Modelisation and Dynamical Analysis of Genetic Regulatory Networks

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

2. S G1 G2 M Solving the puzzle: the role of mathematical modelling

3. 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,….) Modelisation of Genetic Regulatory NetworksGenerally, interaction networks are represented by directed graphs: nodesgenesarcs 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,…) • 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 ={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genesG ={g1,g2,...,gn} • arcs interactions (oriented) • labeltype of interaction (-1 repression, +1 activation) + the condition for which the interaction is operating

6. source target type condition 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])

7. Modelling framework (3)A simple illustration Effects of combinations of regulatory actions defined by logical parameters Kj

8. + - 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 acall 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

9. Modelling framework (5)A simple illustration A/ synchronous B/ asynchronous

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

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

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

13. 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).

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

15. Cad Bcd Hbmat Maternal Zygotic gap Kni Gt Hbzyg Kr Multi-level logical model for the Gap module

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

17. Kni Hb Gt Kr 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

18. ++ + +   + + + 0000 0000 0200 +++ ++ 0200 0001 + ++ + [1000] 0100 0001 ++ + + 1200 0300 0210 ++ + + 0110 0101 - [1300] 1001 [0220] [0111] gt hb Kr kni Gap Module - Simulation (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 hb cad Kr gt kni bcd gt

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

20. 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?

21. Focussing on regulatory circuits Regulatorycircuits 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)

22. d a 1110 0010 0111 c b 0100 0110 1001 Example of a 4-genes positivecircuit: synchronous dynamical graph 4 genes positive regulatory circuit Synchronous dynamical graph 0011 1100 0000 0001 0101 1010 1101 1000 1111 1011

23. d 0011 1100 a 1110 0000 0010 0111 c 0101 1010 b 0100 1111 Example of a 4-genes positivecircuit: synchronous dynamical graph k = 4 configuration a d b c + + 0001 k = 2 - + - + 1101 1000 - - 1011 k = 0 1001 0110

24. Example of a 4-genes negativecircuit: synchronous dynamical graph 4 genes negative regulatory circuit Synchronous dynamical graph 1010 0100 0111 k=3 1100 d 0011 a 1011 1000 0101 c b 0000 0001 0010 k=1 0110 1001 1101 1110 1111

25. 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 stationarystates) Negative Circuits: only odd values for k ( periodic behaviour)

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

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

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

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. A compacted view of the asynchronous graph example of the 4-genespositivecircuit k=4 k=2 k=0

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)