Formal Methods in Systems and Synthe2c Biology

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Formal Methods in

Systems and Synthe2c Biology

François Fages Inria Saclay EPI LIFEWARE

France

h>p://lifeware.inria.fr/

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Lifeware Group

Research Scientists

Grégory Batt (Inria researcher, Synthetic Biology PI) Sylvain Soliman (Inria researcher)

Associates

Pascal Hersen (CNRS MSC lab, Researcher) Denis Thieffry (ENS Paris, Professor)

Thierry Martinez (SED, Inria Paris) PostDoc

Chiara Fracassi (Inria, CNRS MSC lab) PhD students

François Bertaux (AMX, Ecole Polytechnique, with EPI Bang) Katherine Chiang (National Taiwan University)

Xavier Duportet (Inria, MIT, startup Pasteur Institute)

Steven Gay (Inria, Post doc Univ. Louvain la Neuve, Belgium) Jean-Baptiste Lugagne (Inria, CORDI-C, CNRS MSC lab)

Artemis Llamosi (CNRS MSC lab, Inria) Jonas Sénizergues (Inria)

Pauline Traynard (Inria, Ecole Polytechnique, with ENS) Engineers

François-Marie Floch (Inria ADT, Engineer)

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Context: Systems Biology

Gain system-level understanding of multi-scale biological processes in terms of their elementary interactions at the molecular level. [Kitano 1999]

Cell mitosis videos [Lodish et al. 03]

Follow-up of Human Genome Project (90s), beyond genomic data:

! Creation of protein-protein interaction databases, RNAs, …

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Roadmap of my 2 courses

Tuesday:

1.  Biochemical reaction systems, hierarchy of semantics 2.  Reaction rate functions

3.  Cell signaling reaction systems, e.g. MAPK as an analog-digital converter 4.  Analog/digital computation with biochemical reactions, e.g. GPAC

Thursday:

1.  Detecting model reduction by subgraph epimorphisms 2.  Boolean model-checking, e.g. Cell Cycle Control

3.  Parameter search in quantitative temporal logic FO-LTL(Rlin) 4.  Robustness and sensitivity analyses

Conclusion on some hot research topics

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Part I - Biochemical reac0ons

•  Binding, complexation: 𝐴+𝐵 →┴ 𝐶

𝑐𝑑𝑘1+𝑐𝑦𝑐𝐵 →┴ 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵

•  Unbinding, decomplexation: 𝐴→┴ 𝐵+𝐶

+ ↔┴

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Biochemical reac2ons

•  Unbinding, decomplexation: 𝐴 →┴ 𝐵+𝐶

•  Transformation, phosphorylation, transport: 𝐴 →┴ 𝐵 (𝐴+𝐸 →┴ 𝐶→┴𝐵+𝐸) 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵 →┴ 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵𝑝

→ ┴

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Biochemical reac2ons

•  Transformation, phosphorylation, transport: 𝐴 →┴ 𝐵 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵 →┴

𝑐𝑑𝑘1𝑐𝑦𝑐𝐵𝑝

•  Gene expression, synthesis: 𝐴 →┴ 𝐴+𝐵

𝐸2𝐹𝑎 →┴ 𝐸2𝐹𝑎+𝑅𝑁𝐴𝑐𝑦𝑐𝐴

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Biochemical reac2ons

•  Transformation, phosphorylation, transport: 𝐴 →┴ 𝐵 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵 →┴

𝑐𝑑𝑘1𝑐𝑦𝑐𝐵𝑝

•  Gene expression, synthesis: 𝐴 →┴ 𝐴+𝐵

𝐸2𝐹𝑎→┴ 𝐸2𝐹𝑎+𝑅𝑁𝐴𝑐𝑦𝑐𝐴

•  Degradation: 𝐴 →┴ _

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That’s it ?

•  Binding, complexation: 𝐴+𝐵 →𝑘.𝐴.𝐵┴ 𝐶

•  Unbinding, decomplexation: 𝐴 →𝑘.𝐴┴ 𝐵+𝐶

•  Transformation, phosphorylation, transport: 𝐴 →𝑣.𝐴/(𝑘+𝐴)┴ 𝐵

𝑐𝑑𝑘1𝑐𝑦𝑐𝐵 →┴ 𝑐𝑑𝑘1𝑐𝑦𝑐𝐵𝑝

•  Gene expression, synthesis: 𝐴 →𝑣.𝐴↑𝑛 /(𝑘+𝐴↑𝑛 )┴ 𝐴+𝐵

•  Degradation: 𝐴 →𝑘.𝐴┴ _

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Biochemical reac2on rates

•  Degradation: 𝐴 →𝑘.𝐴┴ _

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Biochemical reac2on rates

•  Transformation, phosphorylation, transport: 𝐴 →𝑣.𝐴/(𝑘+𝐴)┴ 𝐵 ⁽𝐴+𝐸 →┴

𝐶→┴𝐵+𝐸)

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Biochemical reac2on rates

•  Degradation: 𝐴 →𝑘.𝐴┴ _ or 𝐴 →𝑣.𝐴/(𝑘 +𝐴)┴_

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Continuous semantics: concentrations, continuous time evolution

Ordinary differential equations (ODE) 𝑑𝐴𝑖/𝑑𝑡 =∑𝑟=1↑𝑛▒ f𝑟 𝗑 δ𝑟(𝐴𝑖)↑  

Seman2cs of reac2ons ^A+B →𝒇 ( 𝑨 , 𝑩 ) ┴ C

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Stochastic semantics: numbers of molecules, probability and time of transition

Continuous Time Markov Chain (CTMC) A , B→𝒑(𝑺𝒊), 𝒕(𝑺𝒊)┴ C++, A--, B--

Seman2cs of reac2ons ^A+B →𝒇 ( 𝑨 , 𝑩 ) ┴ C

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Continuous Time Markov Chain (CTMC) A , B→𝒑(𝑺𝒊), 𝒕(𝑺𝒊)┴ C++, A--, B-- Petri net semantics: numbers of molecules A , B → C++, A--, B--

Multiset rewriting

CHAM [Berry Boudol 90] [Banatre Le Metayer 86]

Seman2cs of reac2ons ^A+B →𝒇 ( 𝑨 , 𝑩 ) ┴ C

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Continuous Time Markov Chain (CTMC) A , B→𝒑(𝑺𝒊), 𝒕(𝑺𝒊)┴ C++, A--, B-- Petri net semantics: numbers of molecules A , B → C++, A--, B--

Multiset rewriting

CHAM [Berry Boudol 90] [Banatre Le Metayer 86]

Boolean semantics: presence/absence A ∧ B → C ∧ A/¬A ∧ B/¬B

Asynchronous transition system

Seman2cs of reac2ons ^A+B →𝒇 ( 𝑨 , 𝑩 ) ┴ C

Porquerolles 2016 François Fages

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Abstrac2on Rela2onships

Stochastic traces"

Petri net traces"

abstract!

Boolean traces"

Theory of abstract Interpretation Abstractions as Galois connections

[Cousot Cousot POPL’77]

Thm. Galois connections between the syntactical, stochastic, Petri Net and Boolean trace semantics

[FF Soliman CMSB’06,TCS’08]

ODE traces"

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Abstrac2on Rela2onships

Stochastic traces"

Petri net traces"

abstract!

concrete!

Boolean traces"

Theory of abstract Interpretation Abstractions as Galois connections

[Cousot Cousot POPL’77]

Thm. Galois connections between the syntactical, stochastic, Petri Net and Boolean trace semantics

[FF Soliman CMSB’06,TCS’08]

Reactions"

ODE traces"

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Abstrac2on Rela2onships

Stochastic traces"

ODE traces"

Petri net traces"

abstract!

Boolean traces"

Thm. Under large number conditions the ODE semantics approximates the mean stochastic behavior

[Gillespie 71]"

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Abstrac2on Rela2onships

Stochastic traces"

ODE traces"

Petri net traces"

abstract!

concrete!

Boolean traces"

Thm. Under large number conditions the ODE semantics approximates the mean stochastic behavior

[Gillespie 71]"

Reactions"

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Hybrid Models and Hybrid Simula2ons

Stochastic traces"

ODE traces"

Petri net traces"

abstract!

Boolean traces"

•  Hybrid Boolean-continuous models (hybrid automata)

Boolean gene expression + continuous protein activation

•  Hybrid stochastic-continuous

models (CTMC+ODE)

Stochastic gene expression + continuous protein activation Specification of hybrid simulators

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Conclusion part I:

Hierarchy of Seman2cs/Formalisms

•  The different interpretations of a reaction system can be related by formal abstraction relationships

•  The choice of an interpretation should be guided by the question at

hand: the more abstract the better as long as it is sufficient to answer the question

–  Boolean attractors and model-checking if no quantitative questions (boolean attractors, reachability, path constraints,,…)

–  Petri net invariants if no continuous time dynamics (structural conservation laws P-invariants, accumulation of metabolites, extreme fluxes T-invariants,…) –  Differential model if quantitative questions but no stochastic effects (possibly

extrinsic noise due to different kinetic parameters between cells)

–  Stochastic model if quantitative questions with high cell variability not due to extrinsic noise (i.e. intrinsic noise)

–  Hybrid model if different abstractions in different regimes (e.g. different timescales)

Porquerolles 2016 22

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Part II - Reac0on Rate Func0ons

Mass action law kinetics k*[A] for A => B

k*[A]*[B] for A+B => C

k*[A]^m*[B]^n for m*A + n*B => R Michaelis-Menten kinetics

Vm*[A]/(Km+[A]) for A => B Hill kinetics

Vm*[A]^n/(Km+[A]^n) for A => B

Guldberg and Waage, 1864

Henry 1903,

Michaelis Menten 1913

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Single Enzyma2c Reac2on

•  An enzyme E binds to a substrate S to catalyze the formation of product P:

E+S !^c1 C !^c3E+P E+S "^c2 C

•  Mass action law kinetics ODE:

dE/dt =

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Single Enzyma2c Reac2on

E+S !^c1 C !^c3E+P E+S "^c2 C

dE/dt = -c₁ES+(c₂+c₃)C dS/dt =

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Single Enzyma2c Reac2on

E+S !^c1 C !^c3E+P E+S "^c2 C

dE/dt = -c₁ES+(c₂+c₃)C dS/dt = -c₁ES+c₂C

dC/dt =

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Single Enzyma2c Reac2on

E+S !^c1 C !^c3E+P E+S "^c2 C

dC/dt = c₁ES-(c₃+c₂)C dP/dt =

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Single Enzyma2c Reac2on

E+S !^c1 C !^c3E+P E+S "^c2 C

dC/dt = c₁ES-(c₃+c₂)C dP/dt = c₃C

•  Conservation laws i.e. (M1,…,Mn) s.t. Σⁿ_i=1dMi/dt = 0 so that Σⁿ_i=1Mi = constant):

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Single Enzyma2c Reac2on

E+S !^c1 C !^c3E+P E+S "^c2 C

dC/dt = c₁ES-(c₃+c₂)C dP/dt = c₃C

•  Conservation laws i.e. (M1,…,Mn) s.t. Σⁿ_i=1dMi/dt = 0 so that Σⁿ_i=1Mi = constant):

E+C=constant, S+C+P=constant,

•  Assuming C₀=P₀=0, we get E=E₀-C and P=S₀-S-C

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Exercise 1

•  Connect to BIOCHAM-web on http://lifeware.inria.fr/biocham/online

•  Load_biocham(Trypsin.bc) from MPRI directory: S+E <=> C => E+P

•  Numerical_simulation(1000) and over 0.1 time unit (seconds)

•  Plot the results over time and between species plot(C,S), plot(P,S), …

•  list_kinetics

•  search_conservations

1.  Compare the results to TrypsinQSSA.bc for the simplified reaction S=>P and mathematically justify the rate function

2.  Do the same for TrypsinQE.bc

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Slow/Fast Time Scales

Hydrolysis of benzoyl-L-arginine ethyl ester by trypsin

present(E,1e-8). present(S,1e-5). absent(C). absent(P). E << S

parameter(c1,4e6). parameter(c2,25). parameter(c3,15). c1 >> c2, c3 c1*[E]*[S] for E+S => C. c2*[C] for C => E+S . c3*[C] for C => E+P.

Complex forma2on 5e-9 in 0.1s Product forma2on 1e-5 in 400s

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Michaelis Menten Reduc2on

parameter(E0,1e-8). macro(Vm, c2*E0). macro(Km, (c2+c3)/c1).

Vm*[S]/(Km+[S]) for S ==> P.

Same Product formation in 400 s Oversimplified early trajectory in 0.1s for P Same stable state

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Assume quasi-steady state dC/dt=0=c₁E₀S-(c₁S+c₂+c₃)C Then C = c₁E₀S/(c₁S+c₂+c₃) = E₀S/(K_m+S)

where K_m=(c₂+c₃)/c₁

Quasi-Steady State Approxima2on

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We get dP/dt = -dS/dt = V_mS / (K_m+S)

where V_m= c₃E₀maximum velocity at saturing substrate concentration

Michaelis-Menten quasi-steady state kinetics: V_mS /(K_m+S) for S => P

Quasi-Steady State Approxima2on (QSSA)

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substrate concentration with half maximum velocity We get dP/dt = -dS/dt = V_mS / (K_m+S)

Quasi-Steady State Approxima2on (QSSA)

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substrate concentration with half maximum velocity We get dP/dt = -dS/dt = V_mS / (K_m+S)

C is eliminated, E is supposed constant and acting on only one substrate

However, E is sometimes re-injected as a variable: c₃ES /(K_m+S) for S+E => E+P

Quasi-Steady State Approxima2on (QSSA)

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Times Scales

“Time taken for a significant change”

Lower bound:

Time scale of f(t) ≈ 𝑓𝑚𝑎𝑥 −𝑓𝑚𝑖𝑛/|𝑑𝑓/𝑑𝑡 |𝑚𝑎𝑥  Time scale of C(t) ≈ 5e−8/4e−8/0.01  = 1/80  Formally, supposing S(t)=S₀ we get

dC/dt = c₁(E₀+C₀)S₀- (c₁S₀+c₂+c₃) C

C(t) = (C₀-¯C ) e^-kt + ¯C  where k= c₁S₀+c₂+c₃ and ¯C  = (E₀+C₀)S₀/(K_m+S) Taking k^-1 as time scale of e^-kt(i.e. decrease of e^-1≈1/3 in k^-1 time)

gives in the Trypsin example 1/(10^-5.4.10⁶+25+15)=1/80 =0.0125s

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Validity Condi2on of QSSA (Segel 84)

When the time scale of S is much longer than the time scale of C…

S varies from S₀ to 0

| dS/dt = -c₁(E₀-C)S+ c₂C | is maximal when S=S₀ , C=C₀ Time scale of S ≈ 1/(c₁E₀)

(in the Trypsin example 1/(4.10⁶. 10^-8)≈25s)

Validity condition: c₁E₀ << c₁S₀+c₂+c₃ (e.g. Trypsin: 4.10^-2<<80) i.e. QSSA valid when E₀ << S₀+K_m

in particular when E₀<<S₀ [Briggs and Haldane 1925]

Much better justification under way [Radulescu et al. …] :

•  Find quasi steady states by tropical (min,+) equilibrations (fast reactions)

•  Verify Tikhonov approximation condition in each simplified regime (hybrid automaton of 3 regimes for Michaelis-Menten ?)

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Enzyma2c Compe22ve Inhibi2on

present(En,1e-8). present(S,1e-5). present(I,1e-5). parameter(k3,5e5).

parameter(k1,4e6). parameter(km1,25). parameter(k2,15).

(k1*[E]*[S],km1*[C]) for E+S <=> C. k2*[C] for C => E+P.

k3*[C]*[I] for C+I => CI.

Complex formation 4e-9 in 0.04s Product formation 3e-8 in 3s

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Isosteric Inhibi2on

present(En,1e-8). present(S,1e-5). present(I,1e-5). parameter(k3,5e5).

parameter(k1,4e6). parameter(km1,25). parameter(k2,15).

(k1*[E]*[S],km1*[C]) for E+S <=> C. k2*[C] for C => E+P.

k3*[E]*[I] for E+I => EI.

Complex formation 2.5e-9 in 0.4s Product formation 2.5e-9 in 1000s

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Allosteric Inhibi2on/Ac2va2on

(i*[E]*[I],im*[EI]) for E+I <=> EI. parameter(i,1e7). parameter(im,10).

(i1*[EI]*[S],im1*[CI]) for EI+S <=> CI. parameter(i1,5e6). parameter(im1,5).

i2*[CI] for CI => EI+P. parameter(i2,2).

Complex formation 2e-9 in 0.4s Product formation 1e-5 in 1000s

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Hill Kine2cs (several binding sites for same ligands)

E.g. Dimer enzyme with two promoters:

E+S !^2*c1 C₁ !^c3E+P C₁+S !^c’1 C2 !^2*c’3C₁+P E+S "^c2 C₁ C₁+S "^2*c’2 C2

Let K_m=(c₂+c₃)/c₁and K’_m=(c’₂+c’₃)/c’₁

Non-cooperative if K’_m=K_m Michaelis-Menten rate: V_mS / (K_m+S) where V_m=2*c₃*E₀ hyperbolic velocity vs substrate concentration

Cooperative if K’_m>K_m Hill equation rate: V_mS² / (K_m*K’_m+S²) where V_m=2*c₃*E₀ sigmoid velocity vs substrate concentration

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Conclusion Part II on Reac2on Rate Func2ons

•  Michaelis-Menten kinetics, Hill kinetics of order n and more general kinetics come from reductions of elementary reaction systems with mass action law kinetics

•  and more precisely from projections on the slow dynamics variables

(eliminate fast dynamics species E, C acting as slaves of slow species S)

•  The slow/fast separation of species may be different in different regimes of the system. This can be modeled by a hybrid automaton of piece-wise

reduced systems.

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Part III – Signaling Reac0on Systems How do Cells Sense their Environment ?

Extra cellular ligands:

hormones: insulin, adrenaline, steroids, EGF, neighbor cells: Delta membrane protein

nutriments

light, pressure, … Membrane receptors:

•  Multiple transmembrane α-helices receptors can induce a change of conformation inside

e.g. 7-TMR G protein-coupled, TGFβ, Notch

•  Single α- helix receptors cannot…

e.g. Tyrosine kinases TRK

Multi- α-helix receptors transmit change conformation from outside to inside

But how can a single α-helix receptor transmit the signal from outside to inside ?

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Receptor Tyrosine Kinase RTK Ac2va2on by Complexa2on

One RTK receptor with its ligand is inac2ve

L::exterior + R::membrane <=> (L-R)::membrane.

but two RTK receptors with their ligands can bind together on the

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Cell Signaling from Membrane Receptors to Gene Ac2va2on

5 MAPK (mitogen ac2vated kinase) signaling pathways in budding yeast (Saccharomyces Cerevisiae)

with 5 cell responses

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External Control of Gene Expression through Hog1

Perception – learning – action loop on a microfluidic device:

1.  Microscope, image analysis (cell tracking or population) 2.  Model calibration (kinetic parameter optimization)

3.  Osmotic pressure control (parameter optimization)

[Uhlendorf … Batt Hersen PNAS 109(35) 2012]

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MAPK Signaling Cascade Input/output

•  Input:

RAS activated by the RTK receptor activates RAF (MAPKKK kinase) RAS-GTP + RAF-P14-3-3 =>

RAS-GDP + RAF + P14-3-3

•  Output:

phosphorylated MAPK moves to the nucleus and

phosphorylates a transcription factor which activates some gene expression RAF + … => …

… => MAPK~{T183,Y185}

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MAPK Signaling Cascade Structure

Three levels of double phosphorylations

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S2mulus/Response Curve

similar to Coopera2ve Hill kine2cs of Order 4-5

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Conclusion Part III:

the MAPK cascade is an analog/digital converter

•  MAPK signaling cascade act as a switch by converting graded inputs into switch-like outputs.

•  The stimulus/response curves become progressively steeper (ultrasensitive) as the cascade is descended.

•  No response to small stimuli (noise filtering).

•  The MAPK cascade implements the Hill kinetics of order 4-5 as with a cooperative allosteric enzyme with a cascade of monomeric or dimeric

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Cell-to-Cell Signaling

Xenopus embryonic skin [Gosh Tomlin 01] Pepper-salt effect in a grid of cells

Transmembrane proteins Delta (ligand) and Notch (receptor) are degraded.

Notch production is triggered by high Delta levels in neigboring cells

(if [D::c21]+[D::c23]+[D::c12]+[D::c32]<0.2 then 0 else ka,MA(kd)) for _<=>N::c22.

Delta production is triggered by low Notch concentration in the same cell

(if [N::c22] > 0.5 then 0 else ka,MA(kd)) for _ <=> D::c22.

At steady state, a cell has either the Delta or the Notch phenotype

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Conclusion Part III on Cell-Cell Signaling

•  Cell-to-cell signaling uses transmembrane proteins as ligands

•  Cell-to-cell signaling can be modeled with reactions between neighbors over a grid of cell locations

•  Delta-Notch signaling is reminiscent to the distributed approximate majority algorithm

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Part IV – Analog/Digital Computa0on with Biochemical Reac0ons

•  MAPK signaling = Analog / Digital converter

•  Beyond describing natural circuits, how to understand them ?

•  Why natural circuits have those structures and not others ?

•  What other functions do they have ?

•  Or what other functions could they get by evolution ? Or could loose ?

!  How to implement analog circuits with biochemical reactions ?

!  How to program with biochemical reactions ?

!  First, how to implement an ODE system with a reaction system ?

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Transcrip2on of ODE Models with Reac2ons

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Infering Reac2on Systems from ODEs

[FF Gay Soliman 15 TCS]

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Strict Kine2cs and Posi2ve Systems

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Reac2on System Inference Algorithm

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Inference of Molecules Eliminated by Invariants

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Compu2ng Real Func2ons with Reac2ons

To compute y=f(X) consider

1.  The ODE dy/dt = k*f(X) – k*y at steady state we will have f(X)=y

2.  Biocham load_ODE reactions: k*f(X) for X => X+y and k*y for y =>_

•  Multiplication z=x*y

x*y for x+y => x+y+z z for z => _

•  Addition z=x+y

x for x => x+z y for y => y+z z for z => _

•  Integral z = ∫ x dt

x for x => x+z

•  Square root z=√⁠𝑥  i.e. z²=x, implement dz/dt=x-z²

x for x => x+z z² for z+z => _

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Logical Precondi2ons on Reac2ons?

• Conjunction

• Disjunction

• Negation

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C Compiler into Reac2ons

[Jiang et al 2012, 2013]

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C Compiler into Reac2ons

[Jiang et al 2012, 2013]

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Computable Real Number

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Computable Func2on

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General Purpose Analog Computer [Shannon 41]

Multiplication z=x*y x*y for x+y => x+y+z z for z => _

Integral z = ∫ x dt x for x => x+z Addition z=x+y

x for x => x+z y for y => y+z

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GPAC Generated Func2on Graph

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Church-Turing Thesis for Analog Computa2on

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Purely Analog Characteriza2on of P2me !

[Pouly Bournez Graca 2015]

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Cosine Func2on Graph Genera2on

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Cosine Func2on Graph Genera2on

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Linear Time Invariant Systems

Porquerolles 2016

General Purpose Analog Computer Transfer Functions

Definitions

Definition: Laplace transform

8s 2 R₊, Lf (s) =

Z ₊₁

0

e ^stf (t)dt

! Laplace transform of linear and time invariant systems are rational fractions of R(s).

Definition: Transfer function

The transfer function of a LTI system with input U and output Y is the rational fraction H such that Y(s) = H(s)U(s).

It is said to be strictly proper when its degree is negative.

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Transfer Func2on of Reac2on Impl.

[Jiang et al 2015]

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Compiling Transfer Func2ons into Reac2ons

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Compiling Transfer Func2ons into Reac2ons

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Final Summing Block

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Conclusion Part IV on Analog/Digital

Computa2on with Real Biochemical Reac2ons

•  Biosensor design and implementation in non-living vesicles

[Franck Molina lab CNRS Sys2Diag Montpellier]

•  Implementation of linear I/O systems, PI controllers and simple programs ?

!  Issue of approximation and compositionality

!  Issue of reaction code optimization (number of species and reactions)

•  Comparison of synthetic programs with natural programs ?

! Issues of multiple functions, evolution history and capacity (PhD thesis subject)

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Part V – Detec2ng Model Reduc2ons

See slides P4.pdf

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Part VI – Dynamical Analysis of Reac2on Systems

A Logical Paradigm for Systems Biology:

Biological process model = State Transition System K Biological property = Temporal Logic Formula φ Model validation = Model-checking K, s ⊨? φ Experiment design = Model-checking K, s? ⊨ φ

Model prediction = Model-checking K, s ⊨ φ? Model Inference= Constraint solving K?, s? ⊨ φ

[Lincoln et al. PSB’02] [Chabrier Fages CMSB’03] [Bernot et al. TCS’04] …

Reaction systems: BIOCHAM Temporal Properties:

- Boolean - simulation - qualitative CTL

- Differential - query evaluation - quantitative FO-LTL(R_lin) - Stochastic - reaction learning

- parameter search

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Cell Cycle: ^G0 ^! ^G1 ^! ^Synthesis ^! ^G2 ^! ^Mitosis

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 in Biocham 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}

!  732 reactions, 165 proteins and genes, 532 variables

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

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Symbolic Representa2on by Constraints

Represent sets of states by Boolean constraints True : full set of 2⁵³² states,

False : empty set,

x : set of 2⁵³¹ states where x is present,

x∧¬y : set of 2⁵³⁰ states where x is present and y is absent

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Computa2on 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) ∧ (EF CycA))! 31.8"

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Kripke Seman2cs of CTL*

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

s ⊨ φ if φ is a simple proposition true in s

s ⊨ E φ if there is a path π starting from s such that π ⊨ φ s ⊨ A φ if for every path π starting from s such that π ⊨ φ

π ⊨ φ for a state formula φ if s⊨ φ where s is the first state of π π ⊨ X φ if π¹ ⊨ φ where π¹ is the suffix of π without its first state π ⊨ F φ if ∃ k ≥ 0 such that π^k ⊨ φ where π^k is the k^th suffix of π π ⊨ G φ if ∀ k ≥ 0, π^k ⊨ φ

π ⊨ φ₁ U φ₂ if ∃ k ≥ 0 π^k ⊨ φ₂ ∧ ∀ j < k π^j ⊨ φ₁ π ⊨ φ₁ R φ₂ if ∀ k ≥ 0 π^k ⊨ φ₂ ∨ ∃ j < k π^j ⊨ φ₁

Duality: ¬ Eφ = A ¬ φ , ¬ Fφ = G ¬ φ , ¬ Xφ = X ¬ φ, ¬ (φ₁ U φ₂ )= ¬ φ₁ R ¬ φ₂

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Minimal Set of CTL* Operators

•  Logical connectives: ∨

¬

•  Path quantifier: E “exists”

•  Temporal operators: X “next”

U “un2l”

Abbreviations:

Fφ = true U φ “ﬁnally”

Aφ = ¬ E ¬ φ “always”

Gφ = ¬ F ¬ φ “globally”

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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 φ ≃ {s∈S : s ⊨ φ } [Emerson 90]

Basis of three operators: EX, EG, EU

•  EF φ = E(true U φ)

•  AX φ = ¬ EX ¬ φ

•  AF φ = ¬ EG ¬ φ

•  AG φ = ¬ EF ¬ φ

•  Etc…

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LTL Fragment of CTL*

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

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

•  The LTL formula A(FG φ) is not expressible in CTL

(AF(AG φ) is stronger, AF(EG φ) is weaker)

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

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Biological Proper2es 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 Proper2es 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 Proper2es 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 Proper2es 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 Proper2es 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(¬s₂U s)

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Biological Proper2es 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(¬s₂U s)

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Biological Proper2es in CTL (2/3)

About stability:

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

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Biological Proper2es 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 Proper2es in CTL (2/3)

About stability:

•  Can the cell reach a stable state s? EF(stable(s))

alternance of path quantifiers EFAG φ,

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

•  Must the cell reach a stable state s?

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Biological Proper2es 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 Proper2es 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, Fages-Rizk 07].

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Biological Proper2es 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?

Formal Methods in Systems and Synthe2c Biology