Mathematical modelling of enzymatic reactions, simulation and parameter estimation

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Content: MATHEMATICAL MODELLING OF ENZYMATIC REACTIONS, SIMULATION AND PARAMETER ESTIMATION SUЁ REYYA OЁ ZOЁ G UЁ R January 2005
MATHEMATICAL MODELLING OF ENZYMATIC REACTIONS, SIMULATION AND PARAMETER ESTIMATION A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF APPLIED MATHEMATICS OF THE Middle East Technical University BY SUЁ REYYA OЁ ZOЁ G UЁ R IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN THE DEPARTMENT OF SCIENTIFIC COMPUTING JANUARY 2005
Approval of the Graduate School of Applied Mathematics Prof. Dr. Aydin AYTUNA Director I certify that this thesis satisfies all the requirements as a thesis for the degree of Master of Science. Prof. Dr. BuЁlent KarasoЁzen Head of Department This is to certify that we have read this thesis and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Master of Science.
Examining Committee Members Prof. Dr. BuЁlent KarasoЁzen Prof. Dr. Gerhard Wilhelm Weber Dr. Hakan OЁ ktem Prof. Dr. MuЁnevver Tezer Prof. Dr. Feza Korkusuz
Prof. Dr. BuЁlent KarasoЁzen Supervisor
Abstract MATHEMATICAL MODELLING OF ENZYMATIC REACTIONS, SIMULATION AND PARAMETER ESTIMATION SuЁreyya OЁ zoЁguЁr M.Sc., Department of Scientific Computing Supervisor: Prof. Dr. BuЁlent KarasoЁzen January 2005, 75 pages A deep and analytical understanding of the human metabolism grabbed attention of scientists from biology, medicine and pharmacy. Mathematical Models of metabolic pathways offer several advances for this deep and analytical understanding due to their incompensable potential in predicting metabolic processes and anticipating appropriate interventions when required. This thesis concerns mathematical modelling analysis and simulation of metabolic pathways. These pathways include intracellular and extracellular compounds such as enzymes, metabolites, nucleotides and cofactors. Experimental data and available knowledge on metabolic pathways are used in constituting a mathematical model. The models are either in the form of nonlinear ordinary differential equations (ODE's) or differential algebraic equations (DAE's). These equations are composed of kinetic parameters such as kinetic rate constants, initial rates and concentrations of metabolites. The nonlinear nature of enzymatic reactions and large number of parameters cause trouble in efficient simulation of those iii
reactions. Metabolic engineering tries to simplify these equations by reducing the number of parameters. In this work, an enzymatic system which includes creatine kinase, Hexokinase and Glucose 6-Phosphate Dehydrogenase (CKHK-G6PDH) is modelled in the form of DAE's, solved numerically and the system parameters are estimated. The numerical results are compared with the results from an existing work in literature. We demonstrate that our solution method based on direct solution of the CK-HK-G6PDH system significantly differs from simplified solutions. We also show that a genetic algorithm (GA) for parameter estimation provides much more clear results to the experimental values of the metabolite, especially, with NADPH. Keywords: metabolic engineering, kinetic modelling, biochemical reactions, enzymatic reactions, differential algebraic equations, parameter estimation, optimization, genetic algorithm. iv
OЁ z ENZIMATIK REAKSIYONLARIN MATEMATIKSEL MODELLENMESI, SIMULASYONU VE PARAMETRE KESTIRIMI SuЁreyya OЁ zoЁguЁr YuЁksek Lisans, Bilimsel Hesaplama. Tez YoЁneticisi: Prof. Dr. BuЁlent KarasoЁzen Ocak 2005, 75 sayfa Insan metabolizmasinin detayli ve analitik olarak incelenmesi, biyoloji, tip ve eczacilik alanlarindaki bilim adamlarinin dikkatlerini uЁzerine toplamiёstir. Metabolik patikalarin matematiksel modellemesi, metabolik suЁrecёlerin gelecekteki davraniёslariniin tahmini ve gerektiginde en uygun muЁdahelelerin oЁngoЁruЁlebilmesi icёin kullanilabilirligi nedeniyle bu detayli ve analitik incelemeye yeri doldurulamaz katkilar saglamaktadir. Bu cёaliёsma metabolik patikalarin matematiksel modeli, analizi ve simuЁlasyonu uЁzerinedir. Metabolik patikalar, huЁcre icёi ve huЁcre diёsi enzimler, metabolitler, nuЁkleotidler ve kofaktoЁrler gibi bileёsikleri icёermektedir. Deneysel veriler ve metabolik patikalar hakkindaki bilgiler matematiksel modelin oluёsturulmasinda kullanilmaktadir. Model denklemleri, lineer olmayan adi diferensiyel denklemler ya da cebirsel diferensiyel denklemlerden oluёsmaktadir. Denklemler, kinetik hiz sabitleri, reaksiyonlarin baёslangicё hizlari ve metabolitlerin baёslangicё konsantrasyonlari gibi kinetik parametreleri icёermektedir. Enzimatik reaksiyonlarin dogrusal olmayan dogasi ve fazla sayida parametre icёermesi dogru simule edilebilmelerinde sorun yaratmaktadir. Meta- v
bolik muЁhendislik, parametre sayisini azaltarak model denklemlerini basitleёstirmeye cёaliёsmaktadir. Bu cёaliёsmada uЁcё enzimli kreatin kinaz, hekzokinaz ve glukoz alti fosfat dehidrogenaz sistemi cebirsel diferensiyel denklemler yardimiyla modellenmiёs, sayisal yoЁntemlerle cёoЁzuЁlmuЁёs ve parametre tahmini yapilmiёstir. Sayisal sonucёlarin literatuЁrde yer alan cёaliёsmalarla karёsilaёstirilmasi yapilmiёstir. Bu karёsilaёstirmada CK-HK-G6PDH sisteminin dogrudan cёoЁzuЁlebilmesine dayanan cёoЁzuЁm metodumuzun basitleёstirilmiёs cёoЁzuЁmlerden oldukcёa farkli sonucёlar verdigi goЁsterilmiёstir. Bunlarin yanisira genetik algoritma (GA) ile parametre kestiriminin deneysel verilere oЁzellikle NADPH metabolitinede cёok daha yakin sonucёlar sagladigi goЁsterilmiёstir. Anahtar Kelimeler: metabolik muЁhendislik, kinetik modelleme biyokimyasal modelleme, enzimatik reaksiyonlar, cebirsel diferensiyel denklemler, parametre tahmini, optimizasyon, genetik algoritmalar. vi
To my family vii
Acknowledgments I would like to thank all those people who have helped in the preparation of this study. I am grateful to Prof.Dr. BuЁlent KARASOЁ ZEN, Prof.Dr. Gerhard Wilhelm WEBER and GuЁlcёin Sagdicёoglu Cё elep for patiently guiding, motivating, and encouraging me throughout this study. Special thanks to Dr. Necmettin Yildirim for all his efforts in every phase of my study. I would like to thank him for his help and patience when writing an answer to all of my questions from North Carolina. I would like to thank Prof.Dr. BuЁlent KARASOЁ ZEN who has always encouraged me for my study. I would like to thank him for all his efforts, energy and support. I am very glad that I have met Prof.Dr. Gerhard-Wilhelm Weber who encourages me throughout this study. I would like to thank Dr. Hakan OЁ ktem and GuЁlcёin Sagdicёoglu Cё elep for their valuable suggestions and helps whenever I need. I want to thank my parents without whose encouragement this thesis would not be possible. I am also grateful to Akin AkyuЁz, Yeliz Yolcu, Turgut Hanoymak, Mesut Taёstan and OЁ znur Yaёsar for being with me all the way. I am very glad that I have met Akin AkyuЁz who supports me in every respects and thanks to him for his endless patience throughout this study. Finally, I am indebted to Oktay SuЁruЁcuЁ, AyёseguЁl Iёscanoglu, Derya Altintan, Osman OЁ zguЁr, Selime GuЁrol, Didem Akcёay, Emre Dagli and Baёsak Akteke for their support. viii
Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii OЁ z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv CHAPTER 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Brief History of Metabolic Engineering . . . . . . . . . . . . . . 3 1.2 Modelling of Biochemical Pathways . . . . . . . . . . . . . . . . 4 1.2.1 Aims and Scope of Metabolic Models . . . . . . . . . . . 5 1.2.2 Metabolic Flux Analysis . . . . . . . . . . . . . . . . . . 7 1.2.3 Metabolic Control Analysis . . . . . . . . . . . . . . . . 10 1.2.4 Non-Stationary Mechanistic Models . . . . . . . . . . . . 11 ix
2 Modelling Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1 A Model of Enzyme Kinetics . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Steady State Kinetics . . . . . . . . . . . . . . . . . . . . 12 2.1.2 Enzyme Catalyzed Reactions . . . . . . . . . . . . . . . 15 2.1.3 Kinetics of Multi-substrate Enzymes . . . . . . . . . . . 17 2.1.4 The Stoichiometric Matrix . . . . . . . . . . . . . . . . . 25 2.2 Deriving Conservation Relations . . . . . . . . . . . . . . . . . . 27 2.2.1 Deriving Conservation Relations by Inspection . . . . . . 27 2.2.2 Deriving Conservation Relations by Gaussian Elimination 30 2.2.3 Deriving Conservation Relations by Partioning the Stoichiometric Matrix into Submatrices . . . . . . . . . . . . 32 2.3 Creatine Kinase, Hexokinase and Glucose-6Phosphate Dehydrogenase System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.1 Creatine Kinase System . . . . . . . . . . . . . . . . . . 36 2.3.2 Hexokinase System . . . . . . . . . . . . . . . . . . . . . 37 2.3.3 Glucose 6-Phosphate Dehydrogenase System . . . . . . . 39 2.3.4 Mathematical Modeling of CK-HK-G6PDH System . . . 40 3 Numerical Solution Methods For Solving Differential Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . . . 48 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Differential Index . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3 Numerical Solution Methods to Solve DAE's . . . . . . . . . . . 52 3.3.1 Backward Euler . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.2 Runge-Kutta Method . . . . . . . . . . . . . . . . . . . . 53 x
3.3.3 Backward Differentiation Formula . . . . . . . . . . . . . 54 4 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1.1 Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1.2 Reproduction . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1.3 Crossover . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1.4 Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 GA Adapted to CK-HK-G6PDH Model . . . . . . . . . . . . . . 61 5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.1 Results of Simulation . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 xi
List of Figures 1.1 Example of fluxes for a simple reaction network where stoichiometric relations of fluxes are u = x + y, v = x - z, w = y + z [31]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Typical situations in which stoichiometric MFA fails [32]. . . . . 9 2.1 Schematic explanation of different types of reaction mechanisms (taken from [38]). . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Schematic explanation of different types of reaction mechanisms (taken from [38]). . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Schematic explanation of different types of reaction mechanisms (taken from [38]). . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Explicit reaction scheme for ordered Bi Bi mechanism [38]. . . . 23 2.5 A. skeleton model of glycolysis, B. This simplified representation includes only metabolites (taken from [13]). . . . . . . . . . . . 27 2.6 Creatine Kinase, Hexokinase and Glucose-6Phosphate Dehydro- genase System [38]. . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.1 Illustration of a chromosome where each box represents one gene of chromosome. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 A binary-valued, randomly generated pool. . . . . . . . . . . . . 59 4.3 A crossover 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4 A crossover 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 xii
5.1 Simulation results of ATP, ADP, NADP, 6PGL. . . . . . . . . . 64 5.2 Simulation results of ATP, ADP, NADP, 6PGL [38] (zaman: time, dakika: minute). . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 Simulation results of DGlu6P, Cr. . . . . . . . . . . . . . . . . . 65 5.4 Simulation results of DGlu6P, Cr [38] (zaman: time, dakika: minute). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.5 Simulation results of CrP, DGlu. . . . . . . . . . . . . . . . . . . 66 5.6 Simulation results of CrP, DGlu [38] (zaman: time, dakika: minute). 67 5.7 Experimental values of NADPH and simulation results of NADPH by ode15s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.8 Experimental values of NADPH and simulation results of NADPH [38] (zaman: time, dakika: minute). . . . . . . . . . . . . . . . . 68 5.9 NADPH results after parameter estimation and experimental val- ues of NADPH. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 xiii
List of Tables 2.1 Parameters of CK system (Morrison and James, 1965). . . . . . 46 2.2 Parameters of HK system (Viola et al., 1982). . . . . . . . . . . 46 2.3 Parameters of G6PDH system (Gordon et al., 1995). . . . . . . 47 xiv
Chapter 1 Introduction Metabolic engineering is a challenging research field on nowadays incorporating of modern applied mathematics into bioetechnology, engineering science and pharmacy. Moreover, in medical studies, scientists work on human metabolism to improve the capabilities of some metabolites or enzymes in a metabolic pathways. In industrial applications, metabolic engineering methods are also widely used to develop certain methods for improving functionality of some molecules in a cell. To achieve these goals, a mathematical model of such metabolic systems must be constructed and simulated. Most of the dynamical systems can be approximated by various types of differential algebraic and integral equations involving finite number of variables and parameters. Thus, the future behaviour of the system can be predicted if model parameters and initial states of the variables are available. In particular, ordinary differential equations (ODE's) and differential algebraic equations (DAE's) are popular in modelling of the metabolic pathways. Because of difficulies in solving DAE's, in studies given in literature, the systems are generally reduced to ODE's. In this thesis, three enzymatic system, Creatin Kinase, Hexokinase and Glucose 6 Phosphate Dehydrogenase is modeled by DAE's with 3 differential equations and 6 constraints. These enzymatic systems play important roles in glycolysis, energy metabolism of tissues such as skeletal and cardiac muscle, and neural tissues. Since G6PDH is a rate limiting enzyme in pentose phosphate pathway, predicting or improving the behaviour of the enzymes or metabolites in the pathway is a crucial task 1
in bioinformatics. Modelling by differential algebraic equations considered in this work is also an important issue because of the fact that it is a different, a relatively new type of modelling. Solving the system of differential algebraic equations is generally avoided in most of the published works [26, 27, 38] because of its difficulty. Thus, our work introduces a new approach to the solution of our model problem. Furthermore, we assume more general cases in model and give a more accurate and general solution than the existing one in the literature. We assume that the pathway has three different rate equations rather than same rate equations assumed in steady state. Instead of approximating nonlinear rate equations as a polynomial functions with GroЁbner basis theory, like in [38], they are taken completely as nonlinear and solved numerically. The experimental data are based on Bergmeyer (1982) and it is carried out in Laboratory of Biotechnology Applied and Research Center in AtatuЁrk University in Erzurum [38]. The experimental conditions and the concentration values are given explicitly in [38]. For the estimation problem, a different method is used for our case. Since the experimental data and simulation results are available, one must minimize the difference of these two quantities to find the parameters that fit to the model best. This can be done by several methods, for example, minimizing an approximated objective function which is an easy and common way to solve this problem. Instead of this, a genetic algorithm is used for the optimization part of the problem which is a popular method in most of the estimation problems of metabolic pathways. By this approach, we achieved more accurate and closer results to the empiricial data. In Chapter 1, methods of metabolic engineering and modelling of metabolic pathways are presented. Types of enzymatic reactions, fundamental rules and formulas in biochemistry are discussed in Chapter 2. In modelling of such systems, we must have some conservation relations in a biological manner. We explained this theory of deriving conservation relations in this chapter and at the end of the Chapter 2, CK-HK-G6PDH is introduced. For solving our DAE's we have studied DAE theory and numerical methods for them in Chapter 3. For this numerical solution, Backward Differentiation Formula (BDF) is selected 2
and it is implemented in the software MATLAB 6.1 by using its DAE solver ode15s. Especially, for complex nonlinear dynamical systems like metabolic pathways all parameters of a first principles model might not to be a priori available or the available information might no to be sufficiently reliable. In such cases, estimating the most likely parameters from the empirical observations can be substituted into an optimization problem by utilizing estimation theory. One of the most widely used method is Genetic Algorithms (GAs) for parameter estimation problems. It is introduced in Chapter 4. Finally, the simulation results, parameter estimation results and comparisons with experimental data are given in Chapter 5. 1.1 Brief History of Metabolic Engineering Metabolic capabilities of different microorganisms are widely investigated with a motivation of producing desired compounds in pharmaceutical and chemical industry [23]. From these investigations we get a broad survey on cell structure, substance and enzymes, which gives us an insight into these microorganisms. Knowledge on the behaviour of those substances and enzymes in a cell provides some basic information, which is the foundation for modifying a pathway to improve it according to our benefit. This process is nowadays often referred to as metabolic engineering, a term introduced by Bailey in 1991. This dynamical interface between biology, medicine, pharmacy and mathematics aims at a directed improvement of product formation or cellular properties through the modification of specific biochemical reaction(s) or using DNA technology [14]. Metabolic engineering focusses on developping targeted methods to improve the metabolic capabilities of microorganisms. To achieve this goal, genetic manipulations of cell metabolism are required [31]. Metabolic engineering does not only deal with a modification of genes or pathways in an organism but also offers a deeper understanding of metabolic pathways in a systemic and analytical way [30]. In fact, metabolic engineering is an interdisciplinary area where 3
biochemists, molecular biologists, chemical engineers and mathematicians corporate to handle and solve the problem in both a way based on real nature and by a language and methodology of modern mathematics. It mainly studies metabolic pathways and gene networks in organisms to optimize some desired metabolites, especially, in industrial applications [23]. In metabolic engineering, a metabolic design problem can be formulated by constructing a mathematical model of a metabolic network. This model can be used for anticipating modifications or simplifications in the genotype of microorganisms. The use of these mathematical models of metabolic networks plays a significant role in medical and life science applications also. There may be difficulties with the complex nature of cellular metabolism and regulation. Therefore, it is often necessary to analyze metabolism as a whole system where the most important biological parts are included and represented with their interrelations [23]. 1.2 Modelling of Biochemical Pathways Mathematical modelling is one of the most important clues for metabolic engineering. Here, metabolic models, several computational and simulation tools have been developped with the aim of System Analysis, prediction, design and optimization. However, the model depends on the assumptions, simplifications and data used for constructing it. The goal of metabolic engineering is to find a solution in an easy, efficient and profitable way while improving the capabilities of industrially relevant microorganisms. Modelling tools and simulation software are already an important task. The main issue in our modelling part is to find the best way of representing the biological information and experimental data in the sense of mathematics. There remains an open question about which information and data are applicable, since there are great and strong differences between biology and 4
computational mathematics. Due to the solvability problem, not every detail about a functional knowledge on cellular systems or all physical conditions of the experimental environment can be included in a mathematical model. In particular, biological systems can not be easily decomposed into their unit components. However, a developing number of studies and growing data bases give us an increasing functional knowledge about cellular systems. While metabolic engineering has initially aimed at specific biochemical reactions which, however, seemed to limit the microbial production process, it soon became realized that a more global approach is required, such as integrated pathways or, going one step further, networks of pathways. Since any change in a pathway affects other pathways directly or indirectly, how one should use available kinetic information to manipulate any pathway without affecting other pathways is an essential question [30]. For all the insights we obtain, e.g., on enzyme-catalyzed reactions, individual pathways and integrated metabolic networks, let us construct mathematical methods and use computational methods. Only a few decades ago, mathematical analysis of metabolic systems would have been very difficult or even impossible because of unsolvable nonlinear equations. Today, the increasing computer power available allows simulating systems. However, the existing of a good mathematical model is a must for such a simulation to represent and analyze complex systems like metabolic pathways. This points out the importance of choosing a good mathematical representation and an efficient method of simulation. For closer information we refer to [30]. 1.2.1 Aims and Scope of Metabolic Models Both the aim of modelling of a given complex metabolic system and which modelling will be used should be specified. According to the increasing demand for the precision and quality of the model, the following requirements are listed in [31]: 5
1. Structural understanding: mathematical models are the most exact way of representing the metabolic system, while having unique and objective application. Therefore, they should not allow any ambigious statements. As a conclusion, modelling is a methodology used for giving a structure to a metabolic system or defining our system in a more precise statement. Even if the model is not used for simulation, it provides having a knowledge about important parts of the system. Nowadays, this methodology is used for designing complex software systems [31]. 2. Simulation for exploring the system: In model applications, simulation is used for exploring the mechanism or behaviour of the system. A simulation procedure based on rather crude models is not strong enough to understand the system behaviour explicitly. Thus, such studies cannot directly produce right ideas. However, they provide most probable explanations for the behaviour of the metabolic system. 3. Measured data analysis and evaluation: Reproduction of measured data by a constructed mathematical model is a well established tool in scientific disciplines. For instance, growth nutrient uptake and product formation or concentration of metabolites in the biochemical reactions, which are characterized by kinetic models, are a common procedure in bioprocess development. It should be pointed out that it is only a reproduction of measured data and does not involve in any prediction. So, without further experimental data we cannot say anything about the predictive power of the model. 4. Metabolic system analysis: After constructing a model, mathematical methods can give sensible results for understanding the system's structure and its qualitative behaviour. The most popular methods are parameter estimation and sensitivity analysis. In fact, they are used to understand, which parameters are sensitive to small changes, for understanding the dynamical behaviour of the system and investigation of its oscillating or chaotic behaviour. Nevertheless, to obtain meaningful results, more exact 6
models are required. 5. Predicting future results: The outcome of future experiments can be predicted based on a validated model. However, the validity and predictive power is often restrictive, i.e., obtaining an expected target feature might not be possible. 6. Optimization: After having a model structure with a predictive character, when estimating unknown or desired parameters, minimizing or maximizing, namely, of fluxes or some metabolites, is required. Thus, it is an optimization problem. In general, the following has to be defined before starting to solve our mathematical model: · parameter and concentration range, · external conditions and input, and · physiological modes for which it holds. These definitions construct our model in a systemic way. As a rule, it is better to have the simplest model with reasonable accuracy [31]. Several modelling tools are developed to achieve this, e.g., the mechanistic approach, metabolic control analysis (MCA) and metabolic flux analysis (MFA). In this study after giving basic definitions of these types, we will focus on nonstationary mechanistic models, which we construct our mathematical model by. 1.2.2 Metabolic Flux Analysis MFA is a powerful tool in metabolic engineering and has an important place in bioengineering sciences. If only extracellular flux data are available and intracellular fluxes are to be estimated, metabolic flux analysis is a helpful tool for evaluating these unknown fluxes. Thus, this data analysis provides to extend the 7
Figure 1.1: Example of fluxes for a simple reaction network where stoichiometric relations of fluxes are u = x + y, v = x - z, w = y + z [31]. information about other pathways, which are connected in cellular metabolism and, moreover, it helps to obtain a detailed description of the metabolic state of the cell. Furthermore, MFA helps to understand the effect of genetic manipulations and suggests any other modifications. Here, a basic approach to MFA is established on stoichiometry coefficients of biochemical reactions in the pathway. It obeys the fundemental law of mass conservation and the application of optimization principles. By the measured fluxes, MFA defines the stoichiometric relations and if the measurements are not redundant, then all intracellular metabolic fluxes can be estimated from the experimental data. This rises to solve a classical linear estimation problem. When one applies stoichiometric relations, one should consider forward and backward fluxes [31]. In Figure 1.1, there is an example of fluxes for a simple reaction. The aim of MFA is to identify detailed fluxes and quantify all intracellular and extracellular fluxes over the metabolic network. According to this flux map, possible identifications of genetic manipulations and comparisons of different flux maps can be performed. Furthermore, it is possible to judge or draw conclusion about already manipulated genes in a cellular mechanism [32]. This method is simply based on the given stoichiometry of a system and 8
Figure 1.2: Typical situations in which stoichiometric MFA fails [32]. the only necessary assumption is that the biological system is in a stationary or quasistationary state, which means that the concentrations of metabolites in the pathways do not change over the time. Under these assumptions, intracellular fluxes balance the extracellular fluxes which result from a linear system of equations [32]. Unfortunately, stoichiometric MFA is strongly limited and it fails. In Figure 1.2 taken from [32], a typical situation in which stoichiometric MFA fails. These are: 1. Parallel pathways without any related flux measurement. It is impossible to find a value of any flux from two branch fluxes where none of the branches is coupled to a measurable variable in a parallel metabolic pathway (Figure 1.2a)). 2. Certain metabolic cycles. "`Metabolic cycles which are not coupled to measurable fluxes can not be resolved"' [32]. In Figure 1.2b), flux is not sufficient to determine the fluxes in the metabolic cycle. 3. Bidirectional reaction steps. Bidirectional reactions are the special case of metabolic cycles where reactions occur in both directions at the same time (Figure 1.2c)). 9
4. Split pathways when cofactors are not balanced. When some metabolites are not in balance, pathway splits up (Figure 1.2d)). For example, in the glycolysis, PEP, the citric acid cycle, it is only possible to determine all the fluxes if ATP, NADPH, NADH, ... are balanced together with the other metabolites. Thus, energy producing and consuming reactions and all the conversion reaction with the energy metabolites must be exactly known [32]. 1.2.3 Metabolic Control Analysis MCA requires smaller a number of experiments and parameter values than the other methods. It introduces some control coefficients and elasticitices when getting responses to perturbation or manipulation of metabolic parameters and defining the kinetic properties of metabolic pathways. The mathematical definition of biochemical systems allows us to understand the unexpected experimental results and provides to see the responses of modifications in genes. MCA describes these relations in the form of some indices (elasticities and control coefficients). These indices can be computed from experiments in the steady state. However, the prediction made by the steady state condition is limited to small changes in the parameters [11]. In [9], MCA is defined as the analysis of the sensitivity of metabolites. It is a local linear approach that does not give accurate results if large changes are made in the parameters. It is focussed on the steady state level of the system. However, true steady states never occur in real experiments and the irrelevance of matching of the quasi-steady state to the true steady state always remains. It is mentioned in [9]: "This problem and its consequences for interpreting measures control coefficients or elasticities are largely ignored by modellers and experimenters. For this and other reasons, it is not yet clear what the experimental side of metabolic control analysis will contribute to our understanding of metabolism and its regulation in the long run."' 10
1.2.4 Non-Stationary Mechanistic Models The last method which plays a fundamental role in this thesis is based on solving a time-dependent model. If the model is formed by the short-time behaviour of microorganism under rapidly changing external conditions, timedependent models are required. The general non-stationary model extends the stationary model into a differential equation: x = N · v(x, k, e), where x is the vector of all metabolites, N is the stoichiometric matrix and v is the rate vector of reactions, which depends on metabolites x, the vectors k and e denote the parameters and external metabolites, respectively. In this study, external metabolites are assumed to be constant and are neglected. Mechanistic models have been used for the analysis of metabolic systems to predict future results for a given present data or to define the kinetic behaviour of the pathway with respect to time. This kind of model requires the knowledge about the kinetic parameters like rate constants and rate expressions dependent on the variables such as metabolite concentrations and metaboic fluxes. In order to achieve these parameters, numerous experimental data are required. Moreover, validity of approximated kinetic parameters from experiments is important for the robustness of the model. If one tries to model the metabolic pathway depending on parameter values, one can be faced with a large number of parameters. The concentration of metabolites over time can be computed by solving either DAE's or after eliminating the constraints solving a system of ODE's. It is not an easy task to obtain solutions when the number of equations is quite large, and stiff problems occur in many metabolic systems. In this work, for the simulation of the metabolites we have used the DAE solver ode15s based on DASSL (backward differentiation methods). 11
Chapter 2 Modelling Approaches 2.1 A Model of Enzyme Kinetics Before constructing our kinetic model, some biological properties and biochemical laws and formulas should be known for the robustness of the model. In this section, basic concepts of biochemistry are discussed such as steady state kinetics, Michaelis Menten equation and its derivation, enzyme catalyzed reactions and kinetics of multi substrate enzymes. 2.1.1 Steady State Kinetics In a cell, if the production and degradation of metabolite rates are very rapid and approximately at the same level, then the concentration of a metabolite is at steady state level. This shows that the catalytic activity of an enzyme under steady state conditions in the cell is very important for the understanding of metabolism. Measurement of steady-state reactions are easy, because at a steady state level the rate of the reaction is constant. Steady state levels show more similar properties to metabolic levels. So it is possible to understand the behaviour or function of the enzyme in the reaction by using parameters. Moreover, it is predictable to learn how an enzyme works under different conditions by changing 12
substrate concentrations [35]. In the following, we will describe the Michaelis Menten equation which is fundamental for enzyme kinetics. It can be descibed by the following kinetic mechanism:
E + S k1 ES -k2 P + E. k-1 In the above reaction, E denotes enzyme, S denotes substrate, P denotes product and ES denotes enzyme substrate complex. The ki 's are the rate constants to be determined by the experiments or from literature. Michaelis-Menten equation assumes the following relations:
· The concentration of an enzyme [E] is much smaller than the substrate [S]; i.e., [E] << [S].
· The rate of the reaction is proportional with the initial enzyme concentration; i.e., v [E]0 . · At sufficiently low substrate concentration [S], v increases linearly with [S].
· Basic equation of enzyme kinetics is given by [8]:
v
=
vmax[S] KM + [S]
,
where KM is the Michaelis constant:
KM
=
k-1 + k1
k2 .
(2.1.1) (2.1.2)
We will derive the Michaelis Menten equation by describing the kinetic behaviour of the enzymes. Let the rate of formation of the product P be equal to the initial velocity of
13
the reaction, i.e.: where
d[P ] dt
=
v0,
v0 = k2[ES].
(2.1.3)
Here, the above relations hold. Then, the mass conservations for enzyme and
substrate are:
[E]total = [ES] + [E]free,
(2.1.4)
[S]0 = [S]free + [ES] + [P ], [S]0 = [S]free.
(2.1.5)
The change of enzyme substrate complex [ES] concentration with respect to
time
d[ES] dt
is
equal
to
the
difference
of
rate
of
formation
k1[E][S]
and
rate
of
consumption k-1[ES] - k2[ES]. The velocity of the reaction remains constant
during the initial state of the reaction; thus, we have
d[ES] dt
=
0.
(2.1.6)
From (2.1.6) we get following relation
k-1[ES] + k2[ES] = k1[E][S]. If we gather the terms with [ES], we obtain
(2.1.7)
(k-1 + k2)[ES] = k1[E][S].
Then, dividing both sides to the k1 and [ES],
(k-1 + k2) k1
=
[E][S] [ES]
.
Let call the left hand side term KM and rewrite the previous equation as
KM
=
[E][S] [ES]
.
(2.1.8)
14
Multiplying both sides by [ES] and substituting [E] defined in (2.1.4) to (2.1.8) with collecting [ES] terms together, gives us following relation:
(KM + [S])[ES] = [E][S].
Finally, dividing both sides by KM + [S], we get the following relation:
[ES]
=
[E KM
][S] + [S
]
.
(2.1.9)
In addition to this, vmax occurs when [ES] = [E]total. Substituting (2.1.9) to (2.1.3) to get final rate equation
v0
=
k2[E][S] KM + [S]
.
(2.1.10)
Let us call k2[E] = vmax and, herewith, we rewrite (2.1.3) by
v0
=
vmax[S] KM + [S
]
.
(2.1.11)
2.1.2 Enzyme Catalyzed Reactions If a small amount of enzyme is used and all but one substrate is kept constant, then the rate of the enzymatically catalyzed reaction depends on the substrate concentration and initial rate like in the equation (2.1.1). The typical notation of the enzyme catalyzed reaction with one substrate can be given as
A + E k1 X k3 P + E,
k2
k4
where A is substrate, E is enzyme, X is enzyme-substrate complex and P is product.
15
The kinetic equations consist of
d[A] dt
=
k2[X] - k1[A][E],
d[E] dt
=
(k2 + k3)[X] - (k1[A] + k4[P ])[E],
d[P ] dt
=
k3[X] - k4[P ][E],
with a conservation relation given in [25]:
(2.1.12)
[E] + [X] = [E]total.
(2.1.13)
It is obvious that the derivative of a substrate with respect to time gives the rate. Thus, the rate is a function of compounds c, s (intracellular and extracellular), enzyme concentrations E and kinetic parameters k. However, the enzyme concentration is hidden in the kinetic constants in the parameter vector k; herewith, we can write v as a function of c, s and P , i.e., v = v(c, s, k) [14]. The more general form of (2.1.12) can be written in the form of
d[A] dt
=
v2 - v1,
d[E] dt
=
v2 + v3 - v1 - v4,
d[X ] dt
=
v1 + v4 - v2 - v3,
d[P ] dt
=
v3 - v4,
with a conservation equation [E] + [X] = [E]total. The form of rate equations is as follows:
v1 = k1[A][E], v2 = k2[X], v3 = k3[X], v4 = k4[P ][E].
16
2.1.3 Kinetics of Multi-substrate Enzymes Introduction In previous subsections, only one substrate enzyme kinetics has been discussed. Now, in this subsection, more than one substrate enzyme kinetics will be studied, which possesses more complex mechanisms and reveals a large number of rate equations. Typical enzymes play a role as catalysts in such systems to convert two substrates into product. Two or three substrates are common, but not four. The usual approach while studying an enzyme with more than one substrate is that all substrates are varying during a reaction but one of them is kept constant. By this way it is observed that most reactions obey MichaelisMenten kinetics [8]. So, the remaining substrate is varied in kinetic assays and valuable information about the relationship between enzyme, substrate and inhibitors can be gained [36]. There are some types of reactions involving their variability to bind enzymes and producing a certain product. In the following subsection, the schematic descriptions and rate equations of each of these multi-substrate enzyme reactions, which are mainly divided in sequential and ping pong mechanisms, will be given [8]. Ping Pong Mechanism Definition 2.1.1. If at least one product is released before all of the substrates have been bound, then it is called ping pong mechanism [36]. The process usually starts by binding of the enzyme to the first substrate: E + A (EA). The next reaction is the key of the whole process: (EA) (F P ). 17
In this reaction, a part of the substrate has been removed from substrate A, converting it to product P . The removed section has become covalently bound to the enzyme to create a new form of the enzyme, enzyme F . The first product of the reaction is now released and the second substrate binds: (F P ) F + P, F + B (F B). Now, the stored section of the first substrate is transferred to the second substrate to create the second product, which is then released: (F B) (EQ), (EQ) E + Q. A plotting of these reactions is shown in Figure 2.1. Sequential Mechanisms Definition 2.1.2. If all the substrates bind to the enzyme before the first product is formed, this is called a sequential reaction. Definition 2.1.3. If all substrates bind to enzymes and products are dissociated in an obligatory order, the sequential reaction is called ordered . Definition 2.1.4. If a sequential mechanism has no any obligatory order in binding and releasing, it is called random sequential mechanism. The terminolgy for sequential mechanisms was developed by Cleland (1963) for multisubstrate enzymes according to their number of substrates, products, binding type of substrates to enzymes and dissociation of products from enzymes [38]. According to this terminology, substrate and product numbers are called by Uni, Bi, Ter and Quad. For example, for two substrates and two products, our mechanism is called a Bi Bi mechanism, for one substrate and two products 18
it is called a Uni Bi mechanism. In this study, enzymes in our model follow Ordered Bi Bi and Rapid Equilibrium Random Bi Bi Mechanism. So, in the following sections, these two types of mechanisms will be introduced. Schematic representations of other mechanisms will be given in Figures 2.1, 2.2 and 2.3 [38]. The rate equations are in detail presented in [16].
Ordered Bi Bi Mechanism
In this kind of reaction mechanism, substrates bind to enzymes in order and, then, an enzyme substrate complex is formed. In addition to this, products dissociates in order again. As you will see in Figure 2.1, firstly substrate A binds to enzyme E and forms enzyme substrate complex EA. Afterwards, second substrate B binds to EA and forms second enzyme substrate complex EAB. The products leave the enzyme complex in order, namely, first product P is produced and, then, Q. According to the steady state assumption and conservation mass law, differential equations of the reaction mechanism can be written in the form by the explicit reaction scheme given in Figure 2.4 [38]:
d[EA] dt
=
k1[E][A] + k4[EAB] - [EA](k2 + k3[B]) = 0,
d[EAB] dt
=
k3[EA][B] + k6[EQ][B] - [EAB](k4 + k5) = 0, (2.1.14)
d[EQ] dt
=
k8[E][Q] + k5[EP Q] - [EQ](k7 + k6[P ]) = 0,
and
[E0] - ([E] + [EA] + [EAB] + [EQ]) = 0,
v
=
d[Q] dt
=
k7[EQ]
-
k8[E][Q].
(2.1.15)
The rate equation can be written as a function of internal and external metabolites, and parameters can be determined by some improved methods such as King Altman and Cleland method. In [25], formulas and proofs are given in a
19
A
B
P
Q
E
EA
(EAB -EPQ)
EQ
E
Ordered Bi Bi Mechanism .
A
P
B
Q
E
(EA - FP)
F
(EB -EQ)
E
Ping Pong Mechanism.
A
B
EA
P
Q
EQ
E
(EAB)
E
EB
(EPQ)
EP
A
B
A
Q
P
Random ordered Bi Bi Me chanism.
A
P
Q
E
(EA - EPQ)
EQ
E
Ordered Uni Bi Mechanism.
Figure 2.1: Schematic explanation of different types of reaction mechanisms (taken from [38]).
20
A
B
P
Q
E
EA
EQ
E
Theorell Chance Me chanism.
P
Q
A
EQ
E (EA)
E
(EPQ)
EP
Q
P
Random Ordered Uni Bi Me chanism.
A
B
C
P
Q
E
E
Ordered Ter Bi Mechanism.
A
B
P
C
Q
E
E
Bi Uni Uni Uni Ping Pong Me chanism.
Figure 2.2: Schematic explanation of different types of reaction mechanisms (taken from [38]).
21
A
P
B
C
Q
E
E
Uni Uni Bi Uni Ping Pong Mechanism.
AB C
P QR
AB
P C QR
E
E
Ordered Ter Ter Mechanism .
E
E
Bi Uni Uni Bi Ping Pong Me chanism .
AP
B
QCR
AP
Q BCR
E
E
E
E
Uni Uni Uni Uni Uni Uni Me chanism . Uni Bi Bi Uni Ping Pong Me chanism .
AB
P
QCR
AP
B
CQR
E
EE
E
Bi Bi Uni Uni Me chanism .
Uni Uni Bi Bi Ping Pong Me chanism .
Figure 2.3: Schematic explanation of different types of reaction mechanisms (taken from [38]). 22
k1 E+A k2 k3 EA+B k4 k5 EPQ k6 k7 EQ k8
EA EAB(EPQ) EQ+P E+Q
Figure 2.4: Explicit reaction scheme for ordered Bi Bi mechanism [38]. more detailed way. In [38], they are stated explicitly as follows:
X
=
1+
A KIA
+
KMA B KIAKMB
+
KMQ P KMP KIQ
+
Q KIQ
+
AB KIAKMB
+
KMQ BP KIAKMP KIQ
,
Y
=
KMA BQ KIAKMB KIQ
+
PQ KMP KIQ
+
ABP KIAKMB KIP
+
BP Q KIBKMP KIQ
,
Vmf axAB v = X -+ Y , KIAKM B
Vmr axP Q KM P KIQ
where KI's are inhibition constants and KM 's are Michaelis Menten constants. The explicit form of the parameters is given by Bowden (1979) and stated below [38].
23
Vmfax = k5k7E0/(k5 + k7), Vmr ax = k2k4E0/(k2 + k4), KMA = k5k7/(k1(k5 + k7)), KMB = (k4 + k5)k7/(k3(k5 + k7)), KMP = (k4 + k5)k2/(k2 + k4)k6, KMQ = k2k4/(k8(k2 + k4)), KIA = k2/k1, KIB = (k2 + k4)/k3, KIP = (k5 + k7)/k6, KIQ = k7/k8.
Rapid Equilibrium Random Bi Bi Mechanism
In this mechanism, the enzyme binds to substrates and dissociates from them randomly. A free enzyme can bind either to the first substrate or to the second substrate. Since it is difficult to obtain rate equations with the steady state assumption, a rate equation is derived from equilibrium assumption. According to this assumption, except EAB EP Q, all reactions are assumed to be in the equilibrium state [38].
Definition 2.1.5. In 1913, Lenor Michaelis and Maude Menten, advancing earlier work of the chemist Victor Henri, assumed that k-1 >> k2. If this is true, then the reversible step in the mechanism does achieve an equilibrium and we can write the law of chemical equilibrium for this reversible step and, hence, equate the ratio between the forward (k1) and the reverse (k-1) rate constants with the equilibrium expression [37]:
Ks
=
k-1 k1
=
[E][S] [ES]
,
24
where all variables are the same as in the mechanism fron Section 2.1.2 and Ks is the equilibrium constant. In [37], it is found that rate equation is
v
=
(
k2 Ks
)[E][S],
where k-1 >> k2. The rate equation and kinetic parameters are given in [38]:
where
Vmf axAB - KIAKM B
Vmr axP Q KM P KIQ
v= A
B
P
Q
AB
PQ
1 + + + + + + KIA
KIB
KIP
KIQ
KIA KM B
KM P KIQ
,
Vmfax = CE0 k1, Vmr ax = CE0 k2, KIA = [A][E]/[EA], KIB = [B][E]/[EB], KMA = [EB][A]/[EAB], KIP = [P ][E]/[EP ], KIQ = [Q][E]/[EQ], KMP = [EQ][P ]/[EP Q],
and
[E0] = [E] + [EA] + [EB] + [EAB] + [EP ] + [EQ] + [EP Q], (2.1.16)
dP dt
= v = k1[EAB] - k2[EP Q].
(2.1.17)
2.1.4 The Stoichiometric Matrix The chemical reaction model can be described with the matrix representation which is called a stoichiometric matrix N . Each chemical reaction in the model network has a stoichiometric coefficient which comes before the reactants. The
25
entries of our m Ч n matrix N are constructed by the following rule:
+c, if the reaction produces metabolite Xi Ni,j = -c, if the reaction consumes metabolite Xi


0,
if the
reaction neither
produces
nor consumes
metabolite
Xi,
where m denotes the number of metabolite, n denotes the number of reactions and c is the stoichiometric constant. The stoichiometric matrix N is a tool for deriving conservation relations of a chemical system. It can be reduced to a simple matrix formed by identitiy elements and zero elements to get conservation relations. With the help of the stoichiometric matrix N , the rate vector v and general kinetic equations discussed above, the following equation can be derived:
x = N · v(c, k).
(2.1.18)
Here, x is the vector of all metabolites, N is the stoichiometric matrix and v is the vector of all rates.
26
Figure 2.5: A. skeleton model of glycolysis, B. This simplified representation includes only metabolites (taken from [13]). 2.2 Deriving Conservation Relations The previous chapter gives an introductory idea of modeling tools and rate equations. In this chapter, conservation relations, stoichiometric matrix analysis with conservation relations will be introduced by an example, namely, a skeleton model of mammalian glycolysis which is represented in Figure 2.5 [13]. 2.2.1 Deriving Conservation Relations by Inspection We assume that the external metabolites glucose, lactate, ADP, ATP, Pi, NADP+ and NADPH are fixed parameters and, therefore, are not included in the kinetic model. The model equation of the system in Figure 2.5 is of the 27
following form:
ds1 dt
=
v1 - v2 - v6,
ds2 dt
=
2v2 - v3,
ds3 dt
=
v3 - v4,
ds4 dt
=
v4 - v5,
ds5 dt
=
v5 - v3,
ds6 dt
=
v3 - v5.
(2.2.19) (2.2.20) (2.2.21) (2.2.22) (2.2.23) (2.2.24)
Matrix Formulation by Stoichiometricity A matrix formulation of the kinetic model is derived by the above differential equations. As it is mentioned before, the kinetic model can be written in the form with the stoichiometric matrix and the vector of metabolites, namely,
x = N · v(C, K).
As we noted, the numbers on the right-hand side matrix are stoichiometric coefficients which come before the reactant in the chemical reaction and their signs are decided by the rule mentioned before:


ds1 dt
1 -1 0 0 0 -1
v1


ds2 dt
0
2 -1
0
0
0

v2


ds3 dt
=
0
0
1 -1
0

0
·
v3
.
ds4

dt


0

0
0
1 -1
0


v4


ds5 dt

0

ds6 dt
0
0 -1 01
01 0 -1
0


v5


0
v6
(2.2.25)
Let us consider the fifth and sixth equations
ds5 dt
=
v5 - v3,
(2.2.26)
28
ds6 dt
=
v3 - v5.
It is obvious that
ds5 dt
+
ds6 dt
=
0.
Equation (2.2.28) can be also written as
(2.2.27) (2.2.28)
d dt
(s5
+
s6)
=
0.
(2.2.29)
From Calculus it is obvious that if the derivative of a smooth function over an
interval is zero, then it must be constant.
Thus,
s5 + s6 = T1,
(2.2.30)
where T1 is a constant. We call equation (2.2.30) a conservation equation. In physical manner, this equation states that although the concentrations of s5 and s6 change with the state of the system, the sum of their concentrations must remain constant. The second constraint is not so obvious as the first:
ds3 dt
+
ds4 dt
+
ds5 dt
=
v3
- v4
+ v4
- v5
+ v5
- v3
=
0,
(2.2.31)
so that
s3 + s4 + s5 = T2.
(2.2.32)
This procedure is not obvious for all enzymatic reactions, but it gives complex and big systems. So, we should generalize this for big systems. Note that differential equations which turned into conservation relations are linearly dependent. Namely, one can be written in terms of the others, e.g., by (2.2.28) and (2.2.31),
ds6 dt
=
-
ds5 dt
ds4 dt
=
-
ds3 dt
-
ds5 dt
.
(2.2.33) (2.2.34)
29
2.2.2 Deriving Conservation Relations by Gaussian Elimination
For simple models, conservation relations can be found just by inspection, but it is not so easy for complex systems. A more systematic procedure is needed for big systems. In this section, the way of getting conservation relations provides to partition the fluxes and concentrations into dependent and independent sets. The procedure is formally called Gaussian elimination which we know from Linear Algebra. Gaussian elimination is applied to our stoichiometric matrix; then, (2.2.25) can be rewritten as:



1 -1 0 0 0 -1
v1
100000
ds1 dt



0
2 -1
0
0
0

v2

0
1
0
0
0
0
ds2 dt
0
0
1 -1
0

0
·
v3

=

0
0
1
0
0
0
·
ds3 dt
.

0

0
0
1 -1
0


v4



0

0
0
1
0
0


ds4

dt


0
0 -1
0
1
0


v5


0
0
0
0
1
0

ds5 dt



0 0 1 0 -1 0
v6
000001
ds6 dt
(2.2.35)
While Gaussian elimination is applied to the stoichiometric matrix, same ele-
mentary row operations must simultaneously be applied on the identity matrix.
By performing this we have recorded which multiple of which row was added to
other ones. Hence, we get linearly dependent or independent metabolites. This
means that we only have to handle the following matrices [13]:
R1 R2 R3 R4 R5 R6 s1 s2 s3 s4 s5 s6 S1 1 -1 0 0 0 -1 1 0 0 0 0 0 S2 0 2 -1 0 0 -1 0 1 0 0 0 0 S3 0 0 1 -1 0 0 0 0 1 0 0 0 S4 0 0 0 1 -1 0 0 0 0 1 0 0 S5 0 0 -1 0 1 0 0 0 0 0 1 0 S6 0 0 1 0 -1 0 0 0 0 0 0 1
(2.2.36)
30
In terms of this matrix formulation, Gaussian elimination has to perform row manipulations on the whole matrix until a row reduced echelon form is obtained. In our model, a row reduced echelon form is performed by matlab command rref. However, this command does not pay attention to row substraction. In fact, if row substraction is done during Gaussian process, it is not going to ensure the added elements of the identity matrix being positive. In biological terms, concentrations in the conservation relationships must be added, not substracted. But with rref command of Matlab, we can see which variables are dependent or independent. We should stop the program if concentrations would be substracted. Although subtracting is not a problem for computational purposes, it is not at all equivalent in terms of physcial and biological interpretation. If we apply the above procedure without substracting the rows, then we obtain the following matrices:
R1 R2 R3 R4 R5 R6 s1 s2 s3 s4 s5 s6 S1 1 -1 0 0 0 -1 1 0 0 0 0 0 S2 0 2 -1 0 0 0 0 1 0 0 0 0 S3 0 0 1 -1 0 0 0 0 1 0 0 0 S4 0 0 0 1 -1 0 0 0 0 1 0 0 S5 0 0 0 0 0 0 0 0 1 1 1 0 S6 0 0 0 0 0 0 0 0 0 0 1 1
(2.2.37)
Thus, we have transformed (2.2.35) into the following form:



1 -1 0 0 0 -1
v1
100000
ds1 dt



0
2 -1
0
0
0

v2

0
1
0
0
0
0
ds2 dt
0
0
1 -1
0

0
·
v3

=

0
0
1
0
0
0
·
ds3 dt
.

0

0
0
1 -1
0


v4



0

0
0
1
0
0


ds4

dt


0
0
0
0
0
0


v5


0
0
1
1
1
0

ds5 dt



000000
v6
000011
ds6 dt
(2.2.38)
31
2.2.3 Deriving Conservation Relations by Partioning the Stoichiometric Matrix into Submatrices In this subsection, conservation relations are derived by dividing the stoichiometric matrix N from the model equation (2.1.18) into submatrices. Let N be an m Ч n matrix represented by
N = L · NR,
(2.2.39)
where NR is an m0 Ч r matrix and r is the dimension of rate vector v. Thus, L can be written in the following form:
L = Im0 , L0
(2.2.40)
where Im0 is the m0 Ч m0 identitiy matrix and L0 is an (m - m0) Ч m0 matrix. It is clear that we can divide the vector X by the following subvectors because of the relationships with N :
x = xR1 , xR2
(2.2.41)
where xR1, xR2 are m0 Ч1 and (m-m0)Ч1 column vectors, respectively. Therefore, according to these definitions and by the help of the following theorem, conservation relations can be determined by using the matrix L0 [38, 39].
Theorem 2.2.1. Every conservation relation in multi enzymatic systems can be written by the linear composition of (m - m0) linearly independent conservation relations defined by the following equation:
d dt
(xR2
-
L0xR1)
=
0.
(2.2.42)
32
Proof. We know that
L = Im0 L0
and x = xR1 . xR2
If we put these into (2.1.18) we get
(2.2.43) (2.2.44)

d dt
xR1 xR2
=
Im0 L0
· NRv.
(2.2.45)
Thus,
d[xR1 ] dt
=
NRv
(2.2.46)
and
d[xR2 ] dt
=
L0NRv.
(2.2.47)
If in (2.2.47) we write the left-hand side of (2.2.46) instead of NRv, we have
d[xR2 ] dt
-
L0
d[xR1 dt
]
=
0.
(2.2.48)
Rearranging this equation, we obtain
d dt
(xR2
-
L0xR1)
=
0.
(2.2.49)
In case of logical errors, the following algorithm is proposed in [38]: Algorithm: Let N be an m Ч n matrix and its rank be m0. The rank of N can be computed easily by the Maple command rank. (In this study, Matlab 6.5 is used, and to evaluate rank, the same command rank is used.) · Matrices NR1 and NR2 are evaluated, where NR1 is composed of m0 linearly 33
dependent rows of N and NR2 consists of the remaining rows of N . Thus, the dimension of NR1 and NR2 is m0 Ч r and (m - m0) Ч r, respectively.
· Now, NR1 is divided into submatrices as NR11 and NR12. The matrix NR11 consists of linearly independent columns of NR1. Thus, the inverse of NR11 is an m0 Ч m0 square matrix which can be computed, and NR12 is an m0 Ч (r - m0) matrix.
· Then, NR2 is divided into submatrices as NR21 and NR22, where NR21 consists of columns of NR2 during construction of NR11 matrix, and NR22 is the remaining part of NR2 after subtracting NR21. Thus, the dimension of NR21 is (m - m0) Ч m0 and NR22 is of the type (m - m0) Ч (r - m0). So, our matrix N is of the following form:
N = NR11 NR21
NR12 . NR22
(2.2.50)
· Furthermore, L0 can be computed by the formula : L0 = NR21.(NR22)-1.
(2.2.51)
· Finally, x can be divided as before: x = xR1 , xR2
(2.2.52)
and we compute conservation relations by the equation (2.2.42).
34
CrP CK ADP
Cr DGlu HK ATP
ADP
NAD+P
NADPH
DGlu6P
G6PD
6PGL
. Figure 2.6: Creatine Kinase, Hexokinase and Glucose-6Phosphate Dehydrogenase System [38]. 2.3 Creatine Kinase, Hexokinase and Glucose6Phosphate Dehydrogenase System Up to now, in previous sections, biochemical laws and mechanisms of some types of reactions and how to construct a model by differential equations with deriving conservation relations have been given. In this section, biological information of our main model will be introduced which consists of three enzymes: Creatin Kinase, Hexokinase and Glucose-6Phosphate Dehydrogenase. The overall reaction of this three enzymatic system is illustrated in Figure 2.7 [38] and (2.3.53) in [39]: Creatinphosphate + ADP creatinekinase creatin + AT P, AT P + D - Glucose hexokinase D - Glucose6 - phosphate + ADP, (2.3.53) D-Glucose6-phosphate+N ADP G6P DH D-Gluconate6-phoshate+N ADP H.
35
2.3.1 Creatine Kinase System Creatine Kinase (EC 2.7.3.2) (CK) plays an important role in energy metabolism of tissues such as skeletal and cardiac muscle neural tissues like brain and retina by providing regeneration of ATP. It catalyzes the reversible transfer of the phosphoryl group from phosphocreatine to ADP, to regenerate ATP. There are types of CK isoenzymes differring according to the place of ATP production such as mitochondria and cytosol. "'Brain cytosolic and mitochondrial isozymes are BB-CK and Mia-CK, respectively (Wallimann et al., 1992; Wallimann et al., 1998a)"' [7]. Energy is important for development or maintaining of the functional celebral activities. Because of this reason, change in the level of CK activity may lead to neuronal loss in brain, which occurs in a neurodegenerative pathway (Tomimoto et al., 1993). Recent studies strengthen this hypothesis by showing that CK activity is severely reduced in some neurodegenerative diseases (David et al., 1998; Aksenov et al., 2000). Cytosolic CK's are divided into muscle type (M) and brain type (B). "'CK-MM and CK-BB are expressed at high levels in the skeletal muscle, and in the brain and smooth muscle, respectively, and hybrid CK-MB is found in the cardiac muscle"' [15]. In this study, CK is obtained from a heart of a rat. All experiments are done in the AtatuЁrk University of Applied Biotechnology Research Center. This experimental study can be found in more detail including experimental assays and units in Necmettin Yildirim's research. CK enters the reaction following rapid equilibrium random ordered Bi Bi mechanism. With the help of schematic representation of rapid equilibrium random ordered Bi Bi mechanism presented in Subsection 2.1.3, the rate equations and parameters are introduced by Necmettin Yildirim: 36
Vmfax = CE0 k1, Vmr ax = CE0 k2, KIADP = [ADP ][E]/[EADP ], KICrP = [CrP ][E]/[ECrP ], KIAT P = [AT P ][E]/[EAT P ], KICr = [Cr][E]/[ECr], KMADP = [ECrP ][ADP ]/[EADP CrP ], KMAT P = [ECr][AT P ]/[EAT P Cr],
[E0] = [E]+[EADP ]+[ECrP ]+[EADP CrP ]+[EAT P ]+[ECr]+[EAT P Cr],
dAT P dt
= v = k1[EADP CrP ] - k2[EAT P Cr].
The rate equation is rearranged by Necmettin Yildirim using GroЁbner Basis
method in Maple [38], which is not studied in this research [12]. It has the
following form:
v = - [CrP ] 1 + + + + + + KICrP
Vmf ax[CrP ][ADP ] KICrP KM ADP
Vmr ax[Cr][AT P ] KM Cr KIAT P
[ADP ] [Cr] [AT P ] [CrP ][ADP ]
KIADP
KICr
KIAT P
KICrP KM ADP
[Cr][AT P ] KM Cr KIAT P
.
2.3.2 Hexokinase System
Hexokinase (EC 2.7.1.1) is the first enzyme of the glycolysis reaction catalyzing the conversion of glucose to glucose-6phosphate. It takes phosphate of ATP and binds to the glucose inorder to produce glucose-6phosphate. During taking phosphate from ATP, Hexokinase requires Mg2+, and ATP binds to enzyme with Mg2++ [33]. The enzyme has a low KM for the sugar substrate (about 0.1 mM) and is inhibited by the product of its reaction, G6P [34]. There are four isoforms of this enzyme in mammalian tissue differing with respect to their function significantly. Isoform I is responsible for rate limiting step in gly-
37
colysis in brain and red blood cells. The reaction product Glucose-6phosphate inhibits both isoform I and II (but not IV). Inorganic phosphate Pi, however, dissociates from Glucose-6phosphate only by inhibition of isoform I. "`Thus, among hexokinase isoforms, brain hexokinase exhibits unique regulatory properties in that physiological levels of Pi can reverse inhibition due to physiological levels of Gluc-6-P [1315] "' [1]. In this research, hexokinase from the yeast cell is used in the experiment. It reacts as rapid equilibrium random Bi Bi mechanism. According to this mechanism, as discussed in the previous section, our mathematical model is:
Vmfax = CE0 k1, Vmr ax = CE0 k2, KIAT P = [AT P ][E]/[EAT P ], KIDGlu = [DGlu][E]/[EDGlu], KIADP = [ADP ][E]/[EADP ], KIDGlu6P = [DGlu6P ][E]/[EDGlu6P ], KMAT P = [EDGlu][AT P ]/[EAT P DGlu], KMDGlu6P = [EDGlu6P ][ADP ]/[EADP DGlu6P ],
[E0] = [E]+[EADP ]+[EDGlu6P ]+[EADP DGlu6P ]+[EAT P ]+[EDGlu]+[EAT P DGlu],
dDGlu6P dt
= v = k1[EAT P DGlu] - k2[EADP DGlu6P ].
After rearranging the rate equation by Maple finding GroЁbner basis (for detailed information see [38]) it is reduced to
v = - [DGlu] 1 + + + + + + KIDGlu
Vmf ax[DGlu][AT P ]
KIDGluKM AT P
[AT P ] [DGlu6P ]
KIAT P
KIDGlu6P
Vmr ax[DGlu6P ][ADP ] KM DGlu6P KIADP
[ADP ] [DGlu][AT P ]
KIADP
KIDGluKM AT P
[DGlu6P ][ADP ] KM DGlu6P KIADP
.
38
2.3.3 Glucose 6-Phosphate Dehydrogenase System
Glucose-6-phosphate Dehydrogenase (G6PDH) (EC 1.1.1.49) is the rate limiting enzyme in pentose phosphate pathway (PPP) providing to control the amount of NADPH. It is important since PPP is the only source for NADPH in the erythrocytes. From the recent studies it is found that G6PDH plays a crucial role against antioxidative stress occuring in intracellular metabolic processes caused by harmful radicals in human erythrocytes. It prevents the forming of such harmful radicals by reducing NADP+ to NADPH. Furthermore, it plays a protective role against the reactive oxygen space in nucleatited eukoryatic cells that results with alternative ways to produce NADPH [40]. G6PDH is taken from human erythrocytes during the experiment, and it obeys the ordered Bi Bi mechanism. By the mechanism given in Subsection 2.1.3, the mathematical model and reduced rate equation of this system is given according to [38]:
d[EN ADP +] dt
=
k1[E][N ADP +] + +k4[EN ADP +DGlu6P ]
-[EN ADP +](k2 + k3[DGlu6P ]) = 0,
d[EN ADP +DGlu6P ] dt
=
k3[E][N ADP +][DGlu6P ] + k6[6P GL][N ADP H]
-[EN ADP +DGlu6P ](k4 + k5) = 0,
d[EN ADP H] dt
=
k8[E][N ADP H] + k5[EN ADP +DGlu6P ]
-[EN ADP H](k7 + k6[6P GL]) = 0,
and
[E0] = [EEN ADP +] + [EN ADP +DGlu6P ] + [EN ADP H] + [E],
v
=
d[N ADP H] dt
=
k7[EN ADP H]
-
k8[E][N ADP H],
Vmf ax[N ADP +][DGlu6P ] v = 1 + A + B +-C + D + E . KINADP + KM DGlu6P
Vmr ax[6P GL][N ADP H] KM 6P GLKIN ADP H
39
Here, A,B and C are:
A
=
[N ADP +] KIN ADP +
+
KMNADP +DGlu6P KIN ADP + KMDGlu6P
+
KMNADP H [6P GL] KM6P GLKINADP H
,
B
=
[N ADP H] KINADP H
+
[N ADP +][DGlu6P KIN ADP + KMDGlu6P
]
,
C
=
KMNADP H [N ADP +][6P GL] KINADP + KM6P GLKINADP H
,
D
=
KMNADP +[DGlu6P ][N ADP H] KIN ADP + KMDGlu6P KIN ADP H
+
[6P GL][N ADP KM6P GLKINADP
H H
]
,
E
=
[N ADP +][DGlu6P ][6P GL] KIN ADP + KMDGlu6P KI6P GL
+
[DGlu6P ][6P GL][N ADP KIDGlu6P KM6P GLKIQ
H
]
,
and the included kinetic parameters are
Vmfax = k5k7E0/(k5 + k7), Vmr ax = k2k4E0/(k2 + k4), KMNADP + = k5k7/(k1(k5 + k7)), KMDGlu6P = (k4 + k5)k7/(k3(k5 + k7)), KM6P GL = (k4 + k5)k2/(k2 + k4)k6, KMNADP H = k2k4/(k8(k2 + k4)), KINADP + = k2/k1, KIDGlu6P = (k2 + k4)/k3, KI6P GL = (k5 + k7)/k6, KIQ = k7/k8.
2.3.4 Mathematical Modeling of CK-HK-G6PDH System In this subsection, all DAE's will be introduced with given rate equations depending on concentrations and parameters discussed in previous subsections. Since chemical reactions occur with some definite rate changing with parameters
40
and concentrations over the time, the kinetic behaviour of the reaction can be defined by some ODE's with initial conditions taken during an experiment. As discussed in Chapter 2, differential equations and the stoichiometric matrix of our model are given in the following: Let x be the vector of metabolites in Figure 2.7:


x1
CrP


x2 ADP



x3


DGlu6P




x4


AT P



x=

x5

=

DGlu
,



x6

Cr

x7

N ADP +




x8


6P GL



x9
N ADP H
and v be the rate vector of dimension 3, since we have 3 overall reactions:
v1
v
=

v2

.

v3
41
The stoichiometric matrix of this system can be written according to the definition given in Chapter 2:


-1 0 0

-1
1
0




0
1
-1




1 -1
0



N =
0 -1
0

.


1 0 0




0 0 -1



0
0
1



001
(2.3.54)
By using (2.1.18), all differential equations can be denoted in the vector form:


-v1


-v1 + v2



v2 - v3




v1 - v2

dx dt
=

-v2
.

v1




-v3


v3



v3
(2.3.55)
Conservation Relations Looking for the rank of matrix N in Matlab 6.5 by using the command rank(N), it gives us the value 3. Clearly, when N is subdivided into matrices as discussed by "`Conservation Relations"' in Chapter 2, we will have 6 conservation relations since N is a (9 Ч 3) matrix and 9 - rank(N ) = 6. Thus, we can divide N in
42
the following form:


-1 0 0
NR1
=

-1
1
0

,


0 1 -1
(2.3.56)


1 -1 0


0 -1 0



1
NR2
=

0

0 0
.
0
-1




0
0
1



001
(2.3.57)
It is obvious that NR1 = NR11, since the rank of N is equal to the number of columns. Remember that NR11 consists of the linearly independent columns of NR1. Obviously, as a conseqence of the previous observation, NR2 = NR21. Thus, L0 can be computed by equation (2.2.51):


0 -1 0


1 -1 0



-1 0
L0
=


1 -1

0
.
1




-1
1
-1



-1 1 -1
In Subsection 2.2.3, the vector x is divided into two parts as xR1 and xR2, where the dimension of xR1 is the rank of N (maximal number of linearly independent columns of N ) and xR2 is the remaining vector in x. Thus, we can write x according to our model by two parts:
x1
xR1
=

x2


x3
(2.3.58)
43
and
x4


x5


xR2
=

x6 x7
.

(2.3.59)

x8


x9
By Theorem 2.2.1, it is clear that conservation relations can be stated in the
following form:


x4
0 -1 0



x5


1

d dt
(xR2
-
L0xR1)
=
0
=

d dt
(

x6 x7
-
-1 1


x8


-1
-1 0 -1 1
0


0 1


x1 x2
) = 0

-1

x3


x9
-1 1 -1
=


x4 + x2



x5 - x1 + x2



d

dt

x6 + x1 x7 - x1 + x2 - x3
=0
.



x8 + x1 - x2 + x3



x9 + x1 - x2 + x3
(2.3.60)
Consequently, the function which we differentiated must be a constant and, to be biologically meaningful, these constants must be the initial concentrations
44
of these metabolites because of the law of mass conservation. Thus, we have:



x4 + x2
x40 + x20




x5 - x1 + x2

x50 - x10 + x20





x6 + x1
=
x60 + x10


x7
- x1
+ x2
- x3


x70 - x10 + x20 - x30





x8 + x1 - x2 + x3


x80 + x10 - x20 + x30




x9 + x1 - x2 + x3
x90 + x10 - x20 + x30
=
(2.3.61)


x4 + x2 - x40 - x20



x5 - x1 + x2 - x50 + x10 - x20




x6 + x1 - x60 - x10
=0.

x7 - x1 + x2 - x3 - x70 + x10 - x20 + x30




x8 + x1 - x2 + x3 - x80 - x10 + x20 - x30



x9 + x1 - x2 + x3 - x90 - x10 + x20 - x30
(2.3.62)
Thus, our model consists of three differential equations given in (2.3.63) with
constraints (2.3.62):
dx1 dt
=
v1 ,
dx2 dt
=
v2 ,
dx3 dt
=
v3 ,
(2.3.63)
where v1, v2, v3 are the rate equations of the CK, HK and G6PDH system, respectively, given in Subsections 2.3.1, 2.3.2 and 2.3.3, respectively.
Parameters Used in the Mathematical Model It is known that rate equations depend on both metabolites concentrations and parameters. In the CK-HK-G6PDH model, we have 3 differential equations with 6 linear constraints and 26 parameters such as KM , KI and Vmax values. 45
KMADP KIADP KMAT P KIAT P KMCr KICr KMCrP KICrP Vcfat Vcrat
Parameters of CK System 5, 00x10-2mM 1, 70x10-1mM 4, 80x10-1mM 1, 20x100mM 6, 10x100mM 1, 56x101mM 2, 90x100mM 8, 60x100mM 200min-1 100min-1
Literature Literature Literature Literature Literature Literature Literature Literature Literature Literature
Table 2.1: Parameters of CK system (Morrison and James, 1965).
KMAT P KIAT P KMADP KIADP KMDGlu6P KIDGlu6P KMGlu KIGlu Vcfat Vcrat
Parameters of HK System 6, 30x10-2mM 6, 30x10-2mM 2, 30x10-1mM 2, 30x10-1mM 4, 00x10-2mM 6, 70x100mM 1, 00x10-1mM 1, 00x10-1mM 58.82min-1 11764.7min-1
Literature Literature Literature Literature Literature Literature Literature Literature Literature Literature
Table 2.2: Parameters of HK system (Viola et al., 1982).
The initial values of metabolites and all parameters are taken from [38].
46
KMN ADP + KIN ADP + KMDGlu6P KIDGlu6P KM6P GL KI6P GL KMNADP H KINADP H Vcfat Vcrat
Parameters of G6PDH System 6, 10x10-3mM 6, 20x10-3mM 3, 90x10-2mM 8, 90x10-1mM 1, 38x10-1mM 6, 90x10-3mM 3, 90x10-3mM 6, 80x10-3mM 2836.87min-1 42553.19min-1
Literature Literature Literature Literature Literature Literature Literature Literature Literature Literature
Table 2.3: Parameters of G6PDH system (Gordon et al., 1995).
The initial concentrations of the metabolites and enzyme concentrations are
x10 = 2, 55x101mM, x20 = 5, 9x10-1mM, x30 = 0mM, x40 = 0mM, x50 = 1, 13x101mM, x60 = 0mM, x70 = 3, 3x10-1mM, x80 = 0mM, x90 = 0mM, CK0 = 8x10-6mM, HK0 = 3, 4x10-5mM, G6P DH0 = 1, 4x10-5mM.
(2.3.64)
47
Chapter 3 Numerical Solution Methods For Solving Differential Algebraic Equations The model obtained in the previous chapter consists of a system of differential equations with algebraic constraints, i.e., a system of differential algebraic equations (DAE's). In this chapter, we will summarize the existing methods for solving DAE's and apply the MATLAB's DAE solver ode15s to our model. 3.1 Introduction Differential algebraic equations are much more recent than ordinary differential equations. In the recent years, many enginneering, bioinformatics and medical problems are modelled by DAE's. However, difficulties arise, especially, when nonlinear algebraic constants and a lot of parameters exist. Modern application of DAE's can be found in the field of exercise metabolism where they are studied, e.g., in Sports Medicine. Here, we refer to the investigations [26, 27]. In these studies, a DAE model is reduced to the form of ODE's in order to simplify the equations. In the last years, several numerical methods are devised for solving DAE's, see, for example, the monographs [10, 24]. 48
The differences and similarities between ODE's and DAE's can be explained as follows: Consider y(t) and z(t) be the two functions defined on some interval [0, b] and related by the differential equation
y (t) = z(t), 0 t b .
(3.1.1)
To obtain y(t) from z(t), an integration with additional knowledge of y(0), over the interval a t b is needed; and to get z(t) from y(t), only y(t) should be differentiated. The differentiation is an easier process than integration. Indeed, y(t) is usually much more smooth than z(t). For example, z(t) may be bounded but have jump discontinuities. On the one hand, integration is a smoothing, process. On the other hand, differentiation is an anti-smoothing, a roughening process. Solution of ODE's involves integration, thus, it is smoothing, but solution of DAE's involves both integration and differentiation.
Definition 3.1.1. A system of DAE's is called fully implicit if it is of the
form
F (y, y, t) = 0 with y(0) = y0 .
(3.1.2)
Definition 3.1.2. A system of DAE's is called linearly implicit if it of in
the form
Ay + f (y, t) = 0 with y(0) = y0 ,
(3.1.3)
with a singular matrix A.
Definition 3.1.3. A system of DAE's is called semi explicit if it is of the
form
x = f (x, z, t),
(3.1.4)
g(x, z, t) = 0.
(3.1.5)
It
is
also
assumed
that
g z
has
a
bounded
inverse
in
a
neighbourhood
of
the
solution.
In our model CK-HK-G6PDH, equation (2.1.18), the differential part, cor-
49
responds to (3.1.4), and equation (2.3.62) is related to the algebraic part (3.1.5). Here, the vector of differential variables is denoted by x = (x1, x2, x3)T and the vector of algebraic variables is called z = (x4, x5, x6, x7, x8, x9)T . Then, we can write our model as a semi-explicit system of DAE's in the form
M y = f (t, y, p),
(3.1.6)
where y = (x1, x2, x3, x4, x5, x6, x7, x8, x9)T , and p is the vector of parameter. Furthermore, a singular matrix M is given by


100000000


0 1 0 0 0 0 0 0 0



0
0
1
0
0
0
0
0
0




0
0
0
0
0
0
0
0
0



M
=
0
0
0
0
0
0
0
0
0

,


0 0 0 0 0 0 0 0 0




0 0 0 0 0 0 0 0 0



0
0
0
0
0
0
0
0
0



000000000
and f (t, y, p) is of the following form:
(3.1.7)


v1(t, x, p)



v2(t, x, p)




v3(t, x, p)




x4 + x2 - x40 - x20



f (t, y, p) =
x5 - x1 + x2 - x50 + x10 - x20
.



x6 + x1 - x60 - x10




x7 - x1 + x2 - x3 - x70 + x10 - x20 + x30


x8 + x1 - x2 + x3 - x80 - x10 + x20 - x30



x9 + x1 - x2 + x3 - x90 - x10 + x20 - x30
(3.1.8)
Here, x10, ..., x90 denote the initial values of differential and algebraic variables.
50
3.2 Differential Index
The differential index of a DAE is defined to be the number of differentiations needed to transform the DAE into an explicit ODE. For the semi-explicit case, the index is the minimum number of differentiation of (3.1.5) with respect to time t, to obtain the following system of ODE's:
x = f (x, z, t) , z = s(x, z, t) .
(3.2.9) (3.2.10)
Definition 3.2.1. Let f be the once continuously differentiable function. DAE (3.1.2) has the differential index one if and only if a system of variables t, x, x , xЁ uniquely determines the variable x as a function of (x, t):
f (x, x , t) = 0 ,
d dt
f
(x,
x ,
t)
=
x
f
(x,
x ,
t)xЁ
+
x
f
(x,
x ,
t)x
+
t
f
(x,
x , t)
=
0
.
Example 3.2.2. Consider the following DAE problem
x = 3x + 4y , y = 5x + 3 .
Taking the derivative of second equation and substituting x from the first equation in the second one, we obtain a system of pure ODE's. Because the number of differentiations to obtain an ODE was one, the ODE is of index one, too. Definition 3.2.3. Let f be twice continuously differentiable. The DAE (3.1.2) has differential index two if and only if it is not of index one and the equation system of the variables t, x, x , xЁ, .x.. uniquely determines the variable x as a
51
function of (x, t):
f (x, x , t) = 0,
d dt
f
(x,
x ,
t)
=
x
f
(x,
x ,
t)xЁ
+
x
f
(x,
x ,
t)x
+
t
f
(x,
x ,
t)
=
0,
d2 dt2
f
(x,
x ,
t)
=
x
f
(x,
x ,
t).x..
+
.
.
.
.
Example 3.2.4. Consider the system of DAE
x2 = x1 + z1(t), x2 + z2(t) = 0. Differentiating the second equation and substituting it into the first one gives:
x 2 = -z2(t) = x1 + z1(t) = x1 = -z1(t) - z2(t) . Again differentiaing we get an ODE:
x 1 = -z1(t) - zЁ2(t) . Thus, our system has index two. DAE's with higher index are much more complex, because the number of dif- ferentiations needed in order to reduce this system to an ODE increases. Therfore, the numerical methods are devised for DAE's up to index three. Higher order index problems should be reduced to lower index ones. This procedure is called index reduction.
3.3 Numerical Solution Methods to Solve DAE's Numerical methods for DAE's are based either on backward differentiation methods or on Runge-Kutta methods. For an extensive study of these methods see the monographs [10] and [24]. 52
3.3.1 Backward Euler Consider the fully implicit DAE
F (t, y, y ) = 0 .
The idea here is to approximate y and y by a discretization formula. Discretize the state variable y into n equal pieces, i.e., yn. We approximate y by
yn - yn-1 hn
,
where hn is the step size. So, our DAE turns into
F
(tn,
yn,
yn
- yn-1 hn
)
=
0.
(3.3.11)
The system (3.3.11) consists of m nonlinear equations for yn at each time step n [24].
3.3.2 Runge-Kutta Method In general, the s-stage Runge-Kutta method applied to fully implicit DAE is defined by (cf. [24]):
0 = F (ti, Yi, Ki), ti = tn-1 + cih, i = 1, 2, ..., s, s Yi = yn-1 + h aijKj, j=1
(3.3.12)
and s
yn = yn-1 + h biKi.
(3.3.13)
i=1
We assume here that the coefficient matrix A = (aij) is nonsingular and the
method is specified by this matrix A and the vector b having the entries bi. For
the case of a semi-explicit DAE formulated in (3.1.4) and (3.1.5), the formula
53
(3.3.12) is converted in [24]:
Ki = f (ti, Xi, Zi), s Xi = xn-1 + h aijKj, j=1 0 = g(ti, Xi, Zi).
3.3.3 Backward Differentiation Formula
Backward differentiation formula (BDF) is a multistep method that uses information from previous steps to construct higher order approximations. Compared to the Runge-Kutta method, multistep methods require fewer function evaluations per step. Backward Differentiation Formula (BDF) is the most popular multistep method. The difference from the Runge-Kutta method is that f (t, y) is evaluated only at the right end of the current step, (tn, yn). The general BDF method for solving ODE's is given in the form of
k iyn-1 = h0f (tn, yn), i=0
(3.3.14)
where h stands for the step-size, i and 0 are coefficients. Here, k is the order of the method and denotes how many backward steps are used. The method is an implicit one if 0 = 0. If it is implicit, then it is necessary to calculate the function value at (tn, yn) which requires the treatment of a nonlinear system of equations solved usually by a modified Newton method.
Application of BDF method to DAE (3.1.2) gives
F
tn,
yn,
1 0h
k j=0
j yn-j
= 0.
For 0 = 0 = 1 this is simply Euler's method.
(3.3.15)
54
MATLAB's DAE solver ode15s BDF methods are implemented in MATLAB by the function ode15s which works for DAE index up to 2. The important advantage of this case is finding consistent initial values for the DAE system. The user supplies initial values for the system and the code evaluates the initial values again, and finds appropriate and consistent initial conditions. Because of these properties ode15s is chosen to simulate our model problem in this study. The MATLAB DAE solver ode15s can be applied to any semi-explicit system (3.1.6) with a singular matrix M . Matlab ode15s automatically recognizes that the problem is a DAE and computes consistent initial conditions close to given y0. Some parameters used in ode15s are given in the following: [t, Y ] = ode15s(odefun,tspan,y0), or [t, Y ]= ode15s(odefun,tspan,y0,options) odefun: A function that evaluates the right-hand side of the differential equation of the form y = f (t, y) or M y = f (t, y). tspan: This is the time interval of integration [t0 tf ]. The solver takes initial condition at t0 and integrates from t0 to tf . y0: denotes the vector of initial conditions. options: The set of optional parameters that change the default integration properties like error tolerances. Options can be changed using odeset of MATLAB [18]. 55
Chapter 4 Parameter Estimation In most of the real world problems such as considered and tested in engineering, computational biology and medicine, or in bioinformatics, mathematical models, contain a number of parameters. Usually, these parameters are not known exactly. Changing some parameter may affect the solutions strongly, which means that the model is sensitive. Sometimes, it may be hard to evaluate the all parameters by an experiment due to its difficulty of measuring the values. In such cases, several parameter estimation methods like Simplex method, Levenberg-Marquardt (LM) method, or quasi-Newton method are used. All of them are a kind of gradient-based method, or mixture between gradient and Newton's method. From these algortihms, LM is the most popular gradient-based method that converges quadratically to the local minimum. Apart from these methods, there are some parameter estimation algorithms that do not require specific functions to be optimized or evaluated. These are genetic algorithms and derivative free algorithms. In many cases, the function which represents the difference between the simulation and experimental results is not given in explicit form. For a closer information about such model free algorithms, we can refer to statistical learning, e.g., to the valuable book [29]. In bioinformatics studies, one of the most popular optimization method is genetic algorithm (GA) for reducing the error with respect to model parameters. We refer to [19] for the heuristic modelling and optimization algorithms like GA. In 56
this work, our nonlinear continuous parameter estimation problem is:
min ||xsimulation(k) - xexperimental(k)||2 , kl k ku. k
(4.0.1)
Here, xsimulation, xexperimental are the vectors of simulation and experimental results, respectively. Furthermore, k denotes the parameter, kl and ku are the upper and lower bound for parameter k, component-wise regarded, and ||.||2 denotes Euclidean norm.
4.1 Genetic Algorithm Genetic algorithms were firstly introduced by Holland in 1975 [2]. They are based on fittest and natural selection of the given population or mechanism which are simulated by these principles. GA is a random search algorithm which is good for nonlinear problems and its efficiency relies on the fact that it does not require to fix the initial value for the parameter estimation [20]. It is widely applied in chemical engineering, biology and and biochemistry, in short: in biotechnology. Contrary to LM, it finds the global minimum of the error function. In a classical GA, the set of candidate solutions are created randomly; because of this property, it is known as a stochastic algorithm. These randomly created solutions are saved in a pool and all individual ones are in the vector form called chromosomes. After a fitness criterion is applied to the solution space, the new generation is constructed from actual population according to reproduction, mutation and crossover [21]. 4.1.1 Coding One of the fundamental issue of GA is coding the solution space. Typically, each individual solution is coded in strings, sometimes called chromosomes. One of the usual form of the coding is binary coding which consists of the elements 57
Gene 1
Gene 9
1
1
0
1
1
0
0
0
1
Figure 4.1: Illustration of a chromosome where each box represents one gene of chromosome. of the genes. Binary values are given to each gene indicating, for example, genes' functions in the metabolic pathways. In Figure 4.1, we have illustrated the coding of a chromosome. 4.1.2 Reproduction This operator selects the pairs of chromosomes to produce offsprings which are the children of mated pairs. Reproduction occurs randomly where the pairs that obey the fitness criteria higher than the others, have more chance to be selected [21]. Each chromosome is ordered with respect to its fitness in the population where the one with minimum error is going to be the first candidate to be chosen. If this individual fulfills the given conditions, e.g., a tolerance value, then the algorithm chooses this vector of genes. If not, then GA follows the mutation or crossover operators. We illustrate the randomly generated pairs of chromosomes in a pool shown Figure 4.2, where each string represents one chromosome. 4.1.3 Crossover The crossover operator redistributes the genetic characteristics between the individuals [21]. Modification occurs within the two pairs of strings which are cut off from randomly chosen points. Then, these cutted pairs alternate each other. These alterations can be done in two ways: 58
Gene 1
Gene 9
1
1
0
1
1
0
0
1
1
1
1
1
1
0
0
1
0
1
0
1
0
0
0
1
0
1
1
0
1
1
0
1
1
1
0
0
Figure 4.2: A binary-valued, randomly generated pool.
1
1
0
1
1
0
0
0
1
0
0
1
1
1
0
1
1
0
Figure 4.3: A crossover 1. 1. We can replace the first part of the first string by the first part of the second, or the first parts may remain the same but the tails can be replaced by each other. We represent this in Figure 4.3. 2. We can crossover the selected parts of the cut strings. We have illustrated this type in Figure 4.4.
59
1
1
0
1
1
0
0
0
1
0
0
1
1
1
0
1
1
0
Figure 4.4: A crossover 2. 4.1.4 Mutation In nature, some genes of offsprings may have totally different characteristics from their parents because of various factors [21]. Mutation is a process to simulate this natural phenomenon. It selects a random gene in a string and changes its value again randomly. Like in case of the other operators, after chromosomes are mutated, they are going to be tested whether they fit to solution or not, and the algorithm continues until the stopping criterion holds. The following algorithm summarizes the whole steps in a GA. 1. Construct your population by randomly generated vectors of solution candidates (reproduction). 2. Test if your candidates in the pool fit your model. 3. Order them according to their fitness by putting solution with minimum error as a first candidate. 4. Apply mutation and crossover. 5. Test a newly generated pool in step 4, whether it fits or not. 6. Go to step 3. 7. If there is a solution which satisfies your stopping criterion: STOP; if not, go to step 4. 60
4.2 GA Adapted to CK-HK-G6PDH Model
In our parameter estimation problem, we coded the genes differently from the previous section, because our parameter values changes on some intervals [a, b]. In Section 4.1, we have discussed that genes are coded by binary values as opposed to their functionality in pathways. However, here, we are looking for the parameter values in the kinetic model of CK-HK-G6PDH system. These parameters are vmax and the KM , KI values of the chemical reactions in the model pathway. We totally have 26 parameters and 9 unknowns for the metabolite concentrations in the model. So, our chromosomes consist of 26 genes, i.e., they are represented by a vector of dimension 26. Now, GA is adapted to our model in the following form: Reproduction We generate our population pool randomly with the size of a 200 Ч 26 matrix by the Matlab command rand(200,26). Also, we specify the number of iterations to put some limit on the algorithm. After constructing candidates of solutions, the Next Step is to assess them according to their fitness to model, by looking at the error function:
Error = ||xsimulation(k) - xexperiment(k)||2.
(4.2.2)
Here, xsimulation is the vector after simulation with parameter k = (vmax, KM , KI ) within the randomly generated pool, and xexperiment is the experimental value of a metabolite over the time. After finding all error vectors, the errors are ordered in ascending form. If the first chromosome satisfies our stopping criterion, e.g., Error < T olerance, then the algorithm stops. If not, it will continue by the mutation and crossover operations. In this thesis, we could correct the prior simulated NADpH values by a GA, since we have only the Empirical results of NADPH. Mutation The chromosomes and genes that mutate them are chosen randomly. Here, we 61
should be careful by giving random values to genes in chromosomes since we have constraints in our parameters. We must restrict our interval of randomization process. Another restriction in this step is that mutation is not repeated in every iteration, but every second iteration, i.e., mod(2). This criterion is chosen to prevent big errors in parameter values. In fact, we may be far from the exact solution. Crossover In this step, we have randomly chosen the cut off point of every pair of individuals and make the crossover operation to the whole pool. As indicated in Subsection 4.1.3, there are two kinds of crossovering. We have chosen the first type, i.e., interchanging the first parts of the cutted strings. The reason for this choice consists in the fact that big changes in structure of chromosomes may produce large errors. After mutation and crossover, we must look at the fitness of the new candidates modified by GA. As we have mentioned before, we should evaluate the error functions, order them and look if the first string, i.e., first row in the pool satisfies our error tolerance. The algorithm continues up to the maximum iteration. 62
Chapter 5 Results and Discussion In this section, the results of simulation of our DAE system given in Chapter 3 will be discussed. The numerical results obtained using the MATLAB DAE solver ode15s are compared with the experimental and simulation results existing in the literature. After parameter estimation, the model is simulated again and the solutions are improved. 5.1 Results of Simulation In Figures 5.1, 5.3, 5.5, our simulation results are illustrated and in Figure 5.2, 5.4, 5.6, the results of [38] are given, where the reaction rates were assumed to be equal: v1 = v2 = v3. In Subsections 2.3.1, 2.3.2, 2.3.3, nonlinear rate equations are approximated using GroЁbner basis in MAPLE, and the resulting equations are solved numerically in FORTRAN. In fact, we have used different values for reaction rates v1, v2, v3 and solved our model as a DAE system. Our results differ from those given in [38]. However, our simulation results are qualitatively correct and give the tendency of the concentrations over time. Since only the concentration values of NADPH over time are measured in Laboratory of Biotechnology Applied and Research Center in AtatuЁrk University in Erzurum [38], our simulation results are compared with experimental values in Figure 5.1. A bit later, in Figure 5.8, we have the simulation results from [38]. 63
600 ADP ATP NADP 6PGL 500
Concentration(x10 -3 mM)
400
300
200
100
0
0
100
200
300
400
500
600
700
time(min)
Figure 5.1: Simulation results of ATP, ADP, NADP, 6PGL. The nonlinearity of the rate equations may cause the roundoff errors during numerical solutions with MATLAB. Furthermore, MATLAB works with sixteen digits, so there may be truncation errors during the numerical computations. Also the assumptions can cause some differences with the values in literature, since the steady state assumption is ignored. It turns out that our results are different from the ones in [38] before parameter estimation. After parameter estimation, however, our computed results fit very well to the experimental data, as it will be shown in Figure 5.9. In the previous chapter, we have introduced GA's which are widely used in parameter estimation problems. We have estimated our 26 parameters by using experimental data of NADPH and GA. By using estimated parameters, we have simulated our model again to get better and more accurate results. Our objective function during estimation was
min ||x9,resimulated(k) - x9,experimental(k)||2.
(5.1.1)
Here, x9,resimulated(k) is the vector of NADPH concentration after GA, x9,experimental(k)
64
Konsantrasyon (x10 -3 mM)
600 500 400 300 200 100 0 0
6PGL NADP ADP ATP
100
200
300
400
500
600
Zaman(dak)
Figure 5.2: Simulation results of ATP, ADP, NADP, 6PGL [38] (zaman: time, dakika: minute).
Concentration(x10-3mM)
900 800 700 600 500 400 300 200 100 0 0
DGlu6P Cr
100
200
300
400
500
600
700
time(min)
Figure 5.3: Simulation results of DGlu6P, Cr.
65
Konsantrasuyom (x10 - 3 mM)
3000 2500 2000 1500 1000 500 0 0
DGlu6P Cr
100
200
300
400
500
600
Zaman(dak)
Figure 5.4: Simulation results of DGlu6P, Cr [38] (zaman: time, dakika: minute).
Concentration(x10-3mM)
26000 24000 22000 20000 18000 16000 14000 12000 10000 0
CrP DGlu
100
200
300
400
500
600
700
time(min)
Figure 5.5: Simulation results of CrP, DGlu.
66
Konsantrasyon (x10 -3 mM)
27000 25000 23000 21000 19000 17000 15000 13000 11000 9000 0
CrP DGlu
100
200
300
400
500
600
Zaman(dak)
Figure 5.6: Simulation results of CrP, DGlu [38] (zaman: time, dakika: minute).
250 NADP(exp) NADP(sim) 200
Concentration(x10-3mM
150
100
50
0
0
100
200
300
400
500
600
700
time(min)
Figure 5.7: Experimental values of NADPH and simulation results of NADPH by ode15s.
67
CNADPH (x10 -3 mM)
270 240 210 180 150 120 90 60 30 0 0
NADPH_tahmin NADPH_deney
100
200
300
400
500
600
Zaman(dak)
Figure 5.8: Experimental values of NADPH and simulation results of NADPH [38] (zaman: time, dakika: minute). consists of the experimental values of NADPH over the time interval of [0, 647], and k is the vector of parameters. The NADPH concentration after parameter estimation by GA is given in Figure 5.9. It is obvious that our results better fits the experimental values.
5.2 Conclusion We have modelled an enzymatic reaction of a system CK-HK-G6PDH as a system of differential algebraic equations. The algebraic parts consisted of linear constraints. The numerical results obtained using MATLAB's ODE solver are comparable with the results given in the literature. After applying parameter estimation using a genetic algorithm, the new results fit very well the experimental results. An important point of this work consists in the application of DAE approach for the analysis of metabolic pathways. In literature, only in few cases the 68
250
200
Concentration(x10-3mM)
150 NADP(exp) NADP(after parameter est.) 100
050
0
0
100
200
300
400
500
600
700
time(min)
Figure 5.9: NADPH results after parameter estimation and experimental values of NADPH. applicability of DAE's to metabolic engineering are mentioned [26, 27, 38]. In these researches, DAE's are applied to human exercise metabolism, especially, in sports medicine where the concentration of lactate and other metabolites are measured for the prediction of future values of these reactants. Since lactate plays an important role in exercise physiology, the simulation of this product is analyzed by the model form of DAE's, and to reduce the system of equations it is simplified to a system of ODE's. We mention that in sports medicine one aim consists in the scientific development and design of training programs. The insights obtained here, e.g., by metabolic engineering which we are dealing with in this thesis, should also serve for general medicine, for instance, for patients in recovering. Indeed, we have modelled our system by means of DAE's and the experimental data are in vitro, i.e, not in living cell. Thus, in future work, this thesis can hopefully be applied to large systems or human metabolism, like those given in [26, 27], or in further and more general contexts. In this work, we have considered linear constraints only, but the application of a DAE solver for the analysis
69
of metabolic pathways with nonlinear constraints will be useful, too. Another important aspect and subject of this thesis's attention is sensitivity analysis of parameters. The sensitivity analysis for DAE methods [5, 6, 17] can be applied to metabolic pathways in future. Moreover, we have emphasized the role of optimization theory, e.g., by using the methods of genetic algorithms which we approached and used from the viewpoints of statistical learning and heuristics. In fact, our entire fruitful approach should be applied to large systems of those metabolic pathways. GA's can be scientifically located in model free approach, since the parameters are estimated by empirical data and stochastic random numbers. In contrast to classical optimization methods, GA's are applied by us since the objective function is not given explicitly. In fact, this function consists of a random and experimental, i.e., of a discrete set of variables. Further progress in optimization theory and statistical learning is expected by us, and it will be utilized in metabolic engineering for many important fields of medicine, health care and other areas of modern life. 70
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