G
r
G
r
D
k
r + 1
j
r + 1
m
r + 1
G
r
= G
r − 1
D
k
r
j
r
m
r
G
r − 1
Unfortunately, these
straightforward ap-
proaches are impractical in many cases. The rea- son is the well known ‘sign problem’, which is
frequently encountered in theoretical physics ap- plications that deal with diagrammatic expansions
see Rozhdestvensky and Favorsky, 1992. In- deed, the numerator and denominator in Eq. 23
may be infinite sums comprising terms with alter- ing signs. Different signs arise here because of the
rules Eqs. 16 and 17. The numerical values of such sums are typically determined with consider-
able uncertainty and their ratios may be com- pletely ill defined. Such a situation in the quantum
Monte-Carlo simulations of antiferromagnetic systems is described, for example, by Rojdestven-
ski 1995. Some possible ways to deal with this problem are either to concentrate on specific cases
where this sign problem does not appear as in Handscomb, 1964 or to choose a different repre-
sentation of variables as in Rozhdestvensky and Favorsky, 1992. We will discuss, in fact, a combi-
nation of these two approaches in application to the current problem.
4. Two types of branching in metabolic graphs
We start by recognising that, besides the in- evitable loops, the negative signs in Eq. 18 ap-
r
have n
ss
G
r
= n
pp
G
r
= 0. We should note that this does not rule out
branching completely. In fact, there are two possi- ble types of branching in metabolic systems. One
may be called ‘metabolite’ branching, where one metabolite is a substrate or a product of two
reactions in different branches. Several examples of such branching may be found, for instance, in
the dark reactions of photosynthesis, such as transformation of glucose-6-phosphate via two
pathways — glycogen biosynthetic pathway, with glucose-6-phosphate being converted into glucose-
1-phosphate by action of phosphoglucornutase enzyme, and penthose phosphate pathway in
which the same glucose-6-phosphate is converted into 6-phosphogluconolactone by action of G6P
dehydrogenase. Diagrammatically this type of sit- uation is shown in Fig. 4a.
Another branching
type, termed
‘enzyme’ branching, takes place if a certain reaction itself is
a branching point. Specifically, it produces two or more products further used in different metabolic
branches andor utilises two or more substrates coming from different metabolic branches, as
shown in Fig. 4b. Example of such type of branching can also be found in the penthose
phosphate pathway, with glyceraldehyde-3-phos- phate and sedoheptulose-7-phosphate both being
substrates for the reaction catalysed by transaldo- lase enzyme, which results in two products being
erythrose-4-phosphate and fructose-6-phosphate.
Although it substantially narrows the appli- cability of the procedures, the above restriction
allows tackling a few practically interesting situa- tions. These include the simplest case of linear
metabolic chain Fig. 5a, a linear chain coupled to a ‘futile’ cycle Fig. 5b, and two linear chains
coupled by a cycle Fig. 5c.
Fig. 4. Two types of branching in metabolic networks; a ‘metabolite’ branching; b ‘enzyme’ branching.
Fig. 5. a A simple linear metabolic chain; b a linear metabolic chain coupled to a futile cycle; c two simple
metabolic chains coupled via a two-enzyme cycle.
r = 0
G
r
m = 1
where all the terms in the summation are positive. Consequently, Eq. 23 simplifies to
C
ij k, l
C
kl
= D
ij
G
r
pG
r
31
This can now be calculated by means of slightly modified Handscomb method see Handscomb,
1962; Rojdestvenski and Cottam, 2000, as A and B in Eq. 26 become, respectively, the number of
Markov chain steps resulting in graphs starting at i and ending at j, and the total number of steps.
As always, the ultimate practicality of such an approach can only be tested by attempting calcu-
lations for real systems and comparing the Monte-Carlo simulation data with results ob-
tained using other methods, which we plan to do in near future.
6. Discussion and conclusions: metabolic graphs and information theory
