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 Symbiont July 2018 François Fages
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 of 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 Symbiont July 2018 François Fages
Current Implementation in BIOCHAM-4 Symbiont July 2018 François Fages
Tyson Cell Cycle Model Symbiont July 2018 François Fages
Symbiont July 2018 François Fages
On-going Work • Master internship: Aymeric Quesne (Supelec) from July 9 th – 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. 1 st – 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 1 st – Graphical requirements for multistationarity in continuous reaction networks Symbiont July 2018 François Fages
Submitted to a special issue of the Journal of Theoretical Biology 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 Symbiont July 2018 François Fages
Thomas-Soulé’s Necessary Conditions for Multistationarity 𝜵 : product of intervals G: graph of the signs of the Jacobian of F F: ℝ n à ℝ n Thomas’ conjecture [Springer Synergetic 1981] for gene regulatory networks Soulé’s theorem [Complexus 2003] : Assume that � is open and that F has at least two nondegenerate zeroes in � .Then there exists a � � such that G ( a ) contains a positive circuit. Remark. E But always satisfied in the influence graph R 1 R 1 of a reaction network R 2 containing a binary reaction… R 1 R − 1 R 1 R 2 S ES P R − 1 Symbiont July 2018 François Fages
Labelled Influence Graphs of a Reaction Network Computed from the signs of the Jacobian matrix (may be an over-approximation for non mass action law kinetics) E E ∂ v 1 -1 ∂ E R 1 R 1 1 R 1 ∂ v 1 1 ∂ S ∂ v 2 R 2 ∂ ES -1 R 1 R − 1 1 R 2 S ES P R 1 R 2 -1 1 -1 S ES P ∂ v � 1 1 ∂ ES R � 1 R − 1 Figure 1: DSR graph of the enzymatic reaction: S + E < = > ES = > E + P. A hooping is a collection C = { C 1 , ..., C k } of circuits such that, for all i ) j , C i . and C j do not have a common vertex. A circuit is thus a special case of hooping. We let Symbiont July 2018 François Fages
Soliman’s [BMB 2013] Necessary Conditions for Multistability in Reaction Networks Definition 2. The restriction of the system to a species hooping H (noted | H ) is the system where reactions { R i | i ∈ I } not appearing in H are omitted. Theorem 2.2 ([1]) . Let F be any di ff erentiable map from Ω to R n 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. i.e. the stoichiometry matrix built using only the reactions appearing in C is of full rank Symbiont July 2018 François Fages
Conditions on the Labelled Influence Graph 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 R 1 R 1 R 2 R 1 R − 1 R 1 R 2 S ES P Symbiont July 2018 R − 1 François Fages
Checking Acyclicity by Graph Simplification in of 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. Symbiont July 2018 François Fages
Checking Corollary Conditions contracting the graph to check acyclicity by Graph Rewriting • INOUTi( v ) : Remove vertex v if v has exactly i incoming or outgoing We can remove a node and associate all incoming and outgoing arcs edges, and create the incoming-outcoming edges labeled by the product by keeping a track of the reactions and species involved. of the signs and the union of the reactions if, and only if, those labels satisfy the conditions of the corollaries. Symbiont July 2018 François Fages 8
contracting the graph to check acyclicity Graph Rewriting Algorithm ∙ At each step we remove the node with the least incoming or outgoing 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). Symbiont July 2018 François Fages 9
Theoretical and Practical Complexity k 2 n � � Proposition 3.1. The time complexity of Alg. 3 is O where n the number of nodes and k is the maximum degree of the graph. Evaluation results in BioModels: Conditions verified Number Nb of species Computation time of 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. Symbiont July 2018 François Fages
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