Qualitative analysis of regulatory networks

Transcription

1 Qualitative analsis of regulator networks Denis THIEFFRY (LGPD-IBDM, Marseille, France) Introduction Formal tools for the analsis of gene networks Graph-based representation of regulator networks Dnamical formalisation Applications E. coli and bacteriophage transcriptional regulation Conclusions

2 Contents Introduction Formal tools for the analsis of gene networks Graph theoretical tools Dnamical formalisation Applications Transcriptional regulation in E. coli Conclusions and perspectives

3 Molecular interactions graphs: (a) DNA-proteins interactions gene A gene B gene C gene D gene E gene F gene G gene H gene I

4 Molecular interactions graphs: (b) Proteins-proteins interactions (e.g. multimerisation) gene A gene B gene C gene D gene E gene F gene G gene H gene I

5 Molecular interactions graphs : (c) DNA-Proteins multimerisation gene A gene B gene C gene D gene E gene F gene G gene H gene I

6 Graph-based representation of transcriptional regulator networks Linear cascade gene A Diverging cascades gene A gene A Circuit gene B gene B [regulator product] gene B gene C gene C gene E gene D gene E gene D gene C Converging cascades Intertwined circuits gene A gene B gene A gene B gene D gene E gene C gene E? gene D gene C gene D gene E

7 Translating biological questions into graph-theoretical framework Biological questions Identification of regulator cascades identification of the set of interactions linking 2 genes Identification of the shortest sequences of interactions linking 2 genes Identification of all regulator circuits Identification of cross-regulator modules Identification of the most interactive nodes Identification of sub-networks with similar regulator structure (cross-regulator motifs) Comparison of molecular sub-networks with genetic sub-networks Intra ou Inter-specific comparisons of sub-networks Graph-theoretical concepts Partial order Interval between two vertices Shortest patwas between two vertices Elementar and isometric circuits Ccles, blocs, or bi-conne componants Vertices of highest degree Isomorphism Homeomorphism Biggest common sub-graph

8 Partial order gene A Eamples Shortest path between A et H gene A Conne componant (protein-protein graph) gene B gene C gene D gene D gene D gene E gene F gene G gene H gene H Composante fortement connee Circuits gene C gene C gene C gene E gene F gene E gene E gene F gene I gene I gene I

9 Contents Introduction Formal tools for the analsis of gene networks Graph theoretical tools Dnamical formalisation Applications Transcriptional regulation in E. coli Conclusions and perspectives

10 Dnamical formalisation of gene networks A wealth of different formal approaches : Graph analsis Logical equations Nonlinear Ordinar Differential Equations Piecewise Linear Differential Equations Partial Differential Equations Stochastic Equations

11 Boolean formalism : snchronous updating (1) t t t1 = t t1 t1 = t t1 interaction graph logical equations interaction matri () t () t [01] 01 [10] state table [10] [01] attractors

12 Boolean formalism : snchronous updating (2) z t1 = z t t1 = t z t1 = t t1 t1 z t1 t t z t interaction graph logical equations interaction matri (z) t (z) t state table z spontaneous state transitions

13 Boolean formalism : asnchronous updating (1) X = Y = X Y interaction graph logical equations interaction matri XY [01] 01 [10] state table 00 [10] [01] spontaneous state transitions

14 Boolean formalism: asnchronous updating (2) z X = z Y = Z = z X Y Z interaction graph XYZ z state table logical equations interaction matri z spontaneous state transitions

15 Boolean formalism : logical operators : OR OR X = Y = X Y interaction graph logical equations interaction matri XY [11] 11 state table 01 [11] spontaneous state transitions

16 Boolean formalism : logical operators : AND AND X =. Y = X Y interaction graph logical equations interaction matri XY [00] state table [00] 10 spontaneous state transitions

17 Differential formalism : functional positive circuit ' = k F - () - k- ' = k F - () - k- / / interaction graph differential equations Jacobian matri s - '=0 - - '=0 threshold F - (, s ) = Hill coefficient s n s n n - decreasing sigmoid Hill function phase space 0 0 s

18 Differential formalism: NON functional positive circuit ' = k F - () - k- ' = k F - () - k- / / interaction graph differential equations Jacobian matri '=0 - '= s s '=0 '= s parameter constraints 0 s role of non linearit

19 Differential formalism: non-linearit and iteration interaction graph ' = k F - () - k - ' = k F - () - k - differential equations At the stead state ('='=0) = (k /k - )F - () = G - 1 () = (k /k - )F - () = G - 2 () --> = (G - 1 (G - 2 ()) - '=0 Linear F - () - - G 2 - (G 1 - ()) Iteration s '=0 s s 0 s

20 Differential formalism: negative circuit z interaction graph ' = k F - (z) - k - ' = k F - () - k - z' = k zf z- () - k -z z Differential equations z / / / z Jocobian matri G 3 - (G 2 - (G 1 - ()) s Iteration 0 0 s

21 Differential formalism: more interactions ' = k F () k F - () - k - ' = k F () - k - / / interaction graph differential equations Jacobian matri Diversit of dnamical behavior: multistationarit and/or oscillations, depending on parameter values and on non linearit threshold Hill coefficient F (, s ) = n s n n increasing sigmoid Hill function

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23 Asnchronous multilevel logical formalisation René THOMAS, Marcelle KAUFMAN, El Houssine SNOUSSI, Philippe VAN HAM, Jean RICHELLE Multi-level logical variables (concentrations, activities: i = 0, 1, 2 ) and functions (epression level: X i = 0, 1, 2 ) Asnchronous versus snchronous treatment Logical parameters (K i, K i.j, K i.jk, = 0, 1, 2, ) Emphasis on the roles of feedback circuits Thomas (1991). J theor Biol 153: 1-23.

24 Generalised logical formalism Logical variables/functions Protein "present" (=1) Protein «absent» (=0) Gene «off» (X=0) Gene "on" (X=1) t t - «order» «order» time Logical parameters Gene Gene z Gene K z K z. K z. K z. 0 S (1) 1 Real scale Logical scale

25 Gene Logical parameters Gene? Gene z K s Logical parameter values 0 0 K z K z K z K z Logical function AND OR XOR SUM "Additive" Boolean Multi-level

26 Logical parameter values and circuit "functionalit" [Protein ] 1 S (1) 0 K K. X=F() Y=F() K. K Eample: phage λ ci-cro "switch" - - «Functional» 0 S (1) 1 [Protein ] [Protein ] 1 0 K S (1) K. K. Y=F() - X=F() K - NOT «functional» 0 S (1) 1 [Protein ]

27 Regulator circuits Characteristics Positive circuits Negative circuits Number of negative interactions Dnamical propert Even Maimal level Odd Bottom level Biological propert Differentiation Homeostasis Eamples - ci cro - - tat rev

28 Contents Introduction Formal tools for the analsis of gene networks Graph theoretical tools Dnamical formalisation Applications Transcriptional regulation in E. coli Conclusions and perspectives

29 Snthesis of auto-regulated gene circuits Gardner et al. (2000) Nature 403: Elowitz & Leibler (2000) Nature 403: Becskei & Serrano (2000) Nature 405: Construction Inductor 1 R1 P2 P1 R2 GFP P2 R1 P3 R2 P1 R3 P1 R1 GFP Inductor 2 P1 GFP Logical scheme Positive circuit R1 R2 Negative circuit R2 R3 R1 GFP Negative circuit R1 Properties - Stable and eclusive epression of one repressor - Memorisation of induction - Stabilit and robustness against biochemical fluctuations - Cclic epression of the repressors and reporter gene - Transmission of this oscillating behaviour through bacterial divisions - Increased stabilit and decreased variabilit of the repressor epression - Compensation of dosage effects due to the variation of the number of copies Thieffr D (2001). Médecine/Sciences 17:

30 The lambda phage Lambda phage. Left: Overall structure. Right: Alternative developmental pathwas

31 Lambda genomic organisation and transcriptional regulation Schematic representation of the lambda genome organisation. The double line represents DNA, along which blocks correspond to genes (long, clear blocks) or to regulator regions (shorter, shaded or dark blocks). Circled names stand for the regulator products encoded b the viral genome and vertical arrows show their points of actions. P i and T i designate the promoters and terminators, respectivel ; finall, horizontal arrows represent the different transcripts.

32 Main regulator components involved in the control of the lsis-lsogen decision in the bacteriophage lambda. Protease RecA SOS UV integration Xis/Int CI Cro CII N Protease Lon Proteases HFLA/B CIII Q late phage functions

33 The lambda phage switch [Protein] Occupied sites Transcription O L 1 O L 2 O L 3 O R 1 O R 2 O R 3 P L P R P M CI Repressed CI Repressed Repressed Activated CI Repressed Repressed Repressed CRO () Repressed CRO () () Repressed Repressed CRO Repressed Repressed Repressed Sub-regions of O L and O R operators and their occupation b CI and Cro transcription factors, for increasing concentration of these proteins, together with the corresponding effects on transcription from the promoters P L, P R and P M.

34 A simple model for the lambda phage switch ci () (1) cro() (2) X Y 0 0 K. K. 0 1 K K. Regulator Graph 0 2 K K. ci X cro 1 0 K. K. 1 1 K K. 1 2 K K Y cro Interaction Matri General State Table

35 A simple model for the lambda phage switch ci () (1) cro() (2) X Y 0 0 K. K K K. 2 0 K K. K K. 1 0 K. 0 K K. K. K. K. 0 1 K = K = 0

36 A simple model for the lambda phage switch (1) ci () cro() (2) X Y K. 2 0 K K K K K. K = K = 0, K. = 1, K. = 2

37 A simple model for the lambda phage switch ci () (1) cro() (2) X Y K = K = 0, K. = 1, K. = 2 K. =1, K. = 0

38 A simple model for the lambda phage switch: Parameter constraints enabling feedback circuit functionalit ci () (1) cro () (2) X Y 0 0 K. K. 0 1 K K. 0 2 K K. 1 0 K. K. 1 1 K K. 1 2 K K K. 1 K. = 0 K. 1 K = 0

39 A simple model for the lambda phage switch: Parameter constraints enabling feedback circuit functionalit ci () (1) cro () (2) X Y 0 0 K. K. 0 1 K K. K. = 1 K = K K. 1 0 K. K. 1 1 K K. 1 2 K K

40 A simple model for the lambda phage switch: Parameter constraints enabling feedback circuit funcitonalit ci () (1) cro () (2) X Y 0 0 K. K. 0 1 K K. 0 2 K K. 1 0 K. K. 1 1 K K. 1 2 K K K. = 2 K. 1 K. = 2 K 1

41 A simple model for the lambda phage switch: Parameter constraints enabling feedback circuit functionalit Circuit Sign Thresholds K K. K K. K. K. for = 1 negative s (2) or 1-2 (2) for = 0 negative - - (0 or 1) 0 or 1-2 positive s (1),s(1) 0 1 (0) or 2 Simulation circuit combinator analsis

42 A simple model for the lambda phage switch (1) ci () cro() (2) K = K = K. = 0, K. = K. = 1, K. = 2

43 A more comple model 2 2 ci cro cii N - Thieffr & Thomas (1995) Bull Math Biol 57: Lambda regulator graph. Onl the genes directl involved in feedback circuits are taken into account.

44 Take home messages Qualitative versus (often illusor) quantitative aspects Dnamical roles of feedback circuits Fleibilit of the generalised logical formalism Beware of the artefacts of the snchronous updating method

45 Further reading Rem E, Mosse B, Chaouia C, Thieffr D (2003). A description of dnamical graphs associated to elementar regulator circuits. Bioinformatics 19 (Suppl 2): ii172-ii178. Thieffr D & Thomas R (1995). Dnamical behaviour of biological regulator networks--ii. Immunit control in bacteriophage lambda Bull Math Biol 57: Thieffr D, Huerta AM, Perez-Rueda E, Collado-Vides J (1998). From specific gene regulation to genomic networks: a global analsis of transcriptional regulation in Escherichia coli. Bioessas 20: Thomas R (1991). Regulator networks seen as asnchronous automata: a logical description. J theor Biol 153: Thomas R, Thieffr D & Kaufman M (1995). Dnamical behaviour of biological regulator networks. I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state. Bull Math Biol 57: