Simulation of mRNA diffusion in the nuclear environment

Incorporating spatial diffusion into models of bursty stochastic transcription

The dynamics of gene expression are stochastic and spatial at the molecular scale, with messenger RNA (mRNA) transcribed at specific nuclear locations and then transported to the nuclear boundary for export. Consequently, the spatial distributions of these molecules encode their underlying dynamics. While mechanistic models for molecular counts have revealed numerous insights into gene expression, they have largely neglected now-available subcellular spatial resolution down to individual molecules. Owing to the technical challenges inherent in spatial stochastic processes, tools for studying these subcellular spatial patterns are still limited. Here, we introduce a spatial stochastic model of nuclear mRNA with two-state (telegraph) transcriptional dynamics. Observations of the model can be concisely described as following a spatial Cox process driven by a stochastically switching partial differential equation. We derive analytical solutions for spatial and demographic moments and validate them with simulations. We show that the distribution of mRNA counts can be accurately approximated by a Poisson-beta distribution with tractable parameters, even with complex spatial dynamics. This observation allows for efficient parameter inference demonstrated on synthetic data. Altogether, our work adds progress towards a new frontier of subcellular spatial resolution in inferring the dynamics of gene expression from static snapshot data.

Analytic delay distributions for a family of gene transcription models

Models intended to describe the time evolution of a gene network must somehow include transcription, the DNA-templated synthesis of RNA, and translation, the RNA-templated synthesis of proteins. In eukaryotes, the DNA template for transcription can be very long, often consisting of tens of thousands of nucleotides, and lengthy pauses may punctuate this process. Accordingly, transcription can last for many minutes, in some cases hours. There is a long history of introducing delays in gene expression models to take the transcription and translation times into account. Here we study a family of detailed transcription models that includes initiation, elongation, and termination reactions. We establish a framework for computing the distribution of transcription times, and work out these distributions for some typical cases. For elongation, a fixed delay is a good model provided elongation is fast compared to initiation and termination, and there are no sites where long pauses occur. The initiation and termination phases of the model then generate a nontrivial delay distribution, and elongation shifts this distribution by an amount corresponding to the elongation delay. When initiation and termination are relatively fast, the distribution of elongation times can be approximated by a Gaussian. A convolution of this Gaussian with the initiation and termination time distributions gives another analytic approximation to the transcription time distribution. If there are long pauses during elongation, because of the modularity of the family of models considered, the elongation phase can be partitioned into reactions generating a simple delay (elongation through regions where there are no long pauses), and reactions whose distribution of waiting times must be considered explicitly (initiation, termination, and motion through regions where long pauses are likely). In these cases, the distribution of transcription times again involves a nontrivial part and a shift due to fast elongation processes.

Labelling and imaging of single endogenous messenger RNA particles in vivo

RNA molecules carry out widely diverse functions in numerous different physiological processes in living cells. The RNA life cycle from transcription, through the processing of nascent RNA, to the regulatory function of non-coding RNA and cytoplasmic translation of messenger RNA has been studied extensively using biochemical and molecular biology techniques. In this Commentary, we highlight how single molecule imaging and particle tracking can yield further insight into the dynamics of RNA particles in living cells. In the past few years, a variety of bright and photo-stable labelling techniques have been developed to generate sufficient contrast for imaging of single endogenous RNAs in vivo. New imaging modalities allow determination of not only lateral but also axial positions with high precision within the cellular context, and across a wide range of specimen from yeast and bacteria to cultured cells, and even multicellular organisms or live animals. A whole range of methods to locate and track single particles, and to analyze trajectory data are available to yield detailed information about the kinetics of all parts of the RNA life cycle. Although the concepts presented are applicable to all types of RNA, we showcase here the wealth of information gained from in vivo imaging of single particles by discussing studies investigating dynamics of intranuclear trafficking, nuclear pore transport and cytoplasmic transport of endogenous messenger RNA.

Accounting for randomness in measurement and sampling in studying cancer cell population dynamics

Knowing the expected temporal evolution of the proportion of different cell types in sample tissues gives an indication about the progression of the disease and its possible response to drugs. Such systems have been modelled using Markov processes. We here consider an experimentally realistic scenario in which transition probabilities are estimated from noisy cell population size measurements. Using aggregated data of FACS measurements, we develop MMSE and ML estimators and formulate two problems to find the minimum number of required samples and measurements to guarantee the accuracy of predicted population sizes. Our numerical results show that the convergence mechanism of transition probabilities and steady states differ widely from the real values if one uses the standard deterministic approach for noisy measurements. This provides support for our argument that for the analysis of FACS data one should consider the observed state as a random variable. The second problem we address is about the consequences of estimating the probability of a cell being in a particular state from measurements of small population of cells. We show how the uncertainty arising from small sample sizes can be captured by a distribution for the state probability.

The role of dimerisation and nuclear transport in the Hes1 gene regulatory network

Hes1 is a member of the family of basic helix-loop-helix transcription factors and the Hes1 gene regulatory network (GRN) may be described as the canonical example of transcriptional control in eukaryotic cells, since it involves only the Hes1 protein and its own mRNA. Recently, the Hes1 protein has been established as an excellent target for an anti-cancer drug treatment, with the design of a small molecule Hes1 dimerisation inhibitor representing a promising if challenging approach to therapy. In this paper, we extend a previous spatial stochastic model of the Hes1 GRN to include nuclear transport and dimerisation of Hes1 monomers. Initially, we assume that dimerisation occurs only in the cytoplasm, with only dimers being imported into the nucleus. Stochastic simulations of this novel model using the URDME software show that oscillatory dynamics in agreement with experimental studies are retained. Furthermore, we find that our model is robust to changes in the nuclear transport and dimerisation parameters. However, since the precise dynamics of the nuclear import of Hes1 and the localisation of the dimerisation reaction are not known, we consider a second modelling scenario in which we allow for both Hes1 monomers and dimers to be imported into the nucleus, and we allow dimerisation of Hes1 to occur everywhere in the cell. Once again, computational solutions of this second model produce oscillatory dynamics in agreement with experimental studies. We also explore sensitivity of the numerical solutions to nuclear transport and dimerisation parameters. Finally, we compare and contrast the two different modelling scenarios using numerical experiments that simulate dimer disruption, and suggest a biological experiment that could distinguish which model more faithfully captures the Hes1 GRN.

The dynamic pathway of nuclear RNA in eukaryotes

The passage of mRNA molecules from the site of synthesis, through the nucleoplasm and the nuclear pore, en route to the cytoplasm, might appear straightforward. Nonetheless, several decades of detailed examination of this pathway, from high resolution electron microscopy in fixed specimens, through the development of immuno-detection techniques and fluorescence toolkits, to the current era of live-cell imaging, show this to be an eventful journey. In addition to mRNAs, several species of noncoding RNAs travel and function in the nucleus, some being retained within throughout their lifetime. This review will highlight the nucleoplasmic paths taken by mRNAs and noncoding RNAs in eukaryotic cells with special focus on live-cell data and in concurrence with the biophysical nature of the nucleus.