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Register a new userGraph-theoretic analysis of multistationarity using degree theory
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Biochemical mechanisms with mass action kinetics are often modeled by systems of polynomial differential equations (DE). Determining directly if the DE system has multiple equilibria (multistationarity) is difficult for realistic systems, since they are large, nonlinear and contain many unknown parameters. Mass action biochemical mechanisms can be represented by a directed bipartite graph with species and reaction nodes. Graph-theoretic methods can then be used to assess the potential of a given biochemical mechanism for multistationarity by identifying structures in the bipartite graph referred to as critical fragments. In this article we present a graph-theoretic method for conservative biochemical mechanisms characterized by bounded species concentrations, which makes the use of degree theory arguments possible. We illustrate the results with an example of a MAPK network.
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Mathematical modelling has become an established tool for studying the dynamics of biological systems. Current applications range from building models that reproduce quantitative data to identifying systems with predefined qualitative features, such as switching behaviour, bistability or oscillations. Mathematically, the latter question amounts to identifying parameter values associated with a given qualitative feature. We introduce a procedure to partition the parameter space of a parameterized system of ordinary differential equations into regions for which the system has a unique or multiple equilibria. The procedure is based on the computation of the Brouwer degree, and it creates a multivariate polynomial with parameter depending coefficients. The signs of the coefficients determine parameter regions with and without multistationarity. A particular strength of the procedure is the avoidance of numerical analysis and parameter sampling. The procedure consists of a number of steps. Each of these steps might be addressed algorithmically using various computer programs and available software, or manually. We demonstrate our procedure on several models of gene transcription and cell signalling, and show that in many cases we obtain a complete partitioning of the parameter space with respect to multistationarity.
Cancer complexome comprises a heterogeneous and multifactorial milieu that varies in cytology, physiology, signaling mechanisms and response to therapy. The combined framework of network theory and spectral graph theory along with the multilayer anal- ysis provides a comprehensive approach to analyze the proteomic data of seven different cancers, namely, breast, oral, ovarian, cervical, lung, colon and prostate. Our analysis demonstrates that the protein-protein interaction networks of the normal and the cancerous tissues associated with the seven cancers have overall similar structural and spectral properties. However, few of these properties implicate unsystematic changes from the normal to the disease networks depicting difference in the interactions and highlighting changes in the complexity of different cancers. Importantly, analysis of common proteins of all the cancer networks reveals few proteins namely the sensors, which not only occupy significant position in all the layers but also have direct involvement in causing cancer. The prediction and analysis of miRNAs targeting these sensor proteins hint towards the possible role of these proteins in tumorigenesis. This novel approach helps in understanding cancer at the fundamental level and provides a clue to develop promising and nascent concept of single drug therapy for multiple diseases as well as personalized medicine.
Tumor metastasis is one of the main factors responsible for the high fatality rate of cancer. Metastasis can occur after malignant cells transition from the epithelial phenotype to the mesenchymal phenotype. This transformation allows cells to migrate via the circulatory system and subsequently settle in distant organs after undergoing the reverse transition from the mesenchymal to the epithelial phenotypes. The core gene regulatory network controlling these transitions consists of a system made up of coupled SNAIL/miRNA-34 and ZEB1/miRNA-200 subsystems. In this work, we formulate a mathematical model of the core regulatory motif and analyze its long-term behavior. We start by developing a detailed reaction network with 24 state variables. Assuming fast promoter and mRNA kinetics, we then show how to reduce our model to a monotone four-dimensional system. For the reduced system, monotone dynamical systems theory can be used to prove generic convergence to the set of equilibria for all bounded trajectories. The theory does not apply to the full model, which is not monotone, but we briefly discuss results for singularly-perturbed monotone systems that provide a tool to extend convergence results from reduced to full systems, under appropriate time separation assumptions.
We apply tools from real algebraic geometry to the problem of multistationarity of chemical reaction networks. A particular focus is on the case of reaction networks whose steady states admit a monomial parametrization. For such systems we show that in the space of total concentrations multistationarity is scale invariant: if there is multistationarity for some value of the total concentrations, then there is multistationarity on the entire ray containing this value (possibly for different rate constants) -- and vice versa. Moreover, for these networks it is possible to decide about multistationarity independent of the rate constants by formulating semi-algebraic conditions that involve only concentration variables. These conditions can easily be extended to include total concentrations. Hence quantifier elimination may give new insights into multistationarity regions in the space of total concentrations. To demonstrate this, we show that for the distributive phosphorylation of a protein at two binding sites multistationarity is only possible if the total concentration of the substrate is larger than either the total concentration of the kinase or the total concentration of the phosphatase. This result is enabled by the chamber decomposition of the space of total concentrations from polyhedral geometry. Together with the corresponding sufficiency result of Bihan et al. this yields a characterization of multistationarity up to lower dimensional regions.
Alterations in the human gut bacteriome can be associated with human health issues, such as type-2 diabetes and cardiovascular disease. Both external and internal factors can drive changes in the composition and in the interactions of the human gut bacteriome, impacting negatively on the host cells. In this paper, we focus on the human gut bacteriome metabolism and we propose a two-layer network system to investigate its dynamics. Furthermore, we develop an in-silico simulation model (virtual GB), allowing us to study the impact of the metabolite exchange through molecular communications in the human gut bacteriome network system. Our results show that the regulation of molecular inputs can strongly affect bacterial population growth and create an unbalanced network, as shown by the shift in the node weights based on the molecular signals that are produced. Additionally, we show that the metabolite molecular communication production is greatly affected when directly manipulating the composition of the human gut bacteriome network in the virtual GB. These results indicate that our human GB interaction model can help to identify hidden behaviors of the human gut bacteriome depending on the molecular signal interactions. Moreover, the virtual GB can support the research and development of novel medical treatments based on the accurate control of bacterial growth and exchange of metabolites.
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