From Asynchronous STG to regulatory graph Given a Boolean asynchronous dynamics, represented by the state transition graph ( V , E ) . → Local regulatory graph G ( x ) ֒ ε → j if ∃ x s.t. ( x, x j ) ∈ E and ( x i , x i,j ) �∈ E with ε = + if x i � = x j • i and ε = − if x i = x j x j x i,j x j x i,j x x x i x i
From Asynchronous STG to regulatory graph Given a Boolean asynchronous dynamics, represented by the state transition graph ( V , E ) . ֒ → Local regulatory graph G ( x ) ε → j if ∃ x s.t. ( x, x j ) ∈ E and ( x i , x i,j ) �∈ E with ε = + if x i � = x j • i and ε = − if x i = x j + • positive self-regulation i � : • when f i ( x ) � = f i ( x i ) and x i = f i ( x ) • if both ( x, x i ) �∈ E and ( x i , x ) �∈ E
From Asynchronous STG to regulatory graph Given a Boolean asynchronous dynamics, represented by the state transition graph ( V , E ) . ֒ → Local regulatory graph G ( x ) ε → j if ∃ x s.t. ( x, x j ) ∈ E and ( x i , x i,j ) �∈ E with ε = + if x i � = x j • i and ε = − if x i = x j + • positive self-regulation i � : • when f i ( x ) � = f i ( x i ) and x i = f i ( x ) • if both ( x, x i ) �∈ E and ( x i , x ) �∈ E − • negative self-regulation i � : • when f i ( x ) � = f i ( x i ) and x i � = f i ( x ) • if both ( x, x i ) ∈ E and ( x i , x ) ∈ E
Boolean Asynchronous Dynamics Given a Boolean dynamics f : { 0 , 1 } n → { 0 , 1 } n → Asynchronous state transition graph ( V , E ) : ֒ ( x, x i ) ∈ E iff x i � = f i ( x ) Definitions: • Attractors: terminal strongly connected components • Stable states (Fixed points): f ( x ) = x • Cyclic attractors attractors which are not fixed points • Trap cycle cyclical trajectory s.t. d ( x, f ( x )) = 1 for all x of the cycle
Analysis of logical regulatory graphs Identify, in huge STG ( 2 n states), asymptotical behaviours (attractors), properties along trajectories, impact of perturbations...
Analysis of logical regulatory graphs Identify, in huge STG ( 2 n states), asymptotical behaviours (attractors), properties along trajectories, impact of perturbations... Deduce properties from the model, without constructing its dynamics • Identification of all stable states A. Naldi et al (2007) LNCS/LNBI 4695:233-47 • Analysis of regulatory circuits Reduce the size of the model • Model reduction A. Naldi et al (2011) Theor Comp Sc 412(21) : 2207-18 Reduce the size of the dynamics (STG), yet keeping properties of interest • Introduction of priority classes A. Fauré et al (2006) Bioinformatics 22(14) : 124-131 • Projection over input components P . Monteiro, C. Chaouiya (2012) AISC, Springer , Vol. 154 : 259-67 • Hierarchical transition graph D. Berenguier et al (submitted)
Regulatory circuits
Regulatory circuits • Helpful for the analysis of regulatory networks • A key tool in synthetic biology • ...
Engineering of synthetic circuits 3 pioneer articles • Elowitz and Leibler (2000). A synthetic oscillatory network of transcriptional regulators . Nature 403 : 335-338. • Gardner, Cantor and Collins (2000). Construction of a genetic toggle switch in Escherichia Coli . Nature 403 : 339-342. • Becskel and Serrano (2000). Engineering stability in gene networks by autoregulation . Nature 405 : 590-593; All of these works ֒ → report the conception and construction of synthetic networks which induce amazing behaviors of the cells ֒ → rely on feedback regulatory circuits mathematical modelisation
Isolated circuits 1 n 2 3 4
Isolated positive regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 f = x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 }
Isolated positive regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 f = x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } − + Starting at state ( 1 , 1 , 1 , 0) Asynchronous dynamics (one update at a time)
Isolated positive regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 f = x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } Activating gene 4 Asynchronous dynamics �→ Fixed point (1 , 1 , 1 , 1) (one update at a time)
Isolated positive regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 f = x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } − + Starting again at ( 1 , 1 , 1 , 0) Asynchronous dynamics (one update at a time)
Isolated positive regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 f = x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } Disactivating gene 1 Asynchronous dynamics − + → (0 , 1 , 1 , 0) (one update at a time)
Isolated positive regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 f = x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } Disactivating gene 2 Asynchronous dynamics − + → (0 , 0 , 1 , 0) (one update at a time)
Isolated positive regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 f = x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } Activating gene 4 Asynchronous dynamics + − → ( 0 , 0 , 1 , 1) (one update at a time)
Isolated positive regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 f = x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } Disactivating gene 3 Asynchronous dynamics + − → ( 0 , 0 , 0 , 1) (one update at a time)
Isolated positive regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 f = x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } Disactivating gene 4 Asynchronous dynamics �→ Fixed point (0 , 0 , 0 , 0) (one update at a time)
Isolated positive regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 f = x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } Asynchronous dynamics (one update at a time) Multistationarity
Isolated regulatory circuits 1 ε n ε 1 n 2 3 4 Simple modelling: • Each gene is the source of a unique interaction, and the target of a unique interaction ⇒ Boolean case � if ε i = +1 x i • f i +1 ( x ) = 1 − x i if ε i = − 1
Isolated regulatory circuits Complete description of the structure of the synchronous state transition graph: → Disconnected elementary cycles ֒ → Staged structure: ֒ • each stage k gathers all the states having k calls for updating • the states are distributed in cycles according to their configurations
Synchronous state transition graph of isolated positive circuit 1010 0101 1110 0111 1100 0110 1000 0100 1101 1011 1001 0011 0001 0010 1111 0000
Isolated regulatory circuits From the synchronous state transition graph: Complete description of the structure of the asynchronous state transition graph: → Connected graph ֒ → Staged structure: at each stage k , each state has exactly k ֒ successors • either at the same stage k • or at the stage below k − 2
Isolated positive regulatory circuits The state transition graph 1010 0101 1110 0110 0100 0111 1100 0010 1011 0011 1000 1101 1001 0001 1111 0000
Isolated negative regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 = f x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } x : negation
Isolated negative regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 f = x 3 x 2 x 4 x 3 − − + Starting at state ( 1 , 1 , 0) x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } 1 , x : negation
Isolated negative regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 = f x 3 x 2 x 4 x 3 Activating gene 4 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } − → (1 , 1 , 1 , 1) x : negation
Isolated negative regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 = f x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } − → (1 , 0 , 1 , 1) x : negation
Isolated negative regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 = f x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } − → (1 , 0 , 0 , 1) x : negation
Isolated negative regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 = f x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } − → ( 1 , 0 , 0 , 0) x : negation
Isolated negative regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 = f x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } + → (0 , 0 , 0 , 0) x : negation
Isolated negative regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 = f x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } + → (0 , 1 , 0 , 0) x : negation
Isolated negative regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 = f x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } + → (0 , 1 , 1 , 0) x : negation
Isolated negative regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 = f x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } + → ( 0 , 1 , 1 , 1) x : negation
Isolated negative regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 = f x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } − �→ return to (1 , 1 , 1 , 1) x : negation
Isolated negative regulatory circuits 4 3 x 1 x 4 x 2 x 1 1 2 = f x 3 x 2 x 4 x 3 x 1 , x 2 , x 3 , x 4 ∈ { 0 , 1 } x : negation One (trap) cycle: homeostasis
Isolated negative regulatory circuits The state transition graph 0100 1101 0101 1011 0100 1010 0010 0110 0111 0011 0001 1111 0000 1110 1100 1000
Isolated circuits and attractors Theorem Let f be an asynchronous Boolean dynamics such that G ( f ) is an isolated circuit. Then, • if G ( f ) is a positive circuit then f contains two stable states; • if G ( f ) is a negative circuit then f contains an attractive cycle, and has no stable state.
Composition of circuits Flower graphs a d h b c
Composition of circuits Flower graphs a A flower-graph • there is a particular component h such that ε h i ε i h E = { h → i | i � = h }∪{ i → h | i � = h } d h b • all regulations are univocal • h is not self regulated c
Composition of circuits Flower graphs A flower-graph a • there is a particular component h such that ε h i ε i h E = { h → i | i � = h }∪{ i → h | i � = h } • all regulations are univocal d h b • h is not self regulated c Number of stable states?
Stable states and flower graphs We define the state v by: • v h = 0 , � + 0 if h → i • for all i ∈ C \{ h } , v i = − → i 1 if h a d h b c
Stable states and flower graphs → v and v are the only potential stable states ֒
Stable states and flower graphs → v and v are the only potential stable states ֒ a d h b c f h ( v )? f h ( v )?
Stable states and flower graphs → v and v are the only potential stable states ֒ a d h b c f h ( v )? f h ( v )? Let { y ∈ { 0 , 1 } C | y h = 0 and f h ( y ) = 0 } E 0 = { y ∈ { 0 , 1 } C | y h = 1 and f h ( y ) = 1 } E 1 =
Stable states and flower graphs We define the state u by: • u h = 0 , � + 0 if i → h • for all i ∈ C \{ h } , u i = − → h 1 if i a d h b c
Stable states and flower graphs We have • u ∈ E 0 and u ∈ E 1
Stable states and flower graphs We have • u ∈ E 0 and u ∈ E 1 • For all D ⊆ C \{ h } , if u D ∈ E 0 then, ∀F ⊂ D , u F ∈ E 0 (antichain).
Stable states and flower graphs We have • u ∈ E 0 and u ∈ E 1 • For all D ⊆ C \{ h } , if u D ∈ E 0 then, ∀F ⊂ D , u F ∈ E 0 (antichain).
Stable states and flower graphs ֒ → v and v are the only potential stable states • v stable state if and only if v ∈ E 0 • v stable state if and only if v ∈ E 1
Stable states and flower graphs !" !" # %" # %" # $" # $" !" !" no stable state 2 stable states
Stable states and flower graphs !" !" # %" # %" # $" # $" !" !" 1 stable state
Stable states and flower graphs • If the flower graph contains only positive circuits, then v = u and v = u ⇒ 2 stable states
Stable states and flower graphs • If the flower graph contains only positive circuits, then v = u and v = u ⇒ 2 stable states • If the flower graph contains only negative circuits, then v = u and v = u ⇒ no stable state
Stable states and flower graphs • If the flower graph contains a unique negative circuit ( e.g. implying a ) and at least one positive circuit then v = u a and v = u C\ a . Suppose there is no stable state, u a �∈ E 0 and u C\ a �∈ E 1 . Then, u D ∈ E 0 iff a �∈ D → a unique regulator of h , a contradiction. ⇒ there exists at least one stable state
Stable states and flower graphs • If the flower graph contains a unique negative circuit ( e.g. implying a ) and at least one positive circuit then v = u a and v = u C\ a . Suppose there is no stable state, u a �∈ E 0 and u C\ a �∈ E 1 . Then, u D ∈ E 0 iff a �∈ D → a unique regulator of h , a contradiction. ⇒ there exists at least one stable state • If the flower graph contains a unique positive circuit, and at least one negative circuit ⇒ at most one stable state
Composition of circuits Hub graphs t w p a h b f d m n c
Hub-graphs - all the regulations are univocal - there is a particular component h , called the hub, such that: • h is the only component which can have more than one regulator • for all components i ∈ C , there is a path from h to i • h is not self-regulated t w p a h b f d m n c
From hub- to flower-graphs Given a hub-graph, there exists a flower graph • which contains the same number of positive and negative circuits • with the same number of stable states
From hub- to flower-graphs: 2 transformations 1- Transformation T b , with component b such that: b has a unique input a and O ( b ) ∩ O ( a ) = ∅ or { a } b s s ′ s ss ′ a b c a c ⇒ 2- Transformation P b : elimination of the output b
From hub- to flower-graphs Iterations of transformation T · t w p t a a w p d h f h b f c b n d m n m c
From hub- to flower-graphs Iterations of transformation P · t a a w p d h f d h b c b n c m
Stable states and hub-graphs Theorem Let f be an asynchronous Boolean dynamics. If G ( f ) is a hub-graph then f contains at most two stable states. In the particular case f does contain two stable states, they are complementary. More precisely: • if G ( f ) contains only positive circuits then f contains two stable states; • if G ( f ) contains only negative circuits then f has no stable state; • if G ( f ) contains a unique negative circuit and at least one positive circuit, then f contains at least one stable state; • if G ( f ) contains a unique positive circuit and at least one negative circuit, then f contains at most one stable state. Didier-R., Soumis
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