Limit cycles and update schedules in Boolean networks: Inverse - PowerPoint PPT Presentation

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Limit cycles and update schedules in Boolean networks: Inverse Problem. (Results of Luis G´ omez’s Ph.D. Thesis) Advisors: L. Salinas(UdeC), J. Demongeot (UG) and J. Aracena (UdeC). Novembre 2014 Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Summary 1 Definition and Notation 2 Limit Cycle Existence problem 3 Limit Cycle Non Existence problem 4 Feasible Limit Cycle problem Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Boolean Networks A Boolean network is N = ( F , s ), where F = ( f v ) v ∈ V :: { 0 , 1 } n → { 0 , 1 } n , global transition function, V a set of n elements. f v ( x ) := F ( x ) v , ∀ v ∈ V , local activation functions. s : V → { 1 , . . . , n } a deterministic update schedule (parallel,sequential, block-sequential). Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Interaction Digraph G F = ( V , A ) interaction digraph associated to a Boolean Network. ( u , v ) ∈ A if an only if f v depends on x u . Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Interaction Digraph G F = ( V , A ) interaction digraph associated to a Boolean Network. ( u , v ) ∈ A if an only if f v depends on x u . Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Example 1 F : { 0 , 1 } 4 → { 0 , 1 } 4 1 4 f 1 ( x ) := x 3 ∧ x 4 f 2 ( x ) := x 1 ∧ x 3 f 3 ( x ) := ( x 1 ∧ x 2 ) ∨ x 4 3 2 f 4 ( x ) := x 2 Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Example 1 F : { 0 , 1 } 4 → { 0 , 1 } 4 1 4 f 1 ( x ) := x 3 ∧ x 4 f 2 ( x ) := x 1 ∧ x 3 f 3 ( x ) := ( x 1 ∧ x 2 ) ∨ x 4 3 2 f 4 ( x ) := x 2 Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Example 1 F : { 0 , 1 } 4 → { 0 , 1 } 4 1 4 f 1 ( x ) := x 3 ∧ x 4 f 2 ( x ) := x 1 ∧ x 3 f 3 ( x ) := ( x 1 ∧ x 2 ) ∨ x 4 3 2 f 4 ( x ) := x 2 Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Example 1 F : { 0 , 1 } 4 → { 0 , 1 } 4 1 4 f 1 ( x ) := x 3 ∧ x 4 f 2 ( x ) := x 1 ∧ x 3 f 3 ( x ) := ( x 1 ∧ x 2 ) ∨ x 4 3 2 f 4 ( x ) := x 2 Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Example 1 F : { 0 , 1 } 4 → { 0 , 1 } 4 1 4 f 1 ( x ) := x 3 ∧ x 4 f 2 ( x ) := x 1 ∧ x 3 f 3 ( x ) := ( x 1 ∧ x 2 ) ∨ x 4 3 2 f 4 ( x ) := x 2 Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Update Schedule s : V → { 1 , . . . , n } , function. Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Update Schedule s : V → { 1 , . . . , n } , function. s (1) = 1 s (2) = 1 1 2 s ( V ) = { 1 } , parallel. 3 5 s (3) = 1 s (5) = 1 4 s (4) = 1 Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Update Schedule s : V → { 1 , . . . , n } , function. s (1) = 1 s (2) = 2 1 2 s ( V ) = { 1 } , parallel. s ( V ) = { 1 , . . . , n } , sequential. 3 5 s (3) = 3 s (5) = 5 4 s (4) = 4 Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Update Schedule s : V → { 1 , . . . , n } , function. s (1) = 1 s (2) = 1 1 2 s ( V ) = { 1 } , parallel. s ( V ) = { 1 , . . . , n } , sequential. 3 5 s (3) = 2 s (5) = 3 s ( V ) = { 1 , . . . , m } , 1 < m < n , 4 block-sequential. s (4) = 3 Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Iteration Given x = ( x v ) v ∈ V ∈ { 0 , 1 } n , the ( k + 1)-iteration of x by F according to s is given by: x k +1 = f v ( x l u u : u ∈ V ) v Where: � k if s ( v ) ≤ s ( u ) l u = k + 1 if s ( v ) > s ( u ) Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Iteration Given x = ( x v ) v ∈ V ∈ { 0 , 1 } n , the ( k + 1)-iteration of x by F according to s is given by: x k +1 = f v ( x l u u : u ∈ V ) v Where: � k if s ( v ) ≤ s ( u ) l u = k + 1 if s ( v ) > s ( u ) Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Dynamical Behavior We can define f s v ( x ) = f v ( g s v , u ( x ): u ∈ V ) Where: � if s ( v ) ≤ s ( u ) x u g s v , u ( x ) = f s u ( x ) if s ( v ) > s ( u ) F s is the dynamical behavior of N = ( F , s ). Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Dynamical Behavior We can define f s v ( x ) = f v ( g s v , u ( x ): u ∈ V ) Where: � if s ( v ) ≤ s ( u ) x u g s v , u ( x ) = f s u ( x ) if s ( v ) > s ( u ) F s is the dynamical behavior of N = ( F , s ). Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Dynamical Behavior We can define f s v ( x ) = f v ( g s v , u ( x ): u ∈ V ) Where: � if s ( v ) ≤ s ( u ) x u g s v , u ( x ) = f s u ( x ) if s ( v ) > s ( u ) F s is the dynamical behavior of N = ( F , s ). Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Limit Behavior Fixed point: x ∈ { 0 , 1 } n : F s ( x ) = x x k � p � k =0 , x k ∈ { 0 , 1 } n , p > 1: Limit Cycles: C = x k +1 = F s ( x k ) x p ≡ x 0 ∧ LC ( N ): set of limit cycles of N . Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Limit Behavior Fixed point: x ∈ { 0 , 1 } n : F s ( x ) = x x k � p � k =0 , x k ∈ { 0 , 1 } n , p > 1: Limit Cycles: C = x k +1 = F s ( x k ) x p ≡ x 0 ∧ LC ( N ): set of limit cycles of N . Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Limit Behavior Fixed point: x ∈ { 0 , 1 } n : F s ( x ) = x x k � p � k =0 , x k ∈ { 0 , 1 } n , p > 1: Limit Cycles: C = x k +1 = F s ( x k ) x p ≡ x 0 ∧ LC ( N ): set of limit cycles of N . Nice 2014 Limit cycles and update schedules in Boolean networks

Definition and Notation Limit Cycle Existence problem Limit Cycle Non Existence problem Feasible Limit Cycle problem Update Digraph Given a Boolean network ( F , s ), we define the associated labeled digraph G F s = ( G F , lab s ), called update digraph , where lab s : A ( G F ) → {⊖ , ⊕} is defined as: � ⊕ if s ( u ) ≥ s ( v ) lab s ( u , v ) = ⊖ if s ( u ) < s ( v ) ⊕ 2 1 ⊖ ⊖ ⊕ ⊕ 3 ⊖ 4 Example: s ( i ) = i , ∀ i ∈ { 1 , . . . , n } . It was proven in (Aracena, J., Goles, E., Moreira, A., Salinas, L., 2009. Biosystems 97, 1-8) that if two different updates schedules have the same update digraph, then they also have the same dynamical behavior. Nice 2014 Limit cycles and update schedules in Boolean networks