Biomedical Engineering Reference
In-Depth Information
•
Formal logics
, mainly temporal logics, can be used to automatically perform
complex reasonnings about a given discrete model, so that discrete parameter
values can often be
deduced
from behaviors observed
in vivo
.
All in all, discrete models are particularly well suited to perform qualitative
reasonning in a computer aided manner and they help biologists to elucidate open
questions about the functioning of many gene networks. It finally appears that
computer
reasonning
capabilities are at least as useful as
simulation
capabilities.
Discrete modeling is consequently able to provide the biologists with quick helpful
information about open problems (possible behaviors, refutation of hypotheses,
missing variables or missing interactions...); they are able to quickly prune some
inconsistent intervals for the parameters in continuous models; they are also able to
suggest experimental plans
optimized to check a biological hypothesis.
2.3.2
Methodological Developments
2.3.2.1
Rene Thomas' Logical Method
R. Thomas' logical method consists in modeling the qualitative behavior of a gene
network under the form of a finite state transition graph. This state transition graph
is built from the interaction graph of the network together with logical parameters
that describe the combined effects of switch-like interactions.
More precisely, the starting point of Thomas logical method is an
interaction
(or
regulatory
)
graph
G
. The vertices, denoted from 1 to
n
, correspond to genes, and
each arc
i
j
is associated with sign
s
ij
(Fig.
2.9
). If
s
ij
is positive (resp. negative),
it means that the protein encoded by
i
activates (resp. inhibits) the synthesis of the
protein encoded by
j
. For every vertex
i
, we denote by
G
i
the set of
regulators
of
i
,
that is, the set of vertices
j
such that
j
→
→
i
is an arc of
G
, and we denote by
T
i
the
set of vertices regulated by
i
.
The
first step
of the logical method consists in associating with every vertex
i
a
natural number
b
i
, called the
bound
of
i
, such that:
b
i
≤
card(
T
i
),and
b
i
>
0 if
T
i
is not empty. Then,
X
i
=
{
0
,
1
,...,b
i
}
corresponds to the possible
(concentration)
levels
for the protein encoded by
i
,and
X
=
i
X
i
corresponds to the set of
possible
(discrete) states
for the system.
The
second step
consists in associating with each interaction
i
→
j
an integer
t
ij
∈
X
i
,
t
ij
>
0, called the
logical threshold
of the interaction
i
→
j
.It
is required that, for every
i
, and for every integer
l
∈
X
i
,
l>
0, there exists
at least one interaction
i
→
j
such that
t
ij
=
l
(condition C1). Then, at state
x
=(
x
1
,...,x
n
)
∈
X
, we say that a regulator
j
of
i
is a
resource
of
i
if:
x
j
≥
t
ji
and
s
ji
=+(effective activator), or
x
j
<t
ji
and
s
ji
=
−
(ineffective inhibitor).
In other words,
j
is a resource of
i
when its concentration level
x
j
“favors” the
synthesis of the protein encoded by
i
. The set of resources of
i
at state
x
is denoted
by
ω
i
(
x
).SeeFig.
2.10
for an illustration.
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