Dynamic properties of a delayed protein cross talk model

Stability and Hopf bifurcation analysis of lac Operon model with distributed delay and nonlinear degradation rate

We propose a simple model of lac operon that describes the expression of B-galactosidase from lac Z gene in Escherichia coli, through the interaction among several identical mRNA. Our goal is to explore the complex dynamics (i.e. the oscillation phenomenon) of this architecture mediated by this interaction. This model was theoretically and numerically investigated using distributed time delay. We considered the average delay as a bifurcation parameter and the nonlinear degradation rate as a control parameter. Sufficient conditions for local stability were gained by using the Routh-Hurwitz criterion in the case of a weak delay kernel. Then we proved that Hopf bifurcation happened and the direction of the periodic solution was determined using multiple time scale technique. Our results suggest that the interaction among several identical mRNA plays the main role in gene regulation.

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

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

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.

Bifurcation Analysis of a Model Accounting for the 14-3-3s Signalling Compartmentalisation

Bifurcation theory studies the qualitative changes in the phase portrait when we vary the parameters of the system. In this book chapter we adapt and extend a mathematical model accounting for the subcellular localisation of 14-3-3s, a protein involved in cell cycle arrest and the regulation of apoptosis. The model is analysed with analytical tools coming from Lyapunov-Andronov theory, and our analytical calculations predict that soft (reversible) loss of stability takes place.

Modelling biochemical networks with intrinsic time delays: a hybrid semi-parametric approach

Background

This paper presents a method for modelling dynamical biochemical networks with intrinsic time delays. Since the fundamental mechanisms leading to such delays are many times unknown, non conventional modelling approaches become necessary. Herein, a hybrid semi-parametric identification methodology is proposed in which discrete time series are incorporated into fundamental material balance models. This integration results in hybrid delay differential equations which can be applied to identify unknown cellular dynamics.

Results

The proposed hybrid modelling methodology was evaluated using two case studies. The first of these deals with dynamic modelling of transcriptional factor A in mammalian cells. The protein transport from the cytosol to the nucleus introduced a delay that was accounted for by discrete time series formulation. The second case study focused on a simple network with distributed time delays that demonstrated that the discrete time delay formalism has broad applicability to both discrete and distributed delay problems.

Conclusions

Significantly better prediction qualities of the novel hybrid model were obtained when compared to dynamical structures without time delays, being the more distinctive the more significant the underlying system delay is. The identification of the system delays by studies of different discrete modelling delays was enabled by the proposed structure. Further, it was shown that the hybrid discrete delay methodology is not limited to discrete delay systems. The proposed method is a powerful tool to identify time delays in ill-defined biochemical networks.

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.

Quantitative analysis of the dynamic signaling pathway involved in the cAMP mediated induction of l-carnitine biosynthesis in E. coli cultures

L-(-)-carnitine can be synthesized from waste bioprecursors in the form of crotonobetaine. Such biotransformation is carried out in E. coli by the enzymes encoded by operons regulated by the cAMP receptor proteins. Non-phosphorylated sugars, such as glycerol are used as energy and carbon source since glucose inhibits cAMP synthesis. Until now little attention has been paid to the regulatory signaling structure that operates during the transition from a glucose-consuming, non-l-carnitine producing steady state, to a glycerol-consuming l-carnitine producing steady state. In this work we aim to elucidate and quantify the underlying regulatory mechanisms operating in the abolition of the glucose inhibiting effect. For this purpose we make use of the systemic approach by integrating the available information and our own experimentally generated data to construct a mathematical model. The model is built using power-law representation and is used as a platform to make predictive simulations and to assess the consistency of the regulatory structure of the overall process. The model is subsequently checked for quality through stability and a special, dynamic sensitivity analysis. The results show that the model is able to deal with the observed system transient phase. The model is multi-hierarchical, comprising the metabolic, gene expression, signaling and bioreactor levels. It involves variables and parameters of a very different nature that develop in different time scales and orders of magnitude. Some of the most relevant conclusions obtained are: (i) the regulatory interactions among glucose, glycerol and cAMP metabolism are far stronger than those present in the l-carnitine transport, production and degradation processes; (ii) carnitine biosynthesis is very sensitive to the cAMP signaling system since it reacts at very low cAMP receptor concentrations, and (iii) ATP is a critical factor in the transient dynamics. All these model-derived observations have been experimentally confirmed by separate studies. As a whole, the model contributes to our general understanding of the transient regulation through the signal regulatory structure, thus enabling more accurate optimization strategies to be used.

A system biology approach to understand functional activity of cell communication systems

Systems Biology is the quantitative analysis of dynamic interactions among several components of a biochemical system, aiming at an understanding of the behavior of the system as a whole. From an experimental perspective, systems biology is a suitable tool to support the biologist in the process of hypotheses generation and the efficient design of experiments. In this chapter, we discuss the elements of a systems biology methodology based on the interaction between experimental biologists and theoreticians. We, furthermore, show the use of such a methodology in a case study, analyzing receptor and transcription factor modulation affecting the responsiveness of the JAK2/STAT5 pathway.