Svetoslav Nikolov

Integration of sensitivity and bifurcation analysis to detect critical processes in a model combining signalling and cell population dynamics

In this article we present and test a strategy to integrate, in a sequential manner, sensitivity analysis, bifurcation analysis and predictive simulations. Our strategy uses some of these methods in a coordinated way such that information, generated in one step, feeds into the definition of further analyses and helps refining the structure of the mathematical model. The aim of the method is to help in the designing of more informative predictive simulations, which focus on critical model parameters and the biological effects of their modulation. We tested our methodology with a multilevel model, accounting for the effect of erythropoietin (Epo)-mediated JAK2-STAT5 signalling in erythropoiesis. Our analysis revealed that time-delays associated with the proliferation–differentiation process are critical to induce pathological sustained oscillations, whereas the modulation of time-delays related to intracellular signalling and hypoxia-controlled physiological dynamics is not enough to induce self-oscillations in the system. Furthermore, our results suggest that the system is able to compensate (through the physiological-level feedback loop on hypoxia) the partial impairment of intracellular signalling processes (downregulation or overexpression of Epo receptor complex and STAT5), but cannot control impairment in some critical physiological-level processes, which provoke the emergence of pathological oscillations.

A multi-level model accounting for the effects of JAK2-STAT5 signal modulation in erythropoiesis

We develop a multi-level model, using ordinary differential equations, based on quantitative experimental data, accounting for murine erythropoiesis. At the sub-cellular level, the model includes a description of the regulation of red blood cell differentiation through Epo-stimulated JAK2-STAT5 signalling activation, while at the cell population level the model describes the dynamics of (STAT5-mediated) red blood cell differentiation from their progenitors. Furthermore, the model includes equations depicting the hypoxia-mediated regulation of hormone erythropoietin blood levels. Take all together, the model constitutes a multi-level, feedback loop-regulated biological system, involving processes in different organs and at different organisational levels. We use our model to investigate the effect of deregulation in the proteins involved in the JAK2-STAT5 signalling pathway in red blood cells. Our analysis results suggest that down-regulation in any of the three signalling system components affects the hematocrit level in an individual considerably. In addition, our analysis predicts that exogenous Epo injection (an already existing treatment for several blood diseases) may compensate the effects of single down-regulation of Epo hormone level, STAT5 or EpoR/JAK2 expression level, and that it may be insufficient to counterpart a combined down-regulation of all the elements in the JAK2-STAT5 signalling cascade.

Hybrid modeling of the crosstalk between signaling and transcriptional networks using ordinary differential equations and multi-valued logic

A decade of successful results indicates that systems biology is the appropriate approach to investigate the regulation of complex biochemical networks involving transcriptional and post-transcriptional regulations. It becomes mandatory when dealing with highly interconnected biochemical networks, composed of hundreds of compounds, or when networks are enriched in non-linear motifs like feedback and feedforward loops. An emerging dilemma is to conciliate models of massive networks and the adequate description of non-linear dynamics in a suitable modeling framework. Boolean networks are an ideal representation of massive networks that are humble in terms of computational complexity and data demand. However, they are inappropriate when dealing with nested feedback/feedforward loops, structural motifs common in biochemical networks. On the other hand, models of ordinary differential equations (ODEs) cope well with these loops, but they require enormous amounts of quantitative data for a full characterization of the model. Here we propose hybrid models, composed of ODE and logical sub-modules, as a strategy to handle large scale, non-linear biochemical networks that include transcriptional and post-transcriptional regulations. We illustrate the construction of this kind of models using as example a regulatory network centered on E2F1, a transcription factor involved in cancer. The hybrid modeling approach proposed is a good compromise between quantitative/qualitative accuracy and scalability when considering large biochemical networks with a small highly interconnected core, and module of transcriptionally regulated genes that are not part of critical regulatory loops. This article is part of a Special Issue entitled: Computational Proteomics, Systems Biology & Clinical Implications. Guest Editor: Yudong Cai.

A model-based strategy to investigate the role of microRNA regulation in cancer signalling networks.

In this paper we present and discuss a model-based approach to link miRNA translational control with cell signalling networks. MicroRNAs are small regulatory RNAs that are able to regulate the activity and the stability of specific messenger RNA and have been implicated in tumour progression due to their ability to translationally regulate critical oncogenes and tumour suppressors. In our approach, data on protein–protein interactions and miRNA regulation, obtained from bioinformatics databases, are integrated with quantitative experimental data using mathematical modelling. Predictive computational simulations and qualitative (bifurcation) analyses of those mathematical models are employed to further support the investigation of such multifactorial networks in the context of cancer progression. We illustrate our approach with the C-Myc/E2F signalling network, involved in the progression of several tumour subtypes, including colorectal cancer.

Role of inhibitory proteins as modulators of oscillations in NFγkB signalling

The authors discuss the role of the Raf kinase inhibitory protein (RKIP) as a modulator of oscillations in NFB signalling. A mathematical model of the NFB signalling pathway was derived and the Lyapunov-Andronov theory was used to analyse dynamical properties of the system. The analytical results were complemented by predictive numerical simulations. Our results suggest that the nature of oscillations, emerging under sustained stimulation of the system, depends on the interplay between the IB kinase (IKK) stimulation and the inhibitory action of RKIP. The authors found a mathematical relation that defines isoclines in IKK and RKIP levels for which the properties of oscillations are conserved and changes in the stimulation can be compensated by modulating RKIP inhibition. On the other hand, the shifting from the current isocline provokes modulation in either the amplitude (for stronger stimulation) or the frequency (for weaker stimulation).

Model-based investigation of the transcriptional activity of p53 and its feedback loop regulation via 14-3-3Σ

Experiments have recently shown that p53 expression can display oscillations in response to certain stress signals. In this work, mathematical modelling and bifurcation analysis are combined to investigate under which conditions the oscillation of p53 could propagate to its direct downstream transcription targets. The authors analysis suggests that oscillations of p53 will propagate only to proteins with medium-fast mRNA and protein turnover rates. The authors retrieved data concerning the half-life of mRNA and protein for a number of p53-promoted genes and found that, according to their model, most of them are not able to inherit the oscillation of p53 because of their slow turnover rates. However, their analysis indicates that p53 oscillation may actually fine-tune the expression pattern of a protein when it is integrated with a second oscillatory signal. The authors also consider the case of additional regulatory loops affecting p53 oscillations and involving proteins transcriptionally induced by p53. Their results for 14-3-3σ, a protein that targets the p53 inhibitor MDM2 for degradation, suggest that the addition of feedback-loop regulation may modulate basic properties of p53 oscillation and induce quick cessation of them under certain physiological conditions. Moreover, the interplay between DNA damage and 14-3-3σ may induce bistability in the oscillation of p53.

Rheodynamic model of cardiac pressure pulsations

To analyse parametrically (in terms of the qualitative theory of dynamical systems) the mechanical influence of inertia, resistance (positive and negative), elasticity and other global properties of the heart-muscle on the left ventricular pressure, an active rheodynamic model based on the Newtonss principles is proposed. The equation of motion of the heart mass centre is derived from an energy conservation law balancing the rate of mechanical (kinetic and potential) energy variation and the power of chemical energy influx and dissipative energy outflux. A corresponding dynamical system of two ordinary differential equations is obtained and parametrically analysed in physiological conditions. As a result, the following main conclusion is made: in physiological norm, because of the heart electrical activity, its equilibrium state is unstable and around it, mechanical self-oscillations emerge. In case the electrical activity ceases, an inverse phase reconstruction occurs during which the unstable equilibrium state of the system becomes stable and the self-oscillations disappear.

Stability and Andronov-Hopf Bifurcation of a System with Three Time Delays

A general system of three autonomous ordinary differential equations with three discrete time delays is considered. With respect to the delays, we investigate the local stability of equilibria by analyzing the corresponding characteristic equation. Using the Hopf bifurcation theorem, we predict the occurrence of a limit cycle bifurcation for the time delay parameters. Thus, some new mathematical results are obtained. Finally, the above mentioned criteria are applied to a system modelling miRNA regulation.

Dynamics of a miRNA Model with Two Delays

ABSTRACT MicroRNAs are a large class of post-transcriptional regulators of gene expression and they have important regulatory functions in basic cellular processes. In this paper, mathematical modeling and bifurcation analysis are combined to investigate in details the effect of time delays on the dynamics of a feedback system regulated with miRNA. Using the Hopf bifurcation theorem, we predict the occurrence of a limit cycle bifurcation for the time delay parameters. From the accomplished analytical and numerical results, it becomes clear that time delays have a destabilization dynamical role. Moreover, our numerical analysis suggests that amplitude of sustained miRNA oscillations depend essentially on the values of time delays.

Model-Based Phenotypic Signatures Governing the Dynamics of the Stem and Semi-differentiated Cell Populations in Dysplastic Colonic Crypts

Mathematical modeling of cell differentiated in colonic crypts can contribute to a better understanding of basic mechanisms underlying colonic tissue organization, but also its deregulation during carcinogenesis and tumor progression. Here, we combined bifurcation analysis to assess the effect that time delay has in the complex interplay of stem cells and semi-differentiated cells at the niche of colonic crypts, and systematic model perturbation and simulation to find model-based phenotypes linked to cancer progression. The models suggest that stem cell and semi-differentiated cell population dynamics in colonic crypts can display chaotic behavior. In addition, we found that clinical profiling of colorectal cancer correlates with the in silico phenotypes proposed by the mathematical model. Further, potential therapeutic targets for chemotherapy resistant phenotypes are proposed, which in any case will require experimental validation.