Significance: The random nature of gene expression is well established experimentally. Mathematical modeling provides a means of understanding the factors leading to the observed stochasticity. In this article, we extend the classical two-state model of stochastic mRNA dynamics to include a considerable number of salient features of single-cell biology, such as cell division, replication, mRNA maturation, dosage compensation, and growth-dependent transcription. By means of biologically relevant approximations, we obtain expressions for the time-dependent distributions of mRNA and protein numbers. These provide insight into how fluctuations are modified and controlled by complex intracellular processes.

The presence of protein–DNA binding reactions often leads to analytically intractable models of stochastic gene expression. Here we present the linear-mapping approximation that maps systems with protein–promoter interactions onto approximately equivalent systems with no binding reactions. This is achieved by the marriage of conditional mean-field approximation and the Magnus expansion, leading to analytic or semi-analytic expressions for the approximate time-dependent and steady-state protein number distributions. Stochastic simulations verify the method’s accuracy in capturing the changes in the protein number distributions with time for a wide variety of networks displaying auto- and mutual-regulation of gene expression and independently of the ratios of the timescales governing the dynamics. The method is also used to study the first-passage time distribution of promoter switching, the sensitivity of the size of protein number fluctuations to parameter perturbation and the stochastic bifurcation diagram characterizing the onset of multimodality in protein number distributions.

Non-Markovian models of stochastic biochemical kinetics often incorporate explicit time delays to effectively model large numbers of intermediate biochemical processes. Analysis and simulation of these models, as well as the inference of their parameters from data, are fraught with difficulties because the dynamics depends on the system’s history. Here we use an artificial neural network to approximate the time-dependent distributions of non-Markovian models by the solutions of much simpler time-inhomogeneous Markovian models; the approximation does not increase the dimensionality of the model and simultaneously leads to inference of the kinetic parameters. The training of the neural network uses a relatively small set of noisy measurements generated by experimental data or stochastic simulations of the non-Markovian model. We show using a variety of models, where the delays stem from transcriptional processes and feedback control, that the Markovian models learnt by the neural network accurately reflect the stochastic dynamics across parameter space.

Iterative learning control (ILC) has been successfully applied to numerous batch processes over the past decades. Monotonic convergence of tracking error is a desired characteristic that attracts much attention in academia. Many factors arising in industrial practice, such as strong nonlinearity and parameter uncertainty, have challenged the most existing monotonically convergent ILC approaches. This motivates the development of the nonlinear monotonically convergent ILC (NMC-ILC) in this paper. The proposed NMC-ILC is an optimization-based control strategy, in which the original nonlinear process is linearly approximated to reduce the complexity of the optimization problem accompanied. An upper bound of the tracking error is derived by considering the effects of model mismatches that consist of two components: structural mismatch induced by the linearization and parametric mismatch arising from the indetermination of process parameters. The NMC-ILC is then devised by minimizing this upper bound. Numerical experiments are conducted on an ILC-testing benchmark and the velocity control in an injection molding process.

Transcriptional rates are often estimated by fitting the distribution of mature mRNA numbers measured using smFISH (single molecule fluorescence in situ hybridization) with the distribution predicted by the telegraph model of gene expression, which defines two promoter states of activity and inactivity. However, fluctuations in mature mRNA numbers are strongly affected by processes downstream of transcription. In addition, the telegraph model assumes one gene copy but in experiments, cells may have two gene copies as cells replicate their genome during the cell cycle. While it is often presumed that post-transcriptional noise and gene copy number variation affect transcriptional parameter estimation, the size of the error introduced remains unclear. To address this issue, here we measure both mature and nascent mRNA distributions of GAL10 in yeast cells using smFISH and classify each cell according to its cell cycle phase. We infer transcriptional parameters from mature and nascent mRNA distributions, with and without accounting for cell cycle phase and compare the results to live-cell transcription measurements of the same gene. We find that: (i) correcting for cell cycle dynamics decreases the promoter switching rates and the initiation rate, and increases the fraction of time spent in the active state, as well as the burst size; (ii) additional correction for post-transcriptional noise leads to further increases in the burst size and to a large reduction in the errors in parameter estimation. Furthermore, we outline how to correctly adjust for measurement noise in smFISH due to uncertainty in transcription site localisation when introns cannot be labelled. Simulations with parameters estimated from nascent smFISH data, which is corrected for cell cycle phases and measurement noise, leads to autocorrelation functions that agree with those obtained from live-cell imaging.