Intrinsic and extrinsic noise are distinguishable in a synthesis – export – degradation model of mRNA production

Intrinsic and extrinsic noise sources in gene expression, originating respectively from transcriptional stochasticity and from differences between cells, complicate the determination of transcriptional models. In particularly degenerate cases, the two noise sources are altogether impossible to distinguish. However, the incorporation of downstream processing, such as the mRNA splicing and export implicated in gene expression buffering, recovers the ability to identify the relevant source of noise. We report analytical copy-number distributions, discuss the noise sources’ qualitative effects on lower moments, and provide simulation routines for both models.

Population growth affects intrinsic and extrinsic noise in gene expression

Clonal cells of exponentially growing populations vary substantially from cell to cell. The main drivers of this heterogeneity are the population dynamics and stochasticity in the intracellular reactions, which are commonly studied separately. Here we develop an agent-based framework that allows tracking of the biochemical dynamics in every single cell of a growing population that accounts for both of these factors. Apart from the common intrinsic variability of the biochemical reactions, the framework also predicts extrinsic noise arising from fluctuations in the histories of cells without the need to introduce fluctuating rate constants. Instead, these extrinsic fluctuations are explained by cell cycle fluctuations and differences in cell age, which are ubiquitously observed in growing populations. We give explicit formulas to quantify mean molecule numbers, intrinsic and extrinsic noise statistics as measured in two-colour experiments. We find that these statistics may differ significantly depending on the experimental setup used to observe the cells. We illustrate this fact using (i) averages over an isolated cell lineage tracked over many generations as observed in the mother machine, (ii) snapshots of a growing population with known cell ages as recorded in time-lapse microscopy, and (iii) snapshots of unknown cell ages as measured from static images. Our integrated approach applies to arbitrary biochemical networks and generation time distributions. By employing models of stochastic gene expression and feedback regulation, we elucidate that isolated lineages, as compared to snapshot data, can significantly overestimate the mean number of molecules, overestimate extrinsic noise but underestimate intrinsic noise and have qualitatively different sensitivities to cell cycle fluctuations.

Comparing the Effects of Intrinsic and Extrinsic Noise on the Vicsek Model in Three Dimensions

A group of simple individuals may show ordered, complex behavior through local interactions. This phenomenon is called collective behavior, which has been observed in a vast variety of natural systems such as fish schools or bird flocks. The Vicsek model is a well-established mathematical model to study collective behavior through interaction of individuals with their neighbors in the presence of noise. How noise is modeled can impact the collective behavior of the group. Extrinsic noise captures uncertainty imposed on individuals, such as noise in measurements, while intrinsic noise models uncertainty inherent to individuals, akin to free will. In this paper, the effects of intrinsic and extrinsic noise on characteristics of the transition between order and disorder in the Vicsek model in three dimensions are studied through numerical simulation.

Inferring intrinsic and extrinsic noise from a dual fluorescent reporter

Dual fluorescent reporter constructs, which measure gene expression from two identical promoters within the same cell, allow total gene expression noise to be decomposed into an extrinsic component, roughly associated with cell-to-cell fluctuations in cellular component concentrations, and intrinsic noise, roughly associated with inherent stochasticity of the biochemical reactions involved in gene expression [1]. A recent paper by Fu and Pachter presented frequentist statistical estimators for intrinsic and extrinsic noise using data from dual reporters [2]. For comparison, I here present results of a Bayesian analysis of this problem. I show that the orthodox estimators suffer from pathologies such as predicting negative values for a manifestly non-negative quantity, i.e. variance, and show that the Bayesian estimators do not suffer from such pathologies. In addition, I show that the Bayesian analysis automatically identifies that optimal estimates of intrinsic and extrinsic noise depend on a subtle combination of two statistics of the data, allowing for accuracies that are up to twice the accuracy of the orthodox estimators in some parameter regimes.I hope up this little worked out example contrasting orthodox statistical analysis based on ad hoc estimators with estimators resulting from a Bayesian analysis, will be educational for others in the field. I distribute a Mathematica Notebook with this paper that allows users to easily reproduce all results and figures of the paper.

Estimating intrinsic and extrinsic noise from single-cell gene expression measurements

Gene expression is stochastic and displays variation (“noise”) both within and between cells. Intracellular (intrinsic) variance can be distinguished from extracellular (extrinsic) variance by applying the law of total variance to data from two-reporter assays that probe expression of identically regulated gene pairs in single cells. We examine established formulas [Elowitz, M. B., A. J. Levine, E. D. Siggia and P. S. Swain (2002): “Stochastic gene expression in a single cell,” Science, 297, 1183–1186.] for the estimation of intrinsic and extrinsic noise and provide interpretations of them in terms of a hierarchical model. This allows us to derive alternative estimators that minimize bias or mean squared error. We provide a geometric interpretation of these results that clarifies the interpretation in [Elowitz, M. B., A. J. Levine, E. D. Siggia and P. S. Swain (2002): “Stochastic gene expression in a single cell,” Science, 297, 1183–1186.]. We also demonstrate through simulation and re-analysis of published data that the distribution assumptions underlying the hierarchical model have to be satisfied for the estimators to produce sensible results, which highlights the importance of normalization.