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between different regimes can be analyzed by measuring the effect of
perturbations, the sensitivity to initial conditions and damage spreading
(Shmulevich
et al.
2005).
The analysis of information processing through biological networks described
by continuous and binary models has revealed characteristic network topologies
leading to reproducible behavior in the presence of noise (Klemm
et al.
2005).
Robustness in the face of random perturbations is an important feature of
biological networks. Often RBNs fail to reproduce this feature at the biologically
observed connectivity. The extension of RBNs into probabilistic Boolean
Networks (PBNs) incorporate the uncertainty in the regulation of a node by
assigning several Boolean functions/predictors to each node, each with some
probability of being chosen to advance the state of the node to which it belongs.
PBNs are also more robust because the predictors are probabilistically
synthesized so that each predictor's contribution is proportional to its
determinative potential. Another way to introduce robustness is by using
canalizing functions. This type of Boolean function in which one of the input
variables is able to determine the function output regardless of the values of the
other variables leads to orderly behavior (Kauffman
et al.
2004).
4.3. Extensions of the Classical Boolean Framework
Most traditional Boolean models consider time quantized into regular intervals
(time steps) and only two states, expressed/active and not expressed/ inactive
(Thomas 1973; Kauffman 1993; Kauffman, Peterson
et al.
2003). Such models
are known as synchronous Boolean models because the components of the
system are updated simultaneously in the algorithm; assuming that all processes
require same time. Often variations of this approach are required to model the
complexity of biological systems. In an effort to improve the description of
variability in the durations of synthesis and decay processes, several
asynchronous algorithms have been proposed. For example, each node can be
assigned an individual time unit and be updated at multiples of this time unit
.
t
t
1
t
t
=
,
where
t
k
represents the time of the last update of node
k
before time
t
i
. Known
information on the timings of the processes can be included in the model by
setting up inequalities of
γ,
e.g. in bacterial infections the fact that epithelial cells
are activated before dendritic cells can be incorporated as
γ
EC
<
γ
DC
. A variety
of asynchronous and synchronous algorithms have been studied for RBNs.
Figure 4.4 shows their classification according to their updating scheme
(Gershenson 2004).
X
B
(
X
,
X
,...,
X
)
k
i
k
=
The update function can be given as
t
i
1
2
k
i
i
2
k
i
