This article introduces the finite state projection (FSP) method for use in the stochastic analysis of chemically reacting systems. One can describe the chemical populations of such systems with probability density vectors that evolve according to a set of linear ordinary differential equations known as the chemical master equation (CME). Unlike Monte Carlo methods such as the stochastic simulation algorithm (SSA) or tau leaping, the FSP directly solves or approximates the solution of the CME. If the CME describes a system that has a finite number of distinct population vectors, the FSP method provides an exact analytical solution. When an infinite or extremely large number of population variations is possible, the state space can be truncated, and the FSP method provides a certificate of accuracy for how closely the truncated space approximation matches the true solution. The proposed FSP algorithm systematically increases the projection space in order to meet prespecified tolerance in the total probability density error. For any system in which a sufficiently accurate FSP exists, the FSP algorithm is shown to converge in a finite number of steps. The FSP is utilized to solve two examples taken from the field of systems biology, and comparisons are made between the FSP, the SSA, and tau leaping algorithms. In both examples, the FSP outperforms the SSA in terms of accuracy as well as computational efficiency. Furthermore, due to very small molecular counts in these particular examples, the FSP also performs far more effectively than tau leaping methods.
Phenotypic variation is ubiquitous in biology and is often traceable to underlying genetic and environmental variation. However, even genetically identical cells in identical environments display variable phenotypes. Stochastic gene expression, or gene expression ‘noise’, has been suggested as a major source of this variability, and its physiological consequences have been topics of intense research for the last decade. Several recent studies have measured variability in protein and mRNA levels, and have discovered strong connections between noise and gene regulation mechanisms. When integrated with discrete stochastic models, measurements of cell-to-cell variability provide a sensitive ‘fingerprint’ with which to explore fundamental questions on gene regulation. In this review, we highlight several studies that used gene expression variability to develop a quantitative understanding of the mechanisms and dynamics of gene regulation.
SUMMARY Transcription is a stochastic process occurring mostly in episodic bursts. Although the local chromatin environment is known to influence the bursting behavior on long timescales, the impact of transcription factors (TFs)—especially in rapidly inducible systems—is largely unknown. Using fluorescence in situ hybridization and computational models, we quantified the transcriptional activity of the proto-oncogene c-Fos with single mRNA accuracy at individual endogenous alleles. We showed that, during MAPK induction, the TF concentration modulates the burst frequency of c-Fos, whereas other bursting parameters remain mostly unchanged. By using synthetic TFs with TALE DNA-binding domains, we systematically altered different aspects of these bursts. Specifically, we linked the polymerase initiation frequency to the strength of the transactivation domain and the burst duration to the TF lifetime on the promoter. Our results show how TFs and promoter binding domains collectively act to regulate different bursting parameters, offering a vast, evolutionarily tunable regulatory range for individual genes.
Synthetic genetic programs are built from circuits that integrate sensors and implement temporal control of gene expression1–4. Transcriptional circuits are layered by using promoters to carry the signal between circuits. In other words, the output promoter of one circuit serves as the input promoter to the next. Thus, connecting circuits requires physically connecting a promoter to the next circuit. We show that the sequence at the junction between the input promoter and circuit can affect the input-output response (transfer function) of the circuit5–9. A library of putative sequences that might reduce (or buffer) such context effects, which we refer to as ‘insulator parts’, is screened in Escherichia coli. We find that ribozymes that cleave the 5′ untranslated region (5′-UTR) of the mRNA are effective insulators. They generate quantitatively identical transfer functions, irrespective of the identity of the input promoter. When these insulators are used to join synthetic gene circuits, the behavior of layered circuits can be predicted using a mathematical model. The inclusion of insulators will be critical in reliably permuting circuits to build different programs.
Although much has been done to elucidate the biochemistry of signal transduction and gene regulatory pathways, it remains difficult to understand or predict quantitative responses. We integrate single-cell experiments with stochastic analyses, to identify predictive models of transcriptional dynamics for the osmotic stress response pathway in Saccharomyces cerevisiae. We generate models with varying complexity and use parameter estimation and cross-validation analyses to select the most predictive model. This model yields insight into several dynamical features, including multi-step regulation and switch-like activation for several osmosensitive genes. Furthermore, the model correctly predicts the transcriptional dynamics of cells in response to different environmental and genetic perturbations. Since our approach is general, it should facilitate a predictive understanding for signal-activated transcription of other genes in other pathways or organisms.
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