Abstract
Experiments show that proteins are translated in sharp bursts; similar bursty phenomena have been observed for protein import into compartments. Here we investigate the effect of burstiness in protein expression and import on the stochastic properties of downstream pathways. We consider two identical pathways with equal mean input rates, except in one pathway proteins are input one at a time and in the other proteins are input in bursts. Deterministically the dynamics of these two pathways are indistinguishable. However the stochastic behavior falls in three categories: (i) both pathways display or do not display noiseinduced oscillations; (ii) the nonbursty input pathway displays noiseinduced oscillations whereas the bursty one does not; (iii) the reverse of (ii). We derive necessary conditions for these three cases to classify systems involving autocatalysis, trimerization and genetic feedback loops. Our results suggest that single cell rhythms can be controlled by regulation of burstiness in protein production.
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Introduction
Noise manifests itself and influences the dynamics of cellular systems at various spatial scales^{1,2,3,4,5}. Of particular interest and acknowledged importance are concentration fluctuations stemming from the random timing of unimolecular and bimolecular chemical events. The ratio of the standard deviation to the mean of these fluctuations roughly scales as the inverse square root of the average number of molecules^{6}, hence its importance to the dynamics of intracellular pathways since many chemical species are present in low numbers per cell^{7}.
Given a particular biochemical pathway of interest, noise can be further categorized as that coming from sources external to the pathway and that originating from the individual reactions constituting the pathway. A ubiquitous source of external noise is the mechanism by which molecules are input or injected into a biochemical pathway. The classical model for this is a Poisson process in which a single molecule is injected at random points in time. However, numerous experimental studies over the past decade have shown that such a description is often inaccurate^{8,9,10,11,12}.
Injection events have at least two physical interpretations for models of intracellular dynamics; injection can describe protein expression when modelling a biochemical pathway in the cytosol, whereas for pathways in membranebound subcellular compartments injection events can describe transport of molecules into the compartment by diffusive or active transport. A number of studies have confirmed that protein expression occurs in sharp and random bursts^{8,9}. The bursts are found to be exponentially distributed and the expression events are temporally uncorrelated. The origin of these bursts can be explained by a simple mechanism. For bacteria and yeast, the lifetime of mRNA is typically short compared to that of proteins. In its short lifetime, each mRNA is translated into a number of protein molecules leading to random uncorrelated bursty events of protein production^{10}. Although such protein expression is the best studied example of burstiness in protein production, it is not the only one. It has recently been found that protein translocation to the nucleus in response to an extracellular stimulus in budding yeast also occurs in sharp bursts^{11}; indeed these bursts may be even more influential than those in protein expression since the mean size of the translocation bursts are estimated to be hundreds of molecules whereas those stemming from protein expressions are of the order of few tens or less^{9,12}.
It is interesting to ponder what effects burstiness in protein production has on the steadystate properties and dynamics of the downstream biochemical pathway into which it feeds. Intuitively, a bursty input mechanism introduces a larger degree of noise to the downstream pathway than a nonbursty one. Indeed, this increase in noise has been quantified in very simple scenarios where the downstream pathway involves protein decay via a firstorder process; for a bursty production mechanism, it was found that the Fano factor (variance of number fluctuations divided by the mean of molecule numbers) is equal to 1 plus the mean burst size, whereas for a nonbursty mechanism the Fano factor is 1^{12,13}. It is expected that this noise amplification occurs for all species' concentrations in more complex downstream pathways; from this point of view, bursting appears to be deleterious to the precise orchestration of cellular function. Consequently one might expect the cell to have developed downstream mechanisms to reduce or control such unwanted noise.
In this article we challenge this notion by demonstrating the nonintuitive effects of bursty inputs on noiseinduced concentration oscillations. We compare the stochastic properties of two identical biochemical pathways, in one of which the protein is produced via a nonbursty input mechanism and in the other via a bursty input mechanism where the number of molecules per burst are distributed according to a general probability distribution. The mean rates of protein production are chosen to be the same in the two pathways and hence, according to the deterministic rate equation formalism, the two systems are characterized by the same steadystate concentrations. However, we show that there exist pronounced differences in the noiseinduced oscillations generated by bursty and nonbursty systems and that the crucial nondimensional parameter distinguishing the two is the sum of the mean and the Fanofactor of the burst size probability distribution. By deriving necessary conditions for noiseinduced oscillations in the two systems we demonstrate a method of classifying simple biochemical circuits by their response to input bursting.
Results
A general framework for assessing the effects of bursty protein production
In this section we introduce the twosystem setup which we will use to study the effects of bursting on the fluctuations in downstream pathways.
Consider a two species system in which both species are injected into a pathway and subsequently interact via a number R of downstream reactions. The nonbursty version of this system can be schematically represented as:
where X_{i} denotes species i, s_{ij} and r_{ij} (i = 1, 2) are the integer stoichiometric coefficients and h_{j} and k_{j} are the associated rate constants of the j^{th} input and j^{th} processing (downstream) reaction respectively.
A bursty input version of this system can also be envisaged as follows:
where q_{i}(m) is the probability that the input burst size is m for species X_{i} and is a proportionality constant such that is an input rate constant. The integer constants M_{1} and M_{2} are the maximum burst sizes for species X_{1} and X_{2} respectively.
In the limit of large molecule numbers, the time evolution of the mean concentrations for the two systems is given by the conventional rate equations:
where the first terms describe the input reactions and the second terms describe the processing reactions. The vector is the concentration vector for bursty (b) and nonbursty, i.e., singlemolecule input (s) systems. The processing rates are given by where S_{ij} = r_{ij} − s_{ij} are the elements of the stoichiometric matrix. The factors μ_{1} and μ_{2} are the mean input burst size for species X_{1} and X_{2} respectively, i.e., . Note that the input rates h_{i} and may be constants, e.g., when modelling diffusive transport, or functions of the concentrations, e.g., when modeling repression or activation of gene transcription.
If the two systems have the same initial conditions and if the mean number of molecules injected per unit time is the same, i.e., if , , then they are indistinguishable from measurements of their mean concentrations, i.e., for all times. Given this condition, it can be shown (see Methods) using the linearnoise approximation (LNA)^{14} that the timeevolution equations for the probability distribution of concentration fluctuations about the mean concentration solution of the above rate equations are given by:
where is the noise about the mean concentration of species X_{i}, is the second moment of the distribution of bursts, is the Jacobian matrix (describing linear stability) of the two rate equations above and . Note that since the bursty and nonbursty input systems have the same vector of mean concentrations and the same Jacobian (under the condition of equal mean input rates), we have denoted these as and J respectively, for both systems.
An inspection of the FokkerPlanck equations, Eqs. (5) and (6), shows that they have the same drift terms but different diffusion terms. This implies that while the bursty and nonbursty systems have the same mean concentrations, their fluctuation properties are different. The crucial set of nondimensional parameters determining the differences in the fluctuations between the bursty and nonbursty input systems are:
where is the variance of the probability distribution of the bursts in species X_{i}. When η_{i} = 1 then the differences between the FokkerPlanck equations for the bursty and nonbursty input systems vanish. As expected, this occurs in the limit that the variance approaches zero and the mean burst size is one. As discussed in the Introduction, experiments show that the mean burst size is larger than one and hence we shall exclusively consider η_{i} > 1. The implication of equation (7) is that the larger is η_{1} − 1, the larger are the expected differences in the fluctuation properties of the downstream pathways in the two systems. For example for a Poissonian distribution of bursts, it is found η_{i} − 1 = μ_{i} whereas for a geometric distribution of bursts we have η_{i} − 1 = 2μ_{i} and hence we expect the burstinessinduced effects to be more prominent for systems with the latter burst input.
We finish this section by noting that we now have a convenient analytical setup with which to understand the effects of burstiness on the fluctuations of a downstream pathway. In the next sections we use the FokkerPlanck equations to understand how the fluctuations from bursty input change the downstream pathway's ability to generate noiseinduced oscillations.
Necessary conditions for bursty input alteration of the oscillatory properties of the downstream pathway
In the setup described in the previous section, the bursty and nonbursty input systems are indistinguishable from a rate equation perspective and hence it follows that the deterministic dynamics of the two systems, including their ability to generate deterministic oscillations (limit cycles) are one and the same. However it is well appreciated that noise can induce oscillations in systems whose rate equations predict none. Given that the noise in the bursty and nonbursty input systems is different, it is plausible that the noiseinduced oscillations displayed by both systems can also be different. In what follows we use the FokkerPlanck equations of the last section to probe this question.
We consider a general two variable FokkerPlanck equation with linear drift and diffusion coefficients of the form:
One can use this equation to derive an equation for the power spectrum of the fluctuations in the number of molecules of species X_{i} () and this is found to be^{15}:
where , α_{2} is the same as α_{1} but with 1 and 2 interchanged, β_{i} = D_{ii}, p = [Det(J)]^{2} and q = [Tr(J)]^{2} − 2Det(J). Here Tr and Det refer to the matrix trace and determinant respectively. The power spectrum of the fluctuations in a given species is the Fourier transform of the autocorrelation function of the fluctuations of that species. Hence for a system in steadystate conditions, a peak in the power spectrum of a species at some frequency ω indicates a noiseinduced oscillation of the same frequency in the concentration fluctuations of that species^{16}. The sharpness of the peak indicates the quality of the oscillation; see Ref. 15 for a detailed discussion of quality measures for noisy oscillators.
By comparing Eqs. (5) and (6) with the general form equation (8), we can deduce that the power spectrum of the fluctuations in the bursty and nonbursty input systems are given by equation (9) with and respectively. These two spectra we shall refer to as and respectively.
By differentiating and with respect to ω, we can find the sufficient conditions for the power spectra to have a maximum, i.e., for the two systems to exhibit noiseinduced oscillations. If q < 0, it can be shown that both and display a peak in their power spectrum; hence in this case burstiness does not lead to any qualitative change in the oscillatory properties of the downstream pathway. However for q > 0 the situation is more interesting. The positive q condition describes downstream pathways which, when parameterized, are far from a Hopf bifurcation^{17}; the equilibrium is described by a node (which satisfies Tr[J]^{2} > 4Det(J)) or by a focus close to the nodefocus borderline in phase space (which satisfies 2Det(J) < Tr[J]^{2} ≤ 4Det(J)). In this case the conditions for noiseinduced oscillations in the concentration of species X_{1} in the bursty and nonbursty input systems are different and given by
respectively. The parameter θ_{1} is a function of the Jacobian elements only and is given by:
The conditions for noiseinduced oscillations in species X_{2} are as above but with 1 and 2 interchanged. Note that although not explicitly shown, the elements of the D and J matrices in Eqs. (10)–(12) are functions of the mean concentration vector .
Hence for q > 0 we can identify three distinct cases: (i) , (ii) and (iii) . These are illustrated in Fig. 1. For case (i) either both systems display no oscillations or they both show noiseinduced oscillations. For case (ii), there is the possibility of a special regime () in which the nonbursty input system displays noiseinduced oscillations but the bursty input system does not. For case (iii), there is the possibility of a special regime () in which the nonbursty input system displays no oscillations but the bursty input system exhibits noiseinduced oscillations. Hence burstiness has no effect in case (i), may cause destruction of noiseinduced oscillations in case (ii) and may promote noiseinduced oscillations in case (iii).
Note that is only a necessary condition for the destruction of noiseinduced oscillations by burstiness in the input reactions; sufficient conditions ensue when we further have () which may not be always possible to satisfy. Similarly should be construed as a necessary condition for the creation of noiseinduced oscillations by burstiness in the input reactions.
By inspection of Eqs. (10)–(11), we can make further specific statements regarding the importance of burstiness in the input reactions to the oscillatory dynamics of the two species pathway:

If the species X_{2} does not activate or inhibit X_{1}, i.e., J_{12} = 0, then and hence burstiness in the inputs of X_{1} or X_{2} do not cause a qualitative change in the oscillatory dynamics of X_{1}. Thus it is clear that bursting on its own is insufficient to affect oscillatory dynamics, rather an interplay of bursting with a downstream pathway featuring the interaction of two or more species is required.

For all other (i.e., J_{12} ≠ 0) pathways, an increase in the input burstiness of species X_{2} (for example by increasing the variance of the burst fluctuations at constant burst size mean) always increases the term . Thus, since it is possible to induce the condition but not the condition , bursting in species X_{2} may destroy noiseinduced oscillations in species X_{1} but can never promote noiseinduced oscillations in species X_{1}.

For pathways that obeys the condition , an increase in the input burstiness of species X_{1} decreases the term . Thus, since it is possible to induce the condition , bursting in species X_{1} for these pathways may promote noiseinduced oscillations in species X_{1}. An exemplary class of such pathways are those in which the reactions are stoichiometrically uncoupled () but kinetically coupled (J_{12} ≠ 0), i.e., in each reaction there is only a net change in the number of molecules of one species yet the kinetics of the two species are coupled (see the Applications section for examples)^{18}.
In the next section we investigate the effects of input bursting in exemplary biochemical circuits, in particular verifying our theoretical prediction that burstiness in the input reactions can both promote and destroy noiseinduced oscillations far from the Hopf bifurcation. We conclude here by highlighting a simple four point recipe, provided in the Methods section, which can be used to calculate the necessary conditions derived in this section for any two species biochemical circuit.
Applications
Modified brusselator
Here we consider a modified form of the Brusselator^{19}, which was introduced by Tyson and Kauffman in an early attempt to model dynamics within the process of mitosis. The nonbursty reaction scheme for this model is:
As previously explained, the bursty input version of this scheme having the same mean concentrations as the nonbursty version involve replacing the input reaction by the set of reactions , , , ......., , where M_{1} is some positive integer representing the maximum input burst size, q_{1}(m) is the probability of an input burst of size m in species X_{1} and is the mean burst size.
The quantities (for i = 1, 2) which determine the necessary conditions for promotion or destruction of noiseinduced oscillations by burstiness can be computed by following a four step recipe (see the Methods section). Here we simply state the results:
where and Λ_{2} = k_{2}/k_{3} are nondimensional parameters of the system. Thus we have and for η_{1} > 1, i.e., for all possible distributions of the burst size with mean burst size greater than 1. These are Case III and Case II in Fig. 1 respectively, implying necessary conditions for bursting in the input to promote noiseinduced oscillations in species X_{1} and for it to destroy noiseinduced oscillations in species X_{2}.
We investigated these predicted phenomena in further detail as follows. We chose the burst size distribution such that it was geometric with a mean burst size μ_{1} = 12 (and hence η_{1} = 25; see equation (7) and the discussion which follows it) and varied Λ_{1} and Λ_{2} over the range 10^{−3} to 10^{3}. The geometric distribution is the discrete analog of the exponential distribution which has been measured in experiments^{9} and has also been predicted from theory^{13}. For each parameter set we deduced the nature of the stable steadystate from linear stability analysis (focus, i.e., Tr[J] < 0, Det[J] > 0 and Tr[J]^{2} < 4Det(J) or node, i.e., Tr[J] < 0, Det[J] > 0 and Tr[J]^{2} > 4Det(J)^{17}) and also checked if there is a peak at some nonzero frequency in the LNA power spectrum as given by equation (9) (which implies noiseinduced oscillations). The results for both species X_{1} and X_{2} are shown in Fig. 2. The red regions in Figs. 2 (a) and (b) denote the regions in parameter space where there are noiseinduced oscillations in species X_{1} for nonbursty and bursty input systems respectively. Similarly the blue regions in Figs. 2 (c) and (d) denote the regions in parameter space where there are noiseinduced oscillations in species X_{2} for nonbursty and bursty input systems respectively. Notice that in accordance with the predictions based on the necessary conditions discussed in the previous paragraph, we find that the burstiness in the input reaction promotes noiseinduced oscillations in X_{1} (increased area of red region in Fig. 2 (b) compared to Fig. 2(a)) and destroys noiseinduced oscillations in X_{2} (decreased area of blue region in Fig. 2 (d) compared to Fig. 2 (c)). One also notices that the changes mainly occur in regions of parameter space characterized by a node and not by a focus (dotted region), which is consistent with the earlier prediction that burstiness has an important effect in systems far from the Hopf bifurcation.
In Fig. 3 (a) and (c) we show the power spectra calculated from the LNA and from stochastic simulations for two points in Λ_{1}–Λ_{2} space for which the LNA analysis of Fig. 2 predicted that burstiness in the input reaction should promote and destroy noiseinduced oscillations respectively. The simulations confirm the predicted phenomena by showing that the spectra of X_{1} and X_{2} exhibit the appearance and disappearance of a peak at a nonzero frequency respectively, when bursting in the input reaction is turned on. It is also shown that the phenomena are more pronounced for geometric burst size distributions rather than for Poisson ones of the same mean burst size; this is since given the same mean, the width of the former distribution is larger than that of the latter. In Fig. 3 (b) and (d) we show the quality factor of the noiseinduced oscillations as a function of the mean burst size μ_{1}. The quality factor is defined as , where is the frequency at which maximum power is obtained and Δω^{99%} is the difference of the frequencies at which the power takes 99% of its maximum value; this measure was introduced in^{15} and shown to be highly reflective of the rhythmicity visible in a time series of noiseinduced oscillations. For these farfrom Hopf oscillations the maximum possible value of Q^{99%} is ≈ 5 (Ref 15). Of particular interest is the saturation observed in Fig. 3 (b) which implies that there are limits to how much burstiness in the input reaction can improve the quality of noiseinduced oscillations. This also means that in the limit of large burst sizes the quality of noiseinduced oscillations is independent of the precise type of the burst size distribution.
Other simple circuits
Next we report the results of a detailed investigation of the effect of burstiness on the oscillatory properties of 8 biochemical pathways. The reaction schemes for the latter including their rate equations and the nondimensional parameters characterizing the steadystate behavior are shown in Table I. Note that for the gene circuits we have set some of the rate constants to 1; the model's behavior can then be described by at most three nondimensional parameters which considerably simplifies our analysis.
Next we used the four point calculational recipe in the Methods section to obtain the quantity for each of these 8 pathways. The sign of this quantity determines which of the three cases shown in Fig. 1 each pathway falls into and hence constitutes necessary conditions for promotion and destruction of noiseinduced oscillations in species X_{1} by bursting. The expressions for are shown in the second column of Table II. It is simple to determine the sign of this quantity since all constants a_{1} to a_{8} are positive, as are Λ_{1}, Λ_{2} and Λ_{3} and η_{1}, η_{2} > 1 (mean burst size is greater than 1). If the sign can take negative values then Case III is possible; if the sign can take positive values then Case II is possible. Which case can be displayed by each pathway is shown in columns 4 and 6 of Table II. Notice that 6 out of 8 pathways can display Case III behavior, i.e., bursting may induce oscillations; 6 out of 8 pathways can display Case II behavior, i.e., bursting may destroy oscillations; 4 out of 8 pathways can display both Case II and Case III behavior, i.e., bursting may promote or destroy oscillations.
As we have shown the simple necessary conditions are very easily determined in practice and give a quick indication of whether input burstiness could cause a qualitative change in oscillatory behaviour in a system. However these conditions are not sufficient by themselves to prove that the systems do actually display the burstinessinduced destruction or promotion of noiseinduced oscillations. As previously explained and shown in Fig. 1, one needs to further determine if θ_{1} falls in the correct range of values. This is considerably more involved analytically and hence we determine it numerically by an extensive parameter scan.
The parameter scan algorithm involved the following steps. We randomly picked 10^{5} sets of nondimensional parameters (the Λ_{i}'s in Table I are uniformly distributed in logspace over the range [10^{−3}, 10^{3}] and the burstiness parameters η_{i} are uniformly distributed integers in the range [1, 25]) for which the system has a steadystate. For each of the models, this chosen range of nondimensional kinetic parameters falls within the biologically relevant range for subcellular processes (see Methods). For each parameter set we calculated the quantities q, θ_{1}, and . If q < 0 then for this parameter set both bursty and nonbursty input systems display noiseinduced oscillations. If q > 0, then θ_{1}, and are used to obtain which case and which particular region of the case shown in Fig. 1 describes the system's behavior for the chosen parameter set. These classifications are recorded for each parameter set.
Information regarding whether the sufficient conditions were found to be satisfied or not is reported in columns 5 and 7 of Table II. A more detailed classification is shown in bar chart form in Fig. 4 for six of the eight pathways in Table I. Note that the two remaining pathways (One Gene Model B and Two Gene Model C) are similar in behaviour to Two Gene Model A and hence not shown in the latter figure. At least one of Case II or Case III behaviour was possible for each model. The necessary conditions for bursts destroying or promoting noiseinduced oscillations were also sufficient, with three exceptions: One Gene Model B, Two Gene Model A and Two Gene Model C. Interestingly, these three exceptions are unique among the models in that they are the only ones which cannot exhibit noiseinduced oscillations for any parameter choices for both bursty and nonbursty systems.
We noticed that the effect of burstiness on each system was strongly linked to two main features: (a) whether burstiness is possible in one or both species; and (b) the pathway's feedback motif, as described by the signs of the offdiagonal elements of the Jacobian matrix (column 3 in Table II). The three exceptional models (One Gene Model B, Two Gene Model A and Two Gene Model C) which never exhibited noiseinduced oscillations and for which the necessary conditions for bursts destroying or promoting noiseinduced oscillations did not also translate to sufficient conditions, all feature either mutual promotion or mutual inhibition between the two species.
Models with negative feedback, whereby one species promotes the other and that species inhibits the first (indicated by different signs on the offdiagonals of J) were sensitive to burstiness destroying or promoting noiseinduced oscillations. When burstiness is possible in both species (Autocatalysis and Two Gene Model B), necessary conditions for both BIO and BDO can be satisfied and BIO and BDO were indeed observed. Therefore, our results suggest that the combination of a negative feedback motif and burstiness in both species allows a wide range of burstinginduced oscillatory behaviour. When burstiness is only possible in one of the species, the matching of necessary and sufficient conditions is again observed, but here the asymmetry of the Jacobian is important; when J_{12} is positive (Brusselator) the necessary conditions indicate that burstiness tends to destroy noiseinduced oscillations in X_{1}, but when J_{12} is negative (Trimerization and One Gene Model A) the necessary conditions indicate that noise tends to promote noiseinduced oscillations in X_{1}.
Although the regions of parameter space for which bursts promote or destroy noiseinduced oscillations is quite small in some models, e.g., Two Gene Model B, we note that this region can be considerably enlarged by choosing a smaller range for the burstiness parameters (e.g. if η_{1} and η_{2} are fixed to 25 and 2 respectively rather than the range [1, 25] used in our parameter scan). The fact that a large proportion of the considered pathways display burstiness alteration of the noisy oscillatory dynamics suggests that such phenomena may be common in many biochemical systems.
Discussion
In this paper we have shown using the LNA that burstiness in the input reactions can have a considerable impact on the oscillatory properties of the downstream pathway. In particular we showed that for two identical pathways, one with bursty and one with nonbursty input, the two pathways may differ in their ability to produce noiseinduced oscillations. We derived necessary conditions for the burstiness to promote oscillations and for the burstiness to destroy oscillations and confirmed the existence of these phenomena using stochastic simulations. Our work is the first to investigate the effect of burstiness on the noisy oscillatory dynamics of biochemical pathways; previous work focused on deriving expressions for the steadystate distributions (or the moments) of protein concentrations in the presence of bursting^{10,20,21,22} and on elucidating the link between circuit architecture and the susceptibility to bursts in gene expressions^{23}.
We note that our analysis is based on the LNA which is a good approximation when describing pathways involving small levels of noise, i.e., pathways characterized by a sufficiently large number of molecules. This is not always the case since a number of species inside cells occur in small molecule numbers^{7}. Our theory can be extended to account for such cases by considering higherorder terms than the LNA in the system size expansion of the master equation^{24,25,26}. Preliminary investigations show that in nonbursty systems if the LNA predicts a peak in the power spectrum of fluctuations for systems far from the Hopf bifurcation then the spectrum calculated from stochastic simulations shows a peak even if the molecule numbers are very small (see Fig. 6 in^{15}); the quality of the oscillations may, however, be lower than that predicted by the LNA. Hence we expect the consideration of terms of higher order than the LNA to have little or no effect on the necessary conditions derived in this paper since these are specifically for the existence or nonexistence of a peak in the power spectrum.
Our results can also be interpreted in the context of single cell rhythms, as follows. Rate equation models are typically constrained to experimental measurements from an ensemble of cells, e.g.^{27}. They are only accurate models of singlecell pathway dynamics when the cells are dynamically noncoupled and each cell is characterized by negligible noise. Hence, experimentally observed oscillating concentrations correspond to rate equation predictions of limit cycles and experimentally observed constant concentrations correspond to rate equation predictions of steadystate conditions in single cells. However, when noise is nonnegligible each realization of the stochastic simulation algorithm provides the behavior of a particular cell in the ensemble^{28} and the power spectrum provides information of the rhythmicity present at the singlecell level. From this point of view, the noiseinduced oscillations described in this article correspond to the biological scenario where ensemble level experiments suggest that cells are nonoscillatory while in reality nonsynchronized rhythms are present in each cell. This phenomenon has been experimentally observed by comparing ensemble and single cell measurements^{29}. In this context, our bursty and nonbursty systems correspond to two independent populations of noncoupled cells, in one of which the oscillatory pathway has a bursty production of proteins and in the other it does not. At the ensemble level both populations appear nonoscillatory and indistinguishable (both their rate equations have the same steadystate) but at the single cell level they are sometimes distinguishable since bursts can either promote or destroy single cell rhythms (noiseinduced oscillations).
We finish by discussing how our theoretical results could be experimentally tested. The procedure consists of three parts: (i) the synthetic engineering of one of the pathways considered in this paper in a single cell; (ii) the variation of burstiness in protein production at fixed protein production rates; (iii) the measurement of single cell power spectra of protein fluctuations in the synthetic pathway. Points (i) and (iii) have been done in various contexts; see for example^{30,31}. Point (ii) is the subtlest of the three as it requires regulation of burstiness at the gene level. A rate equation analysis of the the standard linear model of gene expression^{8} leads to the conclusion that whenever the mRNA lifetime is much shorter than that of proteins (the typical case in bacteria and yeast), the overall rate of protein production is equal to the product of the transcription and translation rates divided by the rate of mRNA degradation while the burst size is equal to the translation rate divided by the mRNA degradation rate. Thus one can conclude that by varying the transcription and translation rates independently, it is possible to increase the burst size at a constant overall rate of protein production and to hence test our predictions. A method to achieve this has been reported in Ref 8 and hence it follows that the results of our theory can be tested by currently available experimental techniques.
Methods
Stochastic analysis via the Linearnoise approximation (LNA)
The stochastic dynamics of any chemical system in a wellmixed compartment are described by chemical master equations^{14} which for the nonbursty and bursty systems shown in schemes (1) and (2) take the respective form:
where P^{b/s}(n_{1}, n_{2}, t) is the probability that there are n_{1} molecules of species X_{1} and n_{2} molecules of species X_{2} at time t for the bursty input (b) and nonbursty, i.e., singlemolecule input (s) systems, Ω is the volume of the compartment in which the downstream pathway operates, S_{ij} = r_{ij} − s_{ij} are the elements of the stoichiometric matrix, is the step operator which when it acts on some function w(n_{i}) gives w(n_{i} + j) and is the microscopic rate function for the j^{th} processing reaction which is given by^{14}:
Note that the first term in each of the master equations above describes the input reactions while the second term describes the processing reactions.
These master equations are typically unsolvable except in special cases (see for example^{32}) or for the case where all processing reactions are firstorder^{33}, a very restrictive assumption given that most interactions inside a cell involve the binding of two molecules. We circumvent this problem by using the LNA of the master equation, a well known technique^{14} which approximates the master equation by a FokkerPlanck equation with linear drift and diffusion coefficients. This approximation is valid for an arbitrarily complex monostable reaction system provided the fluctuations about the mean concentrations are quite small, i.e., provided the molecule numbers are not too small. The general formalism has been described in^{34}; here we simply state the relevant results when the LNA is applied to the master equations (16) and (17).
Within the LNA, the time evolution of the mean concentrations for the two systems is given by the conventional rate equations, Eqs. (3) and (4), in the main text. The LNA analysis also shows that the FokkerPlanck equations describing the probability distribution of concentration fluctuations about the mean concentration solution of the above rate equations are given by:
where is the noise about the mean concentration of species X_{i}, is the second moment of the distribution of bursts, , are the Jacobian matrices (describing linear stability) of the two rate equations, Eqs. (3) and (4) and .
Enforcing the condition of equal mean input rates for the bursty and nonbursty systems, i.e., , Eqs. (19)–(20) simplify to Eqs. (5)–(6) in the main text.
Four step recipe for calculating the necessary conditions for burstinessinduced effects in a twospecies system
Step 1
By comparison of the particular system under study with the general form of the nonbursty input system described by scheme (1), one deduces the stoichiometric coefficients s_{ij} and r_{ij} and constructs the elements of the stoichiometric matrix S_{ij} = r_{ij} − s_{ij} of the downstream pathway where i varies between 1 and 2 and j varies between 1 and the total number R of downstream reactions.
Step 2
Write down the rate equations , where . Solve these equations with the time derivative set to zero to obtain the steadystate concentrations .
Step 3
Calculate the elements of the Jacobian matrix . Calculate the elements of the diffusion matrix of the downstream pathway . Calculate the elements of the diffusion matrices of the downstream path for bursty and nonbursty input: and .
Step 4
Calculate using their definitions in Eqs. (10)–(11) and from the sign of this quantity identify which of the three cases illustrated in Fig. 1 the system under study falls in.
Four step recipe applied to the modified Brusselator
Here we show in detail the steps of the calculational recipe as applied to the modified Brusselator studied in the Results section.
Step 1
Comparison of the modified Brusselator reaction scheme (13) with that in (1) shows that the stoichiometric coefficients are: s_{11} = 1, s_{21} = 2, r_{11} = 0, r_{21} = 3 for the first downstream reaction 2X_{2} + X_{1} → 3X_{2}; s_{12} = 1, s_{22} = 0, r_{12} = 0, r_{22} = 1 for the second downstream reaction X_{1} → X_{2} and s_{13} = 0, s_{23} = 1, r_{13} = 0, r_{23} = 0 for the third downstream reaction X_{2} → Ø. Hence the stoichiometric matrix of the downstream pathway reads:
Step 2
Next one uses the stoichiometric information of Step 1 to write the functions and and hence the rate equations are d_{t}ϕ_{1} = h_{1} + g_{1} and d_{t}ϕ_{2} = g_{2} which have a steadystate solution ϕ_{2} = h_{1}/k_{3} and .
Step 3
Using the functions g_{1} and g_{2} obtained in Step 2, the steadystate concentration solutions also obtained in Step 2 and the stoichiometric information obtained in Step 1, we can now calculate the three relevant matrices:
where and Λ_{2} = k_{2}/k_{3} are nondimensional parameters of the system.
Step 4
Using the three matrices calculated in the previous step and the definitions in Eqs. (10)–(11) we can finally calculate the two quantities relevant to deduce the necessary conditions:
Calculation of numerical power spectra of the modified Brusselator
Firstly, SBML reaction models were created describing the nonbursty input and bursty input versions of the modified Brusselator (reaction scheme (13)). The upper burst size for Poisson or Geometric burst distributions is unbounded, meaning that to be exact an infinite number of input reactions is required in the simulation. For the chosen distributions (with mean burst size equal to 12 molecules) we truncated this to a maximum input burst size of 160 molecules. A simple python script was used to generate such a large reaction scheme. After parameterizing the models with the values given in the legend of Fig. 3, the models were simulated using the exact stochastic simulation implementation in the freely available software iNA (intrinsic noise analyser)^{35}.
For a single realization of the stochastic simulation algorithm, the number of molecules n_{i}(t) of species i over some time interval T was regularly sampled at L discrete points separated by Δt, such that T = (L − 1)Δt. The time interval T has to be chosen much larger than the time taken for the simulations to reach steadystate. Subsequently the steadystate mean is subtracted such that one is left with a time series of fluctuations about the mean. The power spectrum estimate (the periodogram) is then obtained by a discrete Fourier transform of this time series (See Appendix C of Ref. 15 for further details). The choices of sampling parameters for the spectra in Fig. 3a and Fig. 3c were Δt = 0.1059 s, L = 600 and Δt = 0.5296 s, L = 600 respectively. Since the variance of the spectrum estimate is known to be high, the final numerical power spectral density estimates plotted in Fig. 3 were obtained by averaging over 2000 periodograms, each corresponding to an independent realization of the stochastic simulation algorithm.
Use of biologically relevant parameter ranges
Parameters for the 8pathway numerical investigation
In our numerical investigation of the eight biochemical pathways we used the range [10^{−3}, 10^{3}] for each of nondimensional parameters Λ_{i}. As we now show, this range falls within the biologically relevant ranges for each of the circuits. The pathways feature zeroth order, unimolecular, bimolecular and trimolecular reactions. The range of the rate constants for each of these reactions is as follows.

1
Input reaction (zeroth order): We infer this rate as lying in the range [10^{−16} M s^{−1}, 10^{−10} M s^{−1}] from the rate of protein production in mammalian cells, as calculated from the product of mRNA copy numbers (1–1000) and the rate of production of proteins per mRNA per hour (1–1000)^{36} factored by Avogadro's constant N_{A} and the typical volume of a mouse fibroblast NIH3T3 cell (Ω ≈ 3 × 10^{−12} litres)^{37}.

2
Unimolecular reaction (first order): In the same study^{36} protein half lives were found to vary from 1–1000 hours, from which the range of protein degradation rates can be calculated via k = ln(2)/t_{1/2} to be of the order [10^{−4} s^{−1}, 10^{−1} s^{−1}].

3
Bimolecular reaction (second order): For bimolecular reaction rates we use the wide range of enzymesubstrate association rates which are roughly given by the measured range of the ratio of the catalytic rate to the MichaelisMenten constant: [10^{3} M^{−1}s^{−1}, 10^{7} M^{−1}s^{−1}]^{38}.

4
Trimolecular reaction (third order): The unit of the rate constant for such a reaction is M^{−2}s^{−1}. Hence we can estimate this rate by dividing the bimolecular rate constant by a concentration. Using the range of bimolecular rate constants stated above and the range of typical intracellular concentrations (nanomolar to micromolar), we estimate the range of trimolecular rate constants to be [10^{9} M^{−2}s^{−1}, 10^{16} M^{−2}s^{−1}].
Using the above ranges one can calculate the range of the nondimensional parameters in Table I. For example for the first nondimensional parameter of the autocatalysis reaction, Λ_{1}, the input rate constants h_{1} and h_{2} could vary to give the range of possible values:
and for the second nondimensional parameter we have the range:
Hence one can see that the range used in our simulations for nondimensional parameters Λ_{i} ∈ [10^{−3}, 10^{3}] is a subset of the ranges calculated using typical rate constant values.
Note that in the models where genes are explicitly shown (such as One Gene Model A) the relevant input rate parameters k_{0} or were chosen such that the typical input range h_{i} ∈ [10^{−16} M s^{−1}, 10^{−10} M s^{−1}] describes the basal expression rate where is the molar concentration of a single gene.
Parameters for the modified Brusselator model
For input rates to the modified Brusselator model (Fig. 3) we considered the alternative scenario in which proteins are transported into the cell, instead of being expressed in the cell. Transporter proteins are known to transport a large range of ions and molecules across the cell membrane at rates typically in the range [10^{2} molecules s^{−1}, 10^{4} molecules s^{−1}]^{39}. Converting our input h_{i} parameters into these units requires us to write input rates as h_{i}Ω N_{A}, where Ω is the volume of a cell and N_{A} is Avogadro's constant. The input rates used in our study of 1000 and 500 molecules s^{−1} were chosen to fall within this biologically relevant range. The volume Ω was chosen to be equal to 3 × 10^{−15} litres, i.e., roughly that of a typical bacterial cell. Unimolecular and trimolecular rate constants were motivated as for the other models discussed above.
Equipment and settings
Image generation

Fig. 2 was created using Maple (to draw the correct boundary lines), with axis labels, area patterns/colours and numerical annotations added using Inkscape.

Fig. 3 was created using Maple, with axis labels and legend annotations added in Inkscape.

Fig. 4 was created using Maple, with axis labels, bar chart colours and text annotations added using Inkscape. The circuit diagrams were also added using Inkscape.
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Acknowledgements
D. L. K. Toner and R. G. acknowledge support by SULSA (Scottish Universities Life Science Alliance).
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D.L.K.T. and R.G. jointly developed the modelling framework and circuit classification system and wrote the main manuscript text. D.L.K.T. performed the simulations and prepared the figures. Both authors read and approved the final manuscript.
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Toner, D., Grima, R. Effects of bursty protein production on the noisy oscillatory properties of downstream pathways. Sci Rep 3, 2438 (2013). https://doi.org/10.1038/srep02438
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DOI: https://doi.org/10.1038/srep02438
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