Biology Reference
In-Depth Information
In general asynchronous updating
[38]
only one node
is randomly selected and updated at each time step. The
states of the inputs to the node that is being updated
are from earlier time steps, which means
or experimental evidence. If an intracellular substance is
known to be present under all conditions, it can be initial-
ized to be in the 1, or ON, state. Initial states can also be
assigned based on the question of interest. For example, one
can implement a gene knockout by the initial and sustained
OFF state of that particular gene. If there is insufficient
experimental information concerning the concentration or
activity level of an element, one can also randomize the
initial state of that node. Therefore, a large number of runs
of dynamic simulations can be carried out with each run
randomizing the initial states of the nodes that cannot be
predetermined
[10]
. The system will sample different
potentially viable initial states and henceforth take varying
routes in the state space.
Starting from the initial state, the system will evolve as
time progresses and should eventually settle down into an
attractor, that is, a time-invariant steady state or an (ordered
or unordered) repetition of a certain finite set of states.
These attractor states have been proposed as representa-
tions of cell fates going back to the 1940s work of C.
Waddington
[45,46]
and later by S. Kauffman
[6]
.The
complete set of states that can potentially reach a certain
attractor through an updating scheme forms the basin of
attraction of that attractor. Since synchronous updating is
deterministic, the basins of attraction for different attractors
will be distinct, whereas under asynchronous updating the
basins of attraction for different attractors could share states
and be partly overlapping with each other, as illustrated by
the example in
Figure 10.3 [39]
.
s
k
j
t
1for1
n.
In random order asynchronous updating
[38,39]
, at each
time step a random permutation of the node labels (from
1 to N) is first generated, and the updating is carried out
in that sequence in that time step. The random permutation
of 1 to N is regenerated at each time step. Consider node
i with n inputs and its updating:
V
i
;
t
¼
j
F
i
ð
V
k
1
;s
k
1
;
V
k
2
;s
k
2
;.;
V
k
n
;s
kn
Þ
If node k
j
comes before node i in the random sequence,
meaning that by the time node i is being updated node k
j
has already been updated at the current time step to its latest
state,
t should be used. Otherwise, if node k
j
comes
after node i in the random sequence, meaning that at the
time node i is being updated the state of node k
j
has not
been touched yet at the current time step,
s
k
j
¼
s
k
j
¼
t
1
should be used.
In deterministic asynchronous updating
[40]
, every
node is associated with an intrinsic timescale
g
i
. The
updating of node i only takes place if the current time t is
a multiple of
g
i
. One can clearly see that
g
i
is the effective
'pace' of each reaction. In other words,
V
i
;
t
¼
F
i
ð
V
k
1
;s
k
1
;
V
k
2
;s
k
2
;.;
V
k
n
;s
k
n
Þ
;
t
¼
c
g
i
c is a positive integer
;
:
When the updating of node i takes place, the states of its
input nodes should be taken from the latest available time
step, namely
Steady-State Analysis
The system will have the same steady states (fixed points)
under both synchronous and asynchronous updating, owing
to the fact that a steady state repeats itself infinitely, making
the order in which the nodes are updated irrelevant. Boo-
leanNet
[41]
provides functions to detect the steady states
of the system. CellNetAnalyzer
[42]
is also able to probe
the steady states of the system, which are called 'logical
steady states' in the package.
Before we illustrate in a step-by-step manner how to
determine the steady states of a Boolean network, we
note that there are two main focuses encountered in
dynamic analysis of biologicalsystems:determiningthe
attractors of the whole system (of all nodes), and
determining the attractors of a small set of designated
output nodes of the Boolean network. The second focus
is necessarily a subset of the first. The output-oriented
analysis is often not only simpler, but also more relevant
to signal transduction pathway-related research, where no
more than several inputs (signals) are considered and the
system's response is usually characterized by a single
output node. In modeling gene regulatory networks,
usually the attractors and the dynamic sequence of the
1.
Several software packages can be used to carry out
dynamic simulations based on Boolean network modeling
on biological systems. An open-source Python package,
BooleanNet
[41]
, is available online at
http://code.google.
com/p/booleannet/
.
The input to the program is a text file
containing all the Boolean transfer functions, and all of the
updating schemes can be selected and readily implemented.
Other software packages include CellNetAnalyzer
(MATLAB package, [42]), GINsim
[43]
, and BoolNet
(R package,
[44]
).
s
k
1
;.s
k
n
t
Network Initialization
The system state is represented by an N-dimensional
vector, for a system of N nodes (elements) (see earlier). To
perform dynamic simulations of the system under the
Boolean network framework, one needs to specify the first
state in the state sequence (trajectory), namely the initial
state of the system. The initial state of each node is deter-
mined such that it is consistent with known biological facts
