A Disrete Approa h to Mo del Gene Regulatory Net w orks - PowerPoint PPT Presentation

A Dis rete Approa h to Mo del Gene Regulatory Net w orks and the Use of F ormal Logi to Prop ose New W et Exp erimen ts Gilles Bernot Univ ersit y of Ni e sophia antipolis , I3S lab oratory A kno wledgmen ts : Observability Gr oup of the Epigenomi s Pro je t 1

Men u 1. Sim ulation vs. V alidation 2. F ormal Metho ds for the Mo delling A tivit y 3. Gene Regulatory Net w orks & T emp oral Logi 4. P edagogi al example: Pseudomonas aeruginosa 2

Mathemati al Mo dels and Sim ulation 1. Rigorously en o de sensible kno wledge in to mathemati al form ulae 2. � Some parameters are w ell de�ned, e.g. from bio hemi al kno wledge � Some parameters are limited to some in terv als � Some parameters are a priori unkno wn 3. P erform lot of sim ulations, ompare results with kno wn b eha viours, and prop ose some redible v alues of the unkno wn parameters whi h pro du e a eptable b eha viours 4. P erform additional sim ulations re�e ting no v el situations 5. If they predi t in teresting b eha viours, prop ose new biologi al exp erimen ts 6. Simplify the mo del and try to go further 3

Mathemati al Mo dels and V alidation �Brute for e� sim ulations are not the only w a y to use a omputer. W e an o�er omputer aided en vironmen ts whi h help: � to a v oid mo dels that an b e �tuned� ad libitum � to v alidate mo dels with a reasonable n um b er of exp erimen ts � to de�ne only mo dels that ould b e exp erimen tally refuted � to pro v e refutabilit y w.r.t. exp erimen tal apabilities Observability issues: Observability Gr oup , Epigenomi s Pro je t. 4

Men u 1. Sim ulation vs. V alidation 2. F ormal Metho ds for the Mo delling A tivit y 3. Gene Regulatory Net w orks & T emp oral Logi 4. P edagogi al example: Pseudomonas aeruginosa 5

F ormal Logi : syn tax/seman ti s/dedu tion satisfa tion Syntax Semantics Formulae M | = ϕ Models green=Mathematics cyan=Computer red=Computer Science correctness completeness Φ ⊢ ϕ Deduction 6 proved=satisfied proof Rules

Computer Aided Elab oration of Mo dels F rom biologi al kno wledge and/or biologi al h yp otheses, it omes: � prop erties: �Without stimulus, if gene x has its b asal expr ession level, then it r emains at this level.� � mo del s hemas: 2 1 1 + + 2 + x y + x y � � 1 1 . . . F ormal logi and formal mo dels allo w us to: � v erify h yp otheses and he k onsisten y � elab orate more pre ise mo dels in remen tally � suggest new biologi al exp erimen ts to e� ien tly redu e the n um b er of p oten tial mo dels 7

The T w o Questions 2 1 + + x y � and = 1 . . . 1. Is it p ossible that Φ and M ? Consisten y of kno wledge and h yp otheses. Means to sele t mo dels b elonging to the s hemas that satisfy Φ . Φ = { ϕ 1 , ϕ 2 , · · · , ϕ n } M 2. If so, is it true in vivo that Φ and M ? Compatibilit y of one of the sele ted mo dels with the biologi al ob je t. Require to prop ose exp erimen ts to v alidate (or ( ∃ ? M ∈ M | M | = ϕ ) refute ) the sele ted mo del(s). Computer aided pr o ofs and validations 8 →

Men u 1. Sim ulation vs. V alidation 2. F ormal Metho ds for the Mo delling A tivit y 3. Gene Regulatory Net w orks & T emp oral Logi 4. P edagogi al example: Pseudomonas aeruginosa 9

Multiv alued Regulatory Graphs � + x y x y � + 0 1 2 x 1 2 τ 1 τ 2 x y x 10

Regulatory Net w orks (R. Thomas) ( x,y ) Image (0,0) 2 + 1 (0,1) + (1,0) 1 � Basal lev el : K x (1,1) ( K x,y , K y ) helps : K x,x (2,0) ( K x , K y ) Absen t y helps : K x,y x y (2,1) Both : K x,xy ( K x,xy , K y ) K y ( K x,x , K y ) x K y,x ( K x,xy , K y,x ) ( K x,x , K y,x ) 11

State Graphs ( x,y ) Image (0,0) ( K x,y , K y ) =(2,1) (0,1) ( K x , K y ) =(0,1) (1,0) ( K x,xy , K y ) =(2,1) y (1,1) ( K x,x , K y ) =(2,1) (2,0) ( K x,xy , K y,x ) =(2,1) 1 (0,1) (1,1) (2,1) (2,1) ( K x,x , K y,x ) =(2,1) Time has a tree stru ture: (0,0) (1,0) (2,0) 0 x 0 1 2 (0,1) (0,0) (1,1) (2,1) (1,0) 12 (2,0) (2,1)

CTL = Computation T ree Logi A toms = omparaisons : (x=2) (y > 0) . . . Logi al onne tiv es: ( ϕ 1 ∧ ϕ 2 ) T emp oral onne tiv es: made of 2 hara ters �rst hara ter se ond hara ter = for A ll path hoi es = ne X t state ( ϕ 1 = ⇒ ϕ 2 ) · · · = for some F uture state = there E xist a hoi e = for all future states ( G lobally) A X = U n til F AX ( y = 1) : the on en tration lev el of y b elongs to the in terv al 1 in all states dire tly follo wing the onsidered initial state. E G EG ( x = 0) : there exists at least one path from the onsidered initial U state where x alw a ys b elongs to its lo w er in terv al. 13

Question 1 = Consisten y 1. Dra w all the sensible regulatory graphs with all the sensible threshold allo ations. It de�nes M . 2. Express in CTL the kno wn b eha vioural prop erties as w ell as the onsidered biologi al h yp otheses. It de�nes Φ . 3. Automati ally generate all the p ossible regulatory net w orks deriv ed from M a ording to all p ossible parameters K ... . Our soft w are plateform SMBioNet handles this automati ally . 4. Che k ea h of these mo dels against Φ . SMBioNet uses mo del he king to p erform this step. 5. If no mo del surviv e to the previous step, then re onsider the h yp otheses and p erhaps extend mo del s hemas. . . 6. If at least one mo del surviv es, then the biologi al h yp otheses are onsisten t. P ossible parameters K ... ha v e b een indire tly established. No w Question 2 has to b e addressed. 14

Theoreti al Mo dels ↔ Exp erimen ts CTL form ulae are satis�ed (or refuted) w.r.t. a set of paths from a giv en initial state � They an b e tested against the p ossible paths of the theoreti al mo dels ( M | = Model Checking ϕ ) � They an b e tested against the biologi al exp erimen ts ( Biological _ Object | = Experiment ϕ ) CTL form ulae link theoreti al mo dels and biologi al ob je ts together 15

Question 2 = V alidation 1. Among all p ossible form ulae, some are �observ able� i.e., they express a p ossible result of a p ossible biologi al exp erimen t. Let Obs b e the set of all observ able form ulae. 2. Let Λ b e the set of theorems of Φ and M . is the set of exp erimen ts able to v alidate the surviv ors of Question 1. Unfortunately it is in�nite in general. 3. T esting framew orks from omputer s ien e aim at sele ting a �nite subsets of these observ able form ulae, whi h maximize the Λ ∩ Obs han e to refute the surviv ors. 4. These subsets are often to o big, nev ertheless these testing framew orks an b e suitably applied to regulatory net w orks. 16

Men u 1. Sim ulation vs. V alidation 2. F ormal Metho ds for the Mo delling A tivit y 3. Gene Regulatory Net w orks & T emp oral Logi 4. P edagogi al example: Pseudomonas aeruginosa 17

Example : ytoto xi it y ( P.aeruginosa ) T erminology ab out phenot yp e mo di� ation: Geneti mo di� ation: inheritable and not rev ersible (m utation) Epigeneti swit h: inheritable and rev ersible A daptation: not inheritable and rev ersible The biologi al questions (Janine Guespin): is ytoto xi it y in Pseudomonas aeruginosa due to an epigeneti swit h ? [ → ysti �brosis℄ 18

Cytoto xi it y in P. aeruginosa (Janine Guespin and Mar eline Kaufman) + ExsD ExsA + � + Epigeneti h yp othesis = The p ositiv e feedba k ir uit is fun tional, with a ytoto xi toxicity stable state and the other one is not ytoto xi . An external signal (in the ysti �brosis' lungs) ould swit h ExsA from its lo w er stable state to the higher one. → → 19

Consisten y of the Hyp othesis + ExsD ExsA + � + One CTL form ula for ea h stable state: toxicity Question 1, onsisten y: pro v ed b y Mo del Che king 10 mo dels among the 712 mo dels are extra ted b y SMBioNet (ExsA = 2) = ⇒ AXAF (ExsA = 2) (ExsA = 0) = ⇒ AG ( ¬ (ExsA = 2)) Question 2: and in vivo ? . . . → 20