Logical modelling of cellular decisions Denis Thieffry (thieffry@ens.fr) Contents Introduction Logical modelling T-helper cell differentiation MAPK network Porquerolles, June 25th, 2013
Cell proliferation, differentiation or death... How are decisions taken?
Key biological questions How does a cell decide which differentiation pathway to follow? When and to what extend cells become committed? To what extend and how is it possible to force cell to change their differentiation states? => Investigations using dynamical modelling
Dynamical modelling Why ? • To gain rigourous, global, functional understanding of the (complex) underlying networks • To predict the behaviour of the system in novel situations • To design novel experiments How ? • Regulatory charts/maps/graphs (CellDesigner, Cytoscape) • Qualitative modelling: Boolean / multilevel discrete networks • Quantitative modelling: ODE, PDE, Stochastic equations
Boolean networks - Stuart Kauffman (1969) x t + 1 = B ( x t ) The Boolean vector x represents the state of the system Random connections , nodes with predefined degree Canalizing Boolean functions Focus on asymptotic behaviour Two types of attractors : stable states and (simple) cycles Deterministic behaviour (only one possible following state)
Kinetic logic - René Thomas (1973) X i ( image or logical function ) specifies whether gene i is currently transcribed X = B ( x ) x i ( logical variable ) denotes the presence (above a threshold of the functional product of gene i Gene i switched ON Gene i switched OFF 1 0 X i 1 x i 0 t Delay d OFF Delay d ON
Logical modelling of regulatory networks A A graph describes the interactions [1] between genes or regulatory products [2] C Discrete levels of expression associated to each regulatory component and interaction B Logical rules/parameters K A = 2 IFF (C=0) K B = 1 IFF (A=1) K C = 1 IFF (B=1) AND (C=0) K A = 0 otherwise K B = 0 otherwise K C = 0 otherwise C A B C=0 1 0 2 1 C C C 2 0 0 0 1 0 0 1 0 Decision trees
Logical modelling of regulatory networks A A graph describes the interactions [1] between genes or regulatory products [2] C Discrete levels of expression associated to each regulatory component and interaction B Logical rules/parameters K A = 2 IFF (C=0) K B = 1 IFF (A=1) K C = 1 IFF (B=1) AND (C=0) K A = 0 otherwise K B = 0 otherwise K C = 0 otherwise C A B 0 2 C=0 1 1 C 2 0 0 1 1 0 Decision diagrams
Logical state transition graphs A Regulatory graph + Logical rules => simulations / dynamical analysis [1] [2] C Asynchronous updating (R Thomas) B ABC C ↓ C ↑ A ↑ State B ↓ transition B ↓ graph Stable state
Logical state transition graphs A [1] [2] Synchronous updating (S Kauffman) C B ABC + Logical rules A ↑ C ↑ Cycle State B ↓ C ↓ transition graph Cycle C ↓ A ↑ Stable state
Logical state transition graphs A Mixed a/synchronous updating: 2 priority classes: [1] (1) synchronous fast decays [2] (2) synchronous slow syntheses C B ABC + Logical rules A ↑ C ↑ B ↓ C ↓ A ↑ B ↑ B ↓ State B ↓ transition graph A ↑ Stable state Fauré et al (2006) Bioinformatics 22 : e124-31
GINsim ( G ene I nteraction N etworks sim ulation ) Aurélien NALDI Fabrice LOPEZ Duncan BERENGIER Claudine CHAOUIYA analysis toolbox core simulator State transition graph GINML parser user interface Regulatory graph graph simulation editor graph analysis Naldi et al (2009) BioSystems 97 : 134-9 Available at http://ginsim.org Chaouiya et al (2013) Meth Mol Biol 804 : 463-79
Development of dynamical analysis tools Decision diagrams • Identification of attractors • State transition graph compression • Analysis of regulatory circuits • Model reduction Priority classes • Mixed a/synchronous simulations Petri nets • Standard Petri nets • Coloured Petri nets Model checking • Verification of dynamical properties (temporal logic) Logical programming • Attractor identification and reachability analysis
Efficient identification of stable states C => 2 stable states : 001 et 110 A B K A = 1 IFF A K B =1 IFF A & !C K C =1 IFF !A A A A 0 C 1 1 0 1 0
Efficient identification of stable states C => 2 stable states : 001 et 110 A B K A K B K C A A A A * * B B 1 1 B B stable 1 0 C C C C C C stable 0 0 0 1 0 0 1 0 1 unstable Stability condition Naldi, Chaouiya & Thieffry (2007) LNCS 4695 : 233-47.
Coping with the exponential growth of logical state transition graphs Attractor identification Compaction of state transition graphs Model reduction Temporisation (e.g. priorities, delays, etc.) Model checking
Bacteriophage lambda: regulatory graph 2 3 1 2 2 3 1 2 Thieffry & Thomas (1995)
Phage lambda model : logical rules Node => target value Logical Rule CI => 2 !Cro | CII CI => 0 Otherwise Cro => 3 !CI & !Cro Cro => 2 !CI & Cro Cro => 0 CI CII => 1 !CI & !Cro & N CII => 0 Otherwise N => 1 !CI & !Cro N => 0 Otherwise Thieffry & Thomas (1995)
Lambda phage model: state transition graph (STG) Lysogeny ( only CI expressed) [CI, Cro, CII, N]
Lambda phage model: state transition graph (STG) coloration according to strongly connected components cyclic attractor for lysis lysogeny only CRO only CI expressed (homeostatically) expressed
Lambda phage model: graph of strongly connected components (SCCG) lysogeny lysis
Lambda phage model: hierarchical state transition graphs (HTG) transient pathways CI ↓ Cro ↑ Cro ↑ CI ↑ CII ↓ N ↓ Cro ↓ CII ↓ transient cycle N ↓ CI ↑ CI ↓ Cro ↑ Cro ↓ CII ↓ N ↓ stable state cyclic attractor lysogeny lysis only CI expressed only CRO (homeostatically) expressed HTG computation on the fly using Tarjan algorithm + decision diagrams
Content of HTG components (schemata) Component Type Number of states Schemata CI Cro CII N i-7 transient paths 7 0 0 0 * 0 0 1 1 1 0 * * i-3 transient paths 3 2 0 0 1 2 0 1 * ct-31 transient cycles 31 0 1 0 * 0 1 1 1 0 2-3 0 1 0 2-3 1 * 1 1 * * 1 2-3 0 1 1 2-3 1 * 2 1-3 * * ct-2 transient cycle 2 1 2-3 0 0 ss-2000 Stable state 1 2 0 0 0 ca-2 cyclic attractor 1 0 2-3 0 0
Regulatory circuits: dynamics in isolation Positive circuit Negative circuit A A B D D B C C attracting cycle stable states Remy et al (2003) Bioinformatics 10 : ii172-8
Regulatory circuits & Thomas' rules A positive regulatory circuit is necessary to generate multiple stable states or attractors A negative regulatory circuit is necessary to generate sustained oscillatory behaviour Thomas R (1988). Springer Series in Synergics 9 : 180-93. Mathematical theorems and demonstrations: In the differential framework: • Thomas (+, 1994), Plathe et al. (±, 1995), Snoussi (±, 1998), Gouzé (±, 1998), Cinquin & Demongeot (+, 2002), Soulé (+, 2003). In the discrete framework: Aracena et al. (+, 2001), Remy et al. (±, 2005), Richard & Comet (+, 2005).
Regulatory circuit functionality Circuit properties depends on the effect of A on B A If A alone is able to switch OFF B: In the presence of A : → only one stable state with {A,B,C,D}= 1011 B C In the absence of A : → two stable states 0100 and 0011 The positive cross-inhibitory circuit involving B and C is thus functional D only in the absence of A Development of a computational algorithm enabling the analysis of the functionality of regulatory circuits in the discrete case => GINsim
CD4+ T-helper cell differentiation Multiple signalling pathways Various transcriptional factors Specific expression patterns (TFs and lymphokines)
Logical modelling of Th activation Klamt S et al (2006). BMC Bioinformatics 7 : 56.
Logical modelling of Th1/Th2 cell differentiation Mendoza L (2006). BioSystems 84 : 101-14.
Towards a comprehensive, modular logical model of the Th differentiation network $% $%& ( $%& ' !"# !"#" ILR = 1 IFF IL AND ILR1 AND ILR2 Yamoka et al (2004)
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