Application of fuzzy-logic models for metabolic control analysis

ARTICLE IN PRESS

Journal of Theoretical Biology 245 (2007) 391–399 www.elsevier.com/locate/yjtbi

Application of fuzzy-logic models for metabolic control analysis Ezequiel Franco-Lara, Dirk Weuster-Botz Lehrstuhl fu¨r Bioverfahrenstechnik, Technische Universita¨t Mu¨nchen, Boltzmannstr. 15, 85748 Garching, Germany Received 29 November 2005; received in revised form 19 October 2006; accepted 23 October 2006 Available online 27 October 2006

Abstract A priori information or valuable qualitative knowledge can be incorporated explicitly to describe enzyme kinetics making use of fuzzylogic models. Although restricted to linear relationships, it is shown that fuzzy-logic augmented models are not only able to capture nonlinear features of enzyme kinetics but also allow the proper mathematical treatment of metabolic control analysis. The explicit incorporation of valuable qualitative knowledge is crucial, particularly when handling data estimated from in vivo kinetics studies, since this experimental information is scarce and usually contains measurement errors. Therefore, data-driven techniques, such as the one presented in this work, form a serious alternative to established kinetics approaches. r 2006 Elsevier Ltd. All rights reserved. Keywords: Metabolic control analysis; Fuzzy-logic; Enzyme kinetics

1. Introduction Due to the intrinsic complexity of their reaction network and regulation, biochemical systems are not totally well understood from a mechanistic point of view. This is mainly due to the fact that, even when the stoichiometry is generally well characterized, the synthesis of mechanistic mathematical models for kinetics is governed by a rather restrictive compromise between the model’s simplicity and its implicit development costs. To tackle the complex problem of modelling biochemical systems, differential equations describing the metabolite balances and formal algebraic kinetic approaches accounting for individual reaction phenomena (substrate limitation, inhibition, etc.) are commonly applied. The final outcome is a complex scheme in the form of non-linear differential-algebraic equations. Furthermore, the modelling procedure requires not only the solving of the optimization problem for estimating the most adequate kinetic parameters for the mathematical Corresponding author. Present address: Institut fur Bioverfahren¨ stechnik, Technische Universita¨t Braunschweig, GauXstr. 17, 38106 Braunschweig, Germany. Tel.: +49 531 391 7656; fax: +49 531 391 7652. E-mail addresses: [email protected], [email protected] (E. Franco-Lara), [email protected] (D. Weuster-Botz).

0022-5193/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2006.10.023

description, but above all a profound knowledge about reaction mechanisms and interactions involved in the biochemical system. These are not only difficult to elucidate and formulate mathematically, but are even more difficult to validate experimentally. Therefore, when considering the case of a biochemical system which is only partially understood, but for which some a priori knowledge or valuable qualitative data might be available, it would be very advantageous to incorporate this information explicitly into a proper model. Lee et al. (1999) proposed a combination of formal enzyme kinetics with fuzzy-logic approaches as an alternative method to integrate qualitative knowledge. Fuzzy models were developed as a simple and generic solution for complex tasks without a complicated mathematical form (Biewer, 1997). Their functioning is based on an inference mechanism or reasoning which makes use of an expert understanding of the system’s behaviour. Applied to biochemical systems, the development of fuzzy-logic approaches could be viewed as an intermediary evolution step between partial empirical to full mechanistic modelling. The methodology presents the additional advantage of a simultaneous reduction of its development time due to its simplified conception. The use of fuzzy-logic approaches can also be extended to metabolic control analysis (MCA) of biochemical

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E. Franco-Lara, D. Weuster-Botz / Journal of Theoretical Biology 245 (2007) 391–399

reaction networks. MCA has been extensively used to quantify regulation behaviour of complex biological systems (Kacser and Burns, 1973; Heinrich and Rapoport, 1974; Fell, 1997; Stephanopoulos et al., 1998). Although the fuzzy-logic approaches presented here are restricted to linear fuzzy structures, they are able to capture non-linear features of enzyme kinetics, allowing the proper mathematical treatment of MCA. 2. Fuzzy-logic basics Fuzzy-logic sets and basic operations were conceived by Zadeh (1965) and their theory can be seen as an extension to the classical set theory for representing commonsense knowledge in terms of vague concepts. In the classical set theory, a set A can be defined as a collection of objects or elements x of the variable X with a common property P: A ¼ fx 2 X jx has property Pg.

(1)

In the case of the fuzzy set theory, an affiliation of the variable X to the property P can be established through a function fA(X) describing the membership degree of the elements x to the set A. Considering the example presented in Fig. 1, element x1 is not a member of the set A, element x2 is a partial member of the set A, whereas element x3 is a full member of the set A. To create a functional relationship, the independent variable X is mapped into the dependent variable Y by means of semantic rules in form of ‘‘ifythen’’ statements (inference mechanism). The mapping delivers the corre-

sponding element y of the dependent variable Y contained in a fuzzy set B analogous to that of X. Fig. 2 depicts the components involved in the setting up of the fuzzy-logic functional relationship. In the following section, the practical implementation of an enzyme kinetics approach will be discussed considering the modelling of the kinetic rate of a reaction v, with real values lying in the range of 0–0.3 mM s1 as function of the metabolite concentration S, lying in the range of 0–0.2 mM. Assuming that three different domains of the metabolite concentration can be described with linguistic terms, their corresponding ranges might be represented by the membership sets, An(S), where nA[Low, Medium, High] (see Fig. 3). If the correlated reaction rates can also be described by three linguistic sets, the corresponding membership set is Bn(v), where nA[Low, Medium, High]. In order to complete the model, three semantic rules might be used to form the inference database from which the functional relationship between the variables is derived. In this case:

  

Rule 1: If S is Low then v is Low. Rule 2: If S is Medium then v is Medium. Rule 3: If S is High then v is High.

Fig. 4 depicts the graphical interpretation of the three rules and their overlap, which covers the whole range of validity of the resulting verbal model. With these components, the fuzzy system can be employed for kinetic reaction rate modelling using the Takagi–Sugeno approach (Takagi and Sugeno, 1985). This consists of series of linear approximations of the dependent variable v for every variable S belonging to the fuzzy set An such that If S 2 An then vn ¼ an S þ bn ,

Fig. 1. Membership function of a fuzzy system. Element x1 is not a member of the fuzzy set A, element x2 is a partial member of the fuzzy set A, while element x3 is a full member of the fuzzy set A.

(2)

where vn is the linear approximation of the reaction rate for the associated set Bn, being an and bn linear coefficients. The entire range of the reaction rate variable v can be calculated from the independent concentration variable S for all membership functions, fAn(S) from P P fAn ðSÞvn f ðSÞðan S þ bn Þ n P ¼ n An , (3a) v¼ P n fAn ðSÞ n fAn ðSÞ

Fig. 2. Basic components and functioning principle of a fuzzy system.

ARTICLE IN PRESS E. Franco-Lara, D. Weuster-Botz / Journal of Theoretical Biology 245 (2007) 391–399

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Fig. 3. Basic components of a fuzzy variable and its relationships.

Fig. 4. Graphical interpretation of the fuzzy rules and their resulting cause–effect relationship through membership function association.

that is, P  P  fAn ðSÞan fAn ðSÞbn n n v¼ P Sþ P , n fAn ðSÞ n fAn ðSÞ

(3b)

Eq. (3) can be interpreted as the fuzzy local linear model approximation of the non-linear system, as shown in the example on Fig. 5 for an ideal allosteric kinetic.

ARTICLE IN PRESS E. Franco-Lara, D. Weuster-Botz / Journal of Theoretical Biology 245 (2007) 391–399

394

Fig. 5. Takagi–Sugeno function approximation of an allosteric reaction rate.

This kind of approximation can also be extended for saturation, inhibition and saturation-inhibitiontype enzyme kinetics as well as for multiple variables. Considering any vector, S, the Takagi–Sugeno model results in a quasi-linear system. For illustration purposes, reversible kinetics with the forward reaction v+(S1), and the backward reaction v(S2) has been taken into account: P P  ðS1 Þðan S1 þ bn Þ f ðS 1 Þan þ n fAn P ¼ Pn An v ¼ S1 n fAn ðS 1 Þ n fAn ðS 1 Þ P   f ðS1 Þbn þ Pn An , ð4aÞ n fAn ðS 1 Þ 

v ¼

P

ðS Þða S m fAm P 2 m 2

þ bm Þ

f ðS 2 Þ P m Am  fAm ðS 2 Þbm m P þ . m fAm ðS 2 Þ

P  fAm ðS2 Þam m P ¼ S2 m fAm ðS 2 Þ

The net reaction rate, vnet(S) ¼ (S1, S2), is given by P  ðS 1 ÞðaTn S1 þ bTn Þ þ  n fAn P vnet ¼ v  v ¼ n fAn ðS 1 Þ P  ðS ÞðaT S þ bTm Þ m fAm P 2 m 2  . m fAm ðS 2 Þ

ð4bÞ

ð4cÞ

The reaction rate vnet in Eq. (4c) is linear with respect to the vector parameters, aTn , aTm , bTn and bTm , but pseudo-linear in S (see Appendix A.1 for details of its derivation). This approach can be interpreted as the bi-plane or bi-linear approximation of the actual non-linear reaction rate, vnet, and can be extended to more than two independent variables delivering hyper-plane or multi-linear approximation. It is important to remark that the proposed methodology and formulae are valid only so long as the associated membership functions and local linear approximations (inference mechanism) can properly map the variables. For the characterization of the memberships, there is a great variety of useful mathematical functions which can be employed: triangular, trapezoidal, Gaussian, etc. (Biewer, 1997; Passino and Yurkovich, 1998). However, there exists no guideline, which favours the use of one of these

functions over another. Especially in the case where the fuzzy system is built using the technique of ‘‘learning from examples’’ (LFE) (Passino and Yurkovich, 1998), it must be stated clearly that the choice might influence the prediction capabilities. That is, even when intuition plays a primary role by the design guidelines in the extraction of information from data, the local approximations (inferences) might be regarded as the most appropriate starting point in the search for the best global approach. Further improvements can be made by setting proper values for the parameters of the membership functions through optimization routines (Bremermann and Anderson, 1989; Passino and Yurkovich, 1998). 3. Determination of flux control coefficients from fuzzy-logic approaches Metabolic control analysis can be regarded as a linear perturbation theory for enzymatic kinetics taking place in metabolic networks (Stephanopoulos et al., 1998). Considering linear dependence kinetics with respect to a single metabolite concentration, Si, for any reaction, vj, one obtains vj ¼ b0;j þ b1;ij Si ,

(5)

where b1,ij and b0,j are kinetic constants due to the linearization of the reaction rate vj, as function of the metabolite concentration Si. Applying a balance to all metabolites the equation for a reaction network system in steady-state and matrix notation becomes dS ¼ Nv ¼ Nðb0 þ b1 SÞ ¼ 0, (6) dt where S ¼ (S1, S2,y, SI)T, v ¼ (v1, v2,y, vk,y, vJ)T, b0 ¼ (b0,1, b0,2,y, b0,k,y, b0,J)T, b1 ¼ (b1,ij) the matrix with the first-order kinetic constants b1,ij, and N the (I, J)-sized stoichiometric matrix, with elements nij representing the stoichiometric coefficient of metabolite i in reaction j. Making use of the linear kinetics of Eq. (5), the unscaled elasticity coefficient matrix for all reactions, vk, referred to the substrates Si is obtained from (Reder, 1988): 0 1 1 0 qv 1 qv 1 S1 ::: 1SI ::: qS1 qS 1 I C B C B B B ::: C ::: C ::: E ¼ B ::: ¼ C B C @ A @ qv A qvJ J JS1 ::: JSI ::: qS 1 qS I 0 1 b1;11 ::: b1;I1 B C ::: ::: C ¼ b1 , ¼B ð7Þ @ A b1;1J ::: b1;IJ where the fuzzy logic first-order kinetic constants, b1,ik, are given by P an fAn;k ðS i Þ (8) b1;ik ¼ Pn n fAn;k ðS i Þ

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Eq. (7) can be further incorporated into the general formalism for MCA to calculate the unknown flux and concentration control coefficients. As demonstrated by Reder (1988), if the stoichiometric matrix presents conserved moieties, i.e. if the rank r of N is smaller than J, the dependencies can be removed resulting in a reduced stoichiometric matrix, NR, with r independent rows such that: " # Id (9) N ¼ LNR ¼ NR , L0 where L is the so-called (metabolite) link matrix (size: (Jr)  r), Id an r  r identity matrix and L0 a matrix correlating the metabolite dependencies. Combining the MCA relations derived by Reder (1988), Letellier et al. (1991) and Heinrich and Schuster (1996) with the fuzzy logic approximations given by Eqs. (7) and (8), the unscaled substrate control coefficients (see Appendix A.2) can be estimated from C S ¼ LðNR ELÞ1 NR ¼ LðNR b1 LÞ1 NR .

(10a)

Further, the unscaled flux control coefficients (see Appendix A.2) can be estimated from: C J ¼ Id  ELðNR ELÞ1 NR ¼ Id  b1 LðNR b1 LÞ1 NR . (10b) If the reaction network does not contain conserved moieties, i.e. it is of maximum rank (r ¼ J), L reduces to an identity matrix of size J  J. In this case Eqs. (10a) and (10b) can be simplified to: C S ¼ ðNb1 Þ1 N

(11a)

and J

1

J

S

C ¼ Id  b1 ðNb1 Þ N ! C ¼ Id þ b1 C .

(11b)

Furthermore, the summation and connectivity theorems can be derived using Eqs. (10) or Eqs. (11) and (12), considering that at the steady-state condition given by Eq. (6), when Nv ¼ 0 (or NRv ¼ 0, when conserved moieties exist) there exists a matrix K, called the kernel or nullspace of N, such that (Reder, 1988; Ehlde and Zacchi, 1997): NK ¼ 0

or

NR K ¼ 0,

(12)

where each column of K represents a particular nontrivial solution of Eq. (12). The summation relationships between the substrate/flux control coefficients are obtained by post-multiplication by K of the Eqs. (10a) and (10b), respectively, combined with Eq. (12): C S K ¼ LðNR b1 LÞ1 NR K ! C S K ¼ 0,

(13a)

C J K ¼ ðId  b1 LðNR b1 LÞ1 NR ÞK ! C J K ¼ K.

(13b)

the Eqs. (10a) and (10b), respectively, combined with Eq. (12): C S b1 L ¼ LðNR b1 LÞ1 NR b1 L ! C S b1 L ¼ L,

(14a)

C J b1 L ¼ ðId  b1 LðNR b1 LÞ1 NR Þb1 L ! C J b1 L ¼ 0. (14b) Finally, Eqs. (13) and (14) can be combined to generate a single control matrix equation such that: 2 3 CJ     K 0 .. 6 7 ¼ . (15) 4    5 K . b1 L 0 L S C This equation is the general MCA matrix representation for any network of reactions, whose kinetics can be described by a linear fuzzy-logic approximation contained in the b1 matrix. 4. Discussion For illustration purposes MCA was performed according to the reaction scheme shown in Fig. 6, which forms the simplest structural branched system (Kacser and Burns, 1973; Heinrich and Rapoport, 1974; Stephanopoulos et al., 1998). The balance for the intermediate X is given by: dX =dt ¼ v1  v2  v3 .

(16)

All rates were modelled using reversible Michaelis-Menten kinetics described by (Snoep et al., 2004):    V maxi ð1=K is Þ s  ðp K eqi Þ   , vi ¼  1 þ ðs=K is Þ þ ðp K ip Þ

(17)

where s and p are the substrate and product concentrations of the respective reactions i (see Appendix A.3 for details and model parameters). Snoep et al. (2004) have discussed the case of ‘sensitive’ conditions with s ¼ 1 and of ‘saturating’ conditions, where s ¼ 10. Here the steady-state concentration of X has been kept constant at 0.05 mM, whereas the flux values are: J1 ¼ 1.0, J2 ¼ 1.0  104, J3 ¼ 1.0 units/mg cell protein. For the single branched reaction the matrix of flux and concentration control coefficients can be calculated solving Eq. (18) for the net rates: 2 J 3 2 v 31 2 3 C v11 C Jv21 C Jv31 1 0 X1 1 0 0 6 7 6 v 7 7 6 CJ2 CJ2 CJ2 7 6 0 1 X2 7 6 4 0 1 0 5. 6 v1 v2 v3 7 ¼ 4 5 4 5 v J1 J  J 2 X3 0 0 1 CX CX CX J v1

Similarly, the connectivity relationships between the substrate/flux control coefficients and the elasticity coefficients are obtained by post-multiplication by b1L of

395

v2

v3

3

3

(18) Considering the ‘sensitive’ condition and the elasticity coefficients which are derived from Eq. (17), the flux and

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Fig. 6. Singled-branched reaction network (Snoep et al., 2004). The surfaces represent the net reaction rate as function of the corresponding substrates for each of the three reversible reactions v1, v2 and v3, respectively. The grids cut the surfaces along the substrate combinations, where the net reactions equal zero.

concentration control coefficients become: 2 J 3 2 3 C v11 C Jv21 C Jv31 0:4102 0 0:5898 6 7 7 6 CJ2 CJ2 CJ2 7 6 6 v1 v2 v3 7 ¼ 4 0:5251 1:0000 0:5251 5. 4 5 0:5254 0 0:5254 CX CX CX v v v 1

2

3

(19) For ‘saturating’ condition the flux and concentration control coefficients are: 2 J 3 2 3 C v11 C Jv21 C Jv31 0:5087 0 0:4913 6 7 7 6 CJ2 CJ2 CJ2 7 6 0:9992 16:2821 5. 6 v1 v2 v3 7 ¼ 4 16:2826 4 5 16:2907 0:0008 16:2902 CX CX CX v1 v2 v3 (20) Through the estimation of the flux control coefficients, MCA gives a clear guideline as to which enzymes need to be modified, that is, which is the potential site for a targetoriented genetic manipulation, which would exert major control of the system. For example, intuition might advise that for an augment in the flux J2, the amount of enzyme for reaction v2 should be incremented. That might be true in case of the ‘sensitive’ condition where C Jv22 ð¼ 1:0000Þ is the biggest flux control coefficient. However, in case of the ‘saturating’ condition, an increase of the amount of enzyme for reaction v1 can be deduced, since C Jv12 ð¼ 16:2826Þ is the

biggest coefficient, immediately pointing to the enzyme of choice for manipulation. Fig. 7 depicts the results obtained after fitting a Takagi–Sugeno model to the Eq. (17) (v1). A total of four trapezoidal membership functions were used to model the reversible Michaelis–Menten kinetics. Fig. 8 depicts the four membership functions used to fit the model. Similar approximations were used for the rest of the variables (data not shown). Using the fuzzy-logic approaches for all vi, the Takagi–Sugeno approximation gives for the flux and concentration control coefficients under ‘sensible’ condition: 2

C Jv11

6 6 CJ2 6 v1 4 CX v1

C Jv21 C Jv22 CX v2

C Jv31

3Fuzzy

7 C Jv32 7 7 5 X C v3

2

0:4050

6 ¼ 4 0:4912 0:5002

0 1:0000 0

0:5950

3

7 0:4912 5. 0:5002 (21)

And finally for the ‘saturating’ conditions: 2 J 3Fuzzy 2 C v11 C Jv21 C Jv31 0:4897 0 6 7 6 6 CJ2 CJ2 CJ2 7 ¼ 4 14:7841 1:0000 6 v1 v2 v3 7 4 5 X X X 15:0621 0 C v1 C v2 C v3

0:5103

3

7 14:7841 5. 15:0621 (22)

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Fig. 7. Two different angles of the Takagi-Sugeno approximation for a reversible reaction rate. Continuous surface corresponds to the approximation, while the mesh represents the original Michaelis–Menten model.

Fig. 8. Trapezoidal membership functions used for the Takagi–Sugeno depicted in Fig. 7.

Under ‘saturating’ conditions v2,net and v3,net can be easily modelled, since they form plateaus with constant values where bi-lineal approximations are quite accurate (data not shown). As can be asserted, the flux and concentration control coefficients obtained with the fuzzy-logic approach are close to those obtained with classical algebraic approaches and can be as well interpreted. 5. Conclusion A methodology for metabolic control analysis was developed based on fuzzy-logic techniques. Its functioning is supported by inference mechanisms employing basic understanding of the biochemical system behaviour. Such a technique allows the explicit incorporation of valuable qualitative knowledge. This characteristic is crucial, particularly when handling data estimated from in vivo kinetics studies, with which the authors have been working to date. Since this experimental information is scarce and contain measurement errors, data-driven techniques, as presented in this work, represent a viable alternative to established kinetics approaches. However, data-driven techniques must still be regarded as an intermediary step between the evolution from partial empirical to full mechanistic modelling. Although the approximation of the enzyme kinetics is restricted to linear

relationships, it is shown how the fuzzy-logic approaches are able to capture the non-linear features of the system through their membership functions regardless of if these are linear or not. Such a conceptually simple approach—besides being of general application—reduces the complexity of the kinetic description and can be further refined using standard optimization algorithms. Since the fuzzy-logic models use linear approximations, it has been possible to further demonstrate how the cumbersome calculation of elasticity coefficients can be highly simplified. Acknowledgment The authors of this work would like to thank Mrs. AnneMarie Fleury for her advice in the language correction of this manuscript. Appendix A A.1. Derivation of Eq. (4c) Considering Eq. (4a): P  P  fAn ðS1 Þan fAn ðS1 Þbn þ n n v ¼ P S1 þ P . n fAn ðS 1 Þ n fAn ðS 1 Þ

(A.1)

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A.3. Example for the calculation of v2 (Eq. (20)) using a fuzzy-logic model

Expanding for 3 membership functions (n ¼ 3):   fA1 a1 þ fA2 a2 þ fA3 a3 þ v ¼ S1 f þ fA2 þ fA3  A1  fA1 b1 þ fA2 b2 þ fA3 b3 þ . fA1 þ fA2 þ fA3

ðA:2Þ

Rearranging in vector notation: 20 1 0 131 0 a1 b1 Ba C B b C7C Bf þ f þ f 6 B A1 A2 A3 4@ 2 AS 1 þ @ 2 A5C C B a3 b3 C B þ C: B v ¼B C f þ f þ f A1 A2 A3 C B C B A @

( m2 ðSÞ ¼

And with aT ¼ [a1 a2 a3]T and bT ¼ [b1 b2 b3]T Eq. (4c) is derived. A.2. Unscaled coefficients and their transformation to scaled coefficients Besides considering the vector of substrates, S, and the vector of reactions v, a vector of parameters, p, is contemplated to define the unscaled control coefficients, where pj is a specific parameter of the step j with a reaction rate of vj. The concentration control coefficient is given by:

(A.5)

where Ji is the flux i, defined by Ji ¼ vi at steady-state conditions. The normalized substrate/flux control coefficients, C0 S and C0 J can be obtained from S

J

C 0 ij ¼ C Jij

vj , Si

(A.6)

vj vi

(A.7)

and the fuzzy logic-based elasticity matrix: 0 qv

1 S1 qS1 v1 B

::: E ¼B @ qv S 0

J

1

qS1 vJ

(A.10)

vþ 1;1 ðSÞ ¼ 3:2S,

(A.11)

vþ 1;2 ðSÞ ¼ 1:65S þ 0:70.

(A.12)

For the backward reaction: ( 1  X =0:45 if X o0:45; m1 ðX Þ ¼ 0 if X X0:45; ( m2 ðX Þ ¼

X =0:45

if So0:45;

1

if SX0:45

(A.13)

(A.14)

with the associated local linear models: v 1;1 ðX Þ ¼ 13S,

(A.15)

v 1;2 ðSÞ ¼ 2:1S þ 5.

(A.16)

A.4. Parameters used for the kinetic model of Eq. (20)

qJ i =qpj , qvj =qpj

C 0 ij ¼ C Sij

if SX0:5

1

(A.4)

and the flux control coefficient by: C Jij ¼

S=0:5 if So0:5;

with the associated local linear models: (A.3)

qS i =qpj C Sij ¼ qvj =qpj

To approximate v2 of Eq. (20), four trapezoidal membership functions, two per independent variable (S, X) were used. For the forward reaction ( 1  S=0:5 if So0:5; m1 ðSÞ ¼ (A.9) 0 if SX0:5;

::: :::

qv1 S I qS I v1

1

0

C B ::: C ¼ B A @ qv S J

I

qS I vJ

b1;11 S1 b1;11 S1 þb0;1

:::

::: b1;1J S1 b1;1J S1 þb0;J

References b1;I1 S I b1;I1 S I þb0;1

::: :::

The following parameters values were considered for the simulations. For ‘sensitive’ conditions: s ¼ 1; p1 ¼ 0; p2 ¼ 0; Vmax1 ¼ Vmax3 ¼ 4.5606; Vmax2 ¼ 0.1; K1s ¼ 1; K1x ¼ K3x ¼ 0.178; K2x ¼ 100; K2p ¼ K3p ¼ 1; Keq1 ¼ 0.1; Keq3 ¼ 1000; Keq2 ¼ 10. For ’saturating’ conditions the parameter values used were: s ¼ 10; p1 ¼ 0; p2 ¼ 0; Vmax1 ¼ Vmax3 ¼ 1.03221; Vmax2 ¼ 0.1; K1s ¼ 0.01; K1x ¼ K3x ¼ 0.0016; K2x ¼ 100; K2p ¼ K3p ¼ 1; Keq1 ¼ 1000; Keq3 ¼ 1000; Keq2 ¼ 10; metabolite concentrations in mM, binding constant for substrates and products in mM and Vmax values in units/mg cell protein. Steadystate concentration of X under the two conditions was 0.05 mM and the flux values: J1 ¼ 1.00, J2 ¼ 1.00  10–4, J3 ¼ 1.00 units mg–1 cell protein (Snoep et al., 2004).

b1;IJ S I b1;IJ S I þb0;J

1 C C. A

(A.8)

Biewer, B., 1997. Fuzzy-Methoden. Praxisrelevante Rechenmodelle und Fuzzy-Programmiersprachen. Springer, Berlin. Bremermann, H.J., Anderson, R.W., 1989. An Alternative to Backpropagation: A Simple Rule for Synaptic Modification for Neural Net Training and Memory. Internal Report, Department of Mathematics, University of California, Berkley.

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