Symbolic Versus Numerical Computation and Visualization of Parameter Regions for Multistationarity of Biological Networks

We investigate models of the mitogenactivated protein kinases (MAPK) network, with the aim of determining where in parameter space there exist multiple positive steady states. We build on recent progress which combines various symbolic computation methods for mixed systems of equalities and inequalities. We demonstrate that those techniques benefit tremendously from a newly implemented graph theoretical symbolic preprocessing method. We compare computation times and quality of results of numerical continuation methods with our symbolic approach before and after the application of our preprocessing.Comment: Accepted into Proc. CASC 201

Identifying the parametric occurrence of multiple steady states for some biological networks

We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic descriptions of the multistationarity regions of parameter space, and compare symbolic results to numerical methods. The biological networks studied are models of the mitogen-activated protein kinases (MAPK) network which has already consumed considerable effort using special insights into its structure of corresponding models. Our main example is a model with 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment. The model also imposes positivity conditions on all variables and parameters. We apply combinations of symbolic computation methods designed for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition, as well as a simplification technique adapted from Gaussian elimination and graph theory. We are able to determine multistationarity of our main example over a 2-dimensional parameter space. We also study a second MAPK model and a symbolic grid sampling technique which can locate such regions in 3-dimensional parameter space.Comment: 60 pages - author preprint. Accepted in the Journal of Symbolic Computatio

Testing Binomiality of Chemical Reaction Networks Using Comprehensive Gröbner Systems

We consider the problem of binomiality of the steady state ideals of biochemical reaction networks. We are interested in finding polynomial conditions on the parameters such that the steady state ideal of a chemical reaction network is binomial under every specialisation of the parameters if the conditions on the parameters hold. We approach the binomiality problem using Comprehensive Gr\"obner systems. Considering rate constants as parameters, we compute comprehensive Gr\"obner systems for various reactions. In particular, we make automatic computations on n-site phosphorylations and biomodels from the Biomodels repository using the grobcov library of the computer algebra system Singular

Qualitative modeling in computational systems biology

The human body is composed of a large collection of cells,\the building blocks of life". In each cell, complex networks of biochemical processes contribute in maintaining a healthy organism. Alterations in these biochemical processes can result in diseases. It is therefore of vital importance to know how these biochemical networks function. Simple reasoning is not su±cient to comprehend life's complexity. Mathematical models have to be used to integrate information from various sources for solving numerous biomedical research questions, the so-called systems biology approach, in which quantitative data are scarce and qualitative information is abundant. Traditional mathematical models require quantitative information. The lack in ac- curate and su±cient quantitative data has driven systems biologists towards alternative ways to describe and analyze biochemical networks. Their focus is primarily on the anal- ysis of a few very speci¯c biochemical networks for which accurate experimental data are available. However, quantitative information is not a strict requirement. The mutual interaction and relative contribution of the components determine the global system dy- namics; qualitative information is su±cient to analyze and predict the potential system behavior. In addition, mathematical models of biochemical networks contain nonlinear functions that describe the various physiological processes. System analysis and parame- ter estimation of nonlinear models is di±cult in practice, especially if little quantitative information is available. The main contribution of this thesis is to apply qualitative information to model and analyze nonlinear biochemical networks. Nonlinear functions are approximated with two or three linear functions, i.e., piecewise-a±ne (PWA) functions, which enables qualitative analysis of the system. This work shows that qualitative information is su±cient for the analysis of complex nonlinear biochemical networks. Moreover, this extra information can be used to put relative bounds on the parameter values which signi¯cantly improves the parameter estimation compared to standard nonlinear estimation algorithms. Also a PWA parameter estimation procedure is presented, which results in more accurate parameter estimates than conventional parameter estimation procedures. Besides qualitative analysis with PWA functions, graphical analysis of a speci¯c class of systems is improved for a certain less general class of systems to yield constraints on the parameters. As the applicability of graphical analysis is limited to a small class of systems, graphical analysis is less suitable for general use, as opposed to the qualitative analysis of PWA systems. The technological contribution of this thesis is tested on several biochemical networks that are involved in vascular aging. Vascular aging is the accumulation of changes respon- sible for the sequential alterations that accompany advancing age of the vascular system and the associated increase in the chance of vascular diseases. Three biochemical networks are selected from experimental data, i.e., remodeling of the extracellular matrix (ECM), the signal transduction pathway of Transforming Growth Factor-¯1 (TGF-¯1) and the unfolded protein response (UPR). The TGF-¯1 model is constructed by means of an extensive literature search and con- sists of many state equations. Model reduction (the quasi-steady-state approximation) reduces the model to a version with only two states, such that the procedure can be visual- ized. The nonlinearities in this reduced model are approximated with PWA functions and subsequently analyzed. Typical results show that oscillatory behavior can occur in the TGF-¯1 model for speci¯c sets of parameter values. These results meet the expectations of preliminary experimental results. Finally, a model of the UPR has been formulated and analyzed similarly. The qualitative analysis yields constraints on the parameter values. Model simulations with these parameter constraints agree with experimental results

Applications of MATLAB in Science and Engineering

The book consists of 24 chapters illustrating a wide range of areas where MATLAB tools are applied. These areas include mathematics, physics, chemistry and chemical engineering, mechanical engineering, biological (molecular biology) and medical sciences, communication and control systems, digital signal, image and video processing, system modeling and simulation. Many interesting problems have been included throughout the book, and its contents will be beneficial for students and professionals in wide areas of interest