MasterMPRIC2-19ComputationalMethodsinSystemsandSyntheticBiology InﬂuenceSystemsvsReactionSystems

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Reaction/Influence systems Expressive Power Positive/Negative Semantics TSCC Algorithm Examples

Influence Systems vs Reaction Systems

Master MPRI C2-19

Computational Methods in Systems and Synthetic Biology

François Fages

Inria Saclay - Ile de France, Lifeware team, France

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Example: Birth-Death Processes

Lotka–Volterra model of a proliferating preyA and a predatorB Reaction system with kinetics: Influence system with forces:

k1 * A * B for A + B = >2* B . k2 * A for A = >2* A . k3 * B for B = > _ .

k1 * A * B for A , B - < A . k1 * A * B for A , B - > B . k2 * A for A - > A . k3 * B for B - < B .

same ODEs:

dA/dt=−k1∗A∗B+k2∗A dB/dt=k1∗A∗B−k3∗B

What is the expressive power of influence systems compared to reaction systems ?

Are there differences in their respective

Boolean / Petri Net / Stochastic / Differential semantics ? Are they interpreted differently in software tools ?

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Example: Birth-Death Processes

Lotka–Volterra model of a proliferating preyA and a predatorB Reaction system with kinetics: Influence system with forces:

k1 * A * B for A + B = >2* B . k2 * A for A = >2* A . k3 * B for B = > _ .

k1 * A * B for A , B - < A . k1 * A * B for A , B - > B . k2 * A for A - > A . k3 * B for B - < B .

same ODEs:

dA/dt=−k1∗A∗B+k2∗A dB/dt=k1∗A∗B−k3∗B

What is the expressive power of influence systems compared to reaction systems ?

Are there differences in their respective

Boolean / Petri Net / Stochastic / Differential semantics ? Are they interpreted differently in software tools ?

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Outline

1 Well-formed Reaction and Influence Systems

2 Compared Expressive Power

3 Boolean Semantics without/with Negation in Conditions

4 Enumeration Algorithm for Boolean Attractors

5 Examples

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Well-formed Reaction Systems

Syntax distinguishing inhibitors from reactants (modifiers in SBML) The reactants increase the reaction rate, the inhibitors decrease it.

Definition

A reaction(R,M,P,f) with reactants R, inhibitors M, productsP, rate functionf, also noted R/M ⇒^f P, over molecular species {x₁, . . . ,x_s}is well formedif the following conditions hold:

1 f(x1, . . . ,x_s) is partially differentiable, non-negative onR^s+;

2 xi ∈R if and only if ∂f/∂xi(~x)>0for some ~x∈R^s+;

3 xi ∈M if and only if∂f/∂xi(~x)<0 for some~x ∈R^s₊.

4 x_i ∈R implies f(x₁, . . .x_s) =0 wheneverx_i =0 (positivity) Proposition (positivity)

Condition 4 ensures that the concentrations remain positive.

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Well-formed Reaction Systems

Syntax distinguishing inhibitors from reactants (modifiers in SBML) The reactants increase the reaction rate, the inhibitors decrease it.

Definition

A reaction(R,M,P,f) with reactantsR, inhibitorsM, productsP, rate functionf, also noted R/M ⇒^f P, over molecular species {x₁, . . . ,x_s}is well formedif the following conditions hold:

1 f(x1, . . . ,x_s) is partially differentiable, non-negative onR^s+;

2 x_i ∈R if and only if ∂f/∂x_i(~x)>0for some ~x∈R^s+;

3 xi ∈M if and only if∂f/∂xi(~x)<0 for some~x ∈R^s₊.

4 xi ∈R implies f(x1, . . .xs) =0 wheneverxi =0 (positivity) Proposition (positivity)

Condition 4 ensures that the concentrations remain positive.

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Hierarchy of Semantics {R

_i

/M

_i

⇒

^fⁱ

P

_i

}

_i∈I

Differential semantics: ^dx_dt^j =P

i(P_i(j)−R_i(j))×f_i Sustained oscillations between the numbers of preys and predators

Stochastic semantics: ~x −→^f_Sⁱ ~x⁰ if~x ≥Ri, ~x⁰ =~x−Ri+Pi

Petri Net semantics: ~x −→_D ~x⁰ if~x≥Ri, ~x⁰ =~x−Ri +Pi

Positive Boolean semantics: ~x−→_B ~x⁰ if~x ⊇Set(R_i)

~x⁰ =~x\C ∪Set(P_i) for all C ∈ P(Set(R_i)).

Related by approximation[Gillespie 77]or abstract interpretation Galois connections[F- Soliman 08 TCS]

Boolean / Petri Net / Stochastic semantics with negation: add the condition~x∩Mi =∅ to the transitions.

Stochastic semantics no longer related to differential semantics Petri Net semantics reachability becomes undecidable

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Hierarchy of Semantics {R

_i

/M

_i

⇒

^fⁱ

P

_i

}

_i∈I

i(P_i(j)−R_i(j))×f_i

Almost sure extinction of the prey and the predator :-(

Petri Net semantics: ~x −→_D ~x⁰ if~x≥Ri, ~x⁰ =~x−Ri +Pi

~

x⁰ =~x\C ∪Set(P_i) for all C ∈ P(Set(R_i)).

Boolean / Petri Net / Stochastic semantics with negation: add the condition~x∩M_i =∅ to the transitions.

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Hierarchy of Semantics {R

_i

/M

_i

⇒

^fⁱ

P

_i

}

_i∈I

Petri Net semantics: ~x −→_D ~x⁰ if~x≥Ri, ~x⁰ =~x−Ri+Pi

~

biocham: generateCTL.

reachable(not P) reachable(not R) reachable(steady(P)) reachable(stable(R)) reachable(stable(not P)) reachable(stable(not R)) checkpoint(R,not P) checkpoint(P,not R)

no oscillation (absence=extinction)

Boolean / Petri Net / Stochastic semantics with negation: add the condition~x∩Mi =∅ to the transitions.

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Hierarchy of Semantics {R

_i

/M

_i

⇒

^fⁱ

P

_i

}

_i∈I

Petri Net semantics: ~x −→_D ~x⁰ if~x≥Ri, ~x⁰ =~x−Ri+Pi

~

If a behavior is not possible in the Boolean semantics it is not possible in the PN and stochastic semantics for any kinetics.

Boolean / Petri Net / Stochastic semantics with negation: add the condition~x∩M_i =∅ to the transitions.

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Hierarchy of Semantics {R

_i

/M

_i

⇒

^fⁱ

P

_i

}

_i∈I

i(Pi(j)−Ri(j))×fi

Petri Net semantics: ~x −→_D ~x⁰ if~x≥R_i, ~x⁰ =~x−R_i+P_i Positive Boolean semantics: ~x−→_B ~x⁰ if~x ⊇Set(R_i)

~x⁰ =~x\C ∪Set(P_i) for all C ∈ P(Set(R_i)).

Boolean / Petri Net / Stochastic semantics with negation:

add the condition~x∩M_i =∅to the transitions.

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Influence Systems with Forces P

_i

/N

_i

→

^fⁱ

t

_i

, P

_j

/N

_j

−

fj

t

_j

Definition

An influence(P,N,t, σ,f)with sources P, inhibitorsM, targett, signσ and forcef, also noted P/N →^f t,P/N−

f t, over molecular species{x₁, . . . ,xs}is well formedif :

1 f(x1, . . . ,xs) is a partially differentiable function, non-negative on R^s+;

2 xi ∈P if and only if σ= +(resp. −) and∂f/∂xi(~x)>0 (resp. <0) for some value~x∈R^s+;

3 xi ∈N if and only if σ= + (resp.−) and ∂f/∂xi(~x)<0 (resp. >0) for some value~x∈R^s+;

4 t ∈P ifσ=−.

5 xi ∈P implies f(x1, . . . ,xs) =0 wheneverxi =0 (positivity)

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Hierarchy of Semantics {(P

_i

, N

_i

, t

_i

σ

_i

, f

_i

)}

_i_∈I

Differential semantics: ^dx_dt^k =P

(Pi,Ni,xk,σi,fi)σ_if_i

Stochastic semantics: ~x −→^f_Sⁱ^,tⁱ ~x⁰ if~x ≥P_i, ~x⁰=~x σ_i A_i Petri Net semantics: ~x −→_D ~x⁰ if~x≥P_i, ~x⁰ =~x σ_i A_i Positive boolean semantics: ~x−→_B ~x⁰if~x≥P_i, ~x⁰ =~x σ_i A_i Boolean/PN/stochastic semantics with negation in the enabling conditions: add the condition~x∩N_i =∅.

The positive semantics traces over-approximate the negative semantics traces.

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Negative Boolean Semantics à la Thomas

Necessary conditions for multistability and oscillations[Thomas 73 JTB, Rémy Ruet Thieffry 08 AAM, Ruet 16 MFCS]

Functional semantics: ~x⁰ defined by a functionφ(~x), not a relation. No true non-determinism.

Asynchronous semantics by interleaving.

A→^f B, A−

g B cannot be represented: no self-loop, all steady states are (stable) terminal states.

Even more striking in Thomas’s multilevel setting:

no PN transitions from (1,1)−→_PN (1,0) and to(1,2) (no logical parameter forB).

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Expressive Power I/III

Theorem

Any (well-formed) influence system with forces can be represented by a (well-formed) reaction system with kinetics, with the same boolean, discrete, stochastic and differential semantics.

Proof.

Just represent a positive influencef forP/N →t by a catalytic synthesis reactionf forP/N ⇒P+t.

and a negative influencef forP/N−t,

by an active degradation reactionf forP+t/N⇒P.

The converse does not hold, e.g.C ⇒A+B has a

Boolean/PN/stochasticnon-unitary transition (0,0,1)−→(1,1,0)

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Expressive Power I/III

Theorem

Any (well-formed) influence system with forces can be represented by a (well-formed) reaction system with kinetics, with the same boolean, discrete, stochastic and differential semantics.

Proof.

Just represent a positive influencef forP/N →t by a catalytic synthesis reactionf forP/N ⇒P+t.

and a negative influencef forP/N−t,

by an active degradation reactionf forP+t/N⇒P. The converse does not hold, e.g.C ⇒A+B has a

Boolean/PN/stochasticnon-unitary transition (0,0,1)−→(1,1,0)

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Expressive Power II/III

Theorem

Under thedifferential semantics, (well-formed) influence systems and reaction systems have thesame expressive power.

Proof.

For each reactionR/M ⇒^f P, consider the influences:

f ∗(P(i)−R(i))for Set(R)/Set(M)→xi whenP(i)−R(i)>0

f ∗(R(i)−P(i))for Set(R)/Set(M)−x_i when P(i)−R(i)<0 The differential semantics collect the same termsf ∗(P_i−R_i).

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Expressive Power III/III

Proposition

Any unitary boolean transition system(resp. without negative conditions) can be represented by an influence system under the negative (resp. positive) boolean semantics.

Proof.

Since a unitary boolean transitions −→_BN s⁰ changes at most one species, sayx_i, from s tos⁰, it can be represented by the influence ({x |s(x) =1}, {x |s(x) =0},x_i,(x_i =1)).

Corollary

The asynchronous boolean transition systems definable in Thomas’s setting are theunitary boolean transitions without self-loops.

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Properties of Positive Boolean Semantics I/III

Proposition (Monotonicity)

In the positive boolean semantics of an influence system

v1 ≤v2, v1 −→v₁⁰ imply∃v₂⁰ ≥v₁⁰ s.t. v2 −→v₂⁰ v1

v2

v₁⁰

v₂⁰

≤ ≤

Proof.

Since there is no negation in the enabling conditions, any influence that is enabled inv₁ is also enabled inv₂.

Boolean analogue of monotonic ODE systems[Angeli Sontag 03]

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Properties of Positive Boolean Semantics II/III

Proposition (Greatest element)

Any attractor(i.e. TSCC of the state transition graph) of a positive influence systemhas a greatest element.

Proof.

By contradiction with incomparable maximal elementsv1,v2∈C, considering a path fromv₁ to v₂. It includes a transition v₃−→v₄ withv3 ≤v1 andv4 6≤v2. By previous Prop. we get v1 −→v₁⁰ with v4 ≤v₁⁰ sinceC is terminal. Nowv1 is comparable tov₁⁰ since direct influence which leads to contradiction.

Corollary

To enumerate the attractors of a positive influence system, it is enough to check the SCCs of states that haveno strictly increasing transition(TSCC greatest element).

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Properties of Positive Boolean Semantics III/III

Proposition

Given an influence system, there isat least one TSCC of its negative semantics in each TSCC of its positive semantics.

Proof.

The positive semantics only adds transitions by enabling more influences, it can therefore only merge TSCCs.

To compute the attractors of an influence system under the negative semanticsà laThomas

1 enumerate first the greatest elements of the TSCCs of the positive semantics

2 then look for attractors in the negative semantics.

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TSCC Attractor Enumeration Algorithm (in Biocham v4)

functionEnumerateSolutions(Constraints)

Iteratively solve by SAT/CP the CSP defined byConstraints returnThe set of solutions

end function

procedure list_tscc_candidates

Constraints← {P∧ ¬N =⇒ t |(P,N,+,t,f)∈I}

. Enabled positive influences must not change the state Candidates ←EnumerateSolutions(Constraints)

forC ∈Candidates do

if C has no strictly decreasing transitionthen C is astable steady state

else if C has a non-reversible strictly decreasing transitionthen C isnotin a TSCC

else

C’s SCC must be explored to check if it is a TSCC end if

end for end procedure

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Influence Model of p53/Mdm2 DNA Damage Repair System

Original reaction model

[Ciliberto et al. 07 PLos]

Simplified influence model

[Abou-Jaoudé et al. 09 JTB]

C N

P D

multilevels ignored basal activation added

b i o c h a m : l o a d _ b i o c h a m ( k a u f m a n ) . b i o c h a m : l i s t _ i n f l u e n c e s . [ 0 ] P−> C

[ 1 ] C−> N [ 2 ] N−< P [ 3 ] P−< N [ 4 ] P−< D [ 5 ] D−< N

b i o c h a m : l i s t _ t s c c _ c a n d i d a t e s . [ C−0 ,D−0 ,N−0 ,P−0] s t a b l e [ C−0 ,D−0 ,N−1 ,P−0] s t a b l e [ C−1 ,D−0 ,N−1 ,P−0] s t a b l e [ C−1 ,D−0 ,N−1 ,P−1] n o t t e r m i n a l [ C−0 ,D−1 ,N−0 ,P−0] s t a b l e [ C−0 ,D−1 ,N−1 ,P−0] n o t t e r m i n a l [ C−1 ,D−1 ,N−1 ,P−0]

[ C−1 ,D−1 ,N−1 ,P−1] n o t t e r m i n a l b i o c h a m : add_biocham ( k a u f m a n 2 ) . b i o c h a m : l i s t _ i n f l u e n c e s . [ 0 ] P−> C

[ 1 ] C−> N [ 2 ] N−< P [ 3 ] P−< N [ 4 ] P−< D [ 5 ] D−< N [ 6 ] _−> P [ 7 ] _−> D [ 8 ] C−< C

b i o c h a m : l i s t _ t s c c _ c a n d i d a t e s . [ C−1 ,D−1 ,N−1 ,P−1]

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Influence Model of the Mammalian Circadian Clock

Simplified model[Comet Bernot et al. 12]

Genes Complexes Light

_ / L−> L . L−< L . _ / G , PC−> G . G , PC−< G . G / PC , L−> PC .

PC / G−< PC . PC , L−< PC .

Thomas’s neg. boolean semantics Positive boolean semantics (1,0,0)→(1,1,0) only whenL=0.

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Conclusion

Reaction systems and influence systems have thesame expressive power under the differential semantics.

Influence systems are less expressiveunder the

Boolean/PN/stochastic positive or negative semantics.

This explains subtle discrepancies between the precise boolean semanticsof influence and/or reaction systems and in

modeling tools like Biocham, GinSim, ...

Thepositive boolean semanticsis justified by abstractionof concrete quantitative semantics (in the physical world we cannot directly test the absence only the presence...)

Thefunctional negative boolean semantics à la Thomas have nice necessary conditions for multi-stability and oscillations TheTSCC algorithm for computing attractors in the positive boolean semantics can be used toprune the search spacefor efficiently enumerating the attractors à la Thomas.

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Conclusion

Thefunctional negative boolean semantics à la Thomas have nice necessary conditions for multi-stability and oscillations

TheTSCC algorithm for computing attractors in the positive boolean semantics can be used toprune the search spacefor efficiently enumerating the attractors à la Thomas.

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Conclusion

Thefunctional negative boolean semantics à la Thomas have nice necessary conditions for multi-stability and oscillations TheTSCC algorithm for computing attractors in the positive boolean semanticscan be used to prune the search spacefor efficiently enumerating the attractors à la Thomas.

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