Statistical learning of biological networks: a brief overview - PowerPoint PPT Presentation

Statistical learning of biological networks: a brief overview Florence d’Alché–Buc IBISC CNRS, Université d’Evry, GENOPOLE, Evry, France Email: florence.dalche@ibisc.fr Statistical learning of biological networks: a brief overview 1 / 30

Biological networks Statistical learning of biological networks: a brief overview Introduction 2 / 30

Motivation Identify and understand complex mechanisms at work in the cell Biological networks signaling pathways gene regulatory networks protein-protein interaction networks metabolic pathways Use experimental data and prior knowledge AND statistical inference to unravel biological networks and predict their behaviour Statistical learning of biological networks: a brief overview Introduction 3 / 30

How to learn biological networks from data ? Data-mining approaches : extract co-expressed patterns and/or co-regulated patterns, reduce dimension [large scale data, often preliminary to more accurate modelling or prediction] Modeling approaches : model the network behavior, can be used to simulate and predict the network as a system [smaller scale data] Predictive approaches : predict (only) edges in an unsupervised or supervised way [large or medium scale data] Statistical learning of biological networks: a brief overview Introduction 4 / 30

Learning (biological) networks Statistical learning of biological networks: a brief overview Introduction 5 / 30

Outline Introduction 1 Supervised Predictive approaches 2 Modelling approaches 3 Conclusion 4 Statistical learning of biological networks: a brief overview Supervised Predictive approaches 6 / 30

Supervised learning of the regulation concept Instance Problem 1 (transcriptional regulatory networks) :Training sample S = { ( w i = ( v i , v ′ i ) , y i ) , i = 1 ... n } where w i are pairs of components v i and v ′ i (think transcription factor and potential regulee) and y i ∈ Y indicates if there is v i is a transcription factor for v ′ i . We wish to be able to predict new regulations. Reference : Qian et al. 2003, Bioinformatics. In symbolic machine learning, this corresponds to the framework of relational learning classically associated with inductive logic programming (ILP) and more recently to statistical ILP : The predicate interaction(X,Y) can be learned from labeled examples Statistical learning of biological networks: a brief overview Supervised Predictive approaches 7 / 30

Supervised learning of interactions From a known network where each vertex is described by some input feature vector x , predict the edges involving new vertices described by their input feature vector Statistical learning of biological networks: a brief overview Supervised Predictive approaches 8 / 30

Supervised prediction of protein-protein interaction network Instance Problem 2 (protein-protein interaction networks) : Training sample S = { ( w i = ( v i , v ′ i ) , y i ) , i = 1 ... n } where w i are couples of components v i and v ′ i (think proteins) and y i ∈ Y indicates if there is an edge or not between v i and v ′ i . We wish to predict interactions for test and training input data Noble et al. in 2005 (SVM) with kernel combination Further studied by Biau and Bleakley 2006, Bleakley et al. 2007 Statistical learning of biological networks: a brief overview Supervised Predictive approaches 9 / 30

Similarity or kernel learning In the case of non oriented graphs, a similarity between components can be learnt instead of a classification function Yamanishi and Vert’s work (2005) first introduced this kind of approach We proposed a new way of formulating the problem as regression in output space endowed with a kernel (Geurts et al. 2006,2007) Statistical learning of biological networks: a brief overview Supervised Predictive approaches 10 / 30

Supervised learning with output (kernel) feature space Suppose we have a learning sample LS = { x i = x ( v i ) , i = 1 , . . . , N } drawn from a fixed but unknown probability distribution and an additional information provided by a Gram matrix K = k ij = k ( v i , v j ) , for i , j = 1 , . . . , N } that expresses how much objects v i , i = 1 ... n are close to each other. Let us call respectively φ the implicit output feature map and k the positive definite kernel defined on V × V such that < φ ( v ) , φ ( v ′ ) > = k ( v , v ′ ) . From a learning sample { ( x i , K ij | i = 1 , . . . , N , j = 1 , . . . , N } with x i ∈ X ,find a function f : X → F that minimizes the expectation of some loss function ℓ : F × F → IR over the joint distribution of input/output pairs: E x ,φ ( v ) { ℓ ( f φ ( x ) , φ ( v )) } Statistical learning of biological networks: a brief overview Supervised Predictive approaches 11 / 30

Application to supervised inference of edges in a graph 1 For objects v 1 , ..., v N , let us assume we have : feature vectors x ( v i ) , i = 1 ... N and a Gram matrix K defined as K i , j = k ( v i , v j ) . The kernel k reflects the proximity between objects v , as vertices in the known graph. Reminder: kernel k is a positive definite (similarity) function. For such function, there exists a function φ called a feature map : V → F such that k ( v , v ′ ) = � φ ( v ) , φ ( v ′ ) � . Statistical learning of biological networks: a brief overview Supervised Predictive approaches 12 / 30

Supervised inference of edges in a graph Use a machine learning method that can infer a function h : X → F to get for a given x ( v ) , an approximation of φ ( v ) and get an approximation g ( x ( v ) , x ( v ′ )) = � h ( x ( v )) , h ( x ( v ′ )) � of the kernel value between v and v ′ described by their input feature vectors x ( v ) and x ( v ′ ) Connect these two vertices if g ( x ( v ) , x ( v ′ )) > θ (by varying θ we get different tradeoffs between true positive and false positive rates) Statistical learning of biological networks: a brief overview Supervised Predictive approaches 13 / 30

A kernel on graph nodes Diffusion kernel (Kondor and Lafferty, 2002): The Gram matrix K with K i , j = k ( v i , v j ) is given by: K = exp ( − β L ) where the graph Laplacian L is defined by:  d i the degree of node v i if i = j ;  L i , j = − 1 if v i and v j are connected ; 0 otherwise .  Statistical learning of biological networks: a brief overview Supervised Predictive approaches 14 / 30

Interpretability: rules and clusters (an example with a protein-protein network) Statistical learning of biological networks: a brief overview Supervised Predictive approaches 15 / 30

Network completion and function prediction for yeast data Statistical learning of biological networks: a brief overview Supervised Predictive approaches 16 / 30

Challenges and limitations in supervised predictive approaches Semi-supervised learning or even transductive learning Issue : unbalanced distribution of positive and negative examples local approach (the graph is not seen as a single variable) data (labeled examples) are not i.i.d. : regulations are not independent Statistical learning of biological networks: a brief overview Supervised Predictive approaches 17 / 30

Outline Introduction 1 Supervised Predictive approaches 2 Modelling approaches 3 Conclusion 4 Statistical learning of biological networks: a brief overviewModelling approaches 18 / 30

Graphical models : from simple interactions models to complex ones Graphical Gaussian Model model estimation: estimating partial correlation as a measure of conditional independency (classified as graph prediction in my terminology) Bayesian networks estimation: modelling directed interactions Dynamic Bayesian Networks estimation: modelling directed interactions through time State-space models estimation : modelling observed and hidden dynamical processes as well Statistical learning of biological networks: a brief overviewModelling approaches 19 / 30

Focus on state-space models Goal: Quantitative models (easier to learn, encompass mechanistic models : biological relevance) Taking into account time Some variables are not measured: assumption of an hidden process Linear Gaussian models: parameters encapsulate network structure (Perrin et al. 03, Rangel et al. 04) Nonlinear models (more biologically relevant): the structure is encapsulated in the form of the transition function (Nachman 04, Rogers et al. 06, Quach et al. 07) x ( t k + 1 ) = F ( x ( t k ) , u ; θ ) + ǫ h ( t k ) y ( t k ) = H ( x ( t k ) , u ( t k ); θ ) + ǫ ( t k ) Statistical learning of biological networks: a brief overviewModelling approaches 20 / 30

System of Ordinary Differential Equations (ODE) d x = f ( x ( t ) , u ( t ); θ ) dt Let us focus on gene regulatory networks x ( t ) : state variables at time t protein concentrations mRNA concentrations f : the form of f encodes the nature of interactions (and their structure) linear/nonlinear models Michaelis-Menten kinetics Mass action kinetics ... θ : parameter set (kinetic parameters, rate constants,...) u ( t ) : input variables at time t Statistical learning of biological networks: a brief overviewModelling approaches 21 / 30