Formal approaches Static Graph v.s. Dynamic Behaviour to model - PDF document

Formal approaches Static Graph v.s. Dynamic Behaviour to model Difficulty to predict the result of combined regulations Difficulty to measure the strength of a given regulation gene regulatory networks Example of “competitor” circuits Gilles Bernot + University of Nice sophia antipolis , I3S laboratory, France antiAlgU AlgU + — + Positive v.s. Negative circuits mucus Even v.s. Odd number of “—” signs Multistationarity v.s. Homeostasy Acknowledgments : René Thomas, Snoussi, . . . , Soulé, Richard Functional circuits “pilot” the behaviour Observability Group of the Epigenomics Project 4 1 Menu Mathematical Models and Validation 1. Modelling biological regulatory networks “Brute force” simulations are not the only way to use a computer. 2. Discrete framework for biological regulatory networks We can offer computer aided environments which help: 3. Temporal logic and Model Checking for biology • to avoid models that can be “tuned” ad libitum 4. Computer aided elaboration of formal models • to validate models with a reasonable number of experiments 5. Pedagogical example: Pseudomonas aeruginosa • to define only models that could be experimentally refuted 6. Some current research topics • to prove refutability w.r.t. experimental capabilities 7. An extension to delays Observability issues: Observability Group , Epigenomics Project. 5 2 Formal Logic: syntax/semantics/deduction Mathematical Models and Simulation 1. Rigorously encode sensible knowledge into ODEs for instance Syntax 2. • A few parameters are approximatively known Semantics M | = ϕ • Some parameters are limited to some intervals Formulae Models • Many parameters are a priori unknown green=Mathematics satisfaction 3. Perform lot of simulations, compare results with known behaviours, and propose some credible values of the unknown cyan=Computer parameters which produce acceptable behaviours red=Computer Science correctness 4. Perform additional simulations reflecting novel situations completeness Φ ⊢ ϕ 5. If they predict interesting behaviours, propose new biological experiments Deduction proved=satisfied 6. Simplify the model and try to go further proof Rules 6 3

State Graphs Menu y ( x,y ) Focal Point (0,0) ( K x,y , K y ) =(2,1) 1 (0,1) (1,1) (2,1) 1. Modelling biological regulatory networks (0,1) ( K x , K y ) =(0,1) (1,0) ( K x,xy , K y ) =(2,1) 2. Discrete framework for biological regulatory networks (1,1) ( K x,x , K y ) =(2,1) 3. Temporal logic and Model Checking for biology (0,0) (1,0) (2,0) 0 (2,0) ( K x,xy , K y,x ) =(2,1) 4. Computer aided elaboration of formal models (2,1) ( K x,x , K y,x ) =(2,1) x 0 1 2 5. Pedagogical example: Pseudomonas aeruginosa y 6. Some current research topics “desynchronization” − → 1 (0,1) (1,1) (2,1) 7. An extension to delays by units of Manhattan distance 0 (0,0) (1,0) (2,0) x 0 1 2 10 7 Menu Multivalued Regulatory Graphs 1. Modelling biological regulatory networks x y 2. Discrete framework for biological regulatory networks 3. Temporal logic and Model Checking for biology — + x y 4. Computer Aided elaboration Of Formal models 5. Pedagogical example: Pseudomonas aeruginosa 6. Some current research topics x 7. An extension to delays τ 1 τ 2 1 2 — + x y 0 1 2 x 11 8 Time has a tree structure y Regulatory Networks (R. Thomas) 1 (0,1) (1,1) (2,1) As many possible state graphs as possible parameter sets. . . ( x,y ) Focal Point (huge number) (0,0) (1,0) (2,0) (0,0) ( K x,y , K y ) 0 2 + 1 (0,1) ( K x , K y ) x y + x 0 1 2 (1,0) ( K x,xy , K y ) 1 — From an initial state: Basal level : K x K y (1,1) ( K x,x , K y ) (0,1) x helps : K x,x K y,x (2,0) ( K x,xy , K y,x ) Absent y helps : K x,y (0,0) (2,1) ( K x,x , K y,x ) (1,1) (2,1) Both : K x,xy (1,0) (2,0) (2,1) 12 9

CTL to encode Biological Properties CTL = Computation Tree Logic Common properties: 2 1 Atoms = comparaisons : (x=2) (y > 0) . . . + x y “functionality” of a sub-graph + Logical connectives: ( ϕ 1 ∧ ϕ 2 ) ( ϕ 1 = ⇒ ϕ 2 ) · · · — Special role of “feedback loops” 1 Temporal modalities: made of 2 characters – positive: multistationnarity (even number of — ) first character second character – negative: homeostasy (odd number of — ) y y A = for A ll path choices X = ne X t state (0,1) (1,1) (2,1) (0,1) (1,1) (2,1) F = for some F uture state E = there E xist a choice G = for all future states ( G lobally) (0,0) (1,0) (2,0) (0,0) (1,0) (2,0) x x U = U ntil  AX ( y = 1) : the concentration level of y belongs to the interval 1 in all ( x = 2) = ⇒ AG ( ¬ ( x = 0))  Characteristic properties: states directly following the considered initial state. ( x = 0) = ⇒ AG ( ¬ ( x = 2))  EG ( x = 0) : there exists at least one path from the considered initial They express “the positive feedback loop is functional” state where x always belongs to its lower interval. (satisfaction of these formulae relies on the parameters K ... ) 16 13 Temporal Connectives of CTL neXt state: EXϕ : ϕ can be satisfied in a next state AXϕ : ϕ is always satisfied in the next states eventually in the Future: Model Checking EFϕ : ϕ can be satisfied in the future AFϕ : ϕ will be satisfied at some state in the future Efficiently computes all the states of a state graph which satisfy a Globally: given formula: { η | M | = η ϕ } . EGϕ : ϕ can be an invariant in the future Efficiently select the models which globally satisfy a given formula. AGϕ : ϕ is necessarilly an invariant in the future Until: E [ ψUϕ ] : there exist a path where ψ is satisfied until a state where ϕ is satisfied A [ ψUϕ ] : ψ is always satisfied until some state where ϕ is satisfied 17 14 Semantics of Temporal Connectives Model Checking for CTL t+1 t+1 t+k ϕ ϕ ϕ ϕ ϕ Computes all the states of a theoretical model which satisfy a given t t t t ϕ ϕ ϕ ϕ formula: { η | M | = η ϕ } . ϕ ϕ ϕ Idea 1: work on the state graph instead of the path trees. ϕ ϕ Idea 2: check first the atoms of ϕ and then check the connectives of EX ϕ AX ϕ EF ϕ AF ϕ ϕ with a bottom-up computation strategy. Idea 3: (computational optimization) group some cases together ... ... ... ... using BDDs (Binary Decision Diagrams). ... ... ... ... ... ... ... Example : ( x = 0) = ⇒ AG ( ¬ ( x = 2)) ... ... ... Obsession: travel the state graph as less as possible EG ϕ AG ϕ E [ ψUϕ ] A [ ψUϕ ] 18 15

Computer Aided Elaboration of Models ( x = 0) = ⇒ AG ( ¬ ( x = 2)) From biological knowledge and/or biological hypotheses, it comes: y y • properties: “Without stimulus, if gene x has its basal expression level, then it remains at this level.” • model schemas: x x 2 1 1 + 2 + x y x y + + — — z ¬ ( x = 2) . . . z 1 1 x=0 x=2 and AG ( ¬ ( x = 2)) ? Formal logic and formal models allow us to: . . . one should travel all the paths from any green box and check • verify hypotheses and check consistency if successive boxes are green: too many boxes to visit . • elaborate more precise models incrementally Trick: AG ( ¬ ( x = 2)) is equivalent to ¬ EF ( x = 2) • suggest new biological experiments to efficiently reduce the start from the red boxes and follow the transitions backward. number of potential models 22 19 The Two Questions 2 1 Theoretical Models ↔ Experiments + x y + — Φ = { ϕ 1 , ϕ 2 , · · · , ϕ n } and M = CTL formulae are satisfied (or refuted) w.r.t. a set of paths from a . . . 1 given initial state 1. Is it possible that Φ and M ? Consistency of knowledge and hypotheses. Means to select • They can be tested against the possible paths of the theoretical models ( M | = Model Checking ϕ ) models belonging to the schemas that satisfy Φ . ( ∃ ? M ∈ M | M | = ϕ ) • They can be tested against the biological experiments ( Biological _ Object | = Experiment ϕ ) 2. If so, is it true in vivo that Φ and M ? Compatibility of one of the selected models with the biological object. Require to propose experiments to validate or refute CTL formulae link theoretical models and biological objects together the selected model(s). → Computer aided proofs and validations 23 20 Question 1 = Consistency Menu 1. Draw all the sensible regulatory graphs with all the sensible threshold allocations. It defines M . 2. Express in CTL the known behavioural properties as well as 1. Modelling biological regulatory networks the considered biological hypotheses. It defines Φ . 2. Discrete framework for biological regulatory networks 3. Automatically generate all the possible regulatory networks 3. Temporal logic and Model Checking for biology derived from M according to all possible parameters K ... . 4. Computer aided elaboration of formal models Our software plateform SMBioNet handles this automatically. 5. Pedagogical example: Pseudomonas aeruginosa 4. Check each of these models against Φ . 6. Some current research topics SMBioNet uses model checking to perform this step. 7. An extension to delays 5. If no model survive to the previous step, then reconsider the hypotheses and perhaps extend model schemas. . . 6. If at least one model survives, then the biological hypotheses are consistent. Possible parameters K ... have been indirectly established. Now Question 2 has to be addressed. 24 21