Attractor identification and quantification in asynchronous discrete - - PowerPoint PPT Presentation

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 = {gi}i=0,...,n is a set of regulatory components Max : G → N∗ associates a maximum level Mi to each component gi S =

gi ∈G Di: is the state space, where Di = {0, . . . , Max(gi)}

∀gi : Ki : S → Di is the regulatory function specifying the behaviour of gi

2

Background

Discrete modelling: logical formalism

(Thomas and d’Ari, Biological Feedback 1989)

Logical regulatory graph (LRG) R= (G, K)

G = {gi}i=0,...,n is a set of regulatory components Max : G → N∗ associates a maximum level Mi to each component gi S =

gi ∈G Di: is the state space, where Di = {0, . . . , Max(gi)}

∀gi : Ki : S → Di is the regulatory function specifying the behaviour of gi

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) ∈ S2 denote transitions between states

2

Background: Toy example (Boolean)

K0(v) = 1 if v0 = 1 ∨ v1 = 0 ∨ v2 = 1 K1(v) = 1 if v0 = 0 ∨ v2 = 0 K2(v) = 1 if v0 = 1 ∧ v1 = 1 g0 g1 g2 = ⇒

000 001 010 011 100 101 110 111

3

Problem

Attractors

Correspond to asymptotic behaviours where: all gene levels are maintained Stable state long-lasting oscillating behaviour Complex attractor

000 001 010 011 100 101 110 111

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

000 001 010 011 100 101 110 111

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

metastases T2-T4 Ta T1 Carcinome in situ Urothelium normal

Bladder ¡cancer ¡pathway ¡

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

Mainly in Growth factor signaling cascades and Control

• f cell cycle entry (G1/S)

Patients data (copy nb and mutations)

Tumour samples: 163 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 ¡ literature? ¡ ¡ CIT ¡ CIT ¡sup ¡ CIT ¡inv ¡ 1 ¡ 1.1 ¡ co-­‑occurrence ¡ FGFR3 ¡mut ¡-­‑ ¡ PIK3CA ¡mut* ¡ yes ¡ 0.012 ¡ 0.063 ¡ 1 ¡ 1.2 ¡ co-­‑occurrence ¡ FGFR3 ¡mut ¡-­‑ ¡ CDKN2A ¡ homozygous ¡ del ¡ yes ¡(in ¡ invasive ¡ tumours) ¡ 0.017 ¡ 0.4 ¡ 0.0067 ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ 2 ¡ 2.1 ¡ exclusivity ¡ FGFR3 ¡mut ¡-­‑ ¡ RAS ¡mut ¡ yes ¡ 0.01 ¡ 0.001 ¡ 1 ¡ 2.2 ¡ exclusivity ¡ FGFR3 ¡mut ¡-­‑ ¡ E2F3 ¡ampl ¡ no ¡ 0.059 ¡ 0.42 ¡ 1 ¡ 2.3 ¡ exclusivity ¡ FGFR3 ¡mut ¡-­‑ ¡ CCND1 ¡ampl ¡ no ¡ 0.031 ¡ 0.004 ¡ 1 ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ 3 ¡ ¡ ¡ exclusivity ¡ FGFR3 ¡mut ¡-­‑ ¡ TP53 ¡mut ¡ yes ¡and ¡ no! ¡ 0.0014 ¡ 0.304 ¡ 1 ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ 4 ¡ ¡ ¡ co-­‑occurrence ¡ TP53 ¡mut ¡and ¡ amplificaJon ¡de ¡ E2F3 ¡ no ¡ 5.03E-­‑05 ¡ 0.135 ¡ 0.006 ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ 5 ¡ ¡ ¡ co-­‑occurrence ¡ E2F3 ¡ampl ¡-­‑ ¡RB1 ¡ homozygous ¡ del ¡ yes ¡ 1 ¡ 1 ¡ NA ¡

Patients data (copy nb and mutations)

Tumour samples: 163 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

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 E2F1 1 ( (!(CHEK1_2:2 & ATM:2) & (RAS | E2F3:1 | E2F3:2)) | (CHEK1_2:2 & ATM:2 & !RAS & E2F3:1)) & !RB1 & !RBL2 2 (RAS | E2F3:2) & CHEK1_2:2 & ATM:2 & !RB1 & !RBL2 E2F3 1 RAS&!RB1 & !CHEK1_2:2 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 Apoptosis 1 !E2F1 & TP53 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 Ki 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

How to compact the dynamics?

STG SCC graph

20

Hierarchical Transition Graphs

SCC graph HTG σ(C) = {C ′ ∈ Scc, C ′ attractor or complex, s.t. C C ′}

σ gathers trivial transient SCCs from which the same attractors and complex SCCs are reachable

21

Hierarchical Transition Graphs

A hierarchical representation of network dynamics Algorithm on the fly to compact the state transition graph: directly generate compressed representation HTG emphasizes relevant transient and asymptotic dynamical behaviours Easily recovering of basins of attraction → localisation of crucial decisions

• D. Berenguier et al (2013) Chaos 23(2)

22

Bladder Cancer model: Looking for attractors

Hierarchical Transition Graph... not computable!

23

Bladder Cancer model: Looking for attractors

Hierarchical Transition Graph... not computable! ֒ → How to reduce the model?

23

Model reduction

• A. Naldi et al (2011) Theor Comp Sc 412(21):2207-18

Reduce a model by (iteratively) hiding components (non auto-regulated) G3 selected for reduction

G0 G1 G3 G2

G0 → 1 if G0 is present OR G3 is absent fG0(x) = (xG0 = 1)∨ (xG3 = 0) fG1(x) = (xG2 = 1) ∧ (xG0 = 0) fG2(x) = (xG0 = 1) ∧ (xG1 = 0) fG3(x) = (xG2 = 1)

24

Model reduction

• A. Naldi et al (2011) Theor Comp Sc 412(21):2207-18

Reduce a model by (iteratively) hiding components (non auto-regulated) G3 selected for reduction

G0 G1 G3 G2

G0 → 1 if G0 is present OR G3 is absent fG0(x) = (xG0 = 1)∨ (xG3 = 0) fG1(x) = (xG2 = 1) ∧ (xG0 = 0) fG2(x) = (xG0 = 1) ∧ (xG1 = 0) fG3(x) = (xG2 = 1) G0 function revised

G0 G1 G2

G0 → 1 if G0 is present OR G2 is absent (condition for which g3 absent) f

r

G0(x) = (xG1 = 1)∨ (xG2 = 0)

24

Model reduction

• A. Naldi et al (2011) Theor Comp Sc 412(21):2207-18

Reduce a model by (iteratively) hiding components (non auto-regulated) G3 selected for reduction

G0 G1 G3 G2

G0 → 1 if G0 is present OR G3 is absent fG0(x) = (xG0 = 1)∨ (xG3 = 0) fG1(x) = (xG2 = 1) ∧ (xG0 = 0) fG2(x) = (xG0 = 1) ∧ (xG1 = 0) fG3(x) = (xG2 = 1) G0 function revised

G0 G1 G2

G0 → 1 if G0 is present OR G2 is absent (condition for which g3 absent) f

r

G0(x) = (xG1 = 1)∨ (xG2 = 0)

Reduction amounts to consider that the component is faster

24

Impact of the reduction on dynamical properties

Exactly the same stable states Elementary cycles are conserved, new ones might appear Complex attractors might be split, new ones might appear Any new (non-trivial) attractor proceeds from a SCC of the original STG Emergence of new complex attractors indicate the existence of robust (transient) oscillations in the original model Some, well-characterized, transitions are lost, hence reachability is (generally) not conserved If an attractor is reachable in the reduced model, it is reachable in the

• riginal one
• A. Naldi et al (2011) Theor Comp Sc 412(21):2207-18

25

Bladder Cancer model: Looking for attractors

A ”drastic” reduction:

26

Bladder Cancer model: Looking for attractors

HTG of the reduced model contains 20 stables states, and 5 terminal SCCs ... What about the ”original” model?

27

Attractors of the Bladder cancer model

EGFR_stimulus FGFR3_stimulus DNA_damage Growth_inhibitor Proliferation Apoptosis Growth_arrest EGFR FGRF3 HRAS E2F1 E2F3 CyclinD1 CylinE1 CylcinA CDC25A P16INK4a RB1 RBL2 p21CIP ATM CHEK1 MDM2 TP53 p14ARF PTEN PI3K AKT GRB2 SRPY Number of states 1 1 1

1

1 1 1 1 1

1

1 1 1 1

1

1 1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 * * * * * 0/1 * * * * * * * * * * *

184320

* * * * * * * * * * * * * * * * * * * * 0/1 * * * * * * * * * * * * * * * 0/1 * * * 1 * * * * * * * 1 1 1 * * 0/1 1 1 * * * * * *

512

1 1 1 1 * * 1 1 1 1 1 1 1 * *

16

1 1 1 1 1 * * 1 1 1 1 1 1 1 * *

16

* * 0/1 1 1 1 1 1 1 1 * *

32

1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

1 1 1 1 1 1 1 1 1 1 1 1 1

1

! 28

Circuit analysis

29

Bladder cancer model: Circuit analysis

691 circuits, 12 are functional: 3 negative and 9 positive Cyclical attractors seem generated by the negative circuit EGFR/GRB2

30

Bladder cancer model: Circuit analysis

Stable states FGFR3 over-expressed mutant:

Stable states for FGFR3 over expressed mutant

no more cyclical attractor: when blocking FGFR3 to 1, we force the system to leave the functionality context....

31

Model perturbations

Systematic analysis of mutants (single, double, triple, ...)

What would the cancer cell mutate to increase uncontrolled proliferation? What are the mutations that could eliminate Proliferation and reinforce Apoptose?

32

Model perturbations

Systematic analysis of mutants

33

Co-occurence PIK3CA/FGFR3 alterations

Network topology

PI3K is downstream of FGFR3 − → not enough to explain

34

Co-occurence PIK3CA/FGFR3 alterations

Network topology

PI3K is downstream of FGFR3 − → not enough to explain

Simulations of mutants

Appearance of the E2F1-dependent apoptosis (when DNAdam is ON, and p16 OFF )

= ⇒ It is advantageous to the cancer cell to mutate PIK3CA in a FGFR3 mutated cell.

35

Predictions

Context

Is there an additional gene mutation, deletion or amplification where this co-occurrence could be significant? Which third alteration would lead to loss of Growth Arrest phenotype (equivalent to all checkpoints OFF)?

FGFR3 ¡E1, ¡p16INK4a ¡KO, ¡p21CIP ¡KO ¡ ¡ FGFR3 ¡E1, ¡p16INK4a ¡KO, ¡ATM ¡E1 ¡ ¡ FGFR3 ¡E1, ¡p16INK4a ¡KO, ¡ATM ¡E2 ¡ ¡ FGFR3 ¡E1, ¡p16INK4a ¡KO, ¡TP53 ¡E1 ¡ ¡ FGFR3 ¡E1, ¡p16INK4a ¡KO, ¡PI3K ¡E1 ¡ ¡ FGFR3 ¡E1, ¡p16INK4a ¡KO, ¡AKT ¡E1 ¡ ¡ FGFR3 ¡E1, ¡p16INK4a ¡KO, ¡GRB2 ¡E1 ¡ ¡ FGFR3 ¡E1, ¡p16INK4a ¡KO, ¡SPRY ¡KO ¡

36

Prediction of the model

Triple mutant FGFR3 E1 & PIK3 E1 & p16 KO DNA damage ON : only apoptosis (both types) DNA damage OFF : only proliferation → Uncontrolled growth, Very invasive tumours

37

Prediction of the model

Triple mutant FGFR3 E1 & PIK3 E1 & p16 KO DNA damage ON : only apoptosis (both types) DNA damage OFF : only proliferation → Uncontrolled growth, Very invasive tumours

Verification in the data

162 tumours 12 tumors are FGFR3 & PIK3CA-mutated: 4 of them have an homozygote deletion of CDKN2A the 2 invasive tumours among the 12 are CDKN2A deleted

37

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

38

Problem

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

000 001 010 011 100 101 110 111

39

Problem

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

Central question

What is the probability of reaching an attractor from a given portion of the state space?

000 001 010 011 100 101 110 111

39

Problem

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

Central question

What is the probability of reaching an attractor from a given portion of the state space?

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)

000 001 010 011 100 101 110 111

39

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´ erenguier et al., Chaos 2013)

FireFront

(Mendes, Monteiro et al., ECCB 2014 submitted)

Monte Carlo simulations

Boolnet

(M¨ ussel et al., Bioinformatics 2010) 40

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´ erenguier 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) 40

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´ erenguier 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) 40

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

41

Approach: Quasi-exact (FireFront algorithm)

α =

1 16

max iterations = 10 Start exploration from given initial condition v1, with unitary probability Iteration = 1 F = {v1} N = ∅ A = ∅

v1 1 v2 v3 v4 v5 v6 v7 v8

42

Approach: Quasi-exact (FireFront algorithm)

α =

1 16

max iterations = 10 Carry probability to successors dividing it by the number of successors – current firefront Iteration = 2 F = {v2, v5} N = ∅ A = ∅

v1 v2

1 2

v3 v4 v5

1 2

v6 v7 v8

42

Approach: Quasi-exact (FireFront algorithm)

α =

1 16

max iterations = 10 States with no successors are attractors and accumulate probability Iteration = 3 F = {v3, v4, v6} N = ∅ A = {v7}

v1 v2 v3

1 4

v4

1 4

v5 v6

1 4

v7

1 4

v8

42

Approach: Quasi-exact (FireFront algorithm)

α =

1 16

max iterations = 10 States with no successors are attractors and accumulate probability Iteration = 4 F = {v1, v3, v4, v6} N = ∅ A = {v7, v8}

v1

1 8

v2 v3

1 4

v4

1 8

v5 v6

1 8

v7

1 4

v8

1 8 42

Approach: Quasi-exact (FireFront algorithm)

α =

1 16

max iterations = 10 States with no successors are attractors and accumulate probability Iteration = 5 F = {v1, v2, v3, v4, v5, v6} N = ∅ A = {v7, v8}

v1

1 16

v2

1 16

v3

1 8

v4

1 8

v5

1 16

v6

1 16

v7

1 4

v8

1 4 42

Approach: Quasi-exact (FireFront algorithm)

α =

1 16

max iterations = 10 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 Iteration = 6 F = {v1, v2, v3, v4, v6} N = {v5} A = {v7, v8}

v1

1 16

v2

1 16

v3

1 16

v4

3 32

v5

1 32

v6

3 32

v7

9 32

v8

5 16 42

Approach: Quasi-exact (FireFront algorithm)

α =

1 16

max iterations = 10 States in the neglected set still accumulate probability and can be moved back to the firefront Iteration = 7 F = {v3, v5} N = {v1, v2, v4, v6} A = {v7, v8}

v1

3 64

v2

1 32

v3

1 8

v4

1 32

v5

1 16

v6

3 64

v7

5 16

v8

11 32 42

Approach: Quasi-exact (FireFront algorithm)

α =

1 16

max iterations = 10 States in the neglected set still accumulate probability and can be moved back to the firefront Iteration = 8 F = {v4, v6} N = {v1, v2} A = {v7, v8}

v1

3 64

v2

1 32

v3 v4

1 8

v5 v6

5 64

v7

5 16

v8

13 32 42

Approach: Quasi-exact (FireFront algorithm)

α =

1 16

max iterations = 10 States in the neglected set still accumulate probability and can be moved back to the firefront Iteration = 9 F = {v1, v3, v6} N = {v2} A = {v7, v8}

v1

7 64

v2

1 32

v3

5 64

v4 v5 v6

1 16

v7

5 16

v8

13 32 42

Approach: Quasi-exact (FireFront algorithm)

α =

1 16

max iterations = 10 Execution halts when the firefront is empty or the maximum number of iterations is reached Iteration = 10 F = {v2, v3} N = {v4, v5} A = {v7, v8}

v1 v2

11 128

v3

1 16

v4

5 128

v5

7 128

v6 v7

5 16

v8

57 128 42

Approach: Quasi-exact (FireFront algorithm)

α =

1 16

max iterations = 10 Execution halts when the firefront is empty or the maximum number of iterations is reached Iteration = 10 F = {v2, v3} N = {v4, v5} A = {v7, v8} Residual =

31 128

v1 v2

11 128

v3

1 16

v4

5 128

v5

7 128

v6 v7

5 16

v8

57 128 42

Approach: Quasi-exact (FireFront algorithm)

The maximum number of iterations and the α parameters control the running time and the precision

43

Approach: Quasi-exact (FireFront algorithm)

The maximum number of iterations and the α parameters control the running time and the precision Cannot directly identify complex attractors Large transient cycles may take too long to distribute probability “Wide” STGs may hurry every state to the neglected set

Lowering α may help, but the #states in the firefront grows very fast

43

Approach: Stochastic (Monte Carlo algorithm)

Exploration starts at a given initial state v1 Next state is picked at random from set of successors (random walk) Exploration stops when a stable state is reached Repeat for n simulations Number of trajectories towards an attractor measures its probability

44

Approach: Stochastic (Monte Carlo algorithm)

Exploration starts at a given initial state v1 Next state is picked at random from set of successors (random walk) Exploration stops when a stable state is reached Repeat for n simulations Number of trajectories towards an attractor measures its probability

Problems

May get stuck in large transients Is not able to identify complex attractors (unless they are already known)

44

Approach: Stochastic (Avatar algorithm)

Intuition

Modified Monte Carlo simulation When a cycle is detected, the STG is re-wired to remove the cycle – new incarnation of the STG

Transitions between cycle members are replaced by transitions to the cycle exits Equivalent to performing a random walk over Markov chains (proven)

45

Approach: Stochastic (Avatar algorithm)

Q0 =            

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 1 1            

v1 v2 v3 v4 v5 v6 v7 v8

1 2 1 2 46

Approach: Stochastic (Avatar algorithm)

Q0 =            

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 1 1            

v1 v2 v3 v4 v5 v6 v7 v8

1 2 1 2 46

Approach: Stochastic (Avatar algorithm)

Q0 =            

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 1 1            

v1 v2 v3 v4 v5 v6 v7 v8

1 2 1 2 46

Approach: Stochastic (Avatar algorithm)

Q0 =            

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 1 1            

v1 v2 v3 v4 v5 v6 v7 v8

1 2 1 2 46

Approach: Stochastic (Avatar algorithm)

Q0 =            

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 1 1            

v1 v2 v3 v4 v5 v6 v7 v8

46

Approach: Stochastic (Avatar algorithm)

Q1 =            

8 15 1 15 4 15 2 15 1 15 2 15 8 15 4 15 2 15 4 15 1 15 8 15 4 15 8 15 2 15 1 15 1 2 1 2

1 1 1            

v1 v2 v3 v4 v5 v6 v7 v8

8 15 1 15 4 15 2 15 46

Approach: Stochastic (Avatar algorithm)

Q1 =            

8 15 1 15 4 15 2 15 1 15 2 15 8 15 4 15 2 15 4 15 1 15 8 15 4 15 8 15 2 15 1 15 1 2 1 2

1 1 1            

v1 v2 v3 v4 v5 v6 v7 v8

46

Approach: Stochastic (Avatar algorithm)

Q1 =            

8 15 1 15 4 15 2 15 1 15 2 15 8 15 4 15 2 15 4 15 1 15 8 15 4 15 8 15 2 15 1 15 1 2 1 2

1 1 1            

v1 v2 v3 v4 v5 v6 v7 v8

And so forth...

46

Approach: Stochastic (Avatar algorithm)

The number of simulation runs controls the running time and precision

47

Approach: Stochastic (Avatar algorithm)

The number of simulation runs controls the running time and precision Huge transients and complex attractors may exhaust memory (when they correspond to an entire portion of a very large state space) Very large cycles may not be easily re-wired (cycle re-wiring requires a matrix inversion step)

47

Optimizations & additional features

FireFront and Avatar

An oracle may be provided to identify a known complex attractor

48

Optimizations & additional features

Avatar

Prior to cycle re-wiring a phase of τ-expansion is performed Complex attractors identified in one run are used to create an oracle to identify member states in subsequent simulation runs Large transients re-wired in one run are also carried to subsequent runs

48

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

49

Co-occurence PIK3CA/FGFR3 alterations

Prolifera)on ¡ 58% ¡ Growth_arrest ¡ 42% ¡

FGRF3 ¡E1 ¡(DNA ¡damage=0) ¡

Prolifera)on ¡ 44% ¡ Growth_arrest ¡ 44% ¡ Null ¡ 12% ¡

PI3K ¡E1 ¡(DNA ¡damage=0) ¡

Prolifera)on ¡ 65% ¡ Growth_arrest ¡ 35% ¡

FGFR3 ¡E1 ¡& ¡PI3K ¡E1 ¡(DNA ¡damage=0) ¡

Probability of Proliferation is increased and probability of Growth Arrest is decreased in the double mutant

50

Prediction of the model

A third deletion of CDKN2A (p16INK4a + p14ARF) eliminates growth arrest in absence of DNA damage

FGFR3 ¡E1 ¡& ¡PI3K ¡E1 ¡& ¡CDKN2A=0 ¡(DNA ¡damage=0) ¡ ¡ 51

Mutual exclusivity TP53/FGFR3 alterations

Prolifera)on ¡ 14% ¡ Apoptosis ¡ 19% ¡ Growth_arrest ¡ 61% ¡ Osc ¡Prolif-­‑GA ¡ 6% ¡

TP53 ¡KO ¡(all ¡input ¡configura5ons) ¡

Prolifera)on ¡ 29% ¡ Apoptosis ¡ 50% ¡ Growth_arrest ¡ 21% ¡

FGFR3 ¡E1 ¡(all ¡input ¡configura5ons) ¡

Prolifera)on ¡ 29% ¡ Apoptosis ¡ 25% ¡ Growth_arrest ¡ 46% ¡

FGRF3 ¡E1 ¡& ¡TP53 ¡KO ¡(all ¡input ¡configura5ons) ¡

TP53 mutation in a FGFR3-mutated context has very little impact on Proliferation (⇒ mutual exclusivity) Mutating FGFR3 in a TP53-mutated context increase the Proliferation probability (⇒ advantage): rare event

52

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

53

Conclusions

Challenge

Characterize and quantify the attractors in the context of discrete asynchronous dynamics The difficulty lies in the size and structure of the state spaces

54

Conclusions

Challenge

Characterize and quantify the attractors in the context of discrete asynchronous dynamics The difficulty lies in the size and structure of the state spaces There is no ideal solution The structure of the state space is unknown a priori We propose two approaches to tackle the problem

54

Conclusions

Best approach to use depends on the structure of the STG The number and size of transient cycles have an impact on both FireFront and Avatar

55

Conclusions

Best approach to use depends on the structure of the STG The number and size of transient cycles have an impact on both FireFront and Avatar

FireFront

Fast and quasi-exact for STGs which are not too “wide”

55

Conclusions

Best approach to use depends on the structure of the STG The number and size of transient cycles have an impact on both FireFront and Avatar

FireFront

Fast and quasi-exact for STGs which are not too “wide”

Avatar

Well-suited to deal with cycles (complex attractors and transients) Rare attractors may need many simulation runs to be found

55

Thank you!

Availability

http://compbio.igc.gulbenkian.pt/nmd/node/59 http://ginsim.org

Collaborators

Instituto Gulbenkian de Ciˆ encia, Oeiras, PT Institut Curie, Paris Jorge Carneiro Laurence Calzone Claudine Chaouiya Fran cois Radvanyi Nuno Mendes Sandra Rebouissou Pedro Monteiro