Modelisation and Dynamical Analysis of Modelisation and Dynamical Analysis of Genetic Regulatory NetworksGenetic Regulatory Networks
Claudine ChaouiyaDenis Thieffry
LGPD Laboratoire de Génétique et
Physiologie du Développement Marseille
Brigitte MosséElisabeth Remy
IMLInstitut de Mathématiques de
LuminyMarseille
S
M
G2G1
Solving the puzzle: the role of mathematical modellingSolving the puzzle: the role of mathematical modelling
Modelisation of Genetic Regulatory NetworksModelisation of Genetic Regulatory Networks
Generally, interaction networks are represented by directed graphs:nodes genes arcs interactions (oriented)
Discrete-state approach the node assumed to have a small number of discrete states the regulatory interactions described by « logical » functions
(Thomas et al, Mendoza,….)
Continuous-state approach level of expression assumed to be continuous fonction of time evolution within a cell modeled by differential equation
(Reinitz & Sharp, von Dassow,…)
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)
SummarySummary
Modelling framework
Biological application
Focussing on isolated regulatory circuits
Conclusions and perspectives
Modelling frameworkModelling 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)Modelling framework (2)A simple illustrationA simple illustration
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])
T1 T 2
T4
T5
T3
g1
g3
g2
source target type condition
Nodes Values
g1 0, 1, 2
g2 0, 1
g3 0, 1
Modelling framework (3)Modelling framework (3)A simple illustrationA simple illustration
Nodes Values Parametersdefault value is 0
g1 0, 1, 2 K1{ } =2
g2 0, 1 K2{T1}=K2{T3}=K2{T1,T3}=1
g3 0, 1 K3{T4}=1
T1 T 2
T4
T5
T3
g1
g3
g2
Effects of combinations of regulatory actions
defined by logical parameters Kj
Modelling framework (4)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
xj denotes that Kj(x) < xj call to decrease
Two dynamics: Synchronous: 100 210
Asynchronous: 100
Dynamical behaviour of the system represented by oriented graphs
Dynamical Graphs
nodes states of the system arcs transitions between two "consecutive" states
+
-
200
110
++
++
Modelling framework (5)Modelling framework (5)A simple illustrationA simple illustration
Nodes Values Parametersdefault value is 0
g1 0, 1, 2 K1{ } =2
g2 0, 1 K2{T1}=K2{T3}=K2{T1,T3}=1
g3 0, 1 K3{T4}=1
+ +1 0 0
_ _
2 0 1 _ +2 1 0
0 0 0+
1 1 1_
2 0 0
_ _2 1 1
_ +2 1 0
_ _2 0 1
_ + _1 0 1
+ +1 1 0
0 0 0
0 0 1 _
1 0 0+ +
0 1 1
+A/ synchronous B/ asynchronous
T1 T 2
T4
T5
T3
g1
g3
g2
Source: Wolpert et al. (1998)
D. melanogaster :D. melanogaster :from embryofrom embryo
to adult to adult
Source: Wolpert et al. (1998)
3 cross-regulatory modules initiating segmentation
Gap Pair-ruleSegment-polarity
Anterior-Anterior-posterior posterior
patterning in patterning in DrosophilaDrosophila
Multipleasynchronous
transitions
Input:Initial maternal gradients
BCD
HBmat
BCD
HBmat
BCD
CAD CAD
Head Trunk Telson
Output:For expression patterns for the gap genes
BCD
HB
BCD
HB
BCD
CAD CAD
GT KR
KNI
KR GT
HB
Simulation of the Gap ModuleSimulation of the Gap Module
Simultaneous labelling of HB, KR & GT Proteins in Drosophila embryo before the onset of gastrulation (Reinitz , personal communication).
Patterns of gene expression (mRNAs or proteins)Patterns of gene expression (mRNAs or proteins)
Gt
Bcd
Hbzyg
Hbmat
Cad
KrKni
Maternal
Zygotic gap
Collaboration withLucas SANCHEZ (CIB, Madrid)Gap ModuleGap 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
Multi-level logical model for the Gap module
Gt
Bcd
Hbzyg
Hbmat
Cad
Kr
Kni
Maternal
Zygotic gap
Simultaneous labelling of HB, KR & GT Proteins in Drosophila embryo
Patterns of gene expression (mRNAs or proteins)Patterns of gene expression (mRNAs or proteins)
Source: Reinitz , personal communication
gt
gthb
Kr knibcdcad
Gt KGt{Ø}=1, KGt{T3,T8}=1 Hb KHb{Ø}=3, KHb{T4}=3 Kr KKr{T5}=2, KKr{T5,T10}=1, KKr{T1,T5}=1
T10
Gt Hb
Kr
Kni
T2
T1
T3
T5 T6
T7
T8T9
T4
Region A
gt
hb
bcd
Regulatory graph
Patterns observed in region A Asynchronous dynamical graph
Parametrisation
Logical modelling of the GAP moduleLogical modelling of the GAP moduleSource : Sanchez & Thieffry 2001
gt, hbzyg, Kr, kni
Bcd=3, hbmat=2, cad=0 Bcd=2, hbmat=2, cad=0 Bcd=1, hbmat=0, cad=1 Bcd=0, hbmat=0, cad=2
0000
[1000] 0001+
1001-
+ +
0001
0000++ +
[0111]
0100
0110 0101
+ ++
+ +++
++
[0220]
0200+ +
0210+
[1300]
0200+++
1200 0300++ +
gt hb Kr kni
Gap Module - SimulationGap Module - Simulation ( ( gtgt,, hbhbzygzyg,, Kr,Kr, kni kni ))
gt
gthb
Kr knibcdcad
4 trunk domainsAnterior pole Posterior pole
Final state (GT, HB, KR, KNI) Genetic background
A B C D Observations/predictions
Wildtype 1300 0220 0111 1000
Bicoid 0001 0001 0001 1000 loss of GT in region A
loss of HB in ABC and of KR in BC KNI expands anteriorly into region AB
Hunchbackmat 1300 0220 0111 1000 no significant effect
caudal 1300 0220 0120 0000 increase of KR in region C
loss of KNI in region C loss of GT in region D
giant 0300 0220 0111 0001 KNI expands posteriorly into D
Krüppel 1300 1200 1100 1000 GT expands into regions B and C
Loss of KNI in region C knirps 1300 0220 0120 1000 increase of KR in region C Hunchbackmat&zyg
1000
1000
1000
1000
GT expands into regions B and C loss of KR in regions B and C
loss of KNI in region C giant-Krüppel 0300 0200 0101 0001 KNI expands posteriorly into region D Krüppel-knirps 1300 1200 1100 1000 GT expands into regions B and C giant-knirps 0300 0220 0120 0000 increase of KR in region C
Simulation of maternal and gap loss-of-function mutationsSimulation of maternal and gap loss-of-function mutations
Focussing on regulatory circuitsFocussing on regulatory circuitsMotivationsMotivations
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 circuitsFocussing on regulatory circuitsRegulatory circuits are simple structures and play a crucial role in the dynamics of biological systems :
Characteristics Positive circuits Negative circuits
Number of repressions Even 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 caseonly one set of parameters leads to an "interesting" behaviour (functional circuit)
Example of a 4-genes Example of a 4-genes positivepositive circuit:circuit: synchronous dynamical graphsynchronous dynamical graph
01101001
10001101
0001
1011
11000011
10100101
0000
1111
01110010
1110
0100
d
cb
a
4 genes positive
regulatory circuit
Synchronous dynamical graph
a
d
c
b
10001101
0001
1011
10100101
0000
1111
01110010
1110
0100
01101001
k = 4
k = 2
k = 0
11000011
d
cb
a Example of a 4-genes Example of a 4-genes positivepositive circuit:circuit: synchronous dynamical graphsynchronous dynamical graph
+
+
+
--
- -
+
configuration
k=3
0111 0100
10001011
0011
k=10010
1110 1101
0001
01101001
0000
1111
1100
1010
0101
Example of a 4-genes Example of a 4-genes negativenegative circuit:circuit: synchronous dynamical graphsynchronous dynamical graph
d
cb
a
4 genes negative
regulatory circuit
Synchronous dynamical graph
General case: the synchronous General case: the synchronous dynamical graphdynamical graph
Stage k - gathers all the states having k calls for updating - states are distributed in cycles according to their configurations
Constituted of disconnected elementary cycles Staged structure
Positive Circuits: only even values for k ( multi-stable behaviour : for k=0 stationary states)
Negative Circuits: only odd values for k ( periodic behaviour)
k=4
k=2
k=0
Example of a 4-genes positive circuit: the asynchronous dynamical graph
The synchronous version
k=4
k=2
k=0
Example of a 4-genes positive circuit: the asynchronous dynamical graph
The synchronous version
k=3
k=1
The synchronous version Example of a 4-genes negative circuit: the asynchronous dynamical graph
0010
1110 1101
01101001
1111
00010000
General case: the asynchronous General case: the asynchronous dynamical graphdynamical 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
k=4
k=2
k=0
A compacted view of the asynchronous graphA compacted view of the asynchronous graphexample of the 4-genes positive circuit
Conclusions and PerspectivesConclusions 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)
