Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol163.Issue2.Feb2000:

Synergism analysis of biochemical systems. II. Tensor

formulation and treatment of stoichiometric constraints

Armindo Salvador

a,b,*

a

Grupo de Bioquõmica e Biologia Teoricas, Instituto de Investigacßa~o Cientõ®ca Bento da Rocha Cabral, Ccß. Bento da Rocha Cabral, 14 P-1250 Lisbon, Portugal

b

Department of Microbiology and Immunology, The University of Michigan, 5641 Medical Sciences Building II, Ann Arbor, MI 48109-0620, USA

Received 26 March 1999; received in revised form 18 October 1999; accepted 18 October 1999

Abstract

The previous paper outlined a conceptual and mathematical framework for synergism analysis of kinetic models. Though the formalism presented there is adequate for studying simple models, the analysis of large-scale models bene®ts from the more e€ective formulation achieved in this work. The present for-mulation is based on simple tensor operations and takes advantage of the analogy between the formalisms for synergism and log-synergism analysis presented before. Well-known relationships of ®rst-order sensi-tivity analysis and new relationships for (log-)synergism coecients of various steady-state properties are cast in the new formal setting. The formalism is then extended to models that are subject to constraints between variables, ¯uxes and/or parameters. This treatment, which generalises Reder_Õs concept of link matrices, is applied to networks that include moiety conservation cycles [C. Reder, Metabolic control theory: a structural approach, J. Theor. Biol. 135 (1988) 175]. It is also used to take advantage of ¯ux conservation at steady-state to simplify synergism analysis. Issues of numerical e€ectiveness are brie¯y discussed, and the theory illustrated with the study of synergistic behaviour in the metabolism of reactive oxygen species and of a scheme of dynamic channelling. Ó _{2000 Elsevier Science Inc. All rights reserved.}

Keywords:Bioinformatics; BST; Kinetic models; Metabolic networks; Sensitivity analysis; System analysis

1. Introduction

The need to integrate large collections of kinetic data and the availability of powerful com-putational resources led to a resurgent interest in large-scale integrative models of metabolic

*_{Tel.: +1-734 763 5558; fax: +1-734 764 3562.}

E-mail address:a.salvador@mail.telepac.pt (A. Salvador).

processes [1±5]. Owing to mathematical complexity, closed-form analytical steady-state solu-tions of these models are seldom available, while extensive numerical simulasolu-tions are impractical for analysing responses to large sets of parameters. Di€erential sensitivity analysis [6±9] ± as conveyed to biochemistry by various formalisms [10±15] ± usually becomes the method of choice in these circumstances, owing to various factors. First, the concept of sensitivity is adequate for describing the control properties of metabolic networks [16], and the values of the sensitivities can be used as indexes of metabolic performance [17] or model robustness [18]. Second, the sensitivities condense this information into scalar quantities, making the characterisation of the steady-state behaviour of whole systems more intelligible. Third, computer implementation of di€erential sensitivity analysis is relatively straightforward, and even detailed analyses of large-scale models require modest computational resources by present standards.

The formalism presented in the previous paper [19] extends the ®rst two features (physical meaning and condensed information) to multifactorial analyses, and re®nes the characteri-sation of responses to single-parameter perturbations. Ease of implementation and compu-tational economy drove the gradual replacement of earlier scalar formulations [10±15] of sensitivity analysis of biochemical systems by matrix formulations [11,20±24]. The ensuing gains in formal conciseness and numerical e€ectiveness argue for similar improvements of the methodology for synergism analysis [19]. Yet, while the ®rst-order sensitivities of a set of properties to a set of parameters are logically accommodated in a matrix, the (log-)syner-gism coecients for the response of a set of properties to all single and dual perturbations of a set of parameters should be accommodated in a rank-3 tensor. The general results of [19] are thus recast in a tensor formulation (Section 3), and formulas for non-normalised, relative- and log-synergism coecients of various properties are derived (Appendix B). All the relevant expressions make use of only the simplest tensor operations, which are ex-plained in Appendix A. The theory is illustrated by the analysis of a simple model of the metabolism of reactive oxygen species in the mitochondrial matrix of rat hepatocytes (Section 3.1).

The calculation of systemic generalised-synergism coecients of systems that include moiety conservation cycles requires a special procedure. The concept of link matrix [25] permitted overcoming the corresponding problem in ®rst-order sensitivity analysis of metabolic networks. Upon multiplication of a vector of a suitably chosen subset of `independent' sensitivities of a system, link matrices yield a vector of sensitivities that can be expressed as linear combinations of the former non-normalised [25] or logarithmic [26] sensitivities. Section 4.1 extends this concept to synergism analysis and to general constraints between variables, ¯uxes and/or parameters that may apply to biochemical systems. This general framework replaces the concept of link matrix by that of link tensor, with a similar meaning but applying also to synergism analysis. The technique is then applied (a) to synergism analysis of systems that include moiety conservation cycles (Section 4.2.1) and (b) to simplifying synergism analysis of ¯uxes by accounting for ¯ux con-servation at steady-state (Section 4.2.2).

2. Notation and conventions

The notation and conventions introduced in the previous papers are adopted here, with ex-tensions as described below.

The term `generalised-synergism' will be used to denote a measure of deviation from a pre-de®ned pattern of behaviour. This includes non-normalised synergisms, relative synergisms and log-synergisms [19]. The di€erential measures of steepness of deviation from the prede®ned pat-terns will be denoted by `generalised-synergism coecients'. To simplify language, where there is no ambiguity the term `sensitivities' will be used to refer to both ®rst-order sensitivities and generalised-synergism coecients (second-order sensitivities).

It seems natural to represent the generalised-synergisms of all single- and dual-parameter perturbations with respect to a vector P(k_{) of nP functions of a vector} k _{of p parameters as a}

nP´p´p tensor. For instance,

W_Pk_o;Dk_{† ˆ fWP}

i…ko;Dk…j;k††g; i2 f1;2;. . .;nPg; j;k 2 f1;2;. . .;pg:

When the generalised-synergism coecients are similarly arranged, the calculation of the systemic coecients requires only simple tensor operations. These operations and the corresponding no-tation are explained in Appendix A. Additional nono-tation is:Ik, the k´_{kidentity matrix; 0ijk,}

thei´_j´_{knull tensor; ker(A), the kernel of matrixA;x}n_{, diag(x), the diagonal matrix whose main}

diagonal is formed by the elements of x. To avoid ambiguity, one must distinguish matrix-rank (the number of linearly independent rows of a matrix) from tensor-rank (the number of indexes necessary to specify an element of a tensor). This notation casts Eq. (10) of [19] in tensor form as

W_Pk_o;Dk_† 1

2 W

‰2Š_P;k;k†:

2 D

k

Dk_;

similar expressions apply for the other generalised-synergisms. IfP, k_{A and} k_{B are vectors, the following relationships hold:}

S…P;k_{† ˆP}nÿ1_J…P;k_† kn_{; …1†}

W‰2Š…P;k_A;k_{B† ˆP}nÿ1 _J‰2Š _P;k_A;k_{B†_}

2k

n

A

kn

B; …2†

S‰2Š…P;k_A;k_{B† ˆJ}‰2Š_{logjPj; logj}k_{Aj; logj}k_{Bj†; …3†}

S‰2Š…P;k_A;k_{B† ˆW}‰2Š_P;k_A;k_{B† ÿ I}

nP2_S…P;kA†

S…P;k_{B† ‡S…P;}k_{A† I}

nAS…kA;kB†; …4†

withnP and nA standing for the dimensions of the vectors Pand kA, respectively.

The J and S operators also follow the chain rule when operating over composite functions. That is, ifFº_F(P(k),k) andF and Pare both di€erentiable, then

J…F;k_{† ˆJ}k…F;P† J…P;k† ‡JP…F;k† …5†

and

This shared property between J and S allows developing the formalisms for the Cartesian and logarithmic spaces in parallel. To avoid redundancy, the symbol O is used for the operator in relationships that are formally similar for various sensitivity operators. Unless otherwise stated, in such relationships O can be J, S or W. For ®rst-order sensitivities, the operators W (relative sensitivities) and S (logarithmic sensitivities) have the same meaning, but this identity does not hold for generalised-synergism coecients [19].

The symbolsV‡ andUstand for aggregated rates of production of a metabolite and turnover

numbers, respectively.

3. The tensor formulation of synergism analysis

To clarify the relationships of the tensor formulation with the previous formulation [19], and to examine the mathematical assumptions upon which the formalism relies, the basic formulas for synergism analysis are re-derived in tensor form.

An autonomous kinetic model can be expressed as

_

XˆNv…X0;X;k†; …7†

with N the stoichiometric matrix and X0 the initial concentrations of the internal species. The

¯uxes, v, may have various meanings. Each process could be an aggregate of all reactions pro-ducing or consuming each internal species, an enzyme-catalysed transformation, an elementary reaction, etc. Though the initial conditions,X0, determine the steady-state to which the system will

converge if (7) allows coexisting stable states, they lack any other in¯uence on the steady-state. So, if (a) the kinetic model is structurally stable, (b) the operating point is not poised at a bifurcation, and (c)v is di€erentiable around the operating point, one obtains

J…X_;k_{† ˆN …J}k…v;X† J…X;k† ‡JX…v;k†† ˆ0; …8†

or, by normalising the sensitivities,

Nvn …Sk…v;X† S…X;k† ‡SX…v;k†† ˆ0: …9†

If (d) the matrix NJk…v;X† is non-singular, one can solve (8) for the matrix of systemic

sensi-tivities:

J…X;k_{† ˆ}^J…X;v† JX…v;k† …10†

with

^

J…X;v† ˆ ÿ…NJk…v;X††ÿ

1

N: …11†

Furthermore, if (d0_{) the matrix Nv}n_S

k…v;X† is non-singular, then solving (9) for the systemic

logarithmic sensitivities gives

S…X;k_{† ˆS}^_{X;v† SX…v;}k_{† …12†}

with

^

S…X;v† ˆ ÿ NvnSk…v;X†

ÿ ÿ1

We call systemic sensitivities of the type O^…P;v†, which cannot be obtained through explicit di€erentiation of the steady-state solution, `implicit sensitivities', to distinguish them from the systemic parameter sensitivities,O…P;k_†.

If the model includes moiety conservation cycles, conditions (d) and (d0_{) are violated, making}

the treatment presented in Section 4.2.1 necessary. To compute ®rst-order sensitivities of prop-erties that are functions of concentrations (derived propprop-erties) one applies expressions (5) and (6). Table 1 lists various such relationships. There is a direct correspondence between these and similar relationships in biochemical systems theory [30], metabolic control analysis [22±24] and/or ¯ux-oriented theory [21].

If v is twice-di€erentiable around the operating point and k_{A and} k_{B are two vectors of}

pa-rameters, the application of chain rule (5) to Eq. (8) gives

N ‰…J‰_:;:2Š…v;X;X†2_J…X;kA††J…X;kB† ‡J:;:‰2Š…v;X;kA†

T1;3;2 _J…X;k B†

‡J‰_:;:2Š…v;X;k_{B†_}

2J…X;kA† ‡JkA;kB…v;X†:J ‰2Š_X;k

A;kB† ‡J‰

2Š

:;:…v;kA;kB†Š ˆ0: …14†

The solutions of (14) and of its counterpart for logarithmic co-ordinates share the form

O‰2Š…X;k_A;k_{B† ˆO}^_{X;v† …O}~‰2Š

‰XŠ;‰XŠ…v;kA;kB† ‡O~

‰2Š

‰v;XŠ;‰v;XŠ…v;kA;kB††: …15† The subscripts of O~‰_‰2_XŠ_Š;‰XŠ and O~‰_‰2_v;Š_XŠ;‰v;XŠ indicate variables that are allowed ®rst-order variations (linear, forJ, or power-law for W and S) with the parameters. So

~

O‰_‰2_XŠ_Š;‰XŠ…v;k_A;k_{B† ˆ …O}‰2Š

…k_†;k_†…v;X;X†2_O…X;kA†† O…X;kB† ‡O‰

2Š

…k_†;X†…v;X;kA†T1;3;2 O…X;kB†

‡O‰2kŠ†;…X†…v;X;kB†2_O…X;kA† ‡O

‰2Š

…X†;…X†…v;kA;kB† …16†

represents the generalised-synergism coecients that would be observed for rates of reaction if the concentrations responded linearly (forJ) or as a power-law (for W and S) to single-parameter perturbations. These magnitudes are considered as semi-intrinsic generalised-synergism coe-cients, which take the®rst-ordersystemic responses of the concentrations into account but assume that the deviations of each ¯ux from the ®rst-order approximations do not interact.

The term O~‰_‰2_v;Š_XŠ;‰v;XŠ…v;k_A;k_{B† is null whenOstands forJ or W, or takes the value} ~

S‰_‰2_v;Š_XŠ;‰v;XŠ…v;k_A;k_{B† ˆ I}

r2_S…v;kA†

ÿ

S…v;k_{B† ÿS…v;}k_{A† I}

pAS…

k_A;k_{B†: …17†}

Table 1

Systemic ®rst-order sensitivities of derived steady-state propertiesa

Cartesian space Logarithmic space

^

O…v;v† ˆOk…v;X† O^…X;v† ‡I ^

J…V‡;v† ˆN‡J^…v;v† S^…V‡;v† ˆSX;k…V‡;v† ^S…v;v†

ˆVnÿ_‡1N‡vn^S…v;v†

^

J…U;v† ˆXnÿ1 J^…V‡;v† ÿUn^J…X;v†

^

S…U;v† ˆS^…V‡;v† ÿS^…X;v†

O…v;k† ˆO^…v;v† OX…v;k†

J(V‡,k)ˆN‡áJ(v,k) S…V‡;k† ˆSX;k…V‡;v† S…v;k†

ˆVnÿ1

‡ N‡vnS…v;k†

J…U;k_{† ˆ}Xnÿ1

J…V‡;k† ÿUnJ…X;k†

ÿ

S…U;k_{† ˆ}S…V‡;k† ÿS…X;k†

a

The elements of O~‰_‰2_v;Š_XŠ;‰v;XŠ…v;k_A;k_{B† are also semi-intrinsic generalised-synergism coecients. As} ^

S…X;v† S…v;k_{A† ˆ}0, for log-synergism coecients (15) simpli®es to

S‰2Š…X;k_A;k_{B† ˆS}^_{X;v† bS}~‰2Š

‰XŠ;‰XŠ…v;kA;kB† ‡ Ir2_S…v;kA†

ÿ

S…v;k_{B†c: …18†}

Systemic generalised-synergism coecients for ¯uxes, turnover numbers and other derived steady-state properties, PP…X…k_†;k_†, can be computed from the systemic concentration generalised-synergism coecients through

O‰2Š…P;k_A;k_{B† ˆ …O}‰2Š

…k_†;k_†…P;X;X†2_…O;X;kA†† O…X;kB† ‡O

‰2Š

…k_†;X†…P;X;kA†

T1;3;2_O…X;k B†

‡Ok…P;X† O‰2Š…X;kA;kB† ‡O

‰2Š

…k_†;X†…P;X;kB†2_O…X;kA†

‡O‰2_XŠ_†;…X†…P;k_A;k_{B†: …19†}

By consideringk_{a vector ofrreaction-speci®c parameters such thatO:…v;}k_{† ˆI, one can use (15)}

to compute intrinsic [O^‰2Š…X;v;v†] and mixed [intrinsic-parameter, O^‰2Š…X;v;k_†] generalised-syn-ergism coecients. Appendix B presents the derivation of systemic, intrinsic and mixed genera-lised-synergism coecients for various steady-state properties.

3.1. Example

Below the tensor formulation is applied to the synergism analysis of a simple model of the metabolism of reactive oxygen species in the mitochondrial matrix of rat hepatocytes (Fig. 1). Superoxide (Oáÿ

2 ) is produced as subproduct of the respiratory chain (reaction 1), and its

dis-mutation ± quickly catalysed by superoxide dismutase (SOD, reaction 2) ± produces hydrogen peroxide (H2O2). Though the latter is mostly reduced to water through a

glutathione-peroxidase-catalysed reaction (reaction 4, GPx), a small fraction reacts with Fe2‡ _{(reaction 3), producing the}

extremely reactive hydroxyl radical (HOá_{), which damages proteins and DNA (reaction 6).}

Su-peroxide may also damage proteins and DNA ± though with much lower rate constants than HOá

± and, once protonated, reacts also with unsaturated lipids in bilayers. The estimation of the parameters (Table 2) is discussed in Ref. [5]. The present model takes X1 ˆ

‰Oáÿ

2 Š; X2ˆ ‰H2O2Š and X3ˆ ‰HOአas dependent variables, and X4ˆ ‰O2Š; X5 ˆ ‰SODŠ; X6 ˆ

‰Fe2‡_{Š and X}

7 ˆ ‰GPxŠ as parameters of interest (vectork). The rate vd ˆv5‡v6ˆk5X1‡k6X3 is

a coarse index of mitochondrial damage caused by superoxide generation. We are interested in evaluating the type of interaction and degree of synergism and/or antagonism between the various

antioxidant defences and pro-oxidant agents as regards vd. For that we compute the log- and relative-synergism coecients of vd, through the steps described below:

1. Compute the steady-state concentrations and rates. This givesX1 ˆ3:110ÿ10M; X2 ˆ1:5

10ÿ7_{M; X}

3 ˆ9:910ÿ20M; v1 ˆ1:610ÿ5M sÿ1; v2 ˆ8:010ÿ6M sÿ1; v3 ˆ3:110ÿ10

M sÿ1; v4 ˆ8:010ÿ6M sÿ1; v5 ˆ1:810ÿ9M sÿ1; v6ˆ3:110ÿ10M sÿ1.

2. Compute the tensors of intrinsic sensitivities and log-synergism coecients by di€erentiating the rate expressions according to the de®nitions of the respective operators [19] and replacing the operating values. This yields

S‰2_XŠ_†;…X†…v;k_;k_{† ˆ}0644:

3. Compute the implicit sensitivities, using (13):

S…X;v† ˆ

1:0 ÿ1:0 0 0 ÿ1:110ÿ4 ₀

1:0 1:210ÿ4 _ÿ3:910ÿ5 _{ÿ1:0 ÿ1:210}ÿ4 ₀

1:0 1:210ÿ4 _{0 ÿ1:0 ÿ1:210}ÿ4 _ÿ1:0

2

4

3

5:

4. Compute the systemic ®rst-order sensitivities (log gains) of concentrations and ¯uxes, using (12) and the expression for S(v, k) in Table 1:

S…X;k_{† ˆ}

1:0 ÿ1:0 0 0

1:0 1:210ÿ4 _ÿ3:910ÿ5 _ÿ1:0

1:0 1:210ÿ4 _{1:0 ÿ1:0}

2

4

3

5;

Table 2

Rate expressions and parameters considered in the example (see Fig. 1) for identi®cation of the reactions. Parameter estimations are discussed in [5]

Rate expression Parameters

v1ˆk1‰O2Š k1ˆ0:16sÿ1; ‰O2Š ˆ1:010ÿ4M (in lipid bilayers)

v2ˆk2‰SODŠ‰O2áÿŠ k2ˆ4:7109Mÿ1sÿ1; ‰SODŠ ˆ1:110ÿ5M

v3ˆk3‰Fe2‡Š‰H2O2Š k3ˆ2:0104Mÿ1sÿ1; ‰Fe2‡Š ˆ1:010ÿ7M

v4ˆ ‰

GPxŠ

…/1=‰H2O2Š†‡…/2=‰GSHŠ† /1ˆ4:810

ÿ8_{M s; /}

2ˆ2:510ÿ5M s; ‰GPxŠ ˆ2:510ÿ6M; ‰GSHŠ ˆ1:110ÿ2M

v5ˆk5‰O2áÿŠ k5ˆ5:9 sÿ1

6. Compute the systemic log-synergism coecients, through (18), which yields

To compute the systemic relative-synergism coecients one may just apply expression (4):

W‰2Š…X;k_;k_{† ˆ}

7. Compute the logarithmic gains of the rate of damage, through (6):

S…vd;k† ˆ ‰1:0 ÿ0:86 0:14 ÿ0:14Š:

8. Finally, by applying expression (19) obtain the log-synergism coecients of the total rate of damage (vd) and ± again through (4) ± the corresponding relative-synergism coecients:

S‰2Š…vd;k;k† ˆ

antagonisms (negative relative-synergism coecients) between pro-oxidants and antioxidant de-fences should hold. The strongest antagonism predicted is between oxygen and SOD

‰W‰2Š…vd;X4;X5† ˆ ÿ0:86Š and the strongest synergism between SOD and GPx

‰W‰2Š…vd;X5;X7† ˆ0:25Š. Second, combined changes of oxygen concentration and any of the other

factors should yield nearly multiplicative responses (log-synergism coecients near 0). Joint perturbations of [SOD] and [GPx] yield a slightly supra-multiplicative response

‰S‰2Š…vd;X5;X7† ˆ0:12Š, whereas perturbations of [Fe2‡] combined with [SOD] or [GPx] yield

slightly sub-multiplicative responses. Third, vd is nearly proportional to [O2] (unit log gain and

both log- and relative-synergism coecients near 0) and varies almost linearly with [Fe2‡_]

(rela-tive-synergism coecients near 0). The response to [SOD] is clearly supra-linear

‰W‰2Š…vd;X5;X5† ˆ1:7Š and slightly supra-power-law ‰S‰2Š…vd;X5;X5† ˆ0:12Š, indicating

diminish-ing returns of increasdiminish-ing the concentration of this enzyme. The same holds for the response to [GPx].

4. Treatment of models subject to constraints

4.1. Conversion between sensitivities of transformed and non-transformed models ± the general case

4.1.1. Transformations of concentrations and parameters

Consider a metabolic process described by the kinetic model X_ ˆNv…X;k_†. In some cases ± e.g., in presence of moiety conservation or micro-reversibility constraints ± it is desirable to describe the behaviour of the system in terms of transformed vectors of variables (X_N) and pa-rameters (k_{N) instead of the `natural' concentrations and parameters (Xand} k_{). Usually, it is not}

too restrictive to assume that the transformation ofX…k_†intoXN…kN†is such thatXandkcan be expressed as double-di€erentiable functions of XN and kN, respectively: kk…kN†; XX…XN…kN…k††;kN…k††. The chain rule of di€erentiation then gives

O…X;k_{† ˆ}XX

OO…XN;kN† ‡X k

O† PO; …20†

O…X;k_{N† ˆ}XX

OO…XN;kN† ‡X k

O; …21†

O‰2Š…X;k_;k_{† ˆ}XX;X

O‰2Š2_O…XN;kN†† O…XN;kN†

n

‡XX;k

O‰2Š2_O…XN;kN†

‡XX

OO

‰2Š_X

N;kN;kN† ‡XX;k

T_1;3;2

O‰2Š O…XN;kN† ‡Xk_O;‰k2ŠŠ2_PO

o

P_O

‡ XX

OO…XN;kN†

ÿ

‡Xk

O

P

O‰2Š; …22†

O‰2Š…X;k_N;k_{N† ˆ}XX;X

O‰2Š2_ O…XN;kN†† O…XN;kN† ‡ …XX_O;‰2kŠ2_ O…XN;kN† ‡XXO

O‰2Š…XN;kN;kN† ‡ …XX;

kT1;3;2

where

call the latter rank-2 and rank-3 tensors (X:

:, P:, etc.) `link tensors', by similarity to RederÕs [25]

link matrices.

By considering vectors of reaction-speci®c parameters to which the rates of reaction are pro-portional one can derive formulas for the conversion of implicit and mixed sensitivities:

^

4.1.2. Transformations of ¯uxes

If a transformation changes the set of rates of reaction, one can write the modi®ed kinetic model as

with the ¯ux link tensors de®ned as

Vv

The following expressions allow calculating the sensitivities for thetransformedset of parameters:

By replacing (20), (22), (29) and (30) into the pertinent expressions in Section 3 and Appendix B.2, one obtains relationships between systemic sensitivities of derived magnitudes of the original system and systemic sensitivities of the transformed system.

4.2. Applications

4.2.1. Treatment of structural conservation relationships

4.2.1.1. Derivation of the link tensors. If the stoichiometric matrixNhasnrows and rankn0 <n,

then there arenÿn0 independent structural conservation relationships, which are time-invariant

linear combinations of dynamical concentrations. Identifying each of these linear combinations with a new parameter, one can expressnÿn0 variables (sub-vector XE, below) as function of the latter parameters and of the other variables involved in the conservation relationships. So, once the variables in XE are substituted into the kinetic model, the subset ofn0 di€erential equations

corresponding to the remaining n0 dynamical variables (vector XN) is sucient to describe the kinetics of the system. The systemic sensitivities of the latter variables can be found by applying the approach in Section 3 to the transformed system, and the sensitivities of the original variables are then computed through (21) and (23), by using the link tensors derived in Appendix C

…34†

and

…35†

forO standing forJ or W; or

S‰2Š…X;k_N;k_{N† ˆ}XX

S S

‰2Š_X

N;kN;kN† ‡ …X_SX‰;2XŠ2_ S…XN;kN† S…XN;kN† ‡XX;

k

S‰2Š2_ S…XN;kN†

‡XX;k

T_1;3;2

S‰2Š S…XN;kN† ‡Xk_S;‰k2Š: …36†

This procedure extends the synergism analysis described in [19] to systems subject to structural conservation relationships.

For computing the systemic sensitivitiesO…XN;kN†and O‰2Š…XN;kN;kN†, note that the matrices NNJk_N…v;XN† and NN vnSk_N…v;XN† are non-singular. (NN is formed by then0 rows ofNthat

correspond to the variables inXN.) Hence,

^

and

^

S…XN;v† ˆ ÿ…NNvnSkN…v;XN††

ÿ1

NNvn …38†

with

OkN…v;XN† ˆOk…v;X† X

X

O: …39†

The intrinsic sensitivities of the ¯uxes to the transformed set of parameters are also straightfor-wardly obtained from the corresponding sensitivities to the original parameters

…40†

In turn, the intrinsic second-order sensitivities for the original system and those of the trans-formed system can be related through

O‰2kŠN†;…kN†…v;XN;XN† ˆOk…v;X† X

X;X O‰2Š ‡ …O

‰2Š

…k†;…k†…v;X;X†2_ X

X O† X

X

O; …41†

O‰2kŠ_N†;XN†…v;XN;

k_{N† ˆOk…v;X†} XX;k

O‰2Š ‡ …O ‰2Š

…k_†;XN;k†…v;X;XE†2_X

X O†X~

k

OO…kA;kN†

‡ …O‰2kŠ_†;X†…v;X;k†2_X

X O† P

(

; …42†

and

O‰_X2Š_N;XN…v;k_N;k_{N† ˆOX}

N;k…v;XE† …X~

k;k

O‰2Š2_O…kA;kN†† O…kA;kN†

‡ ‰O‰2_XŠ

N;k†;…XN;k†…v;XE;XE†2_ …X~

k

OO…kA;kN††Š X~kOO…kA;kN†

‡ ‰O‰2_XŠ

N;k†;…X†…v;XE; k_{†_}

2…X~

k

OO…kA;kN††Š P

(

‡ …O‰2_XŠ_N;k†;…X†…v;XE;k†

T1;3;2

_

2P

(

† X~k

OO…kA;kN†

‡ …O‰2_XŠ_†;…X†…v;k_;k_{†_}

2P

(

† P(_:43†

Here, . All the intrinsic sensitivities of the transformed system might as well be obtained through di€erentiation of the rate expressions at the operating point, after replacing the eliminated variables. The systemic sensitivities of the transformed system follow from applying expression (15) or (18) to (41)±(43).

As the best algorithms for inversion of annnmatrix are of ordern3 _{[27], the computational}

cost of the calculation of the systemic ®rst-order sensitivities for large-scale models is dominated by the matrix inversion. Cascante et al. [26] presented a strategy that requires the inversion of an

rr matrix for obtaining simultaneously the ¯ux control coecients and the concentration control coecients of a system. Yet, by calculating the systemic ¯ux sensitivities through O…v;k_{N† ˆO}k…v;X†:O…X;k_N† ‡OXN…v;kN†, one obtains the same result by inverting a smaller

(n0n0) matrix. Any kinetic model with a non-trivial steady-state hasr>nPn0. If all processes

much higher than 2n. The computational cost of the two alternative strategies for computing the systemic sensitivities may then di€er by orders of magnitude.

4.2.1.2. Example. Scheme 1 represents a simple mechanism of dynamic metabolic channelling [28]. X1 stands for the substrate of the pathway,X3 for free intermediate,X6 andX7 for free enzymes,

X2 and X4 forX6:X1 and X7:X3 complexes (respectively), X5 for X6:X3:X7 complex. Reactions 7

and 9 are the reverse of reactions 6 and 8, respectively. All reactions follow mass action kinetics, and a physiologically plausible dimensionless operating point is considered:

k1 ˆX10ˆX11ˆ1; k2 ˆ0:35; k3ˆ3:5; k4ˆ100; k5ˆ2; k6 ˆk7 ˆk9 ˆ104; k8 ˆ105: The

di-mensionless concentrations are then

Xˆ9:5110ÿ1 _4:8310ÿ1 _1:0210ÿ1 _5:0010ÿ1 _8:3110ÿ3 _5:0910ÿ1 _4:9210ÿ1T

:

The dimensionless ¯uxes are

vˆ‰1 0:169 0:831 0:831 1 484: 483: 500:2 500:0ŠT;

so that 49% of the ®rst enzyme and 51% of the second are in the free form, and 83% of the ¯ux is channelled.

To ®nd the systemic sensitivities of the concentrations, one proceeds as follows:

(1) Find the conservation matrix and the conservation relationships, through the procedure in Appendix C or following [29]. As the stoichiometric matrix

Nˆ

1 0 0 0 0 ÿ1 1 0 0

0 ÿ1 ÿ1 0 0 1 ÿ1 0 0

0 1 0 0 0 0 0 ÿ1 1

0 0 0 1 ÿ1 0 0 1 ÿ1

0 0 1 ÿ1 0 0 0 0 0

0 1 0 1 0 ÿ1 1 0 0

0 0 ÿ1 0 1 0 0 ÿ1 1

2

6 6 6 6 6 6 6 6 4

3

7 7 7 7 7 7 7 7 5

hasnˆ7 rows and rank n0 ˆ5, there are two conservation relationships. As follows from

LT _ˆ 0 1 0 0 1 1 0

0 0 0 1 1 0 1

;

they are

X2‡X5‡X6ˆX10; X4‡X5‡X7 ˆX11;

whereX10 andX11 are new parameters accounting for the total concentration of each enzyme.

(2) Choose the concentration variables to eliminate and compute the link tensors accordingly, through the procedure in Appendix C. Eliminating X6 and X7 and considering

k_{N ˆ‰k1 k9 X10 X11Š}T _{it follows from (C.1) and (C.3):}

~ Xk

S ˆ X10

X6 0 0 X11

X7

" #

; …44†

…45†

For the second-order link tensors, (C.6) and (C.10) give

(C.7) gives

…47†

and (C.8) gives

…48†

(3) Compute the systemic (implicit and parameter) ®rst-order sensitivities of the transformed system. Start by the intrinsic ¯ux sensitivities, through (39) and (40); then replaceSkN…v;XN† into

(38), to ®nd

^

S…XN;v† ˆ

3:63 ÿ0:336 ÿ1:61 ÿ0:0436 ÿ1:64 ÿ1 0:988 0 0 1:86 ÿ0:171 ÿ0:829 ÿ0:0140 ÿ0:843 0 0 0 0 2:03 ÿ0:00257 0:00257 ÿ0:0169 ÿ2:01 0 0 ÿ1 1:00

1 0 0 0 ÿ1 0 0 0 0

0:826 ÿ0:169 0:169 ÿ0:997 0:171 0 0 0 0

2

6 6 6 6 4

3

7 7 7 7 5

;

…49†

and apply (12) to computeS…XN;kN†.

…50†

HereS…XN;k† ˆS^…XN;v†, as kis the vector of rate constants.

(5) Compute the systemic log-synergism coecients of the transformed system. As the reactions follow mass action kinetics, the intrinsic log-synergism coecients of the original system are null. So, (41)±(43) simplify to

(6) Compute the systemic log-synergism coecients for the original set of variables, by re-placingS‰2Š…XN;kN;kN† into (23)

…52†

Note the high log-synergism coecients, even as compared to the ®rst-order sensitivities. The synergism coecients (not shown) reach even higher values. Such high log- and relative-synergism coecients indicate strong deviations from both power-law/multiplicative and linear/ additive behaviour [19]. Hence, descriptions of this system that are based on ®rst-order approx-imations either in Cartesian or logarithmic co-ordinates may be very inaccurate and may over-look, for instance, the high synergism between both enzymes.

4.2.2. Steady-state constraints between ¯uxes

4.2.2.1. Derivation of the link tensors. Suppose that the (nr) stoichiometric matrix of a kinetic model has matrix-rankn0 <r. IfNis full rank (n0 ˆn), one can rearrange it as ,

whereNEis a square sub-matrix formed byn0 linearly independent columns (corresponding ton0

eliminated rates of reaction) andNDis formed by the remaining columns. Ifn0 <n, Nshould be

replaced for the matrix NN of the previous section. After the concomitant rearrangement of the vector of reaction rates, one ®nds, at steady-state,

which leads to vEˆV~vJvN, with V~vJ ˆ_ Jk…vE;vN† ˆ ÿNÿE1ND. So, vˆVvJ vN, with

…53†

It also follows that Vk

Oˆ0; V k;k

O‰2Š ˆ0; V

v;k

O‰2Š ˆ0; V

v;v

J‰2Š ˆ0; V

v;v

W‰2Š ˆ0.

The non-trivial logarithmic link tensors are

…54†

with

Because this transformation does not change the variables nor the parameters, theXand Plink tensors forJ, W andS are trivial. So, the relationships between the systemic ¯ux sensitivities of the original system and those of the transformed system simplify to

O…v;k_{† ˆ}Vv

4.2.2.2. Example. Consider the example in Section 4.2.1.2. The stoichiometric matrix of the transformed system is

The analysis ofNN or of Scheme 1 shows that one can express ®ve of the steady-state ¯uxes as a linear combination of the remaining four steady-state ¯uxes. Choosing to eliminate

Hence, according to (54),

This work presents a tensor formulation for synergism analysis of kinetic models. Like matrix formulations for ®rst-order sensitivity analysis [11,20±24,26], the tensor formulation permits a concise formal representation and a more e€ective computation of sets of second-order sensitivities. The tensor operations involved are not more complex than conventional matrix multiplication.

First-order sensitivity analysis is intrinsic to synergism analysis, as the computation of gener-alised-synergism coecients (Eq. (15)) requires knowing the ®rst-order sensitivities. Furthermore, because the formalism of (generalised-)synergism analysis integrates operations in both Cartesian and logarithmic space, it embeds the mathematical structure that underlies current approaches [10±15] to sensitivity analysis of metabolic networks. To avoid combinatorial explosion of new symbols, this article and its companion [19] use a new notation. This notation conveys immediate information about the mathematical meaning of the sensitivities being represented, makes it easier to remember relationships, requires a lower number of di€erent symbols, and is more readily generalisable for higher orders or other coordinate systems.

As synergism analyses of large models are computationally demanding, it is important to optimise the computational strategies. For a model withninternal metabolites,rreactions andp parameters, with n;r;p1 and p>n, straight application of the formulas (15) through (18) would require rnp…n‡3p†multiplications and rnp…n‡3p† sums. However, ifk_Aˆk_{Bthe matrix}

and rnp…n‡2p† sums ± an improvement of about 30%. The nature of the rate expressions may also permit simpli®cations where both log- and relative-synergism coecients are of interest. As

~

of those rate expressions, it is more e€ective to compute ®rst the systemic log-synergism coe-cients through (15) and obtaining then the systemic relative-synergism coecoe-cients by applying (4). Flux and concentration conservation relationships permit additional numerical shortcuts in sensitivity and synergism analysis. For formal simplicity, the link tensors in Section 4 refer to conversions between full sets of transformed and non-transformed variables, ¯uxes or parameters; but often the transformations a€ect only a small subset of variables, ¯uxes and/or parameters. Then it is preferable to deal separately with the subsets of sensitivities that are changed by the transformations, and to use the appropriate linksub-tensors instead of the full tensors. Important gains in numerical e€ectiveness of the treatment of moiety conservation cycles also ensue from applying the procedure in Section 4.2.1.1 instead of that in [26]. Algorithms for generalised-synergism analysis are implemented in the software package PARSYS [32,33].

The tensor formulation was illustrated by the study of interactions of pro-oxidant agents and anti-oxidant defences in a simple model of the metabolism of reactive oxygen species in the mi-tochondrial matrix. This analysis highlighted (a) SODs importance in antagonising the e€ects of oxygen activation and (b) the synergistic e€ect of SOD and GPx activities in decreasing the rate of damage to mitochondrial components by oxygen radicals. The analysis also indicates that al-though increasing the concentration of either SOD or GPx alone decreases the rate of damage, there are diminishing returns from increasing each of these concentrations separately. Despite these conclusions being accessible through examination of the analytical steady-state solution of the model, the generalised-synergism coecients remain convenient indexes of relevance of multifactorial e€ects.

As more data becomes available, the need for large-scale integrative approaches to biochem-istry is growing. Present models with such aims (as for instance [1±5]) include dozens of variables and hundreds of parameters. Methods that provide condensed, physically meaningful, informa-tion about systemic responses to such large sets of parameters will likely contribute towards a more integrated understanding of how biochemical processes are coordinated.

Acknowledgements

The author is grateful to Dr Albert Sorribas for helpful discussions and to Rui Alves for critical review of the manuscript. PRAXIS-XXI grants BD/3457/94 and BPD/11763/97 are gratefully acknowledged. FCT contributed to support GBBT through `Fundo de Apoio a Comunidade Cientõ®ca'.

Appendix A. Tensors

Tensors generalise the concept of vectors (rank-1 tensors) and matrices (rank-2 tensors). A rank-3 tensor can be considered as a vector of matrices, a rank-4 tensor as a matrix of matrices, etc. So, for instance, a 222 tensor,A, can be written as

Aˆ fai;j;kg ˆ

a1;1;1 a1;1;2

a1;2;1 a1;2;2

a2;1;1 a2;1;2

a2;2;1 a2;2;2

8

> > <

> > :

9

> > =

> > ;

Transposing a tensor corresponds to exchanging its indexes. A rank-ntensor can be transposed inn! ways, which are distinguished by the indexes of the transposition symbol:TTi₁;i₂;...;in _{is the tensor}

ob-tained by turning the ®rst index ofTinto thei1th index, the second index into thei2th, etc. For instance

AT1;3;2 _ˆ

Contracted products are the generalisation of matrix dot products. A tensor admits contracted products over each of its indexes. The index is indicated under the dot: T_iB is the contracted

product over theith index ofTand the ®rst index ofB. For simplicity, the indication of the index is omitted in contracted products over the last index of the right hand tensor. For instance, ifBis 22, the contracted products ofA byB over the third and over the second indexes of Aare

ABˆA3_ Bˆ

respectively. The multiplication of Bby Aover the ®rst index of Ais

BAˆ

In this work, these types of contracted products arise from the di€erentiation of matrix products which, for B and C matrices of compatible dimensions, follows the rule J…BC;k_{† ˆ}

J…B;k_{†_}

2C‡BJ…C;k†.

A special rank-3 tensor used in this work isI_k, which has unit main diagonal elements and null non-diagonal elements. For instance

Appendix B. Systemic parameter, implicit and mixed generalised-synergism coecients

B.1. Implicit and mixed generalised-synergism coecients of concentrations

By consideringk_{a vector ofrreaction-speci®c parameters such thatO:…v;}k_{† ˆI, (15) becomes}

^

O‰2Š…X;v;v† ˆO^…X;v† …O~‰_‰2_XŠ_Š;‰XŠ…v;v;v† ‡O~‰_‰2_v;Š_XŠ;‰v;XŠ…v;v;v†† …B:1†

with

~

O‰2_XŠ_†;…X†…v;v;v† ˆ …O‰2kŠ†;…k†…v;X;X†2_ O^…X;v†† O^…X;v† …OˆJ;W† …B:2†

or

~

S‰_‰2_v;Š_XŠ;‰v;XŠ…v;v;v† ˆ …I_r2_S^…v;v†† S^…v;v† ÿS^…v;v† I

r: …B:3†

By considering k_A as the vector of reaction-speci®c parameters and k_B as the vector of generic parameters in (15), one ®nds

^

O‰2Š…X;v;k_{† ˆ}O^…X;v† …O~‰_‰2_XŠ_Š;‰XŠ…v;v;k_{† ‡}O~‰_‰2_v;Š_XŠ;‰v;XŠ…v;v;k_††B:4†

with

~

O‰_‰2_XŠ_Š;‰XŠ…v;v;k_{† ˆ …O}‰2Š

…k_†;k_†…v;X;X†2_O^…X;v†† O…X;k† ‡O

‰2Š

…k_†;X†…v;X;k† O^…X;v† …OˆJ;W†

…B:5†

or

~

S‰_‰2_v;Š_XŠ;‰v;XŠ…v;v;k_{† ˆ …I}

r_{2_}S^…v;v†† S…v;k†: …B:6†

B.2. Systemic generalised-synergism coecients of derived steady-state magnitudes

Consider a steady-state propertyPP X …k_†;k_{†that is a twice-di€erentiable function ofXand} k. The replacement of (15) into (19) gives

O‰2Š…P;k_A;k_{B† ˆO}k…P;X† O^…X;v† …O~‰

2Š

‰XŠ;‰XŠ…v;kA;kB† ‡O~

‰2Š

‰v;XŠ;‰v;XŠ…v;kA;kB†† ‡O~

‰2Š

‰XŠ;‰XŠ…P;kA;kB†

…B:7†

and

~

O‰_‰2_XŠ_Š;‰XŠ…P;k_A;k_{B† ˆ …O}‰2Š

…k_†;k_†…P;X;X†2_O…X;kA†† O…X;kB† ‡O

‰2Š

…k_†;X†…P;X;kA†

T1;3;2_O…X;k B†

‡O‰2_kŠ_†;…X†…P;X;k_{B†_}

2O…X;kA† ‡O‰

2Š

O‰2Š…v;k_A;k_{B† ˆO}k…v;X† O^…X;v†:…O~‰

2Š

‰XŠ;‰XŠ…v;kA;kB† ‡O~

‰2Š

‰v;XŠ;‰v;XŠ…v;kA;kB†† ‡O~

‰2Š

‰XŠ;‰XŠ…v;kA;kB†

ˆO^…v;v† O~_‰‰2_XŠ_Š;‰XŠ…v;k_A;k_{B† ‡ …O}^_{v;v† ÿI† O}~‰2Š

‰v;XŠ;‰v;XŠ…v;kA;kB†: …B:9†

By di€erentiatingO^…v;v†(Table 1) with respect to a vector of generic parameters or of reaction-speci®c parameters one can ®nd the mixed generalised-synergism coecients

^

O‰2Š…v;v;k_{† ˆ}O‰2_kŠ_†;…k†…v;X;X†2_O^…X;v††:O…X;k† ‡O

‰2Š

…k†;…X†…v;X;k†2_O^…X;v†

‡Ok…v;X† O^‰2Š…X;v;k†; …B:10†

and the intrinsic generalised-synergism coecients

^

O‰2Š…v;v;v† ˆ …O‰2kŠ†;…k†…v;X;X†2_O^…X;v††:O^…X;v† ‡Ok…v;X† O^‰2Š…X;v;v†: …B:11†

If a derived property Pdepends explicitly on the rates (PP v X …k_†;k_†;Xk_†;k_{†), the systemic}

generalised-synergism coecients can be expressed as

O‰2Š…P;k_A;k_{B† ˆO}^_P;v†:O~‰2Š

‰XŠ;‰XŠ…v;kA;kB† ‡Ok…P;X† O^…X;v† O~‰

2Š

‰v;XŠ;‰v;XŠ…v;kA;kB†

‡O~‰_‰2_v;Š_XŠ;‰v;XŠ…P;k_A;k_B†;B:12†

with

~

O‰_‰2_v;Š_XŠ;‰v;XŠ…P;k_A;k_{B† ˆ …O}‰2Š

…X;k_†;X;k_†…P;v;v†2_O…v;kA†† O…v;kB†

‡ …O‰2_XŠ_;k†;…v;k†…P;v;X†2_O…v;kA†† O…X;kB†

‡ …O‰2_XŠ_;k†;…v;k†…P;v;X†T1;3;2

_

2O…X;kA†† O…v;kB†

‡ …O‰2_VŠ_;k_†;v;k_†…P;X;X†2_O…X;kA†† O…X;kB†

‡O‰2_XŠ_;k†;…v;X†…P;v;kB†2_O…v;kA† ‡O

‰2Š

…X;k†;…v;X†…P;v;kA†

T1;3;2

O…v;k_{B† ‡O}‰2Š

…v;k_†;v;X†…P;X;kB†2_O…X;kA†

‡O‰2_vŠ_;k†;…v;X†…P;X;kA†

T1;3;2 _O…X;k B† ‡O

‰2Š

…v;X†;…v;X†…P;kA;kB†: …B:13†

One can obtain mixed and intrinsic systemic generalised-synergism coecients of properties that are functions of the rates of reaction by di€erentiating

^

O…P;v† ˆOX;k…P;v† O^…v;v† ‡Ov;k…P;X† O^…X;v†

with respect to a vector of generic parameters or of reaction-speci®c parameters, respectively. They take the forms

^

O‰2Š…P;v;k_{† ˆ}O‰2_XŠ_;k†;…X;k†…P;v;v† O…v;k_{† ‡}O‰2_XŠ_;k†;…v;k†…P;v;X† O…X;k_†

‡O‰2_XŠ_;k_†;v;X†…P;v;k††2_O…v;v† ‡ …O

‰2Š

…X;k_†;v;k_†…P;v;X†

T1;3;2_O…v;k†

‡O‰2_vŠ_;k†;…v;k†…P;X;X† O…X;k† ‡O

‰2Š

…v;k†;…v;X†…P;X;k††2_O^…X;v†

and

^

O‰2Š…P;v;v† ˆ …O‰2_XŠ_;k†;…v;k†…P;v;v† O^…v;v† ‡O

‰2Š

…X;k†;…v;k†…P;v;X† O^…X;v††2_O^…v;v†

‡ …O‰2_XŠ_;k_†;v;k_†…P;v;X†

T1;3;2_O^_{v;v† ‡O}‰2Š

…v;k_†;v;k_†…P;X;X†

O^…X;v††2_O^…X;v† ‡OX;k…P;v† O^‰2Š…v;v;v† ‡Ov;k…P;X†

O^‰2Š…X;v;v†; …B:15†

respectively.

Relationships between systemic generalised-synergism coecients of various derived steady-state properties are shown below. The systemic generalised-synergism coecients of the aggre-gated ¯uxes of production (V_‡) follow from applying (5) or (6) to O…V‡;v†:

J‰2Š…V‡;kA;kB† ˆN‡J‰2Š…v;kA;kB†; …B:16†

W‰2Š…V‡;kA;kB† ˆSX;k…V_‡;v† W‰2Š…v;k_A;k_B†; …B:17†

S‰2Š…V‡;kA;kB† ˆ…R2_S…v;kA†† S…v;kB† ‡SX;k…V‡;v† S‰2Š…v;kA;kB† …B:18† with

SX;k…V_‡;v† ˆVnÿ_‡ 1N_‡vn and

RˆS‰2_XŠ_;k_†;X;k_†…V‡;v;v† ˆSX;k…V‡;v† I

r ÿ …I

n2_SX;k…V_‡;v†† SX;k…V_‡;v†: …B:19†

Similar operations onO…U;v† yield:

J‰2Š…U;k_A;k_{B† ˆ}Xnÿ1 fJ‰2Š…V_‡;k_A;k_{B† ÿ}UnJ‰2Š…X;k_A;k_{B† ÿ}I_n2_J…U;kA†† J…X;kB†

ÿ …I_n2_J…X;kA†† J…U;kB†g; …B:20†

W‰2Š…U;k_A;k_{B† ˆW}‰2Š_V‡;k_A;k_{B† ÿW}‰2Š_X;k_A;k_{B† ÿ …I}

p2_S…U;kA†† S…X;kB†

ÿ …I_p2_S…X;kA†† S…U;kB†; …B:21†

S‰2Š…U;k_A;k_{B† ˆS}‰2Š_V‡;k_A;k_{B† ÿS}‰2Š_X;k_A;k_{B†: …B:22†}

The corresponding relationships between mixed generalised-synergism coecients of concen-trations, rates of reaction, aggregated ¯uxes and turnover numbers are formally similar to (B.16)±(B.18) and (B.20)±(B.22). The same holds for the cognate relationships between intrinsic generalised-synergism coecients.

Appendix C. Link tensors for systems subject to concentration conservation relationships

From the de®nition (Section 4.2.1.1) it follows that any linear combination of concentrations of internal species, `T<sub>X, such that `</sub>T_{X_ ˆ0 for any vv…X;k), is a structural conservation}

re-lationship. The latter condition is ful®lled if and only if NT`ˆ0. Therefore, a set of nÿn0

L ± the so-calledconservation matrix± is a n …nÿn0†matrix whose columns form a basis for

ker(NT_{). The physiological interpretation of these conservation relationships is frequently}

com-plicated by the occurrence of negative elements in L. However, as any linear combination of columns of L_{still belongs to ker(N}T_{), one can often rearrange the conservation relationships as}

non-negative linear combinations of concentrations. (For a systematic procedure see [29].) In this form, the conservation relationships represent total concentrations of conserved moieties. The choice of conservation matrix is irrelevant for the derivations below.

By assigning a vector k_A ofnÿn0 new parameters to the conserved magnitudes and by

reor-dering the rows of L and the variables in X, one can write

Here,XEand XN stand for the sub-vectors ofXcorresponding to thenÿn0 eliminated variables

and to the remainingn0 variables, respectively; andLEandLN stand for the corresponding…nÿ

n0† …nÿn0† and n0 …nÿn0† sub-matrices of L. So, assuming that LE is non-singular, the values of the eliminated variables at any time can be calculated as XEˆLT

ÿ1

E kAÿLT

ÿ1 E L

T

NXN. This permits identifying:

~ XX

J ˆ_ JkN…XE;XN† ˆ ÿL

Tÿ1 E L

T N; X~

k

J ˆ_ JXN…XE;kA† ˆL

Tÿ1

E …C:1†

and

~ XX

S ˆ_ SkN…XE;XN† ˆX nÿ1 E X~

X J X

n

N;X~

k

S ˆ_ SXN…XE;kA† ˆX nÿ1 E X~

k

Jk

n

A: …C:2†

So, considering and , it follows

…C:3†

and

…C:4†

From the de®nition of k_{N it follows}

…C:5†

All these link matrices only depend on the (constant) stoichiometric coecients of the kinetic model. Therefore, the second-order link tensors X:;:

O‰2Š and PO‰2Š, for Ostanding for J or W, are

null, whereas the second-order logarithmic link tensors are

XX;k

Here,pN stands for the number of parameters in the transformed system, and

~

S‰2Š are null except those in the sub-tensor

~

Likewise, all the elements of Xk;k

S‰2Š are null except those in the sub-tensor

~

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