Biology Reference
In-Depth Information
Output*
NOT Input if the input suppresses the output.
When there are two or more input nodes, their states need to
be combined via Boolean functions in a way that is
consistent with available knowledge. A biochemical
synthesis reaction A
¼
combinations of input states of a certain node i and assign
output states to each of such combinations. This is called
the truth table of a certain Boolean rule, which is equivalent
to the rule itself. If node i has m input nodes, then the truth
table will have 2
m
rows.
þ
B
C can naturally be represented
/
by the Boolean rule: C*
A AND B. The statement 'B
induces the synthesis of C only in the absence of A' can be
represented by the Boolean rule C*
¼
Updating Schemes and Incorporating Time
We have now explained the network structure and the
Boolean transfer functions that govern the state transitions
of nodes. These two pieces comprise the major steps of the
model. The next step, discretization of the continuous time
stream into steps, is also non-trivial. There are different
choices in terms of the number of nodes to be updated at
each time step and the order in which the updating of those
nodes is to be carried out, and these choices may affect the
results.
Assume that the total number of nodes in the network is
N. Starting from the general expression of Boolean rules
established based on biological knowledge as described in
the previous section, we will look at each updating scheme
in detail.
Suppose a node i has n input nodes and the Boolean
transfer function of node i can be written as follows:
V
i
;
t
¼
F
i
ð
V
k
1
;s
k
1
;
V
k
2
;s
k
2
;.;
V
k
n
;s
kn
Þ
where V
i
;
t
is the state of node i at time step t, 1
NOTA AND B, since
both conditions (stimulus B as well as the absence of
inhibitory factor A) need to be satisfied in order for C to be
synthesized. On the contrary, if for instance two inputs
function largely independently, do not exhibit synergy and
can substitute one another, then the Boolean OR function
should be used, e.g., C*
¼
A OR B. The information of
dependencies of input nodes can be obtained through
literature search.
If in the system shown on
Figure 10.2
both the pres-
ence of Input and the absence of B are required for the
activation of A, then the Boolean function of A can hence
be written as:
¼
A
¼
Input AND
ð
NOT B
Þ
Similarly, if either the absence of A or the presence of
Input is sufficient to activate C, this leads to:
C
¼ð
NOT A
Þ
OR Input
The Boolean transfer functions of the system on
Figure 10.2
are summarized in
Table 10.2
. These rules
govern the state transitions of the nodes in the system.
The more inputs a node has, the more possible ways
there are to combine all the input states. One needs to be
particularly careful in such cases in order to obtain the rule
that is able to generate dynamic simulation results that best
fit the existing experimental data. It is likely that multiple
'trial and error' processes will have to take place before an
optimal solution is found.
Instead of describing the state transition relationship
with compact Boolean rules, one can also list all possible
i
t
denote the time step of the last update of the state of nodes
k
1
;.;
k
1
;.;
k
n
N are the node indices, and
s
k
1
;.s
k
n
;
k
n
. Two major types of updating exist, synchronous
[36]
and asynchronous
[7]
.
For synchronous updating every node is updated once at
each time step, using the states of the input nodes at earlier
time steps as inputs to the transfer functions. In other
words,
n. The fact that the input
states to each and every transfer function are from earlier
time steps makes the order in which all nodes are updated at
each time step irrelevant, since all inputs have already been
fixed before the updating.
Synchronous updating has a simple and straightforward
formalism and leads to reproducible state changes.
However, it overlooks the differences of timescales on
which biological processes are taking place, which can
range wildly from milliseconds for protein phosphorylation
and post-translational modifications to hundreds of seconds
for transcription and transcriptional regulations
[37]
.
Asynchronous updating
[7]
, which provides more detailed
tracking of timescales and temporal orders
[10
e
12]
,is
devised in order to account for the diversity in duration of
biological processes.
We will cover three types of asynchronous updating:
general asynchronous updating
[38]
, random order asyn-
chronous updating
[38,39]
, and deterministic asynchronous
updating
[40]
.
s
k
j
t
1, where 1
j
TABLE 10.2
Boolean Rules (Transfer Functions) of the
System Shown in
Figure 10.2
, for Specific Assumptions
on the Combination of Multiple Inputs.
Node
Boolean Rule (Transfer Function)
Input
d
A
A*
¼
Input
AND
(
NOT
B)
B
B*
¼
A
C
C*
¼
(
NOT
A)
OR
Input
Output
Output*
¼
C
