Out-of-equilibrium gene expression fluctuations in the presence of extrinsic noise

Cellular processes are subjected to stochastic fluctuations. These fluctuations (or noise) can lead to phenotypic differences even in genetically identical cells sharing the same history and environment.

Noise can often be detrimental for the cell since it affects the precision and reliability of several processes, for example related to signalling. Indeed, a high noise level has been associated to partial or complete loss of cellular functions [1–3], and there is evidence of evolutionary selection against cellular noise [4, 5]. On the other hand, molecular noise can be beneficial in various circumstances [6]. It can be exploited to drive genetically identical cells to different cell fates in multi-cellular organisms [7–13], or it can induce the phenotypic diversification at the basis of bet-hedging strategies that protect microbial cell populations from sudden environmental changes [14–20].

Focusing specifically on the gene expression process, two possible sources of fluctuations can be defined, i.e. intrinsic and extrinsic noise. Intrinsic noise arises from the inherently stochastic nature of the molecular reactions involved in the transcription, translation, and degradation of messenger RNAs (mRNAs) and proteins. Extrinsic noise is instead the result of fluctuations in global cellular factors such as the concentration of key macromolecules (e.g. ribosomes and polymerases) involved in the process performing catalytic/enzymatic activity [21, 22]. These extrinsic fluctuations can also arise from cell-to-cell differences in metabolic states [23], cellular signalling [24, 25], cell-cycle stage [26–28], and other relevant phenotypic traits associated with cell physiology (i.e. cell growth rate, doubling time, volume, etc) [29–35].

Noise propagation through gene regulation is another key source of gene expression noise. In fact, the number of regulatory inputs of a gene is correlated with its expression variability in E. coli [36] and eukaryotes [37, 38], and the topology of the regulatory network can play a crucial role in the extent of noise propagation [39, 40].

A large amount of theoretical and experimental work has focused on disentangling the contributions to protein fluctuations from intrinsic and extrinsic sources (see for example [41–43]). On the other hand, what are precisely the dominant sources of extrinsic noise, and thus how they have to be correctly inserted in effective models of gene expression are still open questions, even if extrinsic noise actually seems the main noise source for sufficiently highly expressed genes [38, 44, 45].

The specific definition of what is considered extrinsic noise depends on the system one wants to explicitly model (a single gene, a small genetic circuits, the whole cell). We focus on a single gene and, from a modelling standpoint, extrinsic noise can be defined as fluctuations of the parameters of the expression process such as degradation and production rates. All the possible biological sources of extrinsic noise listed above can affect in complex ways one or more parameters. For example, growth rate fluctuations directly impact the dilution rate of proteins through volume fluctuations, but at the same time can change the protein production rates by affecting the concentration of key enzymes, such as ribosomes, that are closely coupled with cell growth [46]. The problem is to pinpoint which are the parameters that are most affected in the biological system of interest, in order to design the correct minimal model of the stochastic process of gene expression.

This work focuses on this problem by looking at the consequences of extrinsic noise on the out-of-equilibrium dynamics of gene expression. This regime represents the dynamical approach to a steady state, during which the reactions of molecule production and degradation are not yet balanced. Specifically, we will provide analytical expressions supported by simulations that characterize how the protein level and its fluctuations evolve during gene activation and inactivation in the presence of extrinsic fluctuations on different parameters. On top of the theoretical interest of this analysis, the results have immediate practical applications. In fact, the different dynamic profiles of protein noise that we characterized naturally provide an experimentally accessible method to distinguish between different extrinsic-noise scenarios.

2.1. An effective model of stochastic gene expression with extrinsic noise

The 'standard model' of stochastic gene expression takes into account messenger RNA and protein production and degradation as first-order chemical reactions [47–50]. After activation, the gene is transcribed by RNA polymerases in mRNAs with a fixed rate km; each mRNA (m) is in turn translated into proteins (p) with rate kp. Proteins and mRNAs are also removed at specific constant rates (γm and γp ). Figure 1 schematically represents this set of reactions.

Figure 1. Model of the gene expression process with extrinsic noise. Extrinsic noise is included in a basic two-step model of stochastic gene expression by introducing the cellular factor z(t). z(t) can affect any parameter of the model: transcription and translation rates (km and kp), as well as degradation rates for mRNAs and proteins (γm and γp ). The table on the left lists the possible reactions for mRNAs m, proteins p and for the cellular factor z with the respective propensity functions that set the event frequencies.

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The degradation rate of mRNA molecules sets their average lifetime ($1/\gamma_$), which is typically short compared to the average residence time of proteins ($1/\gamma_$). Especially in microorganisms, the average lifetime of mRNAs is just few minutes [44]. On the other hand, the rate γp is mainly set by dilution due to cell growth and division [44, 50–52], and thus the protein lifetime is essentially set by the cell doubling time $\tau_p = \ln(2)/\gamma_p$ [53].

The master equation describing this simple two-step model of gene expression (figure 1) can be solved assuming that the promoter is activated at time t = 0 and the initial number of mRNAs and proteins is zero [50]. In particular, the dynamics of the average protein numbers is described by

Equation (1)

where $p_ = k_}/\gamma_}$ is the average protein level at steady state. The noise can be quantified by the coefficient of variation squared $CV^2_p = \sigma^2_p / \langle p \rangle ^2 \equiv \eta^2$, where $\sigma^2_p$ is the variance of the protein number. The coefficient of variation CVp represents the relative fluctuations, and it is an intuitive and dimensionless measure that can be directly compared with experimental values.

For stable proteins, the timescale separation between the dynamics of mRNAs and proteins ($\gamma_m \gg \gamma_p$) can be used to derive a compact expression for the time evolution of the intrinsic noise [50, 54] and for its equilibrium value:

Equation (2)

$B_p = k_p/\gamma_m$ is the protein burst size, i.e. the average number of proteins produced by a single mRNA during its lifetime. As the expression at steady state shows, burstiness introduces an amplification factor with respect to Poisson noise. We simulate realistic and relatively high levels of expression (pss = 2000 proteins in most examples we will describe) with sufficiently low protein burst sizes (few units). The goal is to focus on genes for which the intrinsic contribution to expression noise is not dominant with respect to the extrinsic part, at least at steady state. Large-scale experimental studies in both E. coli and yeast have suggested that this is the typical case for highly expressed genes by looking at the scaling of protein noise with the average protein level [38, 44]. Interestingly, a similar scaling seems to hold also for mRNA fluctuations [55].

We will also consider the dynamics of a gene at steady state that is inactivated at the transcriptional level. In this case, the average protein evolution in time is given by

Equation (3)

So far we have only included intrinsic fluctuations since all the rates were constant. To include extrinsic fluctuations, we introduce a generic cellular factor z that can affect production or degradation rates (figure 1). For example, fluctuations in RNA polymerases or ribosomes will be captured by a direct action of the factor z on the production rates km or kp. More generally, the cellular factor z can capture the consequences that fluctuations in cell physiology can have on gene expression by modulating the affected rates.

Extrinsic fluctuations often have a lifetime that is not negligible and can be comparable to the cell cycle and to protein half-life [21, 44, 56]. Therefore, extrinsic noise is typically referred to (and modelled as) 'colored' noise [43], which is a noise with a characteristic timescale. In our context, this timescale depends on the effective (and often unknown) source of extrinsic fluctuations.

To explore the role of both extrinsic fluctuations strength and timescale, we model the dynamics of the cellular factor as a bursty birth-and-death process with constant production and degradation rates (kz and γz ). With this modelling choice, production events happen at a constant rate kz (as in a Poisson process) and the burst size is bz, a random variable sampled by a geometric probability distribution with average burst size Bz [42]. Tuning kz, γz and Bz, the extent (CVz) and the timescale ($\tau_z = \ln(2)/\gamma_z$) of extrinsic fluctuations can be independently modulated.

We can thus introduce extrinsic noise on any biochemical rate by multiplying its value to the cellular factor. In other words, we can substitute any parameter θ of the system with $\theta \rightarrow \theta(t) = \theta \frac$. In this way, the average parameter value is still θ, i.e. the value set in the absence of extrinsic fluctuations, but it fluctuates according to z.

The intrinsic noise ηInt will be defined as the variability associated with the model of stochastic gene expression with constant parameters. Instead, the extrinsic noise ηExt can be quantified as the difference between the total measured protein variability and the intrinsic part.

2.2. Assessment of the time evolution of protein cell-to-cell variability

The definition of intrinsic and extrinsic noise naturally implies the decomposition $\eta^2 = \eta^2_ + \eta^2_$ [57]. This decomposition is also valid out-of-equilibrium and for each parameter θ affected by extrinsic fluctuations of strength defined by CVz and timescale by τz . Therefore, we can write

Equation (4)

We are interested in the dynamics of gene-expression noise approaching a steady state. In the case of fluctuations of the production rates km or kp, it is possible to derive the exact transient protein noise expression by solving the corresponding system of ordinary differential equations. The details of the calculation can be found in the appendix, but the main result is that for extrinsic noise acting on production rates we can provide an analytical estimate of the time evolution of protein noise.

Unfortunately, the same approach cannot be applied when extrinsic fluctuations affect the dilution rate γp . Nevertheless, an approximate expression for the steady-state protein noise $\eta_^2(CV_z, \tau_z)$ can still be calculated as a function of the timescale and strength of extrinsic noise. In order to provide also an expression for the out-of-equilibrium noise, we can assume that the time dependence in the noise expression can be factorized as

Equation (5)

$(t)}$ explicitly captures the time-dependent part of the impact of θ fluctuations on gene expression noise, while $(CV_z, \tau_z)}$ is time independent and takes into account the features of extrinsic noise. Assuming this factorization, an intuitive although approximate expression can be provided using the framework of sensitivity analysis [58]. Basically, we can first evaluate the fluctuations due to the extrinsic factor acting on the parameter θ at steady state, and this is possible for every choice of θ. We can then further assume that the effect of the fluctuating parameter θ on the protein level is predominantly set by the first-order dependency of the average protein dynamics on θ. In other words, we can estimate the protein variance in time using the propagation of uncertainty as

Equation (6)

$\sigma^2_(CV_z, \tau_z)$ is the variance of the parameter θ, which is given by the variance of the extrinsic factor z. The relation can be rephrased for the coefficient of variation as

Equation (7)

where, by definition, $CV_^2 = CV_z^2$ at any given time.

This estimate provides the functional dependency of $(t)}$, while the time-independent noise $CV^2_z$ can be included in the factor $(CV_z, \tau_z)}$ of the decomposition in equation (5). The approximation considers p(t) and θ as continuous variables and neglects the impact of the strength and timescale of extrinsic fluctuations on the time dependent factor.

From the expression of p(t), it is easy to show that $\lim_(t)}$ = 1 for any parameter θ. This crucial observation implies that, in order to have consistency at steady state, the factor $(CV_z, \tau_z)}$ has to be equal to the extrinsic noise at steady state $\eta_^2$.

Therefore, we finally have explicitly defined the two factors of equation (5) as

Equation (8)

As discussed above, analytical expressions can be calculated for the total protein noise at steady state $\eta_^2$ as a function of the strength and the timescale of extrinsic fluctuations (see equations (A15), (A18) and (A22)). Analogously, the intrinsic noise $ \eta_^2$ can be calculated [50]. Therefore, the extrinsic part can be extracted with a simple subtraction $\eta_^2 (CV_z, \tau_z) = \eta_^2 - \eta_^2$

All the analytical expressions have been tested with extensive numerical simulations using the exact Gillespie algorithm [59] as detailed in the appendix (section 'Stochastic simulation specification'). In the following sections the analytical curves will always be supported by and compared to numerical results. More specifically, for each model configuration, we generated $5\times10^3$ trajectories simulating the reactions reported in figure 1. In order to estimate the confidence interval of our measurements of noise, we first calculated the protein coefficient of variation values for 5 independent sets of 103 trajectories. The variability of the different values obtained is comparable or smaller than the symbol size used in the figures. We also calculated the interval corresponding to the 95% confidence level for the coefficient of variation values obtained with the bootstrap method [60]. Again, the error interval is consistently smaller than the marker symbols and thus not explicitly shown in the figures.

3.1. Extrinsic fluctuations of the protein dilution rate alter the average protein dynamics

The first result of this work is to provide analytical expressions for the protein dynamics and its variability in the presence of extrinsic fluctuations acting on different gene expression parameters. These analytical results are detailed in the Methods section and in the appendix and will be discussed in the following sections.

Here, we focus on the expected value of the protein level at steady state for different extrinsic noise sources. The different expressions we obtained show that extrinsic fluctuations acting on production rates (i.e. translation rate kp and transcription rate km) do not alter the average protein level pss described by equation (1) at equilibrium, but fluctuations of the dilution rate γp can significantly change it (equation (A21)). When the extrinsic noise acts on the dilution rate, the system of differential equations describing the protein moment dynamics is not closed, but an approximate moment-closure technique (see appendix, section 'Protein dilution rate fluctuations', equation (A20)) can be used to estimate the expected protein level at steady state as

Equation (9)

The expression above indicates how the protein level depends on the cellular factor z and how it is different by the value $p_ = k_}/\gamma_}$ without extrinsic noise. In particular, it increases with the fluctuation strength CVz and has a sigmoidal dependence on the fluctuation timescale τz . These analytical predictions are well supported by stochastic simulations (appendix figure A2).

Therefore, the noise-induced alteration of the average behaviour, also known as deviant effect [40, 61], generates a discrepancy with respect to the classic deterministic prediction pss , and this discrepancy grows with the level of extrinsic fluctuations. Thus, deviations from the expected average protein value could in principle be used in experimental settings as hallmarks of large extrinsic fluctuations acting on degradation rates. However, this observation would practically require the knowledge of the process parameter values, which are often not known.

This work aims to characterize the complex interplay between extrinsic fluctuations and protein dynamics in single cells. We will show that this characterization is instrumental to define easily measurable signatures of the possible dominant sources of extrinsic noise in a system, even when the specific parameter values are not known. However, in order to do so, we need to compare the expected single-cell protein dynamics with extrinsic fluctuations acting on different parameters with a fixed common average steady-state value. As explained in detail in the appendix (section 'A mathematically controlled comparison') and shown in figure A3, we implemented this classic 'mathematically controlled comparison' [52, 62] by taking into account the deviant effects and constraining the average steady-state protein value for any magnitude and timescale of extrinsic fluctuations, independently of which the noisy parameter is.

3.2. The dynamics of cell-to-cell variability is strongly dependent on the dominant source of extrinsic noise

This section describes the single-cell expression dynamics when a gene is activated in the presence of extrinsic fluctuations. The main observation is that the protein noise dynamics is qualitatively different depending on which parameter is predominantly affected by extrinsic fluctuations. In fact, even if the average protein dynamics is constrained to be the same (figure 2(A)), as explained in the previous section, the protein probability densities are significantly different at short times depending on the extrinsic noise source (figure 2(B)). The out-of-equilibrium protein fluctuations are higher for noise affecting production rates rather than degradation rates. This distinction cannot be done at equilibrium since the probability densities progressively collapse as the protein level approaches the steady state.

Figure 2. Cell-to-cell variability for different sources of extrinsic noise. We compare the time evolution of expression variability during the activation dynamics when extrinsic fluctuations affect a single rate of production (km, kp) or of degradation (γm or γp ). Even if the extrinsic noise properties are fixed (CVz = 0.3, $\tau_z/\tau_p$ = 1, $\langle z \rangle$ = 1000 copies), different protein noise profiles $\eta(t)$ can be observed depending on the fluctuating parameter. (A) Thanks to the controlled comparison, the average protein level approaches a fixed steady state following equation (1), independently of the source of extrinsic noise. (B) The protein probability densities are qualitatively different in the presence of fluctuations of km or γp . The mean protein levels at different times are reported as vertical lines and correspond to the horizontal lines in (A). (C) The total gene expression noise, quantified by the coefficient of variation, is reported as a function of time during the transient regime (time $\in[0;6]\tau_p$). The continuous grey line represents the variability without extrinsic noise CVz = 0, i.e. only the intrinsic noise described by equation (2). The horizontal blue line marks the approximate prediction for the steady-state expression variability under fluctuations of γp , to which the values of the simulations asymptotically tend. The dashed lines correspond to the theoretical predictions (equations (10) and (14)), which are well compatible with the simulation results (symbols).

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Figure 2(C) explicitly reports the dynamics of protein expression noise, which shows different behaviors depending on the dominant source of extrinsic noise. While we used the CV2 as the noise measure in our mathematical expressions, the coefficient of variation ($\eta(t) = CV$) is reported in figure 2(C) and in the following figures as a more intuitive measure. The trends in the presence of fluctuations of the production rates (i.e. km or kp) are qualitatively and quantitatively similar. The simulations are well explained by the exact formula (fully derived in the appendix sections 'Transcription burst frequency fluctuations' and 'Translation rate fluctuations') for the time-evolution of protein noise in the presence of transcription rate fluctuations (red dashed line in figure 2):

Equation (10)

The corresponding steady state limit is given by

Equation (11)

The time dependence of equation (10) is analogous to the one of intrinsic fluctuations, given by equation (2) and corresponding to the grey line in figure 2(C). The presence of extrinsic noise essentially increases the relaxation point to the higher noise value predicted by equation (11) without changing the monotonous decreasing trend.

This result can be intuitively understood by considering the factorization proposed in equation (5). Since $\partial \langle p(t) \rangle / \partial k_m = \langle p(t) \rangle/ k_m$ and $\partial \langle p(t) \rangle / \partial k_p = \langle p(t) \rangle/ k_p$, the factors $F_(t)$ and $F_(t)$ do not depend on time, thus explaining why $\eta_(t)$ and $\eta_(t)$ qualitatively follow the intrinsic noise profile. Indeed, the extrinsic fluctuations uniformly affect gene expression during the activation dynamics by shifting the total noise in protein level to higher values.

Note that the predicted steady-state levels of fluctuations are all similar in this setting. In principle, fluctuations on different production rates can produce slightly different levels of steady-state protein noise. In fact, fluctuations of the translation rate (and thus of the protein burst size) lead to higher protein noise, as the comparison between their analytical expressions in equations (11) and (A18) shows. However, the additional noise term has a factor $1/p_$ that makes it negligible for highly expressed genes. Analogously, as explained in details in the appendix (section 'Protein dilution rate fluctuations'), the estimated value of $ \eta_^2 $ for fluctuations of the degradation rate has a particularly compact form when the timescale of extrinsic fluctuations approximately matches the intrinsic one (as in the example considered):

Equation (12)

The Taylor expansion of the extrinsic contribution for small $CV_z^2$ is $CV_z^2/2$, precisely as in the case of fluctuations on production rates (equation (11)).

On the other hand, the protein noise dynamics $\eta(t)$ displays a qualitatively different and non-monotonic trend for fluctuations of degradation rates (γm or γp ). In particular, extrinsic fluctuations on γp do not affect the noise dynamics at short times. The total noise is dominated by the intrinsic part as the simulations (blue circles) precisely lay on the intrinsic noise theoretical prediction (grey line) for

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