Analysis of Metabolic Dynamics in Saccharomyces cerevisiae

Transcription

1 Analysis of Metabolic Dynamics in Saccharomyces cerevisiae SA14X Spring 12 - Degree Project in Engineering Physics, First Level Royal Institute of Technology (KTH), Department of Mathematics Alexander Axelsson, alax@kth.se Maria Bånkestad, bmaria@kth.se Viktor Kull, vkull@kth.se Jesper Welander, jeswel@kth.se Supervisors: Hildur Æsa Oddsdóttir, Xiaoming Hu May 21, 212

2 Abstract This thesis conducts a study on the stability of steady states in the glycolysis of in silico models of Saccharomyces cerevisiae. Such uncontrolled models could reach unstable steady states that are unlikely to occur in vivo. Little work has previously been done to examine stability of such models. The glycolysis is modeled as a system of nonlinear differential equations. This is done by using rate equations describing the rate of change in concentration of each metabolite involved in glycolysis. By linearizing this system around different equilibria and calculating the eigenvalues of the associated jacobian matrices the stability of the steady states can be determined. Additionally perturbation analysis adds further insight into the stability of the steady state. Given the large range of possible initial conditions which result in different steady states, a physiologically feasible one, as well as the environment around it, is chosen to be the subject of this study. A steady state is stable if all the eigenvalues of the Jacobian matrix are negative. The model created by Teusink et al, and expanded upon by Pritchard et, for the glycolysis in S.cerevisiae is used as the primary model of the study. The steady state does not have strictly negative eigenvalues: Two of them are very close to zero, with one positive, within error tolerance of our numerical methods. This means that linear analysis cannot determine whether the steady state is stable. The whole nonlinear system has to be considered. After performing perturbation analysis we conclude that the steady state is most likely stable in the Lyapunov sense.

3 Contents 1 Introduction 4 2 The biology of the cell The cell Metabolism S. cerevisiae Stability analysis Introduction Stability analysis of a linear system Linearization Stability analysis of a nonlinear system Perturbation analysis Systems biology Intro Modeling of chemical pathways Analysis of metabolic systems Method Modeling of glycolysis in S. cerevisiae Implementation Results and Conclusions 32 2

4 6.1 Results Steady state Stability analysis Conclusions Discussion 48 8 Reference list 5 9 Appendices Appendix 1: The model of Theobald et al Appendix 2: Pritchard Appendix 3: PHC Appendix 4: Abbreviations

5 1 Introduction The best test of our understanding of cells will be to make quantitative predictions about their behaviour and test them. This will require detailed simulations of the biochemical processes taking place within [cells]. Hartwell et al. [1] Systems biology is a relatively modern branch of biology where one focuses the application of dynamic systems theory on biological reactions at a molecular level. Having defined a well-functioning model of a cell, it is possible to study the dynamic properties of the system to draw conclusions on the general properties of the cell without experimentation, or even draw conclusions about reactions that cannot be measured or otherwise estimated experimentally. The aim of this thesis is to simulate the glycolysis in the yeast cell of Saccharomyces cerevisiae and to determine the stability of some found steady states. Understanding how S. cerevisiae functions is of great interest since it is used for baking and brewing, and glycolysis in particular is important since the process is common to all cells. It is also the only source of energy for cells under anaerobic conditions. To simulate the glycolysis, we utilize a system modelling these reaction in S. cerevisiae found by Teusink et al. and extended by Pritchard et al. This system models the concentration of 17 metabolic compounds, where the rate of production or utilization is given by approximations of the enzyme productions. The system further depends on the concentrations of 8 external compounds, which are assumed to be constant in the analysis of the system in this thesis. The glycolysis steady state is found using ODE-solvers in MATLAB, where the inital values used are as found by Pritchard et al. The steady state values found are analyzed by calculating the eigenvalues of the Jacobian matrix. The eigenvalues indicate the stability properties of the system. The stability is further analyzed by perturbing the concentration of one or more of the metabolites in the steady state for both the linearized and the original system. Sections two, three and four of this thesis provide the reader with the necessary background to understand the results. Section five discusses the glycolysis models selected and the useage of MATLAB in the simulation. Section six presents the results in form of tables and plots with comments. Then follows a general discussion. The thesis is concluded with an appendix in which the so called rate expressions are given. There is also some informa- 4

6 tion about the program PHClab and the efforts and failures concerning its use. Lastly a list of abbreviations is provided. 5

7 2 The biology of the cell 2.1 The cell The cell is the basic unit of life. There are two main classes of cells: Eukaryotic cells and prokaryotic cells. This article deals with the the eukaryotic cell of Saccharomyces cerevisiae which is discussed in more detail later. Prokaryotic cells are often a singular organism while eukaryotic cells are part of a larger multicelluar organism. The words Prokaryote and Eukaryote stem from greek and roughly translate to before nucleus and true nucleus. This hints at their defining properties: Eukaryotes are cells containing a nucleus, while prokaryotes do not. Apart from the nucleus eukaryotes often contain other compartments with highly specialized functions, called organelles. Eukaryotic cells are surrounded by a plasma membrane and contain a fluid substance called the cytoplasm holding a number of structures called organelles. The organelle at our focus is the mitochondria, being important in the process called energy metabolism. [2] Metabolism is the general term for the breakdown of complex compounds into building blocks and energy units or the reverse process: building complex compounds from building blocks and energy units. [3] Energy metabolism involves many steps during which the original compound is gradually broken down into intermediate compounds, producing high-energy compounds like ATP in the process. The intermediate compounds along with the products are called metabolites. The original compound in this article is glucose and the whole reaction chain is called glycolysis. 2.2 Metabolism The subject matter of this thesis is the metabolic reaction of glycolysis, which will be defined below. For this reason, that will be the focus of the following introduction on the metabolism, however, we will begin by defining the metabolism in broader terms. The building blocks in the metabolism are called metabolites: They are small molecules with a variety of functions, such as acting as signals, being used to fuel reactions, to stimulate or inhibit enzymes or catalyzing other reactions. They are used in nearly all common biochemical processes. Most structures in the cell are created by elements from three broad classes: Amino acids, Lipids and Carbohydrates.[4] 6

8 Amino acids are made up by a amine group, a carboxylic acid group and finally a side-chain which is specific to each different amino acid. Their main use is to be connected into large chains, proteins, of up to thousands of individual amino acids. The proteins in turn have a large variety of uses. Lipids, often called fats, are a broad group of molecules which are either hydrophobic (meaning they repel water molecules) or amphiphilic (meaning they attract both lipids and water molecules). Their main uses are to act as a storage for energy, building up cell membrane or acting as signals. Finally carbohydrates are molecules made from carbon, hydrogen, and oxygen: Their namesake is the fact that the ratio of hydrogen to oxygen is usually two to one (H2O). They are also known as saccharides and are often categorized depending on their length: Monosaccharides, consisting of a single sugar; disaccharides, which are made up by two monosaccharides; oligosaccharides and polysaccharides, which both are made up from a large chain of saccharides where polysaccharides are the larger of the two. [4] The other important types of metabolites are Nucleotides, Coenzymes, minerals and cofactors. Nucleotides are not essential in the scope of the thesis, but they are what makes up the DNA and RNA. Minerals and cofactors are inorganic elements. Coenzymes are molecules which help to transfer chemical groups between different reactions. An enzyme adds the group to the coenzyme, and it is then moved to another enzyme where the group is removed from the coenzyme to produce the substrate. Coenzymes are thus cycled through repeatedly, to move chemical groups within the cell. One of the more prominent coenzymes is adenosine triphosphate (ATP), which is the carrier of energy in the cell. When ATP breaks one of its bonds with a phosphate group and reverts into adenosine biphosphate (ADP), energy is released. Other common coenzymes are vitamins, with some minor modifications. For example Nicotinamide adenine dinucleotide (NADH) comes from vitamin B. NADH and NAD (its reduced form) can be used to move electrons within the cell: NAD working to oxidize and NADH to reduce.[5] The cell metabolism is the set of reactions providing energy and necessary chemical compounds to the cell, and their subsequent usage. These reactions can be divided into two general parts, catabolic and anabolic reactions. The former contains the reactions that break down complex compounds into simple molecules, and the latter the inverse, that is the synthesis of macromolecules from more simple molecules [6]. 7

9 A cell gets its energy from oxidative reactions, or in general terms, reactions where electrons are removed. Two examples are the generation of energy from glucose, or similarly from fatty acids, by letting the carbon and hydrogen react with oxygen. The former reaction, the conversion of glucose to energy is called glycolysis. The glycolysis is the first stage in the cell s energy synthesis by breakdown of glucose: Converting one glucose molecule to two molecules of pyruvate. This process occurs in all cells[6], and is the only metabolic reaction able to generate energy without oxygen. Hence, it is vital for all cells under anaerobic (no oxygen) circumstances, or in cases with cells that lack mitochondria such as the cells in the lens of the eye [6]. Glycolysis takes place in the cytosol for eykaryotes such as S. cerevisiae and uses two molecules of ATP to produce four molecules of ATP. Hence, the reaction results in a net gain of 2 ATP molecules, which can later be broken down, generating free energy [2]. In particular, 36 kcal energy is released for one consumed mole (about.18 kg) of glucose [7]. In condensed form, the reaction can be described as: Glucose(C 6 H 12 O 6 ) 2ATP 2Pyruvate(C 3 H 4 O 3 ) 4ATP. (1) In more detail, Cohen identifies 4 specific phases of Glycolysis [7]: 1. The conversion of one molecule of glucose into two molecules of triose phosphate, consuming two ATP molecules in the process. 2. Dehydrogenation of the triose phosphates into 2-phosphoglycerate, reducing NAD to NADH and generating two molecules of ATP, one for each phosphate molecule. 3. Converting the 2-phosphoglycerate to pyruvate, generating one ATP in each conversion. 4. Reducing pyruvate to regenerate NAD. In a yeast such as S. cerevisiae, pyruvate is reductively decarboxylated to ethanol. The glycolytic pathway is outlined in Figure 1. A detailed description of each step in the process, apart from the figure, is deemed excessively tedious for the scope of this thesis. Hence, only the main important points will be mentioned. Erlansson and Albertsson find the conversion of Glucose into Glucose-6-Phosphate (Glu6P) vital for two reasons. Firstly Glu6P cannot be transported across the cell membrane, hence the molecule is contained for subsequent steps in the reaction. Secondly, while Glucose is a stable 8

10 compond, Glu6P is more metabolically active, i.e. more prone to reacting with other compounds [6]. The usage of ATP in this reaction, and the conversion of Fructose-6-Phosphate (Fru6P) into Fructose-1,6-Biphosphate (Fru16P) is necessary to overcome the activation energy of the reaction. Hence, there is a net loss of energy in the cell for this first step. The conversion of Fru6P into Fru16P is close to irreversible, hence regulating the rate of glycolysis [6]. The extraction of energy is performed by two reactions, the breakdown of 1,3-biphosphateglycerate and the breakdown of phosphoenolpyruvate into pyruvate. The latter is the last step of the glycolysis. Since two molecules of triose phosphate are created for each glucose, and these in turn generate two ATP each, there is a net gain of two ATP from the glycolytic reactions. Some of the enzymes are inhibited or stimulated by the existence of other compounds. Notably for the rate limiting steps of glycolysis, Hexokinase, the enzyme allowing conversion of Glucose into Glu6P is normally inhibited by the existence of its product (Glu6P). Phosphofructokinase, the enzyme converting Fru6P into Fru16bP is stimulated by AMP and inhibited by ATP and citric acid, and pyruvatekinase, the enzyme allowing conversion of Phosphoenolpyruvate into pyruvate, is inhibited by ATP. Further details of the glycolysis do however differ slightly between species. In S. cerevisiae, for example, the conversion of glucose is not inhibited by the prevalence of the product of the reaction, Glucose-6-Phosphate. In many other other organisms, however, it is inhibited[8]. The specific reaction for the glycolysis used in this thesis, as found by Teusink et al. [9], and extended by Pritchard and Kell [1], is outlined in Figure 1 below. 2.3 S. cerevisiae The yeast Saccharomyces cerevisiae is central to this thesis. S. cerevisiae is a yeast cell, that is a eukaryote and part of the fungi kingdom. The cell is, among others, used as the yeast in baking and brewing. Notably, it has been extensively studied to understand biological pathways and reactions. For this reason, the equations and rates describing the metabolites in the cell are well documented e.g. by Rizzi et al[11] or by the Saccharomyces Genome Database. [12] S. cerevisiae is most commonly grown under aerobic conditions, on sugars such as glucose and maltose. Growth is also possible, but more limited under anaerobic conditions. [13] Furthermore, growth requires a nitrogen source 9

11 such as ammonia or most amino acids, and a source of phosphorus. The most basic description of the glycolysis process is that the yeast cell consumes the glucose, and produces ethanol. [14] The cell absorbs Glucose from the external environment and produces Carbon-Dioxide and Ethanol, as well as acetic acid and glycerol. To simulate In Vivo (meaning the same conditions as a natural environment) conditions we approximate the diffusion of the extracellular metabolites, constantly feeding the system glucose and removing carbon-dioxide and ethanol. [11] At the short timespan of our study we can safely ignore changes in enzyme concentration and focus on the change of metabolites. [11] 1

12 Figure 1: The glycolysis cycle in Saccharomyces cerevisae. 11

13 3 Stability analysis 3.1 Introduction Stability analysis of systems of ordinary differential equations (ODEs) can be difficult. A nonlinear system can not be solved exactly, which means that it is almost impossible to find a steady state solution ( dx = ) analytically. What one can do is to run the system with different initial conditions and find a stationary solution this way. If a steady state solution is found, the next challenge is to investigate its stability. In this section the theory of stability analysis will be explained. The first step in stability analysis on a nonlinear system is to linearize the system around a specific point, generally a stationary point. The linearly approximated system makes it possible to apply the same theory as that of stability analysis of a linear system. This section will begin by explaining the theory of stability analysis for a linear system and will be followed by the theory of linearization of a nonlinear system. Finally the theory of stability analysis around a steady state solution of a linearized system will be explained. 3.2 Stability analysis of a linear system A system of ODEs can be written on the form dx = f(t, x), (2) where x is a vector with the dependent variables and f(t, x) is a vector of functions, depending on x. x = (x 1, x 2..., x n ) T, f(t, x) = (f 1 (t, x), f 2 (t, x)..., f n (t, x)) T For a homogeneous linear system, the vector of functions can be written as f(x) = Ax, where A is a matrix. The system of ODEs then becomes dx = Ax x() = x, (3) where x is the value of x at t =. This system is by definition stable if the solution is bounded on the interval [, ) [15] 12

14 The solution to equation (3), can be written as [16]: x(t) = e At x (4) where e At is the exponential matrix defined as, [17] e ta = 1 ta t2 2! A2 t3 3! A3... = t n A n. (5) n! To understand the stability concepts for a linear system, the general case is first considered. If the A-matrix is diagonalizable, the general solution can be written as, [17] x(t) = c 1 e λ 1t s 1... c n e λ nt s n = n= n c k e λkt s k, (6) where λ k are the eigenvalues of A, s k the corresponding eigenvectors and c k, constants depending on the initial condition x. The eigenvalues can be written as λ k = µ k iω k and e λ kt = e µ ktiω k t. The size of this term e µ ktiω k t = e µ kt, so the size of the solution will depend mainly on the real part µ k, of the eigenvalue λ k. k=1 If one of the eigenvalues λ p >, then the term of the solution c p e λ p s p will be indefinite as t reaches infinity and this will lead to an unstable system. If one of the eigenvalues λ p =, then the term of the solution c p e λ p s p will be finite as t reaches infinity but its limit as t will. If the eigenvalues λ p <, then the term of the solution c p e λp s p will be finite as t reaches infinity the limit as t will =.[17] There are different kinds of stabilities to consider. [15] Definition 1. Consider a homogeneous system dx = ax, x() = x. This system is stable if the solution is bounded on the interval [, ] for all initial values x. It is also asymptotically stable if x(t) when t for all x. Theorem 1. For a homogenous system dx it holds that = Ax, where A is diagonalizable max 1 k n Re λ k < : The system is asymptotically stable. 13

15 max 1 k n Re λ k = : The system is stable. max 1 k n Re λ k > : The system is unstable. Now the linear system with a non diagonalizable A-matrix will be considered. A non diagonalizable matrix is a matrix where one or more eigenvalues are multiples with corresponding linearly dependent eigenvectors. Consider a system such as Equation 3, with an A-matrix with repeated eigenvalues, λ 1 = λ 2 =... = λ n. When considering the solution to the system on the form as in Equation 6 there will be linear combinations of the linearly dependent terms in the solution of the form (t t 2..., t n 1 )e λ 1.[17] For large t the contributions of these t-factors will be small in comparison to the exponential term, so the same theory as for the diagonalizable matrices can mostly be applied. To summarize this chapter, the stability of the linear system can be determined by calculating the eigenvalues λ i of the A-matrix. There are four different outcomes: µ i = Re(λ i ) < : The system is asymptotically stable. µ i = and µ i is simple: The system is asymptotically stable. µ i = and µ i is not simple: System might not be stable, further analysis is needed. µ i > for one or multiple λ i : System is unstable.[18] 3.3 Linearization To linearize the nonlinear system dx = f(t, x), is to take the first term of the taylor expansion around a fixed point x.[18] The variables are set to x = x x where x is a small perturbation around the fixed point x. d(x x) = f(x x). (7) 14

16 Taylor expansion of the right hand side of the equations gives dx d x = f(x ) f x (x ) x O( x 2 ), (8) and since dx d x = f(x ), the resulting system of equations is f x (x ) x. (9) The jacobian matrix is defined as f x (x ) = J(x ) = f 1 x 1 f 2 x 1. f n x 1 f 1 x 2 f 2 x 2. f n x 2 f 1 x n f 2 x n.... f n x n. (1) The nonlinear system can now be approximated by the linear system d x = J(x ) x, (11) in a small area around the point x. The theory of stability analysis of a linear system can be applied on the linearized system in a limited area around the point x.[18] 3.4 Stability analysis of a nonlinear system When considering stability of a nonlinear system of ODEs, only local stability in a limited area can be concerned. The most common is to look at the stability near a stationary point. A stationary point x s is a point which satisfies the algebraic equation f(x s ) = (12) The definition of stability for a nonlinear system around the stationary point x s is as follows:[16] Definition 2. If the nonlinear system dx = f(t, x) has a stationary point at x = x s, that is f(x s ) =, then that stationary point is said to be Lyapunov stable if: ϵ >, δ(ϵ) >, such that x() x s < δ(ϵ) x(t) x s < ϵ, t. 15

17 If x(t) x s as t for all x() x s sufficiently small, then the stationary point is asymptotically stable. In other words, a stationary point is stable if the system has the initial condition in an area close to the stationary point, the solution will stay close to this stationary point as t. If the solution approaches the steady state x s as t then the stationary point is asymptotically stable. The nonlinear system can be written as: [19] dx = J(x s)x g(x) (13) where g(x) are the terms of order higher then one. The first order approximation of the system is, dx = J(x s )x, which is also called the linearized system around x s. The stability principles can only be applied locally for a first order approximation of a nonlinear system [19]. It means that one only can analyze the system at a specific point and a limited area around it. Most common is that the analyzed point is the stationary point. For the eigenvalues of the jacobian, at the stationary point, λ i, the following can be said: µ i = Re(λ i ) < : The stationary point is asymptotically stable. µ i = for one or multiple λ i : The higher order terms of Equation 13 has to be consider to determine stability for the stationary point. µ i > for one or multiple λ i : The stationary point is unstable. [19] 3.5 Perturbation analysis The stability of a stationary point of a nonlinear system is sometimes impossible to determine only by calculating the eigenvalues. For example, if some of the eigenvalues are zero then the system can both be stable and unstable. To investigate the stability, further analysis of the stationary point is necessary. One method to investigate if the stationary point is stable is by doing perturbation analysis around the point[18]. The principle of this method is to 16

18 disturb the system by sending in small perturbations when the system is in the steady state. By investigating the behavior of the concentrations after the disruption, the stability of the system can be determined. If the disruption is small then the solution should stay in an area close to the steady state for the stationary point to be stable (see Definition 2). The system will be asymptotically stable if the the system returns to the stationary point after the disruption. 17

19 4 Systems biology 4.1 Intro Systems biology is a relatively new area of study where one instead of looking at the individual molecules, studies whole systems of biological components and how they interact. In systems biology one integrates mathematical modeling into the biology to make it possible to investigate larger networks of reaction cycles in the cell. One example of a biological system within a cell is the reaction between the metabolites of the metabolic network. A metabolic network consists of reactions where metabolites transform into other types of metabolites. [3] The concentrations of the different metabolites and their rates of change build up the network. The time dependence of the concentrations can be modeled with differential equations. The study of stationary states is important when modeling a dynamic system. It is a state of the system where the value of the components do not change in time; they have found an equilibrium. If a steady state is found, it becomes easier to do mathematical investigations and manipulations on the system by linearizing the system at this point. One can for example investigate if the modeled network is reasonable. 4.2 Modeling of chemical pathways The following is an adaptation from the book by Klipp et al.[3] This section discusses some of the basic principles behind modeling a chemical pathway. It gives motivation for the appearance of the mathematical expressions encountered in later sections. A subset of reactions from the first part of glycolysis in yeast is used as a first example: 18

20 Figure 2: A schematic representation of the first part of glycolysis in yeast. Abbreviations: AT P : adenosine triphosphate; ADP : adenosine diphosphate; GLCi : internal glucose; Glu6P : glucose-6-phosphate; F ru6p : fructose-6-phosfate; F ru16p : fructose-1,6-biphosphate. Let GLCi, Glu6P, F ru6p and F ru16p denote the concentration of glucose, glucose-6-phosphate, fructose-6-phosfate and fructose-1,6-biphosfate respectively (unit: moles per liter). Also let V 1,..., V 5 denote the rates of the reactions, that is how fast each reaction proceeds (unit: moles per liter and second). With this notation it is easy to obtain expressions for the time derivatives of the concentrations. An arrow pointing at a reactant in figure 1 above gives a positive contribution the the concentration of the reactant. An arrow pointing away from a reactant gives a negative contribution to the concentration of the reactant. This implies, dglu6p df ru6p df ru16p = V 1 V 2 V 3 (14) = V 3 V 4 (15) = V 4 V 5. (16) The rates V 1,..., V 5 themselves depend on the concentrations Glu6P, F ru6p and F ru16p. A reaction is more likely to happen when the concentration of the reactants is high. The following is generally assumed: The concentrations are spatially homogenous and the reaction rate is proportional to the concentration of the reactants to the power of the molecularity. In a reaction like A B 2C, (17) the molecularity of A and B is one while it is two for C. Let V = V V be the rate of this reaction where V is the rate in direction from left to right 19

21 and V is the rate in direction from right to left. With the assumptions made an expression for V is given by V = V V = k A B k C 2, (18) where k and k are the so called rate constants. For the reaction (17) it holds that da = db = V (19) dc = 2V (2) Using the expression for V in the last two equations gives, da = db = k C 2 k A B (21) dc = 2k A B 2k C 2 (22) The first basic steps in modeling a simple chemical reaction chain have now been illustrated. The equations (14), (15) and (16) could thus be expressed dglu6p df ru6p df ru16p = k 1 GLCi AT P k 2 Glu6P AT P k 3 Glu6P k 4 F ru6p (23) = k 3 Glu6P k 4 F ru6p k 5 F ru6p AT P (24) = k 5 F ru6p AT P k 6 F ru16p, (25) where k 1,..., k 5 are the rate constants. Making assumptions on the enzyme kinetics can simplify the simulation of a system. Enzyme kinetics describes the relation between enzyme and substrate concentrations and the reaction rate. One of the more common models of enzyme kinetics is M ichaelis-m enten kinetics described below. Consider the reaction E S ES E P (26) Here E is the enzyme, S the substrate and ES the enzyme-substrate complex. P is the product. The ODE system for this reaction is obtained in the way described previously. ds = k 1E S k 2 ES (27) 2

22 des = k 1 E S (k 2 k 3 )ES (28) de = k 1E S (k 2 k 3 )ES (29) dp = k 3ES (3) The assumption is that a steady state is reached during the course of reaction, where the concentration of ES is constant. des = (31) This is valid if S(t = ) E. The rate V in this reaction is given by V = dp = ds (32) Using the ODE system an expression for V can be derived. If (28) is added to (29) the result is des de = (33) This reflects the fact that the enzyme is not consumed during the reaction. Let E total = E ES. This together with (28) and (31) yield ES = E totals S k 2k 3 k 1. (34) The fact that V = dp together with (31) gives the expression for V : V = k 3E total S S k 2k 3 k 1 = V maxs S K m (35) where V max is the rate attained when the enzyme is saturated with substrate (let S grow in the expression). For S = K m, V = V max 2. K m is called the Michaelis constant and the expression for V is the expression for Michaelis- Menten kinetics. 21

23 Enzymes are indeed involved in the glycolysis example described first in this section. For example V 3 is facilitated by the enzyme phosphoglucoisomerase. Applying the steps above on the glycolysis example yields [3] V 1 = V max,1 AT P GLCi 1 AT P K AT P,1 GLCi AT P GLCi K GLCi,1 K AT P,1 K GLCi,1 (36) V 2 = k 2 AT P Glu6P (37) V 3 = V f max,3 K Glu6P,3 Glu6P V max,3 r K F ru6p,3 F ru6p 1 Glu6P K Glu6P,3 F ru6p (38) K F ru6p,3 V 4 = V max,4 (F ru6p ) 2 K F ru6p,4 (1 κ( AT P AMP )2 ) F ru6p 2 (39) V 5 = k 5 F ru16p (4) This concludes the demonstration of the derivation of rate expressions and the rudimentary discussion on modeling. 4.3 Analysis of metabolic systems Due to its size, the metabolic system of the subject matter of this thesis is relatively unfeasible to thoroughly explain in every step in text. For this reason, the analysis of metabolic systems will be focused on a smaller system instead. The same methods will be utilized as for the larger system later. This text will continue to focus on the system given by Equations (14) - (16), with the rate equations given by (36)-(4), i.e. the first few steps of the glycolysis. Initially, the dynamics of the system will be investigated by finding the steady state of the system. We solve the steady state numerically using MATLAB. The resulting concentrations of metabolites in the steady state, and a graph over their respective concentrations over time can be found below. It is often useful to analyze the steady state of the system. Both since it simplifies the analysis and as it allows linearization to portray the behaviour of the real system rather accurately in the vicinity of this point. We will linearize the aforementioned system by first finding its Jacobian matrix. With f 1, f 2, f 3 equaling the sum of the rate equations as in 14-16, the first row of the Jacobian, that is the partial derivatives of the first rate eqution 22

24 Table 1: Concentration of metabolites in the steady state, found by using ode15s in MATLAB. Concentrations are reported in mmol / l and to 4 d.p. Metabolite Steady state conc. Glu6P Fru6P.9365 Fru16P Gluc6P Fruc6P Fruc16P2 Concentration (mmol / l) Time (min) Figure 3: Concentrations of metabolites over time describing the rate of change of Glucose-6-Phosphate, will be given by: f 1 Glu6P = k 2 AT P f 1 F ru6p = f 1 F ru16p V r max,3 F ru6p K F ru6p,3 V f max,3 Glu6P K Glu6P,3 ( K F ru6p Glu6P,3 K F ru6p,3 Glu6P K Glu6P,3 1 V f max,3 ( ) (41) K F ru6p Glu6P,3 K F ru6p,3 Glu6P K Glu6P,3 1 V r max,3 K F ru6p,3 ( F ru6p K F ru6p,3 V r max,3 F ru6p K F ru6p,3 ) Glu6P K Glu6P,3 1 V f max,3 Glu6P K Glu6P,3 ( K F ru6p F ru6p,3 K F ru6p,3 Glu6P K Glu6P,3 1 ) 2 ) 2 (42) = (43) For the second row in the Jacobian, the partial derivatives of the rate of 23

25 change of Fructose-6-Phosphate, we have: f 2 Glu6P = V r max,3 F ru6p K F ru6p,3 K Glu6P,3 ( F ru6p K F ru6p,3 f 2 F ru6p = f 2 F ru16p V f max,3 Glu6P K Glu6P,3 V f max,3 Glu6P K Glu6P,3 1 ( ) (44) K F ru6p Glu6P,3 K F ru6p,3 Glu6P K Glu6P,3 1 ) 2 2 V max,4 F ru6p ) F ru6p 2 K F ru6p,4 ( κ AT P 2 AMP V max,4 F ru6p 3 ( ( ) ) K κ AT P 2 2 F ru6p,4 1 F ru6p AMP 2 2 V r max,3 F ru6p K F ru6p,3 V f max,3 Glu6P K Glu6P,3 ( K F ru6p F ru6p,3 K F ru6p,3 Glu6P K Glu6P,3 1 V r max,3 ( ) (45) K F ru6p F ru6p,3 K F ru6p,3 Glu6P K Glu6P,3 1 =. (46) Lastly, the third row concerning the rate of change of Fructose-1,6-biphosphate, is given by: f 3 = (47) Glu6P f 3 F ru6p = 2 V max,4 F ru6p ( ) K κ AT P 2 F ru6p,4 1 F ru6p AMP V max,4 F ru6p 3 ( ( ) ) K κ AT P 2 2 (48) F ru6p,4 1 F ru6p AMP 2 2 f 3 F ru16p = k 5. (49) ) 2 To analyze the steady state, we then find the eigenvalues of the Jacobian matrix. The results are reported below. As all eigenvalues are negative and have no imaginary parts, the system is stable in equilibrium. Having shown a stable system, however, it could be interesting to show a system which does not contain purely negativ eigenvalues as well. To achieve a system that oscillates in the steady state, we 24

26 Table 2: Eigenvalues of the steady state Jacobian matrix Eigenvalue modify the system slightly to force imaginary eigenvalues. For example, assuming that the production of Fru6P is inhibited by the availability of Fru1,6bP, the product from the catabolic reaction of Fru6P by the enzyme PFK. The biological interpretation being that the cell limits the production of Fru6P if there is plenty of Fru16P available, since there could be sufficient for use in subsequent steps in the glycolysis. The new differential equations for the concentrations of the metabolites are given by: where dglu6p df ru6p df ru16p =V 1 V 2 V 3 V 6 (5) =V 3 V 4 (51) =V 4 V 5 (52) V 6 = k 6 F ru16p (53) The Jacobian of this new system changes only in f 1 F ru16p, which is given by f 1 F ru16p = k 6. (54) Let us assume k 6 = 1. The coefficient is likely far too large to be biologically feasible but will make the oscillations visible when plotted. The eigenvalues of this new system will be: Since there are imaginary parts to the equations, the steady state of this system will oscillate. The time course plot of this system is displayed below. 25

27 Table 3: Eigenvalues of the steady state Jacobian matrix of the oscillatory system Eigenvalue i i 3 Gluc6P Fruc6P Fruc16P2 2.5 Concentration (mmol / l) Time (min) Figure 4: Concentrations of metabolites over time in the oscillatory system As seen in the figure, the system oscillates quite heavily for about the first minute. Seeing as the eigenvalues are negative, however, these oscillations decrease over time, and the system will converge to a certain steady state as time goes towards infinity, albeit with very minor oscillations. We can further analyze the stability the system by perturbing the steady state. The perturberation will be modeled by using the steady state values from earlier as initial values for a new time course simulation, but increasing the vlaue for Fructose-6-Phosphate by one percent. The resulting figure on the difference between the steady state and the perturbed system is shown 26

28 below. 6 x 1 3 Gluc6P Fruc6P Fruc16P2 Steady state 4 Perturbation ( mmol / l ) Time ( min ) Figure 5: Time course plot of the system where Fru6P is perturbed by 1% For small perturburations, the linearized system will closely approximate the non-linarized system. For this reason, these figures are omitted. From the above figure, it is evident that the system converges to the original steady state when perturbed. As concluded by the eigenvalue analysis, this system is thus stable. 27

29 5 Method 5.1 Modeling of glycolysis in S. cerevisiae This thesis adopts glycolysis models from two different scientific articles. One by Theobald et al.[14] and the other by Teusink et al.[9] with parameters from Pritchard et al. Both articles provide complete sets of equations, including rate equations, making them useful for the purposes of this thesis. Initially the model of Theobald et al. was the only one studied. As the rate equations were put into MATLAB some of them were found ambiguous and not all constants were given in the article. This necessitated some guesswork and approximations. This however caused other problems, as the model never reached the steady states reported in the article. Either some concentrations reached negative values or MATLAB stopped the integration because some expression became imaginary or the system becoming singular. For the model to reach feasible steady states som extra conditions needed to be imposed. These problems lead to a search for an alternative model. Such was found in the article of Teusink et al. The results of the model of Theobald et al. are covered briefly in Appendix 1.1. A good fit of the model to experimental data under certain conditions alone was not satisfactory for Teusink et al.; they also strove for mechanistically correct rate equations. Teusink et al. regarded their set of kinetic parameters complete and consistent, having experimentally determined them all under the same conditions.[9] Pritchard et al. uses the model of Teusink et al. with minor modifications in some rate equations and with some of the kinetic parameters different. Pritchard et al. numerically estimated V max values that made the model reach steady state values closer to the in vivo steady state values found by Teusink et al.[1] These improved V max values were used in this thesis together with the model of Teusink et al. The kinetic parameters favoured by Pritchard et al. were used too. The model created by Teusink et al. describes anaerobic conditions: This means that oxygen flow is irrelevant to the model. The yeast is also considered to be nongrowing, meaning the cell in itself does not expand and as such there is no need to model the growth of other components in the metabolism. Further we make use of the improved model developed by Pritchard et al. which describes a glucose depressed enviroment. This gives us a simulation closer to In Vivo conditions. 28

30 The external glucose is set to a constant, and the inflow of it into the cell is modeled by v HXT (the hexose transport). Similarly the other endpoints of the branches of the glycolysis are considered to be constant. Trehalose, glycerol and glycogen are set to experimentally determined values. Similarly the C CO2 and C Etoh were both set to constant values.[1] A more detailed description of the rate equations are given in appendix 1.2. The model is given by the following diffential equations:[1, 9] dc GLCi dc Glu6P dc F ru6p dc F ru16p dc DHAP dc GAP dc Gri23P dc Gri3P dc Gri2P dc P EP dc P yr dc AcAld dc AMP dc ADP dc AT P dc NAD dc NADH =v HXT v HK =v HK v P GI v glycogen 2 v trehalose =v P GI v P F K =v P F K v ALD =v ALD v T P I v Gro3P DH =v ALD v T P I v GAP DH =v GAP DH v P GK =v P GK v P GM =v P GM v ENO =v ENO v P Y K =v P Y K v P DC =v P DC v ADH v succinate =v AK =v HK v AT P ase v glycogen v trehalose v P F K v P GK v P Y K 2 v AK =v P GK v P Y K v AK v HK v glycogen v trehalose v P F K v AT P ase =v Gro3P DH v ADH 3 v succinate v GAP DH =v GAP DH 3 v succinate v Gro3P DH v ADH 29

31 5.2 Implementation When describing the glycolysis as a system it becomes possible to perform a numerical analysis of it: The system was modeled in MATLAB and examined. It was written as a large multi-variable function, which was solved through the use of a ode solver. The solver used for the system was ode15s which is used for stiff ordinary differential equations. By varying the starting values one can examine whether the system reaches different steady states. Physiologically feasible values are chosen as initial values. When a stationary point has been found, small perturbations from the stationary point are plotted versus time. In these plots the stationary concentrations are represented by y =. If the perturbation graphs do not tend to zero as the initial perturbations tend to zero, the system is not stable in the Lyapunov sense. After finding a stationary point of the system a linear approximation of the system can be created by calculating the Jacobian matrix around the stationary point. The eigenvalues of the Jacobian are calculated to examine whether the stationary point is stable or not according to chapter 3. To further analyze the stability of the stationary point, perturbation analysis is performed on the system around the stationary point. There are two ways the perturbation analysis is done. The first way is by running the system around the stationary point, with the initial values slightly different from the stationary point. The second disturbance is done by running the system at the stationary point and after a certain time, disturb the system by a pulse of one of the substance in the system. The first step in the process is to model the system in MATLAB. In the model, 25 different metabolites are used, and each is given a differential equation. The MATLAB function returns a vector with the 25 values of their time derivatives, by the use of the equations. Through the use of the ode15s solver we can run a simulation of the system, for a given timespan and initial conditions. By making use of the estimates in [1] we have a good guess of starting values for the system. After finding a time span which reaches a steady state we added an additional marginal to ensure that it is in fact a steady state. Finally the stationary point is retrieved from the values at the endpoint of the simulation. Through the use of symbolic computation, where the concentrations are 3

32 typed as variables, MATLAB is used to determine the Jacobian of the system. After having calculated the Jacobian symbolically we evaluate it, using the stationary point we previously retrieved. This gives us a matrix with numerals, from which we can easily extract the eigenvalues with MATLAB functions. The stationary point is analyzed further through studing the effects of perturbations. This is done by allowing the initial condition to differ slightly from the stationary point when running the system. The system is also disrupted by a incoming pulse of one of the concentrations. This is done with a gaussian pulse to meet the requirements of the MATLAB solver ode15s. Finally we study how the individual concentrations act when affected by perturbations of different sizes. After pertubing the subject concentration with a set value the system is solved with the solver, the original steady state is subtracted from the new steady state. This procedure is repeated in 1 steps, with perturbations varying from.1 to.1 mmol/l. 31

33 6 Results and Conclusions 6.1 Results This section presents the results of the simulation of the system in MATLAB and the results of the stability analysis of the steady state. The steady state resulting from our chosen initial conditions has 16 eigenvalues with negative real parts, and one that is slighly positive. The positive value is of magnitude Furthermore, it has 4 eigenvalues with imaginary parts. The imaginary eigenvalues causes decreasing oscillations when reaching the steady state, and it appears to be stable over reasonable time periods. The positive eigenvalue is a point of concern, as it implies that the system is not stable on very large timescales, it is however within the error tolerence of our solver. This coupled with the other eigenvalue close to zero means we need to examine the system more closely. While the system appears stable, it does not return to the same steady state for all changes. For some concentration, the value of one or more of the metabolites is changed by a small amount between the stationary points. Based on the results of the stability of the steady state stability, it is concluded that the system responds as it should to properly model the glycolysis, and could be considered biologically feasible. Hence, it is also suitable to use as a basis for further or extended simulation of S. cerevisiae. A biologicaly feasible starting point is chosen as the initial condition Steady state In Figure 6 the simulation is plotted. In the simulation the initial values of the concentrations as described in Table 4 are used. After about one minute in simulated time the system has reached a steady state. The value of the concentrations at the steady state can also be found in Table 4. 32

34 Concentrations ( mmol / l ) Concentrations versus time GLCi Glu6P Fru6P Fru16P DHAP GAP Gri23P Gri3P Gri2P PEP PYR AcAld AMP ADP ATP NAD NADH Time (min) Figure 6: The simulation of the system with initial condition as described in Table 4. 33

35 Below is a table containing the initial values and the steady state values for each metabolite. Table 4: Initial values with steady state values Metabolite Initial value Steady state value GLCi Glu6P Fru6P Fru16P DHAP GAP Gri23P Gri3P Gri2P PEP PYR AcAld AMP ADP ATP NAD NADH

36 6.1.2 Stability analysis In this section the results from the stability analysis for the simulation will be presented. The results consist of tables with the eigenvalues of the jacobians and simulations of the behavior around the steady states. The eigenvalues of the jacobian evaluated in the stationary point can be found in Table 5. The eigenvectors corresponding to the two eigenvalues which are close to zero follows in Table 6 and 7. The steady state is tested with small perturbations δx around the stationary point. This is achieved by allowing the system to run from the steady state with a perturbation of δx from the stationary point. In these simulations will represent the unperturbed stationary point for all concentrations. Figure 7, Figure 8 and Figure 9 are simulating the behavior of the different concentrations when the initial perturbation of δx =.1 for all concentrations. The behavior of the concentration of ATP with the initial perturbation δx =.1,.1,.1 and.5 for all concentrations is simulated in Figure 1. In this simulation, the behavior of the linearized solution around the stationary point is also simulated with the same initial values as for the original system. The behavior of Fru16P and DHAP around the stationary point when disturbed by a gaussian pulse of Fru16P at t =.1 min is simulated in Figure 11. The behavior of the concentration of GLCi after different initial perturbation δglci =.1,.5,.1,.1,.5 is simulated. The rest of the concentrations have the unperturbed stationary point as the initial value. Simulations of the behavior of ATP and ADP after different sizes of perturbation of ATP at t = are illustrated in Figure 13. The initial perturbations of ATP, δat P =.1,.5,.1,.1,.5 and (.1mmol/l). The rest of the concentrations begin at their values in the unperturbed stationary point. Finally in Figure 14 to Figure 22 we simulate the system for perturbations of the subject metabolite varying between.1 and.1, and plot the difference between the steady state the pertubed system reaches and the unpertubed steady state. 35

37 Table 5: The eigenvalues associated to the simulation Eigenvalue i i i i Furthermore, according to MATLAB the rank of the Jacobian is

38 Table 6: The eigenvector corresponding to the 16th eigenvalue Eigenvector Table 7: The eigenvector corresponding to the 17th eigenvalue Eigenvector

39 x 1 3 x ATP AMP Steady state 6 Fru16P DHAP Steady state Perturbation ( mmol / l ) Time ( min ) Time ( min ) x 1 3 x PEP PYR Steady state 6 NAD NADH Steady state Perturbation ( mmol / l ) Time ( min ) Time ( min ) Figure 7: Simulation of the system around the stationary point with initial perturbation δx =.1. The plots are for ATP, AMP, Fru16p, DHAP, PEP, PYR, NAD and NADH 38

40 x 1 3 x Glu6P Fru6P Steady state 6 GLCe GLCi Steady state Perturbation ( mmol / l ) Time ( min ) Time ( min ) x 1 3 x GAP Gri23P Steady state 6 Gri3P Gri2P Steady state Perturbation ( mmol / l ) Time ( min ) Time ( min ) Figure 8: Simulation of the system around the stationary point, with initial perturbation δx =.1. The plots are for Glu6P, Fru6P, GLCl, CLCi, GAP, Gri23P, Gri3P and Gri2P. x AcAld ADP Steady state Perturbation ( mmol / l ) Time ( min ) Figure 9: Simulation of the system around the stationary point, with initial perturbation δx =.1. The plots are for AcAld and ADP. 39