VERIFICATION OF ROBUSTNESS PROPERTY IN CHEMICAL REACTION NETWORKS Ph.D. Student Supervisors Roberta Gori Lucia Nasti Paolo Milazzo
HELLO! Nov 2016 Nov 2018 Mar 2019 Oct 2019 Started PhD thesis INRIA MPI PhD submission Modelling, Simulation and Verification of Biological Systems Group SUPERVISED BY: ROBERTA GORI AND PAOLO MILAZZO Lucia Nasti - University of Pisa
PUBLICATIONS Publications presented in the thesis: R. Gori, P. Milazzo and L. Nasti Towards an Efficient Verification Method for Monotonicity Properties of ➤ Chemical Reaction Networks. (BIOINFORMATICS 2019). L. Nasti, R. Gori and P. Milazzo, Formalizing a Notion of Concentration Robustness for Biochemical ➤ Networks . STAF Workshops 2018: 81-97 R. Gori, P. Milazzo and L. Nasti and F. Poloni, Efficient analysis of Chemical Reaction Networks ➤ Dynamics based on Input-Output monotonicity (submitted). L. Nasti and C. Zechner. Verification and analysis of Robustness in Becker-Döring equations (in ➤ preparation). Other publications: L. Nasti and P. Milazzo, A computational model of internet addiction phenomena in social networks . ➤ International Conference on Software Engineering and Formal Methods, 86-100. L. Nasti and P. Milazzo, A Hybrid Automata model of social networking addiction. Journal of Logical ➤ and Algebraic Methods in Programming. Volume 100, November 2018, Pages 215-229. R. Gori, P. Milazzo and L. Nasti, A survey of gene regulatory networks modelling methods: from ODEs, to ➤ Boolean and bio-inspired models (in preparation). Lucia Nasti - University of Pisa
OUTLINE ➤ What is robustness ? ➤ Formalisation: CRN and Petri Nets ➤ Why and how to study monotonicity in CRN? ➤ Results: Sufficient conditions and Tools ➤ Applications: Becker-Döring equations ➤ Future work Lucia Nasti - University of Pisa
BACKGROUND ➤ A cell is a very complex system ➤ Chemical reaction networks ( pathways ) govern the basic cell’s activities ➤ To examine the structure of the cell as a whole, we can design multiscale and predictive models Lucia Nasti - University of Pisa
CHEMICAL REACTIONS Stoichiometric coefficient H 2 2 H 2 O 2 t k Reactants Rate Products (A) O 2 ➤ Kinetic rate: rate of a reaction H 2 ➤ Reactant: chemical species that is 2 H 2 O 2 consumed t k ➤ Product: chemical species that is created (B) O 2 ➤ Stoichiometric coefficient: the number of species involved in the reaction ➤ Concentrations: [A], [B], [C], [D] Lucia Nasti - University of Pisa
<latexit sha1_base64="ENpOEVqFeBJrdHtk2IEHaJCKs=">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</latexit> <latexit sha1_base64="gBxhmtXcDOQodizOHF5R8wlZzrU=">AB/XicbVDLSsNAFJ34rPUVHzs3g0VwVTIi6EaodeOygn1AmobJdNIOnUzCzESofgrblwo4tb/cOfOGmz0NYDFw7n3Mu9wQJZ0o7zre1tLyurZe2ihvbm3v7Np7+y0Vp5LQJol5LDsBVpQzQZuaU47iaQ4CjhtB6Ob3G8/UKlYLO71OKFehAeChYxgbSTfPpQ+uhr5GZq414Pu3WvF5R9u+JUnSngIkEFqYACDd/+6vZjkZUaMKxUi5yEu1lWGpGOJ2Uu6miCSYjPKCuoQJHVHnZ9PoJPDFKH4axNCU0nKq/JzIcKTWOAtMZYT1U814u/ue5qQ4vYyJNVUkNmiMOVQxzCPAvaZpETzsSGYSGZuhWSIJSbaBJaHgOZfXiStsypyqujuvFKrF3GUwBE4BqcAgQtQA7egAZqAgEfwDF7Bm/VkvVjv1sesdckqZg7AH1ifP8mXlCE=</latexit> <latexit sha1_base64="4KZqhWCqFJFQWQpqBRn2u9u9I7A=">AB/nicbVDLSsNAFL2pr1pfUXHlZrAIbixJEXQjFOvCZQX7gDQNk8m0HTp5MDMRSij4K25cKOLW73Dn3zhts9DWAxcO59zLvf4CWdSWda3UVhZXVvfKG6WtrZ3dvfM/YOWjFNBaJPEPBYdH0vKWUSbilO4mgOPQ5bfuj+tRvP1IhWRw9qHFC3RAPItZnBCsteaR8KrXIy87tydO3e0R59btBcgzy1bFmgEtEzsnZcjR8MyvbhCTNKSRIhxL6dhWotwMC8UIp5NSN5U0wWSEB9TRNMIhlW42O3+CTrUSoH4sdEUKzdTfExkOpRyHvu4MsRrKRW8q/uc5qepfuRmLklTRiMwX9VOVIymWaCACUoUH2uCiWD6VkSGWGCidGIlHYK9+PIyaVUrtlWx7y/KtZs8jiIcwmcgQ2XUIM7aEATCGTwDK/wZjwZL8a78TFvLRj5zCH8gfH5A2cGlHc=</latexit> CHEMICAL KINETICS ➤ Law of mass action: reaction rate is proportional to the reactants product Lucia Nasti - University of Pisa
ROBUSTNESS PROPERTY ➤ Robustness: A fundamental feature of complex evolving systems, for which the behaviour of the system remains essentially constant, despite the presence of internal and external perturbations. Lucia Nasti - University of Pisa
ROBUSTNESS IN LITERATURE ➤ In [Kitano, 2007]: Robustness is the ability of a system to maintain specific functionalities against perturbations. ➤ In [Rizk et al. , 2008]: The robustness of a system is measured as the distance of the system behaviour under perturbations from its reference behaviour expressed as temporal logic formula. Lucia Nasti - University of Pisa
OUR PROPOSED WORK ➤ New formal definition of robustness, namely initial concentration robustness ➤ Our new definition is able to analyse all the chemical species involved in the CRN ➤ Our new robustness notion can be proved by performing simulations Lucia Nasti - University of Pisa
INITIAL CONCENTRATION ROBUSTNESS 550 500 450 400 Initial Concentrations of [P] 350 Concentrations concentrations [S] at steady state 300 250 S 1 200 S 2 150 S 3 S 4 100 P 1 P 2 50 P 3 P 4 0 0 500 1000 1500 Time Lucia Nasti - University of Pisa
INITIAL CONCENTRATION ROBUSTNESS 20 18 16 14 Initial Concentrations of [A] Concentrations concentrations [B] 12 at steady state 10 A 1 A 2 8 A 3 A 4 6 B 1 B 2 B 3 4 B 4 2 0 2 4 6 8 10 12 14 16 18 20 Time Lucia Nasti - University of Pisa
CONTINUOUS PETRI NETS FORMALISM DEFINITION A continuous Petri net N is a quintuple: H 2 2 H 2 O 2 t k N=<P , T, F, C, m 0 > (A) O 2 where: H 2 ➤ P is the set of continuous places , conceptually species 2 H 2 O 2 t k ➤ T is the set of continuous transitions , that consume and produce (B) O 2 species ➤ F ⊆ ( P ⨉ T ) ∪ ( T ⨉ P ) ➛ ℝ ≥ 0 represents the set of arcs in terms of a function giving the weight of the arc as result: a weight equal to 0 means that the arc is not present ➤ C : F ➛ ℝ ≥ 0 is a function, which associates each transition with a rate ➤ m 0 is the initial marking , that is the initial distribution of tokens (representing resource instances) among places. A marking is defined formally as m : P ➛ ℝ ≥ 0 Lucia Nasti - University of Pisa
<latexit sha1_base64="RuQJ9+8/fQZFb/L9VzbgNgPcdHo=">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</latexit> <latexit sha1_base64="T1gIoSy0yaKG2Oo/1KJUgKwUj7E=">ACFnicbZBNS8MwGMdTX+d8q3r0EhyCBy2tCHocevE4wb1AW0apVtYknZJKhtln8KLX8WLB0W8ije/jW3Xg24+kPDj/38ekucfxIwqbdvfxtLyuraemWjurm1vbNr7u23VJRITJo4YpHsBEgRgVpaqoZ6cSIB4w0g6GN7nfiBS0Ujc60lMfI76goYUI51JXfNsD0qoCtOIfehN0pQD9IwnFEVCo+RERwXN7eqXbNmW3ZRcBGcEmqgrEbX/PJ6EU4ERozpJTr2LH2UyQ1xYxMq16iSIzwEPWJm6FAnCg/LdawuNM6cEwktkRGhbq74kUcaUmPMg6OdIDNe/l4n+em+jwyk+piBNBJ49FCYM6gjmGcEelQRrNskAYUmzv0I8QBJhnSWZh+DMr7wIrXPLsS3n7qJWvy7jqIBDcAROgAMuQR3cgZoAgwewTN4BW/Gk/FivBsfs9Ylo5w5AH/K+PwBeEOdCw=</latexit> FORMAL DEFINITION OF ROBUSTNESS: AUXILIARIES CONCEPTS ➤ Definition 1 (Intervals) . We define the interval domain Moreover we say that ➤ Definition 2 (Interval marking). An interval marking is a function m [ ] : P ➛ I. We call M [ ] the domain of all interval markings. Input S R 2 Output ES R 3 P R 1 E + + R 21 R 23 Lucia Nasti - University of Pisa
MY NEW FORMAL DEFINITION OF ABSOLUTE ROBUSTNESS ➤ Definition 3 ( 𝒷 -Robustness) . A Petri net PN with output place O is defined as ɑ -robust with respect to a given marking m [ ] iff ∃ k ∈ ℝ such that ∀ m ∈ m [ ] , the marking m’ ss corresponding to the steady state reachable from m , is such that � � m ′ ss ( O ) ∈ 2 , k + α k − α 2 , Observations: ➤ the wider are the intervals of the initial interval marking, the more robust is the network ➤ the smaller is the value of 𝒷 , the more robust is the network Lucia Nasti - University of Pisa
EXAMPLE OF APPLICATION OF ABSOLUTE ROBUSTNESS: TOY MODEL Given a set of chemical reactions: Applying our definition: ➤ with A as output we obtain: m’(A)= [33, 51] → ɑ =18 ➤ with B as output we obtain: m’(B)= [22, 45] → ɑ =23 Lucia Nasti - University of Pisa
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