## Abstract

Recent analyses with high-resolution single-molecule experimental methods have shown highly irregular and variable bursting of mRNA in a wide range of organisms. Noise in gene expression is thought to be beneficial in cell fate specifications, as it can lay a foundation for phenotypic diversification of isogenetic cells in the homogeneous environment. However, because the stability of proteins is, in many cases, higher than that of mRNAs, noise from transcriptional bursting can be considerably buffered at the protein level, limiting the effect of noisy mRNAs at a more global regulation level. This raises a question as to what constructive role noisy mRNAs can play in the system-level dynamics. In this study, we have addressed this question using the computational models that extend the conventional transcriptional bursting model with a post-transcriptional regulation step. Surprisingly, by comparing this stochastic model with the corresponding deterministic model, we find that intrinsic fluctuations can substantially increase the expected mRNA level. Because effects of a higher mRNA level can be transmitted to the protein level even with slow protein degradation rates, this finding suggests that an increase in the protein level is another potential effect of transcriptional bursting. Here, we show that this striking steady state increase is caused by the asynchronous nature of molecular reactions, which allows the transcriptional regulation model to create additional modes of qualitatively distinct dynamics. Our results illustrate non-intuitive effects of reaction asynchronicity on system dynamics that cannot be captured by the traditional deterministic framework. Because molecular reactions are intrinsically stochastic and asynchronous, these findings may have broad implications in modelling and understanding complex biological systems.

## 1. Introduction

The degree of stochasticity displayed in the regulation of biological processes can change based on its network structure as well as other network properties [1–5]. Depending on the characteristics of the underlying reaction network, stochastic effects can play a crucial role in shaping dynamics and functions of a biological system [6–8]. Importantly, such dynamic properties may not be accurately described by continuous-deterministic approximations based on mass-action kinetics or classical chemical kinetics (CCK) [9,10] as it neglects the underlying fluctuations [11]. In order to characterize how dynamical properties are shaped from the network properties given the underlying fluctuations *in silico*, then, it may be necessary to use a discrete-stochastic modelling approach based on stochastic chemical kinetics (SCK) [12–15].

Gene regulation processes are particularly susceptible to such fluctuations as the molecule counts of participating species (e.g. *cis*-regulatory units and transcription factors) can be very low. Recent advances in single-molecule experimental methods have made it possible to observe the expression of mRNA in greater detail, enabling discoveries of high variabilities in mRNA levels in a wide range of organisms such as prokaryotes [16–18], primitive eukaryotes [19,20] and higher eukaryotes [21,22]. Furthermore, previous studies have reported that such bursting can be seen in the expression of genes with widely different transcription kinetics [18,22,23]. These suggest that transcriptional bursting is a general biological phenomenon—rather than a property of a specific model system with a specific range of gene expression kinetics. Without relying on any external factors, this phenomenon can be captured by a simple model of mRNA regulation, where transcription is initiated by a two-state promoter that can randomly switch between an active state and an inactive state (figure 1*a*) [3,24]. Although the underlying mechanism of the stochastic promoter-state transition involved in transcriptional bursting is not decisively known in general, chromatin remodelling [25,26] was experimentally suggested as a cause of stochastic gene expression in some eukaryotes [3,24,27].

While the stochastic two-state promoter model can display irregular, pulsatile gene transcription, the mean time evolution of the mRNA level in this SCK model is accurately captured by the corresponding CCK model as all of the reactions have first-order kinetics [28]. This implies that the average protein count, at the steady state regime, can be accurately captured by the corresponding CCK model. Furthermore, effects of noise in mRNAs can be considerably suppressed at the protein level with a slow protein degradation rate (figure 1*a*). Thus, this reaction model suggests that transcriptional bursting may not be particularly useful for facilitating phenotypic variations when the stability of a protein is high [21,29].

This observation raises a question as to what effects, if any, the fluctuations in the mRNA regulation can have on protein regulation and more global function given that protein half-life may be very long compared with stochastic fluctuations in mRNA expression level. To address this question, we created a gene expression process model that extends the two-state promoter transcriptional bursting model by including a control of mRNA degradation (figure 1*b*). This post-transcriptional control can represent, for example, conformational changes of such mRNA decapping enzymes as Dcp2 [30, 31]. Because post-transcriptional controls can play a crucial role in gene regulation [32], we investigated whether or not effects of the fluctuations on the system dynamics can also be altered by considering the control of mRNA stability. The potential importance of post-transcriptional controls in the system-level dynamics has been explored in different contexts in previous theoretical studies [33–35]. For example, a previous study has shown that stochastic effects in a two-gene negative feedback network with a highly cooperative transcriptional repression process can be reduced in both magnitude and frequency, leading to a more coordinated cell-to-cell response by considering the mRNA degradation control step [35].

Here, we explore the role of transcriptional bursting in a more general context by extending the two-state promoter model to consider the expression of a single gene without any feedback loops. Our analysis shows that, unlike the two-state promoter model, our gene expression model can result in steady state behaviours dramatically different from what is predicated by the CCK model. More specifically, we find that the fluctuations in our model can make the expected steady state of mRNA higher, which, in turn, can substantially increase the expected protein level (figure 1*b*).

While this novel finding suggests that another potential constructive role of underlying fluctuations in mRNA regulation is to increase the expected protein level, how this phenomenon emerges by including a post-transcriptional regulation in the conventional transcriptional bursting model is unclear. To this end, we developed several computational models to specifically analyse how the expected steady state mRNA level can be increased. Our analysis shows that the origin of the steady state gain in the mRNA level in our stochastic gene expression model is the underlying asynchronicity of molecular reactions, which cannot be captured by the traditional deterministic framework. The asynchronous effect in the core part of the mRNA regulatory machinery allows the generation of additional modes of qualitatively distinct dynamics, which in turn can substantially influence the steady state level of mRNA. In particular, we find that, owing to the specific network structure of the gene expression model that we studied, the expected steady state mRNA level can be inevitably increased by this asynchronous effect. By showing that large deviations between stochastic and deterministic steady state predictions are possible with a much simpler mechanism than has previously been considered in gene expression processes, our results imply that similar effects could be a far more widespread phenomenon than is currently appreciated. This observation may have important implications for realistic modelling and better understanding of complex biological reaction systems in general.

## 2. Results

### 2.1. Gene expression reaction model

In this study, we extend the two-state promoter gene expression model (figure 1*a*) by including a catalytic mRNA degradation (figure 1*b*). The mRNA degradation catalyst has two functionally different states: one is an active state in which the catalyst can degrade mRNA; and the other one is an inactive state in which the catalyst is incapable of degrading mRNA. From this specification, we constructed a reaction model to understand what dynamic behaviours can arise from the given structure of the reaction network. Our reaction model is as follows:
2.1
2.2
2.3
2.4
2.5Reaction (2.1) models the regulation of an mRNA, *M*, that is constitutively expressed from an active promoter, *A*, and is down-regulated by the presence of a degradative agent, *B*. Reaction (2.2) captures the state transition of the promoter between an active state (*A*) and an inactive state (*A*^{c}). The functional changes in the degradation catalyst are described by reaction (2.3), where *B* is the active state and *B*^{c} is the inactive state. Reaction (2.4) models the regulation of protein *P*, which is synthesized using *M* as a template and is degraded at a rate proportional to its molecule count, and reaction (2.5) describes the basal degradation of *M*.

### 2.2. Steady state gain via fluctuations

To test if the added mRNA degradation control in our reaction model can alter the role the underlying fluctuations play in the steady state behaviour of gene expression, we simulated the corresponding SCK model using the stochastic simulation algorithm (SSA) [36]. The parameter values of our gene expression model are set mainly based on the recent quantitative gene expression results on mouse fibroblasts [22] (for detail see the electronic supplementary material, §1). Our simulation results show that, similar to that of the two-state gene expression model (figure 1*a*), *M*(∞), the mRNA count in the stationary regime, has a long-tail distribution with a high variance (figure 1*b*). However, unlike that of the two-state gene expression model, 〈*M*(∞)〉 can be higher than what is predicted by the CCK model (figure 1*b*). This steady state gain in mRNA can lead to a substantially increased steady state level of the protein. Indeed, our simulation results show that, while the high noise level and the long-tail distribution seen in *M*(∞) is not observed in *P*(∞), 〈*P*(∞)〉 for the SCK model can be substantially higher than that for the CCK model (figure 1*b*). Our simulation results also show that an increase in the steady state of the protein level becomes substantial as the stability of the protein increases (electronic supplementary material, §2 and figure S1).

### 2.3. Simplified reaction model

In order to gain insights into how the expected mRNA level is increased with the fluctuations, we constructed a simplified reaction model of the gene expression process so that we can extensively analyse what properties of the reaction network give rise to this phenomenon. Thus, in this simplified model, we included a minimal reaction set that we believed could give rise to higher expected mRNA levels with the fluctuations by extending the two-state promoter model. In our analysis of this model, we set the values of the parameters so that the steady state mRNA level is within a biologically relevant range [18,22,23]. Our simplified model has the following reactions: 2.6 2.7 2.8

With reaction (2.6) alone, *M*(*t*) follows a simple birth–death process, which is governed by the following master equation:
2.9where *p*_{1}(*m*,*t*)d*t* is defined as the probability that *M* = *m* in the time interval [*t*,*t* + d*t*). Setting equation (2.9) equal to 0 and solving for *p*_{1}(*m*,*t*) can give us the stationary distribution of this process as follows:
2.10which is the Poisson distribution with mean *k*_{1}*A*/(*k*_{2}*B*). Thus, 〈*M*(∞)〉 without reactions (2.7) and (2.8) can be obtained by solving for the steady state of the CCK model (electronic supplementary material, §3).

The master equations of reactions (2.7) and (2.8) are identical, and have the following form:
2.11where *S* and *S*^{c} are the active form and the inactive form of a molecular species, respectively, *p*_{2}(*s*,*t*)d*t* is the probability that *S* = *s* in the time interval [*t*,*t* + d*t*), and *S*_{T} is *S*(0) + *S*^{c}(0). By solving this finite birth–death process for the stationary distribution of *S*, we can obtain:
2.12where *q* = *k*_{p}/(*k*_{p} + *k*_{d}), which is the binomial distribution with *n* = *S*_{T} and *p* = *q*. Thus, the stationary mean of *A* and *B* is equal to *S*_{T}*q*, which can be accurately predicted by the steady state of the CCK model (electronic supplementary material, §3). In this study, we set *A*(0) = *B*(0) = 1 and *A*^{c}(0) = *B*^{c}(0) = 0 for analysis. Note that these constraints are chosen to make our analysis simpler to give broader conclusions about any potential effects of underlying fluctuations on the steady state behaviour.

### 2.4. Steady state analysis

The mean evolution of our SCK model is expressed by the following set of differential equations (electronic supplementary material, §4):
2.13
2.14
2.15Here, we are interested in obtaining *M*_{∞}, the value of 〈*M*(∞)〉. By setting equations (2.13)–(2.15) to 0 and using the relation 〈*B*(*t*)*M*(*t*)〉 = 〈*B*(*t*)〉×〈*M*(*t*)〉 + Cov(*B*(*t*),*M*(*t*)), we can derive the following expression for *M*_{∞}:
2.16However, the exact solution of *M*_{∞} becomes complicated because of the presence of Cov(*B*(∞),*M*(∞)), which essentially comes from the quadratic moment involved in equation (2.13). If we assume that there is no fluctuation in the system (i.e. Cov(*B*(*t*),*M*(*t*)) = 0), then the expression of *M*_{∞} in equation (2.16) becomes simplified. Indeed, this approximation is essentially that made by CCK [28], from which we can write down the expression of *M*_{E}, the value of 〈*M*(∞)〉 given by our CCK model, as follows:
2.17By assuming there is no fluctuation in the system, the steady state of 〈*M*(*t*)〉 thus depends only on two parameters: *k*_{1} and *k*_{2}.

### 2.5. Deviation from the steady state behaviour of the CCK model

Using the simplified SCK model, we first investigated the cases in which the conformational change reactions of *A* and *B* in the simplified model are in the fast time-scale. Our analytical study suggests that *M*_{E} can accurately predict *M*_{∞} when the fast-scale conformational change reaction assumption is valid (electronic supplementary material, §5). This result is in agreement with the basis of the stochastic rapid equilibrium approximation [37, 38], which can substantially lower the computational cost of the SSA.

We next considered cases in which the fast-scale reaction (2.7) assumption is no longer valid, and simulated the SCK model using the SSA (see §4). Our simulation results show that *M* in the stationary regime can exhibit very infrequent, large bursts and frequent, small bursts, implying a long-tail distribution of *M*(∞) (figure 2*a*). Furthermore, the simulation results show that deviant effects of *M*_{∞} can become so substantial that the prediction from the CCK model becomes quantitatively and qualitatively misleading when the rapid equilibrium assumption does not hold (figure 2*b*). This indicates that, unlike *M*_{E}, which is a function of just *k*_{1} and *k*_{2}, *M*_{∞} does in fact depend on the values of *k*_{p} and *k*_{d}, and inactivation of *A* and *B* can substantially change the steady state behaviour of *M*.

In order to gain more insights into how *M*_{∞} depends on the two parameters, we performed simulation of the SCK model for various combinations of *k*_{p} and *k*_{d} values (see §4). These parameter combinations are in a biologically realistic range in terms of mRNA abundances [18,23,39], which includes parameter regions that have been shown to give transcriptional bursting in some mammalian cells [21,22]. Our results indicate that, while the dependency of *M*_{∞} on *k*_{p} and *k*_{d} appears to be rather complex, we can find several overall trends (figure 2*c*). For instance, overall, *M*_{∞} is a decreasing function of *k*_{p} that reaches a plateau when *k*_{p} is very small. On the other hand, *M*_{∞} appears to be a function of *k*_{d} that has minimum at *M*_{E} when *k*_{d} is either very high or very low, and has maximum at *k*_{d} = *k*_{p}. Our simulation results also suggest that *M*_{∞} never goes below what is predicted by *M*_{E} (i.e. *k*_{1}/*k*_{2}). In other words, underlying fluctuation appears to increase the mean value of *M* in the stationary regime (figure 2*d*).

### 2.6. Role of discreteness in deviant dynamics

One difference between the SCK model and the CCK model is that, while the former is a discrete-state process, the latter is a continuum process. Thus, we analysed whether or not discreteness plays a role in the deviant effect. To do this, we constructed the chemical Langevin equation (CLE) [40] as a continuous-stochastic approximation of our SCK model, and simulated for various combinations of *k*_{p} and *k*_{d} values (see §4). Our simulation results show that the continuous approximation still gives rise to deviations, albeit the magnitude of which is not as substantial as with the discrete-stochastic SCK model (figure 3 and the electronic supplementary material, figures S2 and S3). Also, the shape of the steady state distribution of *M* for each parameter combination is found to resemble what is predicted from the simulation of the SCK model. Our findings suggest that, while it is not the source of the deviant effects in our model, discreteness can substantially amplify the deviation.

### 2.7. Effects of unsynchronized conformational change reactions on deviation

While *A*(*t*) and *B*(*t*) have the same time-dependent and steady state probability mass function, the conformational change reaction events of *A* and *B* may not be coordinated well, as they are statistically independent. This reaction asynchronicity allows the vector of random variables, **V** ≡ [*A*,*B*], to traverse a Markov chain with four states, namely [1, 1], [1, 0], [0, 1] and [0, 0] (figure 4*a*). To test if the unsynchronized *A* and *B* play a role in deviations of *M*_{∞} from what is predicated by *M*_{E}, we constructed a hypothetical discrete-stochastic model where transitions of *A* and *B* are modified so that they are exactly coordinated (i.e. for all *t* ≥ 0, *A*(*t*) = *B*(*t*)). In this model, **V** can be represented by a Markov chain with only two states, namely [1, 1] and [0, 0] (figure 4*b*). Note that, while this modification changes Cov(*A*(∞),*B*(∞)) from 0 to *q*, it does not have any effect on the time-dependent and steady state probability distributions of *A* and *B*. Also note that this modification has no effects on *M*_{E}, as the CCK model is not capable of capturing reaction asynchronicity of *A* and *B*.

We simulated this modified SCK model using the SSA to estimate the steady state probability distribution and *M*_{∞}. Our simulation result shows that, like the CCK model, the hypothetical synchronization of the *A* and *B* reactions in the SCK model makes *M*_{∞} independent of *k*_{p} and *k*_{d} (figure 3 and the electronic supplementary material, figure S4). Furthermore, it shows that steady state distribution of *M* for each parameter combination has a shape that resembles *p*_{1}(*m*,∞) (i.e. a Poisson distribution with *λ* = *k*_{1}/*k*_{2}). The steady state behaviour of *M* in the modified SCK model can also be analytically derived by letting *p*_{4}(*a*,*b*) be the joint probability mass function of *A* and *B* in the stationary regime, *p*_{5}(*m*) be the probability mass function of *M* in the stationary regime, and *p*_{6}(*m* | *a*, *b*) be the conditional steady state probability mass function of *M* given *A* = *a* and *B* = *b* in the stationary regime. Then, *p*_{5}(*m*) of the modified model can be given by:
2.18where . Thus, while the steady state behaviour of the SCK model may not be captured by the CCK model, that of the modified SCK model is accurately described by the CCK model. This indicates that the discrepancy between *M*_{∞} and *M*_{E} arises solely from the deviations between the SCK model and the modified SCK model, that is, the reaction asynchronicity of *A* and *B*.

In order to understand if this causal relationship can also be applied to the continuous-stochastic model, we modified the CLE to have the synchronized *A* and *B* reactions. Our simulation result from the modified CLE indicates that, even with the continuous approximation, the reaction asynchronicity alone can give rise to the deviant effect (figure 3).

### 2.8. The cause of stochastic steady state gain

We next turn our attention to analysis of the network property that increases the steady state of *M* via introduction of fluctuations in the system. Because unsynchronized *A* and *B* via reaction asynchronicity is the main cause of the deviant effects, the stochastic steady state gain cannot occur without this reaction asynchronicity. However, the conformational switching reactions in our model setting are symmetric and do not seem to have a capacity to bias—one way or another—the directionality of the level of *M*. Thus, our initial hypothesis is that the steady state gain results from the mRNA synthesis and degradation reactions affected by the asynchronous mechanism.

To this end, we derived an analytical expression for the condition under which *M*_{∞} ≥ *M*_{E} is true as follows (for detail see the electronic supplementary material, §6):
2.19where *w* ≡ 1/(*k*_{p} + *k*_{d}) is the average pausing time of **V** in states [0, 1] and [1, 0] per visit. This condition states that, in our simple gene expression model, the average mRNA synthesis count is always greater than or equal to the average mRNA degradation count in the same time window, given that the mean mRNA count is initially set to *M*_{E}. Thus, this can give insights into the underlying cause for the emergence of a negative value of Cov(*B*(∞),*M*(∞)), which is necessary for *M*_{∞} > *M*_{E} as shown in equation (2.16).

Here, when *w* is very small, the left-hand side is equal to the right-hand side, in which case there is no discrepancy in the steady states of the stochastic model and the deterministic model (for detail see the electronic supplementary material, §6). That is, when the sum of *k*_{p} and *k*_{d} has a large value, *M*_{∞} can be accurately captured by *M*_{E}. Furthermore, this condition shows that in order to have *M*_{∞} ≫ *M*_{E}, the average pausing time for *w* must be large. However, this does not mean that whenever *w* is large, we have deviations. Condition (2.19) further explains why *M*_{∞} ≈ *M*_{E} when *k*_{p} is large (figure 2*c*,*d*). When the rate of the activation reaction of *A* and *B* is comparable with that of the transcription reaction (i.e. *k*_{p} ≈ *k*_{1}), *w* is very small relative to the transcription rate. In addition, when the activation reaction of *A* and *B* is much faster than the inactivation reaction of *A* and *B* (i.e. *k*_{p} ≫ *k*_{d}), both *A* and *B* are almost always active, limiting the effects of asynchronous states.

To gain further insights into the effect of the asynchronous mechanism, we partitioned simulation results into the four 〈*M*(∞) | *a*, *b*〉 values, the stationary mean of *M* conditional on *A* = *a* and *B* = *b*, for various combinations of *k*_{p} and *k*_{d} (figure 4*c*). Our analysis shows that there can be a high skewness in the four conditional steady states, where 〈*M*(∞) | 1, 0〉 is the highest and 〈*M*(∞) | 0, 1〉 is the lowest for each parameter set. It also indicates that *M*_{∞} is bounded by 〈*M*(∞) | 1, 1〉 and 〈*M*(∞) | 0, 0 〉, where 〈*M*(∞) | 1, 1〉 is the lower bound and 〈*M*(∞) | 0, 0〉 is the upper bound. We observed that 〈*M*(∞) | 0, 0〉 is the asymptote of *M*_{∞} as *k*_{p} becomes smaller than *k*_{d}, while 〈*M*(∞) | 1, 1〉 is the asymptote of *M*_{∞} as *k*_{d} becomes smaller than *k*_{p}. This result is consistent with our prediction, since when *k*_{p} ≫ *k*_{d} the system spends a large portion of time in **V** = [1, 1], while when *k*_{d} ≫ *k*_{p} the system spends a large portion of time in **V** = [0, 0]. From expression (2.19), a lower bound and an upper bound of 〈*M*(∞) | 1, 1〉 can be expressed by *M*_{E} and *M*_{∞}, respectively. This also means that *M*_{∞} is a lower bound of 〈*M*(∞) | 0, 0〉 because it is an upper bound 〈*M*(∞) | 1, 1〉. Taken together, these observations indicate that a complex interplay of mRNA synthesis and degradation made possible by the asynchronous mechanism underpins condition 2.19, which, in turn, results in a non-positive value of Cov(*B*(∞),*M*(∞)).

## 3. Discussion

Randomness is an inevitable feature of biological processes. Assuming that fluctuations are subject to selective pressure, biological systems must have evolved to feature molecular reaction interactions that can—to some degree—control and exploit such underlying uncertainties to better meet their objectives [8,41]. For example, previous theoretical studies showed that underlying fluctuations can induce bistable oscillation even when the corresponding deterministic model is monostable [42,43]. Another example of interesting stochastic effects is *stochastic focusing* where sensitivity of signal-response systems can be amplified with nonlinear, hyperbolic inhibition reactions, whose mean rates become different from the corresponding fluctuation-free, deterministic reaction rates [44]. A qualitative prediction of stochastic focusing was later performed with analysis of the *curvature–covariance* effect, which examined the deviation of the mean of reaction rate from the corresponding deterministic reaction rate based on the contribution of concentration covariances and the curvature of its kinetic function [45].

A particularly interesting example of stochastic effects is random phenotypic variations of isogenic populations. Indeed, with the recent advances in single-molecule and single-cell experimental techniques, we are able to appreciate, in greater detail, the role the fluctuations play in phenotypic diversifications of isogenic cells in homogeneous environments [46–48]. To diversify cell types, therefore, high transcriptional bursting can be very advantageous when the effect is transmitted to the protein level. However, because a protein's stability is often much higher than that of mRNA, effects of transcriptional bursting are largely buffered at the protein level. Furthermore, widespread observations of transcriptional bursting [18,22,49] suggest that such noisy mRNA molecules may not always be used to facilitate random phenotypic variations.

Here, we have reported another potential role of the fluctuations in the mRNA regulation by considering a post-transcriptional control. Surprisingly, we find that the expected mRNA level can be substantially increased in the presence of the underlying fluctuations. This effect could propagate to the protein level even at a slow protein degradation rate, suggesting that the fluctuations in the mRNA regulation can affect regulations at a more global level. Another implication of this finding is that quantitative measures such as the Fano factor may not accurately estimate the full effects of the intrinsic fluctuations. The strength of intrinsic fluctuations in the regulation of mRNAs is often quantified using the Fano factor [3,24,50]. Assuming mRNA is constitutively synthesized and the turnover rate is proportional to mRNA level, the steady state distribution of this mRNA is found to be Poissonian. Thus, treating the Fano factor of this process as the baseline, the noise strength of a particular mRNA regulation can be measured by comparing its Fano factor with that of the baseline. However, because the expected mRNA level can be substantially increased by the intrinsic fluctuations, such a measure may underestimate the full effects of the fluctuations.

An increase in the mRNA abundance in our stochastic model is a consequence of a negative covariance of the mRNA degradation enzyme and the mRNA. As in the stochastic focusing case, hence, the deviation of the mean of reaction rates from the corresponding fluctuation-free, deterministic reaction rates can lead to an apparent systems-level discrepancy between the stochastic treatment and the hypothetical fluctuation-free treatment of our gene expression system. To unravel how the underlying fluctuations increase the steady state mRNA level, we developed simple computational models to specifically dissect the mechanism of this striking noise effect. Our analysis shows that the steady state gain in mRNA level is caused by underlying asynchronicity of molecular reactions. We find that the asynchronous effect generates additional modes of dynamics in the stationary regime that are qualitatively distinct, and that the longer time the system spends in these asynchronous dynamic modes, the wider the deviation becomes between the stochastic model and the deterministic model. These results illustrate non-intuitive effects of reaction asynchronicity on system dynamics that cannot be captured by the traditional deterministic framework. Because molecular reactions are intrinsically stochastic and asynchronous, such an asynchronous effect could be seen in a wide range of biological systems. Dynamical effects of asynchronous mechanisms on many biological systems can, however, be widely different from what we observed in the gene expression process we have studied. This is because an increase in the expected gene expression level that we observed was directly tied to the specific network structure of the transcription regulation process. In different structures of biochemical networks, reaction asynchronicity may give rise to entirely different dynamical effects. This observation may provide useful guidance for future modelling studies, by showing a new way in which deterministic models of inherently stochastic systems may yield substantially inaccurate results even for outputs that are not themselves apparently stochastic.

## 4. Methods

### 4.1. Simulations of the full gene expression model

Each simulation was run for 60 000 min. For steady state distribution plots, data were collected from 10 000 min for every 0.1 min. The simulations of the full SCK model were carried out using the direct method [51] implemented in iBioSim [52].

### 4.2. Estimation of mean time evolution of the simplified SCK model

The mean time evolution of the SCK model shown in figure 2*a* is estimated by generating 400 000 SSA trajectories and calculating the sample mean from them.

### 4.3. Estimation of the steady state distribution and its mean

We generated a sample trajectory of the system, and uniformly sampled at 1 000 000 time points from the trajectory. We discarded the first 5000 samples, and the rest were used to estimate the steady state distribution as well as its mean value.

### 4.4. Simulation of the CLE

From our simplified SCK model, we derived the following CLE for use in implementing an Euler–Maruyama simulation of the Langevin model:
and
where *𝒩*_{j} (0, 1) is a unit normal random variable that is statistically independent for each *j*. In the simulation, we set the time step *τ* to be 10^{−4}.

### 4.5. Simulation tools

All of the simulations except for the ones for the full gene expression model are implemented in custom C code in order to achieve high efficiency.

## Acknowledgements

H.K. is supported by a Lane Fellowship through the Ray and Stephanie Lane Center for Computational Biology. R.S. is supported by US National Institutes of Health awards 1R01AI076318 and 1R01CA140214.

- Received November 5, 2011.
- Accepted December 12, 2011.

- This journal is © 2012 The Royal Society