Overview of work at Inria and implementation in BIOCHAM-4 in the - - PowerPoint PPT Presentation

Overview of work at Inria and implementation in BIOCHAM-4 in the context of SYMBIONT

François Fages, Sylvain Soliman Adrien Baudier, Aymeric Quesne, Eleonore Bellot http://lifeware.inria.fr Inria Saclay, Palaiseau, France

Kickoff meeting, Bonn, March 2018

Overview of talk

• Tropical equilibration problem solver based on Constraint Logic

Programming

– Implementation in BIOCHAM-4 and evaluation in BioModels – Extension of the tropical solver with inequalities and scaling intervals [master internship Aymeric Quesne from July 9th] – Extension of the solver to integrate consistency constraints such as Tikhonov hyperbolicity constraints [PhD thesis of Eleonore Bellot from Sep. 1st]

• A graphical method for multistationarity analysis and its evaluation in

BioModels

– Thomas-Soulé necessary conditions for continuous influence networks – Soliman’s necessary conditions for continuous reaction networks – Graph rewriting algorithm [master internship of Adrien Baudier] – Evaluation in BioModels

Symbiont July 2018 François Fages

Constraint-based Methods

Constraint Programming (CP) provides efficient algorithms to solve many “practical” instances of NP-hard problems. Unlike “generate and test”, CP uses constraints (over different domains) actively to prune the search space a priori during search by applying

• domain-filtering algorithms (e.g. interval arithmetic) associated to each

constraint for over-approximating the set of solutions

• and (complete) search heuristics to enumerate the set of solutions
• Redundant constraints for better pruning
• Global constraints are n-ary relations with domain-filtering algorithms (of

any kind, graph-theoretic, linear relaxation, symbolic computation, etc) to provide better pruning than their decomposition with elementary constraints Pros: general method to handle a wide variety of constraints and domains Cons: does not provide a symbolic representation of the set of all solutions

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CP in Symbiont

Tropical algebra constraints (ℤ, +, min, =, ≤)

Soliman F Radulescu 2014 Algorithms for Molecular Biology

Can provide numerical solutions to tropical constraint satisfaction problems out

• f reach of current symbolic methods

Can make use of symbolic methods for domain filtering algorithms for global constraints (tractable subproblems) à Potentially able to deal with, and enforce, extra consistency constraints (e.g. Tikhonov hyperbolicity condition) for guaranteeing the correctness of model reduction a priori during search

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Current Implementation in BIOCHAM-4

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Tyson Cell Cycle Model

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On-going Work

• Master internship: Aymeric Quesne (Supelec) from July 9th

– Add inequality constraints (ℤ, +, min, =, ≤) to our solver – Define the tropical abstraction 𝛽 by intervals: 𝛽(x+y)≠min(𝛽(x), 𝛽(y)) in general 𝛽(x+y)∈[ min(𝛽(x), 𝛽(y)) - 1 , min(𝛽(x), 𝛽(y) ] e.g. ∊=0.1 𝛽(0.4)=1 𝛽(0.4+0.4)=0 – Study the properties of 𝛽 with respect to the elimination of linear invariants

• PhD thesis: Eléonore Bellot (master AIV, master Physics) from Sep. 1st

– Add consistency constraints such as Tikhonov hyperbolicity constraints to our constraint solver – Compute tropical equilibration solutions that lead a priori to correct reductions – Add matching constraints for chaining model reductions

• Master internship of Adrien Baudier (Centrale Lyon) since April 1st

– Graphical requirements for multistationarity in continuous reaction networks

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Submitted to a special issue of the Journal of Theoretical Biology

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Necessary Conditions for Multistationarity in Reaction Networks and their Verification in BioModels in memoriam of Ren´ e Thomas

Adrien Baudier, Fran¸ cois Fages, Sylvain Soliman

Inria Saclay ˆ Ile-de-France, Palaiseau, France

F:ℝnà ℝn 𝜵: product of intervals G: graph of the signs of the Jacobian of F Thomas’ conjecture [Springer Synergetic 1981] for gene regulatory networks Soulé’s theorem [Complexus 2003] : But always satisfied in the influence graph

• f a reaction network

containing a binary reaction…

Thomas-Soulé’s Necessary Conditions for Multistationarity

Assume that is open and that Fhas at least two nondegenerate zeroes in .Then there exists a such that G(a) contains a positive circuit. Remark.

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E S ES P R1 R1 R−1 R−1 R2 R2 R1 R1

Labelled Influence Graphs of a Reaction Network

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. A hooping is a collection C = {C1, ..., Ck} of circuits such that, for all i ) j, Ci and Cj do not have a common vertex. A circuit is thus a special case of hooping. We let

Computed from the signs of the Jacobian matrix (may be an over-approximation for non mass action law kinetics)

E S ES P R1 R1 R−1 R−1 R2 R2 R1 R1 E R1 S ES R2 P R1

∂v1 ∂E

• 1

∂v1 ∂S

• 1

1

∂v1 ∂ES

• 1

1

∂v2 ∂ES

• 1

1 1 1

Figure 1: DSR graph of the enzymatic reaction: S + E <=> ES => E + P.

Soliman’s [BMB 2013] Necessary Conditions for Multistability in Reaction Networks

Theorem 2.2 ([1]). Let F be any differentiable map from Ω to Rn cor- responding to a biochemical reaction system. If Ω is open and F has two nondegenerate zeroes in Ω then there exists some a in Ω such that:

• 1. The reaction-labelled influence graph G of F at point a contains a pos-

itive circuit C;

• 2. There exists a hooping H in G, such that C is subcycle of H with

(Y 0 − Y )|H of full rank.

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Definition 2. The restriction of the system to a species hooping H (noted |H) is the system where reactions {Ri | i ∈ I} not appearing in H are omitted.

i.e. the stoichiometry matrix built using only the reactions appearing in C is of full rank

Conditions on the Labelled Influence Graph

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Corollary 2.3. A necessary condition for the multistationarity of a biochem- ical reaction system is that there exists a positive cycle in its influence graph, using at most once each reaction.

Corollary 2.4. A necessary condition for the multistationarity of a biochem- ical reaction system is that there exists a positive cycle in its influence graph, not using both forward and backward directions of any reversible reaction. Corollary 2.5. A necessary condition for the multistationarity of a biochem- ical reaction system is that there exists a positive cycle in its influence graph, not using all species involved in a conservation law.

E S ES P R1 R1 R−1 R−1 R2 R2 R1 R1

Checking Acyclicity by Graph Simplification in

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• f O(e log(n))

A graph is acyclic if and only if it reduces to the empty graph with the following graph reduction rules:

• IN0(v): Remove vertex v and all associated edges if v has no incoming

edge.

• OUT0(v): Remove vertex v and all associated edges if v has no out-

going edge.

• IN1(v): Remove vertex v if v has exactly one incoming edge and con-

nect this edge to all the outgoing edges of v.

• OUT1(v): Remove vertex v if v has exactly one outgoing edge and

connect all incoming edges to it.

Checking Corollary Conditions by Graph Rewriting

• INOUTi(v): Remove vertex v if v has exactly i incoming or outgoing

edges, and create the incoming-outcoming edges labeled by the product

• f the signs and the union of the reactions if, and only if, those labels

satisfy the conditions of the corollaries.

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contracting the graph to check acyclicity

We can remove a node and associate all incoming and outgoing arcs by keeping a track of the reactions and species involved.

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Graph Rewriting Algorithm

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contracting the graph to check acyclicity

∙ At each step we remove the node with the least incoming or

• utgoing arcs.

∙ When creating a new arc, its sign is the product of the sign of the considered arcs. ∙ We apply our rules on the labels of the arcs to immediately remove arcs that could give incoherent loops. ∙ When we have a self-loop, we check its sign and continue if it is negative. ∙ If it is positive, we can conclude or continue if we want to enumerate all possible cycles in the graph. ∙ This allows us to determine that 105 models of BioModels cannot present multistationarity (over 506 models).

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Theoretical and Practical Complexity

Proposition 3.1. The time complexity of Alg. 3 is O

• k2n

where n the number of nodes and k is the maximum degree of the graph.

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Conditions verified Number Nb of species Computation time

• f graphs

avg. max.

• avg. (s)

max (s) All graphs 506 21.24 430 No positive circuit 48 3.42 18 < 0.01 < 0.01

• Cor. 2.3 2.4 2.5

105 6.22 46 < 0.01 < 0.01

• Cor. 2.3 2.4 2.5 2.6

160 8.23 54 < 0.01 0.05

• Cor. 2.3 2.4 2.5 2.6 2.7

180 8.38 54 5.90 980.1

Table 1: Analysis of 506 sanitized models from the curated branch of the BioModels

• repository. The table reports the proportion of graphs for which it was possible to rule-out

multistationarity using Thomas’s positive circuit condition and using the refined conditions expressed in the corollaries described above.

Evaluation results in BioModels:

Sign Change Condition

Corollary 2.6. A necessary condition for the multistationarity of a biochem- ical reaction system is that there exist positive cycles fulfilling condition 2 of Theorem 2.2 in the influence graph corresponding to its Jacobian, and in any graph obtained from it choosing a set of species and by reversing the sign

• f all arcs that have as target some species belonging to that set.

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changing the sign

Changing the signs (E) S E ES P R1 R1 R1 R−1 R2 R−1 R1 R2

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changing the sign

Changing the signs (E) S E ES P R1 R1 R1 R−1 R2 R−1 R1 R2

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Sign Change Algorithm by Gaussian Elimination

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finding sign changes

K M MK Mp MpK Mpp R1 R1 R3 R3 R1 R−1 R2 R−1 R1 R2 R3 R−3 R−3 R4 R3 R4 In 2 : xK xMK 1 xK xMpK 1 xK xMK xMp

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solving the system

+ : K

R1

− → MK

R2

− → K + : K

R3

− → MpK

R4

− → K − : K

R1

− → MK

R2

− → Mp

R3

− → K In Z/2Z :      xK + xMK = 1 xK + xMpK = 1 xK + xMK + xMp = 0 If the system has no solutions, then, there is no change of sign that will give no positive loop. We can solve the system using a Gaussian elimination. + : K − MK − : MK − MpK + : Mp      xK + xMK = 1 xMK + xMpK = 0 xMp = 1 Adding two equations is the same as taking the symmetric difference between the sets of species. This can be updated for each loop we find.

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To change the sign of x or not ? x=1/0 Sum on positive/negative circuits = 1/0

Target Swapping Condition

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Corollary 2.7. A necessary condition for the multistationarity of a biochem- ical reaction system is that there exist positive cycles fulfilling condition 2 of Theorem 2.2 in the influence graph corresponding to its Jacobian, and in any graph obtained from it by choosing a permutation of the species and by rewiring the arcs’ target according to the permutation.

swapping targets

Changing targets (E↔P) S E ES P R1 R1 R1 R1 R1 R−1 R2 R1 R−1 R2 R−1 R1 R2

19

swapping targets

Changing targets (E↔P) S E ES P R1 R1 R1 R1 R1 R−1 R2 R1 R−1 R2 R−1 R1 R2

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Target Swapping Algorithm (on remaining circuits)

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problems of swapping targets

∙ There are too many swaps to check explicitly the solutions. ∙ Failure of the Gaussian resolution gives us a set of loops that present a contradiction. To remove the contradiction, a swap needs to be done with at least one species in the set of loops. We don’t need to swap between two species that are not concerned by the contradiction. ∙ Although this highly reduces the number of swaps to check, there are still too many.

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Performance Evaluation in BioModels

Conditions verified Number Nb of species Computation time

• f graphs

avg. max.

• avg. (s)

max (s) All graphs 506 21.24 430 No positive circuit 48 3.42 18 < 0.01 < 0.01

• Cor. 2.3 2.4 2.5

105 6.22 46 < 0.01 < 0.01

• Cor. 2.3 2.4 2.5 2.6

160 8.23 54 < 0.01 0.05

• Cor. 2.3 2.4 2.5 2.6 2.7

180 8.38 54 5.90 980.1

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Beyond our expectations…