Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol163.Issue2.Feb2000:
Mathematical Biosciences 163 (2000) 131±158
www.elsevier.com/locate/mbs
Synergism analysis of biochemical systems. II. Tensor
formulation and treatment of stoichiometric constraints
Armindo Salvador a,b,*
Grupo de Bioquõmica e Biologia Te
oricas, Instituto de Investigacßa~o Cientõ®ca Bento da Rocha Cabral,
Ccß. Bento da Rocha Cabral, 14 P-1250 Lisbon, Portugal
Department of Microbiology and Immunology, The University of Michigan, 5641 Medical Sciences Building II,
Ann Arbor, MI 48109-0620, USA
a
b
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 eective formulation achieved in this work. The present formulation 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 sensitivity analysis and new relationships for (log-)synergism coecients 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 RederÕs 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 eectiveness 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 computational 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: [email protected] (A. Salvador).
0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 9 9 ) 0 0 0 5 7 - 7
132
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
processes [1±5]. Owing to mathematical complexity, closed-form analytical steady-state solutions of these models are seldom available, while extensive numerical simulations are
impractical for analysing responses to large sets of parameters. Dierential 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 dierential 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 characterisation of responses to single-parameter perturbations. Ease of implementation and computational 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 eectiveness 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-)synergism coecients 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 coecients of various properties are derived (Appendix B). All
the relevant expressions make use of only the simplest tensor operations, which are explained 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 coecients 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 conservation at steady-state (Section 4.2.2).
Often, sensitivity and synergism analyses of kinetic models subject to conservation relationships
can be performed with less computation than necessary for other kinetic models of the same
dimensionality. This computational advantage is especially important for large-scale integrative
models (for instance, [1±5], see Section 5).
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
133
2. Notation and conventions
The notation and conventions introduced in the previous papers are adopted here, with extensions as described below.
The term `generalised-synergism' will be used to denote a measure of deviation from a prede®ned pattern of behaviour. This includes non-normalised synergisms, relative synergisms and
log-synergisms [19]. The dierential measures of steepness of deviation from the prede®ned patterns will be denoted by `generalised-synergism coecients'. To simplify language, where there is
no ambiguity the term `sensitivities' will be used to refer to both ®rst-order sensitivities and
generalised-synergism coecients (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,
WP ko ; Dk fWPi ko ; Dk j;k g;
i 2 f1; 2; . . . ; nP g; j; k 2 f1; 2; . . . ; pg:
When the generalised-synergism coecients are similarly arranged, the calculation of the systemic
coecients requires only simple tensor operations. These operations and the corresponding notation are explained in Appendix A. Additional notation is: Ik , the k ´ k identity matrix; 0i j k ,
the i ´ j ´ k null tensor; ker(A), the kernel of matrix A; xn , 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
1
2
W P; k; k: Dk Dk;
WP ko ; Dk
2
2
similar expressions apply for the other generalised-synergisms.
If P, kA and kB are vectors, the following relationships hold:
S P; k Pnÿ1 J P; k kn ;
n
n
W2 P; kA ; kB Pnÿ1 J2 P; kA ; kB 2_ kA kB ;
S2 P; kA ; kB J2 log jPj; logjkA j; log jkB j;
S2 P; kA ; kB W2 P; kA ; kB ÿ InP 2_ S P; kA S P; kB S P; kA InA S kA ; kB ;
1
2
3
4
with nP and nA standing for the dimensions of the vectors P and kA , respectively.
The J and S operators also follow the chain rule when operating over composite functions.
That is, if F º F(P(k),k) and F and P are both dierentiable, then
J F; k Jk F; P J P; k JP F; k
5
S F; k Sk F; P S P; k SP F; k:
6
and
134
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
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 coecients [19].
The symbols V and U stand 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
X_ N v 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 producing 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 steady-states, they lack any other in¯uence on the steadystate. So, if (a) the kinetic model is structurally stable, (b) the operating point is not poised at a
bifurcation, and (c) v is dierentiable around the operating point, one obtains
_ k N Jk v; X J X; k JX v; k 0;
J X;
8
or, by normalising the sensitivities,
N vn Sk v; X S X; k SX v; k 0:
9
If (d) the matrix N Jk v; X is non-singular, one can solve (8) for the matrix of systemic sensitivities:
^
J X; k J X;
v JX v; k
10
^
J X;
v ÿ N Jk v; Xÿ1 N:
11
with
Furthermore, if (d0 ) the matrix N vn Sk v; X is non-singular, then solving (9) for the systemic
logarithmic sensitivities gives
^
S X; k S X;
v SX v; k
12
ÿ
ÿ1
^
S X;
v ÿ N vn Sk v; X
N vn :
13
with
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
135
^
We call systemic sensitivities of the type O P;
v, which cannot be obtained through explicit
dierentiation 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 properties that are functions of concentrations (derived properties) 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 ¯uxoriented theory [21].
If v is twice-dierentiable around the operating point and kA and kB are two vectors of parameters, the application of chain rule (5) to Eq. (8) gives
2
2
N J:;:
v; X; kA T1;3;2 J X; kB
v; X; X2_ J X; kA J X; kB J:;:
2
2
v; kA ; kB 0:
v; X; kB 2_ J X; kA JkA ;kB v; X:J2 X; kA ; kB J:;:
J:;:
14
The solutions of (14) and of its counterpart for logarithmic co-ordinates share the form
2
2
^
~
~
v O
O2 X; kA ; kB O X;
X;X v; kA ; kB Ov;X;v;X v; kA ; kB :
15
~ 2
~ 2
The subscripts of O
X;X and Ov;X;v;X indicate variables that are allowed ®rst-order variations
(linear, for J, or power-law for W and S) with the parameters. So
~ 2 v; kA ; kB O2 v; X; X _ O X; kA O X; kB O2 v; X; kA T1;3;2 O X; kB
O
2
X;X
k; k
k; X
2
2
O k; X v; X; kB 2_ O X; kA O X; X v; kA ; kB
16
represents the generalised-synergism coecients that would be observed for rates of reaction if the
concentrations responded linearly (for J) or as a power-law (for W and S) to single-parameter
perturbations. These magnitudes are considered as semi-intrinsic generalised-synergism coecients, which take the ®rst-order systemic responses of the concentrations into account but assume
that the deviations of each ¯ux from the ®rst-order approximations do not interact.
~ 2
The term O
v;X;v;X v; kA ; kB is null when O stands for J or W, or takes the value
ÿ
~ 2
S
17
v;X;v;X v; kA ; kB Ir 2_ S v; kA S v; kB ÿ S v; kA IpA S kA ; kB :
Table 1
Systemic ®rst-order sensitivities of derived steady-state propertiesa
Cartesian space
Logarithmic space
^ v Ok v; X O X;
^
O v;
v I
^ ; v N J v;
^ v
J V
^ ; v ÿ Un J X;
^
^
v
J U;
v Xnÿ1 J V
^ v OX v; k
O v; k O v;
J(V ,k) N áJ(v,k)
a
J U; k Xnÿ1 J V ; k ÿ Un J X; k
ÿ
N is obtained from N by setting the negative elements to 0.
^ ; v SX ;k V ; v S v;
^ v
S V
n ^
Vnÿ1
N v S v; v
^
^
^ ; v ÿ S X;
v
S U;
v S V
S V ; k SX;k V ; v S v; k
n
Vnÿ1
N v S v; k
S U; k S V ; k ÿ S X; k
136
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
~ 2
The elements of O
v;X;v;X v; kA ; kB are also semi-intrinsic generalised-synergism coecients. As
^
S X; v S v; kA 0, for log-synergism coecients (15) simpli®es to
ÿ
^
~ 2 v; kA ; kB I _ S v; kA S v; kB c:
S2 X; kA ; kB S X;
v bS
18
r2
X;X
Systemic generalised-synergism coecients for ¯uxes, turnover numbers and other derived steadystate properties, P P X k; k, can be computed from the systemic concentration generalisedsynergism coecients through
2
2
O2 P; kA ; kB O k; k P; X; X2_ O; X; kA O X; kB O k; X P; X; kA T1;3;2 O X; kB
2
Ok P; X O2 X; kA ; kB O k; X P; X; kB 2_ O X; kA
2
O X; X P; kA ; kB :
19
By considering k a vector of r reaction-speci®c parameters such that O: v; k I, one can use (15)
^ 2 X; v; k] generalised-syn^ 2 X; v; v] and mixed [intrinsic-parameter, O
to compute intrinsic [O
ergism coecients. Appendix B presents the derivation of systemic, intrinsic and mixed generalised-synergism coecients 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 dismutation ± quickly catalysed by superoxide dismutase (SOD, reaction 2) ± produces hydrogen
peroxide (H2 O2 ). Though the latter is mostly reduced to water through a glutathione-peroxidasecatalysed 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). Superoxide 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
á
Oáÿ
2 ; X2 H2 O2 and X3 HO as dependent variables, and X4 O2 ; X5 SOD; X6
Fe2 and X7 GPx as parameters of interest (vector k). The rate vd v5 v6 k5 X1 k6 X3 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
Fig. 1. Simple scheme of the metabolism of reactive oxygen species in the matrix of rat liver mitochondria. Reactions 5
and 6 account for damage to mitochondrial components. The abbreviations are: GPx, `classic' glutathione peroxidase,
GSH, glutathione, SOD, Mn-dependent superoxide dismutase.
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
137
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
v1
v2
v3
v4
k1 O2
k2 SODOáÿ
2
k3 Fe2 H2 O2
/1 =H2 O2GPx
/2 =GSH
v5 k5 Oáÿ
2
v6 k6 HOá
Parameters
k1 0:16sÿ1 ; O2 1:0 10ÿ4 M (in lipid bilayers)
k2 4:7 109 Mÿ1 sÿ1 ; SOD 1:1 10ÿ5 M
k3 2:0 104 Mÿ1 sÿ1 ; Fe2 1:0 10ÿ7 M
/1 4:8 10ÿ8 M s; /2 2:5 10ÿ5 M s; GPx 2:5 10ÿ6 M; GSH 1:1 10ÿ2 M
k5 5:9 sÿ1
k6 3:1 109 sÿ1
antioxidant defences and pro-oxidant agents as regards vd . For that we compute the log- and
relative-synergism coecients of vd , through the steps described below:
1. Compute the steady-state concentrations and rates. This gives X1 3:1 10ÿ10 M; X2 1:5
10ÿ7 M; X3 9:9 10ÿ20 M; v1 1:6 10ÿ5 M sÿ1 ; v2 8:0 10ÿ6 M sÿ1 ; v3 3:1 10ÿ10
M sÿ1 ; v4 8:0 10ÿ6 M sÿ1 ; v5 1:8 10ÿ9 M sÿ1 ; v6 3:1 10ÿ10 M sÿ1 .
2. Compute the tensors of intrinsic sensitivities and log-synergism coecients by dierentiating
the rate expressions according to the de®nitions of the respective operators [19] and replacing
the operating values. This yields
2
S X; X v; k; k 0644 :
3. Compute the implicit sensitivities, using (13):
2
3
1:0
ÿ 1:0
0
0
ÿ 1:1 10ÿ4
0
S X; v 4 1:0 1:2 10ÿ4 ÿ 3:9 10ÿ5 ÿ 1:0 ÿ 1:2 10ÿ4
0 5:
ÿ4
ÿ4
1:0 1:2 10
0
ÿ 1:0 ÿ 1:2 10
ÿ 1:0
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:
2
3
1:0
ÿ 1:0
0
0
S X; k 4 1:0 1:2 10ÿ4 ÿ 3:9 10ÿ5 ÿ 1:0 5;
1:0 1:2 10ÿ4
1:0
ÿ 1:0
138
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
3
0
0
7
0
0
7
1:0
ÿ 1:0 7
7:
ÿ 3:8 10ÿ5 3:9 10ÿ5 7
7
5
0
0
1:0
ÿ 1:0
ÿ
~ 2 v; k; k, through (16), and I _ S v; k :S v; k
5. Compute S
2
r
X;X
2
1
6 1:0
6
6 1:0
S v; k 6
6 1:0
6
4 1:0
1:0
0
1:1 10ÿ4
1:2 10ÿ4
1:1 10ÿ4
ÿ 1:0
1:2 10ÿ4
2
3
0 0 0
0 0 07
7
0 0 05
0 0 0
2
0
0
1:1 10ÿ4
6 1:1 10ÿ4 ÿ 1:1 10ÿ4 0
6
4
0
0
0
0
0
0
2
7:4 10ÿ3
1:2 10ÿ4
1:0
6 1:2 10ÿ4 ÿ 1:2 10ÿ4
1:2 10ÿ4
6
ÿ4
4
1:0
1:2 10
ÿ 3:9 10ÿ5
ÿ4
ÿ1:0
ÿ 1:2 10
ÿ 1:0
Ir 2_ S v; k:S v; k 2
ÿ7
ÿ4
ÿ2:8 10
1:1 10
ÿ 3:8 10ÿ5
>
>
>
ÿ4
ÿ4
>
6
>
ÿ 1:1 10
ÿ 4:4 10ÿ9
> 6 1:1 10
>
ÿ5
ÿ9
>
4
>
ÿ3:8 10
ÿ 4:4 10
3:8 10ÿ5
>
>
>
ÿ9
ÿ9
>
3:9 10ÿ5
4:4
ÿ 1:5 10
>
3
2 10
>
>
>
0
ÿ 1:0 0 0
>
>
>
6 ÿ1:0
>
2:0
0 07
>
7
6
>
>
4 0
>
0
0 05
>
>
>
>
0
0
0 0
>
>
2
>
ÿ3
ÿ4
>
>
7:4
10
1:2
10
1:0
>
> 6
>
>
1:2 10ÿ4
1:2 10ÿ4 ÿ 1:2 10ÿ4
>
6
>
>
4
>
1:0
1:2 10ÿ4
ÿ 3:9 10ÿ5
>
:
ÿ4
ÿ1:0
ÿ 1:2 10
ÿ 1:0
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
0
60
6
40
0
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
3
>
>
>
0
>
>
>
7
>
07
>
>
>
5
>
0
>
>
>
>
0
3 >
>
>
>
ÿ 1:0
>
>
ÿ4 7 >
ÿ 1:2 10 7 >
>
>
5 >
>
ÿ 1:0
>
>
=
2:0
3
:
3:9 10ÿ5
>
>
>
>
4:4 10ÿ9 7
>
7>
ÿ9 5 >
>
>
ÿ 1:5 10
>
>
>
ÿ5
>
ÿ 3:9 10
>
>
>
>
>
>
>
>
>
>
>
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>
>
>
>
>
3 >
>
>
>
ÿ 1:0
>
ÿ4 7 >
>
ÿ 1:2 10 7 >
>
>
>
5
>
ÿ 1:0
>
;
2:0
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
139
6. Compute the systemic log-synergism coecients, through (18), which yields
3
2
8
9
0
0
0 0
>
>
>
>
>
>
6 0 ÿ 1:1 10ÿ4 0 0 7
>
>
>
>
7
6
>
>
>
>
5
4
>
>
0
0
0
0
>
>
>
>
>
>
>
>
0
0
0
0
>
>
>
>
3
2
>
>
ÿ3
ÿ7
ÿ3
ÿ7
>
>
7:4
10
8:5
10
ÿ
7:4
10
5:7
10
>
>
>
>
>
>
>
>
>
ÿ7
>
>
>
ÿ 6:5 10ÿ11
3:9 10ÿ5
ÿ 3:9 10ÿ5 3 >
>
>
2 5:7 10ÿ3
>
>
>
ÿ7
ÿ3
ÿ7
>
>
>
7:4
10
8:5
10
ÿ
7:4
10
5:7
10
>
>
>
>
>
ÿ7
ÿ4
ÿ7
ÿ11 7 >
6
>
>
8:5
10
ÿ
1:2
10
ÿ
8:5
10
ÿ
6:5
10
>
>
7
6
>
>
>
>
ÿ3
ÿ7
ÿ3
ÿ5
5
4
>
>
ÿ7:4
10
ÿ
8:5
10
7:3
10
3:9
10
>
>
:
;
ÿ7
ÿ11
ÿ5
ÿ5
5:7 10
ÿ 6:5 10
3:9 10
ÿ 3:9 10
To compute the systemic relative-synergism coecients one may just apply expression (4):
3
2
8
9
0
ÿ 1:0 0 0
>
>
>
>
>
>
7
6 ÿ1:0
>
>
2:0
0
0
>
>
7
6
>
>
>
>
5
4
>
>
0
0
0
0
>
>
>
>
>
>
>
>
0
0
0
0
>
>
>2
>
3
>
>
ÿ2
ÿ4
ÿ5
>
>
1:5
10
1:2
10
ÿ
1:0
ÿ
4:0
10
>
>
>
>
>
>
>
>
>
ÿ5
>
>
>
ÿ 4:5 10ÿ9
7:8 10ÿ5
3:0 10ÿ9 3 >
>
>
2ÿ4:0 10ÿ2
>
>
ÿ4
>
>
>
>
1:2
10
ÿ
1:0
1:0
1:5
10
>
>
>
>
>
>
ÿ4
ÿ4
ÿ4
ÿ4
6
7
>
>
1:2
10
ÿ
2:3
10
ÿ
1:2
10
1:2
10
>
>
6
7
>
>
>
>
ÿ4
4
5
>
>
ÿ1:0
1:2
10
2:0
ÿ
1:0
>
>
:
;
ÿ4
ÿ5
1:0
1:2 10
ÿ 1:0
ÿ 3:9 10
7. Compute the logarithmic gains of the rate of damage, through (6):
S vd ; k 1:0 ÿ 0:86 0:14 ÿ 0:14 :
8. Finally, by applying expression (19) obtain the log-synergism coecients of the total rate of
damage (vd ) and ± again through (4) ± the corresponding relative-synergism coecients:
3
2
1:1 10ÿ3 9:0 10ÿ4 ÿ 2:0 10ÿ3 9:0 10ÿ4
6 9:0 10ÿ4
0:12
ÿ 0:12
0:12 7
7;
S2 vd ; k; k 6
ÿ3
4 ÿ2:0 10
ÿ 0:12
0:13
ÿ 0:12 5
9:0 10ÿ4
0:12
ÿ 0:12
0:12
3
2
0:14
ÿ 0:14
2:1 10ÿ3 ÿ 0:86
6 ÿ0:86
1:7
ÿ 0:25
0:25 7
7:
W2 vd ; k; k 6
ÿ3
4 0:14
ÿ 0:25 2:9 10
ÿ 0:14 5
ÿ0:14
0:25
ÿ 0:14
0:29
Analysis of the latter pair of matrices yields the following inferences. First, positive synergisms
between pro-oxidant agents (O2 , Fe2 ) and between antioxidant defences (SOD, GPx), as well as
140
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
antagonisms (negative relative-synergism coecients) between pro-oxidants and antioxidant defences should hold. The strongest antagonism predicted is between oxygen and SOD
W2 vd ; X4 ; X5 ÿ0:86 and the strongest synergism between SOD and GPx
W2 vd ; X5 ; X7 0:25. Second, combined changes of oxygen concentration and any of the other
factors should yield nearly multiplicative responses (log-synergism coecients near 0). Joint
perturbations of [SOD] and [GPx] yield a slightly supra-multiplicative response
S2 vd ; X5 ; X7 0:12, 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 coecients near 0) and varies almost linearly with [Fe2 ] (relative-synergism coecients near 0). The response to [SOD] is clearly supra-linear
W2 vd ; X5 ; X5 1:7 and slightly supra-power-law S2 vd ; X5 ; X5 0:12, indicating diminishing returns of increasing 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_ N v 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 (XN ) and parameters (kN ) instead of the `natural' concentrations and parameters (X and k). Usually, it is not
too restrictive to assume that the transformation of X k into XN kN is such that X and k can be
expressed as double-dierentiable functions of XN and kN , respectively: k k kN ;
X X XN kN k; kN k. The chain rule of dierentiation then gives
O X; k XXO O XN ; kN XkO PO ;
20
O X; kN XXO O XN ; kN XkO ;
21
n
;X
X ;k
O2 X; k; k XXO2
2_ O XN ; kN O XN ; kN XO2 2_ O XN ; kN
o
T
;k 1;3;2
O XN ; kN XOk;k2 2_ PO PO
XXO O2 XN ; kN ; kN XXO2
ÿ
XXO O XN ; kN XkO PO2 ;
22
;k
X
O2 X; kN ; kN XXO2;X 2_ O XN ; kN O XN ; kN XXO2
2_ O XN ; kN XO
T1;3;2
;k
O2 XN ; kN ; kN XXO2
O XN ; kN XOk;k2 ;
23
141
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
where
XXO OkN X; XN ;
XkO OXN X; kN ;
;X
O2 X; XN ; XN ;
XXO2
PO O kN ; k;
XXO2;k O2 X; XN ; kN ;
XOk;k2 O2 X; kN ; kN ;
24
PO2 O2 kN ; k; k
and O stands for J, W or S. Whereas O XN ; kN and O2 XN ; kN ; kN refer to the properties of the
;X
;k
; XXO2
, XOk;k2 and PO2 refer to the properties of the transformation. We
system, XXO ; XkO ; PO ; XXO2
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 proportional one can derive formulas for the conversion of implicit and mixed sensitivities:
^ N ; v;
^
25
O X;
v XX O X
O
^ 2 X; v; v XX 2;X _ O X
^ N ; v O X
^ 2 XN ; v; v;
^ N ; v XX O
O
O
O 2
26
i
h
;k ^
X ^ 2
^ 2 X; v; k XX 2;X _ O X
^ N ; v O XN ;k XX 2
O
O
X
;v;k
PO :
O X
;
v
X
_
N
N
O
O 2
O 2
27
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
X_ N NN vN XN ; kN ;
28
and, considering v v vN ; k, ®nd
O v; k VvO O vN ; kN PO VkO ;
^ 2 v; k; k Vv;v2 _ O vN ; kN PO O vN ; kN PO Vv O2 vN ; kN ; kN _ PO
O
O
2
O 2
29
T1;3;2
PO VvO O vN ; kN PO2 Vv;k
O vN ; kN PO Vv;k
O2
O2 2_
O vN ; kN PO VOk;k2 ;
with the ¯ux link tensors de®ned as
VvO Ok v; vN ; VkO OvN v; k;
2
Vv;v
O k; k v; vN ; vN ;
O2
2
Vv;k
O k; vN v; vN ; k;
O2
2
VOk;k2 O vN ; vN v; k; k:
30
31
The following expressions allow calculating the sensitivities for the transformed set of parameters:
(
O v; kN VvO O vN ; kN VkO PO ;
O2 v; k; kN Vv;v
O v
;
k
O vN ; kN VvO O2 vN ; kN ; kN
_
N
N
2
O 2
(
T1;3;2
VOk;k2 2_ O vN ; kN PO VOv;k2
(
(
(
(
VOk;k2 2_ PO PO Vk PO2 ;
(
with P O k; kN and PO2 O2 k; kN ; kN :
(
2_ PO
32
O vn ; kN
33
142
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
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 matrix N has n rows and rank n0 < n,
then there are n ÿ n0 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 express n ÿ n0 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 of n0 dierential equations
corresponding to the remaining n0 dynamical variables (vector XN ) is sucient 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
for O standing for J or W; or
S2 X; kN ; kN XXS S2 XN ; kN ; kN XXS 2;X 2_ S XN ; kN S XN ; kN XSX;k
_ S XN ; kN
2 2
XSX;k
2
T1;3;2
S XN ; kN XSk;k2 :
36
This procedure extends the synergism analysis described in [19] to systems subject to structural
conservation relationships.
For computing the systemic sensitivities O XN ; kN and O2 XN ; kN ; kN , note that the matrices
NN JkN v; XN and NN vn SkN v; XN are non-singular. (NN is formed by the n0 rows of N that
correspond to the variables in XN .) Hence,
^ N ; v ÿ NN Jk v; XN ÿ1 NN
J X
N
37
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
143
and
^ N ; v ÿ NN vn Sk v; XN ÿ1 NN vn
S X
N
38
OkN v; XN Ok v; X XXO :
39
with
The intrinsic sensitivities of the ¯uxes to the transformed set of parameters are also straightforwardly 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 transformed system can be related through
2
2
O kN ; kN v; XN ; XN Ok v; X XXO2;X O k; k v; X; X2_ XXO XXO ;
41
2
2
;k
~ k O kA ; kN
O kN ; XN v; XN ; kN Ok v; X XXO2
O k; XN ;k v; X; XE 2_ XXO X
O
(
2
O k; X v; X; k2_ XXO P;
42
and
2
~ k;k2 _ O kA ; kN O kA ; kN
OXN ;XN v; kN ; kN OXN ;k v; XE X
O 2
2
~ k O kA ; kN X
~ k O kA ; kN
O XN ;k; XN ;k v; XE ; XE 2_ X
O
O
(
2
~ k O kA ; kN P
O XN ;k; X v; XE ; k2_ X
O
(
2
~ k O kA ; kN
O XN ;k; X v; XE ; kT1;3;2 2_ P X
O
2
(
(
O X; X v; k; k2_ P P:
43
. All the intrinsic sensitivities of the transformed system might as well be
Here,
obtained through dierentiation 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 an n n matrix are of order n3 [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
r r matrix for obtaining simultaneously the ¯ux control coecients and the concentration
control coecients of a system. Yet, by calculating the systemic ¯ux sensitivities through
O v; kN Ok v; X:O X; kN OXN v; kN , one obtains the same result by inverting a smaller
(n0 n0 ) matrix. Any kinetic model with a non-trivial steady-state has r > n P n0 . If all processes
are considered reversible, then r > 2n; and, as cycles are prevalent in metabolic networks, r can be
144
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
much higher than 2n. The computational cost of the two alternative strategies for computing the
systemic sensitivities may then dier 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 and X7 for free enzymes,
X2 and X4 for X6 :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 X10 X11 1; k2 0:35; k3 3:5; k4 100; k5 2; k6 k7 k9 104 ; k8 105 : The dimensionless concentrations are then
T
X 9:51 10ÿ1 4:83 10ÿ1 1:02 10ÿ1 5:00 10ÿ1 8:31 10ÿ3 5:09 10ÿ1 4:92 10ÿ1 :
The dimensionless ¯uxes are
v 1
0:169 0:831 0:831 1
484: 483: 500:2
T
500:0 ;
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
3
2
1
0
0
0
0 ÿ1
1
0
0
6 0 ÿ1 ÿ1
0
0
1 ÿ1
0
07
7
6
7
60
1
0
0
0
0
0
ÿ1
1
7
6
6
N 60
0
0
1 ÿ1
0
0
1 ÿ1 7
7
60
0
1 ÿ1
0
0
0
0
07
7
6
40
1
0
1
0 ÿ1
1
0
05
0
0 ÿ1
0
1
0
0 ÿ1
1
has n 7 rows and rank n0 5, there are two conservation relationships. As follows from
0 1 0 0 1 1 0
T
;
L
0 0 0 1 1 0 1
Scheme 1.
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
145
they are
X2 X5 X6 X10 ;
X4 X5 X7 X11 ;
where X10 and X11 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
kN k1 k9 X10 X11 T it follows from (C.1) and (C.3):
~k
X
S
"
X10
X6
0
0
X11
X7
#
;
44
45
For the second-order link tensors, (C.6) and (C.10) give
46
146
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
(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 replace SkN v; XN into
(38), to ®nd
3
2
3:63
ÿ 0:336
ÿ 1:61 ÿ 0:0436 ÿ 1:64 ÿ 1 0:988
0
0
6 1:86
ÿ 0:171
ÿ 0:829 ÿ 0:0140 ÿ 0:843
0
0
0
0 7
7
6
^
6
0
0
ÿ 1 1:00 7
S XN ; v 6 2:03 ÿ 0:00257 0:00257 ÿ 0:0169 ÿ 2:01
7;
4 1
0
0
0
ÿ1
0
0
0
0 5
0:826
ÿ 0:169
0:169
ÿ 0:997
0:171
0
0
0
0
49
and apply (12) to compute S XN ; kN .
(4) Compute the systemic ®rst-order sensitivities of the original set of concentrations, through
(21)
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
147
50
^ N ; v, as k is the vector of rate constants.
Here S XN ; k S X
(5) Compute the systemic log-synergism coecients of the transformed system. As the reactions
follow mass action kinetics, the intrinsic log-synergism coecients of the original system are null.
So, (41)±(43) simplify to
2
S kN ; kN v; XN ; XN Sk v; X XXS 2;X ;
2
S kN ; XN v; XN ; kN Sk v; X XXS 2;k ;
and
2
~ k;k2 _ S kA ; kN S kA ; kN ;
S XN ; XN v; kN ; kN SXN ;k v; XE X
2
S
respectively. The replacement of these expressions into (18) yields S2 XN ; kN ; kN . For conciseness, only the log-synergism coecients for kC k1 X10 X11 T are shown
82
39
9:51
ÿ 3:49 ÿ 8:57 >
>
>
>
>
>
> 4 ÿ3:49
>
>
1:90
3:13 5 >
>
>
>
>
>
>
> ÿ8:57
>
3:13
7:45
>
>
2
3
>
>
>
>
>
>
1:58
0
ÿ
1:43
>
>
>
>
>
>
4
5
>
>
0
0
0
>
>
>
>
>
>
>
>
ÿ1:43
0
1:14
>
> 2
>
3 >
>
>
>
>
2:08 0 ÿ 2:08
<
=
2
4 0
S XN ; kC ; kC
0
0 5
:
51
>
>
>
>
ÿ2:08
2:07
>
>
>
>
2 0
3
>
>
>
>
>
>
0
0
0
>
>
>
>
>
>
4
5
>
>
0
0
0
>
>
>
>
>
>
>
>
0
0
0
>
>
2
3
>
>
>
>
>
>
ÿ0:501
0
0:644
>
>
>
>
>
>
4
5
>
>
0
0
0
>
>
>
>
:
;
0:644 0 ÿ 0:927
148
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
(6) Compute the systemic log-synergism coecients for the original set of variables, by replacing S2 XN ; kN ; kN into (23)
52
Note the high log-synergism coecients, even as compared to the ®rst-order sensitivities. The
relative-synergism coecients (not shown) reach even higher values. Such high log- and relativesynergism coecients indicate strong deviations from both power-law/multiplicative and linear/
additive behaviour [19]. Hence, descriptions of this system that are based on ®rst-order approximations either in Cartesian or logarithmic co-ordinates may be very inaccurate and may overlook, 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 (n r) stoichiometric matrix of a kinetic
,
model has matrix-rank n0 < r. If N is full rank (n0 n), one can rearrange it as
where NE is a square sub-matrix formed by n0 linearly independent columns (corresponding to n0
eliminated rates of reaction) and ND is formed by the remaining columns. If n0 < n, N should 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,
~ v vN , with V
~ v _ Jk vE ; vN ÿNÿ1 ND . So, v Vv vN , with
which leads to vE V
J
J
E
J
53
It also follows that VkO 0; VOk;k2 0; Vv;k
0;
O2
The non-trivial logarithmic link tensors are
Vv;v
0;
J 2
Vv;v
0.
W 2
54
55
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
149
with
~ v vnÿ1 V
~ v vnN ;
V
S
J
~ v:
~ v I ÿ I _ V
~v V
~ v;v2 V
V
2
S
r
r
ÿr
S
S
S
0
0
56
57
Because this transformation does not change the variables nor the parameters, the X and P link
tensors for J, W and S 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 VvO O vN ; k
and
O2 v; k; k VvO O2 vN ; k; k
for J or W, or
S vN ; k S vN ; k VvS S2 vN ; k; k:
S2 v; k; k Vv;v
S 2 2_
4.2.2.2. Example. Consider the example in Section 4.2.1.2. The stoichiometric matrix of the
transformed system is
2
1
60
6
NN 6
60
40
0
0
0
0
ÿ1 ÿ1
0
1
0
0
0
0
1
0
1 ÿ1
0
0
0
ÿ1
0
ÿ1
1
0
0
0
1
ÿ1
0
0
0
3
0
0
0
07
7
ÿ1
17
7:
1 ÿ1 5
0
0
The analysis of NN 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
v3 ; v4 ; v5 ; v7 and v9 , one ®nds
2
0
6 ÿ1
6
NE 6
6 1
4 0
0
0
0
0
1
ÿ1
0
0
0
ÿ1
0
3
1
0
ÿ1
07
7
0
17
7;
0 ÿ1 5
0
0
2
1
60
6
ND 6
60
40
0
0
ÿ1
1
0
0
ÿ1
1
0
0
0
3
0
07
7
17
7
ÿ1 5
0
from which follows
v3 v1 ÿ v2 ;
v4 v1 ÿ v2 ;
v5 v1 ;
v7 v6 ÿ v1 ;
v9 v8 ÿ v2 :
150
Hence, according to
2
1
6 0
6 v1
6 v ÿv
6 1 v1 2
6 v ÿv
6 1 2
v
VS 6
6 1
6 0
6
1
6 ÿ v vÿv
6 6 1
4 0
0
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
(54),
0
1
2
ÿ v1vÿv
v2 2
ÿ v1 ÿv2
0
0
0
0
2
ÿ v8vÿv
2
0
0
0
0
0
1
v6
v6 ÿv1
0
0
0
0
0
0
0
0
0
1
v8
v8 ÿv2
3
7
7
7
7
7
7
7:
7
7
7
7
7
5
The relationship S v; kN VvS S vN ; kN permits computing the sensitivities of all ¯uxes from
the knowledge of the sensitivities of v1 ; v2 ; v6 and v8 .
5. Discussion
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 eective 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 generalised-synergism coecients (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 dierent 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 with n internal metabolites, r reactions and p
parameters, with n; r; p 1 and p > n, straight application of the formulas (15) through (18)
would require rnp n 3p multiplications and rnp n 3p sums. However, if kA kB the matrix
O2 Pi ; kA ; kB of generalised-synergism coecients for each scalar property Pi is symmetric (i.e.,
O2 Pi ; kj ; kk O2 Pi ; kk ; kj , and so a tensor O2 P; k; k has at most nP p p ÿ 1=2 dierent
2
2
coecients. The symmetry of the tensors O2 X; k; k, O k; k v; X; X, O X; X v; k; k and
2
O k; X v; X; k2_ O X; l allows computing O2 X; k; k with about rnp n 4p=2 multiplications
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 coecients are of interest. As
~ 2
W
v;X;v;X v; kA ; kB 0, the systemic relative-synergism coecients may be easier to compute
through (15) than the corresponding log-synergism coecients. However, generalised mass action
[31] rate expressions yield null intrinsic log-synergism coecients. So, if the model includes many
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
151
of those rate expressions, it is more eective to compute ®rst the systemic log-synergism coecients through (15) and obtaining then the systemic relative-synergism coecients 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 aect 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 link sub-tensors instead of the full tensors. Important
gains in numerical eectiveness 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 generalisedsynergism 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 mitochondrial matrix. This analysis highlighted (a) SODs importance in antagonising the eects of
oxygen activation and (b) the synergistic eect of SOD and GPx activities in decreasing the rate of
damage to mitochondrial components by oxygen radicals. The analysis also indicates that although 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 coecients remain convenient indexes of relevance of
multifactorial eects.
As more data becomes available, the need for large-scale integrative approaches to biochemistry 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, information 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 2 2 2 tensor, A, can be written as
9
8
a1;1;1 a1;1;2 >
>
>
>
< a
a1;2;2 =
1;2;1
:
A fai;j;k g
>
>
> a2;1;1 a2;1;2 >
;
:
a2;2;1 a2;2;2
152
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
Transposing a tensor corresponds to exchanging its indexes. A rank-n tensor can be transposed in n!
ways, which are distinguished by the indexes of the transposition symbol: TTi1 ;i2 ;...;in is the tensor obtained by turning the ®rst index of T into the i1 th index, the second index into the i2 th, etc. For instance
9
8
a
a
>
>
1;1;1
1;2;1
>
>
< a
=
a
1;1;2
1;2;2
T1;3;2
A
:
a
a2;2;1 >
>
>
>
: 2;1;1
;
a2;1;2 a2;2;2
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_i B is the contracted
product over the ith index of T and the ®rst index of B. For simplicity, the indication of the index
is omitted in contracted products over the last index of the right hand tensor. For instance, if B is
2 2, the contracted products of A by B over the third and over the second indexes of A are
9
8
a1;1;1 b1;1 a1;1;2 b2;1 a1;1;1 b1;2 a1;1;2 b2;2 >
>
>
>
< a b a b
=
a
1;2;1 1;1
1;2;2 2;1
1;2;1 b1;2 a1;2;2 b2;2
A B A3_ B
a b a2;1;2 b2;1 a2;1;1 b1;2 a2;1;2 b2;2 >
>
>
>
: 2;1;1 1;1
;
a2;2;1 b1;1 a2;2;2 b2;1 a2;2;1 b1;2 a2;2;2 b2;2
and
9
8
a1;1;1 b1;1 a1;2;1 b2;1 a1;1;2 b1;1 a1;2;2 b2;1 >
>
>
>
< a b a b
a1;1;2 b1;2 a1;2;2 b2;2 =
1;1;1 1;2
1;2;1 2;2
T1;3;2
T1;3;2
;
B
A2_ B A
> a2;1;1 b1;1 a2;2;1 b2;1 a2;1;2 b1;1 a2;2;2 b2;1 >
>
>
;
:
a2;1;1 b1;2 a2;2;1 b2;2 a2;1;2 b1;2 a2;2;2 b2;2
respectively. The multiplication of B by A over the ®rst index of A is
9
8
a
b
a
b
a
b
a
b
>
>
1;1
1;1;1
1;2
2;1;1
1;1
1;1;2
1;2
2;1;2
>
>
< b a
=
b
1;1 1;2;1 b1;2 a2;2;1
1;1 a1;2;2 b1;2 a2;2;2
BA
:
b2;2 a2;1;1 b2;1 a1;1;2 b2;2 a2;1;2 >
b a
>
>
>
: 2;1 1;1;1
;
b2;1 a1;2;1 b2;2 a2;2;1 b2;1 a1;2;2 b2;2 a2;2;2
In this work, these types of contracted products arise from the dierentiation of matrix products
which, for B and C matrices of compatible dimensions, follows the rule J B C; k
J B; k2_ C B J C; k.
A special rank-3 tensor used in this work is Ik , which has unit main diagonal elements and null
non-diagonal elements. For instance
39
82
1
0
0
>
>
>
>
>
40 0 05>
>
>
>
>
>
>
>
>
>
>
0
0
0
>
>
>
>
2
3
>
>
>
< 0 0 0 >
=
4
5
0 1 0
I3
:
>
>
>
>
0
0
0
>
>
>
2
3>
>
>
>
>
>
0 0 0 >
>
>
>
>
>
4
5
>
>
0 0 0 >
>
>
:
;
0 0 1
153
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
Appendix B. Systemic parameter, implicit and mixed generalised-synergism coecients
B.1. Implicit and mixed generalised-synergism coecients of concentrations
By considering k a vector of r reaction-speci®c parameters such that O: v; k I, (15) becomes
^ 2 X; v; v O X;
^
~ 2 v; v; v O
~ 2
O
v O
X;X
v;X;v;X v; v; v
B:1
~ 2 v; v; v O2 v; X; X _ O X;
^
^
O
v O X;
v
2
X; X
k; k
B:2
with
O J; W
or
^
~ 2
^
^
S
v;X;v;X v; v; v Ir 2_ S v; v S v; v ÿ
www.elsevier.com/locate/mbs
Synergism analysis of biochemical systems. II. Tensor
formulation and treatment of stoichiometric constraints
Armindo Salvador a,b,*
Grupo de Bioquõmica e Biologia Te
oricas, Instituto de Investigacßa~o Cientõ®ca Bento da Rocha Cabral,
Ccß. Bento da Rocha Cabral, 14 P-1250 Lisbon, Portugal
Department of Microbiology and Immunology, The University of Michigan, 5641 Medical Sciences Building II,
Ann Arbor, MI 48109-0620, USA
a
b
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 eective formulation achieved in this work. The present formulation 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 sensitivity analysis and new relationships for (log-)synergism coecients 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 RederÕs 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 eectiveness 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 computational 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: [email protected] (A. Salvador).
0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 9 9 ) 0 0 0 5 7 - 7
132
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
processes [1±5]. Owing to mathematical complexity, closed-form analytical steady-state solutions of these models are seldom available, while extensive numerical simulations are
impractical for analysing responses to large sets of parameters. Dierential 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 dierential 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 characterisation of responses to single-parameter perturbations. Ease of implementation and computational 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 eectiveness 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-)synergism coecients 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 coecients of various properties are derived (Appendix B). All
the relevant expressions make use of only the simplest tensor operations, which are explained 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 coecients 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 conservation at steady-state (Section 4.2.2).
Often, sensitivity and synergism analyses of kinetic models subject to conservation relationships
can be performed with less computation than necessary for other kinetic models of the same
dimensionality. This computational advantage is especially important for large-scale integrative
models (for instance, [1±5], see Section 5).
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
133
2. Notation and conventions
The notation and conventions introduced in the previous papers are adopted here, with extensions as described below.
The term `generalised-synergism' will be used to denote a measure of deviation from a prede®ned pattern of behaviour. This includes non-normalised synergisms, relative synergisms and
log-synergisms [19]. The dierential measures of steepness of deviation from the prede®ned patterns will be denoted by `generalised-synergism coecients'. To simplify language, where there is
no ambiguity the term `sensitivities' will be used to refer to both ®rst-order sensitivities and
generalised-synergism coecients (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,
WP ko ; Dk fWPi ko ; Dk j;k g;
i 2 f1; 2; . . . ; nP g; j; k 2 f1; 2; . . . ; pg:
When the generalised-synergism coecients are similarly arranged, the calculation of the systemic
coecients requires only simple tensor operations. These operations and the corresponding notation are explained in Appendix A. Additional notation is: Ik , the k ´ k identity matrix; 0i j k ,
the i ´ j ´ k null tensor; ker(A), the kernel of matrix A; xn , 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
1
2
W P; k; k: Dk Dk;
WP ko ; Dk
2
2
similar expressions apply for the other generalised-synergisms.
If P, kA and kB are vectors, the following relationships hold:
S P; k Pnÿ1 J P; k kn ;
n
n
W2 P; kA ; kB Pnÿ1 J2 P; kA ; kB 2_ kA kB ;
S2 P; kA ; kB J2 log jPj; logjkA j; log jkB j;
S2 P; kA ; kB W2 P; kA ; kB ÿ InP 2_ S P; kA S P; kB S P; kA InA S kA ; kB ;
1
2
3
4
with nP and nA standing for the dimensions of the vectors P and kA , respectively.
The J and S operators also follow the chain rule when operating over composite functions.
That is, if F º F(P(k),k) and F and P are both dierentiable, then
J F; k Jk F; P J P; k JP F; k
5
S F; k Sk F; P S P; k SP F; k:
6
and
134
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
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 coecients [19].
The symbols V and U stand 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
X_ N v 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 producing 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 steady-states, they lack any other in¯uence on the steadystate. So, if (a) the kinetic model is structurally stable, (b) the operating point is not poised at a
bifurcation, and (c) v is dierentiable around the operating point, one obtains
_ k N Jk v; X J X; k JX v; k 0;
J X;
8
or, by normalising the sensitivities,
N vn Sk v; X S X; k SX v; k 0:
9
If (d) the matrix N Jk v; X is non-singular, one can solve (8) for the matrix of systemic sensitivities:
^
J X; k J X;
v JX v; k
10
^
J X;
v ÿ N Jk v; Xÿ1 N:
11
with
Furthermore, if (d0 ) the matrix N vn Sk v; X is non-singular, then solving (9) for the systemic
logarithmic sensitivities gives
^
S X; k S X;
v SX v; k
12
ÿ
ÿ1
^
S X;
v ÿ N vn Sk v; X
N vn :
13
with
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
135
^
We call systemic sensitivities of the type O P;
v, which cannot be obtained through explicit
dierentiation 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 properties that are functions of concentrations (derived properties) 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 ¯uxoriented theory [21].
If v is twice-dierentiable around the operating point and kA and kB are two vectors of parameters, the application of chain rule (5) to Eq. (8) gives
2
2
N J:;:
v; X; kA T1;3;2 J X; kB
v; X; X2_ J X; kA J X; kB J:;:
2
2
v; kA ; kB 0:
v; X; kB 2_ J X; kA JkA ;kB v; X:J2 X; kA ; kB J:;:
J:;:
14
The solutions of (14) and of its counterpart for logarithmic co-ordinates share the form
2
2
^
~
~
v O
O2 X; kA ; kB O X;
X;X v; kA ; kB Ov;X;v;X v; kA ; kB :
15
~ 2
~ 2
The subscripts of O
X;X and Ov;X;v;X indicate variables that are allowed ®rst-order variations
(linear, for J, or power-law for W and S) with the parameters. So
~ 2 v; kA ; kB O2 v; X; X _ O X; kA O X; kB O2 v; X; kA T1;3;2 O X; kB
O
2
X;X
k; k
k; X
2
2
O k; X v; X; kB 2_ O X; kA O X; X v; kA ; kB
16
represents the generalised-synergism coecients that would be observed for rates of reaction if the
concentrations responded linearly (for J) or as a power-law (for W and S) to single-parameter
perturbations. These magnitudes are considered as semi-intrinsic generalised-synergism coecients, which take the ®rst-order systemic responses of the concentrations into account but assume
that the deviations of each ¯ux from the ®rst-order approximations do not interact.
~ 2
The term O
v;X;v;X v; kA ; kB is null when O stands for J or W, or takes the value
ÿ
~ 2
S
17
v;X;v;X v; kA ; kB Ir 2_ S v; kA S v; kB ÿ S v; kA IpA S kA ; kB :
Table 1
Systemic ®rst-order sensitivities of derived steady-state propertiesa
Cartesian space
Logarithmic space
^ v Ok v; X O X;
^
O v;
v I
^ ; v N J v;
^ v
J V
^ ; v ÿ Un J X;
^
^
v
J U;
v Xnÿ1 J V
^ v OX v; k
O v; k O v;
J(V ,k) N áJ(v,k)
a
J U; k Xnÿ1 J V ; k ÿ Un J X; k
ÿ
N is obtained from N by setting the negative elements to 0.
^ ; v SX ;k V ; v S v;
^ v
S V
n ^
Vnÿ1
N v S v; v
^
^
^ ; v ÿ S X;
v
S U;
v S V
S V ; k SX;k V ; v S v; k
n
Vnÿ1
N v S v; k
S U; k S V ; k ÿ S X; k
136
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
~ 2
The elements of O
v;X;v;X v; kA ; kB are also semi-intrinsic generalised-synergism coecients. As
^
S X; v S v; kA 0, for log-synergism coecients (15) simpli®es to
ÿ
^
~ 2 v; kA ; kB I _ S v; kA S v; kB c:
S2 X; kA ; kB S X;
v bS
18
r2
X;X
Systemic generalised-synergism coecients for ¯uxes, turnover numbers and other derived steadystate properties, P P X k; k, can be computed from the systemic concentration generalisedsynergism coecients through
2
2
O2 P; kA ; kB O k; k P; X; X2_ O; X; kA O X; kB O k; X P; X; kA T1;3;2 O X; kB
2
Ok P; X O2 X; kA ; kB O k; X P; X; kB 2_ O X; kA
2
O X; X P; kA ; kB :
19
By considering k a vector of r reaction-speci®c parameters such that O: v; k I, one can use (15)
^ 2 X; v; k] generalised-syn^ 2 X; v; v] and mixed [intrinsic-parameter, O
to compute intrinsic [O
ergism coecients. Appendix B presents the derivation of systemic, intrinsic and mixed generalised-synergism coecients 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 dismutation ± quickly catalysed by superoxide dismutase (SOD, reaction 2) ± produces hydrogen
peroxide (H2 O2 ). Though the latter is mostly reduced to water through a glutathione-peroxidasecatalysed 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). Superoxide 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
á
Oáÿ
2 ; X2 H2 O2 and X3 HO as dependent variables, and X4 O2 ; X5 SOD; X6
Fe2 and X7 GPx as parameters of interest (vector k). The rate vd v5 v6 k5 X1 k6 X3 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
Fig. 1. Simple scheme of the metabolism of reactive oxygen species in the matrix of rat liver mitochondria. Reactions 5
and 6 account for damage to mitochondrial components. The abbreviations are: GPx, `classic' glutathione peroxidase,
GSH, glutathione, SOD, Mn-dependent superoxide dismutase.
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
137
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
v1
v2
v3
v4
k1 O2
k2 SODOáÿ
2
k3 Fe2 H2 O2
/1 =H2 O2GPx
/2 =GSH
v5 k5 Oáÿ
2
v6 k6 HOá
Parameters
k1 0:16sÿ1 ; O2 1:0 10ÿ4 M (in lipid bilayers)
k2 4:7 109 Mÿ1 sÿ1 ; SOD 1:1 10ÿ5 M
k3 2:0 104 Mÿ1 sÿ1 ; Fe2 1:0 10ÿ7 M
/1 4:8 10ÿ8 M s; /2 2:5 10ÿ5 M s; GPx 2:5 10ÿ6 M; GSH 1:1 10ÿ2 M
k5 5:9 sÿ1
k6 3:1 109 sÿ1
antioxidant defences and pro-oxidant agents as regards vd . For that we compute the log- and
relative-synergism coecients of vd , through the steps described below:
1. Compute the steady-state concentrations and rates. This gives X1 3:1 10ÿ10 M; X2 1:5
10ÿ7 M; X3 9:9 10ÿ20 M; v1 1:6 10ÿ5 M sÿ1 ; v2 8:0 10ÿ6 M sÿ1 ; v3 3:1 10ÿ10
M sÿ1 ; v4 8:0 10ÿ6 M sÿ1 ; v5 1:8 10ÿ9 M sÿ1 ; v6 3:1 10ÿ10 M sÿ1 .
2. Compute the tensors of intrinsic sensitivities and log-synergism coecients by dierentiating
the rate expressions according to the de®nitions of the respective operators [19] and replacing
the operating values. This yields
2
S X; X v; k; k 0644 :
3. Compute the implicit sensitivities, using (13):
2
3
1:0
ÿ 1:0
0
0
ÿ 1:1 10ÿ4
0
S X; v 4 1:0 1:2 10ÿ4 ÿ 3:9 10ÿ5 ÿ 1:0 ÿ 1:2 10ÿ4
0 5:
ÿ4
ÿ4
1:0 1:2 10
0
ÿ 1:0 ÿ 1:2 10
ÿ 1:0
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:
2
3
1:0
ÿ 1:0
0
0
S X; k 4 1:0 1:2 10ÿ4 ÿ 3:9 10ÿ5 ÿ 1:0 5;
1:0 1:2 10ÿ4
1:0
ÿ 1:0
138
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
3
0
0
7
0
0
7
1:0
ÿ 1:0 7
7:
ÿ 3:8 10ÿ5 3:9 10ÿ5 7
7
5
0
0
1:0
ÿ 1:0
ÿ
~ 2 v; k; k, through (16), and I _ S v; k :S v; k
5. Compute S
2
r
X;X
2
1
6 1:0
6
6 1:0
S v; k 6
6 1:0
6
4 1:0
1:0
0
1:1 10ÿ4
1:2 10ÿ4
1:1 10ÿ4
ÿ 1:0
1:2 10ÿ4
2
3
0 0 0
0 0 07
7
0 0 05
0 0 0
2
0
0
1:1 10ÿ4
6 1:1 10ÿ4 ÿ 1:1 10ÿ4 0
6
4
0
0
0
0
0
0
2
7:4 10ÿ3
1:2 10ÿ4
1:0
6 1:2 10ÿ4 ÿ 1:2 10ÿ4
1:2 10ÿ4
6
ÿ4
4
1:0
1:2 10
ÿ 3:9 10ÿ5
ÿ4
ÿ1:0
ÿ 1:2 10
ÿ 1:0
Ir 2_ S v; k:S v; k 2
ÿ7
ÿ4
ÿ2:8 10
1:1 10
ÿ 3:8 10ÿ5
>
>
>
ÿ4
ÿ4
>
6
>
ÿ 1:1 10
ÿ 4:4 10ÿ9
> 6 1:1 10
>
ÿ5
ÿ9
>
4
>
ÿ3:8 10
ÿ 4:4 10
3:8 10ÿ5
>
>
>
ÿ9
ÿ9
>
3:9 10ÿ5
4:4
ÿ 1:5 10
>
3
2 10
>
>
>
0
ÿ 1:0 0 0
>
>
>
6 ÿ1:0
>
2:0
0 07
>
7
6
>
>
4 0
>
0
0 05
>
>
>
>
0
0
0 0
>
>
2
>
ÿ3
ÿ4
>
>
7:4
10
1:2
10
1:0
>
> 6
>
>
1:2 10ÿ4
1:2 10ÿ4 ÿ 1:2 10ÿ4
>
6
>
>
4
>
1:0
1:2 10ÿ4
ÿ 3:9 10ÿ5
>
:
ÿ4
ÿ1:0
ÿ 1:2 10
ÿ 1:0
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
0
60
6
40
0
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
3
>
>
>
0
>
>
>
7
>
07
>
>
>
5
>
0
>
>
>
>
0
3 >
>
>
>
ÿ 1:0
>
>
ÿ4 7 >
ÿ 1:2 10 7 >
>
>
5 >
>
ÿ 1:0
>
>
=
2:0
3
:
3:9 10ÿ5
>
>
>
>
4:4 10ÿ9 7
>
7>
ÿ9 5 >
>
>
ÿ 1:5 10
>
>
>
ÿ5
>
ÿ 3:9 10
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
3 >
>
>
>
ÿ 1:0
>
ÿ4 7 >
>
ÿ 1:2 10 7 >
>
>
>
5
>
ÿ 1:0
>
;
2:0
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
139
6. Compute the systemic log-synergism coecients, through (18), which yields
3
2
8
9
0
0
0 0
>
>
>
>
>
>
6 0 ÿ 1:1 10ÿ4 0 0 7
>
>
>
>
7
6
>
>
>
>
5
4
>
>
0
0
0
0
>
>
>
>
>
>
>
>
0
0
0
0
>
>
>
>
3
2
>
>
ÿ3
ÿ7
ÿ3
ÿ7
>
>
7:4
10
8:5
10
ÿ
7:4
10
5:7
10
>
>
>
>
>
>
>
>
>
ÿ7
>
>
>
ÿ 6:5 10ÿ11
3:9 10ÿ5
ÿ 3:9 10ÿ5 3 >
>
>
2 5:7 10ÿ3
>
>
>
ÿ7
ÿ3
ÿ7
>
>
>
7:4
10
8:5
10
ÿ
7:4
10
5:7
10
>
>
>
>
>
ÿ7
ÿ4
ÿ7
ÿ11 7 >
6
>
>
8:5
10
ÿ
1:2
10
ÿ
8:5
10
ÿ
6:5
10
>
>
7
6
>
>
>
>
ÿ3
ÿ7
ÿ3
ÿ5
5
4
>
>
ÿ7:4
10
ÿ
8:5
10
7:3
10
3:9
10
>
>
:
;
ÿ7
ÿ11
ÿ5
ÿ5
5:7 10
ÿ 6:5 10
3:9 10
ÿ 3:9 10
To compute the systemic relative-synergism coecients one may just apply expression (4):
3
2
8
9
0
ÿ 1:0 0 0
>
>
>
>
>
>
7
6 ÿ1:0
>
>
2:0
0
0
>
>
7
6
>
>
>
>
5
4
>
>
0
0
0
0
>
>
>
>
>
>
>
>
0
0
0
0
>
>
>2
>
3
>
>
ÿ2
ÿ4
ÿ5
>
>
1:5
10
1:2
10
ÿ
1:0
ÿ
4:0
10
>
>
>
>
>
>
>
>
>
ÿ5
>
>
>
ÿ 4:5 10ÿ9
7:8 10ÿ5
3:0 10ÿ9 3 >
>
>
2ÿ4:0 10ÿ2
>
>
ÿ4
>
>
>
>
1:2
10
ÿ
1:0
1:0
1:5
10
>
>
>
>
>
>
ÿ4
ÿ4
ÿ4
ÿ4
6
7
>
>
1:2
10
ÿ
2:3
10
ÿ
1:2
10
1:2
10
>
>
6
7
>
>
>
>
ÿ4
4
5
>
>
ÿ1:0
1:2
10
2:0
ÿ
1:0
>
>
:
;
ÿ4
ÿ5
1:0
1:2 10
ÿ 1:0
ÿ 3:9 10
7. Compute the logarithmic gains of the rate of damage, through (6):
S vd ; k 1:0 ÿ 0:86 0:14 ÿ 0:14 :
8. Finally, by applying expression (19) obtain the log-synergism coecients of the total rate of
damage (vd ) and ± again through (4) ± the corresponding relative-synergism coecients:
3
2
1:1 10ÿ3 9:0 10ÿ4 ÿ 2:0 10ÿ3 9:0 10ÿ4
6 9:0 10ÿ4
0:12
ÿ 0:12
0:12 7
7;
S2 vd ; k; k 6
ÿ3
4 ÿ2:0 10
ÿ 0:12
0:13
ÿ 0:12 5
9:0 10ÿ4
0:12
ÿ 0:12
0:12
3
2
0:14
ÿ 0:14
2:1 10ÿ3 ÿ 0:86
6 ÿ0:86
1:7
ÿ 0:25
0:25 7
7:
W2 vd ; k; k 6
ÿ3
4 0:14
ÿ 0:25 2:9 10
ÿ 0:14 5
ÿ0:14
0:25
ÿ 0:14
0:29
Analysis of the latter pair of matrices yields the following inferences. First, positive synergisms
between pro-oxidant agents (O2 , Fe2 ) and between antioxidant defences (SOD, GPx), as well as
140
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
antagonisms (negative relative-synergism coecients) between pro-oxidants and antioxidant defences should hold. The strongest antagonism predicted is between oxygen and SOD
W2 vd ; X4 ; X5 ÿ0:86 and the strongest synergism between SOD and GPx
W2 vd ; X5 ; X7 0:25. Second, combined changes of oxygen concentration and any of the other
factors should yield nearly multiplicative responses (log-synergism coecients near 0). Joint
perturbations of [SOD] and [GPx] yield a slightly supra-multiplicative response
S2 vd ; X5 ; X7 0:12, 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 coecients near 0) and varies almost linearly with [Fe2 ] (relative-synergism coecients near 0). The response to [SOD] is clearly supra-linear
W2 vd ; X5 ; X5 1:7 and slightly supra-power-law S2 vd ; X5 ; X5 0:12, indicating diminishing returns of increasing 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_ N v 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 (XN ) and parameters (kN ) instead of the `natural' concentrations and parameters (X and k). Usually, it is not
too restrictive to assume that the transformation of X k into XN kN is such that X and k can be
expressed as double-dierentiable functions of XN and kN , respectively: k k kN ;
X X XN kN k; kN k. The chain rule of dierentiation then gives
O X; k XXO O XN ; kN XkO PO ;
20
O X; kN XXO O XN ; kN XkO ;
21
n
;X
X ;k
O2 X; k; k XXO2
2_ O XN ; kN O XN ; kN XO2 2_ O XN ; kN
o
T
;k 1;3;2
O XN ; kN XOk;k2 2_ PO PO
XXO O2 XN ; kN ; kN XXO2
ÿ
XXO O XN ; kN XkO PO2 ;
22
;k
X
O2 X; kN ; kN XXO2;X 2_ O XN ; kN O XN ; kN XXO2
2_ O XN ; kN XO
T1;3;2
;k
O2 XN ; kN ; kN XXO2
O XN ; kN XOk;k2 ;
23
141
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
where
XXO OkN X; XN ;
XkO OXN X; kN ;
;X
O2 X; XN ; XN ;
XXO2
PO O kN ; k;
XXO2;k O2 X; XN ; kN ;
XOk;k2 O2 X; kN ; kN ;
24
PO2 O2 kN ; k; k
and O stands for J, W or S. Whereas O XN ; kN and O2 XN ; kN ; kN refer to the properties of the
;X
;k
; XXO2
, XOk;k2 and PO2 refer to the properties of the transformation. We
system, XXO ; XkO ; PO ; XXO2
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 proportional one can derive formulas for the conversion of implicit and mixed sensitivities:
^ N ; v;
^
25
O X;
v XX O X
O
^ 2 X; v; v XX 2;X _ O X
^ N ; v O X
^ 2 XN ; v; v;
^ N ; v XX O
O
O
O 2
26
i
h
;k ^
X ^ 2
^ 2 X; v; k XX 2;X _ O X
^ N ; v O XN ;k XX 2
O
O
X
;v;k
PO :
O X
;
v
X
_
N
N
O
O 2
O 2
27
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
X_ N NN vN XN ; kN ;
28
and, considering v v vN ; k, ®nd
O v; k VvO O vN ; kN PO VkO ;
^ 2 v; k; k Vv;v2 _ O vN ; kN PO O vN ; kN PO Vv O2 vN ; kN ; kN _ PO
O
O
2
O 2
29
T1;3;2
PO VvO O vN ; kN PO2 Vv;k
O vN ; kN PO Vv;k
O2
O2 2_
O vN ; kN PO VOk;k2 ;
with the ¯ux link tensors de®ned as
VvO Ok v; vN ; VkO OvN v; k;
2
Vv;v
O k; k v; vN ; vN ;
O2
2
Vv;k
O k; vN v; vN ; k;
O2
2
VOk;k2 O vN ; vN v; k; k:
30
31
The following expressions allow calculating the sensitivities for the transformed set of parameters:
(
O v; kN VvO O vN ; kN VkO PO ;
O2 v; k; kN Vv;v
O v
;
k
O vN ; kN VvO O2 vN ; kN ; kN
_
N
N
2
O 2
(
T1;3;2
VOk;k2 2_ O vN ; kN PO VOv;k2
(
(
(
(
VOk;k2 2_ PO PO Vk PO2 ;
(
with P O k; kN and PO2 O2 k; kN ; kN :
(
2_ PO
32
O vn ; kN
33
142
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
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 matrix N has n rows and rank n0 < n,
then there are n ÿ n0 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 express n ÿ n0 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 of n0 dierential equations
corresponding to the remaining n0 dynamical variables (vector XN ) is sucient 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
for O standing for J or W; or
S2 X; kN ; kN XXS S2 XN ; kN ; kN XXS 2;X 2_ S XN ; kN S XN ; kN XSX;k
_ S XN ; kN
2 2
XSX;k
2
T1;3;2
S XN ; kN XSk;k2 :
36
This procedure extends the synergism analysis described in [19] to systems subject to structural
conservation relationships.
For computing the systemic sensitivities O XN ; kN and O2 XN ; kN ; kN , note that the matrices
NN JkN v; XN and NN vn SkN v; XN are non-singular. (NN is formed by the n0 rows of N that
correspond to the variables in XN .) Hence,
^ N ; v ÿ NN Jk v; XN ÿ1 NN
J X
N
37
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
143
and
^ N ; v ÿ NN vn Sk v; XN ÿ1 NN vn
S X
N
38
OkN v; XN Ok v; X XXO :
39
with
The intrinsic sensitivities of the ¯uxes to the transformed set of parameters are also straightforwardly 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 transformed system can be related through
2
2
O kN ; kN v; XN ; XN Ok v; X XXO2;X O k; k v; X; X2_ XXO XXO ;
41
2
2
;k
~ k O kA ; kN
O kN ; XN v; XN ; kN Ok v; X XXO2
O k; XN ;k v; X; XE 2_ XXO X
O
(
2
O k; X v; X; k2_ XXO P;
42
and
2
~ k;k2 _ O kA ; kN O kA ; kN
OXN ;XN v; kN ; kN OXN ;k v; XE X
O 2
2
~ k O kA ; kN X
~ k O kA ; kN
O XN ;k; XN ;k v; XE ; XE 2_ X
O
O
(
2
~ k O kA ; kN P
O XN ;k; X v; XE ; k2_ X
O
(
2
~ k O kA ; kN
O XN ;k; X v; XE ; kT1;3;2 2_ P X
O
2
(
(
O X; X v; k; k2_ P P:
43
. All the intrinsic sensitivities of the transformed system might as well be
Here,
obtained through dierentiation 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 an n n matrix are of order n3 [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
r r matrix for obtaining simultaneously the ¯ux control coecients and the concentration
control coecients of a system. Yet, by calculating the systemic ¯ux sensitivities through
O v; kN Ok v; X:O X; kN OXN v; kN , one obtains the same result by inverting a smaller
(n0 n0 ) matrix. Any kinetic model with a non-trivial steady-state has r > n P n0 . If all processes
are considered reversible, then r > 2n; and, as cycles are prevalent in metabolic networks, r can be
144
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
much higher than 2n. The computational cost of the two alternative strategies for computing the
systemic sensitivities may then dier 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 and X7 for free enzymes,
X2 and X4 for X6 :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 X10 X11 1; k2 0:35; k3 3:5; k4 100; k5 2; k6 k7 k9 104 ; k8 105 : The dimensionless concentrations are then
T
X 9:51 10ÿ1 4:83 10ÿ1 1:02 10ÿ1 5:00 10ÿ1 8:31 10ÿ3 5:09 10ÿ1 4:92 10ÿ1 :
The dimensionless ¯uxes are
v 1
0:169 0:831 0:831 1
484: 483: 500:2
T
500:0 ;
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
3
2
1
0
0
0
0 ÿ1
1
0
0
6 0 ÿ1 ÿ1
0
0
1 ÿ1
0
07
7
6
7
60
1
0
0
0
0
0
ÿ1
1
7
6
6
N 60
0
0
1 ÿ1
0
0
1 ÿ1 7
7
60
0
1 ÿ1
0
0
0
0
07
7
6
40
1
0
1
0 ÿ1
1
0
05
0
0 ÿ1
0
1
0
0 ÿ1
1
has n 7 rows and rank n0 5, there are two conservation relationships. As follows from
0 1 0 0 1 1 0
T
;
L
0 0 0 1 1 0 1
Scheme 1.
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
145
they are
X2 X5 X6 X10 ;
X4 X5 X7 X11 ;
where X10 and X11 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
kN k1 k9 X10 X11 T it follows from (C.1) and (C.3):
~k
X
S
"
X10
X6
0
0
X11
X7
#
;
44
45
For the second-order link tensors, (C.6) and (C.10) give
46
146
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
(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 replace SkN v; XN into
(38), to ®nd
3
2
3:63
ÿ 0:336
ÿ 1:61 ÿ 0:0436 ÿ 1:64 ÿ 1 0:988
0
0
6 1:86
ÿ 0:171
ÿ 0:829 ÿ 0:0140 ÿ 0:843
0
0
0
0 7
7
6
^
6
0
0
ÿ 1 1:00 7
S XN ; v 6 2:03 ÿ 0:00257 0:00257 ÿ 0:0169 ÿ 2:01
7;
4 1
0
0
0
ÿ1
0
0
0
0 5
0:826
ÿ 0:169
0:169
ÿ 0:997
0:171
0
0
0
0
49
and apply (12) to compute S XN ; kN .
(4) Compute the systemic ®rst-order sensitivities of the original set of concentrations, through
(21)
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
147
50
^ N ; v, as k is the vector of rate constants.
Here S XN ; k S X
(5) Compute the systemic log-synergism coecients of the transformed system. As the reactions
follow mass action kinetics, the intrinsic log-synergism coecients of the original system are null.
So, (41)±(43) simplify to
2
S kN ; kN v; XN ; XN Sk v; X XXS 2;X ;
2
S kN ; XN v; XN ; kN Sk v; X XXS 2;k ;
and
2
~ k;k2 _ S kA ; kN S kA ; kN ;
S XN ; XN v; kN ; kN SXN ;k v; XE X
2
S
respectively. The replacement of these expressions into (18) yields S2 XN ; kN ; kN . For conciseness, only the log-synergism coecients for kC k1 X10 X11 T are shown
82
39
9:51
ÿ 3:49 ÿ 8:57 >
>
>
>
>
>
> 4 ÿ3:49
>
>
1:90
3:13 5 >
>
>
>
>
>
>
> ÿ8:57
>
3:13
7:45
>
>
2
3
>
>
>
>
>
>
1:58
0
ÿ
1:43
>
>
>
>
>
>
4
5
>
>
0
0
0
>
>
>
>
>
>
>
>
ÿ1:43
0
1:14
>
> 2
>
3 >
>
>
>
>
2:08 0 ÿ 2:08
<
=
2
4 0
S XN ; kC ; kC
0
0 5
:
51
>
>
>
>
ÿ2:08
2:07
>
>
>
>
2 0
3
>
>
>
>
>
>
0
0
0
>
>
>
>
>
>
4
5
>
>
0
0
0
>
>
>
>
>
>
>
>
0
0
0
>
>
2
3
>
>
>
>
>
>
ÿ0:501
0
0:644
>
>
>
>
>
>
4
5
>
>
0
0
0
>
>
>
>
:
;
0:644 0 ÿ 0:927
148
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
(6) Compute the systemic log-synergism coecients for the original set of variables, by replacing S2 XN ; kN ; kN into (23)
52
Note the high log-synergism coecients, even as compared to the ®rst-order sensitivities. The
relative-synergism coecients (not shown) reach even higher values. Such high log- and relativesynergism coecients indicate strong deviations from both power-law/multiplicative and linear/
additive behaviour [19]. Hence, descriptions of this system that are based on ®rst-order approximations either in Cartesian or logarithmic co-ordinates may be very inaccurate and may overlook, 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 (n r) stoichiometric matrix of a kinetic
,
model has matrix-rank n0 < r. If N is full rank (n0 n), one can rearrange it as
where NE is a square sub-matrix formed by n0 linearly independent columns (corresponding to n0
eliminated rates of reaction) and ND is formed by the remaining columns. If n0 < n, N should 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,
~ v vN , with V
~ v _ Jk vE ; vN ÿNÿ1 ND . So, v Vv vN , with
which leads to vE V
J
J
E
J
53
It also follows that VkO 0; VOk;k2 0; Vv;k
0;
O2
The non-trivial logarithmic link tensors are
Vv;v
0;
J 2
Vv;v
0.
W 2
54
55
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
149
with
~ v vnÿ1 V
~ v vnN ;
V
S
J
~ v:
~ v I ÿ I _ V
~v V
~ v;v2 V
V
2
S
r
r
ÿr
S
S
S
0
0
56
57
Because this transformation does not change the variables nor the parameters, the X and P link
tensors for J, W and S 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 VvO O vN ; k
and
O2 v; k; k VvO O2 vN ; k; k
for J or W, or
S vN ; k S vN ; k VvS S2 vN ; k; k:
S2 v; k; k Vv;v
S 2 2_
4.2.2.2. Example. Consider the example in Section 4.2.1.2. The stoichiometric matrix of the
transformed system is
2
1
60
6
NN 6
60
40
0
0
0
0
ÿ1 ÿ1
0
1
0
0
0
0
1
0
1 ÿ1
0
0
0
ÿ1
0
ÿ1
1
0
0
0
1
ÿ1
0
0
0
3
0
0
0
07
7
ÿ1
17
7:
1 ÿ1 5
0
0
The analysis of NN 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
v3 ; v4 ; v5 ; v7 and v9 , one ®nds
2
0
6 ÿ1
6
NE 6
6 1
4 0
0
0
0
0
1
ÿ1
0
0
0
ÿ1
0
3
1
0
ÿ1
07
7
0
17
7;
0 ÿ1 5
0
0
2
1
60
6
ND 6
60
40
0
0
ÿ1
1
0
0
ÿ1
1
0
0
0
3
0
07
7
17
7
ÿ1 5
0
from which follows
v3 v1 ÿ v2 ;
v4 v1 ÿ v2 ;
v5 v1 ;
v7 v6 ÿ v1 ;
v9 v8 ÿ v2 :
150
Hence, according to
2
1
6 0
6 v1
6 v ÿv
6 1 v1 2
6 v ÿv
6 1 2
v
VS 6
6 1
6 0
6
1
6 ÿ v vÿv
6 6 1
4 0
0
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
(54),
0
1
2
ÿ v1vÿv
v2 2
ÿ v1 ÿv2
0
0
0
0
2
ÿ v8vÿv
2
0
0
0
0
0
1
v6
v6 ÿv1
0
0
0
0
0
0
0
0
0
1
v8
v8 ÿv2
3
7
7
7
7
7
7
7:
7
7
7
7
7
5
The relationship S v; kN VvS S vN ; kN permits computing the sensitivities of all ¯uxes from
the knowledge of the sensitivities of v1 ; v2 ; v6 and v8 .
5. Discussion
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 eective 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 generalised-synergism coecients (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 dierent 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 with n internal metabolites, r reactions and p
parameters, with n; r; p 1 and p > n, straight application of the formulas (15) through (18)
would require rnp n 3p multiplications and rnp n 3p sums. However, if kA kB the matrix
O2 Pi ; kA ; kB of generalised-synergism coecients for each scalar property Pi is symmetric (i.e.,
O2 Pi ; kj ; kk O2 Pi ; kk ; kj , and so a tensor O2 P; k; k has at most nP p p ÿ 1=2 dierent
2
2
coecients. The symmetry of the tensors O2 X; k; k, O k; k v; X; X, O X; X v; k; k and
2
O k; X v; X; k2_ O X; l allows computing O2 X; k; k with about rnp n 4p=2 multiplications
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 coecients are of interest. As
~ 2
W
v;X;v;X v; kA ; kB 0, the systemic relative-synergism coecients may be easier to compute
through (15) than the corresponding log-synergism coecients. However, generalised mass action
[31] rate expressions yield null intrinsic log-synergism coecients. So, if the model includes many
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
151
of those rate expressions, it is more eective to compute ®rst the systemic log-synergism coecients through (15) and obtaining then the systemic relative-synergism coecients 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 aect 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 link sub-tensors instead of the full tensors. Important
gains in numerical eectiveness 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 generalisedsynergism 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 mitochondrial matrix. This analysis highlighted (a) SODs importance in antagonising the eects of
oxygen activation and (b) the synergistic eect of SOD and GPx activities in decreasing the rate of
damage to mitochondrial components by oxygen radicals. The analysis also indicates that although 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 coecients remain convenient indexes of relevance of
multifactorial eects.
As more data becomes available, the need for large-scale integrative approaches to biochemistry 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, information 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 2 2 2 tensor, A, can be written as
9
8
a1;1;1 a1;1;2 >
>
>
>
< a
a1;2;2 =
1;2;1
:
A fai;j;k g
>
>
> a2;1;1 a2;1;2 >
;
:
a2;2;1 a2;2;2
152
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
Transposing a tensor corresponds to exchanging its indexes. A rank-n tensor can be transposed in n!
ways, which are distinguished by the indexes of the transposition symbol: TTi1 ;i2 ;...;in is the tensor obtained by turning the ®rst index of T into the i1 th index, the second index into the i2 th, etc. For instance
9
8
a
a
>
>
1;1;1
1;2;1
>
>
< a
=
a
1;1;2
1;2;2
T1;3;2
A
:
a
a2;2;1 >
>
>
>
: 2;1;1
;
a2;1;2 a2;2;2
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_i B is the contracted
product over the ith index of T and the ®rst index of B. For simplicity, the indication of the index
is omitted in contracted products over the last index of the right hand tensor. For instance, if B is
2 2, the contracted products of A by B over the third and over the second indexes of A are
9
8
a1;1;1 b1;1 a1;1;2 b2;1 a1;1;1 b1;2 a1;1;2 b2;2 >
>
>
>
< a b a b
=
a
1;2;1 1;1
1;2;2 2;1
1;2;1 b1;2 a1;2;2 b2;2
A B A3_ B
a b a2;1;2 b2;1 a2;1;1 b1;2 a2;1;2 b2;2 >
>
>
>
: 2;1;1 1;1
;
a2;2;1 b1;1 a2;2;2 b2;1 a2;2;1 b1;2 a2;2;2 b2;2
and
9
8
a1;1;1 b1;1 a1;2;1 b2;1 a1;1;2 b1;1 a1;2;2 b2;1 >
>
>
>
< a b a b
a1;1;2 b1;2 a1;2;2 b2;2 =
1;1;1 1;2
1;2;1 2;2
T1;3;2
T1;3;2
;
B
A2_ B A
> a2;1;1 b1;1 a2;2;1 b2;1 a2;1;2 b1;1 a2;2;2 b2;1 >
>
>
;
:
a2;1;1 b1;2 a2;2;1 b2;2 a2;1;2 b1;2 a2;2;2 b2;2
respectively. The multiplication of B by A over the ®rst index of A is
9
8
a
b
a
b
a
b
a
b
>
>
1;1
1;1;1
1;2
2;1;1
1;1
1;1;2
1;2
2;1;2
>
>
< b a
=
b
1;1 1;2;1 b1;2 a2;2;1
1;1 a1;2;2 b1;2 a2;2;2
BA
:
b2;2 a2;1;1 b2;1 a1;1;2 b2;2 a2;1;2 >
b a
>
>
>
: 2;1 1;1;1
;
b2;1 a1;2;1 b2;2 a2;2;1 b2;1 a1;2;2 b2;2 a2;2;2
In this work, these types of contracted products arise from the dierentiation of matrix products
which, for B and C matrices of compatible dimensions, follows the rule J B C; k
J B; k2_ C B J C; k.
A special rank-3 tensor used in this work is Ik , which has unit main diagonal elements and null
non-diagonal elements. For instance
39
82
1
0
0
>
>
>
>
>
40 0 05>
>
>
>
>
>
>
>
>
>
>
0
0
0
>
>
>
>
2
3
>
>
>
< 0 0 0 >
=
4
5
0 1 0
I3
:
>
>
>
>
0
0
0
>
>
>
2
3>
>
>
>
>
>
0 0 0 >
>
>
>
>
>
4
5
>
>
0 0 0 >
>
>
:
;
0 0 1
153
A. Salvador / Mathematical Biosciences 163 (2000) 131±158
Appendix B. Systemic parameter, implicit and mixed generalised-synergism coecients
B.1. Implicit and mixed generalised-synergism coecients of concentrations
By considering k a vector of r reaction-speci®c parameters such that O: v; k I, (15) becomes
^ 2 X; v; v O X;
^
~ 2 v; v; v O
~ 2
O
v O
X;X
v;X;v;X v; v; v
B:1
~ 2 v; v; v O2 v; X; X _ O X;
^
^
O
v O X;
v
2
X; X
k; k
B:2
with
O J; W
or
^
~ 2
^
^
S
v;X;v;X v; v; v Ir 2_ S v; v S v; v ÿ