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

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.

Dynamic bistable switches enhance robustness and accuracy of cell cycle transitions

Bistability is a common mechanism to ensure robust and irreversible cell cycle transitions. Whenever biological parameters or external conditions change such that a threshold is crossed, the system abruptly switches between different cell cycle states. Experimental studies have uncovered mechanisms that can make the shape of the bistable response curve change dynamically in time. Here, we show how such a dynamically changing bistable switch can provide a cell with better control over the timing of cell cycle transitions. Moreover, cell cycle oscillations built on bistable switches are more robust when the bistability is modulated in time. Our results are not specific to cell cycle models and may apply to other bistable systems in which the bistable response curve is time-dependent.

Many systems in nature show bistability, which means they can evolve to one of two stable steady states under exactly the same conditions. Which state they evolve to depends on where the system comes from. Such bistability underlies the switching behavior that is essential for cells to progress in the cell division cycle. A quick switch happens when the cell jumps from one steady state to another steady state. Typical of this switching behavior is its robustness and irreversibility. In this paper, we expand this viewpoint of the dynamics of the cell cycle by considering bistable switches which themselves are changing in time. This gives the cell an extra layer of control over transitions both in time and in space, and can make those transitions more robust. Such dynamically changing bistability can appear very naturally. We show this in a model of mitotic entry, in which we include a nuclear and cytoplasmic compartment. The activity of a crucial cell cycle protein follows a bistable switch in each compartment, but the shape of its response is changing in time as proteins are imported into and exported from the nucleus.

Oscillatory Dynamics of p53-Mdm2 Circuit in Response to DNA Damage Caused by Ionizing Radiation

Although the dynamical behavior of the p53-Mdm2 loop has been extensively studied, the understanding of the mechanism underlying the regulation of this pathway still remains limited. Herein, we developed an integrated model with five basic components and three ubiquitous time delays for the p53-Mdm2 interaction in response to DNA damage following ionizing radiation (IR). We showed that a sufficient amount of activated ATM level can initiate the p53 oscillations with nearly the same amplitude over a wide range of the ATM level; a proper range of p53 level is also required for generating the oscillations, for too high or too low levels it would fail to generate the oscillations; and increased Mdm2 level leads to decreased amplitude of the p53 oscillation and reduced expression of the p53 activity. Moreover, we found that the negative feedback loop formed between p53 and nuclear Mdm2 plays a dominant role in determining the p53 dynamics, whereas when interaction strength of the negative feedback loop becomes weaker, the positive feedback loop formed between p53 and cytoplasmatic Mdm2 can induce different types of dynamics. Furthermore, we demonstrated that the total time delay required for protein production and nuclear translocation of Mdm2 can induce p53 oscillations even when the p53 level is at a certain stable high steady state or at a certain stable low steady state. In addition, the two important features of the oscillatory dynamics-amplitude and period-can be controlled by such time delay. These results are in agreement with multiple experimental observations and may enrich our understanding of the dynamics of the p53 network.

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

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

Oscillations in well-mixed, deterministic feedback systems: Beyond ring oscillators

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I present a way of breaking down regulatory networks to find Hopf bifurcations. This helps find optimal conditions for oscillations in dynamical systems models of these networks.

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In a model of negative auto-regulation of a gene by its dimeric protein, it is optimal for the monomer to degrade faster than the mRNA and the mRNA to degrade faster than the dimer.

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Adding a weak positive feedback loop to a repressilator increases the probability of oscillations.

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The optimal degradation rate of species in the sub-loop is higher than that of species outside it.

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The opposite is true for a negative feedback sub-loop or a very strong positive feedback sub-loop.

A ring oscillator is a system in which one species regulates the next, which regulates the next and so on until the last species regulates the first. In addition, the number of the regulations which are negative, and so result in a reduction in the regulated species, is odd, making the overall feedback in the loop negative. In ring oscillators, the probability of oscillations is maximised if the degradation rates of the species are equal. When there is more than one loop in the regulatory network, the dynamics can be more complicated. Here, a systematic way of organising the characteristic equation of ODE models of regulatory networks is provided. This facilitates the identification of Hopf bifurcations. It is shown that the probability of oscillations in non-ring systems is maximised for unequal degradation rates. For example, when there is a ring and a second ring employing a subset of the genes in the first ring, then the probability of oscillations is maximised when the species in the sub-ring degrade more slowly than those outside, for a negative feedback subring. When the sub-ring forms a positive feedback loop, the optimal degradation rates are larger for the species in the sub-ring, provided the positive feedback is not too strong. By contrast, optimal degradation rates are smaller for the species in the sub-ring, when the positive feedback is very strong. Adding a positive feedback loop to a repressilator increases the probability of oscillations, provided the positive feedback is not too strong, whereas adding a negative feedback loop decreases the probability of oscillations. The work is illustrated with numerical simulations of example systems: an autoregulatory gene model in which transcription is downregulated by the protein dimer and three-species and four-species gene regulatory network examples.

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.

Mathematical model for transport of DNA plasmids from the external medium up to the nucleus by electroporation

We propose a mathematical model for the transport of DNA plasmids from the extracellular matrix up to the cell nucleus. The model couples two phenomena: the electroporation process, describing the cell membrane permeabilization to plasmids and the intracellular transport enhanced by the presence of microtubules. Numerical simulations of cells with arbitrary geometry, in 2D and 3D, and a network of microtubules show numerically the importance of the microtubules and the electroporation on the effectiveness of the DNA transfection, as observed by previous biological data. The paper proposes efficient numerical tools for forthcoming optimized procedures of cell transfection.

Cell shape and the microenvironment regulate nuclear translocation of NF-κB in breast epithelial and tumor cells

Although a great deal is known about the signaling events that promote nuclear translocation of NF-κB, how cellular biophysics and the microenvironment might regulate the dynamics of this pathway is poorly understood. In this study, we used high-content image analysis and Bayesian network modeling to ask whether cell shape and context features influence NF-κB activation using the inherent variability present in unperturbed populations of breast tumor and non-tumor cell lines. Cell–cell contact, cell and nuclear area, and protrusiveness all contributed to variability in NF-κB localization in the absence and presence of TNFα. Higher levels of nuclear NF-κB were associated with mesenchymal-like versus epithelial-like morphologies, and RhoA-ROCK-myosin II signaling was critical for mediating shape-based differences in NF-κB localization and oscillations. Thus, mechanical factors such as cell shape and the microenvironment can influence NF-κB signaling and may in part explain how different phenotypic outcomes can arise from the same chemical cues.

Is the Cell Nucleus a Necessary Component in Precise Temporal Patterning?

One of the functions of the cell nucleus is to help regulate gene expression by controlling molecular traffic across the nuclear envelope. Here we investigate, via stochastic simulation, what effects, if any, does segregation of a system into the nuclear and cytoplasmic compartments have on the stochastic properties of a motif with a negative feedback. One of the effects of the nuclear barrier is to delay the nuclear protein concentration, allowing it to behave in a switch-like manner. We found that this delay, defined as the time for the nuclear protein concentration to reach a certain threshold, has an extremely narrow distribution. To show this, we considered two models. In the first one, the proteins could diffuse freely from cytoplasm to nucleus (simple model); and in the second one, the proteins required assistance from a special class of proteins called importins. For each model, we generated fifty parameter sets, chosen such that the temporal profiles they effectuated were very similar, and whose average threshold time was approximately 150 minutes. The standard deviation of the threshold times computed over one hundred realizations were found to be between 1.8 and 7.16 minutes across both models. To see whether a genetic motif in a prokaryotic cell can achieve this degree of precision, we also simulated five variations on the coherent feed-forward motif (CFFM), three of which contained a negative feedback. We found that the performance of these motifs was nowhere near as impressive as the one found in the eukaryotic cell; the best standard deviation was 6.6 minutes. We argue that the significance of these results, the fact and necessity of spatio-temporal precision in the developmental stages of eukaryotes, and the absence of such a precision in prokaryotes, all suggest that the nucleus has evolved, in part, under the selective pressure to achieve highly predictable phenotypes.

The pharmacodynamics of the p53-Mdm2 targeting drug Nutlin: the role of gene-switching noise

In this work we investigate, by means of a computational stochastic model, how tumor cells with wild-type p53 gene respond to the drug Nutlin, an agent that interferes with the Mdm2-mediated p53 regulation. In particular, we show how the stochastic gene-switching controlled by p53 can explain experimental dose-response curves, i.e., the observed inter-cell variability of the cell viability under Nutlin action. The proposed model describes in some detail the regulation network of p53, including the negative feedback loop mediated by Mdm2 and the positive loop mediated by PTEN, as well as the reversible inhibition of Mdm2 caused by Nutlin binding. The fate of the individual cell is assumed to be decided by the rising of nuclear-phosphorylated p53 over a certain threshold. We also performed in silico experiments to evaluate the dose-response curve after a single drug dose delivered in mice, or after its fractionated administration. Our results suggest that dose-splitting may be ineffective at low doses and effective at high doses. This complex behavior can be due to the interplay among the existence of a threshold on the p53 level for its cell activity, the nonlinearity of the relationship between the bolus dose and the peak of active p53, and the relatively fast elimination of the drug.

P53 is an antitumor gene regulating vital cellular functions such as repair of DNA damage, cellular suicide, and cell proliferation: in many tumors p53 is lowly expressed and/or mutated. Drugs targeting the biomolecular network of p53 are becoming important. The network includes the key proteins Mdm2 and PTEN, whose production is regulated by p53, and which, in turn, enact positive and negative feedbacks on p53. Drug Nutlin, inhibiting the p53 inhibitor Mdm2, might be important for tumors where p53 is underproduced but unmutated. We investigate the cellular mechanism of action of Nutlin. The basic concept of our mathematical model is that the experimentally observed cell-to-cell variability of Nutlin efficacy is caused by the randomness of gene activation/deactivation of Mdmd2 and PTEN. Indeed, the abundance/scarceness of p53 regulates the probability that the relative genes are active or inactive. The model reproduced the experimental cell-specific response to different doses of Nutlin (dose-response curves) in some types of tumor cells. Much clinical research focus on 'metronomic' drug delivery regimens, where instead of giving large doses with long intervals, smaller doses are frequently delivered. In our simulations, dose-splitting of Nutlin produced a response generally worse than the response to a single dose.

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.

Reaction-diffusion systems for spatio-temporal intracellular protein networks: A beginner's guide with two examples

Spatio-temporal dynamics of a variety of proteins is, among other things, regulated by post-translational modifications of these proteins. Such modifications can thus influence stability and biochemical activities of the proteins, activity and stability of their upstream targets within specific signalling pathways. Commonly used mathematical tools for such protein–protein (and/or protein-mRNA) interactions in single cells, namely, Michaelis–Menten and Hill kinetics, yielding a system of ordinary differential equations, are extended here into (non-linear) partial differential equations by taking into account a more realistic spatial representation of the environment where these reactions occur. In the modelling framework under consideration, all interactions occur in a cell divided into two compartments, the nucleus and the cytoplasm, connected by the semipermeable nuclear membrane and bounded by the impermeable cell membrane. Passive transport mechanism, modelled by the so-called Kedem–Katchalsky boundary conditions, is used here to represent migration of species throughout the nuclear membrane. Nonlinear systems of partial differential equations are solved by the semi-implicit Rothe method. Examples of two spatial oscillators are shown. Namely, these are the circadian rhythm for concentration of the FRQ protein in Neurospora crassa and oscillatory dynamics observed in the activation and regulation of the p53 protein following DNA damage in mammalian cells.

The p53 protein and its molecular network: modelling a missing link between DNA damage and cell fate

Various molecular pharmacokinetic-pharmacodynamic (PK-PD) models have been proposed in the last decades to represent and predict drug effects in anticancer chemotherapies. Most of these models are cell population based since clearly measurable effects of drugs can be seen much more easily on populations of cells, healthy and tumour, than in individual cells. The actual targets of drugs are, however, cells themselves. The drugs in use either disrupt genome integrity by causing DNA strand breaks, and consequently initiate programmed cell death, or block cell proliferation mainly by inhibiting factors that enable cells to proceed from one cell cycle phase to the next through checkpoints in the cell division cycle. DNA damage caused by cytotoxic drugs (and also cytostatic drugs at high concentrations) activates, among others, the p53 protein-modulated signalling pathways that directly or indirectly force the cell to make a decision between survival and death. The paper aims to become the first-step in a larger scale enterprise that should bridge the gap between intracellular and population PK-PD models, providing oncologists with a rationale to predict and optimise the effects of anticancer drugs in the clinic. So far, it only sticks at describing p53 activation and regulation in single cells following their exposure to DNA damaging stress agents. We show that p53 oscillations that have been observed in individual cells can be reconstructed and predicted by compartmentalising cellular events occurring after DNA damage, either in the nucleus or in the cytoplasm, and by describing network interactions, using ordinary differential equations (ODEs), between the ATM, p53, Mdm2 and Wip1 proteins, in each compartment, nucleus or cytoplasm, and between the two compartments. This article is part of a Special Issue entitled: Computational Proteomics, Systems Biology & Clinical Implications.

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.

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.