Attractor identification and quantification in asynchronous discrete - PowerPoint PPT Presentation

Attractor identification and quantification in asynchronous discrete dynamics Nuno D. Mendes 2 , 3 Pedro T. Monteiro 1 , 3 Jorge Carneiro 1 Elisabeth Remy 4 Claudine Chaouiya 1 1 Instituto Gulbenkian de Ciˆ encia, Oeiras, PT 2 Instituto de Biologia Experimental T´ ecnologica, Oeiras, PT 3 INESC-ID, Lisboa, PT 4 Institut de Math´ ematiques de Luminy, Marseille, FR 3rd CoLoMoTo meeting, Lausanne, Switzerland April 18, 2014

Outline 1 Introduction 2 Methods 3 Results 4 Conclusions and Prospects 1

Background Discrete modelling: logical formalism (Thomas and d’Ari, Biological Feedback 1989) 2

Background Discrete modelling: logical formalism (Thomas and d’Ari, Biological Feedback 1989) Logical regulatory graph (LRG) R = ( G , K ) G = { g i } i =0 ,..., n is a set of regulatory components Max : G → N ∗ associates a maximum level M i to each component g i S = � g i ∈G D i : is the state space, where D i = { 0 , . . . , Max ( g i ) } ∀ g i : K i : S → D i is the regulatory function specifying the behaviour of g i 2

Background Discrete modelling: logical formalism (Thomas and d’Ari, Biological Feedback 1989) Logical regulatory graph (LRG) R = ( G , K ) G = { g i } i =0 ,..., n is a set of regulatory components Max : G → N ∗ associates a maximum level M i to each component g i S = � g i ∈G D i : is the state space, where D i = { 0 , . . . , Max ( g i ) } ∀ g i : K i : S → D i is the regulatory function specifying the behaviour of g i State transition graph (STG) The dynamic behaviour of an LRG, is represented by an STG where: nodes are states in S and arcs ( v , w ) ∈ S 2 denote transitions between states 2

Background: Toy example (Boolean) K 0 ( v ) = 1 if v 0 = 1 ∨ v 1 = 0 ∨ v 2 = 1 K 1 ( v ) = 1 if v 0 = 0 ∨ v 2 = 0 K 2 ( v ) = 1 if v 0 = 1 ∧ v 1 = 1 3

Background: Toy example (Boolean) K 0 ( v ) = 1 if v 0 = 1 ∨ v 1 = 0 ∨ v 2 = 1 K 1 ( v ) = 1 if v 0 = 0 ∨ v 2 = 0 K 2 ( v ) = 1 if v 0 = 1 ∧ v 1 = 1 g 2 g 0 g 1 3

Background: Toy example (Boolean) K 0 ( v ) = 1 if v 0 = 1 ∨ v 1 = 0 ∨ v 2 = 1 K 1 ( v ) = 1 if v 0 = 0 ∨ v 2 = 0 K 2 ( v ) = 1 if v 0 = 1 ∧ v 1 = 1 011 111 g 2 010 110 = ⇒ g 0 g 1 001 101 000 100 3

Problem Attractors Correspond to asymptotic behaviours where: all gene levels are maintained Stable state long-lasting oscillating behaviour Complex attractor 011 111 010 110 001 101 000 100 4

Problem Attractors Correspond to asymptotic behaviours where: all gene levels are maintained Stable state long-lasting oscillating behaviour Complex attractor Trajectories quantification The weighted number of trajectories towards an attractor represents the structural biases of the STG Hidden assumption: successor states are equiprobable This assumption can easily be modified introducing weights 011 111 010 110 001 101 000 100 4

Problem Attractors Correspond to asymptotic behaviours where: all gene levels are maintained Stable state long-lasting oscillating behaviour Complex attractor Trajectories quantification The weighted number of trajectories towards an attractor represents the structural biases of the STG Hidden assumption: successor states are equiprobable This assumption can easily be modified introducing weights 011 111 010 110 Central question What is the likelihood of reaching an attractor from a 001 101 given portion of the state space? 000 100 4

Problem Objective Given a (set of) initial condition(s) and, optionally, a (set of) attractor(s), quantify the trajectories towards the attractor(s) Identify/characterize unknown attractor(s) 5

Problem Objective Given a (set of) initial condition(s) and, optionally, a (set of) attractor(s), quantify the trajectories towards the attractor(s) Identify/characterize unknown attractor(s) Size of the State Transition Graphs # States # Components Boolean 3-valued 3 8 27 10 1 024 59 049 20 1 048 576 3 486 784 401 30 1 073 741 824 205 891 132 094 649 40 1 099 511 627 776 12 157 665 459 056 928 801 5

Problem Objective Given a (set of) initial condition(s) and, optionally, a (set of) attractor(s), quantify the trajectories towards the attractor(s) Identify/characterize unknown attractor(s) Size of the State Transition Graphs # States # Components Boolean 3-valued 3 8 27 10 1 024 59 049 20 1 048 576 3 486 784 401 30 1 073 741 824 205 891 132 094 649 40 1 099 511 627 776 12 157 665 459 056 928 801 Challenge Combinatorial explosion! 5

Outline 1 Introduction 2 Methods 3 Results 4 Conclusions and Prospects 6

Attractor characterization approaches Without STG exploration Using OMDDs (Naldi et al. , CMSB 2007) Using SAT (de Jong and Page, IEEE/ACM Trans. Comp. Biol. Bioinf. 2008) Using reduction techniques and network motifs (Za˜ nudo and Albert, PLoS One 2013) With full (reachable) STG exploration Using ROBDDs (Garg et al. , RECOMB 2007) Using HTG (B´ erengier et al. , Chaos 2013) 7

Attractor characterization approaches Without STG exploration Using OMDDs (Naldi et al. , CMSB 2007) Using SAT (de Jong and Page, IEEE/ACM Trans. Comp. Biol. Bioinf. 2008) Using reduction techniques and network motifs (Za˜ nudo and Albert, PLoS One 2013) With full (reachable) STG exploration Using ROBDDs (Garg et al. , RECOMB 2007) Using HTG (B´ erengier et al. , Chaos 2013) FireFront (Mendes, Monteiro et al. , ECCB 2014 submitted) 7

Attractor characterization approaches Without STG exploration Using OMDDs (Naldi et al. , CMSB 2007) Using SAT (de Jong and Page, IEEE/ACM Trans. Comp. Biol. Bioinf. 2008) Using reduction techniques and network motifs (Za˜ nudo and Albert, PLoS One 2013) With full (reachable) STG exploration Using ROBDDs (Garg et al. , RECOMB 2007) Using HTG (B´ erengier et al. , Chaos 2013) FireFront (Mendes, Monteiro et al. , ECCB 2014 submitted) Monte Carlo simulations Boolnet (M¨ ussel et al. , Bioinformatics 2010) 7

Attractor characterization approaches Without STG exploration Using OMDDs (Naldi et al. , CMSB 2007) Using SAT (de Jong and Page, IEEE/ACM Trans. Comp. Biol. Bioinf. 2008) Using reduction techniques and network motifs (Za˜ nudo and Albert, PLoS One 2013) With full (reachable) STG exploration Using ROBDDs (Garg et al. , RECOMB 2007) Using HTG (B´ erengier et al. , Chaos 2013) FireFront (Mendes, Monteiro et al. , ECCB 2014 submitted) Monte Carlo simulations Boolnet (M¨ ussel et al. , Bioinformatics 2010) Avatar (Mendes, Monteiro et al. , ECCB 2014 submitted) 7

Attractor characterization approaches Without STG exploration Using OMDDs (Naldi et al. , CMSB 2007) Using SAT (de Jong and Page, IEEE/ACM Trans. Comp. Biol. Bioinf. 2008) Using reduction techniques and network motifs (Za˜ nudo and Albert, PLoS One 2013) With full (reachable) STG exploration Using ROBDDs (Garg et al. , RECOMB 2007) Using HTG (B´ erengier et al. , Chaos 2013) FireFront (Mendes, Monteiro et al. , ECCB 2014 submitted) Monte Carlo simulations Boolnet (M¨ ussel et al. , Bioinformatics 2010) Avatar (Mendes, Monteiro et al. , ECCB 2014 submitted) Trajectory characterization approach: MaBoSS (Stoll et al. , BMC Syst Biol 2012) 7

Approach: Quasi-exact ( FireFront algorithm) Intuition Explore the STG from an initial condition Divide and carry probability to successor states Accumulate probability in states with no successors – stable states Do not explore states with probability below α The algorithm maintains 3 state sets: F – the current firefront N – the set of neglected states A – the set of attractors 8

Approach: Quasi-exact ( FireFront algorithm) 1 α = max iterations = 10 16 Start exploration from given initial condition v 1 , with unitary probability v 1 1 Iteration = 1 F = { v 1 } v 5 v 4 v 2 v 7 N = ∅ A = ∅ v 6 v 3 v 8 9

Approach: Quasi-exact ( FireFront algorithm) 1 α = max iterations = 10 16 Carry probability to successors dividing it by the number of successors – current firefront v 1 Iteration = 2 F = { v 2 , v 5 } 1 v 5 v 4 1 v 2 v 7 N = ∅ 2 2 A = ∅ v 6 v 3 v 8 9

Approach: Quasi-exact ( FireFront algorithm) 1 α = max iterations = 10 16 States with no successors are attractors and accumulate probability v 1 Iteration = 3 F = { v 3 , v 4 , v 6 } v 5 v 4 1 v 2 v 7 1 4 4 N = ∅ A = { v 7 } v 6 v 3 v 8 1 4 1 4 9

Approach: Quasi-exact ( FireFront algorithm) 1 α = max iterations = 10 16 States with no successors are attractors and accumulate probability v 1 1 8 Iteration = 4 F = { v 1 , v 3 , v 4 , v 6 } v 5 v 4 1 v 2 v 7 1 8 4 N = ∅ A = { v 7 , v 8 } v 6 v 3 v 8 1 1 8 8 1 4 9

Approach: Quasi-exact ( FireFront algorithm) 1 α = max iterations = 10 16 States with no successors are attractors and accumulate probability v 1 1 16 Iteration = 5 F = { v 1 , v 2 , v 3 , v 4 , v 5 , v 6 } 1 v 5 v 4 1 1 v 2 v 7 1 16 8 16 4 N = ∅ A = { v 7 , v 8 } v 6 v 3 v 8 1 1 16 4 1 8 9

Approach: Quasi-exact ( FireFront algorithm) 1 α = max iterations = 10 16 States accumulate probability given by multiple predecessor states States with probability below α are moved to a special set – neglected states – and are no longer explored v 1 1 16 Iteration = 6 F = { v 1 , v 2 , v 3 , v 4 , v 6 } 1 v 5 v 4 3 1 v 2 v 7 9 32 32 16 32 N = { v 5 } A = { v 7 , v 8 } v 6 v 3 v 8 3 5 32 16 1 16 9