A fixed point theorem for Boolean networks expressed in terms of - PowerPoint PPT Presentation

A fixed point theorem for Boolean networks expressed in terms of forbidden subnetworks Adrien Richard University of Nice - France CNRS I3S Laboratory

Contents ◃ 1. Introduction ◃ 2. Robert’s fixed point theorem (1980) ◃ 3. Shih-Dong’s fixed point theorem (2005) ◃ 4. Forbidden subnetworks theorem

An n -dimensional Boolean network is a function f : B n → B n ( B = { 0 , 1 } ) x = ( x 1 , . . . , x i , . . . , x n ) �→ f ( x ) = ( f 1 ( x ) , . . . , f i ( x ) , . . . , f n ( x )) ↑ local transition function The interaction graph of f is the directed graph G ( f ) with vertex set { 1 , . . . , n } and arcs defined by j → i ∈ G ( f ) ⇔ f i depends on x j

Example : f : B 3 → B 3 is defined by : f ( x ) x 000 100 001 000 f 1 ( x ) = x 1 ∨ x 3 010 101 011 001 ⇔ f 2 ( x ) = x 1 ∧ x 3 100 100 f 3 ( x ) = x 2 101 110 110 101 111 111 The interaction graph of f is : G ( f ) 1 2 3

A network f with an update schedule (parallel, sequential, block- sequential, asynchronous...) defines a discrete dynamical system . With the parallel update schedule : x t +1 = f ( x t ) Parallel dynamics 011 f 1 ( x ) = x 1 ∨ x 3 001 010 f 2 ( x ) = x 1 ∧ x 3 f 3 ( x ) = x 2 000 101 100 110 111 For all update schedules : fixed points of f = stable states .

Simple definitions, but complex behaviors : several attractors, long limit cycles, long transient phases... Many applications : biology, sociology, computer science... In particular, from the seminal works of Thomas and Kau ff man (60’s), Boolean networks are extensively used to model gene networks. In this context : ◃◃ - G ( f ) is “known” but f is “unknown” ◃◃ - fixed points of f ≃ cell types What can be said on fixed points of f according to G ( f ) ?

Contents ◃ 1. Introduction ◃ 2. Robert’s fixed point theorem (1980) ◃ 3. Shih-Dong’s fixed point theorem (2005) ◃ 4. Forbidden subnetworks theorem

THEOREM (Robert 1980) If G ( f ) has no cycle, then f has a unique fixed point. More precisely, if G ( f ) has no cycle, then f has a unique fixed point ξ , and the system converges toward ξ (for all update schedules).

THEOREM (Robert 1980) If G ( f ) has no cycle, then f has a unique fixed point. More precisely, if G ( f ) has no cycle, then f has a unique fixed point ξ , and the system converges toward ξ (for all update schedules). Layer 1 f i = cst = 0 or 1 Layer 2 Layer 3

THEOREM (Robert 1980) If G ( f ) has no cycle, then f has a unique fixed point. More precisely, if G ( f ) has no cycle, then f has a unique fixed point ξ , and the system converges toward ξ (for all update schedules). Layer 1 f i = cst = 0 or 1 0 1 0 Layer 2 Layer 3

THEOREM (Robert 1980) If G ( f ) has no cycle, then f has a unique fixed point. More precisely, if G ( f ) has no cycle, then f has a unique fixed point ξ , and the system converges toward ξ (for all update schedules). Layer 1 0 1 0 Layer 2 Only depends on Layer 1 Layer 3

THEOREM (Robert 1980) If G ( f ) has no cycle, then f has a unique fixed point. More precisely, if G ( f ) has no cycle, then f has a unique fixed point ξ , and the system converges toward ξ (for all update schedules). Layer 1 0 1 0 Layer 2 Only depends on Layer 1 0 1 Layer 3

THEOREM (Robert 1980) If G ( f ) has no cycle, then f has a unique fixed point. More precisely, if G ( f ) has no cycle, then f has a unique fixed point ξ , and the system converges toward ξ (for all update schedules). Layer 1 0 1 0 Layer 2 0 1 Layer 3 Only depends on Layer 2

THEOREM (Robert 1980) If G ( f ) has no cycle, then f has a unique fixed point. More precisely, if G ( f ) has no cycle, then f has a unique fixed point ξ , and the system converges toward ξ (for all update schedules). Layer 1 0 1 0 Layer 2 0 1 Layer 3 Only depends on Layer 2 1 0 0

Contents ◃ 1. Introduction ◃ 2. Robert’s fixed point theorem (1980) ◃ 3. Shih-Dong’s fixed point theorem (2005) ◃ 4. Forbidden subnetworks theorem

x i = ( x 1 , . . . , x i , . . . , x n ) Notation : The local interaction graph of f : B n → B n evaluated at state x ∈ B n is the directed graph Gf ( x ) with vertex set { 1 , . . . , n } and such that j → i ∈ Gf ( x ) ⇔ f i ( x ) ̸ = f i ( x j ) ⇓ f i depends on x j ⇕ j → i ∈ G ( f ) Property : ∀ x ∈ B n , Gf ( x ) is a subgraph of G ( f ) . More precisely � Gf ( x ) = G ( f ) x ∈ B n

x i = ( x 1 , . . . , x i , . . . , x n ) Notation : The local interaction graph of f : B n → B n evaluated at state x ∈ B n is the directed graph Gf ( x ) with vertex set { 1 , . . . , n } and such that j → i ∈ Gf ( x ) ⇔ f i ( x ) ̸ = f i ( x j ) ⇓ f i depends on x j ⇕ j → i ∈ G ( f ) Property : ∀ x ∈ B n , Gf ( x ) is a subgraph of G ( f ) . More precisely � Gf ( x ) = G ( f ) x ∈ B n

x i = ( x 1 , . . . , x i , . . . , x n ) Notation : The local interaction graph of f : B n → B n evaluated at state x ∈ B n is the directed graph Gf ( x ) with vertex set { 1 , . . . , n } and such that j → i ∈ Gf ( x ) ⇔ f i ( x ) ̸ = f i ( x j ) ⇓ f i depends on x j ⇕ j → i ∈ G ( f ) Property : ∀ x ∈ B n , Gf ( x ) is a subgraph of G ( f ) . More precisely � Gf ( x ) = G ( f ) x ∈ B n

THEOREM (Shih & Dong 2005) If Gf ( x ) has no cycle ∀ x ∈ B n , then f has a unique fixed point. The proof is more technical. It’s an induction on n that uses the notion of subnetwork (introduced in few slides). Shih-Dong’s theorem generalizes Robert’s one : G ( f ) has no cycle ⇓ ̸⇑ Gf ( x ) has no cycle ∀ x ∈ B n ⇓ f has a unique fixed point

Example : f : B 4 → B 4 is defined by : G ( f ) f 1 ( x ) = x 2 ∧ ( x 3 ∨ x 4) 1 2 f 2 ( x ) = x 3 ∧ x 4 f 3 ( x ) = x 1 ∧ x 2 ∧ x 4 f 4 ( x ) = x 1 ∧ x 2 ∧ x 3 3 3 G ( f ) has 14 cycles, but Gf ( x ) has no cycle ∀ x ∈ B 4 , and f has indeed a unique fixed point : 1011 1001 0110 1110 1000 0111 0100 1101 0011 0010 0101 0000 1111 0001 1010 1100 The condition “ Gf ( x ) has no cycle ∀ x ∈ B n ” doesn’t imply the convergence toward the unique fixed point.

Contents ◃ 1. Introduction ◃ 2. Robert’s fixed point theorem (1980) ◃ 3. Shih-Dong’s fixed point theorem (2005) ◃ 4. Forbidden subnetworks theorem

A subnetwork of f : B n → B n is a network ˜ f : B k → B k obtained from f by fixing n − k components to zero or one, with 1 ≤ k ≤ n . Remark : f is a subnetwork of f Example : f : B 3 → B 3 is defined by f 1 ( x 1 , x 2 , x 3 ) = x 1 ∨ x 3 f 2 ( x 1 , x 2 , x 3 ) = x 1 ∧ x 3 f 3 ( x 1 , x 2 , x 3 ) = x 2 f : B 2 → B 2 obtained by fixing “ x 3 = 1 ” is The subnetwork ˜ ˜ f 1 ( x 1 , x 2 ) = x 1 ∨ 1 = x 1 ˜ f 1 ( x 1 , x 2 ) = x 1 ∧ 1 = x 1

A subnetwork of f : B n → B n is a network ˜ f : B k → B k obtained from f by fixing n − k components to zero or one, with 1 ≤ k ≤ n . Remark : f is a subnetwork of f Example : f : B 3 → B 3 is defined by f 1 ( x 1 , x 2 , x 3 ) = x 1 ∨ x 3 f 2 ( x 1 , x 2 , x 3 ) = x 1 ∧ x 3 f 3 ( x 1 , x 2 , x 3 ) = x 2 f : B 2 → B 2 obtained by fixing “ x 3 = 1 ” is The subnetwork ˜ ˜ f 1 ( x 1 , x 2 ) = x 1 ∨ 1 = x 1 ˜ f 1 ( x 1 , x 2 ) = x 1 ∧ 1 = x 1

Let ˜ f be a subnetwork of f of dimension k ≤ n . There exists an injection h : B k → B n such that ∀ x ∈ B k G ˜ f ( x ) ⊆ Gf ( h ( x )) As a consequence G ( ˜ f ) ⊆ G ( f ) . PROPERTY OF SUBNETWORKS If there exists λ points x ∈ B k such that G ˜ f ( x ) has a cycle, then there exists λ points x ∈ B n such that Gf ( x ) has a cycle of length ≤ k .

Let ˜ f be a subnetwork of f of dimension k ≤ n . There exists an injection h : B k → B n such that ∀ x ∈ B k G ˜ f ( x ) ⊆ Gf ( h ( x )) As a consequence G ( ˜ f ) ⊆ G ( f ) . PROPERTY OF SUBNETWORKS If there exists λ points x ∈ B k such that G ˜ f ( x ) has a cycle, then there exists λ points x ∈ B n such that Gf ( x ) has a cycle of length ≤ k .

Let ˜ f be a subnetwork of f of dimension k ≤ n . There exists an injection h : B k → B n such that ∀ x ∈ B k G ˜ f ( x ) ⊆ Gf ( h ( x )) As a consequence G ( ˜ f ) ⊆ G ( f ) . PROPERTY OF SUBNETWORKS If there exists λ points x ∈ B k such that G ˜ f ( x ) has a cycle, then there exists λ points x ∈ B n such that Gf ( x ) has a cycle of length ≤ k .

Let C be the set of all circular networks , that is, the set of networks f such that G ( f ) is a cycle. PROPERTY OF CIRCULAR NETWORKS If f : B n → B n is a circular network, then it has 0 or 2 fixed points, and Gf ( x ) = G ( f ) is a cycle for all x ∈ B n . According to Robert’s theorem, circular networks are the most simple networks without a unique fixed point. QUESTION : If f has no subnetwork in C , then f has a unique fixed point ?

Let C be the set of all circular networks , that is, the set of networks f such that G ( f ) is a cycle. PROPERTY OF CIRCULAR NETWORKS If f : B n → B n is a circular network, then it has 0 or 2 fixed points, and Gf ( x ) = G ( f ) is a cycle for all x ∈ B n . According to Robert’s theorem, circular networks are the most simple networks without a unique fixed point. QUESTION If f has no subnetwork in C , then f has a unique fixed point ?