BioSystems 56 2000 63 – 73
Diagrammatic approach to calculation of the fluctuation correlation matrix in a metabolic system
I. Rojdestvenski
a,
, M.G. Cottam
b
a
Department of Plant Physiology, Umea Uni6ersity, Umea
90187
, Sweden
b
Department of Physics and Astronomy, Uni6ersity of Western Ontario, London, Ont., Canada N
6
A
3
K
7
Received 3 June 1999; received in revised form 21 January 2000; accepted 10 February 2000
Abstract
We present here a simple diagrammatic approach for the time evolution of the fluctuations in metabolite concentrations around the steady state. A fluctuation correlation matrix is introduced to characterise the response in
the concentrations of metabolites to a singular initial fluctuation in one of the metabolites. We show how the temporal evolution of the correlation matrix can be represented in the form of a series with individual terms
corresponding to pathways on a metabolic graph. The basic properties of such graphs are studied and it is shown how each term in the series can be evaluated. A Monte-Carlo procedure is outlined to calculate the fluctuation correlation
matrix. We discuss various properties of the graphical representation and discuss links to information theory that arise from it. © 2000 Elsevier Science Ireland Ltd. All rights reserved.
Keywords
:
Fluctuation correlation matrix; Metabolite; Monte-Carlo procedure www.elsevier.comlocatebiosystems
1. Introduction
Recently considerable attention has been paid to theoretical studies in the field of metabolic
control and regulation. Since the seminal work by Kacser see Kacser and Burns, 1973, where a
metabolic control analysis via control coefficients was introduced, many papers have been devoted
to developing both theoretical and experimental methods
for calculating
and measuring
the metabolic control coefficients. In particular, con-
siderable effort has been put into studying the properties of the matrix of control coefficients
e.g. Cascante et al., 1996; Giesrch, 1997; Elsner and Giersch, 1998, the ways to introduce time-
scale hierarchies Delgado and Liao, 1995, spatial blocking Brand, 1996; Rohwer et al., 1996; Ain-
scow and Brand, 1998, and ‘asynchronous au- tomata’
metaphorae of
regulatory networks
Thomas, 1991. While many of these works used graphical rep-
resentations for the metabolic pathways, the es- sential part of the analysis was done mainly by
applying matrix algebra. The graphs of metabolic pathways played only an illustrative or qualitative
Corresponding author. Tel.: + 46-70-7195291; fax: + 46- 90-7866676.
E-mail address
:
igor.rojdestvenskiplantphys.umu.se I. Rojdestvenski
0303-264700 - see front matter © 2000 Elsevier Science Ireland Ltd. All rights reserved. PII: S0303-26470000076-9
A metabolism can be described completely if one knows how the concentrations of all the
metabolites in the system vary with time. It is convenient to describe a metabolism schematically
by means of an ordered directed graph, such as the example given in Fig. 1. The vertices that are
depicted by shaded circles denote ‘pools’ of metabolites; they are each labelled by an integer i
where i [1, 2, ..., N]. The vertices that are de- picted as open circles denote metabolic reactions;
likewise they are each labelled by an integer m [1, 2, ..., M]. Here N and M stand, respectively, for
the total number of metabolites in the system and the total number of metabolic reactions in the
{reaction substrates} l
reaction
{reaction products} On the other hand, reaction is at the same time
a function and a material entity represented by the enzyme. This is a common situation when
studying biological systems, which are essentially autopoietic, i.e. represent systems in which certain
relations between their elements are included in these systems as new elements. We may argue
here that, philosophically speaking, in biological systems the difference between the thing and its
representation, formulated by Kirby 1998, often vanishes. Kirby writes, ‘‘Representations denote
classes; things are individual. Representations cannot themselves evolve like certain things;
they can only be changed. A representation does not go out in the world and function; the thing it
represents, or for whom it meditates, does’’. The point is that the representation in Kirby’s notion
does not become part of the described repre- sented system. On the other hand, the above
duality of a reaction, representing a functional relation between material objects metabolites, is,
in turn, represented by another material entity enzyme. Going further up the description hier-
archy, enzyme description is encoded in the genome. Genome thus becomes a description
representation of material objects enzymes, at the same time being itself represented by material
entities DNA, RNA. Moreover, often, as in the case of certain autocatalytic reactions, as well as
for those enzymes which catalyse gene expression
Fig. 1. An example of a metabolic graph. The shaded circles denote indexed metabolite pools and the open circles denote
indexed reactions. The arrows coming into reaction vertices represent reaction substrates, while those coming out of the
reaction vertices represent reaction products. The thick line shows an example of an ‘intuitively defined’ pathway.
In our scheme this ‘material’ aspect of the reaction function is emphasised, with the arrows
describing relations between metabolites and reac- tions i.e. the properties of being either a substrate
or a product. Very often metabolism is also described in terms of ‘pathways’. A common-
sense definition of a pathway is quite obvious, namely it is a sequence of reactions sharing a
common mass flow. An example of such a path- way is given by the thick line running through
Fig. 1.
The time evolution for the metabolic concentra- tions can be conveniently expressed in terms of a
‘time-evolution’ operator. This operator acts on the concentration vector for the initial conditions
i.e. the vector comprising the values of the con- centrations at time t = 0 to produce the concen-
tration vectors at a later time t. Our goal in developing the theory is to write the time-evolu-
tion operator for the system of metabolic equa- tions as a superposition of operators representing
the propagation along all the possible pathways of the given metabolic system. Then the weights
of the pathways would be functions of time, rep- resenting the relative ‘importance’ of the path-
ways at a given moment of time. We shall develop this into a systematic diagrammatic procedure for
evaluating the variations in time of the metabo- lite concentrations.
2. Analysis of the fluctuations about the steady state
