Biology Reference
In-Depth Information
(b) In high-level nets, the model designer can distinguish tokens via their colors. This is a prerequisite
for overcoming the restrictions of low-level nets or METATOOL, i.e., for both detecting hitherto
unknown S-invariants of our model and determining its partial order dynamics, as shown in section
“Defects, effects and invariants”.
The software tool of our choice - for graphical editing, analyzing and executing the net models - is
Design/CPN [Design].
STEADY STATE PATHWAYS, ELEMENTARY MODES
First some Petri net notions are recalled that we will apply to metabolic pathways later on. The
algebraic analysis of Petri nets mainly relies on their invariants. However, in a great number of systems,
like metabolic pathways, deviations from invariants deserve even more attention. Hence the following
definitions [see Genrich, 2002] will turn out to be useful.
Let N be a colored (high-level) Petri net with the sets S of places and T of transitions. (A transition
represents a whole class of transition occurrences, each one determined by a particular
binding of the
variables in the adjacent arc labels by color values). The incidence matrix C of N is an |
| matrix
whose elements c ij are the positive/negative labels of the arcs pointing from/to transition t j to/from place
s i .An S
S
|×|
T
resp.
T-vector
is a vector with an entry for each s
S
resp.
t
T .A marking
of N
assigns a multi-set of tokens (colors) to each place s
S . At a marking M , usually several transitions
are enabled to occur. One transition occurrence leads to a follower marking, enabling other transitions,
and thus defines a partial order among them. The initial marking M 0 of N and all markings reachable
from M 0 are called states of the net system.
The (symbolic) analysis of N is based on multiplying C with vectors of transformations of expressions .
A distribution is a mapping transforming the elements of a color set D into linear combinations (with
integer coefficients) of elements of a not necessarily different set D .A substitution replaces variables
by expressions in colors.
Let y be an S-vector such that, for every place S , the component y s is a combination (list) of
distributions of S , and all the y s have the same range. Then the transpose matrix C T
can be multiplied
by the vector (one column matrix) y . A product is c ij
y s i the application of the distributions of s i to the
arc expression c ij and yields an integer linear combination of tokens from the range color set.
Treating y as a one-column matrix, the product C T
·
y is a T-vector whose entries are integer linear
combinations of colors denoting the marking differences caused by the individual transitions. It is called
the defect of y . A vector consisting only of zero elements 0 ( = 0 arbitrary color) is called null vector
and denoted by O .
The S-vector y is an S-invariant of N iff C T
·
y = O .
An S-invariant represents a state quantity of the net system, i.e., a quantity which, starting from the
initial state, is maintained during the whole life time of the system. It describes a conservation rule ,as
known from many areas in (natural) science. Such a mandatory S-invariant is a valuable means to detect
inconsistencies of a system specification or model.
A
·
process is a partially ordered set of transition occurrences leading from a state M 1 to a state M 2 ,
π M 2 . Ignoring the order of occurrences yields a T-vector x of combinations of transition occurences
which is called the action performed by π . To be precise, every entry of such a T-vector is a combination
of integer weighted substitutions of color variables by expressions in colors, and all variables have to
be substituted by colors of the same set.
M 1
Each substitution corresponds to a binding of the variables
Search WWH ::




Custom Search