Méthode logique multivaluée de René Thomas et logique temporelle

Transcription

1 Méthode logique multivaluée de René Thomas et logique temporelle Gilles Bernot Universit of Nice sophia antipolis, I3S laborator, France

2 Menu 1. Models and formal logic 2. Thomas models for gene networks 3. Gene networks and temporal logic 4. Models for checking biological hpotheses 5. Extracting experiments from models 6. Model Simplifications

3 Mathematical models: what for? Models as Data Base to store biological knowledge Models as design tools for snthetic biolog Models as logical analsis tools of causalit chains Models as guidelines for the choice of experiments For the 2 or 3 last purposes, models can deviate far from biological descriptions while remaining ver useful: Kleenex models...

4 Static Graph v.s. Dnamic Behaviour Difficult to predict the result of combined regulations Difficult to measure the strength of a given regulation Example of competitor circuits Multistationarit? Homeostas? + + Alginate Muc-B + mucus Man underling models 700 qualitative behaviours

5 Mathematical Models and Simulation 1. Rigorousl encode sensible knowledge, into ODEs for instance 2. A few parameters are approximativel known Some parameters are limited to some intervals Man parameters are a priori unknown 3. Perform lot of simulations, compare results with known behaviours, and propose some credible values of the unknown parameters which produce robust acceptable behaviours 4. Perform additional simulations reflecting novel situations 5. If the predict interesting behaviours, propose new biological experiments 6. Simplif the model and tr to go further

6 Mathematical Models and Validation Brute force simulations are not the onl wa to use a computer. There are computer aided environments which help: designing simplified models that can be anaticall solved avoiding models that can be tuned ad libitum validating models with a reasonable number of experiments defining onl models that could be experimentall refuted proving refutabilit w.r.t. experimental capabilities establishing a methodolog: models experiments Operabilit and observabilit issues (Observabilit Group, Epigenomics Project)

7 Formal Logic: sntax/semantics/deduction Sntax Formulae gold=computer M = satisfaction Semantics Models green=mathematics Φ proof can=computer Science Deduction Rules correctness completeness proved=satisfied

8 Menu 1. Models and formal logic 2. Thomas models for gene networks 3. Gene networks and temporal logic 4. Models for checking biological hpotheses 5. Extracting experiments from models 6. Model Simplifications

9 Multivalued Regulator Graphs + z x z x 1 + x τ 1 τ z x

10 Regulator Networks (R. Thomas) x x x x No help : K x x helps : K x,x Absent helps : K x, Both : K x,x K K,x (x,) Focal Point (0,0) (K x,, K ) (0,1) (K x, K ) (1,0) (K x,x, K ) (1,1) (K x,x, K ) (2,0) (K x,x, K,x ) (2,1) (K x,x, K,x )

11 State Graphs (x,) Focal Point (0,0) (K x,,k )=(2,1) (0,1) (K x,k )=(0,1) (1,0) (K x,x, K )=(2,1) (1,1) (K x,x,k )=(2,1) (2,0) (K x,x,k,x)=(2,1) (2,1) (K x,x,k,x)=(2,1) 1 0 (0,1) (1,1) (2,1) (0,0) (1,0) (2,0) x desnchronization b units of Manhattan distance 1 (0,1) (1,1) (2,1) 0 (0,0) (1,0) (2,0) x

12 Example on paper sheets... Do it b ourselves!

13 Menu 1. Models and formal logic 2. Thomas models for gene networks 3. Gene networks and temporal logic 4. Models for checking biological hpotheses 5. Extracting experiments from models 6. Model Simplifications

14 Time has a tree structure (0,1) (1,1) (2,1) (0,0) (1,0) (2,0) from each initial state: x (0,1) As man possible state graphs as possible parameter sets... (huge number) (0,0) (1,0) (1,1) (2,0) (2,1) (2,1)

15 CTL = Computation Tree Logic Atoms = comparaisons : (x=2) (>0)... Logical connectives: ( 1 2 ) ( 1 = 2 ) Temporal modalities: made of 2 characters first character second character A = for All path choices X = next state F = for some Future state E = there Exist a choice G = for all future states (Globall) U = Until AX( = 1) : the concentration level of belongs to the interval 1 in all states directl following the considered initial state. EG(x = 0) : there exists at least one path from the considered initial state where x alwas belongs to its lower interval.

16 Semantics of Temporal Connectives t+1 t EX t AX t+1 t EF t+k t AF EG AG E[ψU] A[ψU]

17 CTL to encode Biological Properties Common properties: functionalit of a sub-graph Special role of feedback loops positive: multistationnarit (even number of ) negative: homeostas (odd number of ) x + 1 (0,1) (1,1) (2,1) (0,1) (1,1) (2,1) (0,0) (1,0) (2,0) (0,0) (1,0) (2,0) Characteristic properties: x { (x = 2) = AG( (x = 0)) (x = 0) = AG( (x = 2)) The express the positive feedback loop is functional (satisfaction of these formulas relies on the parameters K... ) x

18 Model Checking Efficientl computes all the states of a state graph which satisf a given formula: { η M = η }. Efficientl select the models which globall satisf a given formula.

19 Menu 1. Models and formal logic 2. Thomas models for gene networks 3. Gene networks and temporal logic 4. Models for checking biological hpotheses 5. Extracting experiments from models 6. Model Simplifications

20 Computer Aided Elaboration of Models From biological knowledge and/or biological hpotheses, it comes: properties: Without stimulus, if gene x has its basal expression level, then it remains at this level. model schemas: x Formal logic and formal models allow us to: x verif hpotheses and check consistenc elaborate more precise models incrementall suggest new biological experiments to efficientl reduce the number of potential models

21 The Two Questions x Φ = { 1, 2,, n,h} and M = K x K x,x K x,x 1. Is it possible that Φ and M? Consistenc of knowledge and hpotheses. Means to select models belonging to the schemas that satisf Φ. (? M M M = Φ) 2. If so, is it true in vivo that Φ and M? Compatibilit of one of the selected models with the biological object. Require to propose experiments to validate or refute the selected model(s). Computer aided proofs and validations

22 Theoretical Models Experiments CTL formulas are satisfied (or refuted) w.r.t. a set of paths from a given initial state The can be tested against the possible paths of the theoretical models (M = Model Checking ) The can be tested against the biological experiments (Biological_Object = Experiment ) CTL is a bridge between theoretical models and biological objects

23 Menu 1. Models and formal logic 2. Thomas models for gene networks 3. Gene networks and temporal logic 4. Models for checking biological hpotheses 5. Extracting experiments from models 6. Model Simplifications

24 Generation of biological experiments (1) Set of all the formulas: = hpothesis

25 Generation of biological experiments (2) Set of all the formulas: = hpothesis Obs = possible experiments Obs

26 Generation of biological experiments (3) Set of all the formulas: = hpothesis Obs = possible experiments Th() = inferences Obs

27 Generation of biological experiments (4) Set of all the formulas: = hpothesis Obs = possible experiments Th() = inferences S = sensible experiments S Obs

28 Generation of biological experiments (5) Set of all the formulas: = hpothesis Obs = possible experiments Th() = inferences S = sensible experiments Refutabilit: S =? S Obs

29 Generation of biological experiments Set of all the formulas: = hpothesis Obs = possible experiments Th() = inferences S = sensible experiments Refutabilit: S =? Best refutations: Choice of experiments in S?... optimisations S Obs

30 Example: Mucus Production in P. aeruginosa AlgU... MucB... operon + self inducer + AlgU + MucB AlgU mucus Capture: AlgU abstract behaviour membrane MucB

31 How to validate a multistationnarit M: (unknown thresholds) Φ: + + Alginate Muc-B + mucus { (Alginate = 2) = AG(Alginate = 2) (hpothesis) (Alginate = 0) = AG(Alginate < 2) (knowledge) Assume that onl mucus can be observed: Lemma: AG(Alginate = 2) AFAG(mucus = 1) (... formal proof b computer...) To validate: (Alginate = 2) = AFAG(mucus = 1)

32 (Alginate = 2) = AFAG(mucus = 1) A = B true false true true false false true true Karl Popper: to validate = to tr to refute thus A=false is useless experiments must begin with a pulse The pulse forces the bacteria to reach the initial state Alginate = 2. If the state is not directl controlable we need to prove lemmas: General form of a test: (something reachable) = (Alginate = 2) (something reachable) = (something observable)

33 Menu 1. Models and formal logic 2. Thomas models for gene networks 3. Gene networks and temporal logic 4. Models for checking biological hpotheses 5. Extracting experiments from models 6. Model Simplifications

34 Hpothesis driven model simplifications Successive simplified views of the studied biological object: Model M 1 satisfies 1 Model M 2 satisfies 2 Model M 3 satisfies 3... Example: gene removal often preserves the number of attraction basins [Naldi&al.2011]

35 Simplifications via level folding ρ = 2 ρ = 1 ρ = x ρ x = 0 ρ x = 1 ρ x = 2 folding x 2 1 folding x x

36 Simplifications via subgraphs Embeddings of Regulator Networks: x t u z v x t u z v Studied behaviour Preserved behaviour? Necessar and sufficient condition on the local dnamics of the input frontier... Also fusion of genes, etc.

37 Take Home Messages Formalize the hpotheses that motivate the biological research Behavioural properties (Φ) are as much important as models (M) Smbolic parameter identification is essential Modelling is significant onl with respect to the considered experimental reachabilit and observabilit (for refutabilit) Formal proofs can suggest wet experiments Mathematical models are not realit: let s use this freedom! (simplified views of a biological object) Kleenex models help understanding main behaviours