On the convergence of Boolean automata networks without negative cycles Tarek Melliti and Damien Regnault e d’´ IBISC - Universit´ Evry Val d’Essonne, France Adrien Richard I3S - Universit´ e de Nice-Sophia Antipolis, France Sylvain Sen´ e LIF - Universit´ e d’Aix-Marseille, France Gießen, September 19, 2013 Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 1/20
Boolean networks = Finite and heterogeneous CAs on { 0 , 1 } Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 2/20
Boolean networks = Finite and heterogeneous CAs on { 0 , 1 } Classical models for Neural networks [McCulloch & Pitts 1943] Gene regulatory networks [Kauffman 1969, Tomas 1973] Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 2/20
Focus on interaction graphs 5 6 4 7 3 1 8 9 2 10 11 Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 3/20
Focus on interaction graphs 5 6 4 7 3 1 8 9 2 10 11 Question What can be said on the dynamics of a Boolean network according to its interaction graph ? Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 3/20
Focus on interaction graphs 5 6 4 7 3 1 8 9 2 10 11 [Arabidopsis Thaliana] Question What can be said on the dynamics of a Boolean network according to its interaction graph ? Application to gene networks : reliable information on the interaction graph only. Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 3/20
Definitions Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 4/20
Setting There are n components (cells) denoted from 1 to n The set of possible states (configurations) is { 0 , 1 } n The local transition function of component i ∈ [ n ] is any map f i : { 0 , 1 } n → { 0 , 1 } The resulting global transition function is f : { 0 , 1 } n → { 0 , 1 } n , f ( x ) = ( f 1 ( x ) , . . . , f n ( x )) Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 5/20
Setting There are n components (cells) denoted from 1 to n The set of possible states (configurations) is { 0 , 1 } n The local transition function of component i ∈ [ n ] is any map f i : { 0 , 1 } n → { 0 , 1 } The resulting global transition function is f : { 0 , 1 } n → { 0 , 1 } n , f ( x ) = ( f 1 ( x ) , . . . , f n ( x )) We consider the fully-asynchronous updating ֒ → very usual in the context of gene networks [Thomas 73] Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 5/20
Given a map v : N → [ n ] , the fully-asynchronous dynamics is x t +1 v ( t ) = f v ( t ) ( x t ) , x t +1 = x t ∀ i � = v ( t ) i i Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 6/20
Given a map v : N → [ n ] , the fully-asynchronous dynamics is x t +1 v ( t ) = f v ( t ) ( x t ) , x t +1 = x t ∀ i � = v ( t ) i i In practice, non information on v ... → we regroup all the possible asynchronous dynamics under the form of a directed graph Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 6/20
Given a map v : N → [ n ] , the fully-asynchronous dynamics is x t +1 v ( t ) = f v ( t ) ( x t ) , x t +1 = x t ∀ i � = v ( t ) i i In practice, non information on v ... → we regroup all the possible asynchronous dynamics under the form of a directed graph Definition The asynchronous state graph of f , denoted by ASG ( f ) , is the directed graph on { 0 , 1 } n with the following set of arcs: x i | x ∈ { 0 , 1 } n , i ∈ [ n ] , x i � = f i ( x ) � � x → ¯ Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 6/20
Example x f ( x ) ASG ( f ) 000 100 011 111 001 110 010 100 010 110 011 110 100 010 001 101 101 110 110 010 000 100 111 111 Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 7/20
Example x f ( x ) ASG ( f ) 000 100 011 111 001 110 010 100 010 110 011 110 100 010 001 101 101 110 110 010 000 100 111 111 Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 7/20
Example x f ( x ) ASG ( f ) 000 100 011 111 001 110 010 100 010 110 011 110 100 010 001 101 101 110 110 010 000 100 111 111 Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 7/20
Example x f ( x ) ASG ( f ) 000 100 011 111 001 110 010 100 010 110 011 110 100 010 001 101 101 110 110 010 000 100 111 111 Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 7/20
Example x f ( x ) ASG ( f ) 000 100 011 111 001 110 010 100 010 110 011 110 100 010 001 101 101 110 110 010 000 100 111 111 Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 7/20
Example x f ( x ) ASG ( f ) 000 100 011 111 001 110 010 100 010 110 011 110 100 010 001 101 101 110 110 010 000 100 111 111 The attractors of ASG ( f ) are its terminal strong components - Attractor of size one = fixed point - Attractor of size at least two = cyclic attractor Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 7/20
Example x f ( x ) ASG ( f ) 000 100 011 111 001 110 010 100 010 110 011 110 100 010 001 101 101 110 110 010 000 100 111 111 The attractors of ASG ( f ) are its terminal strong components - Attractor of size one = fixed point - Attractor of size at least two = cyclic attractor A path from a state x to a state y is a direct path if its length ℓ is equal to the Hamming distance between x and y (so ℓ ≤ n ). Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 7/20
Definition The interaction graph of f , denoted G ( f ) , is the signed directed graph on { 1 , . . . , n } with the following arcs: - There is a positive arc j → i iff there is a state x such that f i ( x 1 , . . . , x j − 1 , 0 , x j +1 , . . . , x n ) = 0 f i ( x 1 , . . . , x j − 1 , 1 , x j +1 , . . . , x n ) = 1 - There is a negative arc j → i iff there is a state x such that f i ( x 1 , . . . , x j − 1 , 0 , x j +1 , . . . , x n ) = 1 f i ( x 1 , . . . , x j − 1 , 1 , x j +1 , . . . , x n ) = 0 Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 8/20
Definition The interaction graph of f , denoted G ( f ) , is the signed directed graph on { 1 , . . . , n } with the following arcs: - There is a positive arc j → i iff there is a state x such that f i ( x 1 , . . . , x j − 1 , 0 , x j +1 , . . . , x n ) = 0 f i ( x 1 , . . . , x j − 1 , 1 , x j +1 , . . . , x n ) = 1 - There is a negative arc j → i iff there is a state x such that f i ( x 1 , . . . , x j − 1 , 0 , x j +1 , . . . , x n ) = 1 f i ( x 1 , . . . , x j − 1 , 1 , x j +1 , . . . , x n ) = 0 j → i ∈ G ( f ) ⇐ ⇒ f i ( x ) depends on x j Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 8/20
Example Asynchronous State Graph Interaction Graph f ( x ) x G ( f ) ASG ( f ) 000 100 001 110 011 111 010 100 1 2 011 110 010 110 100 010 101 110 001 101 110 010 3 111 111 000 100 Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 9/20
Example Asynchronous State Graph Interaction Graph f ( x ) x G ( f ) ASG ( f ) 000 100 001 110 011 111 010 100 1 2 011 110 010 110 100 010 101 110 001 101 110 010 3 111 111 000 100 Question What can be said on ASG ( f ) according to G ( f ) ? Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 9/20
Results Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 10/20
Theorem [Robert 1980] If G ( f ) has no cycles then 1. f has a unique fixed point 2. ASG ( f ) has no cycles Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 11/20
Theorem [Robert 1980] If G ( f ) has no cycles then 1. f has a unique fixed point 2. ASG ( f ) has no cycles 3. ASG ( f ) has a direct path from every state to the fixed point Melliti, Regnault, Richard , Sen´ e Convergence of Boolean networks without negative cycles Automata 2013 11/20
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