Computational Methods for Systems and Synthetic Biology

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Computational Methods for Systems and Synthetic Biology

François Fages

Inria Saclay – Ile de France

Lifeware project-team

http://lifeware.inria.fr/

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Overview of the Lecture

The Boolean semantics of reaction systems potentially applies to

• Very large systems

• Without any knowledge on the reaction rates

How to query the asynchronous non-deterministic Boolean dynamics ? 1. Example of Kohn’s map of the mammalian cell cycle 2. Computation Tree Logic CTL query language

3. State-based model-checking algorithm 4. Symbolic model-checking algorithm

5. Model reduction preserving CTL properties

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Cell Cycle: G0 àG1àSynthesisàG2àMitosis

G1: CdK4-CycD S: Cdk2-CycA G2,M: Cdk1-CycA

Cdk6-CycD Cdk1-CycB (MPF)

Sir Paul Nurse Nobel prize 2001

for his work on Cyclins

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Mammalian Cell Cycle Control Map ^{[Kohn 99]}

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Kohn’s map detail for Cdk2

Complexation of cdk2 with cycA and cycE Biocham reaction patterns:

cdk2~$P + cycA-$C => cdk2~$P-cycA-$C where $C in {_,cks1} .

cdk2~$P + cycE~$Q-$C => cdk2~$P-cycE~$Q-$C where $C in {_,cks1} .

p57 + cdk2~$P-cycA-$C => p57-cdk2~$P-cycA-$C where $C in {_, cks1}.

cycE-$C =[cdk2~{p2}-cycE-$S]=> cycE~{T380}-$C

where $S in {_, cks1} and $C in {_, cdk2~?, cdk2~?-cks1}

à 732 reactions, 165 proteins and genes, 532 variables

How to analyze a transition system over 2⁵³² states ? and 2²⁵³² sets of states ?

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Kohn’s map detail for Cdk2

Complexation of cdk2 with cycA and cycE Biocham reaction patterns:

cdk2~$P + cycA-$C => cdk2~$P-cycA-$C where $C in {_,cks1} .

cdk2~$P + cycE~$Q-$C => cdk2~$P-cycE~$Q-$C where $C in {_,cks1} .

p57 + cdk2~$P-cycA-$C => p57-cdk2~$P-cycA-$C where $C in {_, cks1}.

cycE-$C =[cdk2~{p2}-cycE-$S]=> cycE~{T380}-$C

where $S in {_, cks1} and $C in {_, cdk2~?, cdk2~?-cks1}

à 732 reactions, 165 proteins and genes, 532 variables

How to analyze a transition system over 2⁵³² states ? and 2²⁵³² sets of states ? Symbolic model-checking ! Represent a set of states by a Boolean constraint

True: full set of 2⁵³² states, False: empty set, M: set of 2⁵³¹ states with M present, MÚ¬N : set of 3.2⁵³⁰ states with M present or N absent, etc.

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Computation Tree Logic CTL

Temporal logics extend classical logic with modal operators for time and non- determinism. Introduced for program verification by [Pnueli 77]

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Kohn’s Map Model-Checking

BIOCHAM NuSMV symbolic model-checker time in seconds [Chabrier Fages 03 cmsb]

Initial state G2 Query: Time:

compiling 29

Reachability G1 EF CycE 2

Reachability G1 EF CycD 1.9

Checkpoint for mitosis complex

¬E (¬Cdc25~{Nterm} U Cdk1~{Thr161}-CycB) 2.2

Oscillations CycA EG ( (EF ¬ CycA) Ù (EF CycA)) 31.8 Osciallations CycB EG ( (EF ¬ CycB) Ù (EF CycB))

false ! (omission of CycB synthesis in Kohn’s map)

6

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Kripke Semantics of CTL*

A Kripke structure K=(S,R) is a set S of states with a total relation RÍSxS The truth of a formula f in a state s or on a path p of K is defined by:

s ⊨ f if f is a proposition true in s

s ⊨ E f if there is a path p starting from s such that p ⊨ f s ⊨ A f if for every path p starting from s such that p ⊨ f

p ⊨ f for a state formula f if s ⊨ f where s is the first state of p p ⊨ X f if p¹ ⊨ f where p¹ is the suffix of p without its first state p ⊨ F f if $ k ≥ 0 such that p^k ⊨ f where p^k is the k^th suffix of p p ⊨ G f if " k ≥ 0, p^k ⊨ f

p ⊨ f₁ U f₂ if $ k ≥ 0 p^k ⊨ f₂ Ù " j < k p^j ⊨ f₁ p ⊨ f₁ R f₂ if " k ≥ 0 p^k ⊨ f₂ Ú $ j < k p^j ⊨ f₁

Duality: ¬ Ef = A ¬ f , ¬ Ff = G ¬ f , ¬ Xf = X ¬ f, ¬ (f₁ U f₂ )= ¬ f₁ R ¬ f₂

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Minimal Set of CTL* Operators

• Logical connectives: Ú

¬

• Path quantifier: E “exists”

• Temporal operators: X “next”

U “until”

Abbreviations:

Ff = true U f “finally”

Af = ¬ E ¬ f “always”

Gf = ¬ F ¬ f “globally”

f₁ R f₂ = ¬ ( ¬ f₁ U ¬ f₂) “release”

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CTL Fragment of CTL*

In CTL, each temporal operator must be preceded by a path quantifier Any CTL formula is thus a state formula

and can be identified to the set of states which satisfy it f ≃ {sÎS : s ⊨ f } [Emerson 90]

Basis of three operators: EX, EG, EU

• EF f = E(true U f)

• AX f = ¬ EX ¬ f

• AF f = ¬ EG ¬ f

• AG f = ¬ EF ¬ f

• Etc…

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LTL Fragment of CTL*

Linear Time Logic (LTL) formulae are of the form Af

where f contains no path quantifier, only temporal operators Basis of two operators: X, U

• The LTL formula A(FG f) is not expressible in CTL AF(AG f) is stronger, e.g. false on p ¬ p p AF(EG f) is weaker, e.g. true on p ¬ p

• The CTL formula EF(AG f) is not expressible in LTL

• LTL and CTL are strict fragments of CTL*

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Biological Properties in CTL (1/3)

About reachability:

• Can the cell produce some protein P? reachable(P)==EF(P)

• Can the cell produce P, Q and not R?

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Biological Properties in CTL (1/3)

About reachability:

• Can the cell produce P, Q and not R? reachable(P^Q^¬R)

• Can the cell always produce P?

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Biological Properties in CTL (1/3)

About reachability:

• Can the cell always produce P? AG(reachable(P)) About pathways:

• Can the cell reach a set s of (partially described) states while passing by another set of states s₂?

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Biological Properties in CTL (1/3)

About reachability:

• Can the cell reach a set s of (partially described) states while passing by another set of states s₂? EF(s₂^EFs)

• Is it possible to produce P without Q ?

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Biological Properties in CTL (1/3)

About reachability:

• Is it possible to produce P without Q ? E(¬Q U P)

• Is set of state s₂ a necessary checkpoint for reaching set of state s?

checkpoint(s₂,s)== ^¬E(¬s₂ U s)

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Biological Properties in CTL (1/3)

About reachability:

• Is it possible to produce P without Q ? E(¬Q U P)

• Is set of state s₂ a necessary checkpoint for reaching set of state s?

checkpoint(s₂,s)== ^¬E(¬s₂ U s)

• Is s₂ always a checkpoint for s? AG(^¬s -> checkpoint(s₂,s))

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Biological Properties in CTL (2/3)

About stability:

• Is a set of states s a stable state? stable(s)== AG(s)

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Biological Properties in CTL (2/3)

About stability:

• Is s a steady state (with possibility of escaping) ? steady(s)==EG(s)

• Can the cell reach a stable state s?

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Biological Properties in CTL (2/3)

About stability:

• Can the cell reach a stable state s? EF(stable(s)) alternance of path quantifiers EFAG f,

not in Linear Time Logic LTL (fragment without path quantifiers)

• Must the cell reach a stable state s?

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Biological Properties in CTL (2/3)

About stability:

• Must the cell reach a stable state s? AF(stable(s))

• What are the stable states?

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Biological Properties in CTL (2/3)

About stability:

• Must the cell reach a stable state s? AF(stable(s))

• What are the stable states? Not expressible in CTL.

needs to combine CTL with search [Chan 00, Calzone-Chabrier-Fages-Soliman 05].

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Biological Properties in CTL (3/3)

About durations:

• How long does it take for a molecule to become activated?

• In a given time, how many Cyclins A can be accumulated?

• What is the duration of a given cell cycle’s phase?

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Biological Properties in CTL (3/3)

About durations:

CTL operators abstract from durations. Time intervals can be modeled in FOL by adding numerical constraints for start times and durations.

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Biological Properties in CTL (3/3)

About durations:

CTL operators abstract from durations. Time intervals can be modeled in FOL by adding numerical constraints for start times and durations.

About oscillations:

• Can the system exhibit a cyclic behavior w.r.t. the presence of P ? oscil(P)== EG((F ¬P) ^ (F P))

temporal operators not preceded by a path operator: CTL* formula approximation in CTL ? oscil(P)== EG((EF ¬P) ^ (EF P))

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Oscillations in CTL ?

EG((EF ¬P) ^ (EF P))

• Necessary but not sufficient condition for oscillations:

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Oscillations in CTL ?

EG((EF ¬P) ^ (EF P))

• Necessary but not sufficient condition for oscillations without fairness:

P ¬ P

• also with weak fairness (no rule stays continuously fireable without being fired)

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Oscillations in CTL ?

EG((EF ¬P) ^ (EF P))

• Necessary but not sufficient condition for oscillations without fairness:

P ¬ P

• also with weak fairness (no rule stays continuously fireable without being fired)

P ¬ P

P,Q

• Needs strong fairness (no rule is infinitely often fireable without being fired)

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MAPK Signaling Cascade Model

RAF + RAFK <=> RAF-RAFK.

RAF-RAFK => RAFK + RAF~{p1}.

RAF~{p1} + RAFPH <=> RAF~{p1}-RAFPH.

RAF~{p1}-RAFPH => RAF + RAFPH.

MEK~$P + RAF~{p1} <=> MEK~$P-RAF~{p1}

where p2 not in $P.

MEK~{p1}-RAF~{p1} => MEK~{p1,p2} + RAF~{p1}.

MEK-RAF~{p1} => MEK~{p1} + RAF~{p1}.

MEKPH + MEK~{p1}~$P <=> MEK~{p1}~$P-MEKPH.

MEK~{p1}-MEKPH => MEK + MEKPH.

MEK~{p1,p2}-MEKPH => MEK~{p1} + MEKPH.

MAPK~$P + MEK~{p1,p2} <=> MAPK~$P-MEK~{p1,p2}

where p2 not in $P.

MAPKPH + MAPK~{p1}~$P <=> MAPK~{p1}~$P-MAPKPH.

MAPK~{p1}-MAPKPH => MAPK + MAPKPH.

MAPK~{p1,p2}-MAPKPH => MAPK~{p1} + MAPKPH.

MAPK-MEK~{p1,p2} => MAPK~{p1} + MEK~{p1,p2}.

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Temporal Logic Querying of MAPK Cascade

MEK~{p1} is a checkpoint for the cascade, i.e. producing MAPK~{p1,p2}

biocham: checkpoint(MEK~{p1} , MAPK~{p1,p2})

!E(!MEK~{p1} U MAPK~{p1,p2}) is True

The PH complexes are not checkpoints

biocham: checkpoint(MEK~{p1}-MEKPH , MAPK~{p1,p2})

!E(!MEK~{p1}-MEKPH U MAPK~{p1,p2}) is false Step 1 rule 15

Step 2 rule 1 RAF-RAFK present Step 3 rule 21 RAF~{p1} present Step 4 rule 5 MEK-RAF~{p1} present Step 5 rule 24 MEK~{p1} present Step 6 rule 7 MEK~{p1}-RAF~{p1} present Step 7 rule 23 MEK~{p1,p2} present Step 8 rule 13 MAPK-MEK~{p1,p2} present Step 9 rule 27 MAPK~{p1} present Step 10 rule 15 MAPK~{p1}-MEK~{p1,p2} present Step 11 rule 28 MAPK~{p1,p2} present

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Automatic Generation of CTL Properties

biocham: add_genCTL.

Ei(reachable(RAF))

Ei(reachable(!(RAF))) Ai(oscil(RAF))

Ei(steady(RAF))

Ai(AG(RAF->checkpoint(RAFK,!(RAF))))

…

Ei(reachable(MAPK~{p1,p2}))

Ei(reachable(!(MAPK~{p1,p2}))) Ai(oscil(MAPK~{p1,p2}))

Ei(steady(!(MAPK~{p1,p2})))

Ai(AG(MAPK~{p1,p2}->checkpoint(MAPKPH,!(MAPK~{p1,p2}))))

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Model Reduction preserving CTL Properties

biocham: reduce_model.

1: deleting RAF-RAFK=>RAF+RAFK

2: deleting RAFPH-RAF~{p1}=>RAFPH+RAF~{p1}

3: deleting MEK-RAF~{p1}=>MEK+RAF~{p1

4: deleting MEKPH-MEK~{p1}=>MEKPH+MEK~{p1}

6: deleting MAPKPH-MAPK~{p1}=>MAPKPH+MAPK~{p1}

7: deleting MEK~{p1}-RAF~{p1}=>MEK~{p1}+RAF~{p1}

8: deleting MEKPH-MEK~{p1,p2}=>MEKPH+MEK~{p1,p2}

9: deleting MAPK~{p1}-MEK~{p1,p2}=>MAPK~{p1}+MEK~{p1,p2}

10: deleting MAPKPH-MAPK~{p1,p2}=>MAPKPH+MAPK~{p1,p2}

The reverse reactions are not necessary to satisfy the CTL specification of reachability, checkpoint and oscillations.

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Tyson 91’s Model of the Cell Cycle

k1 for _=>Cyclin.

k2*[Cyclin] for Cyclin=>_.

k3*[Cyclin]*[Cdc2~{p1}] for

Cyclin+Cdc2~{p1}=>Cdc2-Cyclin~{p1,p2}.

k4p*[Cdc2-Cyclin~{p1,p2}] for

Cdc2-Cyclin~{p1,p2}=>Cdc2-Cyclin~{p1}.

k4*[Cdc2-Cyclin~{p1}]^2*[Cdc2-Cyclin~{p1,p2}] for

Cdc2-Cyclin~{p1,p2}=[Cdc2-Cyclin~{p1}]=>Cdc2-Cyclin~{p1}.

k5*[Cdc2-Cyclin~{p1}] for Cdc2-Cyclin~{p1}=>Cdc2-Cyclin~{p1,p2}.

k6*[Cdc2-Cyclin~{p1}] for Cdc2-Cyclin~{p1}=>Cyclin~{p1}+Cdc2.

k7*[Cyclin~{p1}] for Cyclin~{p1}=>_.

k8*[Cdc2] for Cdc2=>Cdc2~{p1}.

k9*[Cdc2~{p1}] for Cdc2~{p1}=>Cdc2.

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Automatic Generation of CTL Properties

Biocham: add_genCTL.

Ei(reachable(Cyclin)).

Ei(reachable(!(Cyclin))).

Ai(oscil(Cyclin)).

Ei(steady(!(Cyclin))).

…

Ei(reachable(Cyclin~{p1})).

Ei(reachable(!(Cyclin~{p1}))).

Ai(oscil(Cyclin~{p1})).

Ei(steady(!(Cyclin~{p1}))).

Ai(AG(!(Cyclin~{p1})->checkpoint(Cdc2-Cyclin~{p1},Cyclin~{p1}))).

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Model Reduction preserving CTL Properties

biocham: reduce_model.

1: deleting Cyclin=>_

2: deleting Cdc2-Cyclin~{p1,p2}=[Cdc2-Cyclin~{p1}]=>Cdc2-Cyclin~{p1}

3: deleting Cdc2-Cyclin~{p1}=>Cdc2-Cyclin~{p1,p2}

4: deleting Cdc2~{p1}=>Cdc2

Some degradation, phosphorylation, dephosphorylation and the autocatalysis reaction

are not necessary for the CTL specification

of reachability, checkpoints and pseudo-oscillations.

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Basic CTL Model-Checking Algorithm

Dynamic programming algorithm for computing, in a finite Kripke structure K, the set of states satisfying a CTL formula: {sÎK : s ⊨ f }.

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Basic CTL Model-Checking Algorithm

Represent K explicitly as a finite graph and iteratively label the nodes with the subformulas of f that are true in that node:

• Add f to the states satisfying f

• Add EF f (EX f) to

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Basic CTL Model-Checking Algorithm

• Add EF f (EX f) to the (immediate) predecessors of states labeled by f

• Add E(f1 U f2 ) to

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Basic CTL Model-Checking Algorithm

• Add E(f1 U f2 ) to the predecessor states of f2 while they satisfy f1

• Add EG f to

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Basic CTL Model-Checking Algorithm

• Add EG f to the states for which there exists a path leading to a non trivial strongly connected component of the subgraph of states satisfying f.

Space and time in

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Basic CTL Model-Checking Algorithm

Space and time in O(|K|*|f|), CTL model-checking is Ptime-complete

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Basic CTL Model-Checking Algorithm

Space and time in O(|K|*|f|), CTL model-checking is Ptime-complete

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Symbolic CTL Model-Checking Algorithm

• Represent sets of states as a boolean constraint c(V)

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Symbolic CTL Model-Checking Algorithm

• Represent the transition relation as a boolean constraint r(V,V’)

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Symbolic CTL Model-Checking Algorithm

• Represent the transition relation as a boolean constraint r(V,V’) Let

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Symbolic CTL Model-Checking Algorithm

• Represent the transition relation as a boolean constraint r(V,V’) Let

• Least fixed point operator µ(f)=

Ú

ⁱ^fⁱ^(false) greatest fixed point n(f)=

Ù

ⁱ^fⁱ^(true)

• Dynamic programming: for each subformula f of the formula to prove,

associate the constraint [f] which represents the set of states where it is true

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Exercise

Write down the transition constraint for P ¬ P r(P,P’)= {(1,1),(1,0),(0,0)}

Apply the algorithm to 𝜑 = EG((EF ¬P)^(EF P))

• Apply the algorithm to AG(P)

Sub-formulae 𝜓 of 𝜑 Constraint [𝜓] ~ set of states where 𝜓 is true

P P(x)

¬P ¬P(x)

EF P µZ.(P(x) Ú ex(Z)) = false Ú P(x) Ú $x’( r(x,x’)Ù(P(x’) Ú P(x’)) Ú …

= P(x)

EF ¬P µZ.(¬P Ú ex(Z)) = false Ú ¬P(x) Ú $x’( r(x,x’)Ù(¬P(x’) Ú¬P(x’)) Ú …

= ¬P(x) Ú true

= true

EF ¬P Ù EF P P(x)

EG(EF ¬P Ù EF P) nZ.(P Ú ex(Z)) = true Ù P(x) Ù $x’( r(x,x’)Ù(P(x’) ÚP(x’)) Ù… = P(x)

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Ordered Binary Decision Diagrams OBDD

Ordered Binary Decision Diagrams OBDD [Bryant 85] are decision graphs

• With fixed ordering of variables by levels

• And compressed in binary graphs with maximum sharing of common subtrees Example: (x⋁¬y)⋀(y⋁¬z)⋀(z⋁¬x) OBDD(x,y,z) :

(x⋁¬z)⋀(z⋁¬y)⋀(y⋁¬x) has the same OBDD(x,y,z) and is indeed equivalent OBDD provide canonical forms for Boolean formulas OBDD decide both SAT in NP and TAUT in co-NP

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Logical Paradigm for Systems Biology

Use of model-checking algorithms[Lincoln et al. 02] [Chabrier Fages 03] [Bernot et al. 04]…

Biological process model = State Transition System K Biological property = Temporal Logic Formula φ Model validation = model-checking K, s ⊨? φ Model reduction = model-checking K’?⊂K K’, s ⊨ φ Static experiment design = model-checking K, s? ⊨ φ Model functions = true formulae enumeration K, s ⊨ φ?

Model Inference, dyn. exp. design = constraint solving K?, s? ⊨ φ

Generalizations to quantitative temporal logics

• FO-LTL(R_lin) [Rizk, Batt, F, Soliman 09, Donze Maler 12] parameter search, robustness

• SAT modulo ODE [Gao Clarke 2012] formal verification on parameter range

• Mixed logical-analog specification K?, s? ⊨ reachable(stable(y ≈ ¹²

341² )

Computational Methods for Systems and Synthetic Biology