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Many cellular components are present in such low numbers that individual stochastic production and degradation events lead to significant fluctuations in molecular abundances. Although feedback control can, in principle, suppress such…
Ring oscillators are biochemical circuits consisting of a ring of interactions capable of sustained oscillations. The non-linear interactions between genes hinder the analytical insight into their function, usually requiring computational…
We introduce a stochastic model of coupled genetic oscillators in which chains of chemical events involved in gene regulation and expression are represented as sequences of Poisson processes. We characterize steady states by their…
Gene expression is a biochemical process, where stochastic binding and un-binding events naturally generate fluctuations and cell-to-cell variability in gene dynamics. These fluctuations typically have destructive consequences for proper…
Modelling gene regulatory networks not only requires a thorough understanding of the biological system depicted but also the ability to accurately represent this system from a mathematical perspective. Throughout this chapter, we aim to…
Understanding the interplay among processes that occur over different timescales is a challenging issue in the physics of systems regulation. In gene regulation, the timescales for changes in chromatin states can differ from those for…
The molecular evolution in a gene regulatory network is classically modeled by Markov jump processes. However, the direct simulation of such models is extremely time consuming. Indeed, even the simplest Markovian model, such as the…
The modeling of conventional (deterministic) electronic circuits - ones consisting of transistors, resistors, capacitors, inductors, and other traditional electronic components - is a well-established subject. The cycle-to-cycle variability…
Cells may control fluctuations in protein levels by means of negative autoregulation, where transcription factors bind DNA sites to repress their own production. Theoretical studies have assumed a single binding site for the repressor,…
Stochastic reaction-diffusion models can be analytically studied on complex networks using the linear noise approximation. This is illustrated through the use of a specific stochastic model, which displays traveling waves in its…
Many processes in nature such as conformal changes in biomolecules and clusters of interacting particles, genetic switches, mechanical or electromechanical oscillators with added noise, and many others are modeled using stochastic…
There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a…
In order to express specific genes at the right time, the transcription of genes is regulated by the presence and absence of transcription factor molecules. With transcription factor concentrations undergoing constant changes, gene…
We discuss recent developments in the modeling of negative autoregulated genetic networks. In particular, we consider the temporal evolution of the population of mRNA and proteins in simple networks using rate equations. In the limit of low…
In the simplest view of transcriptional regulation, the expression of a gene is turned on or off by changes in the concentration of a transcription factor (TF). We use recent data on noise levels in gene expression to show that it should be…
It is well established that gene expression can be modeled as a Markovian stochastic process and hence proper observables might be subjected to large fluctuations and rare events. Since dynamics is often more than statics, one can work with…
The phage $\lambda$ infection of an \textit{E. coli} cell has become a paradigm for understanding the molecular processes involved in gene expression and cell signaling. This system provides an example of a genetic switch, as cells with…
Stochastic reaction network models are widely utilized in biology and chemistry to describe the probabilistic dynamics of biochemical systems in general, and gene interaction networks in particular. Most often, statistical analysis and…
Within systems biology there is an increasing interest in the stochastic behavior of genetic and biochemical reaction networks. An appropriate stochastic description is provided by the chemical master equation, which represents a continuous…
Biochemical reactions are fundamentally noisy at a molecular scale. This limits the precision of reaction networks, but also allows fluctuation measurements which may reveal the structure and dynamics of the underlying biochemical network.…