Reconstruction and Clustering with Graph optimization and Priors on - - PowerPoint PPT Presentation

Reconstruction and Clustering with Graph optimization and Priors on Gene Networks and Images

Aurélie Pirayre

PhD supervisors: Frédérique BIDARD-MICHELOT IFP Energies nouvelles Camille COUPRIE Facebook A.I. Research Laurent DUVAL IFP Energies nouvelles Jean-Christophe PESQUET CentraleSupélec

July 3th, 2017

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• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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INTRODUCTION

An overview

Gene regulatory networks Signals and images Reconstruction Clustering Our framework Variational Bayes variational Method BRANE HOGMep

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INTRODUCTION Context

Biological motivation

Second generation bio-fuel production

Cellulases from Trichoderma reesei Lignocellulosic Biomass Cellulose Hemi-cellulose Sugar Ethanol Bio-fuels Enzymatic Hydrolysis Pre-treatment Fermentation Mixing with fuels July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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INTRODUCTION Context

Biological motivation

Second generation bio-fuel production

Cellulases from Trichoderma reesei Lignocellulosic Biomass Cellulose Hemi-cellulose Sugar Ethanol Bio-fuels Enzymatic Hydrolysis Pre-treatment Fermentation Mixing with fuels

Improve Trichoderma reseei cellulase production Understand cellulase production mechanisms

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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INTRODUCTION Context

Biological motivation

Second generation bio-fuel production

Cellulases from Trichoderma reesei Lignocellulosic Biomass Cellulose Hemi-cellulose Sugar Ethanol Bio-fuels Enzymatic Hydrolysis Pre-treatment Fermentation Mixing with fuels

Improve Trichoderma reseei cellulase production Understand cellulase production mechanisms ⇒ Use of Gene Regulatory Network (GRN)

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INTRODUCTION GRN overview

What is a Gene Regulatory Network (GRN)?

GRN: a graph G(V, E) V = {v1, . . . , vG}: a set of G nodes (corresponding to genes) E: a set of edges (corresponding to interactions between genes)

v1 v2 v3 vX July 3th, 2017

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INTRODUCTION GRN overview

What is a Gene Regulatory Network (GRN)?

GRN: a graph G(V, E) V = {v1, . . . , vG}: a set of G nodes (corresponding to genes) E: a set of edges (corresponding to interactions between genes)

A gene regulatory network...

v1 v2 v3 vX

... models biological gene regulatory mechanisms

DNA Gene 1 Gene 2 Gene 3 Gene X mRNA Protein TF1 TF2 TF3 TFX ⊕ ⊖ ⊕⊕

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INTRODUCTION Data overview

What biological data can be used?

For a given experimental condition, transcriptomic data answer to: which genes are expressed? in which amount?

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INTRODUCTION Data overview

What biological data can be used?

For a given experimental condition, transcriptomic data answer to: which genes are expressed? in which amount? How to obtain transcriptomic data? Microarray and RNAseq experiments What do transcriptomic data look like? Gene expression data (GED): G genes × S conditions

M =      S conditions

• −0.948

−0.013 . . . −1.308 −0.977 0.737 0.619 . . . −0.141 −0.803 −0.253 −0.175 . . . −0.859 −0.595 3.747 1.115 . . . −0.418 −0.084 1.383 1.184 . . . −0.493 −0.562               G genes

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INTRODUCTION Links between data and GRNs

How to use GED to produce a GRN ?

From gene expression data...

M =     sj . . . gi · · · mi,j    , July 3th, 2017

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INTRODUCTION Links between data and GRNs

How to use GED to produce a GRN ?

From gene expression data...

M =     sj . . . gi · · · mi,j    ,

V = {v1, · · · , vG} a set of vertices (genes) and E a set of edges leading to a complete graph...

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INTRODUCTION Links between data and GRNs

How to use GED to produce a GRN ?

From gene expression data...

M =     sj . . . gi · · · mi,j    ,

V = {v1, · · · , vG} a set of vertices (genes) and E a set of edges Each edge ei,j is weighted by ωi,j leading to a complete weighted graph...

W =     gj . . . gi · · · ωi,j    , July 3th, 2017

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INTRODUCTION Links between data and GRNs

How to use GED to produce a GRN ?

From gene expression data...

M =     sj . . . gi · · · mi,j    ,

V = {v1, · · · , vG} a set of vertices (genes) and E a set of edges Each edge ei,j is weighted by ωi,j leading to a complete weighted graph...

W =     gj . . . gi · · · ωi,j    ,

We look for a subset of edges E∗ reflecting regulatory links between genes to infer a meaningful gene network.

Wi,j =

• 1

if ei,j ∈ E∗

• therwise.

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INTRODUCTION

Difficulties in GRN inference

July 3th, 2017

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INTRODUCTION

Difficulties in GRN inference

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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INTRODUCTION

Difficulties in GRN inference

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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INTRODUCTION

Difficulties in GRN inference

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• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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INTRODUCTION

Our BRANE strategy

What is the subset of edges E∗ reflecting real regulatory links between genes? ⇒ what is the binary adjacency matrix W ∈ {0, 1}G×G?

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INTRODUCTION

Our BRANE strategy

What is the subset of edges E∗ reflecting real regulatory links between genes? ⇒ what is the binary adjacency matrix W ∈ {0, 1}G×G? We note xi,j the binary label of edge presence: xi,j =

• 1

if ei,j ∈ E∗,

• therwise.

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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INTRODUCTION

Our BRANE strategy

What is the subset of edges E∗ reflecting real regulatory links between genes? ⇒ what is the binary adjacency matrix W ∈ {0, 1}G×G? We note xi,j the binary label of edge presence: xi,j =

• 1

if ei,j ∈ E∗,

• therwise.

Classical thresholding: x∗

i,j =

• 1

if ωi,j > λ,

• therwise.

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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INTRODUCTION

Our BRANE strategy

What is the subset of edges E∗ reflecting real regulatory links between genes? ⇒ what is the binary adjacency matrix W ∈ {0, 1}G×G? We note xi,j the binary label of edge presence: xi,j =

• 1

if ei,j ∈ E∗,

• therwise.

Classical thresholding: x∗

i,j =

• 1

if ωi,j > λ,

• therwise.

Given by a cost function for given weights ω:

maximize

x∈{0,1}E

• (i,j)∈V2 ωi,j xi,j + λ(1 − xi,j)

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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INTRODUCTION

Our BRANE strategy

What is the subset of edges E∗ reflecting real regulatory links between genes? ⇒ what is the binary adjacency matrix W ∈ {0, 1}G×G? We note xi,j the binary label of edge presence: xi,j =

• 1

if ei,j ∈ E∗,

• therwise.

Classical thresholding: x∗

i,j =

• 1

if ωi,j > λ,

• therwise.

Given by a cost function for given weights ω:

maximize

x∈{0,1}E

• (i,j)∈V2 ωi,j xi,j + λ(1 − xi,j) ⇔ minimize

x∈{0,1}E

• (i,j)∈V2 ωi,j (1 − xi,j) + λ xi,j

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INTRODUCTION

Our BRANE strategy

BRANE: Biologically Related A priori Network Enhancement Extend classical thresholding Integrate biological priors into the functional to be optimized Enforce modular networks Additional knowledge:

Transcription factors (TFs): regulators Non transcription factors (TFs): targets

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INTRODUCTION

Our BRANE strategy

BRANE: Biologically Related A priori Network Enhancement Extend classical thresholding Integrate biological priors into the functional to be optimized Enforce modular networks Additional knowledge:

Transcription factors (TFs): regulators Non transcription factors (TFs): targets

Method a priori Formulation Algorithm Inference BRANE Cut Gene co-regulatiton Discrete Maximal flow BRANE Relax TF-connectivity Continuous Proximal method Joint inference and clustering BRANE Clust Gene grouping Mixed Alternating scheme

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS

A discrete method: BRANE Cut

We look for a discrete solution for x ⇔ x ∈ {0, 1}E

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS A priori

A priori: modular structure and gene co-regulation

minimize

x∈{0,1}E

• (i,j)∈V2 ωi,jϕ(xi,j − 1) + λi,jϕ(xi,j) + µΨ(xi,j)

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS A priori

A priori: modular structure and gene co-regulation

minimize

x∈{0,1}E

• (i,j)∈V2 ωi,jϕ(xi,j − 1) + λi,jϕ(xi,j) + µΨ(xi,j)

Modular network: favors links between TFs and TFs λi,j =      2η if (i, j) / ∈ T2 2λTF if (i, j) ∈ T2 λTF + λTF

• therwise.

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS A priori

A priori: modular structure and gene co-regulation

minimize

x∈{0,1}E

• (i,j)∈V2 ωi,jϕ(xi,j − 1) + λi,jϕ(xi,j) + µΨ(xi,j)

Modular network: favors links between TFs and TFs λi,j =      2η if (i, j) / ∈ T2 2λTF if (i, j) ∈ T2 λTF + λTF

• therwise.

with:

T: the set of TF indices η > max {ωi,j | (i, j) ∈ V2} λTF > λTF

A linear relation is sufficient: λTF = βλTF with β = |V|

|T |

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS A priori

A priori: modular structure and gene co-regulation

minimize

x∈{0,1}E

• (i,j)∈V2 ωi,jϕ(xi,j − 1) + λi,jϕ(xi,j) + µΨ(xi,j)

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS A priori

A priori: modular structure and gene co-regulation

minimize

x∈{0,1}E

• (i,j)∈V2 ωi,jϕ(xi,j − 1) + λi,jϕ(xi,j) + µΨ(xi,j)

Gene co-regulation: favors edge coupling Ψ(xi,j) =

• (j,j′)∈T2

i∈V\T

ρi,j,j′|xi,j − xi,j′| ρi,j,j′: co-regulation probability

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS A priori

A priori: modular structure and gene co-regulation

minimize

x∈{0,1}E

• (i,j)∈V2 ωi,jϕ(xi,j − 1) + λi,jϕ(xi,j) + µΨ(xi,j)

Gene co-regulation: favors edge coupling Ψ(xi,j) =

• (j,j′)∈T2

i∈V\T

ρi,j,j′|xi,j − xi,j′| ρi,j,j′: co-regulation probability with ρi,j,j′ =

• k∈V\(T ∪{i})

1(min{ωj,j′, ωj,k, ωj′,k} > γ)

|V\T |−1

γ: the (|V| − 1)th of the normalized weights ω

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution

A maximal flow for a minimum cut formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j|xi,j − 1| + λi,j xi,j +

• i∈V\T

(j,j′)∈T2, j′>j

ρi,j,j′|xi,j − xi,j′|

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution

A maximal flow for a minimum cut formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j|xi,j − 1| + λi,j xi,j +

• i∈V\T

(j,j′)∈T2, j′>j

ρi,j,j′|xi,j − xi,j′|

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• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution

A maximal flow for a minimum cut formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j|xi,j − 1| + λi,j xi,j +

• i∈V\T

(j,j′)∈T2, j′>j

ρi,j,j′|xi,j − xi,j′| s t

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• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution

A maximal flow for a minimum cut formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j|xi,j − 1| + λi,j xi,j +

• i∈V\T

(j,j′)∈T2, j′>j

ρi,j,j′|xi,j − xi,j′| s t x1,2 x1,3 x1,4 x2,3 x2.4 x3,4

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• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution

A maximal flow for a minimum cut formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j|xi,j − 1| + λi,j xi,j +

• i∈V\T

(j,j′)∈T2, j′>j

ρi,j,j′|xi,j − xi,j′| s t x1,2 x1,3 x1,4 x2,3 x2.4 x3,4

8 5 10 5 5 1

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution

A maximal flow for a minimum cut formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j|xi,j − 1| + λi,j xi,j +

• i∈V\T

(j,j′)∈T2, j′>j

ρi,j,j′|xi,j − xi,j′| s t x1,2 x1,3 x1,4 x2,3 x2.4 x3,4

v1 v2 v3 v4

8 5 10 5 5 1

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• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution

A maximal flow for a minimum cut formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j|xi,j − 1| + λi,j xi,j +

• i∈V\T

(j,j′)∈T2, j′>j

ρi,j,j′|xi,j − xi,j′| s t x1,2 x1,3 x1,4 x2,3 x2.4 x3,4

v1 v2 v3 v4

8 5 10 5 5 1 ∞ ∞ ∞ ∞

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution

A maximal flow for a minimum cut formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j|xi,j − 1| + λi,j xi,j +

• i∈V\T

(j,j′)∈T2, j′>j

ρi,j,j′|xi,j − xi,j′| s t x1,2 x1,3 x1,4 x2,3 x2.4 x3,4

v1 v2 v3 v4

8 5 10 5 5 1 ∞ ∞ ∞ ∞ η = 6 λTF = 3 λTF = 1

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution

A maximal flow for a minimum cut formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j|xi,j − 1| + λi,j xi,j +

• i∈V\T

(j,j′)∈T2, j′>j

ρi,j,j′|xi,j − xi,j′| s t x1,2 x1,3 x1,4 x2,3 x2.4 x3,4

v1 v2 v3 v4

8 5 10 5 5 1 3 ∞ ∞ ∞ ∞ η = 6 λTF = 3 λTF = 1

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution

A maximal flow for a minimum cut formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j|xi,j − 1| + λi,j xi,j +

• i∈V\T

(j,j′)∈T2, j′>j

ρi,j,j′|xi,j − xi,j′| s t x1,2 x1,3 x1,4 x2,3 x2.4 x3,4

v1 v2 v3 v4

8 5 10 5 5 1 3 ∞ ∞ ∞ ∞ η = 6 λTF = 3 λTF = 1

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution

A maximal flow for a minimum cut formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j|xi,j − 1| + λi,j xi,j +

• i∈V\T

(j,j′)∈T2, j′>j

ρi,j,j′|xi,j − xi,j′| s = 1 t = 0 x1,2 x1,3 x1,4 x2,3 x2.4 x3,4

v1 v2 v3 v4

8 5 10 5 5 1 3 ∞ ∞ ∞ ∞ η = 6 λTF = 3 λTF = 1

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution

A maximal flow for a minimum cut formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j|xi,j − 1| + λi,j xi,j +

• i∈V\T

(j,j′)∈T2, j′>j

ρi,j,j′|xi,j − xi,j′| s = 1 t = 0 1 1 1

v1 v2 v3 v4

8 5 10 5 5 1 3 ∞ ∞ ∞ ∞ η = 6 λTF = 3 λTF = 1

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Cut — NETWORK INFERENCE WITH GRAPH CUTS Formulation and resolution

A maximal flow for a minimum cut formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j|xi,j − 1| + λi,j xi,j +

• i∈V\T

(j,j′)∈T2, j′>j

ρi,j,j′|xi,j − xi,j′| s = 1 t = 0 1 1 1

v1 v2 v3 v4

8 5 10 5 5 1 3 ∞ ∞ ∞ ∞ η = 6 λTF = 3 λTF = 1

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM

A continuous method: BRANE Relax

We look for a continuous solution for x ⇔ x ∈ [0, 1]E

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BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM A priori

A priori: modular structure and TF connectivity

minimize

x∈{0,1}E

• (i,j)∈V2 ωi,jϕ(xi,j − 1) + λi,jϕ(xi,j) + µΨ(xi,j)

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM A priori

A priori: modular structure and TF connectivity

minimize

x∈{0,1}E

• (i,j)∈V2 ωi,jϕ(xi,j − 1) + λi,jϕ(xi,j) + µΨ(xi,j)

Modular network: favors links between TFs and TFs λi,j =      2η if (i, j) / ∈ T2 2λTF if (i, j) ∈ T2 λTF + λTF

• therwise.

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM A priori

A priori: modular structure and TF connectivity

minimize

x∈{0,1}E

• (i,j)∈V2 ωi,jϕ(xi,j − 1) + λi,jϕ(xi,j) + µΨ(xi,j)

Modular network: favors links between TFs and TFs λi,j =      2η if (i, j) / ∈ T2 2λTF if (i, j) ∈ T2 λTF + λTF

• therwise.

TF connectivity: constraint TF node degree Ψ(xi,j) =

• i∈V\T

φ  

j∈V

xi,j − d   φ(·): a convex distance function with β-Lipschitz continuous gradient

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Formulation

A convex relaxation for a continuous formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j(1 − xi,j) + λi,j xi,j + µ

• i∈V\T

φ

• j∈V

xi,j − d

• July 3th, 2017
• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Formulation

A convex relaxation for a continuous formulation

minimize

x∈{0,1}E

• (i,j)∈V2

j>i

ωi,j(1 − xi,j) + λi,j xi,j + µ

• i∈V\T

φ

• j∈V

xi,j − d

• Relaxation and vectorization:

minimize

x∈[0,1]E E

• l=1

ωl(1 − xl) + λl xl + µ

P

• i=1

φ E

• k=1

Ωi,kxk − d

• ,

where Ω ∈ {0, 1}P×E encodes the degree of the P TFs nodes in the complete graph. Ωi,j =

• 1

if j is the index of an edge linking the TF node vi in the complete graph,

• therwise.

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Formulation

Distance function in BRANE Relax

minimize

x∈[0,1]E E

• l=1

ωl(1 − xl) + λl xl + µ

P

• i=1

φ E

• k=1

Ωi,kxk − d

• Choice of φ: node degree distance function, with respect to d

zi =

E

• k=1

Ωi,kxk − d squared ℓ2 norm: φ(z) = ||z||2 Huber function: φ(zi) =

• z2

i

if |zi| ≤ δ 2δ(|zi| − 1

2δ)

• therwise

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BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Resolution

Optimization strategy via proximal methods

Splitting minimize

x∈RE

ω⊤(1E − x) + λ⊤x + µΦ(Ωx − d)

• f2

+ ι[0,1]E(x)

• f1

f1 ∈ Γ0(RE): proper, convex, and lower semi-continuous f2: convex, differentiable with an L−Lipschitz continuous gradient

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BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Resolution

Optimization strategy via proximal methods

Splitting minimize

x∈RE

ω⊤(1E − x) + λ⊤x + µΦ(Ωx − d)

• f2

+ ι[0,1]E(x)

• f1

f1 ∈ Γ0(RE): proper, convex, and lower semi-continuous f2: convex, differentiable with an L−Lipschitz continuous gradient Algorithm 1: Forward-Backward

Fix x0 ∈ RE for k = 0, 1, . . . do Select the index jk ∈ {1, . . . , J} of a block of variables z(jk)

k

= x(jk)

k

− γkA−1

jk ∇jkf2(xk)

x(jk)

k+1 = proxγk−1,Ajk f

(jk) 1

(z(jk)

k

) x(¯

jk) k+1 = x(¯ jk) k

, ¯ jk = {1, . . . , J}\{jk}

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BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Resolution

Optimization strategy via proximal methods

Splitting minimize

x∈RE

ω⊤(1E − x) + λ⊤x + µΦ(Ωx − d)

• f2

+ ι[0,1]E(x)

• f1

f1 ∈ Γ0(RE): proper, convex, and lower semi-continuous f2: convex, differentiable with an L−Lipschitz continuous gradient Algorithm 2: Preconditioned Forward-Backward

Fix x0 ∈ RE for k = 0, 1, . . . do Select the index jk ∈ {1, . . . , J} of a block of variables z(jk)

k

= x(jk)

k

− γkA−1

jk ∇jkf2(xk)

x(jk)

k+1 = proxγk−1,Ajk ,f

(jk) 1

(z(jk)

k

) x(¯

jk) k+1 = x(¯ jk) k

, ¯ jk = {1, . . . , J}\{jk}

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BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM Resolution

Optimization strategy via proximal methods

Splitting minimize

x∈RE

ω⊤(1E − x) + λ⊤x + µΦ(Ωx − d)

• f2

+ ι[0,1]E(x)

• f1

f1 ∈ Γ0(RE): proper, convex, and lower semi-continuous f2: convex, differentiable with an L−Lipschitz continuous gradient Algorithm 3: Block Coordinate + Preconditioned Forward-Backward

Fix x0 ∈ RE for k = 0, 1, . . . do Select the index jk ∈ {1, . . . , J} of a block of variables z(jk)

k

= x(jk)

k

− γkA−1

jk ∇jkf2(xk)

x(jk)

k+1 = proxγk−1,Ajk ,f

(jk) 1

(z(jk)

k

) x(¯

jk) k+1 = x(¯ jk) k

, ¯ jk = {1, . . . , J}\{jk}

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

A mixed method: BRANE Clust

We look for a discrete solution for x and a continuous one for y

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING A priori

A priori: gene grouping and modular structure

maximize

x∈{0,1}E y∈NG

• (i,j)∈V2 f(yi, yj)ωi,jxi,j + λ(1 − xi,j) + Ψ(yi)

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING A priori

A priori: gene grouping and modular structure

maximize

x∈{0,1}E y∈NG

• (i,j)∈V2 f(yi, yj)ωi,jxi,j + λ(1 − xi,j) + Ψ(yi)

Clustering-assisted inference

Node labeling y ∈ NG Weight ωi,j reduction if nodes vi and vj belong to distinct clusters Cost function: f(yi, yj) = β − 1(yi = yj) β

July 3th, 2017

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING A priori

A priori: gene grouping and modular structure

maximize

x∈{0,1}E y∈NG

• (i,j)∈V2 f(yi, yj)ωi,jxi,j + λ(1 − xi,j) + Ψ(yi)

Clustering-assisted inference

Node labeling y ∈ NG Weight ωi,j reduction if nodes vi and vj belong to distinct clusters Cost function: f(yi, yj) = β − 1(yi = yj) β

TF-driven clustering promoting modular structure Ψ(yi) =

• i∈V

j∈T

µi,j1(yi = j)

µi,j: modular structure controlling parameter

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Alternating optimization strategy

maximize

x∈{0,1}n y∈NG

• (i,j)∈V2

β−1(yi=yj) β

ωi,jxi,j + λ(1 − xi,j) +

• i∈V

j∈T

µi,j1(yi = j)

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Alternating optimization strategy

Alternating optimization maximize

x∈{0,1}n y∈NG

• (i,j)∈V2

β−1(yi=yj) β

ωi,jxi,j + λ(1 − xi,j) +

• i∈V

j∈T

µi,j1(yi = j)

July 3th, 2017

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Alternating optimization strategy

Alternating optimization maximize

x∈{0,1}n y∈NG

• (i,j)∈V2

β−1(yi=yj) β

ωi,jxi,j + λ(1 − xi,j) +

• i∈V

j∈T

µi,j1(yi = j) At y fixed and x variable: maximize

x∈{0,1}n

• (i,j)∈V2

β − 1(yi = yj) β ωi,j xi,j + λ(1 − xi,j)

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Alternating optimization strategy

Alternating optimization maximize

x∈{0,1}n y∈NG

• (i,j)∈V2

β−1(yi=yj) β

ωi,jxi,j + λ(1 − xi,j) +

• i∈V

j∈T

µi,j1(yi = j) At y fixed and x variable: maximize

x∈{0,1}n

• (i,j)∈V2

β − 1(yi = yj) β ωi,j xi,j + λ(1 − xi,j) At x fixed and y variable: minimize

y∈NG

• (i,j)∈V2

ωi,j xi,j β 1(yi = yj) +

• i∈V, j∈T

µi,j 1(yi = j)

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Alternating optimization strategy

Alternating optimization maximize

x∈{0,1}n y∈NG

• (i,j)∈V2

β−1(yi=yj) β

ωi,jxi,j + λ(1 − xi,j) +

• i∈V

j∈T

µi,j1(yi = j) At y fixed and x variable: maximize

x∈{0,1}n

• (i,j)∈V2

β − 1(yi = yj) β ωi,j xi,j + λ(1 − xi,j) Explicit form: x∗

i,j =

• 1

if ωi,j >

λβ β−1(yi=yj)

• therwise.

At x fixed and y variable: minimize

y∈NG

• (i,j)∈V2

ωi,j xi,j β 1(yi = yj) +

• i∈V, j∈T

µi,j 1(yi = j)

July 3th, 2017

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Clustering optimization strategy

At x fixed and y variable: minimize

y∈NG

• (i,j)∈V2

ωi,j xi,j β 1(yi = yj) +

• i∈V, j∈T

µi,j1(yi = j)

July 3th, 2017

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Clustering optimization strategy

At x fixed and y variable: minimize

y∈NG

• (i,j)∈V2

ωi,j xi,j β 1(yi = yj) +

• i∈V, j∈T

µi,j1(yi = j) (NP)

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Clustering optimization strategy

At x fixed and y variable: minimize

y∈NG

• (i,j)∈V2

ωi,j xi,j β 1(yi = yj) +

• i∈V, j∈T

µi,j1(yi = j) (NP) discrete problem ⇒ quadratic relaxation T-class problem ⇒ T binary sub-problems

label restriction to T: {s(1), . . . , s(T)} such that s(t)

j

= 1 if j = t and 0 otherwise. Y = {y(1), . . . , y(T)} such that y(t) ∈ [0, 1]G

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Clustering optimization strategy

At x fixed and y variable: minimize

y∈NG

• (i,j)∈V2

ωi,j xi,j β 1(yi = yj) +

• i∈V, j∈T

µi,j1(yi = j) (NP) discrete problem ⇒ quadratic relaxation T-class problem ⇒ T binary sub-problems

label restriction to T: {s(1), . . . , s(T)} such that s(t)

j

= 1 if j = t and 0 otherwise. Y = {y(1), . . . , y(T)} such that y(t) ∈ [0, 1]G

Problem re-expressed as:

minimize

Y

T

• t=1

 

(i,j)∈V2

ωi,j xi,j β

• y(t)

i

− y(t)

j

2 +

• i∈V, j∈T

µi,j

• y(t)

i

− s(t)

j

2  

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Clustering optimization strategy

minimize

Y

T

• t=1

 

(i,j)∈V2

ωi,j xi,j β

• y(t)

i

− y(t)

j

2 +

• i∈V, j∈T

µi,j

• y(t)

i

− s(t)

j

2   This is the Combinatorial Dirichlet problem Minimization via solving a linear system of equations [Grady, 2006]

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Clustering optimization strategy

minimize

Y

T

• t=1

 

(i,j)∈V2

ωi,j xi,j β

• y(t)

i

− y(t)

j

2 +

• i∈V, j∈T

µi,j

• y(t)

i

− s(t)

j

2   This is the Combinatorial Dirichlet problem Minimization via solving a linear system of equations [Grady, 2006] Final labeling: node i is assigned to label t for which y(t)

i

is maximal y∗

i = argmax t∈T

y(t)

i

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Random walker in graphs

We want to obtain the optimal labeling y∗ based on a weighted graph ⇒ Random Walker algorithm

y1 y4 y2 y3 y5 12 3 10 7 6 5 5 9 3 10 July 3th, 2017

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Random walker in graphs

We want to obtain the optimal labeling y∗ based on a weighted graph ⇒ Random Walker algorithm

y1 y4 y2 y3 y5 1 2 3 12 3 10 7 6 5 5 9 3 10 July 3th, 2017

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Random walker in graphs

We want to obtain the optimal labeling y∗ based on a weighted graph ⇒ Random Walker algorithm

y1 y4 y2 y3 y5 1 2 3 12 3 10 7 6 5 5 9 3 10 0.95 0.46 0.01 0.03 0.35 1

y(1)

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Random walker in graphs

We want to obtain the optimal labeling y∗ based on a weighted graph ⇒ Random Walker algorithm

y1 y4 y2 y3 y5 1 2 3 12 3 10 7 6 5 5 9 3 10 0.95 0.46 0.01 0.03 0.35 1 0.01 0.28 0.97 0.02 0.19 1

y(1) y(2)

July 3th, 2017

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Random walker in graphs

We want to obtain the optimal labeling y∗ based on a weighted graph ⇒ Random Walker algorithm

y1 y4 y2 y3 y5 1 2 3 12 3 10 7 6 5 5 9 3 10 0.95 0.46 0.01 0.03 0.35 1 0.01 0.28 0.97 0.02 0.19 1 0.03 0.26 0.02 0.95 0.46 1

y(1) y(2) y(3)

July 3th, 2017

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

Random walker in graphs

We want to obtain the optimal labeling y∗ based on a weighted graph ⇒ Random Walker algorithm

y1 y4 y2 y3 y5 1 2 3 12 3 10 7 6 5 5 9 3 10 0.95 0.46 0.95 0.46 0.01 0.03 0.35 1 0.01 0.28 0.97 0.97 0.02 0.19 1 0.03 0.26 0.02 0.95 0.46 0.95 0.46 1 1 1 2 3 3

y(1) y(2) y(3) y∗ = {1, 1, 2, 3, 3}

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BRANE Clust — NETWORK INFERENCE WITH CLUSTERING Formulation and resolution

hard- vs soft- clustering in BRANE Clust

minimize

Y T

• t=1
• (i,j)∈V2

ωi,j xi,j β

• y(t)

i

− y(t)

j

2 +

• i∈V, j∈T

µi,j

• y(t)

i

− s(t)

j

2

• hard-clustering

soft-clustering # clusters = # TF # clusters ≤ # TF µi,j =

• → ∞

if i = j

• therwise.

µi,j =      α if i = j α1(ωi,j > τ) if i = j and i ∈ T ωi,j1(ωi,j > τ) if i = j and i / ∈ T

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BRANE RESULTS

It’s time to test the BRANE philosophy...

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BRANE RESULTS Methodology

Numerical evaluation strategy

Gene expression data DREAM4 or DREAM5 challenges Gene-gene interaction scores (ND)-CLR or (ND)-GENIE3 Classical thresholding

BRANE edge selection P =

|TP| |TP|+|FP|

R =

|TP| |TP|+|FN|

Reference Precision-Recall curve BRANE Precision-Recall curve

AUPR Ref AUPR BRANE

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BRANE RESULTS DREAM4 synthetic results

BRANE performance on in-silico data

DREAM4 [Marbach et al., 2010]

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BRANE RESULTS DREAM4 synthetic results

BRANE performance on in-silico data

DREAM4 [Marbach et al., 2010]

Network 1 2 3 4 5 Average Gain CLR 0.256 0.275 0.314 0.313 0.318 0.295 BRANE Cut 0.282 0.308 0.343 0.344 0.356 0.327 10.9 % BRANE Relax 0.278 0.293 0.336 0.333 0.345 0.317 7.8 % BRANE Clust 0.275 0.337 0.360 0.335 0.342 0.330 12.2 % GENIE3 0.269 0.288 0.331 0.323 0.329 0.308 BRANE Cut 0.298 0.316 0.357 0.344 0.352 0.333 8.4 % BRANE Relax 0.293 0.320 0.356 0.345 0.354 0.334 8.5 % BRANE Clust 0.287 0.348 0.364 0.371 0.367 0.347 12.8 % Network 1 2 3 4 5 Average Gain ND-CLR 0.254 0.250 0.324 0.318 0.331 0.295 BRANE Cut 0.271 0.277 0.334 0.335 0.343 0.312 5.9 % BRANE Relax 0.270 0.264 0.327 0.325 0.332 0.304 3.1 % BRANE Clust 0.258 0.251 0.327 0.337 0.342 0.303 2.5 % ND-GENIE3 0.263 0.275 0.336 0.328 0.354 0.309 BRANE Cut 0.275 0.312 0.367 0.346 0.368 0.334 7.2 % BRANE Relax 0.276 0.307 0.369 0.347 0.371 0.334 7.3 % BRANE Clust 0.273 0.311 0.354 0.373 0.370 0.336 8.1 % July 3th, 2017

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BRANE RESULTS DREAM4 synthetic results

BRANE performance on in-silico data

DREAM4 [Marbach et al., 2010] CLR GENIE3 ND-CLR ND-GENIE3 BRANE Cut 10.9 % 8.4 % 5.9 % 7.2 % BRANE Relax 7.8 % 8.5 % 3.1 % 7.3 % BRANE Clust 12.2 % 12.8 % 2.5 % 8.1 % BRANE approaches validated on small synthetic data BRANE methodologies outperform classical thresholding First and second best performers: BRANE Clust and BRANE Cut ⇒ Validation on more realistic synthetic data

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BRANE RESULTS DREAM5 synthetic results

BRANE performance on in-silico data

DREAM5 [Marbach et al., 2012]

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BRANE RESULTS DREAM5 synthetic results

BRANE performance on in-silico data

DREAM5 [Marbach et al., 2012]

AUPR Gain AUPR Gain CLR 0.252 GENIE3 0.283 BRANE Cut 0.268 6.3 % BRANE Cut 0.295 4.2 % BRANE Relax 0.272 7.9 % BRANE Relax 0.294 3.8 % BRANE Clust 0.301 19.4 % BRANE Clust 0.336 18.6 % AUPR Gain AUPR Gain ND-CLR 0.272 ND-GENIE3 0.313 BRANE Cut 0.277 1.9 % BRANE Cut 0.317 1.1 % BRANE Relax 0.274 0.6 % BRANE Relax 0.314 0.3 % BRANE Clust 0.289 6.2 % BRANE Clust 0.345 10.2 %

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BRANE RESULTS DREAM5 synthetic results

BRANE performance on in-silico data

DREAM5 [Marbach et al., 2012]

AUPR Gain AUPR Gain CLR 0.252 GENIE3 0.283 BRANE Cut 0.268 6.3 % BRANE Cut 0.295 4.2 % BRANE Relax 0.272 7.9 % BRANE Relax 0.294 3.8 % BRANE Clust 0.301 19.4 % BRANE Clust 0.336 18.6 % AUPR Gain AUPR Gain ND-CLR 0.272 ND-GENIE3 0.313 BRANE Cut 0.277 1.9 % BRANE Cut 0.317 1.1 % BRANE Relax 0.274 0.6 % BRANE Relax 0.314 0.3 % BRANE Clust 0.289 6.2 % BRANE Clust 0.345 10.2 %

BRANE approaches validated on realistic synthetic data and outperform classical thresholding First and second best performer: BRANE Clust and BRANE Cut ⇒ Validation of BRANE Cut and BRANE Clust on real data

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BRANE RESULTS Escherichia coli results

BRANE Clust performance on real data

Escherichia coli dataset

AUPR Gain AUPR Gain CLR 0.0378 5.5 % GENIE3 0.0488 9.8 % BRANE Clust 0.0399 BRANE Clust 0.0536

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BRANE RESULTS Escherichia coli results

BRANE Clust performance on real data

Escherichia coli dataset

AUPR Gain AUPR Gain CLR 0.0378 5.5 % GENIE3 0.0488 9.8 % BRANE Clust 0.0399 BRANE Clust 0.0536

BRANE Clust predictions using GENIE3 weights

July 3th, 2017

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BRANE RESULTS Escherichia coli results

BRANE Clust performance on real data

Escherichia coli dataset

AUPR Gain AUPR Gain CLR 0.0378 5.5 % GENIE3 0.0488 9.8 % BRANE Clust 0.0399 BRANE Clust 0.0536

BRANE Clust predictions using GENIE3 weights BRANE Clust validated on real dataset

July 3th, 2017

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BRANE RESULTS Trichoderma results results

BRANE Cut in the real life

GRN of T. reesei obtained with BRANE Cut using CLR weights

July 3th, 2017

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BRANE RESULTS Trichoderma results results

BRANE Cut in the real life

GRN of T. reesei obtained with BRANE Cut using CLR weights

July 3th, 2017

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BRANE RESULTS Trichoderma results results

BRANE Cut in the real life

GRN of T. reesei obtained with BRANE Cut using CLR weights

July 3th, 2017

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BRANE RESULTS Trichoderma results results

BRANE Cut in the real life

GRN of T. reesei obtained with BRANE Cut using CLR weights

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CONCLUSIONS

It’s time to conclude...

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CONCLUSIONS

Conclusions

Inference: BRANE Cut and BRANE Relax Joint inference and clustering: BRANE Clust

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CONCLUSIONS

Conclusions

Inference: BRANE Cut and BRANE Relax Joint inference and clustering: BRANE Clust The BRA- in BRANE: integrating biological a priori constrains the search of relevant edges The -NE in BRANE: proposed graph inference methods lead to promising results and outperforms state-of-the-art methods

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CONCLUSIONS

Conclusions

Inference: BRANE Cut and BRANE Relax Joint inference and clustering: BRANE Clust The BRA- in BRANE: integrating biological a priori constrains the search of relevant edges The -NE in BRANE: proposed graph inference methods lead to promising results and outperforms state-of-the-art methods ⇒ Average improvements around 10 % ⇒ Biological relevant inferred networks ⇒ Negligible time complexity with respect to graph weight computation

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CONCLUSIONS

Conclusions

Inference: BRANE Cut and BRANE Relax Joint inference and clustering: BRANE Clust The BRA- in BRANE: integrating biological a priori constrains the search of relevant edges The -NE in BRANE: proposed graph inference methods lead to promising results and outperforms state-of-the-art methods ⇒ Average improvements around 10 % ⇒ Biological relevant inferred networks ⇒ Negligible time complexity with respect to graph weight computation Biological a priori relevance for network inference BRANE Clust ≻ BRANE Cut ≻ BRANE Relax

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CONCLUSIONS

Perspectives

From biological graphs...

GRN

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CONCLUSIONS

Perspectives

From biological graphs...

TF-based a priori GRN

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CONCLUSIONS

Perspectives

From biological graphs...

TF-based a priori GRN Clustering

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• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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CONCLUSIONS

Perspectives

From biological graphs...

TF-based a priori GRN Clustering

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• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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CONCLUSIONS

Perspectives

From biological graphs...

TF-based a priori GRN Clustering

Extend TF-based a priori for

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CONCLUSIONS

Perspectives

From biological graphs...

TF-based a priori GRN Clustering

Extend TF-based a priori for GRN, clustering

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• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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CONCLUSIONS

Perspectives

From biological graphs...

TF-based a priori GRN Clustering Gene-gene scores

Extend TF-based a priori for GRN, clustering

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• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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CONCLUSIONS

Perspectives

From biological graphs...

TF-based a priori GRN Clustering Gene-gene scores Normalized gene expression data

Extend TF-based a priori for GRN, clustering

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CONCLUSIONS

Perspectives

From biological graphs...

TF-based a priori GRN Clustering Normalized gene expression data Gene-gene scores

Extend TF-based a priori for GRN, clustering , graph weighting,

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CONCLUSIONS

Perspectives

From biological graphs...

TF-based a priori GRN Clustering Gene-gene scores Normalized gene expression data

Extend TF-based a priori for GRN, clustering , graph weighting, data normalization...

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• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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CONCLUSIONS

Perspectives

From biological graphs...

TF-based a priori GRN Clustering Gene-gene scores Normalized gene expression data

Extend TF-based a priori for GRN, clustering , graph weighting, data normalization... Integrate transcriptomic data treatment

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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CONCLUSIONS

Perspectives

From biological graphs...

TF-based a priori GRN Clustering Gene-gene scores Normalized gene expression data

Extend TF-based a priori for GRN, clustering , graph weighting, data normalization... Integrate transcriptomic data treatment

July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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CONCLUSIONS

Perspectives

From biological graphs...

TF-based a priori GRN Clustering Normalized gene expression data

Extend TF-based a priori for GRN, clustering , graph weighting, data normalization... Integrate transcriptomic data treatment

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• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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CONCLUSIONS

Perspectives

From biological graphs...

Omics data TF-based a priori GRN Clustering Omics data

Extend TF-based a priori for GRN, clustering , graph weighting, data normalization... Integrate transcriptomic data treatment Integrate a priori, omics- data and treatments

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CONCLUSIONS

Perspectives

... to general graphs BRANE-like applications for non biological graphs Coupled edge inference: social networks Node-degree constraint: telecommunication Coupling between inference and clustering: temperature networks, brain networks

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CONCLUSIONS

Perspectives

... to general graphs BRANE-like applications for non biological graphs Coupled edge inference: social networks Node-degree constraint: telecommunication Coupling between inference and clustering: temperature networks, brain networks Topological constraint in graph inference Expected node degree distribution Scale-free networks: webgraphs, financial networks, social networks...

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CONCLUSIONS

Perspectives

... to general graphs BRANE-like applications for non biological graphs Coupled edge inference: social networks Node-degree constraint: telecommunication Coupling between inference and clustering: temperature networks, brain networks Topological constraint in graph inference Expected node degree distribution Scale-free networks: webgraphs, financial networks, social networks... Laplacian-based approach for graph comparison Spectral view of the graph Modularity Local and topological-based criteria

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CONCLUSIONS

Publications

Journal papers — published

• D. Poggi-Parodi, F. Bidard, A. Pirayre, T. Portnoy. C. Blugeon, B. Seiboth, C.P. Kubicek, S. Le Crom and A. Margeot

Kinetic transcriptome reveals an essentially intact induction system in a cellu- lase hyper-producer Trichoderma reesei strain Biotechnology for Biofuels, 7:173, Dec. 2014

• A. Pirayre, C. Couprie, F. Bidard, L. Duval, and J.-C. Pesquet.

BRANE Cut: biologically-related a priori network enhancement with graph cuts for gene regulatory network inference BMC Bioinformatics, 16(1):369, Dec. 2015.

• A. Pirayre, C. Couprie, L. Duval, and J.-C. Pesquet.

BRANE Clust: Cluster-Assisted Gene Regulatory Network Inference Refinement IEEE/ACM Transactions on Computational Biology and Bioinformatics, Mar. 2017.

Journal papers — in preparation

• Y. Zheng, A. Pirayre, L. Duval and J.-C. Pesquet

Joint restoration/segmentation of multicomponent images with variational Bayes and higher-order graphical models (HOGMep) To be submitted to IEEE Transactions on Computational Imaging, Jul. 2017.

• A. Pirayre, D. Ivanoff, L. Duval, C. Blugeon, C. Firmo, S. Perrin, E. Jourdier, A. Margeot and F. Bidard

Growing Trichoderma reesei on a mix of carbon sources suggests links between development and cellulase production To be submitted to BMC Genomics, Jul. 2017. July 3th, 2017

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CONCLUSIONS July 3th, 2017

• Recons. and Clust. with Graph Optim. and Priors on GRN and Images

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HOGMep

HOGMep for non-blind inverse problems

y = H x + n x: unknown signal to be recovered H: known degradation operator n: additive noise y: observations

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HOGMep

HOGMep — Bayesian framework

Estimation of x from the knowledge of the posterior pdf p(x|y) p(x|y) = p(x)p(y|x) p(y) p(x): the marginal pdf encoding information about x p(y|x): the likelihood highlighting the uncertainty in y p(y): the marginal pdf of y

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HOGMep

HOGMep — Variational Bayesian Approximation

q(x): approximation of p(x|y) qopt(x) = argmin

q(x)

KL(q(x) || p(x | y)) Separable distribution: q(x) =

J

• j=1

qj(xj), with qopt

j (xj) ∝ exp

• ln p(y, x)

i=j qi(xi)

• Estimation of the distributions in an iterative manner

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HOGMep

HOGMep — Bayesian formulation

Likelihood prior: p(y | x, γ) = N(Hx, γ−1I) p(z): prior on hidden variables z ⇒ generalized Potts model p(x|z): prior on x conditionally to z ⇒ MEP distribution restricted to Gaussian Scale Mixtures GSM(m, Ω, β) Hyperpriors: p(γ), p(ml) and p(Ωl)

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HOGMep

HOGMep — Bayesian formulation

Likelihood prior: p(y | x, γ) = N(Hx, γ−1I) p(z): prior on hidden variables z ⇒ generalized Potts model p(x|z): prior on x conditionally to z ⇒ MEP distribution restricted to Gaussian Scale Mixtures GSM(m, Ω, β) Hyperpriors: p(γ), p(ml) and p(Ωl)

Joint posterior distribution

p(y | x, γ)

N

• i=1
• p(xi | zi, ui, m, Ω)p(ui | β)
• p(z)p(γ)

L

• l=1

p(ml)p(Ωl)

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HOGMep

HOGMep — VBA strategy

Separable form for the approximation: q(Θ) =

N

• i=1

(q(xi, zi)q(ui)) q(γ)

L

• l=1

(q(ml)q(Ωl)) with q(xi|zi = l) = N(ηi,l, Ξi,l), q(zi = l) = πi,l, q(ml) = N(µl, Λl), q(Ωl) = W(Γl, νl), q(γ) = G(a, b).

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HOGMep

HOGMep — Some restoration results

Restoration Original Degraded DR 3MG VB-MIG HOGMep SNR 6.655 9.467 6.744 12.737 12.895 Original Degraded DR 3MG VB-MIG HOGMep SNR 19.659 18.728 17.188 15.486 19.555

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HOGMep

HOGMep — Some segmentation results

Segmentation ICM ICM SC SC VB-MIG HOGMep ICM ICM SC SC VB-MIG HOGMep

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