Attractor identification and quantification in asynchronous discrete dynamics Elisabeth Remy (Institut de Math´ ematique de Marseille) Workshop Th´ eorie des r´ eseaux bool´ eens et ses applications en biologie November 4, 2014
Outline 1 Introduction 2 Logical model of bladder cancer progression 3 Analysis of the model 4 Towards quantifications... 5 Back to the bladder cancer model 6 Conclusions
Outline 1 Introduction 2 Logical model of bladder cancer progression 3 Analysis of the model 4 Towards quantifications... 5 Back to the bladder cancer model 6 Conclusions 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 asynchronous 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 = 1 011 111 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 Questions identification of the attractors reachability 011 111 010 110 001 101 000 100 4
Problem 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 Logical model of bladder cancer progression 3 Analysis of the model 4 Towards quantifications... 5 Back to the bladder cancer model 6 Conclusions 6
Bladder cancer pathway Bladder ¡cancer ¡pathway ¡ Ta T1 Urothelium � T2-T4 normal � Carcinome in situ metastases Two types of tumours Ta : low grade – recur – low probability invasivness Cis : high grade – muscle invasive Coll. L. Calzone, F. radvanyi (Curie); C. Chaouiya (IGC) 7
Alterations in bladder tumours Reported alterations in bladder tumors Mainly in Growth factor signaling cascades and Control of cell cycle entry (G1/S) 8
Alterations in bladder tumours Reported alterations in bladder tumors Mainly in Growth factor signaling cascades and Control of cell cycle entry (G1/S) How gene alterations are combined in order to promote cancer progression? → explore patterns of genetics alterations (co-occurence or mutual ֒ exclusivity) 8
Alterations in bladder tumours Reported alterations Patients data (copy nb and mutations) Mainly in Growth factor signaling cascades and Control Tumour samples: 163 of cell cycle entry (G1/S) Invasive samples: 89 Non-invasive samples: 74 Tumor stage CIS signature TP53 mutation MDM2 GNL status (BAC CGH) FGFR3 mutation PIK3CA mutation RAS mutation CCND1 GNL status (BAC CGH) CDKN2A GNL status (MLPA) RB1 GNL status (MLPA) E2F3 GNL status (BAC CGH) RBL2 GNL status (BAC CGH) Non-muscle-invasive tumors (Ta/T1) No alteration Hemizygous deletion Muscle-invasive tumors (T2-4) Mutated Homozygous deletion CIS+ Amplification Missing data CIS- Gain 9
Alterations in bladder tumours Found ¡in ¡ ¡ ¡ CIT ¡ CIT ¡sup ¡ CIT ¡inv ¡ literature? ¡ ¡ Patients data (copy nb and mutations) co-‑occurrence ¡ 1.1 ¡ FGFR3 ¡mut ¡-‑ ¡ yes ¡ 0.012 ¡ 0.063 ¡ 1 ¡ PIK3CA ¡mut* ¡ co-‑occurrence ¡ Tumour samples: 163 1 ¡ FGFR3 ¡mut ¡-‑ ¡ yes ¡(in ¡ 1.2 ¡ CDKN2A ¡ invasive ¡ 0.017 ¡ 0.4 ¡ 0.0067 ¡ Invasive samples: 89 homozygous ¡ tumours) ¡ Non-invasive samples: 74 del ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ exclusivity ¡ 2.1 ¡ FGFR3 ¡mut ¡-‑ ¡ yes ¡ 0.01 ¡ 0.001 ¡ 1 ¡ RAS ¡mut ¡ exclusivity ¡ 2.2 ¡ FGFR3 ¡mut ¡-‑ ¡ no ¡ 0.059 ¡ 0.42 ¡ 1 ¡ 2 ¡ E2F3 ¡ampl ¡ Tumor stage exclusivity ¡ CIS signature TP53 mutation 2.3 ¡ FGFR3 ¡mut ¡-‑ ¡ no ¡ 0.031 ¡ 0.004 ¡ 1 ¡ MDM2 GNL status (BAC CGH) FGFR3 mutation CCND1 ¡ampl ¡ PIK3CA mutation RAS mutation CCND1 GNL status (BAC CGH) CDKN2A GNL status (MLPA) ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ RB1 GNL status (MLPA) E2F3 GNL status (BAC CGH) RBL2 GNL status (BAC CGH) exclusivity ¡ Non-muscle-invasive tumors (Ta/T1) No alteration Hemizygous deletion yes ¡and ¡ 3 ¡ ¡ ¡ FGFR3 ¡mut ¡-‑ ¡ 0.0014 ¡ 0.304 ¡ 1 ¡ Muscle-invasive tumors (T2-4) Mutated Homozygous deletion no! ¡ CIS+ Amplification Missing data TP53 ¡mut ¡ CIS- Gain ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ co-‑occurrence ¡ TP53 ¡mut ¡and ¡ 4 ¡ ¡ ¡ no ¡ 5.03E-‑05 ¡ 0.135 ¡ 0.006 ¡ amplificaJon ¡de ¡ E2F3 ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ co-‑occurrence ¡ E2F3 ¡ampl ¡-‑ ¡RB1 ¡ 5 ¡ ¡ ¡ yes ¡ 1 ¡ 1 ¡ NA ¡ homozygous ¡ del ¡ 10
Associations of alterations Co-occurence/Mutual exclusivity of genetic alterations Can we understand mechanistically these alterations and their associations? Topological analysis co-occurring gene alterations − → belong to parallel pathway mutually exclusive gene alterations − → from redundant pathways Use of the mathematical modeling to provide insight on possible mechanisms by which cells become invasive 11
A mathematical model 23 internal components, 4 inputs, 3 outputs 12
A mathematical model Node Value Logical function DNAdamage 0/1 Constant (input) GrowthInhibitors 0/1 Constant (input) EGFR_stimulus 0/1 Constant (input) FGFR3_stimulus 0/1 Constant (input) EGFR 1 (EGFR_stimulus | SPRY) & !FGFR3 & !GRB2 FGFR3 1 FGFR3_stimulus &!EGFR &!GRB2 GRB2 1 (FGFR3 & !GRB2 & !SPRY) | EGFR SPRY 1 RAS RAS 1 EGFR | FGFR3 | GRB2 PI3K 1 GRB2 & RAS & !PTEN AKT 1 PI3K PTEN 1 TP53 CyclinD1 1 (RAS | AKT) & !p16INK4a & !p21CIP p16INK4a 1 GrowthInhibitors & !RB1 p14ARF 1 E2F1 RB1 1 !CyclinD1 & !CyclinE1 & !p16INK4a & !CyclinA RBL2 1 !CyclinD1 & !CyclinE1 p21CIP 1 (GrowthInhibitors | TP53) & !CyclinE1 & !AKT CDC25A 1 (E2F1 | E2F3) & !CHEK1_2 & !RBL2:1 CyclinE1 1 CDC25A & (E2F1 | E2F3) & !RBL2 & !p21CIP CyclinA 1 (E2F1 | E2F3) & CDC25A & !p21CIP & !RBL2 ( (!(CHEK1_2:2 & ATM:2) & (RAS | E2F3:1 | E2F3:2)) | (CHEK1_2:2 & ATM:2 & !RAS & 1 E2F1 E2F3:1)) & !RB1 & !RBL2 (RAS | E2F3:2) & CHEK1_2:2 & ATM:2 & !RB1 & 2 !RBL2 1 RAS&!RB1 & !CHEK1_2:2 E2F3 2 RAS & !RB1 & CHEK1_2:2 ATM 1 DNAdamage & !E2F1 2 DNAdamage & E2F1 CHEK1_2 1 ATM & !E2F1 2 ATM & E2F1 MDM2 1 (TP53 | AKT) & !p14ARF & !ATM TP53 1 ((ATM & CHEK1_2) | E2F1:2) & !MDM2 1 !E2F1 & TP53 Apoptosis 2 E2F1:1|E2F1:2 Proliferation 1 CyclinE1 | CyclinA GrowthArrest 1 p21CIP | RB1 | RBL2 ! 13
Questions Focus on these two associations: • Co-occurence of PIK3CA and FGFR3 mutations • Mutually exclusive mutations of TP53 and FGFR3 14
Outline 1 Introduction 2 Logical model of bladder cancer progression 3 Analysis of the model 4 Towards quantifications... 5 Back to the bladder cancer model 6 Conclusions 15
1) Identification of the stable states x stable state ⇐ ⇒ K ( x ) = x → Efficient algorithm to find all the possible stable states (with no simulation) ֒ Principle: to build a stability condition for each logical parameter K i to combine all these partial conditions 16
Bladder Cancer model: Looking for attractors Identification of 20 stable states The 20 stable states 17
Bladder Cancer model: Looking for attractors → other attractors (cyclical)? ֒ → reachability of the stable states for some initial conditions? ֒ 18
Bladder Cancer model: Looking for attractors → other attractors (cyclical)? ֒ → reachability of the stable states for some initial conditions? ֒ Pb: Full dynamics (STG) not computable How to compact the dynamics? 18
How to compact the dynamics? 19
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