Convex and toric geometry to analyze complex dynamics in chemical reaction systems

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1 Convex and toric geometry to analyze complex dynamics in chemical reaction systems DISSERTATION zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) genehmigt durch die Fakultät für Naturwissenschaften der Otto-von-Guericke-Universität, Magdeburg vorgelegt von Dipl.-Biophys. Anke Sensse geb. am 14. April 1976 in Kronberg im Taunus Gutachter: Prof. Dr. G. Ertl Prof. Dr. R. Friedrich PD Dr. M. Hauser Eingereicht am: 25. Januar 2005 Verteidigt am: 11. Juli 2005

2 Dedicated to the memory of KARIN GATERMANN

3 Table 1: List of mathematical symbols C R Z C[x] I def, tor I deformed V (I) V (I def, tor ) F h, j F P N D A S A N ν ij γ i, γ i κ Jac α i Jac H i v i k i x i0 h i0 E i field of complex numbers field of real numbers ring of integer numbers ring of polynomials with coefficients in the field of complex numbers ideal toric ideal variety of an ideal variety of a deformed toric variety family of polynomials depending on the parameters h, j family of polynomials with a certain sign pattern in the coefficients family of all matrices emerging from the matrix A if it is multiplied by a positive definite diagonal matrix family of all matrices having the same sign pattern as the matrix A stoichiometric matrix stoichiometric coefficient of the i t h-species in the j th -reaction coefficients of the i th -species in the reaction equations kinetic matrix Jacobian matrix coefficient of the characteristic polynomial of the Jacobian part of the convex Jacobian matrix i th -minor of the Hurwitz matrix rate of the i th -reaction rate constant of the i th -reaction steady state concentration of the i t h-species inverse steady state concentration of the i t h-species minimal generating vector of the cone of nonnegative stationary reaction rates Table 2: List of examples Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Example 7 Example 8 Extended Sel kov oscillator Electrooxidation of formic acid (galvanostatic, one current carrier) Electrooxidation of formic acid (galvanostatic, all current carriers) Electrooxidation of formic acid (potentiostatic, one current carrier) Methylene blue sulfide oscillator Ca 2+ -oscillations in the cilia of olfactory sensory neurons Sel kov oscillator (and several extensions) Peroxidase-oxidase reaction system (O2, P er 2+ and P er 3+ subsystems)

4 Contents 1 Introduction 2 2 Methods and theoretical background Basics of chemical reaction kinetics Algebraic structures in the space of reaction rates Set of reaction rates as deformed toric variety Set of stationary reaction rates as a convex cone Intersection of the deformed toric variety and the convex cone Graph theory for reaction networks Stability criteria and the role of feedback-loops Bifurcation analysis in a high-dimensional parameter space Routh-Hurwitz criterion Feedback-loops Electrocatalytic oxidation of formic acid Reaction networks and kinetics Curves of stationary solutions Geometric considerations Stability analysis Intersection function of the curve of stationary solutions and the plane of conservation relations Oscillations Extensions to the model Results, discussions and outlook Sources for instability Destabilizing feedback-loops Realistic examples for different types of instability Activator-inhibitor systems Feedback-loops for bistability and periodicity Minimal bistable oscillators Examples for bistable oscillators ii

5 6.1.3 Regions of bistability and oscillations in parameter space Feedback-loops for chaos Extended bistable oscillators Codimension-two bifurcations Shil nikov s theorem Homoclinic chaos Results, discussion and outlook Enzymatic oxidation of N ADH The peroxidase-oxidase reaction system The PO-reaction as an extended activator-inhibitor system Shil nikov chaos in the P O system Results, discussion and outlook Summary and Outlook Outlook Bibliography 123 1

6 Chapter 1 Introduction Chemical reaction kinetics is a challenging topic which attracts more and more scientific interest, especially, since nonlinear dynamical phenomena have been found in an increasingly large variety of electrochemical and biochemical systems. These systems very often show a high degree of complexity, not only because of the high number of components but also because of the great quantity of parameters to take into account. With regard to this it seems essential to strain an interdisciplinary investigation and to make a whole arsenal of tools and methods work together to solve each problem with its own particular difficulties. The aim of this work, presented as doctorial thesis, is not the addition of some further models to a long list of successful modelings of different reaction systems. This report is primarily concerned with the improvement and refinement of recently introduced methods and algorithms to deal with polynomial systems. It will be demonstrated how they can be combined with older tools from stoichiometric network analysis and a few spontaneous tricks to solve various models for complex reaction systems of intriguing interest. Those are above all the catalytic reaction systems, which naturally enable nonlinear reaction behavior on the one hand and on the other hand, they represent the large majority of systems of scientific and industrial importance. Representatives of these systems can be found in many reaction systems of our environment. In this work two main examples are presented, each of them stemming from another field of chemistry. In an electrocatalytic reaction the catalytic nature is due the electrode in combination with the electric field, in an biochemical reaction sequence the catalysis is usually exhibited by enzymes. The procedures and conceptions, which will be presented, stem from algebraic geometry, a theory hardly applied in the analysis of dynamical systems so far. One pioneering work in theoretical chemistry stems from Bruce Clarke [11], who derived his so-called stoichiometric network analysis by concepts from convex geometry and related the network s topology to the stability of the according kinetic system. The basic idea of this analysis is the transformation of the set of stationary solutions from the concentration space into the space of reaction rates, which represents a convex polyhedral cone. Making use of the special convexity properties of this space Clarke derived the Jacobian matrix and executed the stability analysis in this space. However in this theory, which provides very efficient methods for a fast recognition of unstable networks 2

7 in practice, one important detail was missing. The properties of the mapping from the set of stationary concentrations onto the set of stationary reaction rates and of its inverse have not been elucidated. There was no direct way to transfer the results obtained in the set of stationary reaction rates back to the set of stationary concentrations. This gap between the two spaces has been closed two decades later, when supplementary restrictions for the reaction rates have been found [30]. If the stationary reaction rates satisfy these restrictions in the space of reaction rates, there is a surjective mapping back to the set of stationary concentrations. In order to realize the existence of these restrictions and even for their calculation a huge theoretical background from toric geometry must be introduced. Using mass action rate law the reaction rates represent monomials in the algebraic sense, which offers the way for ideal theoretic considerations. The construction of different ideals, the change of an ideal basis and the choice of a basis with special elimination properties are the procedures to show that the image of the monomial mapping from the concentration space into the space of reaction rates must lie in the variety of a toric ideal and is therefore subjected to further restrictions. The theory of toric varieties has been deeply investigated by Karin Gatermann [28, 30, 32]. She was the first to see the potential of this theory in the application to chemical reaction systems and elucidated the functional relation between the set of stationary concentrations and the set of stationary reaction rates. A central point of this investigation is to use the restriction given by the algebraic structures to define new coordinate systems for the stationary solutions. Exploiting the special properties of the new coordinates one may efficiently reduce the dimension of the problem and thus find an analytic solution where traditional methods fail. The stability criteria can be adapted to the new coordinates which leads to a strong simplification of the the algebraic expressions determining the occurrence of various bifurcations. It is even possible to make justified propositions about the parameter region where they are to be expected, thereby accelerating the research by numerical simulations. Clarke used diagrammatic methods to illustrate his derivations. In recent times they can be replaced by graph theory, which is applied for chemical reaction networks by a large group of authors [44, 96, 42, 28]. Also in this investigation graph theory will be used as additional tool. As an example for the successful application of toric geometry several models for the electrocatalytic oxidation of formic acid will be solved analytically. This reaction system has been thoroughly investigated with experiments and numerical simulations before [88, 86, 45, 47, 58, 75]. However, only the analytic solution of the different systems may elucidate the origins of the arising instability, the influence of galvanostatic and potentiostatic conditions and the role of the current carrying processes on the dynamics. In these example calculations the theory is extended to non mass action rate law. A fundamental notion of this work is feedback-loop, which has been introduced by Clarke [11]. Having a graph theoretic and an equivalent matrix theoretic definition, this structure links the network stability problem to a matrix stability problem. In [11] several theorems on stability involving the existence of feedback-loops are given. Special feedback-loops are identified as sources of instability. 3

8 Markus Eiswirth showed in [18] by algebraic conclusions how feedback-loops interact to produce bistability and oscillations and classified a great part of the known chemical and electrochemical oscillators [87] according to their constellation of feedback-loops. The class of activator-inhibitor systems has been defined, which includes the wide class of autocatalytic oscillators. Its representatives are characterized by hysteretic self-oscillatory reaction behavior. The second part of this investigation deals exactly with this class of systems. Assuming a close relationship between the constellation of feedback-loops and particular system properties the question arises, if there are certain feedback-loop constellations inducing chaotic dynamics in an activator-inhibitor system. Numerical simulations of a collection of minimal activator-inhibitor systems extended by certain feedback-loops give a first impression of what is to be expected from the influence of additional feedback-loops on the dynamics of a bistable oscillatory system. The collection is designed in a way that one may find for more realistic systems corresponding model systems showing similar structures. In this case the theoretic results may be transferred. An example for the occurrence of the model systems in biochemical pathways is the oxidation of NADH by molecular oxygen O 2 catalyzed by the enzyme peroxidase. This represents the famous peroxidase-oxidase reaction, which has also known a long history of experimental investigations ([59], [4], [40], [8]). Several models have been conceived and simulated ([13], [66], [5], [99], [3], [48], [68]). One very successful model which accounts for the experimental findings is the BF SO-model [7]. In the present investigation this model is decomposed in several subsystems corresponding to some of the model systems considered in the previous theoretic analysis. This work is subdivided in two main parts, each consisting of general theory and a case study with detailed example calculations. The first part (chapters 2, 3, 4) deals with the application of algebraic geometry in the analysis of chemical reaction networks. The second part (chapters 5, 6, 7) investigates the relations between certain network structures, realized as feedback-loops, and basic system properties, like instability, oscillations, and chaotic dynamics. In the example parts the formic acid system (chapter 4), the peroxidaseoxidase reaction system (chapter 7) and a whole collection of certain model-systems based on the Sel kov model (chapter 6) are described. Small example systems to illustrate the algorithms in the theory parts are emphasized in the example environment in italics. There are eight of these examples, they are numbered throughout the whole investigation. The second chapter contains an introduction into chemical reaction kinetics as well as the basic concepts of convex and toric geometry necessary to understand the restrictions on the stationary reaction rates given by the convex cone and the toric variety. The properties of these structures and their intersection will be explained. In the third chapter the stability criteria will be adapted to the algebraic structures in the set of reaction rates. Feedback-loops will be defined and their influence on the occurrence of saddle-node and Hopf bifurcations will be elucidated. As a realistic system, a model for the electrooxidative formic acid system is presented in the fourth chapter as well as several modifications to the basic model. For three of 4

9 them the analytic solution will be derived revealing the physical validity of the models in view of their ability to exhibit bistbility. In the fifth chapter we will create minimal unstable networks by special feedback-loops and we will classify them in order to define different sources of instability. An example for each type will be presented. In the following (chapter six) these unstable networks will be extended to account for bistability and oscillations, thereby creating prototypes for the class of activator-inhibitor systems. The properties of these prototypes will be described in detail. A collection of extended activator-inhibitor systems will be checked by numerical simulations for the occurrence of chaotic oscillations in order to identify chaos generating feedback-loops and to define a class of chaotic activator-inhibitor systems. Special attention will be paid on the nature of the arising chaos, i.e. we will try to prove the emergence of Shil nikov chaos. The seventh chapter deals with the transfer of the results concerning chaotic activatorinhibitor systems in an enzymatic reaction sequence. The chaos generating species and feedback-loops of the peroxidase-oxidase reaction system as well as the nature of the arising chaos will be determined. The last chapter contains a concluding summary and an outlook for further investigations. The computation of the examples was supported by Maple software [31]. The numerical simulations and the bifurcation diagrams were calculated with the continuation program auto [14]. Lyapunov exponents were determined using the integrator in [41]. For the calculation of the algebraic structures special algorithms have been applied ([53, 77] for cone generators, [33] for Groebner bases). 5

10 Chapter 2 Methods and theoretical background The main tools for an analytic investigation of chemical reaction networks are developed in convex and toric geometry. This results from the fact that only nonnegative solutions for the concentrations are of interest and from the fact that most of those systems have polynomial structure due to mass action rate law or another power rate law. The set of nonnegative stationary solutions of a polynomial chemical reaction system is described by characteristic algebraic structures which can be considered in the space of reaction rates or in the concentration space. The two main structures defining the set of stationary solutions are a convex polyhedral cone [11] and a deformed toric variety [30]. They are considered in the space of reaction rates. Thereby, one can make use of concepts from stoichiometric network analysis such as the representation of the Jacobian matrix in convex coordinates and the decomposition of the network in extreme subnetworks. The set of nonnegative stationary solutions is confined to the intersection of the convex cone and the deformed toric variety. This set represents a curve in the space of reaction rates. The low-dimensionality of this structure simplifies significantly the stability analysis. A central point of this consideration is the description of the mapping from the curve of stationary reaction rates onto the set of stationary concentrations. It is the key to transfer the stability results obtained in the space of reaction rates to the concentration space. Another tool for the analysis of chemical reaction networks is graph theory, which can illustrate to a certain degree the algebraic derivations. It relates directly relevant network structures to structures in the kinetic equations and in the Jacobian matrix. Therefore several graphs are created and it is shown how the kinetic equations can be set up with the corresponding adjacency and incidence matrices. A graph describing stability is derived. It encodes important structures determining a network s stability, which will be of great importance throughout the whole investigation. 6

11 2.1 Basics of chemical reaction kinetics Usually a chemical reaction system of r reactions and m reacting species is represented by reaction equations. For each reaction, the reactants and the products are assembled on the left and the right side of an arrow. γ 1j S γ mj S m k j γ 1j S γ mj S m, j = 1,..., r. The arrows stand for the reactions. S i are the reacting species. The variable standing for a species S i is denoted by x i. Usually, it denotes the according concentration. Species on the left side of an arrow are called reactants and species on the right side are called products of a reaction. k j on the arrow stands for an experimental constant called rate constant or kinetic constant. These constants encode external factors influencing the velocity of the reaction j and the concentrations of external species which are constant throughout the reaction. The velocity of a reaction j is called reaction rate v j. It is usually modelled by the mass action rate law. For an elementary step the reaction rate is proportional to the probability that the reacting particles strike together. It follows that each of these reaction rates represents a monomial in the variables x multiplied by the rate constant k j. The molecularity of the species i in the reaction j is encoded in the exponent κ ij of the i th -species in the j th -reaction rate. These kinetic exponents are arranged in the kinetic matrix κ. The entries κ ij of this matrix are the derivatives of the logarithm of the j-th reaction rate with respect to the logarithm of the i-th species at steady state x 0, i.e. κ ij = log(v j(x 0, k j )) log(x 0i. This function has been named effective power function by Clarke ) [11]. Using the mass action rate law it corresponds to the kinetic exponent of the i-th species in the j-th reaction. In most of the cases the coefficients γ ij can be chosen such that they equal the kinetic exponents κ ij. The monomials of the reaction rates according to the mass action rate law are formed by v j (k j, x) = k m j i=1 xκ ij. The monomials of the reaction rates are arranged in a vector called velocity vector v(x, k). The net amount of a species i which is consumed or produced in the reaction j is denoted by the stoichiometric coefficient n ij = γ ij γ ij. The stoichiometric coefficients are arranged in the stoichiometric matrix N. The kinetic equations are the product of the stoichiometric matrix and the velocity vector, ẋ = N v(k, x) Example 1 As an example to demonstrate the working of the following theory we consider a network based on the Sel kov model for glycolytic oscillations conceived in 1968 [79]. The presented system includes a third species and two reactions more than the original Sel kov model. Reaction equations: k 4 S 2 k 5 2 S 1 + S 2 k 1 3 S 1 0 k 3 k 2 S 1 k 7 k 6 S 3. 7

12 Kinetic matrix: κ = Stoichiometric matrix and vector of reaction rates: N = Kinetic equation: ; v(k, x) = k 1 x 2 1x 2 k 2 k 3 x 1 k 4 k 5 x 2 k 6 x 1 k 7 x 3 ; ẋ 1 = k 1 x 1 2 x 2 + k 2 k 3 x 1 k 6 x 1 + k 7 x 3 ẋ 2 = k 1 x 2 1 x 2 + k 4 k 5 x 2 ẋ 3 = k 6 x 1 k 7 x 3. A conservation relation results from a conserved quantity in a system, which can be the total amount of adsorption sites on a catalytic surface or the total amount of enzyme species in a reactor. In the equations it appears as a rank defect of the rows of the stoichiometric matrix, i.e. there exist vectors g R m with g N = 0. It follows that g x c = 0, where c is a constant denoting the conserved quantity. 2.2 Algebraic structures in the space of reaction rates In general, one is interested in the stationary reaction behavior, which is observable in experiments, i.e. one investigates the solution set of N v(x, k) = 0. The set of stationary solutions is usually considered in the concentration space, i.e in the variables x R m. However, there are some advantages to consider the set of stationary solutions in the space of reaction rates, i.e in the variables v R r, which will be called reaction rate coordinates. A first advantage is that the Jacobian in the space of reaction rates is of the form Jac = N diag(v) κ t diag(h 0 ) (2.1) with h 0 = x 1 0 being the inverse steady state concentrations. Remark 1 The derivation of the Jacobian matrix in reaction coordinates for a vector field f with steady state solution x 0 proceeds as follows [11]. ẋ = f(x, k), x R m, 0 = f(x 0, k 0 ), h 0i := x 1 0i 8

13 First of all the variable x(t) R m is scaled by the inverse steady state solution h 0 = x 1 0. This has the effect that the scaled time dependent variable w(t) R m, with w(t) = diag(h 0 ) x(t), is at steady state at unity. This transforms the vector field f(x) and its Jacobian matrix D x f ẋ = f(x, k) f(x, k) = N v(x, k) = N diag(k) φ(x) D x f(x) x0 to the vector field f(w) and to the Jacobian D w f ẇ = f(w, k) f(w, k) = diag(h 0 ) N diag(k) φ(diag(x 0 ) w) D w f(w) w0. φ(x) is a vector containing the pure monomials of the reaction rates without the kinetic constants, which are assembled on the diagonal of the matrix diag(k). The derivative of the monomial vector φ with respect to w at steady state has a special form: D w f(w) w0 = diag(h 0 ) N diag(k) d dw φ w 0 d with dw φ w 0 = diag(φ(x 0 )) κ t where κ is the kinetic matrix containing the exponents of the reaction rates. The linearized system in w-coordinates is then ẇ = diag(h 0 ) N diag(k) diag(φ(x 0 )) κ t w(t) Retransformation into the x-variables and multiplication from the left by diag(h 0 ) leads to ẋ = N diag(k) diag(φ(x 0 )) κ t diag(h 0 ) x(t) ẋ = N diag(v(x 0 )) κ t diag(h 0 ) x(t). The Jacobian matrix in reaction rate coordinates is thus Jac(v, h) = N diag(v) κ t diag(h 0 ). Example 1 ( continued) The Jacobian matrix of the example in reaction rate coordinates reads 2 v 1 v 3 v 6 v 1 v 7 Jac(v) = 2 v 1 v 1 v 5 0 diag(h 0) v 6 0 v 7 9

14 Further advantages to consider the space of reaction rates result from the fact that the set of stationary reaction rates is confined to two algebraic structures. The origin of these structures lies in the special properties of the monomial mapping given by the reaction rates and the linear mapping given by the stoichiometric matrix and the concatenation of these mappings at steady state. If x is at steady state, obviously the image v(x, k) of the monomial mapping is in the null space of the stoichiometric matrix N. On the one hand the reaction rates v(x, k) represent the image of a family of monomial mappings R m R r, x v(x, k) from the concentration space into the space of reaction rates, which is parameterized by the rate constants k. Since the number of variables x i is in general smaller than the number of monomials v(x, k), the image of this mapping is not the full reaction space and there must exist some restrictions on v. These restrictions are given by the variety of a deformed toric ideal. They enable a strong reduction of the dimension of the set of stationary reaction rates. (see subsection 2.2.1). On the other hand the nonnegative stationary reaction rates v must lie in the nonnegative kernel of the linear mapping given by the stoichiometric matrix, i.e. v (ker(n) R ). r The intersection set of the kernel of the stoichiometric matrix with the nonnegative orthant of the reaction space represents a convex polyhedral cone. The choice of these generating vectors as new coordinate axes leads to a several advantages in the stability analysis (see subsection 2.2.2). Since the two mappings are linked together at steady state, both structures must be considered as restrictive sets. The intersection set of a convex cone and a toric variety represents a curve, which can be mapped uniquely onto the set of stationary solutions in the concentration space under certain conditions (see 2.2.3). These relationships will be explained in the next subsections Set of reaction rates as deformed toric variety Due to the properties of the monomial mapping from the concentration space onto the space of reaction rates, the monomial reaction rates are not completely independent from each other, i.e. there exist functional relations among them. There is a huge theoretical background for the existence and the importance of these relations. Its relevance for chemical reaction kinetics has been shown in [28, 30, 19]. In order to understand the essence of these relations special concepts of ideal theory must be elucidated, which have not, to date, been commonly used in applied mathematics. Even the application of these relations is not evident and requires clarification of the notions polynomial ideal, variety of an ideal, basis of a polynomial ideal, change of basis, leading monomial, Groebner basis and deformed toric ideal. The procedure can be summarized as follows: The functional relations we are looking for will be of binomial structure. First, we will rewrite the monomial reaction rates in a binomial form in order to generate a binomial ideal. Secondly, we will change the basis of this ideal, i.e. we will find a Groebner basis which spans the same ideal with other basis elements. Some of these new basis elements are the binomials we are looking for. The reaction rates must fulfill these binomials and are confined to the set of zeros. At least 10

15 this fact can be verified without difficulties (see next paragraph of example 1). The ideal generated by the binomials in the reaction rates is a deformed toric ideal, its set of zeros is called deformed toric variety. The carrying-out of these two steps and the necessary definitions and theorems will now be explained in more detail. The reaction rates v i (x, k) rewritten in a suitable binomial form represent basis elements of a polynomial ideal. Definition 1 Let f 1,..., f s be polynomials in C[x]. Then we set { s } I = f 1,..., f s = h i f i : h i C[x]. I is a polynomial ideal. The binomials for our purpose are formed by the reaction rates v i and the corresponding monomials v i (x, k), i.e the ideal under consideration is generated by the basis elements v i v i (x, k). I = v 1 v 1 (x, k),..., v r v r (x, k) C(k)[x, v] Consequently, the binomials are in x and v, the coefficients depending on the rate constants k. Example 1 ( continued) The ideal I spanned by the binomials of the reaction rates is I = v 1 k 1 x 1 2 x 2, v 2 k 2, v 3 k 3 x 1, v 4 k 4, v 5 k 5 x 2, v 6 k 6 x 1, v 7 k 7 x 3 Obviously, the reaction rates lie in the set of zeros of these binomials. The set of zeros of the basis elements of an ideal I is equal to the set of zeros of the whole ideal I. The set of zeros of a polynomial ideal is called variety. Theorem 1 Let f 1...f s be the polynomials in C[x] and let I =< f 1,..., f s > be the ideal they are generating. Then i=1 V (I) = {x C n : f(x) = 0 f I} = V (f 1,..., f s ). The variety of the reaction rate ideal consists of points (x, v). In this variety there are no restrictions on x, but the variables v must lie in the image of the monomial mapping of the reaction rates for given x. These restrictions may be found after a change of the basis of the ideal I. This is possible thanks to a theorem stated and proved by Hilbert, which says that the variety of an ideal is not affected by the choice of the basis. Theorem 2 If {f 1,..., f s } and {g 1,..., g t } are generating the same ideal I in C[x], e.g. if f 1,..., f s = g 1,..., g t, then V (f 1,..., f s ) = V (g 1,..., g t ) = V (I). 11

16 By this property the variety of an ideal is distinguished from a simple set of solutions. The preferred basis for this kind of problem is a Groebner basis. To describe comprehensively the concept of a Groebner basis would largely exceed the length of this work and would not contribute to the aim of this investigation. A detailed description can be found in [12]. Here we will restrict ourselves to the definition and the basics necessary to apply a Groebner basis. A monomial order is a relation which allows to decide for two monomials which one is greater than another and to define in a set of monomials the greatest one, which is called leading monomial. The notation is LM(f i ) for the leading monomial of a polynomial f i. LM(I) denotes the set of all leading monomials of the elements of I. Given an ideal I generated by the basis elements f i, i = 1...n, with the set of leading monomials of this ideal LM(I), one can take the leading monomials as a basis and generate another ideal < LM(I) >. If this ideal is the same as the ideal, which is generated by the leading monomials of the basis elements < LM(f 1 ),..., LM(f n ) >, then the basis is called Groebner basis [12]. Definition 2 Fix a monomial order. A finite subset GB = {g 1,..., gt} of an ideal I is said to be a Groebner Basis if < LM(I) > = < LM(g 1 ), LM(g 2 ),..., LM(g t ) >. The important point is to fix the monomial order. In order to obtain the special relations between the variables, it is crucially necessary to choose an elimination order. The elimination order to obtain the following Groebner basis is a pure lexicographic order. This indication suffices for a procedure call in special mathematical tools (for example Maple: plex), where the algorithms to calculate a Groebner basis are implemented. Definition 3 A monomial order is an order relation between monomials x α in the ring of polynomials C[x 1,..., x m ] with the following properties: (i) For two monomials x α, x β with x α x β either x α > x β or x α < x β holds. (ii) x α > x β x α+γ > x β+γ γ (iii) Each nonempty set of monomials contains a smallest element Definition 4 Let W = w ij Z m,m be a matrix with rows w 1,..., w m and (i) rank(w)=m (ii) For each column j there exists an index of rows i with w ij = 0 for k = 1,...i 1 and w ij > 0. then x α > x β if there exists an i with wk t α = wt k β for k = 1,..., i 1 and wt iα > wiβ. t If furthermore the variables can be subdivided in two groups, x and v such that for all exponents α 0, β 0, γ 0 x α v β > v γ the order is an elimination order. 12

17 Example 1 Compare the monomials x 1, x 2, x 2 1 and x 1 x 2 2 i.e. [ ] [ ] [ ] [ α =, β =, γ =, δ = The pure lexicographic order is defined by the matrix [ 1 0 W plex Z 2,2 = I d = 0 1 ]. ]. According to the pure lexicographic order the leading monomial is x 2 1 x 1 x 2 2 > x 1 > x 2 : and furthermore w1 t α > w1 t β [ ] [ ] > [ 1 0 ] [ > 0 x 1 > x 2 ] w1 t γ > w1 t α [ ] [ ] > [ 1 0 ] [ > 1 x 2 1 > x 1 ] w1 t γ > w1 t δ [ ] [ ] > [ 1 0 ] [ > 1 x 2 1 > x 1 x 2 2 ] w1 t δ = w1 t α [ ] [ ] = [ 1 0 ] [ = 1 w2 t δ > w2 t α [ ] [ ] > [ 0 1 ] [ > 0 x 1 x 2 2 > x 1 ] ] Another monomial order is the graded lexicographic which is defined by the matrix [ ] 1 1 W deglex Z 2,2 =. 1 0 According to this order x 2 1 < x 1 x 2 2. The pure lexicographic order is an elimination order, graded orders are in general no elimination orders. Example 1 ( continued) A Gröbner basis with respect to the pure lexicographic order (plex) is { v 3 + k 3 x 1, k 5 x 2 v 5, k 7 x 3 v 7, v 2 k 2, k 5 k 3 2 v 1 + k 1 v 5 v 3 2, k 6 v 3 + k 3 v 6, v 4 k 4 } The restrictive binomials are {v 2 k 2, k 5 k 3 2 v 1 + k 1 v 5 v 3 2, k 6 v 3 + k 3 v 6, v 4 k 4 } 13

18 Substituting the reaction rates v(x, k) v 1 = k 1 x 1 2 x 2, v 2 = k 2, v 3 = k 3 x 1, v 4 = k 4, v 5 = k 5 x 2, v 6 = k 6 x 1 in these binomials it is obvious that they are confined to the set of zeros. Applying an elimination order it is possible to eliminate some variables during the formation of the Groebner basis. One can thus find in the resulting basis binomials which depend only on the reaction rates v. Since these binomials are within the ideal they vanish on the image of v(x, k). Obviously, they represent the restrictive binomials we are looking for. The ideal which is generated by these binomials is called deformed toric ideal and its variety is called deformed toric variety. The reaction rates v are confined to this deformed toric variety. Definition 5 The ideal def, tor I = { f R[v] f ( v(x) ) 0 } C(k)[v] is called deformed toric ideal and its variety is called deformed toric variety. Example 1 ( continued) The deformed toric ideal is spanned by the last four elements of the Groebner basis. I def, tor = v 2 k 2, k 1 v 5 v 3 2 k 3 2 k 5 v 1, k 6 v 3 k 3 v 6, v 4 k 4 Remark 2 The name toric results from the fact that a toric variety is invariant with respect to the induced representation of the algebraic torus: - A representation of the algebraic torus is given by the torus group T = (C \ 0) m and the mapping C m C m, x t 1 x 1,..., t m x m, t = (t 1,..., t m ) T. - For a given matrix of exponents κ Z m,r there is an induced representation of the algebraic torus. For each t T = (C \ 0) r there is a mapping D(t) : C r C r, v diag ( m i=1 t κ i1 i The deformed toric variety is invariant with respect to D(t). v V (I κ ) D(t) v V (I κ ) m i=1 ) t κ ir i v The word deformed indicates the parameter dependence of the binomials on k. Toric ideals in general are studied in [91], for a general theory of Gröbner bases see [12]. There exist implementations of efficient algorithms for the computation of Gröbner bases of toric ideals in Singular [33], in the library of algorithms [93], and in Cocoa [10]. 14

19 2.2.2 Set of stationary reaction rates as a convex cone As the intersection of the kernel of the stoichiometric matrix kern with the nonnegative orthant of the r-dimensional reaction space R r 0 the set of nonnegative stationary reaction rates represents a convex polyhedral cone. { } K v = {v R r N v = 0, v 0} = (ker{n}) R r 0 = j i E i, j i > 0, i It has been introduced in chemical literature by Clarke, who emphasized the advantages to execute the stability analysis in this cone [11]. A convex cone is spanned by a set of minimal generating vectors E i. Each element can be represented as a convex combination of the generating vectors. The explicit representation of an element of the cone of nonnegative stationary reaction rates is thus a sum over the generating vectors with nonnegative, i.e. convex coefficients j i, v(j) = t i=1 j ie i. Example 1 ( continued) Four generating vectors span the convex polyhedral cone of the nonnegative stationary reaction rates. Consequently, the reaction rates depend on four convex coordinates. E 1 = (0, 1, 1, 0, 0, 0, 0) E 2 = (0, 0, 0, 1, 1, 0, 0) E 3 = (0, 0, 0, 0, 0, 1, 1) E 4 = (1, 0, 1, 1, 0, 0, 0) v(j) = i=1 j 4 j 1 j 1 + j 4 j 2 + j 4 j 2 j 3 j 3 The minimal generating vectors are called extreme currents in chemical literature. Many algorithms for their computation have been invented and implemented, for example the FluxAnalyzer, which has been used for the calculations of some of the models presented in this work [53, 54, 78]. If the minimal generating vectors are linearly independent, the cone of nonnegative stationary reaction rates can be used as a new coordinate system for the set of stationary solutions. The minimal generating vectors represent the coordinate axes with coefficients j i. The new coordinates will be called convex coordinates. If the generating vectors are linearly dependent, the cone must be subdivided into simplicial subcones before calling it a coordinate system. The Jacobian matrix in reaction rate coordinates can be transformed in convex coordinates by a simple substitution of v(j) = t i=1 j ie i into the formula 2.1. The Jacobian matrix is then automatically evaluated at steady state. ( ) Jac(v) = N diag(v) κ t diag(h 0 ) Jac(j) = N diag j i E i κ t diag(h 0 ) i 15

20 This matrix can be decomposed into a part Jac(j) which is independent from the stationary concentrations and the positive definite diagonal matrix diag(h 0 ). ( ) Jac(j, h 0 ) = Jac(j) diag(h 0 ), with Jac(j) = N diag j i E i κ t (2.2) Example 1 ( continued) The Jacobian in convex coordinates reads j 4 j 1 j 3 j 4 j 3 Jac(j) = 2 j 4 j 4 j 2 0 diag(h 0) j 3 0 j 3 The matrix Jac(j) as a product of three matrices, contains many information on stability. The system s stoichiometry is given by the stoichiometric matrix, the structure of the cone of stationary reaction rates is reflected by the convex reaction rates in the diagonal matrix and the kinetic matrix κ encodes the degree of nonlinearity of the reaction rates. It depends only on the convex reaction rates j. Note that the sign pattern of Jac is the same as the sign pattern of the real Jacobian matrix. The convex coordinates have been used by Clarke for the first time and are widely applied in chemistry. In view of the special form of the convex Jacobian matrix at steady state, which allows to apply theorems from matrix stability, they have several advantages for the stability analysis (see next chapter). Furthermore, they provide a possibility to decompose the whole network into subnetworks and to conclude from the stability of the subnetworks on the stability of the whole system: For each extreme current E i an extreme subnetwork is defined by restricting to the support of E i. In turn this defines a subsystem of the original differential system. For combinations of extreme currents the subsystems are defined analogously. The relation between the steady states of the extreme subsystems and the whole system of differential equations is very complicated. It is studied in [32]. The stability of an extreme current E i is defined by the stability of the corresponding subnetwork and is determined by the corresponding Jacobian matrix. The Jacobian can be considered at steady state in the concentration space or in convex coordinates. In convex coordinates one can see that the relation between the Jacobian matrix of the extreme subnetworks and the Jacobian of the whole network is simply linear. Let t be the number of extreme currents. Then the Jacobian can be represented by the following sum. Jac(j) diag(h 0 ) = j 1 Jac(E1 ) diag(h 0 ) + + j t Jac(Et ) diag(h 0 ). The stability of an extreme current E i is determined by the summand of the Jacobian matrix Jac(E i ) diag(h 0 ). If the coordinate j i is larger than the other convex coordinates, the extreme subnetwork E i will predominate and determine the stability of the whole network. The same holds for combinations of extreme currents. There are two different kinds of stability of an extreme current: 16 i

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