´ Equipe Mabios Math´ ematiques et Algorithmiques pour la Biologie des syst` emes Institut de Math´ ematiques de Marseille November 23, 2017
Math´ ematiques et Algorithmique pour la Biologie des Syst` emes (MABIOS) Team members Ana¨ ıs Baudot, CR CNRS, Bioinformatics Alain Gu´ enoche, DR CNRS, Computer science Brigitte Moss´ e, MCU, Mathematics Laurent Tichit, MCU, Computer science ´ Elisabeth Remy, CR CNRS, Biomathematics Alberto Valeolidas, PhD student (A. Baudot/P. Cau, La Timone ) Firas Hammami, PhD student (E. Remy/ F. Barras, P. Mandin LCB) Elva Novoa del Toro, PhD student (A. Baudot/N. Levy) L´ eonard H´ erault, PhD student (E. Remy/E. Duprez CRCM)
MABIOS team - Main issues From genotypes.... to phenotypes
Di ff erent biological networks • Molecular level: Biochemical networks
Di ff erent biological networks • Protein level: Protein interaction networks
Di ff erent biological networks • Gene regulation level: genetic networks O. Sahin et al. (2009). BMC Syst Biol.3(1):1
Di ff erent biological networks • Tissue level: inter-cellular level Lab. of Intercellular Communication Network, Deptartment of Life Science, POSTECH
Di ff erent abstraction levels Molecular level: Biochemical networks B Protein level: Protein interaction networks B Gene regulation level: genetic networks Tissue level: inter-cellular level Higher levels: ecological networks, . . .
MABIOS team - Main issues System biology approaches : focus on interactions MABIOS B A C Dynamical Network Modeling Algorithmic and Applications Mathematics developments Large-scale Network Mining
Biological networks Large-scale network mining Aims: Study protein cellular functioning → Map the landscape of biological processes , → Predict functions for unknown proteins , Idea Tell me who your friends are and I’ll tell you who you are
Biological networks Large-scale network mining Di ffi culties Size of the graphs Heterogeneity of the graphs Algorithmic and mathematic developments → Multiplex framework , → Community-detection algorithms (classification, clustering) , → Network explorations (markov chains) , Keywords : graphs, modularity, classification, communities
Biological networks Dynamical network modelling Motivation Regulatory interaction networks control the cellular processes (e.g. proliferation, apoptosis, di ff erentiation, ...) To get a better understanding of biological process To study the impact of given perturbations, such as disease-induced perturbation
Abstraction, Reduction and Composition Properties of the model? Asymptotical behaviours (attractors) e.g. stable expression patterns Properties along trajectories e.g. transient activation of a component Impact of perturbations e.g. gene knockout Role of input components influence of the environment
Modelling of biological networks
Modelling of biological networks How? Qualitative/Quantitative Deterministic/Stochastic Graph theory Boolean/Logical models Piecewise Linear Di ff erential Equations Nonlinear Ordinary Di ff erential Equations Stochastic Equations Petri Nets . . .
Modelling of biological networks How? Qualitative/Quantitative Deterministic/Stochastic Graph theory Boolean/Logical models Piecewise Linear Di ff erential Equations Nonlinear Ordinary Di ff erential Equations Stochastic Equations Petri Nets . . .
Qualitative modelling Lack of precise quantitative data (concentrations, kinetic parameters...) Regulations are strongly non linear: sigmo¨ ıd functions ∼ step functions ⇢ 1 i present ⇒ Boolean abstraction: x i = 0 i absent
Qualitative modelling Lack of precise quantitative data (concentrations, kinetic parameters...) Regulations are strongly non linear: sigmo¨ ıd functions ∼ step functions ⇢ 1 i present ⇒ Boolean abstraction: x i = 0 i absent Activation Inhibition x A = 1 = f B ( x ) = 1 x A = 1 = f B ( x ) = 0 ⇒ ⇒ x A = 0 = f B ( x ) = 0 x A = 0 = f B ( x ) = 1 ⇒ ⇒
Qualitative modelling Lack of precise quantitative data (concentrations, kinetic parameters...) Regulations are strongly non linear: sigmo¨ ıd functions ∼ step functions ⇢ 1 i present ⇒ Boolean abstraction: x i = 0 i absent Activation Inhibition x A = 1 = f B ( x ) = 1 x A = 1 = f B ( x ) = 0 ⇒ ⇒ x A = 0 = f B ( x ) = 0 x A = 0 = f B ( x ) = 1 ⇒ ⇒ A discrete vector represents the state of the system x = ( x 1 , . . . x n ) (Multi-valued variables) A discrete function f indicates the target state x → f ( x ) Asynchronous, non deterministic dynamics
Modelling of biological networks
A model... and now what??
A model... and now what??
A model... and now what??
The logical formalism 10 Boolean components ⇒ 2 10 = 1 , 024 states Combinatorial explosion: 20 Boolean components ⇒ 2 20 = 1 , 048 , 576 states
The logical formalism 10 Boolean components ⇒ 2 10 = 1 , 024 states Combinatorial explosion: 20 Boolean components ⇒ 2 20 = 1 , 048 , 576 states Algorithmic and mathematic developments Access to the dynamics without generating it reduction methods with conservation of dynamical properties find out mathematical properties (link between topological properties of the regulatory graph and dynamical properties)
Un sujet de stage sur le formalisme logique
Un sujet de stage sur le formalisme logique Dynamiques asynchrones modulo les isom´ etries de l’hypercube Un graphe d’interactions et la dynamique correspondante K 0 ( v ) = 1 if ( v 0 = 1) ∨ ( v 1 = 0) ∨ ( v 2 = 1) K 1 ( v ) = 1 if ( v 0 = 0) ∨ ( v 2 = 0) K 2 ( v ) = 1 if ( v 0 = 1) ∧ ( v 1 = 1) 011 111 g 2 010 110 = ⇒ g 0 g 1 001 101 000 100
Un sujet de stage sur le formalisme logique Dynamiques asynchrones modulo les isom´ etries de l’hypercube Formalisme logique, r` egles logiques K 0 ( v ) = 1 if ( v 0 = 1) ∨ ( v 1 = 0) ∨ ( v 2 = 1) K 1 ( v ) = 1 if ( v 0 = 0) ∨ ( v 2 = 0) K 2 ( v ) = 1 if ( v 0 = 1) ∧ ( v 1 = 1) 011 111 g 2 010 110 = ⇒ g 0 g 1 001 101 000 100
Un sujet de stage sur le formalisme logique Dynamiques asynchrones modulo les isom´ etries de l’hypercube Un graphe d’interactions et la dynamique correspondante K 0 ( v ) = 1 if ( v 0 = 1) ∨ ( v 1 = 0) ∨ ( v 2 = 1) K 1 ( v ) = 1 if ( v 0 = 0) ∨ ( v 2 = 0) K 2 ( v ) = 1 if ( v 0 = 1) ∧ ( v 1 = 1) 011 111 g 2 010 110 = ⇐ g 0 g 1 001 101 000 100
Un sujet de stage sur le formalisme logique Dynamiques asynchrones modulo les isom´ etries de l’hypercube Topologie du graphe dynamique K 0 ( v ) = 1 if ( v 0 = 1) ∨ ( v 1 = 0) ∨ ( v 2 = 1) K 1 ( v ) = 1 if ( v 0 = 0) ∨ ( v 2 = 0) K 2 ( v ) = 1 if ( v 0 = 1) ∧ ( v 1 = 1) 011 111 g 2 010 110 = ⇒ g 0 g 1 001 101 000 100
Un sujet de stage sur le formalisme logique dynamiques asynchrones modulo les isom´ etries de l’hypercube Action des isom´ etries du cube sur le graphe dynamique 100 000 111 110 011 111 101 001 011 010 010 110 110 010 101 100 001 101 111 011 001 000 000 100 Sym´ etrie centrale Rotation
Questions possibles comme objet du stage Identifier les changements e ff ectu´ es sur le graphe d’interactions et les r` egles logiques par action sur la dynamique d’une isom´ etrie en dimension 2, 3, au del` a ?
Questions possibles comme objet du stage Identifier les changements e ff ectu´ es sur le graphe d’interactions et les r` egles logiques par action sur la dynamique d’une isom´ etrie en dimension 2, 3, au del` a ? Classer les dynamiques modulo les isom´ etries de l’hypercube et d´ enombrer les classes (dimension...). Classification moins fine ? Classification plus fine - li´ ee aux modifications du graphe d’interactions ? Examiner le cas particulier des graphes d’interactions pour lesquels les signes des interactions sont enti` erement d´ etermin´ es. D´ enombrement ? Dans ce cas particulier, on connaˆ ıt des liens entre la dynamique et des circuits positifs ou n´ egatifs du graphe d’interactions dits ”fonctionnels”. Chercher des invariants par isom´ etries relatifs ` a ces circuits. Outils : combinatoire, th´ eorie des graphes, syst` emes dynamiques, th´ eorie des groupes (actions de groupes)
Example of research internship Determination of attractors in a Boolean network
Determination of attractors in a boolean network ENS short intership 2015 - Lucas Baudin 1 Study properties of another kind of update strategy: the fully asynchronous one. 2 Computation of attractors in boolean networks Model of Gene Regulatory Network: set of boolean functions NP-hard problem Instead, look for stable subspaces of dynamics, aka trapsets (supersets of attractors), easier to compute and independent of update strategy Implementation of the method on the GINsim software platform and its library LogicalModels (github, Java)
Unfolding of dynamics and trapsets representation [Klarner et al., 2014] Figure: Dynamics (in grey), attractors (in red), and the 4 trapsets S i (outlined in black) Trapset = set of states (hypercubes) encompassing one or several attractors (and their basin of attraction) Set of states ⇔ logical function (e.g. Disjonctive Normal Form)
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