In this chapter basic mathematical methods for the deterministic kinetic modeling of biochemical systems are described. Mathematical analysis methods, the respective algorithms, and appropriate tools and resources, as well as established standards for data exchange, model representations and definitions are presented. The methods comprise time-course simulations, steady state search, parameter scanning, and metabolic control analysis among others. An application is demonstrated using a test case model that describes parts of the extrinsic apoptosis pathway and a small example network demonstrates an implementation of metabolic control analysis.
We can observe the molecular background of complex disease processes with systems biology. Today, drug development plays a central role in the interdisciplinary area of systems biology. Scientists from various disciplines, such as biology, bioinformatics, medical research, chemistry, physics, computer science and mathematics, work together to analyze complex disease processes by studying molecular interaction networks. The aim is to understand and analyze the complex behavior of human diseases. Through this collaborative research we can obtain specific methods and results that can lead to new predictions and assumptions about the observed diseases and processes.
Metabolic pathways like the citric acid cycle and glycolysis, for example, are important processes that may be used to analyze the complex mechanism in living biological systems. The modeling of these pathways provides appropriate structures for observing the behavior of metabolic diseases, i.e. diabetes. Gene regulatory networks describe protein-DNA interactions or indirectly as the interaction between DNA and DNA. Here, we can also examine the regulation of the activity of single genes and the behavior of single molecules, as well as observe the interaction and regulation between different genes and proteins in complex structures. The normal function of single genes is affected in several diseases. A lot of information is available about the functionality of single genes in different diseases and also the roles of mutations in these genes are comprehensively described (Weinberg 1994). Many project studies about this topic are currently underway, and we expect more interesting results in the future (Futreal et al. 2004). Multiprotein complexes are the result of numerous protein-protein interactions. These interactions are essential for physiological processes. In biological networks, diffusion and the molecular transport across cell membranes are also important physiological processes. In different compartments of the cell the function of a protein can change, because the functionality of a protein depends also on the existing targets in the appropriate compartments. For example the p53 protein acts in the nucleus as a transcription factor of apoptosis. The protein Mdm2 in turn can bind to p53 and initiates its ubiquitination and subsequent degradation in the cytoplasm. In the cytoplasm p53 can not act as a transcription factor and instead the degradation of p53 is initialized in this compartment. Signaling pathways play an important role in many diseases. For example the cell signaling mechanism regulates cell proliferation and cell differentiation. The main structures that are responsible for the progression of cancer are interferences in signaling pathways (Cui et al. 2007). The reasons for these inferences in turn are founded in mutated proteins.
Using mathematical models we can describe complex cellular processes. This chapter gives an overview of the possibilities available to analyze these molecular, cellular and physiological processes with mathematical modeling, and outlines how we can integrate experimental data. We demonstrate the utilization of the mathematical model analysis in two examples. Time-course simulation and parameter scanning are applied to a model for the extrinsic apoptosis pathways and metabolic control analysis is applied to a small sample network.
Key Terms in this Chapter
Topology: The topology defines the properties of spaces and maps. Dependent on the described space arbitrary complex structures used.
Euclidean Norm: The Euclidean space denotes an n-dimensional space that can be characterized through Euclidean geometry. An n-dimensional vector describes a point in this space. The Euclidean norm defines the length of a vector and the distance function between two vectors is called an Euclidean metric.
Stoichiometry: The quantitative relationship between the reactant and the product in a chemical reaction is described by the stoichiometry. It can be used to calculate the quantitative amount of the product or the educt of a reaction, where one of both measurable quantities is known.
ODE: An ordinary differential equation is an equation that contains a function and the derivates of this function. It differs from partial differential equations in that, in an ordinary differential equation, the included function depends on only one variable.
Protein Complex: A group of two or more chemically bound proteins formed by stable protein-protein interactions.
Eigenvalues: In linear algebra the equation defines the eigenvalues ? and the corresponding eigenvectors v of a linear transformation represented by a quadratic matrix A. ? is a complex value and v is a complex valued vector. The eigenvalues describe essential characteristics of the linear map.
Kinetics: In this chapter the term kinetic is used in a chemical sense and describes the principles of reaction velocities.
DAE: A differential-algebraic equation (DAE) is a special kind of differential equation and expressed by means of differential algebra. This equation does not necessarily include all dependent variables, and their derivates must not be expressed explicitly.
Object-Oriented: Object-oriented programming uses abstract objects and their interactions to describe the contents and the functionalities of the program, according to its design.
Python: The high-level programming script language Python supports functional, object-oriented and imperative programming paradigms.
Fortran: The name of a programming language derived from FORmula TRANSlation. Fortran was developed in the 1950s and was especially used for numerical programming. The programming language directly supports numerical operations and because of this, optimized compiler calculations can be performed. Up to now, various numerical programming libraries are available and used in mathematical, physical and chemical science.
Bcl2: Is the prototype for a protein family of mammalian genes and the proteins coded by these genes. The name is derived from B-cell lymphoma 2.
C++: The programming language C++ was developed in 1979 by Bjane Stroustrup. It supports object-oriented programming and is an enhancement to the programming language C.
Enzyme: Protein molecules that catalyze chemical reactions. They play an important role in most metabolic processes and are responsible for activation and control of biochemical reactions in living systems.
Linearization: In mathematics, linearization describes the linear approximation of a function at a given point.
Deterministic: An algorithm is called deterministic when the same results are always obtained under identical conditions. In this respect, deterministic mathematical modeling describes systems without any random possibility function. Identical starting conditions always provide the same output.
Experimental Data: Gene expression profiles derived from micro array based experiments with different study foci, i.e. prostate cancer, compound testing or diabetes.