agreement with the numerical simulations when system size Ω→∞ (and when the former

have bounded solutions). However, significant differences were found in the bifurcation line

between the stochastic case and the approximations when Ω is not large. We have computed

delayed conditional averages, and have shown that they are non trivial near the bifurcation.

A multiple scale argument allows an accurate estimate in the weak noise limit, but discrep-

16

ancies remain for Ω small. They can be attributed to the constraint n ≥ 0 in the stochastic

case which is not important in the limit Ω→∞. In both model systems analyzed, feedback

of larger amplitude was required to sustain oscillations when Ω was small, i.e., stochastic

effects stabilize their stationary fixed points.

ACKNOWLEDGMENTS

We are indebted to David Jasnow and Dan Zuckerman for stimulating discussions. We

thank the Minnesota Supercomputing Institute for support.

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