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proteins, transcription factors, cells, and by the relationships between those
components, for example interaction between two proteins, activation of a
transcription factor, or regulation of gene expression by a transcription factor.
Such system can be represented by a network, where nodes of the network
represent components and edges describe the processes connecting those nodes.
The nodes are characterized by binary states, and the relationships among the
nodes are incorporated into updating functions for the nodes' states. Boolean
modeling has been pioneered by S. Kauffman and R. Thomas in the middle of the
20
th
century. Due to the lack of information about the architecture of biological
networks most of the early research focused on the generic properties of
networks governed by Boolean dynamics. The post-genomic revolution brought
with it a resurgence of Boolean modeling and its successful application to a
variety of biological systems.
4.2. Boolean Network Concepts and History
In a Boolean model, each node is assumed to have one of two states, denoted
ON (1) and OFF (0). The ON state can correspond to a high gene expression,
high concentration, open channel, or active transcription factor; the OFF state
corresponds to low expression or concentration, closed channel or inactive
transcription factor. The future state of each node is given by a Boolean
function
B
i
, describing the regulation of node
X
i
by the other nodes,
1
+
=
.
B
i
is a statement whose inputs are the nodes that have
edges directed toward node
i
(i.e. the regulators of
i
), and whose output is 1 (ON)
or 0 (OFF). There are two main frameworks for Boolean models. In the first
B
i
is
based on the Boolean (logical) operators AND, OR and NOT. In the second
framework, called threshold Boolean networks (Derrida 1987; Kuerten 1988)
B
i
is a statement comparing the weighted sum of input signals to a node-specific
threshold value. In both cases the function's output is 1(0) if the corresponding
statement is true (false). In most Boolean models time is assumed to be quantized
into discrete time-steps, thus the future state means the state at the next time-step.
Starting from a given initial condition, the network then produces a dynamical
sequence of network states, eventually reaching a periodic attractor (limit cycle)
or a fixed point (Fig. 4.2). All initial conditions that evolve to a given attractor
constitute its basin of attraction.
Most of the theory of Boolean networks was developed in the context of
Random Boolean Networks (RBNs, also known as N-K models or Kauffman
networks) introduced by S. Kauffman in 1969. RBNs are generic, because one
t
t
t
t
X
B
(
X
,
X
,...,
X
)
i
i
1
2
k
