Diffusion driven oscillations in gene regulatory networks

Biological Rhythms Generated by a Single Activator-Repressor Loop with Inhomogeneity and Diffusion

Common models of circadian rhythms are typically constructed as compartmental reactions of well-mixed biochemicals, incorporating a negative-feedback loop consisting of several intermediate reaction steps essentially required to produce oscillations. Spatial transport of each reactant is often represented as an extra compartmental reaction step. Contrary to this traditional understanding, in this Letter we demonstrate that a single activation-repression biochemical reaction pair is sufficient to generate sustained oscillations if the sites of both reactions are spatially separated and molecular transport is mediated by diffusion. Our proposed scenario represents the simplest configuration in terms of the participating chemical reactions and offers a conceptual basis for understanding biological oscillations and inspiring in vitro assays aimed at constructing minimal clocks.

Noisy Delay Denoises Biochemical Oscillators

Genetic oscillations are generated by delayed transcriptional negative feedback loops, wherein repressor proteins inhibit their own synthesis after a temporal production delay. This delay is distributed because it arises from a sequence of noisy processes, including transcription, translocation, translation, and folding. Because the delay determines repression timing and, therefore, oscillation period, it has been commonly believed that delay noise weakens oscillatory dynamics. Here, we demonstrate that noisy delay can surprisingly denoise genetic oscillators. Specifically, moderate delay noise improves the signal-to-noise ratio and sharpens oscillation peaks, all without impacting period and amplitude. We show that this denoising phenomenon occurs in a variety of well-studied genetic oscillators, and we use queueing theory to uncover the universal mechanisms that produce it.

Theoretical and computational tools to model multistable gene regulatory networks

The last decade has witnessed a surge of theoretical and computational models to describe the dynamics of complex gene regulatory networks, and how these interactions can give rise to multistable and heterogeneous cell populations. As the use of theoretical modeling to describe genetic and biochemical circuits becomes more widespread, theoreticians with mathematical and physical backgrounds routinely apply concepts from statistical physics, non-linear dynamics, and network theory to biological systems. This review aims at providing a clear overview of the most important methodologies applied in the field while highlighting current and future challenges. It also includes hands-on tutorials to solve and simulate some of the archetypical biological system models used in the field. Furthermore, we provide concrete examples from the existing literature for theoreticians that wish to explore this fast-developing field. Whenever possible, we highlight the similarities and differences between biochemical and regulatory networks and 'classical' systems typically studied in non-equilibrium statistical and quantum mechanics.

The ups and downs of biological oscillators: a comparison of time-delayed negative feedback mechanisms

Many biochemical oscillators are driven by the periodic rise and fall of protein concentrations or activities. A negative feedback loop underlies such oscillations. The feedback can act on different parts of the biochemical network. Here, we mathematically compare time-delay models where the feedback affects production and degradation. We show a mathematical connection between the linear stability of the two models, and derive how both mechanisms impose different constraints on the production and degradation rates that allow oscillations. We show how oscillations are affected by the inclusion of a distributed delay, of double regulation (acting on production and degradation) and of enzymatic degradation.

Mathematical Modelling of p53 Signalling during DNA Damage Response: A Survey

No gene has garnered more interest than p53 since its discovery over 40 years ago. In the last two decades, thanks to seminal work from Uri Alon and Ghalit Lahav, p53 has defined a truly synergistic topic in the field of mathematical biology, with a rich body of research connecting mathematic endeavour with experimental design and data. In this review we survey and distill the extensive literature of mathematical models of p53. Specifically, we focus on models which seek to reproduce the oscillatory dynamics of p53 in response to DNA damage. We review the standard modelling approaches used in the field categorising them into three types: time delay models, spatial models and coupled negative-positive feedback models, providing sample model equations and simulation results which show clear oscillatory dynamics. We discuss the interplay between mathematics and biology and show how one informs the other; the deep connections between the two disciplines has helped to develop our understanding of this complex gene and paint a picture of its dynamical response. Although yet more is to be elucidated, we offer the current state-of-the-art understanding of p53 response to DNA damage.

Stability and Oscillation Analysis of a Gene Regulatory Network With Multiple Time Delays and Diffusion Rate

In genetic regulatory networks (GRNs), the diffusion rate of mRNA and protein play a key role in regulatory mechanisms of gene expression, especially in translation and transcription. However, the influence of diffusion rate on oscillatory gene expression is not well understood. In this paper, by considering the diffusion rate of mRNA and protein, a novel GRN is proposed. Then, two basic problems of such network, i.e. stability and oscillation, are solved in detail. Moreover, the properties of oscillation are also investigated. it is found that the total biochemistry reaction time can affect the stability of the positive equilibrium and give rise to the oscillation. The diffusion rate of mRNA and proteins have a major impact on the oscillation properties. Finally, two examples not only verify the theoretical results, but also show that a slight diffusion rate increasing may lead to huge change in oscillatory gene expressions.

Endogenous miRNA sponges mediate the generation of oscillatory dynamics for a non-coding RNA network

Oscillations are crucial to the normal function of living organisms, across a wide variety of biological processes. In eukaryotes, oscillatory dynamics are thought to arise from interactions at the protein and RNA levels; however, the role of non-coding RNA in regulating these dynamics remains understudied. In this work, we show how non-coding RNA acting as microRNA (miRNA) sponges in a conserved miRNA - transcription factor feedback motif, can give rise to oscillatory behaviour, and how to test for this experimentally. Control of these non-coding RNA can dynamically create oscillations or stability, and we show how this behaviour predisposes to oscillations in the stochastic limit. These results, supported by emerging evidence for the role of miRNA sponges in development, point towards key roles of different species of miRNA sponges, such as circular RNA, potentially in the maintenance of yet unexplained oscillatory behaviour. These results help to provide a paradigm for understanding functional differences between the many redundant, but distinct RNA species thought to act as miRNA sponges in nature, such as long non-coding RNA, pseudogenes, competing mRNA, circular RNA, and3' UTRs.

Bridging intracellular scales by mechanistic computational models

The impact of intracellular spatial organization beyond classical compartments on processes such as cell signaling is increasingly recognized. A quantitative, mechanistic understanding of cellular systems therefore needs to account for different scales in at least three coordinates: time, molecular abundances, and space. Mechanistic mathematical models may span all these scales, but corresponding multi-scale models need to resolve mechanistic details on small scales while maintaining computational tractability for larger ones. This typically results in models that combine different levels of description: from a microscopic representation of chemical reactions up to continuum dynamics in space and time. We highlight recent progress in bridging these model classes and outline current challenges in multi-scale models such as active transport and dynamic geometries.

Computational Modelling of Cancer Development and Growth: Modelling at Multiple Scales and Multiscale Modelling

In this paper, we present two mathematical models related to different aspects and scales of cancer growth. The first model is a stochastic spatiotemporal model of both a synthetic gene regulatory network (the example of a three-gene repressilator is given) and an actual gene regulatory network, the NF-[Formula: see text]B pathway. The second model is a force-based individual-based model of the development of a solid avascular tumour with specific application to tumour cords, i.e. a mass of cancer cells growing around a central blood vessel. In each case, we compare our computational simulation results with experimental data. In the final discussion section, we outline how to take the work forward through the development of a multiscale model focussed at the cell level. This would incorporate key intracellular signalling pathways associated with cancer within each cell (e.g. p53-Mdm2, NF-[Formula: see text]B) and through the use of high-performance computing be capable of simulating up to [Formula: see text] cells, i.e. the tissue scale. In this way, mathematical models at multiple scales would be combined to formulate a multiscale computational model.