Modelling transcriptional feedback loops: the role of Gro/TLE1 in Hes1 oscillations

Stochastic switching of delayed feedback suppresses oscillations in genetic regulatory systems

Delays and stochasticity have both served as crucially valuable ingredients in mathematical descriptions of control, physical and biological systems. In this work, we investigate how explicitly dynamical stochasticity in delays modulates the effect of delayed feedback. To do so, we consider a hybrid model where stochastic delays evolve by a continuous-time Markov chain, and between switching events, the system of interest evolves via a deterministic delay equation. Our main contribution is the calculation of an effective delay equation in the fast switching limit. This effective equation maintains the influence of all subsystem delays and cannot be replaced with a single effective delay. To illustrate the relevance of this calculation, we investigate a simple model of stochastically switching delayed feedback motivated by gene regulation. We show that sufficiently fast switching between two oscillatory subsystems can yield stable dynamics.

Systematic comparison of modeling fidelity levels and parameter inference settings applied to negative feedback gene regulation

Quantitative stochastic models of gene regulatory networks are important tools for studying cellular regulation. Such models can be formulated at many different levels of fidelity. A practical challenge is to determine what model fidelity to use in order to get accurate and representative results. The choice is important, because models of successively higher fidelity come at a rapidly increasing computational cost. In some situations, the level of detail is clearly motivated by the question under study. In many situations however, many model options could qualitatively agree with available data, depending on the amount of data and the nature of the observations. Here, an important distinction is whether we are interested in inferring the true (but unknown) physical parameters of the model or if it is sufficient to be able to capture and explain available data. The situation becomes complicated from a computational perspective because inference needs to be approximate. Most often it is based on likelihood-free Approximate Bayesian Computation (ABC) and here determining which summary statistics to use, as well as how much data is needed to reach the desired level of accuracy, are difficult tasks. Ultimately, all of these aspects-the model fidelity, the available data, and the numerical choices for inference-interplay in a complex manner. In this paper we develop a computational pipeline designed to systematically evaluate inference accuracy for a wide range of true known parameters. We then use it to explore inference settings for negative feedback gene regulation. In particular, we compare a detailed spatial stochastic model, a coarse-grained compartment-based multiscale model, and the standard well-mixed model, across several data-scenarios and for multiple numerical options for parameter inference. Practically speaking, this pipeline can be used as a preliminary step to guide modelers prior to gathering experimental data. By training Gaussian processes to approximate the distance function values, we are able to substantially reduce the computational cost of running the pipeline.

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.

Deciphering the Role of Fluctuation Dependent Intercellular Communication in Neural Stem Cell Development

Neural stem cells (NPCs) efficiently communicate in an intercellular manner to govern specific cell fate decisions during the developmental process despite withstanding the fluctuating cellular environment. How these fluctuations from diverse origins functionally affect the precise cell fate decision making remains elusive. By taking a stochastic mathematical modeling approach, we unravel that the transcriptional variability arising within an NPC population due to intermittent cell cycle events significantly influences the neuron to NPC ratio during development. Our model proficiently quantifies the impact of different sources of heterogeneities in maintaining an exact neuron to NPC ratio and predicts plausible experimental ways to fine-tune the development of NPCs. In the future, these modeling insights may lead to better therapeutic avenues to regenerate neurons from NPCs.

Multiple random phosphorylations in clock proteins provide long delays and switches

Theory predicts that self-sustained oscillations require robust delays and nonlinearities (ultrasensitivity). Delayed negative feedback loops with switch-like inhibition of transcription constitute the core of eukaryotic circadian clocks. The kinetics of core clock proteins such as PER2 in mammals and FRQ in Neurospora crassa is governed by multiple phosphorylations. We investigate how multiple, slow and random phosphorylations control delay and molecular switches. We model phosphorylations of intrinsically disordered clock proteins (IDPs) using conceptual models of sequential and distributive phosphorylations. Our models help to understand the underlying mechanisms leading to delays and ultrasensitivity. The model shows temporal and steady state switches for the free kinase and the phosphoprotein. We show that random phosphorylations and sequestration mechanisms allow high Hill coefficients required for self-sustained oscillations.

Limit cycles in models of circular gene networks regulated by negative feedback loops

BACKGROUND

The regulatory feedback loops that present in structural and functional organization of molecular-genetic systems and the phenomenon of the regulatory signal delay, a time period between the moment of signal reception and its implementation, provide natural conditions for complicated dynamic regimes in these systems. The delay phenomenon at the intracellular level is a consequence of the matrix principle of data transmission, implemented through the rather complex processes of transcription and translation.However, the rules of the influence of system structure on system dynamics are not clearly understood. Knowledge of these rules is particularly important for construction of synthetic gene networks with predetermined properties.

RESULTS

We study dynamical properties of models of simplest circular gene networks regulated by negative feedback mechanisms. We have shown existence and stability of oscillating trajectories (cycles) in these models. Two algorithms of construction and localization of these cycles have been proposed. For one of these models, we have solved an inverse problem of parameters identification.

CONCLUSIONS

The modeling results demonstrate that non-stationary dynamics in the models of circular gene networks with negative feedback loops is achieved by a high degree of non-linearity of the mechanism of the autorepressor influence on its own expression, by the presence of regulatory signal delay, the value of which must exceed a certain critical value, and transcription/translation should be initiated from a sufficiently strong promoter/Shine-Dalgarno site. We believe that the identified patterns are key elements of the oscillating construction design.

Alteration in MicroRNA Expression Governs the Nature and Timing of Cellular Fate Commitment

In the central nervous system, the expression level of transcriptional repressor Hes1 (hairy and enhancer of split-1) tightly controls the alternative cell fate commitment during differentiation as well as the time required for such cellular transitions. A microRNA, miR-9, that interacts with Hes1 in a mutually antagonistic manner, influences both the process of lineage specification and timing of differentiation significantly, but the impact of the miR-9 in guiding these events still remains poorly understood. Here, we proposed a stochastic mathematical model of the miR-9/Hes1 double-negative feedback interaction network that at the outset shows how alternative cell fate such as quiescence, progenitor, and neuronal states can be accomplished through fine-tuning the Hes1 dynamics by altering the expression level of miR-9. The model simulations further foretell a correlated variation of the period of oscillation of Hes1, and the time delay observed between Hes1 mRNA and protein as the transcription rate of miR-9 increases during the neural progenitor state attainment. Importantly, the model simulations aided by the systematic sensitivity analysis predict that the timing of differentiation to the neuronal state crucially depends on the negative regulators (miR-9 and Hes6) of the Hes1. Our results indicate that miR-9/Hes1 interaction network can be effectively exploited for an efficient and well-timed neuronal transformation.

Differential interactions between Notch and ID factors control neurogenesis by modulating Hes factor autoregulation

During embryonic and adult neurogenesis, neural stem cells (NSCs) generate the correct number and types of neurons in a temporospatial fashion. Control of NSC activity and fate is crucial for brain formation and homeostasis. Neurogenesis in the embryonic and adult brain differ considerably, but Notch signaling and inhibitor of DNA-binding (ID) factors are pivotal in both. Notch and ID factors regulate NSC maintenance; however, it has been difficult to evaluate how these pathways potentially interact. Here, we combined mathematical modeling with analysis of single-cell transcriptomic data to elucidate unforeseen interactions between the Notch and ID factor pathways. During brain development, Notch signaling dominates and directly regulates Id4 expression, preventing other ID factors from inducing NSC quiescence. Conversely, during adult neurogenesis, Notch signaling and Id2/3 regulate neurogenesis in a complementary manner and ID factors can induce NSC maintenance and quiescence in the absence of Notch. Our analyses unveil key molecular interactions underlying NSC maintenance and mechanistic differences between embryonic and adult neurogenesis. Similar Notch and ID factor interactions may be crucial in other stem cell systems.

Summary: Computational analysis of transcriptome data from neural stem cells reveals key differences in the synergistic interactions between Notch and inhibitor of DNA-binding factors during embryonic and adult neurogenesis.

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.

Time to Decide? Dynamical Analysis Predicts Partial Tip/Stalk Patterning States Arise during Angiogenesis

Angiogenesis is a highly dynamic morphogenesis process; however, surprisingly little is known about the timing of the different molecular processes involved. Although the role of the VEGF-notch-DLL4 signaling pathway has been established as essential for tip/stalk cell competition during sprouting, the speed and dynamic properties of the underlying process at the individual cell level has not been fully elucidated. In this study, using mathematical modeling we investigate how specific, biologically meaningful, local conditions around and within an individual cell can influence their unique tip/stalk phenotype switching kinetics. To this end we constructed an ordinary differential equation model of VEGF-notch-DLL4 signaling in a system of two, coupled endothelial cells (EC). Our studies reveal that at any given point in an angiogenic vessel the time it takes a cell to decide to take on a tip or stalk phenotype may be drastically different, and this asynchrony of tip/stalk cell decisions along vessels itself acts to speed up later competitions. We unexpectedly uncover intermediate "partial" yet stable states lying between the tip and stalk cell fates, and identify that internal cellular factors, such as NAD-dependent deacetylase sirtuin-1 (Sirt1) and Lunatic fringe 1 (Lfng1), can specifically determine the length of time a cell spends in these newly identified partial tip/stalk states. Importantly, the model predicts that these partial EC states can arise during normal angiogenesis, in particular during cell rearrangement in sprouts, providing a novel two-stage mechanism for rapid adaptive behavior to the cells highly dynamic environment. Overall, this study demonstrates that different factors (both internal and external to EC) can be used to modulate the speed of tip/stalk decisions, opening up new opportunities and challenges for future biological experiments and therapeutic targeting to manipulate vascular network topology, and our basic understanding of developmental/pathological angiogenesis.

Unraveling the differential dynamics of developmental fate in central and peripheral nervous systems

Bone morphogenetic protein 2 (BMP2), differentially regulates the developmental lineage commitment of neural stem cells (NSC’s) in central and peripheral nervous systems. However, the precise mechanism beneath such observations still remains illusive. To decipher the intricacies of this mechanism, we propose a generic mathematical model of BMP2 driven differentiation regulation of NSC’s. The model efficiently captures the dynamics of the wild-type as well as various mutant and over-expression phenotypes for NSC’s in central nervous system. Our model predicts that the differential developmental dynamics of the NSC’s in peripheral nervous system can be reconciled by altering the relative positions of the two mutually interconnected bi-unstable switches inherently present in the steady state dynamics of the crucial developmental fate regulatory proteins as a function of BMP2 dose. This model thus provides a novel mechanistic insight and has the potential to deliver exciting therapeutic strategies for neuronal regeneration from NSC’s of different origin.

Linear models of activation cascades: analytical solutions and coarse-graining of delayed signal transduction

Cellular signal transduction usually involves activation cascades, the sequential activation of a series of proteins following the reception of an input signal. Here, we study the classic model of weakly activated cascades and obtain analytical solutions for a variety of inputs. We show that in the special but important case of optimal gain cascades (i.e. when the deactivation rates are identical) the downstream output of the cascade can be represented exactly as a lumped nonlinear module containing an incomplete gamma function with real parameters that depend on the rates and length of the cascade, as well as parameters of the input signal. The expressions obtained can be applied to the non-identical case when the deactivation rates are random to capture the variability in the cascade outputs. We also show that cascades can be rearranged so that blocks with similar rates can be lumped and represented through our nonlinear modules. Our results can be used both to represent cascades in computational models of differential equations and to fit data efficiently, by reducing the number of equations and parameters involved. In particular, the length of the cascade appears as a real-valued parameter and can thus be fitted in the same manner as Hill coefficients. Finally, we show how the obtained nonlinear modules can be used instead of delay differential equations to model delays in signal transduction.

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.

Stochastic oscillations induced by intrinsic fluctuations in a self-repressing gene

Biochemical reaction networks are subjected to large fluctuations attributable to small molecule numbers, yet underlie reliable biological functions. Thus, it is important to understand how regularity can emerge from noise. Here, we study the stochastic dynamics of a self-repressing gene with arbitrarily long or short response time. We find that when the mRNA and protein half-lives are approximately equal to the gene response time, fluctuations can induce relatively regular oscillations in the protein concentration. To gain insight into this phenomenon at the crossroads of determinism and stochasticity, we use an intermediate theoretical approach, based on a moment-closure approximation of the master equation, which allows us to take into account the binary character of gene activity. We thereby obtain differential equations that describe how nonlinearity can feed-back fluctuations into the mean-field equations to trigger oscillations. Finally, our results suggest that the self-repressing Hes1 gene circuit exploits this phenomenon to generate robust oscillations, inasmuch as its time constants satisfy precisely the conditions we have identified.

The role of dimerisation and nuclear transport in the Hes1 gene regulatory network

Hes1 is a member of the family of basic helix-loop-helix transcription factors and the Hes1 gene regulatory network (GRN) may be described as the canonical example of transcriptional control in eukaryotic cells, since it involves only the Hes1 protein and its own mRNA. Recently, the Hes1 protein has been established as an excellent target for an anti-cancer drug treatment, with the design of a small molecule Hes1 dimerisation inhibitor representing a promising if challenging approach to therapy. In this paper, we extend a previous spatial stochastic model of the Hes1 GRN to include nuclear transport and dimerisation of Hes1 monomers. Initially, we assume that dimerisation occurs only in the cytoplasm, with only dimers being imported into the nucleus. Stochastic simulations of this novel model using the URDME software show that oscillatory dynamics in agreement with experimental studies are retained. Furthermore, we find that our model is robust to changes in the nuclear transport and dimerisation parameters. However, since the precise dynamics of the nuclear import of Hes1 and the localisation of the dimerisation reaction are not known, we consider a second modelling scenario in which we allow for both Hes1 monomers and dimers to be imported into the nucleus, and we allow dimerisation of Hes1 to occur everywhere in the cell. Once again, computational solutions of this second model produce oscillatory dynamics in agreement with experimental studies. We also explore sensitivity of the numerical solutions to nuclear transport and dimerisation parameters. Finally, we compare and contrast the two different modelling scenarios using numerical experiments that simulate dimer disruption, and suggest a biological experiment that could distinguish which model more faithfully captures the Hes1 GRN.

Maximum profile likelihood estimation of differential equation parameters through model based smoothing state estimates

Statistical inference for biochemical models often faces a variety of characteristic challenges. In this paper we examine state and parameter estimation for the JAK-STAT intracellular signalling mechanism, which exemplifies the implementation intricacies common in many biochemical inference problems. We introduce an extension to the Generalized Smoothing approach for estimating delay differential equation models, addressing selection of complexity parameters, choice of the basis system, and appropriate optimization strategies. Motivated by the JAK-STAT system, we further extend the generalized smoothing approach to consider a nonlinear observation process with additional unknown parameters, and highlight how the approach handles unobserved states and unevenly spaced observations. The methodology developed is generally applicable to problems of estimation for differential equation models with delays, unobserved states, nonlinear observation processes, and partially observed histories.

Spatial stochastic modelling of the Hes1 gene regulatory network: intrinsic noise can explain heterogeneity in embryonic stem cell differentiation

Individual mouse embryonic stem cells have been found to exhibit highly variable differentiation responses under the same environmental conditions. The noisy cyclic expression of Hes1 and its downstream genes are known to be responsible for this, but the mechanism underlying this variability in expression is not well understood. In this paper, we show that the observed experimental data and diverse differentiation responses can be explained by a spatial stochastic model of the Hes1 gene regulatory network. We also propose experiments to control the precise differentiation response using drug treatment.

The interplay of cis-regulatory elements rules circadian rhythms in mouse liver

The mammalian circadian clock is driven by cell-autonomous transcriptional feedback loops that involve E-boxes, D-boxes, and ROR-elements. In peripheral organs, circadian rhythms are additionally affected by systemic factors. We show that intrinsic combinatorial gene regulation governs the liver clock. With a temporal resolution of 2 h, we measured the expression of 21 clock genes in mouse liver under constant darkness and equinoctial light-dark cycles. Based on these data and known transcription factor binding sites, we develop a six-variable gene regulatory network. The transcriptional feedback loops are represented by equations with time-delayed variables, which substantially simplifies modelling of intermediate protein dynamics. Our model accurately reproduces measured phases, amplitudes, and waveforms of clock genes. Analysis of the network reveals properties of the clock: overcritical delays generate oscillations; synergy of inhibition and activation enhances amplitudes; and combinatorial modulation of transcription controls the phases. The agreement of measurements and simulations suggests that the intrinsic gene regulatory network primarily determines the circadian clock in liver, whereas systemic cues such as light-dark cycles serve to fine-tune the rhythms.

From dynamic expression patterns to boundary formation in the presomitic mesoderm

The segmentation of the vertebrate body is laid down during early embryogenesis. The formation of signaling gradients, the periodic expression of genes of the Notch-, Fgf- and Wnt-pathways and their interplay in the unsegmented presomitic mesoderm (PSM) precedes the rhythmic budding of nascent somites at its anterior end, which later develops into epithelialized structures, the somites. Although many in silico models describing partial aspects of somitogenesis already exist, simulations of a complete causal chain from gene expression in the growth zone via the interaction of multiple cells to segmentation are rare. Here, we present an enhanced gene regulatory network (GRN) for mice in a simulation program that models the growing PSM by many virtual cells and integrates WNT3A and FGF8 gradient formation, periodic gene expression and Delta/Notch signaling. Assuming Hes7 as core of the somitogenesis clock and LFNG as modulator, we postulate a negative feedback of HES7 on Dll1 leading to an oscillating Dll1 expression as seen in vivo. Furthermore, we are able to simulate the experimentally observed wave of activated NOTCH (NICD) as a result of the interactions in the GRN. We esteem our model as robust for a wide range of parameter values with the Hes7 mRNA and protein decays exerting a strong influence on the core oscillator. Moreover, our model predicts interference between Hes1 and HES7 oscillators when their intrinsic frequencies differ. In conclusion, we have built a comprehensive model of somitogenesis with HES7 as core oscillator that is able to reproduce many experimentally observed data in mice.

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.

Oscillabolastic model, a new model for oscillatory dynamics, applied to the analysis of Hes1 gene expression and Ehrlich ascites tumor growth

This paper introduces a new dynamical model, called the oscillabolastic model, to analyze the dynamical behavior of biomedical data when one observes oscillatory behavior. The proposed oscillabolastic model is sufficiently flexible to represent various types of oscillatory behavior. The oscillabolastic model is applied to two sets of data. The first data set deals with the oscillabolastic modeling of Ehrlich ascites tumor cells and the second one is the oscillabolastic modeling of the mean signal intensity of Hes1 gene expression in response to serum stimulation. A generalized oscillabolastic model is also suggested to accommodate cases in which predictor variables other than time are also involved.

Tuning the mammalian circadian clock: robust synergy of two loops

The circadian clock is accountable for the regulation of internal rhythms in most living organisms. It allows the anticipation of environmental changes during the day and a better adaptation of physiological processes. In mammals the main clock is located in the suprachiasmatic nucleus (SCN) and synchronizes secondary clocks throughout the body. Its molecular constituents form an intracellular network which dictates circadian time and regulates clock-controlled genes. These clock-controlled genes are involved in crucial biological processes including metabolism and cell cycle regulation. Its malfunction can lead to disruption of biological rhythms and cause severe damage to the organism. The detailed mechanisms that govern the circadian system are not yet completely understood. Mathematical models can be of great help to exploit the mechanism of the circadian circuitry. We built a mathematical model for the core clock system using available data on phases and amplitudes of clock components obtained from an extensive literature search. This model was used to answer complex questions for example: how does the degradation rate of Per affect the period of the system and what is the role of the ROR/Bmal/REV-ERB (RBR) loop? Our findings indicate that an increase in the RNA degradation rate of the clock gene Period (Per) can contribute to increase or decrease of the period--a consequence of a non-monotonic effect of Per transcript stability on the circadian period identified by our model. Furthermore, we provide theoretical evidence for a potential role of the RBR loop as an independent oscillator. We carried out overexpression experiments on members of the RBR loop which lead to loss of oscillations consistent with our predictions. These findings challenge the role of the RBR loop as a merely auxiliary loop and might change our view of the clock molecular circuitry and of the function of the nuclear receptors (REV-ERB and ROR) as a putative driving force of molecular oscillations.

Stability and bifurcation analysis of models for zebrafish somitogenesis

Notch-Delta signaling is indispensable for somitogenesis, which controls the vertebrate segmentation during embryonic development. Several theoretical models have been proposed to explain this interesting process. In zebrafish somitogenesis, genes her1, her7, delta, and their proteins plays the important roles. However, an auto-repression model with time delay involved only by her1/her7 is able to explain zebrafish somitogenesis. This paper will systematically study the dynamics of this model. Specifically we investigate its stability, bifurcation (oscillation), and stability of oscillation. First, the conditions for both stability and bifurcation are presented based on the linearized model. Then three indices for bifurcation of this nonlinear model are derived by using linear functional operator analysis. Finally, the numerical simulations are carried out to illustrate the theoretical results developed in this study.

DYNAMICS OF A SEGMENTATION CLOCK MODEL WITH DISCRETE AND DISTRIBUTED DELAYS

In this paper, we study the effects of time delays on the dynamics of a segmentation clock model with both discrete and distributed delays. Two cases are considered. The first case corresponds to the model with only distributed delay. The second case involves both discrete and distributed delay. Local stability analysis is carried out for all cases. Numerical simulations are also performed to illustrate the results.

A systems view of the protein expression process

Many biological processes are regulated by changing the concentration and activity of proteins. The presence of a protein at a given subcellular location at a given time with a certain conformation is the result of an apparently sequential process. The rate of protein formation is influenced by chromatin state, and the rates of transcription, translation, and degradation. There is an exquisite control system where each stage of the process is controlled both by seemingly unregulated proteins as well as through feedbacks mediated by RNA and protein products. Here we review the biological facts and mathematical models for each stage of the protein production process. We conclude that advances in experimental techniques leading to a detailed description of the process have not been matched by mathematical models that represent the details of the process and facilitate analysis. Such an exercise is the first step towards development of a framework for a systems biology analysis of the protein production process.

ELECTRONIC SUPPLEMENTARY MATERIAL

The online version of this article (doi:10.1007/s11693-011-9088-1) contains supplementary material, which is available to authorized users.

Spatio-temporal modelling of the NF-κB intracellular signalling pathway: the roles of diffusion, active transport, and cell geometry

The nuclear factor kappa B (NF-κB) intracellular signalling pathway is central to many stressful, inflammatory, and innate immune responses. NF-κB proteins themselves are transcription factors for hundreds of genes. Experiments have shown that the NF-κB pathway can exhibit oscillatory dynamics-a negative feedback loop causes oscillatory nuclear-cytoplasmic translocation of NF-κB. Given that cell size and shape are known to influence intracellular signal transduction, we consider a spatio-temporal model of partial differential equations for the NF-κB pathway, where we model molecular movement by diffusion and, for several key species including NF-κB, by active transport as well. Through numerical simulations we find values for model parameters such that sustained oscillatory dynamics occur. Our spatial profiles and animations bear a striking resemblance to experimental images and movie clips employing fluorescent fusion proteins. We discover that oscillations in nuclear NF-κB may occur when active transport is across the nuclear membrane only, or when no species are subject to active transport. However, when active transport is across the nuclear membrane and NF-κB is additionally actively transported through the cytoplasm, oscillations are lost. Hence transport mechanisms in a cell will influence its response to activation of its NF-κB pathway. We also demonstrate that sustained oscillations in nuclear NF-κB are somewhat robust to changes in the shape of the cell, or the shape, location, and size of its nucleus, or the location of ribosomes. Yet if the cell is particularly flat or the nucleus sufficiently small, then oscillations are lost. Thus the geometry of a cell may partly determine its response to NF-κB activation. The NF-κB pathway is known to be constitutively active in several human cancers. Our spatially explicit modelling approach will allow us, in future work, to investigate targeted drug therapy of tumours.

Delay in feedback repression by cryptochrome 1 is required for circadian clock function

Direct evidence for the requirement of delay in feedback repression in the mammalian circadian clock has been elusive. Cryptochrome 1 (Cry1), an essential clock component, displays evening-time expression and serves as a strong repressor at morning-time elements (E box/E' box). In this study, we reveal that a combination of day-time elements (D box) within the Cry1-proximal promoter and night-time elements (RREs) within its intronic enhancer gives rise to evening-time expression. A synthetic composite promoter produced evening-time expression, which was further recapitulated by a simple phase-vector model. Of note, coordination of day-time with night-time elements can modulate the extent of phase delay. A genetic complementation assay in Cry1(-/-):Cry2(-/-) cells revealed that substantial delay of Cry1 expression is required to restore circadian rhythmicity, and its prolonged delay slows circadian oscillation. Taken together, our data suggest that phase delay in Cry1 transcription is required for mammalian clock function.

A Wnt oscillator model for somitogenesis

We propose a model for the segmentation clock in vertebrate somitogenesis, based on the Wnt signaling pathway. The core of the model is a negative feedback loop centered around the Axin2 protein. Axin2 is activated by beta-catenin, which in turn is degraded by a complex of GSK3beta and Axin2. The model produces oscillatory states of the involved constituents with typical time periods of a few hours (ultradian oscillations). The oscillations are robust to changes in parameter values and are often spiky, where low concentration values of beta-catenin are interrupted by sharp peaks. Necessary for the oscillations is the saturated degradation of Axin2. Somite formation in chick and mouse embryos is controlled by a spatial Wnt gradient which we introduce in the model through a time-dependent decrease in Wnt3a ligand level. We find that the oscillations disappear as the ligand concentration decreases, in agreement with observations on embryos.

Probability distributed time delays: integrating spatial effects into temporal models

Background

In order to provide insights into the complex biochemical processes inside a cell, modelling approaches must find a balance between achieving an adequate representation of the physical phenomena and keeping the associated computational cost within reasonable limits. This issue is particularly stressed when spatial inhomogeneities have a significant effect on system's behaviour. In such cases, a spatially-resolved stochastic method can better portray the biological reality, but the corresponding computer simulations can in turn be prohibitively expensive.

Results

We present a method that incorporates spatial information by means of tailored, probability distributed time-delays. These distributions can be directly obtained by single in silico or a suitable set of in vitro experiments and are subsequently fed into a delay stochastic simulation algorithm (DSSA), achieving a good compromise between computational costs and a much more accurate representation of spatial processes such as molecular diffusion and translocation between cell compartments. Additionally, we present a novel alternative approach based on delay differential equations (DDE) that can be used in scenarios of high molecular concentrations and low noise propagation.

Conclusions

Our proposed methodologies accurately capture and incorporate certain spatial processes into temporal stochastic and deterministic simulations, increasing their accuracy at low computational costs. This is of particular importance given that time spans of cellular processes are generally larger (possibly by several orders of magnitude) than those achievable by current spatially-resolved stochastic simulators. Hence, our methodology allows users to explore cellular scenarios under the effects of diffusion and stochasticity in time spans that were, until now, simply unfeasible. Our methodologies are supported by theoretical considerations on the different modelling regimes, i.e. spatial vs. delay-temporal, as indicated by the corresponding Master Equations and presented elsewhere.

Cellular corepressor TLE2 inhibits replication-and-transcription- activator-mediated transactivation and lytic reactivation of Kaposi's sarcoma-associated herpesvirus

Replication and transcription activator (RTA) encoded by open reading frame 50 (ORF50) of Kaposi's sarcoma-associated herpesvirus (KSHV) is essential and sufficient to initiate lytic reactivation. RTA activates its target genes through direct binding with high affinity to its responsive elements or by interaction with cellular factors, such as RBP-Jkappa, Ap-1, C/EBP-alpha, and Oct-1. In this study, we identified transducin-like enhancer of split 2 (TLE2) as a novel RTA binding protein by using yeast two-hybrid screening of a human spleen cDNA library. The interaction between TLE2 and RTA was confirmed by glutathione S-transferase (GST) binding and coimmunoprecipitation assays. Immunofluorescence analysis showed that TLE2 and RTA were colocalized in the same nuclear compartment in KSHV-infected cells. This interaction recruited TLE2 to RTA bound to its recognition sites on DNA and repressed RTA auto-activation and transactivation activity. Moreover, TLE2 also inhibited the induction of lytic replication and virion production driven by RTA. We further showed that the Q (Gln-rich), SP (Ser-Pro-rich), and WDR (Trp-Asp repeat) domains of TLE2 and the Pro-rich domain of RTA were essential for this interaction. RBP-Jkappa has been shown previously to bind to the same Pro-rich domain of RTA, and this binding can be subject to competition by TLE2. In addition, TLE2 can form a complex with RTA to access the cognate DNA sequence of the RTA-responsive element at different promoters. Intriguingly, the transcription level of TLE2 could be upregulated by RTA during the lytic reactivation process. In conclusion, we identified a new RTA binding protein, TLE2, and demonstrated that TLE2 inhibited replication and transactivation mediated by RTA. This provides another potentially important mechanism for maintenance of KSHV viral latency through interaction with a host protein.

Computational models of the Notch network elucidate mechanisms of context-dependent signaling

The Notch signaling pathway controls numerous cell fate decisions during development and adulthood through diverse mechanisms. Thus, whereas it functions as an oscillator during somitogenesis, it can mediate an all-or-none cell fate switch to influence pattern formation in various tissues during development. Furthermore, while in some contexts continuous Notch signaling is required, in others a transient Notch signal is sufficient to influence cell fate decisions. However, the signaling mechanisms that underlie these diverse behaviors in different cellular contexts have not been understood. Notch1 along with two downstream transcription factors hes1 and RBP-Jk forms an intricate network of positive and negative feedback loops, and we have implemented a systems biology approach to computationally study this gene regulation network. Our results indicate that the system exhibits bistability and is capable of switching states at a critical level of Notch signaling initiated by its ligand Delta in a particular range of parameter values. In this mode, transient activation of Delta is also capable of inducing prolonged high expression of Hes1, mimicking the “ON” state depending on the intensity and duration of the signal. Furthermore, this system is highly sensitive to certain model parameters and can transition from functioning as a bistable switch to an oscillator by tuning a single parameter value. This parameter, the transcriptional repression constant of hes1, can thus qualitatively govern the behavior of the signaling network. In addition, we find that the system is able to dampen and reduce the effects of biological noise that arise from stochastic effects in gene expression for systems that respond quickly to Notch signaling.

This work thus helps our understanding of an important cell fate control system and begins to elucidate how this context dependent signaling system can be modulated in different cellular settings to exhibit entirely different behaviors.

The Notch signaling pathway is an evolutionarily conserved signaling system that is involved in various cell fate decisions, both during development of an organism and during adulthood. While the same core circuit functions in various different cellular contexts, it has experimentally been shown to elicit varied behaviors and responses. On the one hand, it functions as a cellular oscillator critical for somitogenesis, whereas in other situations, it can function as a cell fate switch to pattern developing tissue, for example in the Drosophila eye. Furthermore, malfunctioning of Notch signaling is implicated in various cancers. To better understand the underlying mechanisms that allow the network to function distinctly in different contexts, we have mathematically modeled the behavior of the Notch network, encompassing the Notch gene along with two of its downstream effector transcription factors, which together form a network of positive and negative feedback loops. Our results indicate that the qualitative and quantitative behavior of the system can readily be tuned based on key parameters to reflect its multiple roles. Furthermore, our results provide insights into alterations in the signaling system that lead to malfunction and hence disease, which could be used to identify potential drug targets for therapy.

Delay-induced degrade-and-fire oscillations in small genetic circuits

Robust oscillations have recently been observed in a synthetic gene network composed of coupled positive and negative feedback loops. Here we use deterministic and stochastic modeling to investigate how a small time delay in such regulatory networks can lead to strongly nonlinear oscillations that can be characterized by "degrade-and-fire" dynamics. We show that the period of the oscillations can be significantly greater than the delay time, provided the circuit components possess strong activation and tight repression. The variability of the period is strongly influenced by fluctuations near the oscillatory minima, when the number of regulatory molecules is small.

Bifurcation analysis informs Bayesian inference in the Hes1 feedback loop

Background

Ordinary differential equations (ODEs) are an important tool for describing the dynamics of biological systems. However, for ODE models to be useful, their parameters must first be calibrated. Parameter estimation, that is, finding parameter values given experimental data, is an inference problem that can be treated systematically through a Bayesian framework.

A Markov chain Monte Carlo approach can then be used to sample from the appropriate posterior probability distributions, provided that suitable prior distributions can be found for the unknown parameter values. Choosing these priors is therefore a vital first step in the inference process. We study here a negative feedback loop in gene regulation where an ODE incorporating a time delay has been proposed as a realistic model and where experimental data is available. Our aim is to show that a priori mathematical analysis can be exploited in the choice of priors.

Results

By focussing on the onset of oscillatory behaviour through a Hopf Bifurcation, we derive a range of analytical expressions and constraints that link the model parameters to the observed dynamics of the system. Computational tests on both simulated and experimental data emphasise the usefulness of this analysis.

Conclusion

Mathematical analysis not only gives insights into the possible dynamical behaviour of gene expression models, but can also be used to inform the choice of priors when parameters are inferred from experimental data in a Bayesian setting.

Modeling the dynamics of transcriptional gene regulatory networks for animal development

The dynamic process of cell fate specification is regulated by networks of regulatory genes. The architecture of the network defines the temporal order of specification events. To understand the dynamic control of the developmental process, the kinetics of mRNA and protein synthesis and the response of the cis-regulatory modules to transcription factor concentration must be considered. Here we review mathematical models for mRNA and protein synthesis kinetics which are based on experimental measurements of the rates of the relevant processes. The model comprises the response functions of cis-regulatory modules to their transcription factor inputs, by incorporating binding site occupancy and its dependence on biologically measurable quantities. We use this model to simulate gene expression, to distinguish between cis-regulatory execution of "AND" and "OR" logic functions, rationalize the oscillatory behavior of certain transcriptional auto-repressors and to show how linked subcircuits can be dealt with. Model simulations display the effects of mutation of binding sites, or perturbation of upstream gene expression. The model is a generally useful tool for understanding gene regulation and the dynamics of cell fate specification.

Delayed coupling theory of vertebrate segmentation

Rhythmic and sequential subdivision of the elongating vertebrate embryonic body axis into morphological somites is controlled by an oscillating multicellular genetic network termed the segmentation clock. This clock operates in the presomitic mesoderm (PSM), generating dynamic stripe patterns of oscillatory gene-expression across the field of PSM cells. How these spatial patterns, the clock's collective period, and the underlying cellular-level interactions are related is not understood. A theory encompassing temporal and spatial domains of local and collective aspects of the system is essential to tackle these questions. Our delayed coupling theory achieves this by representing the PSM as an array of phase oscillators, combining four key elements: a frequency profile of oscillators slowing across the PSM; coupling between neighboring oscillators; delay in coupling; and a moving boundary describing embryonic axis elongation. This theory predicts that the segmentation clock's collective period depends on delayed coupling. We derive an expression for pattern wavelength across the PSM and show how this can be used to fit dynamic wildtype gene-expression patterns, revealing the quantitative values of parameters controlling spatial and temporal organization of the oscillators in the system. Our theory can be used to analyze experimental perturbations, thereby identifying roles of genes involved in segmentation.

Dissecting the dynamics of the Hes1 genetic oscillator

Serum stimulation of a number of different mouse cell lines results in sustained oscillations of Hes1, a member of this Hes/Her family of transcription factors. Quantitative time-course expression data obtained in this system provide an excellent opportunity to explore transcriptional oscillations in a relatively simple setting. Simple models of the Hes1 regulatory circuit are capable of generating oscillations that share many features with those observed in mouse fibroblasts, and highlight the central role played by delayed negative feedback. However, taking into account constraints on model parameters imposed by experimental data, these models can only generate oscillations with quite low peak-to-trough expression ratios. To explore the origin of this limitation, we develop a more detailed model of the Hes1 circuit, incorporating nucleo-cytoplasmic transport, Hes1 dimerisation, and differential stability of Hes1 monomers and dimers. We show that differential protein stability can increase the amplitude of Hes1 oscillations, but that the resulting expression profiles do not fully match experimental data. We extend the model by incorporating periodic forcing of the Hes1 circuit by cyclic phosphorylation of the protein Stat3. We show that time delays and differential stability act synergistically in this extended model to generate large amplitude oscillatory solutions that match the experimental data well.

A proposed mechanism for the interaction of the segmentation clock and the determination front in somitogenesis

Background

Recent discoveries in the field of somitogenesis have confirmed, for the most part, the feasibility of the clock and wavefront model. There are good candidates for both the clock (various genes expressed cyclically in the tail bud of vertebrate embryos have been discovered) and the wavefront (there are at least three different substances, whose expression levels vary along the presomitic mesoderm [PSM], that have important effects on the formation of somites). Nevertheless, the mechanisms through which the wavefront interacts with the clock to arrest the oscillations and induce somite formation have not yet been fully elucidated.

Principal Findings

In this work, we propose a gene regulatory network which is consistent with all known experimental facts in embryonic mice, and whose dynamic behaviour provides a potential explanation for the periodic aggregation of PSM cells into blocks: the first step leading to the formation of somites.

Significance

To our knowledge, this is the first proposed mechanism that fully explains how a block of PSM cells can stop oscillating simultaneously, and how this process is repeated periodically, via the interaction of the segmentation clock and the determination front.

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.

The segmentation clock in mice: Interaction between the Wnt and Notch signalling pathways

In the last few years, the efforts to elucidate the mechanisms underlying the segmentation clock in various vertebrate species have multiplied. Early evidence suggested that oscillations are caused by one of the genes under the Notch signalling pathway (like those of the her or Hes families). Recently, Aulehla et al. [Wnt3a plays a major role in the segmentation clock controlling somitogenesis. Dev. Cell 4, 395-406] discovered that Axin2 (a gene under the Wnt3a signalling pathway) also oscillates in the presomitic mesoderm (PSM) of mice embryos and proposed some mechanisms through which the Notch and Wnt3a pathways may interact. They further suggested that a decreasing concentration of Wnt3a along the PSM may be the gradient the segmentation clock interacts with to form somites. These results were reviewed by Rida et al. [A notch feeling of somite segmentation and beyond. Dev. Biol. 265, 2-22], who introduced a complex clockwork comprising genes Hes1, Lfng (under the Notch pathway), and Axin2, as well as their multiple interactions. In the present work we develop a mathematical model based on the Rida et al. review and use it to tackle some of the questions raided by the Aulehla et al. paper: can the Axin2 feedback loop constitute a clock? Could a decreasing Wnt3a signaling constitute the wavefront, where phase is recorded and the spatial pattern laid down? What is the master oscillator?

Bayesian inference for dynamic transcriptional regulation; the Hes1 system as a case study

MOTIVATION

In this study, we address the problem of estimating the parameters of regulatory networks and provide the first application of Markov chain Monte Carlo (MCMC) methods to experimental data. As a case study, we consider a stochastic model of the Hes1 system expressed in terms of stochastic differential equations (SDEs) to which rigorous likelihood methods of inference can be applied. When fitting continuous-time stochastic models to discretely observed time series the lengths of the sampling intervals are important, and much of our study addresses the problem when the data are sparse.

RESULTS

We estimate the parameters of an autoregulatory network providing results both for simulated and real experimental data from the Hes1 system. We develop an estimation algorithm using MCMC techniques which are flexible enough to allow for the imputation of latent data on a finer time scale and the presence of prior information about parameters which may be informed from other experiments as well as additional measurement error.

The 'Marx' of Groucho on development and disease

Groucho proteins are abundant and broadly expressed nuclear factors that lack intrinsic DNA-binding activity but can interact with a variety of DNA-binding proteins. The recruitment of Groucho to specific gene regulatory sequences results in transcriptional repression. In both invertebrates and vertebrates, Groucho family members act as important regulators of several signaling mechanisms, including the Notch, Wingless/Wnt and Dpp/BMP/TGF-beta signaling pathways. Recent studies of embryonic development in several species point to an important role for Groucho in the regulation of multiple patterning and differentiation events. Moreover, a deregulated expression of human Groucho family members is correlated with several neoplastic conditions. Here we focus on the functions of Groucho proteins during body patterning and their implication in tumorigenesis.

Oscillations in the immune system

Oscillations are surprisingly common in the immune system, both in its healthy state and in disease. The most famous example is that of periodic fevers caused by the malaria parasite. A number of hereditary disorders, which also cause periodic fevers, have also been known for a long time. Various reports of oscillations in cytokine concentrations following antigen challenge have been published over at least the past three decades. Oscillations can also occur at the intracellular level. Calcium oscillations following T-cell activation are familiar to all immunologists, and metabolic and reactive oxygen species oscillations in neutrophils have been well documented. More recently, oscillations in nuclear factor kappaB activity following stimulation by tumor necrosis factor alpha have received considerable publicity. However, despite all of these examples, oscillations in the immune system still tend to be considered mainly as pathological aberrations, and their causes and significance remained largely unknown. This is partly because of a lack of awareness within the immunological community of the appropriate theoretical frameworks for describing and analyzing such behavior. We provide an introduction to these frameworks and give a survey of the currently known oscillations that occur within the immune system.