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

PhD supervisors:

Frédérique BIDARD-MICHELOT Camille COUPRIE Laurent DUVAL Jean-Christophe PESQUET

IFP Energies nouvelles Facebook A.I. Research IFP Energies nouvelles CentraleSupélec

July 3th , 2017

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

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I NTRODUCTION

An overview

Gene regulatory networks

Signals and images

Our framework

Variational

Bayes variational

Method

BRANE

HOGMep

Reconstruction

Clustering

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I NTRODUCTION

Context

Biological motivation Second generation bio-fuel production Cellulases from Trichoderma reesei Lignocellulosic Biomass

Pre-

trea

tme n

t

Cellulose Hemi-cellulose

July 3th , 2017

atic ym Enz sis roly Hyd

Sugar F erm ent a

tio

n

Ethanol

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

g xin Mi s uel hf wit

Bio-fuels

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I NTRODUCTION

Context

Biological motivation Second generation bio-fuel production Cellulases from Trichoderma reesei Lignocellulosic Biomass

Pre-

trea

tme n

t

Cellulose Hemi-cellulose

atic ym Enz sis roly Hyd

Sugar F erm ent a

tio

n

Ethanol

g xin Mi s uel hf wit

Bio-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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I NTRODUCTION

Context

Biological motivation Second generation bio-fuel production Cellulases from Trichoderma reesei Lignocellulosic Biomass

Pre-

trea

tme n

t

Cellulose Hemi-cellulose

atic ym Enz sis roly Hyd

Sugar F erm ent a

tio

n

Ethanol

g xin Mi s uel hf wit

Bio-fuels

Improve Trichoderma reseei cellulase production Understand cellulase production mechanisms ⇒ Use of Gene Regulatory Network (GRN) July 3th , 2017

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

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I NTRODUCTION

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

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v3

vX

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

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I NTRODUCTION

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

TF2

TF3

Gene X

mRNA

Protein

July 3th , 2017

TF1

⊕

⊖

TFX ⊕⊕

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

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I NTRODUCTION

Data overview

What biological data can be used? For a given experimental condition, transcriptomic data answer to: which genes are expressed? in which amount?

July 3th , 2017

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

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I NTRODUCTION

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

z −0.948  0.737  M = −0.253  3.747 1.383 

July 3th , 2017

S conditions }| −0.013 . . . −1.308 0.619 . . . −0.141 −0.175 . . . −0.859 1.115 . . . −0.418 1.184 . . . −0.493

{ −0.977 −0.803 −0.595 −0.084 −0.562

          

G genes

What do transcriptomic data look like? Gene expression data (GED): G genes × S conditions

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

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I NTRODUCTION

Links between data and GRNs

How to use GED to produce a GRN ? sj 

From gene expression data...

July 3th , 2017

M=

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

gi

  ···

. . . mi,j

  , 

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I NTRODUCTION

Links between data and GRNs

How to use GED to produce a GRN ? sj 

From gene expression data...

M=

gi

  ···

. . . mi,j

  , 

V = {v1 , · · · , vG } a set of vertices (genes) and E a set of edges

leading to a complete graph...

July 3th , 2017

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

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I NTRODUCTION

Links between data and GRNs

How to use GED to produce a GRN ? sj 

From gene expression data...

M=

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 gj

leading to a complete weighted graph...

July 3th , 2017

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

 W=

gi

  ···

. . . ωi,j

  , 

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I NTRODUCTION

Links between data and GRNs

How to use GED to produce a GRN ? sj 

From gene expression data...

M=

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 gj

leading to a complete weighted graph...

 W=

gi

  ···

. . . ωi,j

  , 

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

July 3th , 2017

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

( Wi,j =

1 0

if ei,j ∈ E ∗ otherwise.

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I NTRODUCTION

Difficulties in GRN inference

July 3th , 2017

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

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I NTRODUCTION

Difficulties in GRN inference

July 3th , 2017

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

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I NTRODUCTION

Difficulties in GRN inference

July 3th , 2017

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

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I NTRODUCTION

Difficulties in GRN inference

July 3th , 2017

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

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I NTRODUCTION

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 ?

July 3th , 2017

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

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I NTRODUCTION

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 ? ( 1 if ei,j ∈ E ∗ , We note xi,j the binary label of edge presence: xi,j = 0 otherwise.

July 3th , 2017

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

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I NTRODUCTION

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 ? ( 1 if ei,j ∈ E ∗ , We note xi,j the binary label of edge presence: xi,j = 0 otherwise. ( ∗ = 1 if ωi,j > λ, Classical thresholding: xi,j 0 otherwise.

July 3th , 2017

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

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I NTRODUCTION

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 ? ( 1 if ei,j ∈ E ∗ , We note xi,j the binary label of edge presence: xi,j = 0 otherwise. ( ∗ = 1 if ωi,j > λ, Classical thresholding: xi,j 0 otherwise. Given by a cost function for given weights ω: maximize E x∈{0,1}

July 3th , 2017

P

ωi,j xi,j + λ(1 − xi,j )

(i,j)∈V2

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

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I NTRODUCTION

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 ? ( 1 if ei,j ∈ E ∗ , We note xi,j the binary label of edge presence: xi,j = 0 otherwise. ( ∗ = 1 if ωi,j > λ, Classical thresholding: xi,j 0 otherwise. Given by a cost function for given weights ω: maximize E x∈{0,1}

July 3th , 2017

P (i,j)∈V2

ωi,j xi,j + λ(1 − xi,j ) ⇔ minimize E x∈{0,1}

P

ωi,j (1 − xi,j ) + λ xi,j

(i,j)∈V2

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

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I NTRODUCTION

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

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I NTRODUCTION

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 BRANE Relax

Gene co-regulatiton TF-connectivity

Discrete Continuous

Maximal flow Proximal method

Joint inference and clustering

BRANE Clust

Gene grouping

Mixed

Alternating scheme

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 discrete method: BRANE Cut We look for a discrete solution for x ⇔ x ∈ {0, 1}E

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

July 3th , 2017

P

ωi,j ϕ(xi,j − 1) + λi,j ϕ(xi,j ) + µΨ(xi,j )

(i,j)∈V2

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

P

ωi,j ϕ(xi,j − 1) + λi,j ϕ(xi,j ) + µΨ(xi,j )

(i,j)∈V2

Modular network: favors links between TFs and TFs   2η λi,j = 2λTF   λTF + λTF

July 3th , 2017

if (i, j) ∈ / T2 if (i, j) ∈ T2 otherwise.

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

P

ωi,j ϕ(xi,j − 1) + λi,j ϕ(xi,j ) + µΨ(xi,j )

(i,j)∈V2

Modular network: favors links between TFs and TFs   2η λi,j = 2λTF   λTF + λTF

if (i, j) ∈ / T2 if (i, j) ∈ T2 otherwise.

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 = July 3th , 2017

|V| |T |

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 E x∈{0,1}

July 3th , 2017

P

ωi,j ϕ(xi,j − 1) + λi,j ϕ(xi,j ) + µΨ(xi,j )

(i,j)∈V2

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 E x∈{0,1}

P

ωi,j ϕ(xi,j − 1) + λi,j ϕ(xi,j ) + µΨ(xi,j )

(i,j)∈V2

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

X 0

ρi,j,j0 |xi,j − xi,j0 | 2

(j,j )∈T i∈V\T

ρi,j,j0 : co-regulation probability

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 E x∈{0,1}

P

ωi,j ϕ(xi,j − 1) + λi,j ϕ(xi,j ) + µΨ(xi,j )

(i,j)∈V2

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

X 0

ρi,j,j0 |xi,j − xi,j0 | 2

(j,j )∈T i∈V\T

ρi,j,j0 : co-regulation probability with X ρi,j,j0 =

1(min{ωj,j0 , ωj,k , ωj0 ,k } > γ)

k∈V\(T ∪{i}) |V\T |−1

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

July 3th , 2017

P

ωi,j |xi,j − 1| + λi,j xi,j +

(i,j)∈V2 j>i

P

ρi,j,j0 |xi,j − xi,j0 |

i∈V\T (j,j0 )∈T2 , j0 >j

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

July 3th , 2017

P

ωi,j |xi,j − 1| + λi,j xi,j +

(i,j)∈V2 j>i

P

ρi,j,j0 |xi,j − xi,j0 |

i∈V\T (j,j0 )∈T2 , j0 >j

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

P

ωi,j |xi,j − 1| + λi,j xi,j +

(i,j)∈V2 j>i

P

s

July 3th , 2017

ρi,j,j0 |xi,j − xi,j0 |

i∈V\T (j,j0 )∈T2 , j0 >j

t

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

P

ωi,j |xi,j − 1| + λi,j xi,j +

(i,j)∈V2 j>i

P

ρi,j,j0 |xi,j − xi,j0 |

i∈V\T (j,j0 )∈T2 , j0 >j

x1,2 x1,3 x1,4 s x2,3

t

x2.4 x3,4 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

P

ωi,j |xi,j − 1| + λi,j xi,j +

(i,j)∈V2 j>i

P

ρi,j,j0 |xi,j − xi,j0 |

i∈V\T (j,j0 )∈T2 , j0 >j

x1,2

s

8 5 5 10 5 1

x1,3 x1,4 x2,3

t

x2.4 x3,4 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

P

P

ωi,j |xi,j − 1| + λi,j xi,j +

(i,j)∈V2 j>i

ρi,j,j0 |xi,j − xi,j0 |

i∈V\T (j,j0 )∈T2 , j0 >j

x1,2

s

8 5 5 10 5 1

x1,3

v1

x1,4

v2

x2,3

v3

x2.4

v4

t

x3,4 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

P

P

ωi,j |xi,j − 1| + λi,j xi,j +

(i,j)∈V2 j>i

ρi,j,j0 |xi,j − xi,j0 |

i∈V\T (j,j0 )∈T2 , j0 >j

x1,2

s

8 5 5 10 5 1

x1,3

v1

x1,4

v2

x2,3

v3

x2.4

v4

∞ ∞ ∞

t

∞

x3,4 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

P (i,j)∈V2 j>i

ρi,j,j0 |xi,j − xi,j0 |

i∈V\T (j,j0 )∈T2 , j0 >j

x1,2

s

P

ωi,j |xi,j − 1| + λi,j xi,j +

8 5 5 10 5 1

η=6 λTF = 3 λTF = 1

x1,3

v1

x1,4

v2

x2,3

v3

x2.4

v4

∞ ∞ ∞

t

∞

x3,4 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

P (i,j)∈V2 j>i

ρi,j,j0 |xi,j − xi,j0 |

i∈V\T (j,j0 )∈T2 , j0 >j

x1,2

s

P

ωi,j |xi,j − 1| + λi,j xi,j +

8 5 5 10 5 1

η=6 λTF = 3 λTF = 1

x1,3

v1

x1,4

v2

x2,3

v3

x2.4

v4

∞ ∞ ∞

t

∞

3 x3,4

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

P (i,j)∈V2 j>i

ρi,j,j0 |xi,j − xi,j0 |

i∈V\T (j,j0 )∈T2 , j0 >j

x1,2

s

P

ωi,j |xi,j − 1| + λi,j xi,j +

8 5 5 10 5 1

η=6 λTF = 3 λTF = 1

x1,3

v1

x1,4

v2

x2,3

v3

x2.4

v4

∞ ∞ ∞

t

∞

3 x3,4

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

P

P

ωi,j |xi,j − 1| + λi,j xi,j +

(i,j)∈V2 j>i

x1,2

s=1

ρi,j,j0 |xi,j − xi,j0 |

i∈V\T (j,j0 )∈T2 , j0 >j

8 5 5 10 5 1

η=6 λTF = 3 λTF = 1

x1,3

v1

x1,4

v2

∞

x2,3

v3

∞

x2.4

v4

∞ ∞

t=0

3 x3,4

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

P

P

ωi,j |xi,j − 1| + λi,j xi,j +

(i,j)∈V2 j>i

η=6 λTF = 3 λTF = 1

1 8

s=1

ρi,j,j0 |xi,j − xi,j0 |

i∈V\T (j,j0 )∈T2 , j0 >j

5 5 10 5 1

1

v1

0

v2

∞

1

v3

∞

∞ ∞

t=0

v4

0 3

0 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

P

P

ωi,j |xi,j − 1| + λi,j xi,j +

(i,j)∈V2 j>i

η=6 λTF = 3 λTF = 1

1 8

s=1

ρi,j,j0 |xi,j − xi,j0 |

i∈V\T (j,j0 )∈T2 , j0 >j

5 5 10 5 1

1

v1

0

v2

∞

1

v3

∞

∞ ∞

t=0

v4

0 3

0 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

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 E x∈{0,1}

July 3th , 2017

P

ωi,j ϕ(xi,j − 1) + λi,j ϕ(xi,j ) + µΨ(xi,j )

(i,j)∈V2

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 E x∈{0,1}

P

ωi,j ϕ(xi,j − 1) + λi,j ϕ(xi,j ) + µΨ(xi,j )

(i,j)∈V2

Modular network: favors links between TFs and TFs   2η λi,j = 2λTF   λTF + λTF

July 3th , 2017

if (i, j) ∈ / T2 if (i, j) ∈ T2 otherwise.

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 E x∈{0,1}

P

ωi,j ϕ(xi,j − 1) + λi,j ϕ(xi,j ) + µΨ(xi,j )

(i,j)∈V2

Modular network: favors links between TFs and TFs   2η λi,j = 2λTF   λTF + λTF

if (i, j) ∈ / T2 if (i, j) ∈ T2 otherwise.

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

X i∈V\T

  X φ xi,j − d j∈V

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

July 3th , 2017

P

ωi,j (1 − xi,j ) + λi,j xi,j +

(i,j)∈V2 j>i

µ

P

φ

i∈V\T

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

P

xi,j − d

j∈V

16 / 45

BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM

Formulation

A convex relaxation for a continuous formulation ! P

minimize x∈{0,1}E

ωi,j (1 − xi,j ) + λi,j xi,j +

(i,j)∈V2 j>i

µ

P

P

φ

xi,j − d

j∈V

i∈V\T

Relaxation and vectorization: minimize x∈[0,1]E

E X

ωl (1 − xl ) + λl xl + µ

l=1

P X

φ

E X

i=1

! Ωi,k xk − d ,

k=1

P×E

where Ω ∈ {0, 1} encodes the degree of the P TFs nodes in the complete graph. ( 1 if j is the index of an edge linking the TF node vi in the complete graph, Ωi,j = 0 otherwise.

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 P

ωl (1 − xl ) + λl xl + µ

l=1

P P

 φ

i=1

E P

 Ωi,k xk − d

k=1

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

E P

Ωi,k xk − d

k=1

squared `2 norm: φ(z) = ( ||z||2 z2 if |zi | ≤ δ Huber function: φ(zi ) = i 1 2δ(|zi | − 2 δ) otherwise

July 3th , 2017

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

17 / 45

BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM

Resolution

Optimization strategy via proximal methods Splitting minimize ω > (1E − x) + λ> x + µΦ(Ωx − d) + ι[0,1]E (x) | {z } x∈RE | {z } f2

f1

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

Resolution

Optimization strategy via proximal methods Splitting minimize ω > (1E − x) + λ> x + µΦ(Ωx − d) + ι[0,1]E (x) | {z } x∈RE | {z } f2

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 (j ) (j ) zk k = xk k − γk A−1 jk ∇jk f2 (xk ) (j )

k xk+1 = proxγ −1 ,A k

(j )

(jk ) jk f1

(zk k )

(j¯k ) (j¯ ) xk+1 = xk k , j¯k = {1, . . . , J}\{jk } July 3th , 2017

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

18 / 45

BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM

Resolution

Optimization strategy via proximal methods Splitting minimize ω > (1E − x) + λ> x + µΦ(Ωx − d) + ι[0,1]E (x) | {z } x∈RE | {z } f2

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 (j ) (j ) zk k = xk k − γk A−1 jk ∇jk f2 (xk ) (j )

k xk+1 = proxγ −1 ,A k

(j )

(jk ) jk ,f1

(zk k )

(j¯k ) (j¯ ) xk+1 = xk k , j¯k = {1, . . . , J}\{jk } July 3th , 2017

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

18 / 45

BRANE Relax — NETWORK INFERENCE AS A RELAXED PROBLEM

Resolution

Optimization strategy via proximal methods Splitting minimize ω > (1E − x) + λ> x + µΦ(Ωx − d) + ι[0,1]E (x) | {z } x∈RE | {z } f2

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 (j ) (j ) zk k = xk k − γk A−1 jk ∇jk f2 (xk ) (j )

k xk+1 = proxγ −1 ,A k

(j )

(jk ) jk ,f1

(zk k )

(j¯k ) (j¯ ) xk+1 = xk k , j¯k = {1, . . . , J}\{jk } July 3th , 2017

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

18 / 45

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

July 3th , 2017

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

19 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

A priori

A priori: gene grouping and modular structure maximize x∈{0,1}E y∈NG

July 3th , 2017

P

f (yi , yj )ωi,j xi,j + λ(1 − xi,j ) + Ψ(yi )

(i,j)∈V2

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

20 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

A priori

A priori: gene grouping and modular structure maximize x∈{0,1}E y∈NG

P

f (yi , yj )ωi,j xi,j + λ(1 − xi,j ) + Ψ(yi )

(i,j)∈V2

Clustering-assisted inference Node labeling y ∈ NG Weight ωi,j reduction if nodes vi and vj belong to distinct clusters Cost function: β − 1(yi 6= yj ) f (yi , yj ) = β

July 3th , 2017

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

20 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

A priori

A priori: gene grouping and modular structure maximize x∈{0,1}E y∈NG

P

f (yi , yj )ωi,j xi,j + λ(1 − xi,j ) + Ψ(yi )

(i,j)∈V2

Clustering-assisted inference Node labeling y ∈ NG Weight ωi,j reduction if nodes vi and vj belong to distinct clusters Cost function: β − 1(yi 6= yj ) f (yi , yj ) = β

TF-driven clustering promoting modular structure

Ψ(yi ) =

X

µi,j 1(yi = j)

i∈V j∈T

µi,j : modular structure controlling parameter July 3th , 2017

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

20 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Alternating optimization strategy maximize n x∈{0,1} y∈NG

July 3th , 2017

P (i,j)∈V2

β−1(yi 6=yj ) ωi,j xi,j β

+ λ(1 − xi,j ) +

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

P

µi,j 1(yi = j)

i∈V j∈T

21 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Alternating optimization strategy Alternating optimization maximize n x∈{0,1} y∈NG

July 3th , 2017

P (i,j)∈V2

β−1(yi 6=yj ) ωi,j xi,j β

+ λ(1 − xi,j ) +

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

P

µi,j 1(yi = j)

i∈V j∈T

21 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Alternating optimization strategy Alternating optimization maximize n x∈{0,1} y∈NG

P (i,j)∈V2

β−1(yi 6=yj ) ωi,j xi,j β

+ λ(1 − xi,j ) +

At y fixed and x variable: X β − 1(yi 6= yj ) maximize ωi,j xi,j β x∈{0,1}n 2

P

µi,j 1(yi = j)

i∈V j∈T

+ λ(1 − xi,j )

(i,j)∈V

July 3th , 2017

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

21 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Alternating optimization strategy Alternating optimization maximize n x∈{0,1} y∈NG

P (i,j)∈V2

β−1(yi 6=yj ) ωi,j xi,j β

+ λ(1 − xi,j ) +

P

µi,j 1(yi = j)

i∈V j∈T

At y fixed and x variable: X β − 1(yi 6= yj ) maximize ωi,j xi,j β x∈{0,1}n 2

+ λ(1 − xi,j )

(i,j)∈V

At x fixed and y variable: X ωi,j xi,j 1(yi 6= yj ) + minimize β y∈NG 2 (i,j)∈V

July 3th , 2017

X

µi,j 1(yi 6= j)

i∈V, j∈T

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

21 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Alternating optimization strategy Alternating optimization maximize n x∈{0,1} y∈NG

P (i,j)∈V2

β−1(yi 6=yj ) ωi,j xi,j β

+ λ(1 − xi,j ) +

P

µi,j 1(yi = j)

i∈V j∈T

At y fixed and x variable: X β − 1(yi 6= yj ) maximize ωi,j xi,j β x∈{0,1}n 2

+ λ(1 − xi,j )

(i,j)∈V

Explicit form:

∗ xi,j

=

( 1 0

if ωi,j >

λβ β−1(yi 6=yj )

otherwise.

At x fixed and y variable: X ωi,j xi,j 1(yi 6= yj ) + minimize β y∈NG 2 (i,j)∈V

July 3th , 2017

X

µi,j 1(yi 6= j)

i∈V, j∈T

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

21 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Clustering optimization strategy At x fixed and y variable: X ωi,j xi,j X minimize 1(yi 6= yj ) + µi,j 1(yi 6= j) β y∈NG 2 (i,j)∈V

July 3th , 2017

i∈V, j∈T

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

22 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Clustering optimization strategy At x fixed and y variable: X ωi,j xi,j X minimize 1(yi 6= yj ) + µi,j 1(yi 6= j) β y∈NG 2 (i,j)∈V

July 3th , 2017

(NP)

i∈V, j∈T

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

22 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Clustering optimization strategy At x fixed and y variable: X ωi,j xi,j X minimize 1(yi 6= yj ) + µi,j 1(yi 6= j) β y∈NG 2 (i,j)∈V

(NP)

i∈V, j∈T

discrete problem ⇒ quadratic relaxation T-class problem ⇒ T binary sub-problems (t)

label restriction to T: {s(1) , . . . , s(T) } such that sj = 1 if j = t and 0 otherwise. Y = {y(1) , . . . , y(T) } such that y(t) ∈ [0, 1]G

July 3th , 2017

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

22 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Clustering optimization strategy At x fixed and y variable: X ωi,j xi,j X minimize 1(yi 6= yj ) + µi,j 1(yi 6= j) β y∈NG 2 (i,j)∈V

(NP)

i∈V, j∈T

discrete problem ⇒ quadratic relaxation T-class problem ⇒ T binary sub-problems (t)

label restriction to T: {s(1) , . . . , s(T) } such that sj = 1 if j = t and 0 otherwise. Y = {y(1) , . . . , y(T) } such that y(t) ∈ [0, 1]G

Problem re-expressed as:   T 2  2 X X ωi,j xi,j  (t) X (t) (t) (t)   minimize yi − yj + µi,j yi − sj Y β 2 t=1

July 3th , 2017

(i,j)∈V

i∈V, j∈T

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

22 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Clustering optimization strategy

minimize Y

T X t=1



  2 2 X ωi,j xi,j  (t) X (t) (t) (t)   yi − yj + µi,j yi − sj β 2 (i,j)∈V

i∈V, j∈T

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

July 3th , 2017

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

23 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Clustering optimization strategy

minimize Y

T X t=1



  2 2 X ωi,j xi,j  (t) X (t) (t) (t)   yi − yj + µi,j yi − sj β 2 i∈V, j∈T

(i,j)∈V

This is the Combinatorial Dirichlet problem Minimization via solving a linear system of equations [Grady, 2006] (t)

Final labeling: node i is assigned to label t for which yi is maximal (t)

y∗i = argmax yi t∈T

July 3th , 2017

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

23 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Random walker in graphs y1

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

6 10 y5

y4 7

10 y3

July 3th , 2017

3 12 5

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

3 5 9

y2

24 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Random walker in graphs 1 y1

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

6 10 y5

y4 7

10 y3 3

July 3th , 2017

3 12 5

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

3 5 9

y2 2

24 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Random walker in graphs 1 y1

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

6 10 y5

3 12 y4

5

7

10 y3 3

3 5 9

y2 2

1

0.95 0.35

0.46

0.03

0.01

0

0

y(1) July 3th , 2017

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

24 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Random walker in graphs 1 y1

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

6 10 y5

7

3

0

0.95

0.01

0.35

0.46

0.03

0.01

0

0

y(1) July 3th , 2017

0.19

y4

10 y3

1

3 12 5 3 5 9

y2 2

0.28

0.02

0.97

0

1

y(2) Recons. and Clust. with Graph Optim. and Priors on GRN and Images

24 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Random walker in graphs 1 y1

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

6 10 y5

7

3

0

0

0.95

0.01

0.03

0.35

0.46

0.03

0.01

0

0

y(1) July 3th , 2017

0.19

0.28

0.02

0.97

0

1

y(2)

0.46

y4

10 y3

1

3 12 5 3 5 9

y2 2

0.26

0.95

0.02

1

0

y(3)

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

24 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

Random walker in graphs 1 y1

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

6 10 y5

3 12 y4

5

7

10 3 5 9

y3

y2

3

1

0

0

0.95

0.01

0.03

0.35

0.46

0.19

0.28

1

0.46

0.26

0.03

0.01

0.02

0.97

0.95

0.02

0

0

0

1

1

0

y(1) July 3th , 2017

y(2)

2

y(3)

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

3

1

3

2

y∗ = {1, 1, 2, 3, 3} 24 / 45

BRANE Clust — NETWORK INFERENCE WITH CLUSTERING

Formulation and resolution

hard- vs soft- clustering in BRANE Clust minimize Y

T P

P

t=1

(i,j)∈V2

ωi,j xi,j β



(t) yi

−

 (t) 2 yj

+

P

µi,j



(t) yi

−

 (t) 2 sj

i∈V, j∈T

hard-clustering

soft-clustering

# clusters = # TF ( → ∞ if i = j µi,j = 0 otherwise.

# clusters ≤ # TF   if i = j α µi,j = α1(ωi,j > τ ) if i 6= j and i ∈ T   ωi,j 1(ωi,j > τ ) if i 6= j and i ∈ /T

July 3th , 2017

!

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

25 / 45

BRANE RESULTS

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

July 3th , 2017

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

26 / 45

BRANE RESULTS

Methodology

Numerical evaluation strategy AUPR Ref Reference Precision-Recall curve Classical thresholding

P=

|TP| |TP|+|FP|

R=

|TP| |TP|+|FN|

July 3th , 2017

AUPR BRANE BRANE Precision-Recall curve

BRANE edge selection Gene-gene interaction scores (ND)-CLR or (ND)-GENIE3 Gene expression data DREAM4 or DREAM5 challenges

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

27 / 45

BRANE RESULTS

DREAM4 synthetic results

BRANE performance on in-silico data DREAM4 [Marbach et al., 2010]

July 3th , 2017

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

28 / 45

BRANE RESULTS

DREAM4 synthetic results

BRANE performance on in-silico data DREAM4 [Marbach et al., 2010] 1

2

3

4

5

Average

Gain

CLR BRANE Cut BRANE Relax BRANE Clust

0.256 0.282 0.278 0.275

0.275 0.308 0.293 0.337

0.314 0.343 0.336 0.360

0.313 0.344 0.333 0.335

0.318 0.356 0.345 0.342

0.295 0.327 0.317 0.330

10.9 % 7.8 % 12.2 %

GENIE3 BRANE Cut BRANE Relax BRANE Clust

0.269 0.298 0.293 0.287

0.288 0.316 0.320 0.348

0.331 0.357 0.356 0.364

0.323 0.344 0.345 0.371

0.329 0.352 0.354 0.367

0.308 0.333 0.334 0.347

8.4 % 8.5 % 12.8 %

Network

1

2

3

4

5

Average

Gain

ND-CLR BRANE Cut BRANE Relax BRANE Clust

0.254 0.271 0.270 0.258

0.250 0.277 0.264 0.251

0.324 0.334 0.327 0.327

0.318 0.335 0.325 0.337

0.331 0.343 0.332 0.342

0.295 0.312 0.304 0.303

5.9 % 3.1 % 2.5 %

ND-GENIE3 BRANE Cut BRANE Relax BRANE Clust

0.263 0.275 0.276 0.273

0.275 0.312 0.307 0.311

0.336 0.367 0.369 0.354

0.328 0.346 0.347 0.373

0.354 0.368 0.371 0.370

0.309 0.334 0.334 0.336

7.2 % 7.3 % 8.1 %

Network

July 3th , 2017

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

28 / 45

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 July 3th , 2017

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

29 / 45

BRANE RESULTS

DREAM5 synthetic results

BRANE performance on in-silico data DREAM5 [Marbach et al., 2012]

July 3th , 2017

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

30 / 45

BRANE RESULTS

DREAM5 synthetic results

BRANE performance on in-silico data DREAM5 [Marbach et al., 2012] CLR BRANE Cut BRANE Relax BRANE Clust

ND-CLR BRANE Cut BRANE Relax BRANE Clust

July 3th , 2017

AUPR

Gain

0.252 0.268 0.272 0.301

6.3 % 7.9 % 19.4 %

AUPR

Gain

0.272 0.277 0.274 0.289

1.9 % 0.6 % 6.2 %

GENIE3 BRANE Cut BRANE Relax BRANE Clust

ND-GENIE3 BRANE Cut BRANE Relax BRANE Clust

AUPR

Gain

0.283 0.295 0.294 0.336

4.2 % 3.8 % 18.6 %

AUPR

Gain

0.313 0.317 0.314 0.345

1.1 % 0.3 % 10.2 %

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

30 / 45

BRANE RESULTS

DREAM5 synthetic results

BRANE performance on in-silico data DREAM5 [Marbach et al., 2012] CLR BRANE Cut BRANE Relax BRANE Clust

ND-CLR BRANE Cut BRANE Relax BRANE Clust

AUPR

Gain

0.252 0.268 0.272 0.301

6.3 % 7.9 % 19.4 %

AUPR

Gain

0.272 0.277 0.274 0.289

1.9 % 0.6 % 6.2 %

AUPR

Gain

0.283 0.295 0.294 0.336

4.2 % 3.8 % 18.6 %

AUPR

Gain

0.313 0.317 0.314 0.345

1.1 % 0.3 % 10.2 %

GENIE3 BRANE Cut BRANE Relax BRANE Clust

ND-GENIE3 BRANE Cut BRANE Relax BRANE Clust

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 July 3th , 2017

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

30 / 45

BRANE RESULTS

Escherichia coli results

BRANE Clust performance on real data Escherichia coli dataset AUPR CLR BRANE Clust

July 3th , 2017

0.0378 0.0399

Gain 5.5 %

GENIE3 BRANE Clust

AUPR

Gain

0.0488 0.0536

9.8 %

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

31 / 45

BRANE RESULTS

Escherichia coli results

BRANE Clust performance on real data Escherichia coli dataset AUPR CLR BRANE Clust

0.0378 0.0399

Gain 5.5 %

GENIE3 BRANE Clust

AUPR

Gain

0.0488 0.0536

9.8 %

BRANE Clust predictions using GENIE3 weights

July 3th , 2017

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

31 / 45

BRANE RESULTS

Escherichia coli results

BRANE Clust performance on real data Escherichia coli dataset AUPR CLR BRANE Clust

0.0378 0.0399

Gain 5.5 %

GENIE3 BRANE Clust

AUPR

Gain

0.0488 0.0536

9.8 %

BRANE Clust predictions using GENIE3 weights

BRANE Clust validated on real dataset July 3th , 2017

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

31 / 45

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

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

32 / 45

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

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

32 / 45

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

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

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C ONCLUSIONS

It’s time to conclude...

July 3th , 2017

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

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C ONCLUSIONS

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

July 3th , 2017

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

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C ONCLUSIONS

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

July 3th , 2017

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

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C ONCLUSIONS

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

July 3th , 2017

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

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C ONCLUSIONS

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

July 3th , 2017

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

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C ONCLUSIONS

Perspectives From biological graphs...

GRN

July 3th , 2017

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

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C ONCLUSIONS

Perspectives From biological graphs...

TF-based a priori

GRN

July 3th , 2017

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

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C ONCLUSIONS

Perspectives From biological graphs...

TF-based a priori

GRN

July 3th , 2017

Clustering

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

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C ONCLUSIONS

Perspectives From biological graphs...

TF-based a priori

GRN

July 3th , 2017

Clustering

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

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C ONCLUSIONS

Perspectives From biological graphs...

TF-based a priori

Extend TF-based a priori for GRN

July 3th , 2017

Clustering

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

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C ONCLUSIONS

Perspectives From biological graphs...

TF-based a priori

GRN

July 3th , 2017

Clustering

Extend TF-based a priori for GRN, clustering

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

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C ONCLUSIONS

Perspectives From biological graphs...

TF-based a priori

GRN

Clustering

Extend TF-based a priori for GRN, clustering

Gene-gene scores

July 3th , 2017

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

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C ONCLUSIONS

Perspectives From biological graphs...

TF-based a priori

Clustering

GRN

Extend TF-based a priori for GRN, clustering

Gene-gene scores

Normalized gene expression data

July 3th , 2017

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

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C ONCLUSIONS

Perspectives From biological graphs...

TF-based a priori

Clustering

GRN

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

Gene-gene scores

Normalized gene expression data

July 3th , 2017

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

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C ONCLUSIONS

Perspectives From biological graphs...

TF-based a priori

Clustering

GRN

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

Gene-gene scores

Normalized gene expression data

July 3th , 2017

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

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C ONCLUSIONS

Perspectives From biological graphs...

TF-based a priori

Clustering

GRN

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

Gene-gene scores

Normalized gene expression data

July 3th , 2017

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

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C ONCLUSIONS

Perspectives From biological graphs...

TF-based a priori

Clustering

GRN

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

Gene-gene scores

Normalized gene expression data

July 3th , 2017

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

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C ONCLUSIONS

Perspectives From biological graphs...

TF-based a priori

GRN

Clustering

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

Normalized gene expression data

July 3th , 2017

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

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C ONCLUSIONS

Perspectives From biological graphs...

TF-based a priori

Clustering

GRN

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

Omics data

Integrate a priori, omics- data and treatments

Omics data

July 3th , 2017

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

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C ONCLUSIONS

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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C ONCLUSIONS

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...

July 3th , 2017

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

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C ONCLUSIONS

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 July 3th , 2017

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

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C ONCLUSIONS

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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C ONCLUSIONS

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

July 3th , 2017

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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 KL(q(x) || p(x | y)) q(x)

Separable distribution: q(x) =

J Y

qj (xj ),

j=1

with

  Q qopt (x ) ∝ exp hln p(y, x)i j j i6=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 −1 I) 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 −1 I) 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  Y

L  Y p(xi | zi , ui , m, Ω)p(ui | β) p(z)p(γ) p(ml )p(Ωl )

i=1

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l=1

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HOGMep

HOGMep — VBA strategy Separable form for the approximation: q(Θ) =

N Y

(q(xi , zi )q(ui )) q(γ)

i=1

L Y

(q(ml )q(Ωl ))

l=1

with q(xi |zi = l) = N (η i,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

SC

SC

VB-MIG

HOGMep

July 3th , 2017

ICM

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

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