1. 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

2. S M G2G1 Solving the puzzle: the role of mathematical modelling

3. Modelisation of Genetic Regulatory Networks Modelisation 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)

4. Summary  Modelling framework  Biological application  Focussing on isolated regulatory circuits  Conclusions and perspectives

5. Modelling framework  A multivalued discrete method G ={g 1,g 2,...,g n } set of genes, regulatory products… for each g i  expression level x i  {0,..., max i } max i is the number of "relevant" levels of expression of g i  Interaction networks represented by labeled oriented graphs, the Regulatory Graphs  nodes  genes G ={g 1,g 2,...,g n }  arcs  interactions (oriented)  label  type of interaction (-1 repression, +1 activation) + the condition for which the interaction is operating

6. Modelling framework (2) A simple illustration Interactions: T 1 = ( g 1, g 2, 1, [1]) T 2 =(g 1,g 2,-1,[2]) T 3 =(g 2,g 2,1,[1]) T 4 =(g 2,g 3,1,[1]) T 5 =(g 3,g 1,-1,[1]) source target type condition NodesValues g1g1 0, 1, 2 g2g2 0, 1 g3g3

7. Modelling framework (3) A simple illustration NodesValues Parameters default value is 0 g1g1 0, 1, 2 K 1 {  } =2 g2g2 0, 1K 2 {T 1 }=K 2 {T 3 }=K 2 {T 1,T 3 }=1 g3g3 0, 1K 3 {T 4 }=1 Effects of combinations of regulatory actions defined by logical parameters Kj

8. Modelling framework (4)  given x=(x 1,x 2,...,x n ) a state, K j (x) precises to which value gene g j should tend if K j (x)  x j gene g j receives a call for updating x j denotes that K j (x) > x j call to increase x j denotes that K j (x) < x j 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 ++

9. Modelling framework (5) A simple illustration NodesValues Parameters default value is 0 g1g1 0, 1, 2 K 1 {  } =2 g2g2 0, 1K 2 {T 1 }=K 2 {T 3 }=K 2 {T 1,T 3 }=1 g3g3 0, 1K 3 {T 4 }=1 A/ synchronous B/ asynchronous

10. Source: Wolpert et al. (1998) D. melanogaster : from embryo to adult

11. Source: Wolpert et al. (1998) 3 cross-regulatory modules initiating segmentation Gap Pair-rule Segment-polarity Anterior- posterior patterning in Drosophila

12. Multiple asynchronous transitions Input: Initial maternal gradients BCD HB mat BCD HB mat BCD CAD HeadTrunkTelson Output: For expression patterns for the gap genes BCD HB BCD HB BCD CAD GTKR KNI KR GT HB Simulation of the Gap Module

13. 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)

14. Gt Bcd Hb zyg Hb mat Cad Kr Kni Maternal Zygotic gap Collaboration with Lucas SANCHEZ (CIB, Madrid) Gap Module

15. Multi-level logical model for the Gap module Gt Bcd Hb zyg Hb mat Cad Kr Kni Maternal Zygotic gap

16. Simultaneous labelling of HB, KR & GT Proteins in Drosophila embryo Patterns of gene expression (mRNAs or proteins) Source: Reinitz, personal communication gt hb Kr kni bcd cad

17. T 10 Gt Hb Kr Kni T2T2 T1T1 T3T3 T5T5 T6T6 T7T7 T8T8 T9T9 T4T4 Region A gt hb bcd Regulatory graph Patterns observed in region A Asynchronous dynamical graph Parametrisation Logical modelling of the GAP module Source : Sanchez & Thieffry 2001 gt, hb zyg, Kr, kni

18. Bcd=3, hb mat =2, cad=0Bcd=2, hb mat =2, cad=0Bcd=1, hb mat =0, cad=1Bcd=0, hb mat =0, cad=2 0000 [1000] 0001 + 1001 - + 0001 0000 ++ + [0111] 0100 01100101 + ++ + ++ [0220] 0200 + 0210 + [1300] 0200 +++ 12000300 +++ gthbKrkni Gap Module - Simulation ( gt, hb zyg, Kr, kni ) gt hb Kr kni bcd cad

19. 4 trunk domains Anterior pole Posterior pole Simulation of maternal and gap loss-of-function mutations

20. 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?

21. Focussing on regulatory circuits Regulatory circuits are simple structures and play a crucial role in the dynamics of biological systems : CharacteristicsPositive circuits Negative circuits Number of repressions Even Odd Dynamical property Biological property DifferentiationHomeostasis 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)

22. Example of a 4-genes positive circuit: synchronous dynamical graph 0110 1001 1000 1101 0001 1011 11000011 1010 0101 0000 1111 0111 0010 1110 0100 d c b a 4 genes positive regulatory circuit Synchronous dynamical graph

23. a d c b 1000 1101 0001 1011 1010 0101 0000 1111 0111 0010 1110 0100 01101001 k = 4 k = 2 k = 0 11000011 d c b a Example of a 4-genes positive circuit: synchronous dynamical graph + + + - - -- + configuration

24. k=3 0111 0100 10001011 0011 k=1 0010 1110 1101 0001 0110 1001 0000 1111 1100 1010 0101 Example of a 4-genes negative circuit: synchronous dynamical graph d c b a 4 genes negative regulatory circuit Synchronous dynamical graph

25. General case: the synchronous dynamical 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)

26. k=4 k=2 k=0k=0 Example of a 4-genes positive circuit: the asynchronous dynamical graph The synchronous version

27. k=4 k=2 k=0k=0 Example of a 4-genes positive circuit: the asynchronous dynamical graph The synchronous version

28. k=3 k=1 The synchronous version Example of a 4-genes negative circuit: the asynchronous dynamical graph 0010 1110 1101 0110 1001 1111 0001 0000

29. 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

30. k=4 k=2 k=0k=0 A compacted view of the asynchronous graph example of the 4-genes positive circuit

31. 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)