1
Discrete dynamic modeling of biological
systems
•
The functional form of regulatory relationships
and kinetic parameters
are often unknown
•
Increasing evidence for
•
robustness to changes in kinetic parameters.
•
bistability (two steady states)
Hypothesis:
the kinetic details of individual interactions are less
important than the organization of the regulatory network
Discrete dynamic models assume that nodes can be characterized by
only a few (minimum two) discrete states.
Discrete models can handle larger networks than continuous models.
Boolean modeling of biological systems
Main assumption: components have two main states :
Expressed
or
not expressed
,
active
or
inactive,
open
or
closed
(ion
channel),
high
or
low
level
.
Denote these states by
ON (1)
or
OFF (0)
The changes in state are given by discrete (logical) rules.
The future state of a regulated node (the output) depends on the
current state of its regulators (inputs), which may or may not include
its own current state.
e.g. If transcription factor is active, gene will be transcribed,
gene will be expressed in the next time step.
Boole logic: based on the operators NOT, AND, OR
Can be defined based on set intersection and union, or inputoutput relations
(gates, truth tables)
Truth tables for Boolean operators
In
Out
In1
In2 Out
0
0
0
In1
In2 Out
000
NOT
AND
OR
01
10
010
100
111
011
101
Out= NOT In
Out= In1 AND In2
Out= In1 OR In2
In
Out
In1
In2
Out
In1
In2
Out
Out= NOT In
Out In1 AND In2
Out In1 OR In2
Out= In1 AND In2
Out= In1 OR In2
Ex. 1 Give examples for the realization of these Boolean rules in a gene regulatory
network.
Ex. 2 Consider a transcription event activated by a transcription factor. Compare the
continuous
and Boolean description of this process.
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From doseresponse curves
to
Boolean switches
•
X – mRNA
•
Y – transcriptional activator
If
ν
is large, the doseresponse
curve becomes a switch
If Y>K
Y
dX/dt>0
If Y<K
Y
dX/dt<0
Hill function
The activation threshold is K
Y
If activation is weak, mRNA
can decay.
Boolean simplification:
X* = Y
Activation:
If Y=ON
X*=ON
Decay:
If Y= OFF X*=OFF
Hybrid models: Boolean regulation
combined with continuous decay
•
Each node is characterized by both a continuous and a Boolean
variable.
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 Fall '09
 Work, Cellular differentiation, Boolean function

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