Modelisation and Dynamical Analysis of Genetic Regulatory Networks

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Modelisation and Dynamical Analysis of Genetic
Regulatory Networks
Claudine Chaouiya Denis Thieffry LGPD
Laboratoire de Génétique et Physiologie du
Développement Marseille
Brigitte Mossé Elisabeth Remy IML Institut de
Mathématiques de Luminy Marseille

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Solving the puzzle the role of mathematical
modelling
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Modelisation of Genetic Regulatory
NetworksGenerally, interaction networks are
represented by directed graphs nodes ? genes
arcs ? interactions (oriented)

• 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,)

• 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,.)

• Other approach PLDE
• level of expression assumed to be continuous
  fonction of time
• Hyp exp. level of gene products follow sigmoid
  regulation functions
• gtThe parameters of the differential
  equations are discrete (de Jong et al)

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Summary

• Modelling framework
• Biological application
• Focussing on isolated regulatory circuits
• Conclusions and perspectives

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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

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Modelling framework (2)A 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)
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Modelling framework (3)A simple illustration
Effects of combinations of regulatory actions
defined by logical parameters Kj
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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) gt xj
  call to increase
• xj denotes that Kj(x) lt 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
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Modelling framework (5)A simple illustration
A/ synchronous
B/ asynchronous
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D. melanogaster from embryo to adult
Source Wolpert et al. (1998)
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Anterior-posterior patterning in Drosophila
3 cross-regulatory modules initiating
segmentation Gap Pair-rule Segment-polarity
Source Wolpert et al. (1998)
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Simulation of the Gap Module
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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).
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Gap Module
Collaboration with Lucas SANCHEZ (CIB, Madrid)
Cad
Bcd
Hbmat
Maternal
Zygotic gap
Kni
Gt
Hbzyg
Kr
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Multi-level logical model for the Gap module
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Patterns of gene expression (mRNAs or proteins)
Simultaneous labelling of HB, KR GT Proteins
in Drosophila embryo
Source Reinitz , personal communication
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Logical modelling of the GAP module
Source Sanchez Thieffry 2001
Regulatory graph
T10
Region A
T2
T7
T3
T4
T8
T5
T6
T9
T1
Parametrisation
Asynchronous dynamical graph
Patterns observed in region A
hb
bcd
gt
gt, hbzyg, Kr, kni
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Gap Module - Simulation ( gt, hbzyg, Kr, kni )
Bcd3, hbmat2, cad0
Bcd2, hbmat2, cad0
Bcd1, hbmat0, cad1
Bcd0, hbmat0, cad2
hb
cad
gt
Kr
kni
bcd
gt
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Simulation of maternal and gap loss-of-function
mutations
4 trunk domains
Anterior pole
Posterior pole
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Focussing on regulatory circuitsMotivations

• 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?

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Focussing on regulatory circuits
Regulatory circuits are simple structures and
play a crucial role in the dynamics of biological
systems
Characteristics
Positive circuits
Negative circuits
Odd
Number of repressions
Even
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)

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Example of a 4-genes positive circuit
synchronous dynamical graph
4 genes positive regulatory circuit
Synchronous dynamical graph
1100
0011
0000
0001
0101
1010
1101
1000
1111
1011
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Example of a 4-genes positive circuit
synchronous dynamical graph
k 4
configuration
a
d
b
c


0001
k 2

-
-

1101
1000
-
-
1011
k 0
0110
1001
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Example of a 4-genes negative circuit
synchronous dynamical graph
4 genes negative regulatory circuit
Synchronous dynamical graph
1010
0100
0111
k3
1100
d
0011
a
1000
1011
0101
c
b
0000
0001
0010
k1
0110
1001
1101
1110
1111
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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 k0 stationary
states) Negative Circuits only odd values for k
(? periodic
behaviour)
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Example of a 4-genes positive circuit the
asynchronous dynamical graph
The synchronous version
k4
k2
k0
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Example of a 4-genes positive circuit the
asynchronous dynamical graph
The synchronous version
k4
k2
k0
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Example of a 4-genes negative circuit the
asynchronous dynamical graph
The synchronous version
k3
0000
0001
0010
k1
0110
1001
1101
1110
1111
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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

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A compacted view of the asynchronous
graph example of the 4-genes positive circuit
k4
k2
k0
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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)