# Pathway dynamics can delineate the sources of transcriptional noise in gene expression

## Summary (2 min read)

### Introduction

- Sometimes it can afford evolutionary advantages, for example, in the context of bet-hedging strategies.
- Such sources of variability contribute extrinsic noise, and reflect the variation in gene expression and transcription activity across the cell population.
- Here the authors develop a widely applicable generalisation (and simplification) of the original dual-reporter approach [4].
- The Telegraph Model The Telegraph model was first introduced in [21], and since then has been widely employed in the literature to model bursty gene expression in eukaryotic cells [22–25].
- Throughout, the authors will refer to the probability mass function p̃T (n; θ) as the Telegraph distribution with parameters θ.

### Identifiability Considerations

- Decoupling the effects of extrinsic noise from experimental measurements has been notoriously challenging.
- In Fig. 2A (middle panel), the authors compare the representation obtained in (4) with the corresponding fixed-parameter negative binomial distribution for two different sets of parameters.
- Thus, the distribution of any instantaneously bursty system with mean burst intensity b can be obtained from one with greater burst frequency, by varying the mean burst intensity θ according to a shifted beta prime distribution.
- Noise on the transcription rate will invariably produce copy number data that is suggestive of a more bursty model.
- The truncated normal distribution is not chosen on the basis of biological relevance, but rather to demonstrate that even a symmetric noise distribution (except for truncation at 0) produces qualitatively similar results to the distributions used in the precise non-identifiability results.

### Resolving Non-identifiability

- The results of the previous section show that additional information, beyond the observed copy number distribution, is required to constrain the space of possible dynamics that could give rise to the same distribution.
- The decomposition applies to dynamic noise [39], and generalises to higher moments in [40].
- The dual-reporter method requires distinguishable measurements of transcripts or proteins from two independent and identically distributed reporter genes integrated into the same cell.
- As the authors show in the next section, there are many situations where the random variable E(X;Z) is precisely the common part of E(Y ;Z) and E(X;Z) (i.e., h(Z′) = E(X;Z)), and the normalised intrinsic contribution to the covariance is either zero or negligible.
- In these cases, the normalised covariance of X and Y will identify precisely the extrinsic noise contribution η2ext to the total noise η2X .

### The Pathway-Reporter Method

- The authors show that for some reporters X and Y belonging to the same biochemical pathway, the covariance of X and Y continues to identify the extrinsic, and subsequently intrinsic, noise contributions to the total noise.
- Thus, again the NDP holds, and the normalised covariance of E(XN ;Z) and E(XP ;Z) will identify the noise on the transcriptional component KN λ(λ+µ) .
- The time series of copy numbers for each of nascent mRNA, mature mRNA and protein broadly follow each other, each with delay from its predecessor (Fig. 4B).
- The parameter KN is given the noise distribution Beta(3, 6), which has a slightly higher coefficient of variation η2 = 0.2.
- The results for the nascent mRNA–protein reporters, case (c), given in Table 4 show comparable performance to dual reporters, with only modest overshoot; even in the worst performing case of λ = 0.5, µ = 1 the result of the pathway reporters is within one standard deviation, in a very tight distribution.

### Discussion

- The ability to extract transcriptional dynamics from measured distributions of mRNA copy numbers is limited.
- It is therefore necessary to collect further information, beyond measurements of the transcripts alone, in order to constrain the number of possible theoretical models of gene activity that could represent the system.
- The authors have developed a theoretical framework for estimating levels of extrinsic noise, which can assist in resolving the non-identifiability problems.
- The dual reporter method of Swain et al. [4] already provides one such approach; but it is experimentally challenging to set up in many systems, and requires strictly identical and independent pairs of gene reporters.
- The authors have exploited this to yield reliable estimates of noise strength, which they are confident will assist in setting better practices for model fitting and inference in the analysis of single-cell data.

### Acknowledgments

- The authors gratefully acknowledge Rowan D. Brackston helpful discussions in the early stages of this research.
- The authors also wish to thank Arjun Raj for providing valuable feedback on this work.
- L.H. and M.P.H.S. were supported by the University of Melbourne DVCR fund.

### Data Availability

- Simulations of the models used in the paper are performed using Gillespie’s Stochastic Simulation Algorithm (SSA) implemented in Julia.
- The simulation code is available in the GitHub repository https://github.com/leham/PathwayReporters.
- The data used in the paper are provided in the supplementary datasets.

### Author Contributions

- L.H. and M.J. conceptualised the research, with support from M.P.H.S.
- All authors provided critical feedback and helped shape the research.

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...(13) Since mRNA tends to be less stable than protein, we have that δp < 1, and often δp 1 (45, 46)....

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