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

Exploring the influence of cytosolic and membrane FAK activation on YAP/TAZ nuclear translocation

Membrane binding and unbinding dynamics play a crucial role in the biological activity of several nonintegral membrane proteins, which have to be recruited to the membrane to perform their functions. By localizing to the membrane, these proteins are able to induce downstream signal amplification in their respective signaling pathways. Here, we present a 3D computational approach using reaction-diffusion equations to investigate the relation between membrane localization of focal adhesion kinase (FAK), Ras homolog family member A (RhoA), and signal amplification of the YAP/TAZ signaling pathway. Our results show that the theoretical scenarios in which FAK is membrane bound yield robust and amplified YAP/TAZ nuclear translocation signals. Moreover, we predict that the amount of YAP/TAZ nuclear translocation increases with cell spreading, confirming the experimental findings in the literature. In summary, our in silico predictions show that when the cell membrane interaction area with the underlying substrate increases, for example, through cell spreading, this leads to more encounters between membrane-bound signaling partners and downstream signal amplification. Because membrane activation is a motif common to many signaling pathways, this study has important implications for understanding the design principles of signaling networks.

Polarization of concave domains by traveling wave pinning

Pattern formation is one of the most fundamental yet puzzling phenomena in physics and biology. We propose that traveling front pinning into concave portions of the boundary of 3-dimensional domains can serve as a generic gradient-maintaining mechanism. Such a mechanism of domain polarization arises even for scalar bistable reaction-diffusion equations, and, depending on geometry, a number of stationary fronts may be formed leading to complex spatial patterns. The main advantage of the pinning mechanism, with respect to the Turing bifurcation, is that it allows for maintaining gradients in the specific regions of the domain. By linking the instant domain shape with the spatial pattern, the mechanism can be responsible for cellular polarization and differentiation.

Importins promote high-frequency NF-κB oscillations increasing information channel capacity

Background

Importins and exportins influence gene expression by enabling nucleocytoplasmic shuttling of transcription factors. A key transcription factor of innate immunity, NF-κB, is sequestered in the cytoplasm by its inhibitor, IκBα, which masks nuclear localization sequence of NF-κB. In response to TNFα or LPS, IκBα is degraded, which allows importins to bind NF-κB and shepherd it across nuclear pores. NF-κB nuclear activity is terminated when newly synthesized IκBα enters the nucleus, binds NF-κB and exportin which directs the complex to the cytoplasm. Although importins/exportins are known to regulate spatiotemporal kinetics of NF-κB and other transcription factors governing innate immunity, the mechanistic details of these interactions have not been elucidated and mathematically modelled.

Results

Based on our quantitative experimental data, we pursue NF-κB system modelling by explicitly including NF-κB–importin and IκBα–exportin binding to show that the competition between importins and IκBα enables NF-κB nuclear translocation despite high levels of IκBα. These interactions reduce the effective relaxation time and allow the NF-κB regulatory pathway to respond to recurrent TNFα pulses of 45-min period, which is about twice shorter than the characteristic period of NF-κB oscillations. By stochastic simulations of model dynamics we demonstrate that randomly appearing, short TNFα pulses can be converted to essentially digital pulses of NF-κB activity, provided that intervals between input pulses are not shorter than 1 h.

Conclusions

By including interactions involving importin-α and exportin we bring the modelling of spatiotemporal kinetics of transcription factors to a more mechanistic level. Basing on the analysis of the pursued model we estimated the information transmission rate of the NF-κB pathway as 1 bit per hour.

Reviewers

This article was reviewed by Marek Kimmel, James Faeder and William Hlavacek.

Electronic supplementary material

The online version of this article (doi:10.1186/s13062-016-0164-z) contains supplementary material.

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.

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.

Regulation of NF-κB Oscillation by Nuclear Transport: Mechanisms Determining the Persistency and Frequency of Oscillation

The activated transcription factor NF-κB shuttles between the cytoplasm and the nucleus resulting in the oscillation of nuclear NF-κB (NF-κB_n). The oscillation pattern of NF-κB_n is implicated in the regulation of gene expression profiles. Using computational models, we previously reported that spatial parameters, such as the diffusion coefficient, nuclear to cytoplasmic volume ratio, transport through the nuclear envelope, and the loci of translation of IκB protein, modified the oscillation pattern of NF-κB_n. In a subsequent report, we elucidated the importance of the “reset” of NF-κB_n (returning of NF-κB to the original level) and of a “reservoir” of IκB in the cytoplasm. When the diffusion coefficient of IκB was large, IκB stored at a distant location from the nucleus diffused back to the nucleus and “reset” NF-κB_n. Herein, we report mechanisms that regulate the persistency and frequency of NF-κB_n oscillation by nuclear transport. Among the four parameters of nuclear transport tested in our spatio-temporal computational model, the export of IκB mRNA from the nucleus regulated the persistency of oscillation. The import of IκB to the nucleus regulated the frequency of oscillation. The remaining two parameters, import and export of NF-κB to and from the nucleus, had virtually no effect on the persistency or frequency. Our analyses revealed that lesser export of IκB mRNA allowed NF-κB_n to transcript greater amounts of IκB mRNA, which was retained in the nucleus, and was subsequently exported to the cytoplasm, where large amounts of IκB were synthesized to “reset” NF-κB_n and drove the persistent oscillation. On the other hand, import of greater amounts of IκB led to an increase in the influx and the efflux of NF-κB to and from the nucleus, resulting in an increase in the oscillation frequency. Our study revealed the importance of nuclear transport in regulating the oscillation pattern of NF-κB_n.

HIF-1α restricts NF-κB-dependent gene expression to control innate immunity signals

Hypoxia and inflammation are intimately linked. It is known that nuclear factor κB (NF-κB) regulates the hypoxia-inducible factor (HIF) system, but little is known about how HIF regulates NF-κB. Here, we show that HIF-1α represses NF-κB-dependent gene expression. HIF-1α depletion results in increased NF-κB transcriptional activity both in mammalian cells and in the model organism Drosophila melanogaster. HIF-1α depletion enhances the NF-κB response, and this required not only the TAK-IKK complex, but also CDK6. Loss of HIF-1α results in an increased angiogenic response in mammalian cancer cells and increased mortality in Drosophila following infection. These results indicate that HIF-1α is required to restrain the NF-κB response, and thus prevents excessive and damaging pro-inflammatory responses.

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.

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.

Mathematical modeling of the Phoenix Rising pathway

Apoptosis is a tightly controlled process in mammalian cells. It is important for embryogenesis, tissue homoeostasis, and cancer treatment. Apoptosis not only induces cell death, but also leads to the release of signals that promote rapid proliferation of surrounding cells through the Phoenix Rising (PR) pathway. To quantitatively understand the kinetics of interactions of different molecules in this pathway, we developed a mathematical model to simulate the effects of various changes in the PR pathway on the secretion of prostaglandin E2 (PGE2), a key factor for promoting cell proliferation. These changes include activation of caspase 3 (C3), caspase 7 (C7), and nuclear factor κB (NFκB). In addition, we simulated the effects of cyclooxygenase-2 (COX2) inhibition and C3 knockout on the level of secreted PGE2. The model predictions on PGE2 in MEF and 4T1 cells at 48 hours after 10-Gray radiation were quantitatively consistent with the experimental data in the literature. Compared to C7, the model predicted that C3 activation was more critical for PGE2 production. The model also predicted that PGE2 production could be significantly reduced when COX2 expression was blocked via either NFκB inactivation or treatment of cells with exogenous COX2 inhibitors, which led to a decrease in the rate of conversion from arachidonic acid to prostaglandin H2 in the PR pathway. In conclusion, the mathematical model developed in this study yielded new insights into the process of tissue regrowth stimulated by signals from apoptotic cells. In future studies, the model can be used for experimental data analysis and assisting development of novel strategies/drugs for improving cancer treatment or normal tissue regeneration.

Apoptosis, or programmed cell death, is known to be important for embryogenesis, tissue homoeostasis, and cancer treatment. Furthermore, researchers have recently observed that apoptosis may promote wound healing and tissue regeneration, and accelerate undesired solid tumor regrowth after chemotherapy/radiation therapy. Mechanisms of apoptosis-induced tissue regrowth are related to a molecular network discovered recently in our lab. To quantitatively understand the kinetics of interactions of different molecules in this network, we developed a mathematical model and validated it by comparing the simulation results to experimental data reported in previous studies. To gain new insights into the process of tissue regrowth after inducing apoptosis, we used the model to simulate the effects of radiation on the production of a key growth stimulating factor, PGE2, in apoptotic cells. Additionally, we simulated how PGE2 production could be altered when cells were treated with different inhibitors. We expect that the new mathematical model can be used in future studies to facilitate design of better approaches to cancer treatment or normal tissue regeneration.

Spontaneous NF-κB activation by autocrine TNFα signaling: a computational analysis

NF-κB is a key transcription factor that regulates innate immune response. Its activity is tightly controlled by numerous feedback loops, including two negative loops mediated by NF-κB inducible inhibitors, IκBα and A20, which assure oscillatory responses, and by positive feedback loops arising due to the paracrine and autocrine regulation via TNFα, IL-1 and other cytokines. We study the NF-κB system of interlinked negative and positive feedback loops, combining bifurcation analysis of the deterministic approximation with stochastic numerical modeling. Positive feedback assures the existence of limit cycle oscillations in unstimulated wild-type cells and introduces bistability in A20-deficient cells. We demonstrated that cells of significant autocrine potential, i.e., cells characterized by high secretion of TNFα and its receptor TNFR1, may exhibit sustained cytoplasmic-nuclear NF-κB oscillations which start spontaneously due to stochastic fluctuations. In A20-deficient cells even a small TNFα expression rate qualitatively influences system kinetics, leading to long-lasting NF-κB activation in response to a short-pulsed TNFα stimulation. As a consequence, cells with impaired A20 expression or increased TNFα secretion rate are expected to have elevated NF-κB activity even in the absence of stimulation. This may lead to chronic inflammation and promote cancer due to the persistent activation of antiapoptotic genes induced by NF-κB. There is growing evidence that A20 mutations correlate with several types of lymphomas and elevated TNFα secretion is characteristic of many cancers. Interestingly, A20 loss or dysfunction also leaves the organism vulnerable to septic shock and massive apoptosis triggered by the uncontrolled TNFα secretion, which at high levels overcomes the antiapoptotic action of NF-κB. It is thus tempting to speculate that some cancers of deregulated NF-κB signaling may be prone to the pathogen-induced apoptosis.

Modeling the intra- and extracellular cytokine signaling pathway under heat stroke in the liver

Heat stroke (HS) is a life-threatening illness induced by prolonged exposure to a hot environment that causes central nervous system abnormalities and severe hyperthermia. Current data suggest that the pathophysiological responses to heat stroke may not only be due to the immediate effects of heat exposure per se but also the result of a systemic inflammatory response syndrome (SIRS). The observation that pro- (e.g., IL-1) and anti-inflammatory (e.g., IL-10) cytokines are elevated concomitantly during recovery suggests a complex network of interactions involved in the manifestation of heat-induced SIRS. In this study, we measured a set of circulating cytokine/soluble cytokine receptor proteins and liver cytokine and receptor mRNA accumulation in wild-type and tumor necrosis factor (TNF) receptor knockout mice to assess the effect of neutralization of TNF signaling on the SIRS following HS. Using a systems approach, we developed a computational model describing dynamic changes (intra- and extracellular events) in the cytokine signaling pathways in response to HS that was fitted to novel genomic (liver mRNA accumulation) and proteomic (circulating cytokines and receptors) data using global optimization. The model allows integration of relevant biological knowledge and formulation of new hypotheses regarding the molecular mechanisms behind the complex etiology of HS that may serve as future therapeutic targets. Moreover, using our unique modeling framework, we explored cytokine signaling pathways with three in silico experiments (e.g. by simulating different heat insult scenarios and responses in cytokine knockout strains in silico).

Mathematical modelling and computational study of two-dimensional and three-dimensional dynamics of receptor-ligand interactions in signalling response mechanisms

Cell signalling processes involve receptor trafficking through highly connected networks of interacting components. The binding of surface receptors to their specific ligands is a key factor for the control and triggering of signalling pathways. But the binding process still presents many enigmas and, by analogy with surface catalytic reactions, two different mechanisms can be conceived: the first mechanism is related to the Eley-Rideal (ER) mechanism, i.e. the bulk-dissolved ligand interacts directly by pure three-dimensional (3D) diffusion with the specific surface receptor; the second mechanism is similar to the Langmuir-Hinshelwood (LH) process, i.e. 3D diffusion of the ligand to the cell surface followed by reversible ligand adsorption and subsequent two-dimensional (2D) surface diffusion to the receptor. A situation where both mechanisms simultaneously contribute to the signalling process could also occur. The aim of this paper is to perform a computational study of the behavior of the signalling response when these different mechanisms for ligand-receptor interactions are integrated into a model for signal transduction and ligand transport. To this end, partial differential equations have been used to develop spatio-temporal models that show trafficking dynamics of ligands, cell surface components, and intracellular signalling molecules through the different domains of the system. The mathematical modeling developed for these mechanisms has been applied to the study of two situations frequently found in cell systems: (a) dependence of the signal response on cell density; and (b) enhancement of the signalling response in a synaptic environment.

Inference of dynamic networks using time-course data

Cells execute their functions through dynamic operations of biological networks. Dynamic networks delineate the operation of biological networks in terms of temporal changes of abundances or activities of nodes (proteins and RNAs), as well as formation of new edges and disappearance of existing edges over time. Global genomic and proteomic technologies can be used to decode dynamic networks. However, using these experimental methods, it is still challenging to identify temporal transition of nodes and edges. Thus, several computational methods for estimating dynamic topological and functional characteristics of networks have been introduced. In this review, we summarize concepts and applications of these computational methods for inferring dynamic networks and further summarize methods for estimating spatial transition of biological networks.

Modelling c-Abl Signalling in Activated Neutrophils: the Anti-inflammatory Effect of Seliciclib

When mammalian tissues are infected by bacteria or fungi, inflammatory cytokines are released that cause circulating neutrophils to invade the infected tissue. The cytosolic tyrosine kinase, c-Abl, in these tissue neutrophils is activated by TNFα. c-Abl then phosphorylates STAT transcription factors, which results in production of the antiapoptotic protein Mcl-1. The normally short-lived tissue neutrophils are then unable to enter apoptosis. c-Abl also causes release of reactive oxygen species (ROS) from the mitochondria of the activated neutrophils. These ROS, and ROS generated by NADPH oxidase, are bactericidal agents of the innate immune system. In some inflammatory diseases, such as chronic obstructive pulmonary disease (COPD), the invading neutrophils become permanently activated, and the resulting ROS overproduction causes severe tissue damage. The cyclin-dependent kinase inhibitor, seliciclib, blocks transcription through inhibition of cdk9. This results in a relatively rapid decline of antiapoptotic Mcl-1 transcripts in activated neutrophils, an increase in neutrophil apoptosis, and less ROS leakage and oxidative damage. We present here a model of neutrophil kinetics that simulates the principal pathways of c-Abl signalling and use it to explore possible treatment options for inflammatory lung disease.

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.

A spatial physiological model for p53 intracellular dynamics

In this paper we design and analyse a physiologically based model representing the accumulation of protein p53 in the nucleus after triggering of ATM by DNA damage. The p53 protein is known to have a central role in the response of the cell to cytotoxic or radiotoxic insults resulting in DNA damage. A reasonable requirement for a model describing intracellular signalling pathways is taking into account the basic feature of eukaryotic cells: the distinction between nucleus and cytoplasm. Our aim is to show, on a simple reaction network describing p53 dynamics, how this basic distinction provides a framework which is able to yield expected oscillatory dynamics without introducing either positive feedbacks or delays in the reactions. Furthermore we prove that oscillations appear only if some spatial constraints are respected, e.g. if the diffusion coefficients correspond to known biological values. Finally we analyse how the spatial features of a cell influence the dynamic response of the p53 network to DNA damage, pointing out that the protein oscillatory dynamics is indeed a response that is robust towards changes with respect to cellular environments. Even if we change the cell shape or its volume or better its ribosomal distribution, we observe that DNA damage yields sustained oscillations of p53.

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.