Equipe Mabios Math ematiques et Algorithmiques pour la Biologie - - PowerPoint PPT Presentation

´ 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

Different biological networks

• Molecular level: Biochemical networks

Different biological networks

• Protein level: Protein interaction networks

Different biological networks

• Gene regulation level: genetic networks
• O. Sahin et al. (2009). BMC Syst Biol.3(1):1

Different biological networks

• Tissue level: inter-cellular level
• Lab. of Intercellular Communication Network, Deptartment of Life Science, POSTECH

Different 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

Algorithmic and Mathematics developments Applications Dynamical Network Modeling Large-scale Network Mining

A B C

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 Difficulties

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, differentiation, ...) 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 Differential Equations Nonlinear Ordinary Differential Equations Stochastic Equations Petri Nets . . .

Modelling of biological networks How?

Qualitative/Quantitative Deterministic/Stochastic Graph theory Boolean/Logical models Piecewise Linear Differential Equations Nonlinear Ordinary Differential 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 ⇒ Boolean abstraction: xi = ⇢ 1 i present i absent

Qualitative modelling

Lack of precise quantitative data (concentrations, kinetic parameters...) Regulations are strongly non linear: sigmo¨ ıd functions ∼ step functions ⇒ Boolean abstraction: xi = ⇢ 1 i present i absent

Activation

xA = 1 = ⇒ fB(x) = 1 xA = 0 = ⇒ fB(x) = 0

Inhibition

xA = 1 = ⇒ fB(x) = 0 xA = 0 = ⇒ fB(x) = 1

Qualitative modelling

Lack of precise quantitative data (concentrations, kinetic parameters...) Regulations are strongly non linear: sigmo¨ ıd functions ∼ step functions ⇒ Boolean abstraction: xi = ⇢ 1 i present i absent

Activation

xA = 1 = ⇒ fB(x) = 1 xA = 0 = ⇒ fB(x) = 0

Inhibition

xA = 1 = ⇒ fB(x) = 0 xA = 0 = ⇒ fB(x) = 1 A discrete vector represents the state of the system x = (x1, . . . xn) (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

Combinatorial explosion: 10 Boolean components ⇒ 210 = 1, 024 states 20 Boolean components ⇒ 220 = 1, 048, 576 states

The logical formalism

Combinatorial explosion: 10 Boolean components ⇒ 210 = 1, 024 states 20 Boolean components ⇒ 220 = 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 K0(v) = 1 if (v0 = 1) ∨ (v1 = 0) ∨ (v2 = 1) K1(v) = 1 if (v0 = 0) ∨ (v2 = 0) K2(v) = 1 if (v0 = 1) ∧ (v1 = 1) g0 g1 g2 = ⇒

000 001 010 011 100 101 110 111

Un sujet de stage sur le formalisme logique

Dynamiques asynchrones modulo les isom´ etries de l’hypercube

Formalisme logique, r` egles logiques K0(v) = 1 if (v0 = 1) ∨ (v1 = 0) ∨ (v2 = 1) K1(v) = 1 if (v0 = 0) ∨ (v2 = 0) K2(v) = 1 if (v0 = 1) ∧ (v1 = 1) g0 g1 g2 = ⇒

000 001 010 011 100 101 110 111

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 K0(v) = 1 if (v0 = 1) ∨ (v1 = 0) ∨ (v2 = 1) K1(v) = 1 if (v0 = 0) ∨ (v2 = 0) K2(v) = 1 if (v0 = 1) ∧ (v1 = 1) g0 g1 g2 ⇐ =

000 001 010 011 100 101 110 111

Un sujet de stage sur le formalisme logique

Dynamiques asynchrones modulo les isom´ etries de l’hypercube

Topologie du graphe dynamique K0(v) = 1 if (v0 = 1) ∨ (v1 = 0) ∨ (v2 = 1) K1(v) = 1 if (v0 = 0) ∨ (v2 = 0) K2(v) = 1 if (v0 = 1) ∧ (v1 = 1) g0 g1 g2 = ⇒

000 001 010 011 100 101 110 111

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

000 001 010 011 100 101 110 111 111 110 101 100 011 010 001 000

Sym´ etrie centrale

001 101 011 111 000 100 010 110

Rotation

Questions possibles comme objet du stage Identifier les changements effectu´ 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 effectu´ 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 Si (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)

Prime implicants graph

Trapset hierarchy represented by a prime implicant graph Implicant of a boolean function F: conjunction of literals P such as F = 1 if P = 1 Prime implicant = implicant that cannot be covered by a more general implicant

[Klarner et al., 2014]

Figure: Colors indicate 3 different stable and consistent arc sets: Blue = 0102 cyclic attractor, Red = 11120314 stable state, Green = 11 trapset

Finding maximal trapsets from prime implicants

Finding minimal set of consistent prime implicants: NP-hard. Quine-McCluskey algorithm (efficient heuristic):

1 find all prime implicants 2 extract a consistent set of essential prime implicants

⇒ Implementation of an Integer Linear Programming (glpk) program to enumerate and find the trapsets