Inverse problems on the blocksequential operator in Boolean networks - PowerPoint PPT Presentation

Inverse problems on the blocksequential operator in Boolean networks Julio Aracena † , Luis Cabrera-Crot ∗ , Adrien Richard ◦ and Lilian Salinas + on, Chile. 1 ∗ PhD Student in Computer Science, U. of Concepci´ † Department of Mathematical Engineering, U. of Concepci´ on, Chile. + Department of Computer Science, U. of Concepci´ on, Chile. ote d´ ◦ CRNS and Universit´ e Cˆ Azur, France International Workshop on Boolean Networks January 9 th , 2020 1Funded by CONICYT-PFCHA/Doctorado Nacional/2016-21160885

Motivation Contents Motivation 1 Algorithm 2 Work in progress 3 Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 2 / 31

Motivation Motivation Complex Systems (L. Mendoza and E. Alvarez, 1998) Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 3 / 31

Motivation Motivation Complex Systems Boolean networks f : { 0 , 1 } n → { 0 , 1 } n (L. Mendoza and E. Alvarez, 1998) f 1 ( x ) = x 4 f 2 ( x ) = x 1 ∧ x 2 f 3 ( x ) = x 2 ∨ x 3 f 4 ( x ) = x 3 ∧ x 4 Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 3 / 31

Motivation Motivation Complex Systems Boolean networks f : { 0 , 1 } n → { 0 , 1 } n Interaction Graph (L. Mendoza and E. Alvarez, 1998) f 1 ( x ) = x 4 2 1 f 2 ( x ) = x 1 ∧ x 2 f 3 ( x ) = x 2 ∨ x 3 f 4 ( x ) = x 3 ∧ x 4 3 4 Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 3 / 31

Motivation Block-sequential schedule Definition A block-sequential schedule is an ordered partition of the components of a Boolean network which defines the order in which the states of the network are updated in one unit of time. Examples s 1 = { 3 , 4 }{ 1 }{ 2 } , s 2 = { 1 , 2 , 3 , 4 } , s 3 = { 2 }{ 3 }{ 4 }{ 1 } . Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 4 / 31

Motivation Labeled digraph Given a interaction graph G and a block-sequential schedule s , a labeled digraph ( G , s ) is a digraph with a labeling function lab s : lab s : A ( G ) → {⊕ , ⊖} lab s ( u , v ) = ⊕ ⇐ ⇒ s ( u ) ≥ s ( v ) G s 1 = { 1 , 2 , 3 , 4 } s 2 = { 2 } { 3 } { 4 } { 1 } s 3 = { 3 , 4 } { 1 } { 2 } s 4 = { 4 } { 1 } { 3 , 2 } ⊕ ⊕ ⊕ 2 1 2 1 2 1 2 1 ⊕ ⊕ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ 3 ⊕ 4 3 ⊖ 4 3 ⊕ 4 3 4 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 5 / 31

Motivation Parallel digraph The parallel digraph is a digraph that represent the real dependence of the components of the Boolean network, according with the block-sequential schedule. Also, is equivalent to the interaction graph of a Boolean network with equal dynamic behavior when is updated in parallel (one only block). s = { 2 } { 3 } { 4 } { 1 } P ( G , s ) G 2 1 2 1 3 4 3 4 Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 6 / 31

Motivation Parallel digraph The parallel digraph is a digraph that represent the real dependence of the components of the Boolean network, according with the block-sequential schedule. Also, is equivalent to the interaction graph of a Boolean network with equal dynamic behavior when is updated in parallel (one only block). s = { 2 } { 3 } { 4 } { 1 } ( G , s ) P ( G , s ) ⊕ 2 1 2 1 ⊕ ⊖ ⊖ ⊖ 3 4 3 4 ⊕ ⊕ Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 6 / 31

Motivation Parallel digraph The parallel digraph is a digraph that represent the real dependence of the components of the Boolean network, according with the block-sequential schedule. Also, is equivalent to the interaction graph of a Boolean network with equal dynamic behavior when is updated in parallel (one only block). s = { 2 } { 3 } { 4 } { 1 } ( G , s ) P ( G , s ) ⊕ 2 1 2 1 ⊕ ⊖ ⊖ ⊖ 3 4 3 4 ⊕ ⊕ Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 6 / 31

Motivation Parallel digraph The parallel digraph is a digraph that represent the real dependence of the components of the Boolean network, according with the block-sequential schedule. Also, is equivalent to the interaction graph of a Boolean network with equal dynamic behavior when is updated in parallel (one only block). s = { 2 } { 3 } { 4 } { 1 } ( G , s ) P ( G , s ) ⊕ 2 1 2 1 ⊕ ⊖ ⊖ ⊖ 3 4 3 4 ⊕ ⊕ Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 6 / 31

Motivation Parallel digraph The parallel digraph is a digraph that represent the real dependence of the components of the Boolean network, according with the block-sequential schedule. Also, is equivalent to the interaction graph of a Boolean network with equal dynamic behavior when is updated in parallel (one only block). s = { 2 } { 3 } { 4 } { 1 } ( G , s ) P ( G , s ) ⊕ 2 1 2 1 ⊕ ⊖ ⊖ ⊖ 3 4 3 4 ⊕ ⊕ Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 6 / 31

Motivation Parallel digraph The parallel digraph is a digraph that represent the real dependence of the components of the Boolean network, according with the block-sequential schedule. Also, is equivalent to the interaction graph of a Boolean network with equal dynamic behavior when is updated in parallel (one only block). s = { 2 } { 3 } { 4 } { 1 } ( G , s ) P ( G , s ) ⊕ 2 1 2 1 ⊕ ⊖ ⊖ ⊖ 3 4 3 4 ⊕ ⊕ Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 6 / 31

Motivation Parallel digraph Can be obtained from the labeled digraph. ∀ ( u , v ) ∈ V ( G ) × V ( G ) , ( u , v ) ∈ A ( P ( G , s )) if and only if: s = { 2 } { 3 } { 4 } { 1 } ( G , s ) P ( G , s ) ⊕ 2 1 2 1 ⊕ ⊖ ⊖ 3 ⊖ 4 3 4 ⊕ ⊕ Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 7 / 31

Motivation Parallel digraph Can be obtained from the labeled digraph. ∀ ( u , v ) ∈ V ( G ) × V ( G ) , ( u , v ) ∈ A ( P ( G , s )) if and only if: ( u , v ) is labeled ⊕ . s = { 2 } { 3 } { 4 } { 1 } ( G , s ) P ( G , s ) ⊕ 2 1 2 1 ⊕ ⊖ ⊖ 3 ⊖ 4 3 4 ⊕ ⊕ Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 7 / 31

Motivation Parallel digraph Can be obtained from the labeled digraph. ∀ ( u , v ) ∈ V ( G ) × V ( G ) , ( u , v ) ∈ A ( P ( G , s )) if and only if: ( u , v ) is labeled ⊕ . ∃ w ∈ V ( G ) , ( u , w ) is labeled ⊕ and exists a path from w to v labeled ⊖ . s = { 2 } { 3 } { 4 } { 1 } ( G , s ) P ( G , s ) ⊕ 2 1 2 1 ⊕ ⊖ ⊖ 3 ⊖ 4 3 4 ⊕ ⊕ Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 7 / 31

Motivation Parallel digraph The function that associates ( G , s ) with the digraph P ( G , s ) is called block-sequential operator and can be constructed in polynomial time. s = { 2 } { 3 } { 4 } { 1 } ( G , s ) P ( G , s ) ⊕ 2 1 2 1 ⊕ ⊖ ⊖ 3 ⊖ 4 3 4 ⊕ ⊕ Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 7 / 31

Motivation Inverse problem Given a digraph P and a block-sequential schedule s , does there exist a digraph G such that P ( G , s ) = P ? s = { 1 } { 2 } P 1 2 Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 8 / 31

Motivation Inverse problem Given a digraph P and a block-sequential schedule s , does there exist a digraph G such that P ( G , s ) = P ? s = { 1 } { 2 } ( G , s ) P 1 2 1 2 Unique solution Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 8 / 31

Motivation Inverse problem Given a digraph P and a block-sequential schedule s , does there exist a digraph G such that P ( G , s ) = P ? s = { 1 } { 2 } P 1 2 Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 8 / 31

Motivation Inverse problem Given a digraph P and a block-sequential schedule s , does there exist a digraph G such that P ( G , s ) = P ? s = { 1 } { 2 } ( G , s ) P 1 2 1 2 1 2 1 2 Multiple solutions Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 8 / 31

Motivation Inverse problem Given a digraph P and a block-sequential schedule s , does there exist a digraph G such that P ( G , s ) = P ? s = { 1 } { 2 } P 1 2 Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 8 / 31

Motivation Inverse problem Given a digraph P and a block-sequential schedule s , does there exist a digraph G such that P ( G , s ) = P ? s = { 1 } { 2 } P 1 2 No solution Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 8 / 31

Algorithm Contents Motivation 1 Algorithm 2 Work in progress 3 Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 9 / 31

Algorithm Results P 1 2 For example, for the schedule { 1 } { 2 } , there are three preimages: ( G ′ , s ) ( G ′′ , s ) ( G , s ) 1 2 1 2 1 2 Luis Cabrera-Crot et al. (U. Concepci´ on) Inverse block-sequential operator IWBN 2020 10 / 31