Decision Diagrams for the
Representation and Analysis of
Logical Models of Genetic Networks
Aurélien Naldi, Denis Thieffry, Claudine Chaouiya
Logical Formalism
 Analytical Algorithms
 T cell differentiation and activation

CMSB - September 2007
http://tagc.univ-mrs.fr
Logical Formalism

2
A
1
Regulatory graph

 Discrete
expression levels
(Boolean or multivalued)
B
C
Genes (A, B, C)

Interactions
 Activity
threshold
 Activations, inhibitions

Dynamical rules
 Target
expression level
depending on regulator levels
Dynamical Rules
2
A
1
B
C

B expressed in presence of:
A at medium level or C (or both)

C expressed in absence of B
Dynamics of B given by the logical function KB
{
1 if  A1∨C
K B=
0 otherwise
}
Representation of Logical Functions
KB as a decision tree
2
A
1
A
B
C
C
C
C
0
1
1
1
0
1
Dynamics of B given by the logical function KB
{
1 if  A1∨C
K B=
0 otherwise
}
Representation of Logical Functions
KB as a decision diagram
2
1
A
A
B
C
C
0
Dynamics of B given by
the logical function KB
{
1 if  A1∨C
K B=
0 otherwise
}
1
Efficient structure
Canonical representation
(for a given ordering of the
decision variables)
Determination of Stable States

Stable state: all expression levels are stable


Analytic method to find all possible stable states

No simulation

No initial condition


Principle

Identify a stability condition for each gene

Combine these partial conditions
Determination of Stable States
C
A∧! C
KB
KC
A
A
C
0
A
B
1
KA
1
0
0
A
A
0
A
SB
1
SC
B
1
A
B
C
1
SA
!A
0
C
1
C
0
0
C
1
0
Determination of Stable States
C

A
Combination of stability conditions
B
A
A
B
*
B
1
C
0
B
C
C
1
A
0
0
C
C
1
0
0
B
0
1
C
0
2 stable states : 001 et 110
Role of Regulatory Circuits
Positive circuit
A
A
B
D
Negative circuit
C
Multistability
D
B
C
Cyclic attractor
Functionality Context
A negative circuit inducing a cyclic behaviour
A
00
01
10
11
B
Functionality Context
C prevents A from activating B
C
A
00
01
10
11
B
The circuit is functional in a given context:
in absence of C

Functionality Context
Functionality context: set of constraints
on the expression levels of regulators
Each interaction has its own context
Context of the circuit: combination of all
interactions contexts
Functionality of an Interaction
In a circuit (...,A,B,C,...), the
functionality of (A,B) depends on:

A

KB

the threshold of (A,B)

the threshold of (B,C)
X
B
Y

C

Functionality: logical function
depending on the regulators of B

(represented as a decision diagram)

Functionality of an Interaction
KB
A
X
A
X
X
Y
B
Y
Y
Y
Y
1
1
1
0
0
1
1
1
C
X
-1
0
0
+1
Y
-1
Y
0
0
+1
Functionality context: Shortcuts
A negative circuit inducing a cyclic behaviour
A
00
01
10
11
B
Functionality context: Shortcuts
The shortcut prevents B from inhibiting A
A
00
01
10
11
B

The circuit is functional in absence of A

A is a member of the circuit
Members of the circuit must be
able to cross their thresholds

Functionality context: Shortcuts
The context of (B,A) introduces
a constraint on the value of A
X
A
B
The circuit is functional when the
value of A does not matter
A
X
0
X
-1
-1
0
-1
X
-1
0
-1
Implementation in GINsim
Tcell activation and differentiation
TCR
Activation
T-bet
Th1
cell
Cellular
response
Th2
cell
Humoral
response
Naive
T helper
cell
GATA-3
TCR Signalling

Circuit analysis:
4 circuits functional among 12

3 positive circuits:
auto-regulations on inputs
➔
8 attractors:
one for each input combination

1 negative circuit:
ZAP70 cCbl (functional in presence of
LCK and TCRphos)
➔
cyclic attractor
for “111” input
Stable states analysis:
7 stable states

S. Klamt et al. (2006)
Th Differentiation
L. Mendoza (2006)
5 functional (positive) circuits among 22

4 stable states:


Th0 (naïve)

Th1 and Th1* (cellular response: IFNg, Tbet)

Th2 (humoral response: IL4, GATA3)
Conclusions



Decision diagrams in the logical formalism

Improved performance of GINsim

Determination of stable states

Functionality context analysis
Prospects


Determination of complex attractors

Further elaboration of circuit analysis

Extension of the Th models

Coupling the two models

Other regulatory components (IL2)

Other differentiation pathways (Treg, Th17)
Current supports
Evolution Rules (2)
2
A
A
1
D
B
0
B
1
{
B
2
2 if  A1∧B
K D= 1 if  A1∧! B∨ A2∧B
0 otherwise i.e. if A ∨ A ∧! B
0
2
0
}
1