How reliable is the linear noise approximation of gene regulatory networks?
- Philipp Thomas^{1, 2},
- Hannes Matuschek^{3} and
- Ramon Grima^{1}Email author
https://doi.org/10.1186/1471-2164-14-S4-S5
© Thomas et al.; licensee BioMed Central Ltd. 2013
Published: 1 October 2013
Abstract
Background
The linear noise approximation (LNA) is commonly used to predict how noise is regulated and exploited at the cellular level. These predictions are exact for reaction networks composed exclusively of first order reactions or for networks involving bimolecular reactions and large numbers of molecules. It is however well known that gene regulation involves bimolecular interactions with molecule numbers as small as a single copy of a particular gene. It is therefore questionable how reliable are the LNA predictions for these systems.
Results
We implement in the software package intrinsic Noise Analyzer (iNA), a system size expansion based method which calculates the mean concentrations and the variances of the fluctuations to an order of accuracy higher than the LNA. We then use iNA to explore the parametric dependence of the Fano factors and of the coefficients of variation of the mRNA and protein fluctuations in models of genetic networks involving nonlinear protein degradation, post-transcriptional, post-translational and negative feedback regulation. We find that the LNA can significantly underestimate the amplitude and period of noise-induced oscillations in genetic oscillators. We also identify cases where the LNA predicts that noise levels can be optimized by tuning a bimolecular rate constant whereas our method shows that no such regulation is possible. All our results are confirmed by stochastic simulations.
Conclusion
The software iNA allows the investigation of parameter regimes where the LNA fares well and where it does not. We have shown that the parametric dependence of the coefficients of variation and Fano factors for common gene regulatory networks is better described by including terms of higher order than LNA in the system size expansion. This analysis is considerably faster than stochastic simulations due to the extensive ensemble averaging needed to obtain statistically meaningful results. Hence iNA is well suited for performing computationally efficient and quantitative studies of intrinsic noise in gene regulatory networks.
Keywords
Background
It is generally accepted that the relative size of molecular fluctuations scales as the inverse square root of the mean molecule numbers [1]. Since the key players of gene regulatory networks are present in amounts as small as one molecule it follows that gene expression is inherently noisy [2, 3]. This molecular noise manifests itself in the copy number variations of transcripts and their proteins among genetically identical cells [4]. The main measures that have been used to quantify these cell-to-cell variations both experimentally and through modeling are the coefficient of variation (CV) and the Fano factor [5–9].
Exact analytical results for these quantities have been derived only for very simple gene regulatory systems [10–12] and hence they are more commonly obtained by means of Monte Carlo simulations using the stochastic simulation algorithm (SSA) [13, 14]. Despite being formally exact with the Chemical Master Equation (CME), in practice, this approach turns out to be computationally expensive mainly due to the considerable amount of sampling required to compute reliable statistical averages. The situation is exacerbated when networks are to be studied over a wide range of parameters. The main analytical tool to address this issue has since been the linear noise approximation (LNA) of the Chemical Master Equation (CME) [15–17] which allows one to approximate the dynamics of the latter by a set of linear stochastic differential equations from which all moments can be computed in closed form. In this approximation, the mean concentrations of the CME are approximated by the solution of the deterministic rate equations (REs) and the probability distribution of the fluctuations is approximated by a Gaussian. Thereby the LNA can give insight into the parametric dependence of the noise whenever the REs admit a unique steady state solution. However unlike the CME, this approximation is valid only in the limit of large molecule numbers and hence the accuracy of its predictions is questionable for intracellular biochemical reaction networks [18, 19]. A handful of theoretical studies access the accuracy of the REs and the LNA predictions by computing finite molecule number corrections to both approximations [20–22], a task which can be carried out analytically only for some simple systems. Hence, to-date, it is unclear how important these corrections are for many gene regulatory networks of interest.
We recently developed intrinsic Noise Analyzer (iNA) [23], the first software package enabling a fluctuation analysis for a broad class of biochemical networks of interest via the LNA and the Effective Mesoscopic Rate Equation (EMRE) approximations of the CME. The latter approximation gives accurate mean concentrations for systems characterized by intermediate to large molecule numbers and is hence more accurate than the conventional REs.
In this article we develop and efficiently implement in iNA, the Inverse Omega Square (IOS) method which gives the variances and covariances of fluctuations about the means calculated by the EMRE method. From these we can calculate the CVs and Fano factors of mRNA and protein fluctuations to an accuracy higher than possible with the LNA. Hence the software iNA provides a means of probing the validity of the LNA for any biochemical network under study. We use the EMRE and IOS methods to study the parametric dependence of the CV and Fano factors of mRNA and protein fluctuations in two examples of stochastic gene regulation involving nonlinear protein degradation, post-transcriptional, post-translational and negative feedback regulation. We show that these results agree with stochastic simulations but in many instances disagree with the LNA results. In particular the LNA predicts that the noise levels can be optimized by tuning a bimolecular rate constant whereas no such regulation is predicted by EMRE/IOS and simulations. It is also found that the LNA significantly underestimates the amplitude and period of noise-induced oscillations in genetic oscillators. Using detailed benchmarks we demonstrate that the present methodology is typically computationally more efficient than stochastic simulations using the SSA.
Results
In this section we describe the results of the novel IOS method implemented in iNA. Its predictions are compared to the RE and LNA approximations of the CME and with exact stochastic simulations using the SSA for two examples of gene regulation. Finally we discuss its computational efficiency. The three methods (LNA, EMRE, IOS) are obtained from the system size expansion of the CME [15] which is applicable to monostable chemical systems. Technical details of the various approximation methods are provided in the section Methods.
Investigating the parametric dependence of the size of molecular fluctuations
where j varies from 1 to R. The mesoscopic state of the system is given by the vector of molecular populations $\overrightarrow{n}=\left({n}_{1},{n}_{2},\dots ,{n}_{N}\right)$ and can be characterized by the probability $P\left(\overrightarrow{n}\right)$ to find the system in a particular configuration $\overrightarrow{n}$. The latter is however very difficult to obtain from analysis and hence intrinsic noise may be more easily characterized in terms of CVs and Fano factors which are defined in the following. We show how these quantifies are calculated using the LNA and higher order approximations implemented in the software iNA.
The above expressions fully characterize the parametric dependence of the size of the fluctuations in terms of the compartment volume Ω, the set of reaction rate constants $\overrightarrow{k}=\left({k}_{1},{k}_{2},\dots ,{k}_{R}\right)$ and the set of initial conditions whose explicit dependence has been omitted here. Note that the leading order contributions in the infinite volume limit, ${C}_{V,i}^{\mathsf{\text{LNA}}}\left(\mathrm{\Omega},\overrightarrow{k}\right)$ and ${F}_{i}^{\mathsf{\text{LNA}}}\left(\overrightarrow{k}\right)$, are given by the LNA's result for the CV and Fano factor which can be shown to scale as Ω^{-1/2} and Ω^{0}, respectively. The factors ${c}_{i}\left(\overrightarrow{k}\right)$ and ${f}_{i}\left(\overrightarrow{k}\right)$ determine the relative corrections to the LNA result and can be obtained from the EMRE and IOS approximations as has been carried out explicitly in the Methods section. It can be argued that the size of these correction terms is proportional to the bimolecular reaction rate constants since the LNA is exact up to second moments for networks composed solely of unimolecular reactions since the propensities are linear functions of the concentrations (see section Methods). Summarizing, this analysis suggests novel correction terms to the CVs and Fano factors that are of order Ω^{-3/2} and Ω^{-1}, respectively, and hence of higher accuracy than the LNA.
Applications
The model has been used to quantify variability in the proteome of E. coli [3, 5, 10], yeast [26] and mammalian cells [27] as well as having being subject to a number of theoretical studies [6, 11]. The LNA's predictions of the first two moments of this model are exact since it is composed of only unimolecular reactions.
Given the complexity of the intracellular biochemistry, it is clear that this simple model cannot fully account for regulation which occurs at transcriptional, post-transcriptional, translational and post-translational stages. These processes typically involve bimolecular reactions with regulatory molecules such as transcription factors, functional RNAs or enzymes. While it is obvious that the CVs and Fano factors of more realistic models will differ from those predicted by the "standard" linear model (6), it is however not immediately clear whether these noise measures are qualitatively different than those obtained from the LNA.
In this section we demonstrate the use of the software iNA, which makes use of the approximation methods described in the previous section, to predict the noise characteristics of two gene regulatory networks involving post-transcriptional regulation by non-coding RNA and negative autoregulation via post-translational modification. Specifically we focus on how well these characteristics are described by the LNA both quantitatively and qualitatively and point out the LNA's limitations using correction terms of the IOS analysis and stochastic simulations provided by iNA.
sRNA mediated post-transcriptional regulation
A large number of functional RNAs called small RNAs (sRNAs) have been found in bacteria which are not actively translated. This non-coding form of RNA is believed to coordinate pathways in response to external stimuli such as stress [28, 29]. To investigate the robustness of critical pathways it is therefore important to understand the impact of intrinsic noise on their regulation.
Note that reactions (7a) are as considered in Ref. [30]. In (7b) we describe the transcription and degradation of sRNA with respective rates k_{0α}and k_{dS}. The parameter α is given by the ratio of sRNA to mRNA transcription and can be used to describe the coordination of the stress response due to tight regulation of sRNA transcription. When sRNA is expressed it binds with its mRNA target at a rate k_{ R } and quickly degrades thereafter. Similar models have been studied in Refs. [31, 32].
Gene expression model with sRNA regulation
paramater | set(i) | set (ii) |
---|---|---|
k_{0}[G] | 0.024min^{-1}µM | 0.0024min^{-1}µM |
k _{dM} | 0.2min^{-1} | 0.2min^{-1} |
k _{ s } | 1.5min^{-1} | 1.5min^{-1} |
k_{-1}, k_{2} | 2min^{-1} | 2min^{-1} |
k _{1} | 400(µM min)^{-1} | 4000(µM min)^{-1} |
k _{dS} | 0.2min^{-1} | 0.2min^{-1} |
k _{ R } | 100(µM min)^{-1} | 1000(µM min)^{-1} |
Genome-wide studies in E. coli revealed that some proteins can be expressed in much lower copy numbers than 60 [3]. We next make use of parameter set (ii) in Table 1 to probe the validity of the LNA under low copy number conditions. The results for the CV are shown in Figure 4 (b). In the absence of sRNA control the protein levels correspond to approximately 6 protein molecules. We observe that for such low copy numbers the LNA is in severe qualitative disagreement with the IOS. In particular, the LNA predicts that there exists a stress level for which the size of the noise is minimized while, in contrast, the IOS predicts the noise level to increase monotonically with stress. The latter is also reproduced by simulations using the SSA in Figure 4 (b) which hence signals the breakdown of the LNA under low copy number conditions.
Gene expression with negative autoregulation
A similar model has been analyzed using the LNA and the EMREs implemented in a previous version of iNA in Ref. [23]. With the present version of iNA the more accurate IOS analysis is available and is used here to investigate the reliability of the LNA estimates of the CVs. This presents a major benchmark for the LNA since the analysis includes fluctuations of a single promoter.
Genetic expression model with negative autoregulation
parameter | Value | parameter | value |
---|---|---|---|
k _{0} | 50$\stackrel{\u0303}{\mathrm{\Omega}}$ (nM) | k _{ s } | 50h^{-1} |
k _{dM} | 5h^{-1} | k _{dp} | 0.5h^{-1} |
k _{1} | 0.5 · (nMh)^{-1} | k _{-1} | 1h^{-1} |
k _{2} | 0.5 · (nMh)^{-1} | k _{-2} | k _{-1} |
k _{3} | 0.5$\stackrel{\u0303}{\mathrm{\Omega}}$ | k _{-3} | 450h^{-1} |
k _{4} | 50 · k3 | k _{-4} | k _{-3} |
k _{ 5 } | 0.25 · (nMh)^{-1} | k _{-5} | 0.5h^{-1} |
k _{ 6 } | 5 · (nMh)^{-1} | k _{-6} | 5h^{-1} |
${k}_{6}^{\prime}$ | 10h^{-1} | ${k}_{5}^{\prime}$ | 0.5h^{-1} |
$\stackrel{\u0303}{\mathrm{\Omega}}$ | 450 · (nM)^{-1} |
This result suggests that the corrections to the LNA are susceptible to the fluctuations in the promoter states. In order to test this hypothesis we investigated the oscillatory dynamics (that is often associated with the presence of a negative feedback loop) as a function of the gene copy number. In rapidly growing E. coli, for instance, the copy number of chromosomal genes located near the origin of DNA replication can be increased by 4-fold over genes located near the terminus [42]. Moreover genes located on plasmids can be present in higher copy numbers than those integrated in the genome. For synthetic circuits the plasmid copy number can also be controlled experimentally [43, 44].
Implementation
iNA is a GUI-based software which at heart is based on the SBML description of stoichiometric reaction networks. With the current release we introduce model manipulation capabilities which are tailored to fit the needs of stochastic modeling, as well as a just-in-time compilation engine that increases the overall execution speed of the analysis.
Model editor capabilities
The software is based on the wide-spread SBML file format [48]. Although in common use, SBML has the shortcoming that it is barely human readable. The present version of iNA supports the compatible format SBML shorthand (SBML-sh) which represents the essential SBML model structure in an easy to read and write description language [49]. Therefore SBML-sh complements the existing SBML functionality by allowing import and export of both formats together with an online SBML-sh editor, see Additional file 4 (c).
Apart from this iNA's GUI also incorporates basic model editing capabilities. Additional file 4 (a) shows the list of reactions which allows to add or edit reactions within a dialog shown in Additional file 4 (b). Within this dialog the propensity of the reaction is either constructed automatically from the statistical formulation of the law of mass action [23, 50, 51] or to be specified by the user.
Performance
The system size expansion of the CME yields a high dimensional system of coupled ODEs of order N^{2} equations for the LNA and N^{3} equations for the IOS analysis. Parameter scans as well as numerical integration of large systems are particularly challenging because of the large number of function evaluations needed to obtain accurate results. iNA's initial release addressed this issue by providing a bytecode interpreter for efficient expression evaluation [23]. The present version improves on this using a just-in-time (JIT) compiler provided by the LLVM infrastructure [30, 52]. This technique provides the means of platform specific code generation for the system size expansion ODEs at runtime mimicking the performance of statically compiled code.
Performance of iNA's time course analysis
method | IOS, LSODA | IOS, Rosenbrock | SSA, single (ens.) |
---|---|---|---|
BCI | 18.2s (18.5s) | 59.5s (59.8s) | 0.04s (0.5h) |
JIT | 0.9s (3.6s) | 13.0s (35.8s) | 0.03s (0.4h) |
Performance of iNA's steady state parameter scan
method | REs | LNA | IOS | SSA per sample (overall) |
---|---|---|---|---|
GiNaC | 0.61s | 1.48s | 39.93s | -- |
BCI | 0.19s | 0.19s | 0.39s | 11ms (16h) |
JIT | 0.17s | 0.17s | 0.28s | 7ms (10h) |
Methods
where ${\widehat{f}}_{j}\left(\overrightarrow{n},\mathrm{\Omega}\right)$ is the probability per unit time and unit volume for the j^{ th } reaction to occur [14] and ${\left({\overrightarrow{\mu}}_{j}\right)}_{i}={r}_{ij}-{s}_{ij}$ is the stoichiometry of species i in the j^{ th } reaction. Note that both approaches are equivalent [51].
Rate equations and the Linear Noise Approximation
The implicit assumption made by ansatz (14) is that in the infinite volume limit the instantaneous concentrations equal the solution of the REs. It can then be shown that the macroscopic rate function of the j^{ th } reaction is obtained from the relation ${f}_{j}\left(\left[\overrightarrow{X}\right]\right)=\underset{\mathrm{\Omega}\to \infty}{\text{lim}}{\widehat{f}}_{j}\left(\left[\overrightarrow{X}\right],\phantom{\rule{2.77695pt}{0ex}}\mathrm{\Omega}\right)$[30]. Since the limit is taken at constant concentration, this implies the large molecule number limit as well.
which are of order Ω^{-1/2} and Ω^{0}, respectively.
It is well known that the means and variances of concentrations predicted by the LNA are exact only for reaction networks involving at most unimolecular reactions. For bimolecular reactions the LNA can be inaccurate if some species are present only in low molecule numbers. This is because for unimolecular reactions the hierarchy of moment equations obtained from the CME is closed, i.e., the n^{ th } moment depends only the (n - 1)^{ th } moment and all lower order moments [54]. The equations for the first moment are given by the REs since the propensities are linear functions of the concentrations. The equations for the second moments are a system of linear equations for the variances and covariances which depends parametrically on the solution of the REs and are equivalent to the LNA result. For bimolecular reactions this is not the case since the hierarchy of moment equations obtained from the CME is not closed, i.e., the equations for the means involve the covariances and similarly the equations for the covariances involve higher moments. A systematic approximation of these equations can be achieved using the system size expansion which yields the REs and the LNA in the limit of large volumes [22]. The latter represent a closed system of equations for the first two moments.
Finite molecule number corrections
Again, the leading order (Ω^{0}) contribution to the mean concentrations, Eq. (19a), is given by the macroscopic REs while the leading order contribution given by the Ω^{-1} term in Eq. (19b) corresponds to the LNA estimate for the variance and the covariance. Including terms to order Ω^{-1} in Eq. (19a) gives the EMRE estimate of the mean concentrations which corrects the solution of the REs [20]. Finally, considering also the Ω^{-2} term in Eq. (19b) gives the IOS (Inverse Omega Squared) estimate of the variance and the covariance. From the form of this higher order contribution it is clear that the variance estimate is centered around the EMRE concentrations and is of higher accuracy than the LNA method.
Again, the leading order contributions are determined by the LNA result, Eq. (17). Note that the factors multiplying Ω^{-1} in Eq. (20) yield the relative corrections to the LNA measures and are denoted by c_{ i } and f_{ i } in the main text. Note also that each of these factors contains a contribution stemming from a change in the variance of concentration fluctuations and another one reflecting the change in the mean of the concentrations. The equations determining the coefficients [ε_{ i }]_{1} and [ε_{ i }ε_{ i }]_{2} needed to compute Eqs. (20) have been derived in Refs. [21] and [22]. As shown therein, the quantities depend only on $\left[\overrightarrow{X}\right]$ and the vector of reaction rate constants $\overrightarrow{k}=\left({k}_{1},{k}_{2},\dots ,{k}_{R}\right)$. Hence the CVs and Fano factors depend parametrically on the reaction rate constants through the solution of the REs together with their initial conditions.
In Additional file 5 we have verified the correctness of iNA's implementation of the IOS for the example of a simple enzyme catalyzed reaction against an analytical result obtained in Ref. [22], Eq. (74) therein. We remark that using the IOS it is also possible to deduce the mean concentrations accurately to order Ω ^{-2} [21] which is superior to the EMRE and hence can be used as an error estimate of the method. However the variance about these concentrations is of Ω^{-3} as can be seen from Eq. (19b) and hence requires to consider higher orders in the system size expansion.
Discussion
In this article we have analyzed the parametric dependence of intrinsic noise in gene regulatory networks by means of average concentrations and variances as well as noise measures such as the CV and the Fano factor. The leading order contributions to the average concentrations and variances of the fluctuations as obtained from the system size expansion are given by the deterministic REs and the LNA respectively. The next to leading order contribution are given by novel terms which we have referred to as the EMRE and the IOS approximations respectively. The relative size of these corrections to the LNA are proportional to the inverse compartment volume and to the size of the bimolecular reaction rate constants. Hence, as we have demonstrated, these higher order terms can be significant for networks involving low copy number of molecules and nonlinear reaction kinetics as is common in gene regulation.
In the case of sRNA regulation we have found that for highly expressed proteins the size of the noise can attain a minimum at intermediate stress levels. This result is in line with the LNA's prediction and may perhaps be advantageous as a mechanism of noise minimization in gene expression. The LNA result predicts that such optimization is indeed possible even for low protein expression levels yet the EMRE/IOS analysis and stochastic simulations show that this is not the case, i.e., the CV in the protein fluctuations increases monotonically with the stress levels.
For the case of gene autoregulation we observed that the LNA reliably describes the CV of mRNA and protein fluctuations when the transcription rate, a first order rate constant, is varied but gives very different results from simulations and the EMRE/IOS analysis when the protein-DNA association constant is varied (a bimolecular rate constant measuring the strength of the feedback loop). In particular in the latter case the LNA predicts a maximum in the CV is achieved as one increases the strength of the negative feedback loop whereas the EMRE/IOS analysis shows that the CV increases monotonically. We also found that the LNA can give considerably misleading results for the amplitude and period of noise-induced oscillations in the expression of a single gene while it becomes increasingly more accurate as the gene copy number is increased.
Hence in summary we have shown by means of these examples that the LNA's predictions regarding the regulation of noise by genetic regulatory networks can be quite different than those obtained by stochastic simulations using the SSA. In contrast the results from the EMRE/IOS methods agree well with those obtained from the SSA for the examples studied here. This is surprising since transcriptional feedback involves transitions between internal states of a gene represented by only one or two copies in a cell. We note that the methods presented can become inaccurate when the noise contribution of the feedback loop dominates. However, our methods enjoy the advantage that they can be computed in a fraction of the time needed to calculate the SSA. Hence the EMRE/IOS analysis tools implemented in iNA 0.4 present a quick means to accurately study the stochastic properties of biochemical reaction networks of intermediate or large size involving many bimolecular reactions.
Availability
Project name: intrinsic Noise Analyzer
Version: 0.4.2
Project home page: http://code.google.com/p/intrinsic-noise-analyzer
Operating systems: platform independent, binaries available for Mac OSX, Linux and Windows
Programming language: C++
License: GNU GPL v2
Declarations
Acknowledgements
Based on "Computation of biochemical pathway fluctuations beyond the linear noise approximation using iNA", by P. Thomas, H. Matuschek and R. Grima which appeared in Bioinformatics and Biomedicine (BIBM), 2012 IEEE International Conference on. ©2012 IEEE [30]. RG gratefully acknowledges support by SULSA (Scottish Universities Life Science Alliance).
Declarations
The publication costs for this article were funded by the corresponding author.
This article has been published as part of BMC Genomics Volume 14 Supplement S4, 2013: Selected articles from the IEEE International Conference on Bioinformatics and Biomedicine 2012: Genomics. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcgenomics/supplements/14/S4.
Authors’ Affiliations
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