Stochastic bifurcation in discrete models of biological oscillators. Abstract

Transcription

1 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 Viñals 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

2 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 trascription 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 repressor 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 reasonable 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 oscillators [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 focused 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

3 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 fluctuations 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 reexamine 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 = {n i }. If delayed reactions with a single delay time τ are allowed, reaction kinetics is described by n i Ω = j g ij A j ( n(t) Ω n(t τ), ) (1) Ω where g ij specifies the stoichiometric coefficient of species i in reaction j, and A j is the rate law for reaction j. Ω is the volume of the system. For generalized mass action rate laws, A j 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 ẋ(t) = ax(t) + bx(t τ), a perturbation expansion in the delay τ is valid only up 3

4 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 g ij i 1)A j ( n Ω, m Ω )P 2(n, t; m, t τ) (2) Here E i denotes the raising operator for species i, E i P (n, t) = P (n+ê i, t), and P 2 (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 P 2. 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 ṅ i (t) = j g ij Ω A j ( n(t) Ω n(t τ), ) (3) Ω If the A j are affine in their arguments, then A j ( n(t), n(t τ) Ω Ω ) = A j( n(t) Ω, n(t τ) ) and Eq. Ω (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 P 2 (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

5 noise limits. A key observation is that the boundary condition that captures the positivity requirement of the discrete model (n i 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 X B 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(e 1)H(n)P 2 (n, t; m, t τ) (5) m=0 where the last term in the right hand side can be rewritten as, m(e 1)H(n)P 2 (n, t; m, t τ) = (E 1) [ m, t τ n, t P (n, t)] m=0 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

6 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 P 2 (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) + Ω 1/2 ξ(t) in Eq. (5) [28]. At leading order, O(Ω 1/2 ), x(t) obeys the deterministic equation, ẋ(t) = A Bx(t) Cx(t τ). (6) Equation (6) has a single fixed point x = A/(B + C) or n = determined by expanding x(t) = x + δe (µ+iω)t. Direct substitution yields, AΩ. Its stability can be B+C µ = B Ce µτ cos(ωτ) ω = Ce µτ sin(ωτ). (7) 6

7 n(t) n(t) t 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 C 2 B 2). The Hopf frequency is given by ω 2 = C 2 B 2. 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 + Ω 1/2 dξ [37]), Π(ξ, t) t = ξ [( Bξ C ) ] dξ τ ξ τ Π 2 (ξ τ, t τ; ξ, t) Π(ξ, t) (A + Bx(t) + Cx(t τ)) 2 Π(ξ, t) ξ 2. (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 ξ τ, t τ ξ, t ) Π(ξ, t)] + A 2 Π(ξ, t) ξ 2. (9) 7

8 This equation has a stationary solution, Π(ξ, t) exp ( Bξ2 2A C ξ dξ ξ τ, t τ ξ, t ). (10) A 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 cos ω(t + τ) + b sin ω(t + τ) ξ = cos(ωτ)ξ + sin(ωτ) ξ ξ (12) ω 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 ξ τ ξ ). (13) If the process is stationary, then ξ τ ξ = ξ τ ξ. This leads to, [ ] ω cos(ωτ) B sin(ωτ) ξ τ ξ = ξ. (14) ω + C sin(ωτ) We then find for the stationary distribution function, ( ) ] Π(ξ, t) exp [ Bξ2 Cξ 2 ω cos(ωτ) B sin(ωτ) 2A. (15) 2A ω + C sin(ωτ) 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 ξ τ ξ ). 8

9 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]: P 2 (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 P 2 (n, m) by calculating the histogram 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

10 Π Analytical SSA Π Analytical SSA n-n * n-n * 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 (C c 18.7). Right: τ = 1, closer to the bifurcation (C c 5.20). The solid line in each plot is the prediction of Eq. (15) using ω = C 2 B 2. 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 = AΩ, with stochastic fluctuations manifested by the width of the distribution. B+C 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 define the bifurcation threshold as the set of parameters at which the distribution goes from unimodal to bimodal. Our results for the bifurcation line C c = C c (B) are shown in Fig. 7. The figure compares the numerically determined threshold for Ω = 1, 10 (A = 100 so that n 10, 100 respectively) 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

11 m n 1e FIG. 3. Stationary joint probability matrix P 2 (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

12 m n 1e 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 C c = The blue line is the conditional average predicted by Eq. (14). m n 1e 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

13 m e n 0.0 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 P 2 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 D 0 = 1, D 1 = 0, then delayed production occurs at rate A. If the site is occupied by a dimer X 2, then D 0 = 13

14 10 C c 5 Ω = 1 Ω = B FIG. 7. Critical values C c at which the two point probability P 2 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, D 1 = 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 D 0 A D 0 + X,X B k D 0 + X 1 k 1 2 D1,D 1 D0 + X 2 (17) X + X k 2 k 2 X 2,X 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 P 2 (n, m) as the line in parameter space separating regions of unimodal versus bimodal distributions. The bifurcation 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 numerically. Given the fixed point (x, x 2, d 0) of Eq. (17), linearization of the governing equation 14

15 m n m n FIG. 8. Joint probability distribution P 2 (n, m) for the dimer feedback system with A = 70 (unimodal distribution) and A = 80 (bimodal distribution). Other parameters of the model were k 1 = 100, k 1 = 1000, k 2 = 200, k 2 = 1000, B = 4, Ω = 1. yields the following equation λ 0 0 B 2k 2 x 2k 2 x 0 0 λ 0 k 2 k 1 d 0 k 2 k 1 d 0 = 0 (18) Ae λτ 0 λ 0 k 1 k 1 x 2 k 1 k 1 x 2 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 A c (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

16 τa c Ω = 1 Ω = τβ FIG. 9. Critical value A c 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 k 1 = 100, k 1 = 1000, k 2 = 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 distribution P 2 (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

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