Transcript A Description of Dynamical Graphs Associated to Elementary

Modelisation and Dynamical Analysis of
Genetic Regulatory Networks
Brigitte Mossé
Elisabeth Remy
Claudine Chaouiya
Denis Thieffry
IML
Institut de Mathématiques de
Luminy
Marseille
LGPD
Laboratoire de Génétique et
Physiologie du Développement
Marseille
Solving the puzzle: the role of mathematical modelling
S
G1
G2
M
Modelisation of Genetic Regulatory Networks
Generally, interaction networks are represented by directed graphs:
nodes  genes
arcs  interactions (oriented)
Continuous-state approach
Discrete-state approach
level of expression assumed to be
continuous fonction of time

the node assumed to have a small
number of discrete states


the regulatory interactions described
by « logical » functions

(Thomas et al, Mendoza,….)
(Reinitz & Sharp, von Dassow,…)
evolution within a cell modeled by
differential equation
Other approach: PLDE
 level of expression assumed to be continuous fonction of time

Hyp: exp. level of gene products follow sigmoid regulation functions
=>The parameters of the differential equations are discrete (de Jong et al)
Summary
 Modelling framework
 Biological application
 Focussing on isolated regulatory circuits
 Conclusions and perspectives
Modelling framework
 A multivalued discrete method
G ={g1,g2,...,gn} set of genes, regulatory products…
for each gi  expression level xi  {0, ..., maxi}
maxi is the number of "relevant" levels of expression of gi
 Interaction networks represented by labeled oriented graphs, the
Regulatory Graphs
 nodes  genes G ={g1,g2,...,gn}
 arcs  interactions (oriented)
 label  type of interaction (-1 repression, +1 activation)
+ the condition for which the interaction is operating
Modelling framework (2)
A simple illustration
g1
T1
T5
T2
T3
Values
g1
0, 1, 2
g2
0, 1
g3
0, 1
source target type condition
T4
g2
Nodes
g3
Interactions:
T1= ( g1, g2, 1, [1])
T2=(g1,g2,-1,[2])
T3=(g2,g2,1,[1])
T4=(g2,g3,1,[1])
T5=(g3,g1,-1,[1])
Modelling framework (3)
A simple illustration
g1
T1
Effects of combinations
of regulatory actions
defined by
logical parameters Kj
T5
T2
T4
g3
g2
T3
Parameters
Nodes
Values
g1
0, 1, 2
K1{ } =2
g2
0, 1
K2{T1}=K2{T3}=K2{T1,T3}=1
g3
0, 1
K3{T4}=1
default value is 0
Modelling framework (4)
 given x=(x1,x2,...,xn) a state, Kj(x) precises to which value gene gj
should tend
if Kj(x)  xj gene gj receives a call for updating
+
xj denotes that Kj(x) > xj
call to increase
x-j
denotes that Kj(x) < xj
Two dynamics:
++
Synchronous: 100
++
call to decrease
210
200
Asynchronous: 100
110
 Dynamical behaviour of the system represented by oriented graphs
Dynamical Graphs
 nodes  states of the system
 arcs  transitions between two "consecutive" states
Modelling framework (5)
A simple illustration
A/ synchronous
B/ asynchronous
++
+
000
++
+
_
_
_
g1
T5
T2
T4
g2
T3
_ +_
101
+
210
201
T1
_
001
100
000
g3
Nodes
Values
g1
g2
g3
0, 1, 2
0, 1
0, 1
100
+
200
_ _
201
+
110
_+
210
_
111
__
211
0 11
Parameters
default value is 0
K1{ } =2
K2{T1}=K2{T3}=K2{T1,T3}=1
K3{T4}=1
D. melanogaster :
from embryo
to adult
Source: Wolpert et al. (1998)
Anteriorposterior
patterning in
Drosophila
3 cross-regulatory
modules initiating
segmentation
Gap
Pair-rule
Segment-polarity
Source: Wolpert et al. (1998)
Simulation of the Gap Module
Head
Trunk
BCD
BCD
Telson
BCD
HBmat HBmat CAD
CAD
Input:
Initial maternal gradients
Multiple
asynchronous
transitions
BCD
BCD
BCD
CAD
HB
HB
GT
KR KR
CAD
HB
KNI
GT
Output:
For expression patterns
for the gap genes
Patterns of gene expression (mRNAs or proteins)
Simultaneous labelling of HB, KR & GT Proteins in Drosophila embryo
before the onset of gastrulation (Reinitz , personal communication).
Gap Module
Collaboration with
Lucas SANCHEZ (CIB, Madrid)
Cad
Bcd
Hbmat
Maternal
Zygotic gap
Gt
Hbzyg
Kni
Kr
Multi-level logical model for the Gap module
gap network
maternal inputs
gt
hb
Kr
kni
bcd
cad
hbmat
gt
0
-1
-1
0
+1
+2
0
hb
0
(+1)
-2
0
+[1...3]
0
(+1)
Kr
-1
+1/-3
0
-1
+1
0
0
kni
-1
-2
0
0
+1
+1
0
Cad
Bcd
Hbmat
Maternal
Zygotic gap
Gt
Hbzyg
Kni
Kr
Patterns of gene expression (mRNAs or proteins)
Simultaneous
labelling of
HB, KR & GT
Proteins in
Drosophila
embryo
Source: Reinitz , personal communication
hb
Kr
bcd
gt
kni
gt
cad
Logical modelling of the GAP module
Source : Sanchez & Thieffry 2001
Kni
Regulatory graph
T2
T3
T10
T7
T4
Hb
Gt
T8
T9
T1
T5
T6
Kr
Patterns observed in region A
Region A
Gt KGt{Ø}=1, K Gt{T3,T8}=1
Hb KHb{Ø}=3, KHb{T4}=3
Kr KKr{T5}=2, KKr{T5,T10}=1, KKr{T1,T5}=1
Parametrisation
Asynchronous dynamical graph
hb
bcd
gt
gt, hbzyg, Kr, kni
Gap Module - Simulation ( gt, hbzyg, Kr, kni )
Bcd=3, hbmat=2, cad=0
+++
0200
++
1200
+
0300
Bcd=2, hbmat=2, cad=0
Bcd=1, hbmat=0, cad=1
Bcd=0, hbmat=0, cad=2
++ +
0000
+ +
0000
+ +
0200
+
0210
++
0001
+ ++
0100
++
0110
+
0001
[1000]
+ +
0101
-
[1300]
gt
hb
Kr
1001
kni
[0220]
[0111]
Kr
kni
hb
bcd
gt
gt
cad
Simulation of maternal and gap loss-of-function mutations
Genetic background
Wildtype
Final state
(GT, HB, KR, KNI)
A
B
C
D
1300
0220
0111
1000
Bicoid
0001
0001
0001
1000
Hunchback mat
1300
0220
0111
1000
caudal
1300
0220
0120
0000
giant
0300
0220
0111
0001
Krüppel
1300
1200
1100
1000
knirps
1300
0220
0120
1000
Hunchback mat&zyg
1000
1000
1000
1000
giant-Krüppel
Krüppel-knirps
giant-knirps
0300
1300
0300
0200
1200
0220
0101
1100
0120
0001
1000
0000
Observations/predictions
loss of GT in region A
loss of HB in ABC and of KR in BC
KNI expands anteriorly into region AB
no significant effect
increase of KR in region C
loss of KNI in region C
loss of GT in region D
KNI expands posteriorly into D
GT expands into regions B and C
Loss of KNI in region C
increase of KR in region C
GT expands into regions B and C
loss of KR in regions B and C
loss of KNI in region C
KNI expands posteriorly into region D
GT expands into regions B and C
increase of KR in region C
4 trunk domains
Anterior pole
Posterior pole
Focussing on regulatory circuits
Motivations
 Dynamical graphs can be very large,
exponential growth of the number of states with the number of
genes
 Problems for storage, visualisation, analysis...
 NP-complete problems (cycles, paths...)
 Reduce the size (development of heuristics)
 Establish formal relation between structural properties of
the regulatory graph and its corresponding dynamical graph
 Establish formal relationship between synchronous and
asynchronous graphs
“Natural” first step: what can be said about the very simple
regulatory graphs?
Focussing on regulatory circuits
Regulatory circuits are simple structures and play a crucial
role in the dynamics of biological systems :
Characteristics
Number of repressions
Positive circuits
Even
Negative circuits
Odd
Dynamical property
Biological property
Differentiation
Homeostasis
Simplified modelling:
each gene is the source of a unique interaction and the
target of a unique interaction  boolean case
only one set of parameters leads to an "interesting"
behaviour (functional circuit)
Example of a 4-genes positive circuit:
synchronous dynamical graph
4 genes
positive
regulatory
circuit
Synchronous dynamical graph
0011
1100
d
a
0000
0001
c
b
1101
1000
0101
1110
1010
1111
1011
1001
0010
0111
0100
0110
d
Example of a 4-genes positive circuit:
synchronous dynamical graph
a
c
b
configuration
0011
a
1100
k=4
d
b
c
++
0001
+1101
0000
- +
1000
0101
-1011
1110
1010
1111
1001
0010
k=2
0111
0100
0110
k=0
Example of a 4-genes negative circuit:
synchronous dynamical graph
4 genes
negative
regulatory
circuit
0111
1010
0100
1100
d
a
Synchronous dynamical graph
0011
1011
c
0101
k=3
1000
b
0010
0000
0001
k=1
0110
1001
1110
1111
1101
General case: the synchronous
dynamical graph
 Constituted of disconnected elementary cycles
 Staged structure
Stage k - gathers all the states having k calls for updating
- states are distributed in cycles according to their
configurations
Positive Circuits: only even values for k ( multi-stable
behaviour : for k=0 stationary states)
Negative Circuits: only odd values for k ( periodic
behaviour)
The synchronous version
Example of a 4-genes positive circuit:
the asynchronous dynamical graph
k=4
k=2
k=0
The synchronous version
Example of a 4-genes positive circuit:
the asynchronous dynamical graph
k=4
k=2
k=0
The synchronous version
Example of a 4-genes negative circuit:
the asynchronous dynamical graph
k=3
0010
0000
0001
0110
k=1
1001
1110
1111
1101
General case: the asynchronous
dynamical graph
 Connected graph
 The staged structure can be conserved
 At stage k, each state has exactly k successors
 either at the same stage k
 or at the stage below k-2
A compacted view of the asynchronous graph
example of the 4-genes positive circuit
k=4
k=2
k=0
Conclusions and Perspectives
 Mathematical analysis
 extension to more complex regulatory networks (intertwined
circuits…)
 deeper understanding of the role of circuits embedded in
regulatory networks
 specification of information about transition delay
 Computational developments
 GINML: a dedicated standard XML format
 GINsim: a software which implements our modelling
framework
 Biological applications
 Drosophila development
 T Lymphocyte differentiation
 progressive increase of network size (~ 30 genes)