13th International Conference on Membrane Computing - MTA Sztaki

V. Manca
expression levels, etc.) denoted by a reaction such as X → Y, meaning that X (in
general all the real variables on the left part of the rule) decreases of a certain
amount, while Y (in general all the right real variables) increases of the same
amount (with respect to some measurement unit). The decreasing/increasing
amount, called flux of the rule, is determined by a regulator (regulation map) f
depending on some variables of the system. In the case that a variable occurs in
a rule with multiplicity k greater than 1, its increasing/decreasing is k times the
value of the flux of the rule.
An MP grammar (see [2-4] for formal definitions) is a set of rules: reaction +
regulator. A reaction is constituted by left variables (decreasing) → right variables
(increasing), each variable with a corresponding multiplicity. A regulator,
is a function providing the flux of the rule, in dependence of the values of some
regulation variables, called tuners of the rule. Given a grammar, when we start
from an initial state (the values of the variables at an initial time), by applying
all the rules of the grammar, we obtain the next state, and so on, for all the subsequent
steps. An MP grammar becomes an MP system when some numerical
values are fixed for the physical interpretation of the time series: the time interval
between two consecutive applications of rules, and other values related to the
quantity units (depending on the physical nature of the variables). In mathematical
terms an MP grammar is specified by: variables, reactions, regulators, and
initial values. Variables which do not occur in reactions, but occur in regulators
are called parameters. Therefore, an MP grammar deterministically generates
a time series for each of its proper variables (different from parameters), which
is determined by its initial state (and by the time series of parameters if they
are present). It is easy to realize that an MP grammar define a system of finite
difference equations which represent the invariant of the dynamical system
generated from the initial state.
An important mathematical aspect of MP grammars is their representation
in linear algebra notation (by means of vectors and matrices). This makes very
efficient the computation of the dynamics generated by an MP grammar, which
provides a particular kind of finite difference recurrent vector equation. Moreover,
an algorithm was discovered, called LGSS (Log Gain Stoichiometric Stepwise
algorithm, see [8,10]) that solves the inverse dynamical problem in terms of
MP grammars.
2 MP analysis of gene expression
In the specific application of MP grammars to breast cancer gene expression,
we started from the time series of gene expressions of a cancer cell under an
effect E that inhibits the cancer growth factor HER2. After standard procedures
of error filtering and data normalization, the expression time series were selected
which show a behavior clearly correlated to the inhibitory effect E. This means
that genes having time series that are constant in time, or with a chaotic shape,
are considered to be scarcely related to E. Therefore, only about one thousand
genes having time series with influenced shapes were selected. Then we clus-
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