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this process corresponds, in principle, to the search for a grammar. In the case of
metabolic processes, we show that what is usually expressed in terms of differen-
tial equations can be formulated in terms of a special class of grammars, and very
often these grammars are directly related to the biochemical mechanisms of the
phenomenon under investigation.
Let us consider a biochemical system. Its state can be expressed by means of a
multiset built on the set of its chemical species (molecule types). But we know that
a chemical reaction is a rule transforming multisets of molecules into other multi-
sets of molecules. This viewpoint implies that the whole dynamics of a biochemi-
cal system, where some reactions are active, corresponds to a grammar of multiset
rewriting rules. Moreover, in order to provide a quantitative description of the ef-
fects produced by the reactions, these rules have to specify the amount of molecules
they transform. If the evolution of a system is considered along a sequence of dis-
crete steps, then a natural manner for expressing quantitative effects of rules can
be expressed by equipping rules with functions (depending on the state of the sys-
tem) which establish the fluxes, that is the quantities of substances transformed in
any evolution step. This perspective of considering biochemical systems explains
the central role of grammars in the description of biochemical dynamics, and con-
sequently, the dynamical inverse problem for a biochemical system reduces to the
search for suitable multiset grammars generating the sequence of states which we
observe through time in the given system.
Given an MP system, the recurrent equation 3.10 of EMA generates the evolution
of substances, according to the MP grammar of the system (starting from an initial
metabolic state, and with the knowledge of parameter evolution in time). In other
words, when regulators are given, the substance variations follow easily from the set
of reactions defined in the system. The flux maps express the logic of an MP system
in terms of its internal states.
Let us assume we “observe” a metabolic system for a number of steps (separated
by some temporal intervals). The observation is represented by one
time series
for
each observed substance, that is, the sequence of quantities of the substance in cor-
respondence to the observation steps.
How can we discover an MP system which provides the dynamics which we
observe? The substances and reactions in question are given by the particular phe-
nomenon we want to describe. Moreover, the stoichiometry can be deduced by a
basic knowledge of the phenomenon, but how do we know the fluxes of matter
transformation that are responsible for the observed evolution? This is the prob-
lem of discovering the flux regulation maps, or simply, the
Regulation Discovery
Problem
. Its solution can be described as the passage from the observation of the
behavior which a system exhibits in time
to the
determination of an internal
logic that provides rules of transformation of its states
. In a metabolic system,
an internal state is naturally representable by a multiset, therefore, we could synthe-
size the way of solving the dynamical inverse problem, for metabolic systems, by
means of the following statement.
