System Reduction of Nonlinear Positive Systems by Linearization and Truncation
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1 System Reduction of Nonlinear Positive Systems by Linearization and Truncation Hanna M. Härdin 1 and Jan H. van Schuppen 2 1 Department of Molecular Cell Physiology, Vrije Universiteit, De Boelelaan 1085, NL-1081 HV Amsterdam, The Netherlands, hanna.hardin@falw.vu.nl 2 CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands, J.H.van.Schuppen@cwi.nl Summary. System reduction is applied to a mathematical model of glycolysis in yeast and to a chain network without and with feedback. The method of system reduction used is linearization of a rational positive system at a steady state, balancing of the local linear system, and truncation of the balanced linear system. For a model of glycolysis in yeast with glucose as input and pyruvate as output, it is shown that a third order linear system locally approximates well the original thirteenth order nonlinear system. Key words: Metabolic networks, rational positive systems, system reduction, balancing, glycolysis in yeast, chain network. 1 Introduction In order to better understand the functioning of cells in living organisms, which is essential for the development of new drugs, system reduction of biochemical systems is an important issue. Biochemical systems can be described by a set of chemical reactions which can be modeled with nonlinear differential equations. The problem of system reduction, also called model reduction, is to determine for a system an approximant system of lower complexity such that the external behavior of the two systems is close in terms of an approximation criterion. System reduction has been developed for linear systems and is routinely used for computations of chip design, aerospace structures, and hydrological models. System reduction is described in the book [1] and in the proceedings [2]. System reduction of nonlinear positive systems requires a novel approach based on the algebraic properties of positive systems. This paper is the first part of a long term project on system reduction of a class of nonlinear positive systems which are models of biochemical reaction networks. The approach discussed in this paper is linearization of a nonlinear
2 2 Härdin and Van Schuppen positive system followed by balancing of the linear system and truncation of the balanced system. There are now several publications on system reduction of mathematical models of biochemical reaction networks, see [7, 8, 10]. The approach of this paper is close to that of [8] yet differs from it in the examples and in the approach. 2 Problem Formulation and Preliminaries Problem 1. Consider a particular class of positive systems which are models of biochemical reaction networks. This is to be called the class of to-beapproximated systems. Select: Another class of systems for biochemical reaction networks to be called the class of approximant systems. An approximation criterion on the external behaviors of both classes of systems. For example, the Hankel norm or an operator norm on the external signals such as the least-squares norm. A complexity criterion on the class of approximant systems. For example, the order of the system which equals the dimension of the state space. Determine a member of the class of approximant systems which best approximates in terms of the approximation criterion on the external behavior and the complexity, the prespecified member of the class of to-be-approximated systems. Approaches to system reduction for positive systems. 1. Local linearization of a positive system followed by a system reduction algorithm for linear systems. The approximant system is then likely not a positive system. This approach is formulated in Section 4 and two examples are provided in Section System reduction of a rational positive system (not necessarily positive) to a nonlinear system, not necessarily positive. 3. System reduction for a linear positive system in which the approximant system has to be a linear positive system of lower complexity. 4. System reduction of a rational positive system to a system in the same class but of lower complexity. 3 Rational Positive Systems Rational positive systems are realistic models of biochemical reaction networks. Attention in this paper is restricted to metabolic networks, see [3, 5].
3 System Reduction 3 Definition 1. A rational positive system for a biochemical reaction network is defined as a dynamical system defined by the equations, ẋ(t) = BDiag(r(x(t), x e (t)))u(t), x(t 0 ) = x 0, (1) or, per component i Z n, m ẋ i (t) = B i,j r j (x(t), x e (t))u j (t) = j=1 [ m p + (B + i,j B i,j ) j (x(t), x e(t)) q j (x(t), x e (t)) p j (x(t), x ] e(t)) u j (t) (2) q j (x(t), x e (t)) j=1 = f i (x(t), x e (t), u(t)), x i (t 0 ) = x i,0, i Z n, z(t) = HDiag(r(x(t), x e (t)))u(t) = h(x(t), x e (t), u(t)). (3) with the notation, n, m Z + = {1, 2,...}, n e, n z N = {0, 1,...}, T = [t 0, ), the time index set, X = R n +, the state set, X e = R n e +, the set of external concentrations, U = R m +, the set of enzyme concentrations, Z = R p is the space of controlled output including flows and concentrations, B Z n m is called the stoichiometric matrix, u : T U and x e : T X e are input functions, x : T X, z : T R n z, H N nz m. The conditions imposed on the system are not listed, see [9]. 4 Procedure for System Reduction The system reduction method of this paper is restricted to the first approach formulated in Section 2: it is based on linearization and on truncation of the local linear system. The reasons for this restricted choice is that the authors want to gain experience with the system reduction algorithms for biochemical reaction networks. The choices for the system reduction problem are: The class of to-beapproximated systems is the class of rational positive systems described in Section 2. The class of approximant systems is the class of ordinary linear systems. Note that the approximant system is thus not positive. The complexity criterion is the order of the linear system. Consider then the following nonlinear system, d dt x(t) = f(x(t), x e(t), u(t)), x(t 0 ) = x 0, (4) z(t) = h(x(t), x e (t), u(t)), (5) of which a rational positive system is a particular case. Further, consider a steady state around which the system will be linearized, i.e. an enzyme input concentration u s R m +, a state x s R n +, an external concentration x e,s R me +, and a flow z s R p +. The approximant of the original nonlinear system locally in the neighborhood of (x s, x e,s, u s ) is represented by the linear system,
4 4 Härdin and Van Schuppen d dt x r(t) = A r x r (t) + B e,r x e (t) + B r u(t) + S r Sf(x s, x e,s, u s ), (6) x s + S 1 S r x r (t) is the local approximation to x(t), t T,(7) z s + C r x r (t) + D e,r x e (t) + D r u(t), (8) is the local approximation to z(t), t T. Below u(t) = 0 t T because it is assumed that the enzyme concentrations are constant and equal the steady input values u s. Algorithm 1 System reduction of a nonlinear positive system by linearization, balancing, and truncation. Data: The functions describing the rational positive system f and h. The steady external concentration x e,s R n r + and the steady enzyme concentration u s R m Compute an approximation to the steady state x s R n + and the steady flow z s R p +, satisfying 0 = f(x s, x e,s, u s ), z s = h(x s, x e,s, u s ). (9) 2. Linearization of the rational positive system: (A, B e, C, D e, f(x s, x e,s, u s ), h(x s, x e,s, u s )), (10) A = D x f(x, x e, u) (xs,x e,s,u s ), B e = D xe f(x, x e, u) (xs,x e,s,u s ), (11) C = D x h(x, x e, u) (xs,x e,s,u s ), D e = D xe h(x, x e, u) (xs,x e,s,u s ). (12) 3. Balancing a) Solve for the controllability and the observability Grammians, AP + P A T = B e B T e, QA + A T Q = C T C. (13) b) Compute the state space isomorphism, P = U 1 D 2 1U T 1, D 1 U T 1 QU 1 D 1 = U 2 D 4 2U T 2, (14) by singular value decompositions where, U i R n n orthogonal matrix, D j R n n diagonal matrix, S = D 2 U T 2 D 1 1 U T 1, D = D 2 2. (15) Then SP S T = S T QS 1 = D. (16) c) Compute the system matrices of the balanced system, A b = SAS 1, B e,b = SB e, C b = CS 1, D e,b = D e, (17) (A b, B e,b, C b, D e,b, f(x s, x e,s, u s ), h(x s, x e,s, u s )). (18) 4. Truncation. Select the dimension of the reduced-order system n r N n and compute S r = (I nr 0), A r = S r A b S T r, B e,r = S r B b, (19) C r = CS T r, D e,r = D e. 5. Output the system matrices of the approximant system (A r, B e,r, C r, D e,r, f(x s, x e,s, u s ), h(x s, x e,s, u s )), (20) x s + S 1 S T r x r, z s + C r x r + D e,r x e. (21)
5 System Reduction 5 5 Examples of System Reduction for Biochemical Reaction Networks 5.1 Chain Network without and with Feedback As the first example we study two closely related systems: a chain of enzyme catalysed reactions with and without feedback, respectively, see Figure 1. Our (a) S X1 X2 X3 X4 X5 P (b) S X1 X2 X3 X4 X5 P Fig. 1. (a) Reaction chain without feedback. A biochemical compound, S, is converted into another, P, in 6 reaction steps. X 1 to X 5 are the intermediate compounds. (b) Reaction chain with feedback. Same chain as in (a) but the fifth intermediate is inhibiting the second reaction. purpose here is to investigate whether feedback in a biological network makes the system more easily reducible. The reaction chain we have modeled consists of 6 reaction steps, modeled with enzyme kinetics [3]. The reactions are modeled as reversible reactions, except for the second which is modeled as irreversible. The second reaction is inhibited by the fifth intermediate compound in the feedback model. The state equations for the systems with and without feedback are the same, except for the difference of the expression of the rate of the second reaction. The systems are on the form x i (t) = r i (x(t), x e (t))u i r i+1 (x(t), x e (t))u i+1, (22) ( ) r5 (x(t), x z(t) = e (t)) (23) r 6 (x(t), x e (t)) where x(t) = (x 1 (t),..., x 5 (t)) T contains the concentrations of the intermediate compounds X 1,...,X 5, x e (t) = (s(t), p(t)) T contains the concentrations of the compounds S and P, and u = (u 1,..., u 6 ) T contains the constant concentrations of the enzymes for the 6 reactions. For the reversible reactions (i.e. for i = 1, 3, 4, 5, 6), the rates r i (x(t), x e (t)) of the enzymes per enzyme unit are given by Equation (24) below, where a and b are the concentrations of substrates and products of each reaction respectively. K i,a, V i,a, K i,b, and V i,a are kinetic constants. In the system without feedback the irreducible reaction follows the ordinary Michaelis-Menten equation (25) and in the system with feedback the rate of the second reaction follows (26). V 2, K 2, K ic, and K iu are kinetic constants. See [3] for details about kinetic modeling.
6 6 Härdin and Van Schuppen r i (x(t), x e (t)) = [ V i,aa K i,a V i,bb K i,b ][1 + a K i,a + b K i,b ] 1, (24) r 2 (x(t), x e (t)) = V 2 x 1 [K 2 + x 1 ] 1, (25) r 2 (x(t), x e (t)) = V 2 x 1 [K 2 (1 + x 5 K ic ) + x 1 (1 + x 5 K iu )] 1. (26) Table 1. Hankel singular values (HSV) and the percentage of each HSV out of the sum of HSVs for the systems without and with feedback, respectively. Without feedback With feedback HSV Percentage HSV Percentage System reduction of the two systems following Algorithm 1 was made. The Hankel singular values, (eig(p Q)), for the two linearized systems are shown in Table 1. These show that for this example a reduced model of the system with feedback, gives a better approximation of the original system, than a reduced model of the system without feedback. The numerical values of the matrices of the linearized and reduced system are not displayed because of lack of space. 5.2 Glycolysis in Yeast The glycolysis is the name of a cellular process in which glucose is converted into different carbon compounds in a number of reaction steps, see Figure 2. This process plays an important role in the metabolism of most organisms, and its function is partly to produce energy-rich molecules and partly to produce carbon compounds that can be used for biosynthesis. Glycolysis has been extensively explored in many organisms and several mathematical models of it have been made, see for example [6, 4]. Because of the complexity of the glycolysis, the models contain many state variables. Therefore it is of interest to find a lower-order model in order to get an understanding of its overall dynamics. Here, a model of glycolysis made by Teusink et al. [4] has been investigated. After changing a few state variables to constants in that model, and certain constants to states, it contains 13 state variables, x 1,..., x 13, each representing the concentration of a carbon compound, except from one, x 5, which represents a module of two interconvertible carbon compounds, see Figure 2. The concentration of glucose on
7 System Reduction 7 Outside the cell Glucose Enzymatic transport through cell membrane Cell membrane Glucose (x1) Inside the cell Glucose 6 phosphate (x2) Glycogen Trehalose Fructose 6 phosphate (x3) Fructose 1,6 biphosphate (x4) Dihydroxyacetone phosphate Glyceraldehyde 3 phosphate x5 Glycerol (x12) 1,3 Biphosphoglycerate (x6) 3 Phosphoglycerate (x7) 2 Phosphoglycerate (x8) Phosphoenolpyruvate (x9) Pyruvate (x10) Acetaldehyde (x11) Ethanol (x13) Succinate Fig. 2. Diagram of glycolysis in yeast as it was described by Teusink et al. [4]. The input and the state variables are indicated. the outside of the cell is used as input. As controlled output z(t), pyruvate, x 10, was chosen, since this compound is directly used for further steps in energy production, and can hence be seen as a measure of energy that is being produced. A reduced model of order 3 was calculated according to Algorithm 1. The order of the reduced system was chosen by investigating the Hankel singular values of the linearized system. The three largest Hankel singular values were , , and , while the remaining 10 were almost 0. The reduced model is (6), (7), and (8), with A r = , B e,r = , C r = ( ), D e,r = 0. The output pyruvate level of the original model, the linearized model, and the reduced model, was simulated as a response to a step in glucose level, see Figure 3. As seen, the approximation error seems to result from the linearization, not from the reduction of the linearized model. However, in a local neighborhood of the steady state, the linearized model is still a good approximation. To sum up, the reduction method based on balancing and truncation has three main drawbacks: (i) The approximant system gives a good approximation of the system behavior only in a local neighborhood of the steady state, but for larger deviations it is not reliable. This is mainly due to the linearization, while the truncation has a minor effect. (ii) The resulting system is not positive. (iii) It is not straightforward to interpret the biological meaning of the state variables of reduced system. The first two drawbacks motivates our
8 8 Härdin and Van Schuppen conc mmol L a time min conc mmol L b time min Fig. 3. Simulation results. The pyruvate levels of the three models (original, linearized, and truncated) as responses to an (a) 10 %, and (b) 50 % change in input glucose level. Dashed line: the original model. Solid line: the linearized and truncated models (indistinguishable). further work with approaches 2-4 in Section 2. The last drawback motivates further analysis of the biological systems, which is ongoing work. References 1. Athanasios C. Antoulas. Approximation of large-scale dynamical systems. Number DC06 in Advances in Design and Control. SIAM, Philadelphia, Peter Benner, Volker Mehrmann, and Danny C. Sorensen. Dimension Reduction of Large-Scale Systems. Number 45 in Lecture Notes in Computational Science and Engineering. Springer-Verlag, Berlin, A. Cornish-Bowden. Fundamentals of enzyme kinetics (3rd. Ed.). Portland Press, B. Teusink etal. Can yeast glycolysis be understood in terms of in vitro kinetics of the constituent enzymes? testing biochemistry. Eur. J. Biochem., 267: , R. Heinrich and S. Schuster. The regulation of cellular systems. Chapman and Hall, New York, F. Hynne, S. Dano, and P.G. Sorensen. Full-scale model of glycolysis in Saccharomyces cerevisiae. Biophys. Chem., 94(1-2): , Roland Krüger and Reinhart Heinrich. Model reduction and analysis of robustness for the Wnt/β-Catenin signal transduction pathway. Genome Informatics, 15: , Wolfram Liebermeister, Ulrike Baur, and Edda Klipp. Biochemical network models simplified by balanced truncation. FEBS Journal, 272: , Jan H. van Schuppen. System theory of rational positive systems for cell reaction networks. In Bart De Moor etal., editor, Proc. MTNS.2004, Leuven, Katholieke Universiteit Leuven. 10. A. Zagaris, H.G. Kaper, and T.J. Kaper. Analysis of the computational singular perturbation reduction method for chemical kinetics. J. Nonlinear Sci., 14:59 91, 2004.
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