Biology Reference
In-Depth Information
result can stem from multiple causes, e.g., photosynthesis
can be carried out under either blue or red light, then the
Boolean equation denoting that will be Blue_light OR
Red_light
time implementation later. As time evolves, the system
state (N-dimensional vector) traverses the state space, and
after a finite-duration transient behavior it settles into an
attractor (stable dynamic behavior). Two types of attractors
are possible: after hitting a certain state, any future updat-
ing results in the same state, hence the system reaches
a 'fixed point' or steady state; or there exists a certain small
set of states of the system G which the system keeps
revisiting; that is, any updating of the system state will
carry it to one of the states that belongs G. Depending on
how updating is implemented, the second type of attractor
of the system (non-steady state) can take on two possible
forms, either an oscillation (a series of system states that
repeat regularly) or a loose attractor (a random sequence of
system states that is generated from a finite pool of states).
See later in the chapter for further details.
Boolean dynamic modeling of a biological system is
comprised of the following steps: reconstruct the network
based on biological knowledge; determine the Boolean
transfer functions; choose an updating scheme; determine
the initial state of the model; analyze the model, including
its attractors and state space; validate the model (repro-
duction of known results); and finally, study novel
scenarios, e.g., robustness against disruptions, make useful
predictions and inferences. We next look at each step in
more detail and elaborate them through examples.
Photosynthesis*. When a certain element or
activity is negatively regulated by another, the NOT rule is
used. The asterisks on the two example Boolean equations
indicate that the specific processes to generate the product
on the right-hand side take a certain amount of time to
complete. We will return to this point later.
The additional information contained in the Boolean
rules (e.g., the conditionality or independence of two edges
incident on the same node) can be integrated into the
network for a more complete representation. Merging
edges may be used to represent the Boolean AND relation
(
Figure 10.1
(B)), and separate edges for OR relations
(
Figure 10.1
(A))
[29]
. There is also a third choice, which
involves an intermediate node, if the Boolean rule is AND
(
Figure 10.1
(C)). Such a node would not exist if the rule
were Boolean OR
[21,30,31]
. So far there is no universal
'standard' for how to map a system of complex relations
onto a graph. The most appropriate choice is the one that
best facilitates the analysis of the particular system under
study.
¼
BOOLEAN NETWORK MODELING
As mentioned in the introduction, a dynamic model char-
acterizes a system's behavior over time. In Boolean
dynamic modeling specifically, both time and the system's
status are discretized. The state of a node can be 0 or 1,
making the state of a system of N nodes an N-dimensional
vector of 0s and 1s. A continuous time stream over a certain
period (e.g., an experiment) is represented by a series of
steps, which are an abstract representation of important
time points at which biochemical events are taking place.
The state of the system at a time step is determined by its
predecessor state (or sometimes several predecessor states
at earlier time steps) through what are called Boolean
transfer functions. The calculation of a node state at a time
step based on system state(s) at earlier step(s) is called
updating. Depending on the updating scheme used,
a number of nodes, ranging from 1 to N, are updated,
thereby obtaining the system state at a new (future) time
step. Given below is an example of a general expression of
a Boolean transfer function of a certain node i. Suppose the
state of node i at time step t is denoted as V
i
;
t
. The transfer
function through which V
i
;
t
Reconstruct the Network Based on Biological
Knowledge
The first step of Boolean dynamic modeling is to represent
the system under study by a network, denoting the relevant
elements of system by nodes and their pairwise relation-
ships by edges. This is accomplished through extensive
literature searches and compilation. Experimental data-
bases available online, such as Transcription Factor Data-
base (TRANSFAC,
[32]
) and Kyoto Encyclopedia of Genes
and Genomes (KEGG,
[33]
), can be used for data mining to
deduce causal relationships between components. Many,
but not all, experiments indicate direct interactions or
regulatory relationships of elements, such as transcrip-
tion factor
e
gene interactions, enzymatic activities and
protein
e
protein interactions. Genetic knockout or phar-
maceutical evidence, such as exogenous application of
a certain chemical, indicates regulatory relationships indi-
rectly. With such relationships further inference and inter-
pretation of experimental results might be needed to obtain
the most proper regulatory relations to represent in the
network
[10]
. For instance, experimental identification of
the change in the activity level of a protein after a certain
stimulus was applied implicates the protein as a potential
downstream responder to that stimulus. Similarly, if the
over-expression of a gene results in the downregulation of
is calculated is given as
follows:
F
i
V
k
1
;s
k
1
;
V
k
n
;s
k
n
V
i
;
t
¼
V
k
2
;s
k
2
;.;
where 1
i
k
1
;.;
k
n
N are the node indices and
;
s
k
1
;.s
k
n
t denote the time step when the state of nodes
k
1
;.;
k
n
was last updated. We will revisit the details of
