Models of genetic control by repression with time delays and spatial effects

Dynamic processes of fate decision in inducible bistable systems

The process of biological fate decision regulated by gene regulatory networks involves numerous complex dynamical interactions among many components. Mathematical modeling typically employed ordinary differential equations and steady-state analysis, which has yielded valuable quantitative insights. However, stable states predicted by theoretical models often fail to capture transient or metastable phenomena that occur during most observation periods in experimental or real biological systems. We attribute this discrepancy to the omission of dynamic processes of various complex interactions. Here, we demonstrate the influence of delays in gene regulatory steps and the timescales of the external induction on the dynamic processes of the fate decision in inducible bistable systems. We propose that steady-state parameters determine the landscape of fate decision. However, during the dynamic evolution along the landscape, the unequal delays of biochemical interactions as well as the timescale of external induction cause deviations in the differentiation trajectories, leading to the formation of new transient distributions that persist long term. Our findings emphasize the importance of considering dynamic processes in fate decision instead of relying solely on steady-state analysis. We provide insights into the interpretation of experimental phenomena and offer valuable guidance for future efforts in dynamical modeling and synthetic biology design.

The dynamic-process characterization and prediction of synthetic gene circuits by dynamic delay model

Differential equation models are widely used to describe genetic regulations, predict multicomponent regulatory circuits, and provide quantitative insights. However, it is still challenging to quantitatively link the dynamic behaviors with measured parameters in synthetic circuits. Here, we propose a dynamic delay model (DDM) which includes two simple parts: the dynamic determining part and the doses-related steady-state-determining part. The dynamic determining part is usually supposed as the delay time but without a clear formula. For the first time, we give the detail formula of the dynamic determining function and provide a method for measuring all parameters of synthetic elements (include 8 activators and 5 repressors) by microfluidic system. Three synthetic circuits were built to show that the DDM can notably improve the prediction accuracy and can be used in various synthetic biology applications.

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.

Operon dynamics with state dependent transcription and/or translation delays

Transcription and translation retrieve and operationalize gene encoded information in cells. These processes are not instantaneous and incur significant delays. In this paper we study Goodwin models of both inducible and repressible operons with state-dependent delays. The paper provides justification and derivation of the model, detailed analysis of the appropriate setting of the corresponding dynamical system, and extensive numerical analysis of its dynamics. Comparison with constant delay models shows significant differences in dynamics that include existence of stable periodic orbits in inducible systems and multistability in repressible systems. A combination of parameter space exploration, numerics, analysis of steady state linearization and bifurcation theory indicates the likely presence of Shilnikov-type homoclinic bifurcations in the repressible operon model.

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.

PhysiBoSS: a multi-scale agent-based modelling framework integrating physical dimension and cell signalling

MOTIVATION

Due to the complexity and heterogeneity of multicellular biological systems, mathematical models that take into account cell signalling, cell population behaviour and the extracellular environment are particularly helpful. We present PhysiBoSS, an open source software which combines intracellular signalling using Boolean modelling (MaBoSS) and multicellular behaviour using agent-based modelling (PhysiCell).

RESULTS

PhysiBoSS provides a flexible and computationally efficient framework to explore the effect of environmental and genetic alterations of individual cells at the population level, bridging the critical gap from single-cell genotype to single-cell phenotype and emergent multicellular behaviour. PhysiBoSS thus becomes very useful when studying heterogeneous population response to treatment, mutation effects, different modes of invasion or isomorphic morphogenesis events. To concretely illustrate a potential use of PhysiBoSS, we studied heterogeneous cell fate decisions in response to TNF treatment. We explored the effect of different treatments and the behaviour of several resistant mutants. We highlighted the importance of spatial information on the population dynamics by considering the effect of competition for resources like oxygen.

AVAILABILITY AND IMPLEMENTATION

PhysiBoSS is freely available on GitHub (https://github.com/sysbio-curie/PhysiBoSS), with a Docker image (https://hub.docker.com/r/gletort/physiboss/). It is distributed as open source under the BSD 3-clause license.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

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.

Stability and Hopf bifurcation analysis for a Lac operon model with nonlinear degradation rate and time delay

In this paper, we construct a discrete time delay Lac operon model with nonlinear degradation rate for mRNA, resulting from the interaction among several identical mRNA pieces. By taking a discrete time delay as bifurcation parameter, we investigate the nonlinear dynamical behaviour arising from the model, using mathematical tools such as stability and bifurcation theory. Firstly, we discuss the existence and uniqueness of the equilibrium for this system and investigate the effect of discrete delay on its dynamical behaviour. Absence or limited delay causes the system to have a stable equilibrium, which changes into a Hopf point producing oscillations if time delay is increased. These sustained oscillation are shown to be present only if the nonlinear degradation rate for mRNA satisfies specific conditions. The direction of the Hopf bifurcation giving rise to such oscillations is also determined, via the use of the so-called multiple time scales technique. Finally, numerical simulations are shown to validate and expand the theoretical analysis. Overall, our findings suggest that the degree of nonlinearity of the model can be used as a control parameter for the stabilisation of the system.

Temporal precision of regulated gene expression

Important cellular processes such as migration, differentiation, and development often rely on precise timing. Yet, the molecular machinery that regulates timing is inherently noisy. How do cells achieve precise timing with noisy components? We investigate this question using a first-passage-time approach, for an event triggered by a molecule that crosses an abundance threshold and that is regulated by either an accumulating activator or a diminishing repressor. We find that either activation or repression outperforms an unregulated strategy. The optimal regulation corresponds to a nonlinear increase in the amount of the target molecule over time, arises from a tradeoff between minimizing the timing noise of the regulator and that of the target molecule itself, and is robust to additional effects such as bursts and cell division. Our results are in quantitative agreement with the nonlinear increase and low noise of mig-1 gene expression in migrating neuroblast cells during Caenorhabditis elegans development. These findings suggest that dynamic regulation may be a simple and powerful strategy for precise cellular timing.

Cells control important processes with precise timing, even though their underlying molecular machinery is inherently imprecise. In the case of Caenorhabditis elegans development, migrating neuroblast cells produce a molecule until a certain abundance is reached, at which time the cells stop moving. Precise timing of this event is critical to C. elegans development, and here we investigate how it can be achieved. Specifically, we investigate regulation of the molecule production by either an accumulating activator or a diminishing repressor. Our results are consistent with the nonlinear increase and low noise of gene expression observed in the C. elegans cells.

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.

Spatio-temporal models of synthetic genetic oscillators

Signal transduction pathways play a major role in many important aspects of cellular function e.g. cell division, apoptosis. One important class of signal transduction pathways is gene regulatory networks (GRNs). In many GRNs, proteins bind to gene sites in the nucleus thereby altering the transcription rate. Such proteins are known as transcription factors. If the binding reduces the transcription rate there is a negative feedback leading to oscillatory behaviour in mRNA and protein levels, both spatially (e.g. by observing fluorescently labelled molecules in single cells) and temporally (e.g. by observing protein/mRNA levels over time). Recent computational modelling has demonstrated that spatial movement of the molecules is a vital component of GRNs and may cause the oscillations. These numerical findings have subsequently been proved rigorously i.e. the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. In this paper we first present a model of the canonical GRN (the Hes1 protein) and show the effect of varying the spatial location of gene and protein production sites on the oscillations. We then extend the approach to examine spatio-temporal models of synthetic gene regulatory networks e.g. n-gene repressilators and activator-repressor systems.

How time delay and network design shape response patterns in biochemical negative feedback systems

BACKGROUND

Negative feedback in combination with time delay can bring about both sustained oscillations and adaptive behaviour in cellular networks. Here, we study which design features of systems with delayed negative feedback shape characteristic response patterns with special emphasis on the role of time delay. To this end, we analyse generic two-dimensional delay differential equations describing the dynamics of biochemical signal-response networks.

RESULTS

We investigate the influence of several design features on the stability of the model equilibrium, i.e., presence of auto-inhibition and/or mass conservation and the kind and/or strength of the delayed negative feedback. We show that auto-inhibition and mass conservation have a stabilizing effect, whereas increasing abruptness and decreasing feedback threshold have a de-stabilizing effect on the model equilibrium. Moreover, applying our theoretical analysis to the mammalian p53 system we show that an auto-inhibitory feedback can decouple period and amplitude of an oscillatory response, whereas the delayed feedback can not.

CONCLUSIONS

Our theoretical framework provides insight into how time delay and design features of biochemical networks act together to elicit specific characteristic response patterns. Such insight is useful for constructing synthetic networks and controlling their behaviour in response to external stimulation.

The role of time delay in adaptive cellular negative feedback systems

Adaptation in cellular systems is often mediated by negative feedbacks, which usually come with certain time delays causing several characteristic response patterns including an overdamped response, damped or sustained oscillations. Here, we analyse generic two-dimensional delay differential equations with delayed negative feedback describing the dynamics of biochemical adaptive signal-response networks. We derive explicit thresholds and boundaries showing how time delay determines characteristic response patterns of these networks. Applying our theoretical analyses to concrete data we show that adaptation to osmotic stress in yeast is optimal in the sense of minimizing adaptation time without causing oscillatory behaviour, i.e., a critically damped response. In addition, our framework demonstrates that a slight increase of time delay in the NF-κB system might induce a switch from damped to sustained oscillatory behaviour. Thus, we demonstrate how delay differential equations can be used to explicitly study the delay in biochemical negative feedback systems. Our analysis also provides insight into how time delay may tune biological signal-response patterns and control the systems behaviour.

Regulation of signal transduction by spatial parameters: a case in NF-κB oscillation

NF-κB is a transcription factor regulating expression of more than 500 genes, and its dysfunction leads to the autoimmune and inflammatory diseases. In malignant cancer cells, NF-κB is constitutively activated. Thus the elucidation of mechanisms for NF-κB regulation is important for the establishment of therapeutic treatment caused by incorrect NF-κB responses. Cytoplasmic NF-κB translocates to the nucleus by the application of extracellular stimuli such as cytokines. Nuclear NF-κB is known to oscillate with the cycle of 1.5-4.5 h, and it is thought that the oscillation pattern regulates the expression profiles of genes. In this review, first we briefly describe regulation mechanisms of NF-κB. Next, published computational simulations on the oscillation of NF-κB are summarised. There are at least 60 reports on the computational simulation and analysis of NF-κB oscillation. Third, the importance of a 'space' for the regulation of oscillation pattern of NF-κB is discussed, showing altered oscillation pattern by the change in spatial parameters such as diffusion coefficient, nuclear to cytoplasmic volume ratio (N/C ratio), and transport through nuclear membrane. Finally, simulations in a true intracellular space (TiCS), which is an intracellular 3D space reconstructed in a computer with organelles such as nucleus and mitochondria are discussed.

Mean field analysis of a spatial stochastic model of a gene regulatory network

A gene regulatory network may be defined as a collection of DNA segments which interact with each other indirectly through their RNA and protein products. Such a network is said to contain a negative feedback loop if its products inhibit gene transcription, and a positive feedback loop if a gene product promotes its own production. Negative feedback loops can create oscillations in mRNA and protein levels while positive feedback loops are primarily responsible for signal amplification. It is often the case in real biological systems that both negative and positive feedback loops operate in parameter regimes that result in low copy numbers of gene products. In this paper we investigate the spatio-temporal dynamics of a single feedback loop in a eukaryotic cell. We first develop a simplified spatial stochastic model of a canonical feedback system (either positive or negative). Using a Gillespie's algorithm, we compute sample trajectories and analyse their corresponding statistics. We then derive a system of equations that describe the spatio-temporal evolution of the stochastic means. Subsequently, we examine the spatially homogeneous case and compare the results of numerical simulations with the spatially explicit case. Finally, using a combination of steady-state analysis and data clustering techniques, we explore model behaviour across a subregion of the parameter space that is difficult to access experimentally and compare the parameter landscape of our spatio-temporal and spatially-homogeneous models.

A minimal spatio-temporal model of the NF-κB signalling pathway exhibits a range of behaviours

In animal cells, the transcription factor NF-κB regulates many stressful, inflammatory, and innate immune responses. Experiments have revealed that, in response to cell stimulation, NF-κB can exhibit oscillatory dynamics where the nature of these dynamics can influence the pattern of NF-κB-dependent gene expression. Oscillations in NF-κB are believed to depend on a negative feedback loop linking NF-κB and one of its downstream products, namely IκBα. This negative feedback loop is enhanced by cell stimulation. However, it also exists in the absence of cell stimulation. Here we propose a minimal spatio-temporal model of the NF-κB signalling pathway, composed of partial differential equations. Through numerical simulations, we find various combinations of behaviours before and during cell stimulation: equilibrium dynamics (rapid convergence to a solution that is everywhere constant) before cell stimulation, followed by oscillatory dynamics during cell stimulation; oscillatory dynamics before and during cell stimulation; oscillatory dynamics before cell stimulation, followed by equilibrium dynamics during cell stimulation; and equilibrium dynamics before and during cell stimulation. In each case, when cell stimulation ceases, the model quickly returns to its pre-stimulation behaviour. All of these different combinations of behaviours occur for similar sets of parameter values. Therefore, our results may help to explain why, in experiments on the NF-κB pathway involving populations of cells, only a certain fraction of the cells exhibit oscillatory dynamics.

Nested autoinhibitory feedbacks alter the resistance of homeostatic adaptive biochemical networks

Negative feedback control is a ubiquitous feature of biochemical systems, as is time delay between a signal and its response. Negative feedback in conjunction with time delay can lead to oscillations. In a cellular context, it might be beneficial to mitigate oscillatory behaviour to avoid recurring stress situations. This can be achieved by increasing the distance between the parameters of the system and certain thresholds, beyond which oscillations occur. This distance has been termed resistance. Here, we prove that in a generic three-dimensional negative feedback system the resistance of the system is modified by nested autoinhibitory feedbacks. Our system features negative feedbacks through both input-inhibition as well as output-activation, a signalling component with mass conservation and perfect adaptation. We show that these features render the system applicable to biological data, exemplified by the high osmolarity glycerol system in yeast and the mammalian p53 system. Output-activation is better supported by data than input-inhibition and also shows distinguished properties with respect to the system's stimulus. Our general approach might be useful in designing synthetic systems in which oscillations can be tuned by synthetic autoinhibitory feedbacks.

Roles of spatial parameters on the oscillation of nuclear NF-κB: computer simulations of a 3D spherical cell

Transcription factor NF-κB resides in the cytoplasm and translocates to the nucleus by application of extracellular stimuli. It is known that the nuclear NF-κB oscillates and different oscillation patterns lead to different gene expression. Nearly forty reports on modeling and simulation of nuclear NF-κB have been published to date. The computational models reported so far are temporal or two-dimensional, and the discussions on spatial parameters have not been involved or limited. Since spatial parameters in cancer cells such as nuclear to cytoplasmic volume (N/C) ratio are different from normal cells, it is important to understand the relationship between oscillation patterns and spatial parameters. Here we report simulations of a 3D computational model for the oscillation of nuclear NF-κB using A-Cell software. First, we found that the default biochemical kinetic constants used in the temporal model cannot replicate the experimentally observed oscillation in the 3D model. Thus, the default parameters should be changed in the 3D model. Second, spatial parameters such as N/C ratio, nuclear transport, diffusion coefficients, and the location of IκB synthesis were found to alter the oscillation pattern. Third, among them, larger N/C ratios resulted in persistent oscillation of nuclear NF-κB, and larger nuclear transport resulted in faster oscillation frequency. Our simulation results suggest that the changes in spatial parameters seen in cancer cells is one possible mechanism for alteration in the oscillation pattern of nuclear NF-κB and lead to the altered gene expression in these cells.

Influence of the nuclear membrane, active transport, and cell shape on the Hes1 and p53-Mdm2 pathways: insights from spatio-temporal modelling

There are many intracellular signalling pathways where the spatial distribution of the molecular species cannot be neglected. These pathways often contain negative feedback loops and can exhibit oscillatory dynamics in space and time. Two such pathways are those involving Hes1 and p53-Mdm2, both of which are implicated in cancer. In this paper we further develop the partial differential equation (PDE) models of Sturrock et al. (J. Theor. Biol., 273:15-31, 2011) which were used to study these dynamics. We extend these PDE models by including a nuclear membrane and active transport, assuming that proteins are convected in the cytoplasm towards the nucleus in order to model transport along microtubules. We also account for Mdm2 inhibition of p53 transcriptional activity. Through numerical simulations we find ranges of values for the model parameters such that sustained oscillatory dynamics occur, consistent with available experimental measurements. We also find that our model extensions act to broaden the parameter ranges that yield oscillations. Hence oscillatory behaviour is made more robust by the inclusion of both the nuclear membrane and active transport. In order to bridge the gap between in vivo and in silico experiments, we investigate more realistic cell geometries by using an imported image of a real cell as our computational domain. For the extended p53-Mdm2 model, we consider the effect of microtubule-disrupting drugs and proteasome inhibitor drugs, obtaining results that are in agreement with experimental studies.

Stochastic delay accelerates signaling in gene networks

The creation of protein from DNA is a dynamic process consisting of numerous reactions, such as transcription, translation and protein folding. Each of these reactions is further comprised of hundreds or thousands of sub-steps that must be completed before a protein is fully mature. Consequently, the time it takes to create a single protein depends on the number of steps in the reaction chain and the nature of each step. One way to account for these reactions in models of gene regulatory networks is to incorporate dynamical delay. However, the stochastic nature of the reactions necessary to produce protein leads to a waiting time that is randomly distributed. Here, we use queueing theory to examine the effects of such distributed delay on the propagation of information through transcriptionally regulated genetic networks. In an analytically tractable model we find that increasing the randomness in protein production delay can increase signaling speed in transcriptional networks. The effect is confirmed in stochastic simulations, and we demonstrate its impact in several common transcriptional motifs. In particular, we show that in feedforward loops signaling time and magnitude are significantly affected by distributed delay. In addition, delay has previously been shown to cause stable oscillations in circuits with negative feedback. We show that the period and the amplitude of the oscillations monotonically decrease as the variability of the delay time increases.

Delay in gene regulatory networks often arises from the numerous sequential reactions necessary to create fully functional protein from DNA. While the molecular mechanisms behind protein production and maturation are known, it is still unknown to what extent the resulting delay affects signaling in transcriptional networks. In contrast to previous studies that have examined the consequences of fixed delay in gene networks, here we investigate how the variability of the delay time influences the resulting dynamics. The exact distribution of “transcriptional delay” is still unknown, and most likely greatly depends on both intrinsic and extrinsic factors. Nevertheless, we are able to deduce specific effects of distributed delay on transcriptional signaling that are independent of the underlying distribution. We find that the time it takes for a gene encoding a transcription factor to signal its downstream target decreases as the delay variability increases. We use queueing theory to derive a simple relationship describing this result, and use stochastic simulations to confirm it. The consequences of distributed delay for several common transcriptional motifs are also discussed.

A spatio-temporal model of Notch signalling in the zebrafish segmentation clock: conditions for synchronised oscillatory dynamics

In the vertebrate embryo, tissue blocks called somites are laid down in head-to-tail succession, a process known as somitogenesis. Research into somitogenesis has been both experimental and mathematical. For zebrafish, there is experimental evidence for oscillatory gene expression in cells in the presomitic mesoderm (PSM) as well as evidence that Notch signalling synchronises the oscillations in neighbouring PSM cells. A biological mechanism has previously been proposed to explain these phenomena. Here we have converted this mechanism into a mathematical model of partial differential equations in which the nuclear and cytoplasmic diffusion of protein and mRNA molecules is explicitly considered. By performing simulations, we have found ranges of values for the model parameters (such as diffusion and degradation rates) that yield oscillatory dynamics within PSM cells and that enable Notch signalling to synchronise the oscillations in two touching cells. Our model contains a Hill coefficient that measures the co-operativity between two proteins (Her1, Her7) and three genes (her1, her7, deltaC) which they inhibit. This coefficient appears to be bounded below by the requirement for oscillations in individual cells and bounded above by the requirement for synchronisation. Consistent with experimental data and a previous spatially non-explicit mathematical model, we have found that signalling can increase the average level of Her1 protein. Biological pattern formation would be impossible without a certain robustness to variety in cell shape and size; our results possess such robustness. Our spatially-explicit modelling approach, together with new imaging technologies that can measure intracellular protein diffusion rates, is likely to yield significant new insight into somitogenesis and other biological processes.

Avoiding transcription factor competition at promoter level increases the chances of obtaining oscillation

Background

The ultimate goal of synthetic biology is the conception and construction of genetic circuits that are reliable with respect to their designed function (e.g. oscillators, switches). This task remains still to be attained due to the inherent synergy of the biological building blocks and to an insufficient feedback between experiments and mathematical models. Nevertheless, the progress in these directions has been substantial.

Results

It has been emphasized in the literature that the architecture of a genetic oscillator must include positive (activating) and negative (inhibiting) genetic interactions in order to yield robust oscillations. Our results point out that the oscillatory capacity is not only affected by the interaction polarity but by how it is implemented at promoter level. For a chosen oscillator architecture, we show by means of numerical simulations that the existence or lack of competition between activator and inhibitor at promoter level affects the probability of producing oscillations and also leaves characteristic fingerprints on the associated period/amplitude features.

Conclusions

In comparison with non-competitive binding at promoters, competition drastically reduces the region of the parameters space characterized by oscillatory solutions. Moreover, while competition leads to pulse-like oscillations with long-tail distribution in period and amplitude for various parameters or noisy conditions, the non-competitive scenario shows a characteristic frequency and confined amplitude values. Our study also situates the competition mechanism in the context of existing genetic oscillators, with emphasis on the Atkinson oscillator.

Dynamical approaches to modeling developmental gene regulatory networks

The network of interacting regulatory signals within a cell comprises one of the most complex and powerful computational systems in biology. Gene regulatory networks (GRNs) play a key role in transforming the information encoded in a genome into morphological form. To achieve this feat, GRNs must respond to and integrate environmental signals with their internal dynamics in a robust and coordinated fashion. The highly dynamic nature of this process lends itself to interpretation and analysis in the language of dynamical models. Modeling provides a means of systematically untangling the complicated structure of GRNs, a framework within which to simulate the behavior of reconstructed systems and, in some cases, suites of analytic tools for exploring that behavior and its implications. This review provides a general background to the idea of treating a regulatory network as a dynamical system, and describes a variety of different approaches that have been taken to the dynamical modeling of GRNs.

Microfluidic devices for measuring gene network dynamics in single cells

The dynamics governing gene regulation have an important role in determining the phenotype of a cell or organism. From processing extracellular signals to generating internal rhythms, gene networks are central to many time-dependent cellular processes. Recent technological advances now make it possible to track the dynamics of gene networks in single cells under various environmental conditions using microfluidic 'lab-on-a-chip' devices, and researchers are using these new techniques to analyse cellular dynamics and discover regulatory mechanisms. These technologies are expected to yield novel insights and allow the construction of mathematical models that more accurately describe the complex dynamics of gene regulation.

Oscillatory Notch-pathway activity in a delay model of neuronal differentiation

Lateral inhibition resulting from a double-negative feedback loop underlies the assignment of different fates to cells in many developmental processes. Previous studies have shown that the presence of time delays in models of lateral inhibition can result in significant oscillatory transients before patterned steady states are reached. We study the impact of local feedback loops in a model of lateral inhibition based on the Notch signaling pathway, elucidating the roles of intracellular and intercellular delays in controlling the overall system behavior. The model exhibits both in-phase and out-of-phase oscillatory modes and oscillation death. Interactions between oscillatory modes can generate complex behaviors such as intermittent oscillations. Our results provide a framework for exploring the recent observation of transient Notch-pathway oscillations during fate assignment in vertebrate neurogenesis.

Intron delays and transcriptional timing during development

The time taken to transcribe most metazoan genes is significant because of the substantial length of introns. Developmentally regulated gene networks, where timing and dynamic patterns of expression are critical, may be particularly sensitive to intron delays. We revisit and comment on a perspective last presented by Thummel 16 years ago: transcriptional delays may contribute to timing mechanisms during development. We discuss the presence of intron delays in genetic networks. We consider how delays can impact particular moments during development, which mechanistic attributes of transcription can influence them, how they can be modeled, and how they can be studied using recent technological advances as well as classical genetics.

Oscillatory expression of Hes family transcription factors: insights from mathematical modelling

Oscillatory expression of the Hes family of transcription factors plays a central role in the segmentation of the vertebrate body during embryonic development. Analogous oscillations in cultured cells suggest that Hes oscillations may be important in other developmental processes, and provide an excellent opportunity to explore the origin of these oscillations in a relatively simple setting. Mathematical and computational modelling have been used in combination with quantitative mRNA and protein expression data to analyse the origin and properties of Hes oscillations, and have highlighted the important roles played by time delays in negative feedback circuits. In this chapter, we review recent theoretical and experimental results, and discuss how analysis of existing models suggests potential avenues for further study of delayed feedback oscillators.

Dynamics of a two-dimensional model of cell tissues with coupled stochastic gene networks

Gene expression and differentiation were shown to be stochastic processes. However, cells in a tissue can coordinate their behavior, including gene expression and differentiation pathways choice. A tissue of coupled cells is modeled as a two-dimensional regular square lattice of identical cells, each a three-dimensional compartment with a gene regulatory network (GRN) and a toggle switch (TS). The dynamics is driven by a delayed stochastic simulation algorithm, nearest neighbor cells are coupled by normally distributed time delayed reactions allowing interchange of proteins, and gene expression is a multiple time delayed reaction. It is defined the coupling strength (C), to characterize the lattice structure as a function of the rate constants of the reactions coupling nearest neighbor cells. Conditions are investigated for the emergence of synchronization and stable differentiation of cells within a tissue. From the time series of the cells GRNs, the tissue dynamical stability (S) is computed from the average toggling period of each GRN. The synchronization of cells' proteins expression levels is measured by their time series entropy (H). It is shown that the tissue goes through various dynamical regimes as C is increased, by measuring H and S . For null C, the cells GRNs toggle asynchronously. For weak C, cells synchronize by regions of space. Increasing C, the tissue becomes homogeneously synchronous. As C is further increased, S goes through a phase transition, from synchronized toggling to stable, where all cells produce one and the same protein. Finally, increasing C even further, a new stable state emerges where both genes of all cells are expressed and bistability is lost. This state, resembling an infinitely long transient, is an emergent behavior not observable in a single TS. The results provide an explanation of how cells with bistable GRNs, inherently stochastic, can synchronize or uniformly differentiate into stable states, by interacting with nearest neighbors.

Effects of coupling strength and space on the dynamics of coupled toggle switches in stochastic gene networks with multiple-delayed reactions

Genetic toggle switches (TSs) are one of the best studied small gene regulatory networks (GRNs), due to their simplicity and relevant role. They have been interpreted as decision circuits in cell differentiation, a process long hypothesized to be bistable, or as cellular memory units. In these contexts, they must be reliable. Once a "decision" is made, the system must remain stable. One way to gain stability is by duplicating the genes of a TS and coupling the two TSs. Using a recent modeling strategy of GRNs, driven by a delayed stochastic simulation algorithm (delayed SSA) that allows modeling transcription and translation as multidelayed reactions, we analyze the stability of systems of coupled TSs. For this, we introduce the coupling strength (C), a parameter to characterize the GRN structure, against which we compare the GRN stability (S). We first show that time delays in transcription, associated to the promoter region release, ensure bistability of a TS, given no cooperative binding or self-activation reactions. Next, we couple two TSs and measure their toggling frequencies as C varies. Three dynamical regimes are observed: (i) for weak coupling, high frequency synchronized oscillations, (ii) for average coupling, low frequency synchronized oscillations, and (iii) for strong coupling the system becomes stable after a transient, in one of two steady states. The system stability, S, goes through a first order phase transition as C increases, in the average coupling regime. After, we study the effects of spatial separation in two compartments on the dynamics of two coupled TSs, where spatial separation is modeled as normally distributed random time delayed reactions. The phase transition of S, as C increases, occurs for lower values of C than when the two TSs are in the same compartment. Finally, we couple weakly and homogeneously several TSs within a single compartment and observe that as the number of coupled TSs increases, the system goes through the phase transition in S, from oscillatory to stable and for C values lower than in the two previous cases.

Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays

Delay-differential equations are commonly used to model genetic regulatory systems with the delays representing transcription and translation times. Equations with delayed terms can also be used to represent other types of chemical processes. Here we analyze delayed mass-action systems, i.e. systems in which the rates of reaction are given by mass-action kinetics, but where the appearance of products may be delayed. Necessary conditions for delay-induced instability are presented in terms both of a directed graph (digraph) constructed from the Jacobian matrix of the corresponding ODE model and of a species-reaction bipartite graph which directly represents a chemical mechanism. Methods based on the bipartite graph are particularly convenient and powerful. The condition for a delay-induced instability in this case is the existence of a subgraph of the bipartite graph containing an odd number of cycles of which an odd number are negative.

Periodicity in piecewise-linear switching networks with delay

Gene regulatory networks and neural networks can be modeled by piecewise-linear switching systems of differential equations, known as Glass networks. These biological networks exhibit delays in regulatory activity, for example, transcription, translation and spatial transport in gene networks, and transmission delays in neural networks. Such delays may be significant in determining their dynamical behavior. Here Glass networks with a discrete delay are introduced and analyzed. Fixed points away from thresholds are straightforward to identify, even in the presence of delays, so the focus of this work is on cyclic patterns of switching. Under a condition that ensures an unambiguous pattern of switching, it is shown by means of a fractional linear mapping that delayed Glass networks have a periodic orbit for all positive finite delays. Furthermore, an algorithm is presented to locate the periodic orbit for a given cycle, to determine whether the periodic orbit is locally asymptotically stable, and to check if it is unique. In addition, the complete dynamics of the two-dimensional delayed Glass network is provided: if there is a cycle of four orthants, then there exists a unique globally stable limit cycle; whereas if there is a black wall, then across the wall there exists a unique limit cycle that is globally stable with respect to the associated orthants. This behavior is in contrast to the non-delayed case, in which spiralling approach to fixed points on threshold boundaries can occur.

p53-Mdm2 loop controlled by a balance of its feedback strength and effective dampening using ATM and delayed feedback

When the genomic integrity of a cell is challenged, its fate is determined in part by signals conveyed by the p53 tumour suppressor protein. It was observed recently that such signals are not simple gradations of p53 concentration, but rather a counter-intuitive limit-cycle behaviour. Based on a careful mathematical interpretation of the experimental body of knowledge, we propose a model for the p53 signalling network and characterise the p53 stability and oscillatory dynamics. In our model, ATM, a protein that senses DNA damage, activates p53 by phosphorylation. In its active state, p53 has a decreased degradation rate and an enhanced transactivation of Mdm2, a gene whose protein product Mdm2 tags p53 for degradation. Thus the p53-Mdm2 system forms a negative feedback loop. However, the feedback in this loop is delayed, as the pool of Mdm2 molecules being induced by p53 at a given time will mark for degradation the pool of p53 molecules at some later time, after the Mdm2 molecules have been transcribed, exported out of the nucleus, translated and transported back into the nucleus. The analysis of our model demonstrates how this time lag combines with the ATM-controlled feedback strength and effective dampening of the negative feedback loop to produce limit-cycle oscillations. The picture that emerges is that ATM, once activated by DNA damage, makes the p53-Mdm2 oscillator undergo a supercritical Hopf bifurcation. This approach yields an improved understanding of the global dynamics and bifurcation structure of our time-delayed, negative feedback model and allows for predictions of the behaviour of the p53 system under different perturbations.

Gene Expression Time Delays and Turing Pattern Formation Systems

The incorporation of time delays can greatly affect the behaviour of partial differential equations and dynamical systems. In addition, there is evidence that time delays in gene expression due to transcription and translation play an important role in the dynamics of cellular systems. In this paper, we investigate the effects of incorporating gene expression time delays into a one-dimensional putative reaction diffusion pattern formation mechanism on both stationary domains and domains with spatially uniform exponential growth. While oscillatory behaviour is rare, we find that the time taken to initiate and stabilise patterns increases dramatically as the time delay is increased. In addition, we observe that on rapidly growing domains the time delay can induce a failure of the Turing instability which cannot be predicted by a naive linear analysis of the underlying equations about the homogeneous steady state. The dramatic lag in the induction of patterning, or even its complete absence on occasions, highlights the importance of considering explicit gene expression time delays in models for cellular reaction diffusion patterning.

Oscillatory expression of Hes1, p53, and NF-kappaB driven by transcriptional time delays

Feedback inhibition of gene expression is a widespread phenomenon in which the expression of a gene is downregulated by its protein product. Feedback in eukaryotic cells involves time delays resulting from transcription, transcript splicing and processing, and protein synthesis. In principle, such delays can result in oscillatory mRNA and protein expression. However, experimental evidence of such delay-driven oscillations has been lacking. Using mathematical modeling informed by recent data, I show that the observed oscillatory expression and activity of three proteins is most likely to be driven by transcriptional delays. Each protein (Hes1, p53, and NF-kappaB) is a component of a short feedback inhibition loop. The oscillatory period is determined by the delay and the protein and mRNA half-lives, but it is robust to changes in the rates of transcription and protein synthesis. In contrast to nondelayed models, delayed models do not require additional components in the feedback loop. These results provide direct evidence that transcriptional delays can drive oscillatory gene activity and highlight the importance of considering delays when analyzing genetic regulatory networks, particularly in processes such as developmental pattern formation, where short half-lives and feedback inhibition are common.

Effects of macromolecular transport and stochastic fluctuations on dynamics of genetic regulatory systems

To predict the dynamics of genetic regulation, it may be necessary to consider macromolecular transport and stochastic fluctuations in macromolecule numbers. Transport can be diffusive or active, and in some cases a time delay might suffice to model active transport. We characterize major differences in the dynamics of model genetic systems when diffusive transport of mRNA and protein was compared with transport modeled as a time delay. Delays allow for history-dependent, non-Markovian responses to stimuli (i.e., "molecular memory"). Diffusion suppresses oscillations, whereas delays tend to create oscillations. When simulating essential elements of circadian oscillators, we found the delay between transcription and translation necessary for oscillations. Stochastic fluctuations tend to destabilize and thereby mask steady states with few molecules. This computational approach, combined with experiments, should provide a fruitful conceptual framework for investigating the function and dynamic properties of genetic regulatory systems.

Oscillations and multiple steady states in a cyclic gene model with repression

In this paper we study the cyclic gene model with repression considered by H. T. Banks and J. M. Mahaffy. Roughly, the model describes a biochemical feedback loop consisting of an integer number G of single gene reaction sequences in series. The model leads to a system of functional differential equations. We show that there is a qualitative difference in the dynamics between even and odd G if the feedback repression is sufficiently large. For even G, multiple stable steady states can coexist while for odd G, periodic orbits exist.

Interaction of spatial diffusion and delays in models of genetic control by repression

A class of models based on the Jacob and Monod theory of genetic repression for control of biosynthetic pathways in cells is considered. Both spatial diffusion and time delays are taken into account. A method is developed for representing the effects of spatial diffusion as distributed delay terms. This method is applied to two specific models and the interaction between the diffusion and the delays is treated in detail. The destabilization of the steady-state and the bifurcation of oscillatory solutions are studied as functions of the diffusivities and the delays. The limits of very small and very large diffusivities are analyzed and comparisons with well-mixed compartment models are made.