Global alignment of protein-protein interaction networks by graph - - PowerPoint PPT Presentation

▶
global alignment of protein protein interaction networks
SMART_READER_LITE
LIVE PREVIEW

Global alignment of protein-protein interaction networks by graph - - PowerPoint PPT Presentation

Mar 29, 2023 229 likes •898 views

Global alignment of protein-protein interaction networks by graph matching methods. Mikhail Zaslavkiy 1 Francis Bach 2 Jean-Philippe Vert 1 1 Mines ParisTech / Institut Curie / INSERM 2 INRIA / Ecole normale superieure de Paris Kyushu University,

slide-1
SLIDE 1

Global alignment of protein-protein interaction networks by graph matching methods.

Mikhail Zaslavkiy 1 Francis Bach2 Jean-Philippe Vert1

1Mines ParisTech / Institut Curie / INSERM 2INRIA / Ecole normale superieure de Paris

Kyushu University, Department of Informatics, July 14, 2009.

JP Vert (ParisTech) Global alignment of PPI networks 1 / 47

slide-2
SLIDE 2

Outline

1 Identification of functional orthologs 2 Algorithm for constrained global network alignment 3 Algorithms for balanced global network alignment 4 Experiments 5 Conclusion

JP Vert (ParisTech) Global alignment of PPI networks 2 / 47

slide-3
SLIDE 3

Outline

1 Identification of functional orthologs 2 Algorithm for constrained global network alignment 3 Algorithms for balanced global network alignment 4 Experiments 5 Conclusion

JP Vert (ParisTech) Global alignment of PPI networks 3 / 47

slide-4
SLIDE 4

Functional orthologs

Species 1 Species 2

f1: MKQALAAADDDDAQ... y1: MDDDDALGLLLLA... f2:MGDXLLMMAALLLL... y2: MHHAAKLLDDAS...

... ...

Definition

Functional orthologs are pairs of proteins directly inherited from a common ancestor and which play functionally equivalent roles.

Our goal

Automatic identification of functional orthologs (useful for annotation transfer)

JP Vert (ParisTech) Global alignment of PPI networks 4 / 47

slide-5
SLIDE 5

Identification of functional orthologs by best-best hit

Species 1 Species 2

f1: MKQDLARIEQFLDALF... y1: MSRLPVLLLLQLLVRGA. . . f2: MSKLKIAVSDSCPDCF... y2: MELAALCRAGLLLALDA. . .

... ...

C=

  y1 y2 f1 10 50 f2 27 10   Cij-BLAST similarity scores Optimal assignment : f1 → y2, f2 → y1

JP Vert (ParisTech) Global alignment of PPI networks 5 / 47

slide-6
SLIDE 6

Limitations of sequence comparison-based methods

y may be the best hit for f, but f may not be the best hit for y... (y1, f ) and (y2, f ) may produce very similar blast scores...

JP Vert (ParisTech) Global alignment of PPI networks 6 / 47

slide-7
SLIDE 7

Clusters of orthologs

Many programs produce clusters of orthologous genes from sequence comparison only (COG, KEGG, Inparanoid, ...) Several genes of each species may be in the same cluster How to find functional orthologs within the clusters?

JP Vert (ParisTech) Global alignment of PPI networks 7 / 47

slide-8
SLIDE 8

Ideas to solve ambiguous functional orthologs

Increase the similarity of similarity scores / phylogenetic approaches Comparison of expression profiles across species Functional orthologs tend to have more conserved protein-protein interactions (PPI) across species

(Bandyopadhyay et al., 2006) JP Vert (ParisTech) Global alignment of PPI networks 8 / 47

slide-9
SLIDE 9

Disambiguation by PPI conservation

Idea: If we know that y∗ and f ∗ are functional orthologs, and there exist interactions f ∗ − f and y∗ − y2. Then the assignment y2 − f is more likely because it conserves one interaction.

JP Vert (ParisTech) Global alignment of PPI networks 9 / 47

slide-10
SLIDE 10

Disambiguation by PPI conservation

Idea: If we know that y∗ and f ∗ are functional orthologs, and there exist interactions f ∗ − f and y∗ − y2. Then the assignment y2 − f is more likely because it conserves one interaction.

??? ???

JP Vert (ParisTech) Global alignment of PPI networks 9 / 47

slide-11
SLIDE 11

Disambiguation by PPI conservation

Idea: If we know that y∗ and f ∗ are functional orthologs, and there exist interactions f ∗ − f and y∗ − y2. Then the assignment y2 − f is more likely because it conserves one interaction.

PPI ??? ??? PPI

JP Vert (ParisTech) Global alignment of PPI networks 9 / 47

slide-12
SLIDE 12

Disambiguation by PPI conservation

Idea: If we know that y∗ and f ∗ are functional orthologs, and there exist interactions f ∗ − f and y∗ − y2. Then the assignment y2 − f is more likely because it conserves one interaction.

PPI PPI

JP Vert (ParisTech) Global alignment of PPI networks 9 / 47

slide-13
SLIDE 13

Extension to PPI networks

JP Vert (ParisTech) Global alignment of PPI networks 10 / 47

slide-14
SLIDE 14

Extension to PPI networks

matchings

JP Vert (ParisTech) Global alignment of PPI networks 10 / 47

slide-15
SLIDE 15

Extension to PPI networks

3 conserved interactions

JP Vert (ParisTech) Global alignment of PPI networks 10 / 47

slide-16
SLIDE 16

Global Network Alignment (GNA)

3 conserved interactions

Given two PPI networks and the all-vs-all sequence similarity matrix, find a global matching that maximizes the number of conserved interactions subject to: Constraint GNA: matchings only occur within clusters of orthologs. Balanced GNA: the mean sequence similarity between matched pairs is as large as possible.

JP Vert (ParisTech) Global alignment of PPI networks 11 / 47

slide-17
SLIDE 17

Complexity of the problems (bad news)

Both problems are NP-hard for general graphs and similarity matrix. Therefore we must use algorithms that approximately optimize the criteria, e.g:

MRF method (Bandyopadhyay et al., MSB 2006) for constrained GNA IsoRank (Singh et al., PNAS 2008) for balanced GNA

We investigate other algorithms for these problems, borrowing ideas from state-of-the-art graph matching algorithms.

JP Vert (ParisTech) Global alignment of PPI networks 12 / 47

slide-18
SLIDE 18

Complexity of the problems (bad news)

Both problems are NP-hard for general graphs and similarity matrix. Therefore we must use algorithms that approximately optimize the criteria, e.g:

MRF method (Bandyopadhyay et al., MSB 2006) for constrained GNA IsoRank (Singh et al., PNAS 2008) for balanced GNA

We investigate other algorithms for these problems, borrowing ideas from state-of-the-art graph matching algorithms.

JP Vert (ParisTech) Global alignment of PPI networks 12 / 47

slide-19
SLIDE 19

Complexity of the problems (bad news)

Both problems are NP-hard for general graphs and similarity matrix. Therefore we must use algorithms that approximately optimize the criteria, e.g:

MRF method (Bandyopadhyay et al., MSB 2006) for constrained GNA IsoRank (Singh et al., PNAS 2008) for balanced GNA

We investigate other algorithms for these problems, borrowing ideas from state-of-the-art graph matching algorithms.

JP Vert (ParisTech) Global alignment of PPI networks 12 / 47

slide-20
SLIDE 20

Outline

1 Identification of functional orthologs 2 Algorithm for constrained global network alignment 3 Algorithms for balanced global network alignment 4 Experiments 5 Conclusion

JP Vert (ParisTech) Global alignment of PPI networks 13 / 47

slide-21
SLIDE 21

Constrained GNA

Problem

Find matchings within the clusters that maximise the number of conserved interactions

JP Vert (ParisTech) Global alignment of PPI networks 14 / 47

slide-22
SLIDE 22

Graph of clusters induced by PPI

JP Vert (ParisTech) Global alignment of PPI networks 15 / 47

slide-23
SLIDE 23

Global optimum

Proposition

If the graph of clusters generated by the PPI has no cycle, then we can find the optimal matching efficiently with a message passing algorithm.

JP Vert (ParisTech) Global alignment of PPI networks 16 / 47

slide-24
SLIDE 24

Global optimum by message passing

(Similar to Viterbi’s algorithm for HMM)

JP Vert (ParisTech) Global alignment of PPI networks 17 / 47

slide-25
SLIDE 25

Global optimum by message passing

(Similar to Viterbi’s algorithm for HMM)

JP Vert (ParisTech) Global alignment of PPI networks 17 / 47

slide-26
SLIDE 26

Global optimum by message passing

(Similar to Viterbi’s algorithm for HMM)

JP Vert (ParisTech) Global alignment of PPI networks 17 / 47

slide-27
SLIDE 27

Global optimum by message passing

(Similar to Viterbi’s algorithm for HMM)

JP Vert (ParisTech) Global alignment of PPI networks 17 / 47

slide-28
SLIDE 28

What if the graph of clusters has cycle?

The message passing method can not be used... Instead we reformulate the constrained GNA problem as a balanced GNA by setting similarity between proteins in different clusters to −∞, and use algorithms for balanced GNA.

JP Vert (ParisTech) Global alignment of PPI networks 18 / 47

slide-29
SLIDE 29

Outline

1 Identification of functional orthologs 2 Algorithm for constrained global network alignment 3 Algorithms for balanced global network alignment 4 Experiments 5 Conclusion

JP Vert (ParisTech) Global alignment of PPI networks 19 / 47

slide-30
SLIDE 30

Balanced GNA

Given two graphs and a matrix of all-vs-all similarities, find a matching P ∈ P that jointly maximizes:

the number of conserved interaction CI(P), the mean similarity of matched pairs S(P).

The trade-off can be found by maximizing over P: min

P∈P F(P) = (1 − α)CI(P) + αS(P) ,

where α ∈ [0, 1] determines the balance between both objectives.

JP Vert (ParisTech) Global alignment of PPI networks 20 / 47

slide-31
SLIDE 31

Balanced GNA

Given two graphs and a matrix of all-vs-all similarities, find a matching P ∈ P that jointly maximizes:

the number of conserved interaction CI(P), the mean similarity of matched pairs S(P).

The trade-off can be found by maximizing over P: min

P∈P F(P) = (1 − α)CI(P) + αS(P) ,

where α ∈ [0, 1] determines the balance between both objectives.

JP Vert (ParisTech) Global alignment of PPI networks 20 / 47

slide-32
SLIDE 32

Existing methods for balanced GNA

min

P∈P F(P) = (1 − α)CI(P) + αS(P) ,

When α = 1 this is an optimal assignment problem efficiently solved by the Hungarian algorithm (Kuhn, 1955). When α < 1 this is a general graph matching problem, usually computationally intractable. Existing algorithms include:

Exact solution by incomplete enumeration (only for small graphs) Spectral methods (Umeyama, 1986; Singh et al., 2008) Relaxations of the problem into a continuous optimization problem (Almohamad and Duffuaa, 1993; Gold and Rangarajan, 1996).

JP Vert (ParisTech) Global alignment of PPI networks 21 / 47

slide-33
SLIDE 33

Existing methods for balanced GNA

min

P∈P F(P) = (1 − α)CI(P) + αS(P) ,

When α = 1 this is an optimal assignment problem efficiently solved by the Hungarian algorithm (Kuhn, 1955). When α < 1 this is a general graph matching problem, usually computationally intractable. Existing algorithms include:

Exact solution by incomplete enumeration (only for small graphs) Spectral methods (Umeyama, 1986; Singh et al., 2008) Relaxations of the problem into a continuous optimization problem (Almohamad and Duffuaa, 1993; Gold and Rangarajan, 1996).

JP Vert (ParisTech) Global alignment of PPI networks 21 / 47

slide-34
SLIDE 34

Relaxation algorithms

min

P∈P F(P)

Embed the discrete set P into a continuous space D Extend the function F(P) to D Minimize F(P) over D Map back the solution to P

JP Vert (ParisTech) Global alignment of PPI networks 22 / 47

slide-35
SLIDE 35

Relaxation algorithms

min

P∈P F(P)

Embed the discrete set P into a continuous space D Extend the function F(P) to D Minimize F(P) over D Map back the solution to P

JP Vert (ParisTech) Global alignment of PPI networks 22 / 47

slide-36
SLIDE 36

Relaxation algorithms

min

P∈P F(P)

Embed the discrete set P into a continuous space D Extend the function F(P) to D Minimize F(P) over D Map back the solution to P

JP Vert (ParisTech) Global alignment of PPI networks 22 / 47

slide-37
SLIDE 37

Relaxation algorithms

min

P∈P F(P)

Embed the discrete set P into a continuous space D Extend the function F(P) to D Minimize F(P) over D Map back the solution to P

JP Vert (ParisTech) Global alignment of PPI networks 22 / 47

slide-38
SLIDE 38

Mathematical formulation

1 3 5 2 4

AG =       1 1 1 1 1 1 1 1 1       P =       1 1 1 1 1       P = permutation matrices (Pij = 1 if i is matched to j) D = doubly stochastic matrices (P ≥ 0, P1N = 1N, 1⊤

NP = 1N)

Classical relaxation: CI(P) = ||AG − AP(H)|| = ||AG − PAHPT||

JP Vert (ParisTech) Global alignment of PPI networks 23 / 47

slide-39
SLIDE 39

Quadratic convex relaxation (QCV)

Minimize F0(P) = ||AGP − PAH||2

F = vec(P)TQvec(P) over D

(convex QP) Project the solution D∗ to P (Hungarian algorithm) Not very good if D∗ is far from P...

JP Vert (ParisTech) Global alignment of PPI networks 24 / 47

slide-40
SLIDE 40

Quadratic convex relaxation (QCV)

Minimize F0(P) = ||AGP − PAH||2

F = vec(P)TQvec(P) over D

(convex QP) Project the solution D∗ to P (Hungarian algorithm) Not very good if D∗ is far from P...

JP Vert (ParisTech) Global alignment of PPI networks 24 / 47

slide-41
SLIDE 41

Quadratic convex relaxation (QCV)

Minimize F0(P) = ||AGP − PAH||2

F = vec(P)TQvec(P) over D

(convex QP) Project the solution D∗ to P (Hungarian algorithm) Not very good if D∗ is far from P...

JP Vert (ParisTech) Global alignment of PPI networks 24 / 47

slide-42
SLIDE 42

Quadratic convex relaxation (QCV)

Minimize F0(P) = ||AGP − PAH||2

F = vec(P)TQvec(P) over D

(convex QP) Project the solution D∗ to P (Hungarian algorithm) Not very good if D∗ is far from P...

JP Vert (ParisTech) Global alignment of PPI networks 24 / 47

slide-43
SLIDE 43

A new concave relaxation

On P we also have: CI(P) = F1(P) = −tr(∆P) − vec(P)T(LG ⊗ LH)vec(P) This is a concave function, therefore its global minimum over D is on P (extreme points) Idea: starting from a ”good solution” on D, we can project to P by gradient ascent (GA) to maximize −F1(P)

JP Vert (ParisTech) Global alignment of PPI networks 25 / 47

slide-44
SLIDE 44

The PATH algorithm

F0(P) = ||AGP − PAH||2

F = vec(P)TQvec(P)

F1(P) = −tr(∆P) − vec(P)T(LG ⊗ LH)vec(P) Fλ(P) = (1 − λ)F0(P) + λF1(P) (Zaslavskyi et al., IEEE PAMI, 2009.)

JP Vert (ParisTech) Global alignment of PPI networks 26 / 47

slide-45
SLIDE 45

Outline

1 Identification of functional orthologs 2 Algorithm for constrained global network alignment 3 Algorithms for balanced global network alignment 4 Experiments 5 Conclusion

JP Vert (ParisTech) Global alignment of PPI networks 27 / 47

slide-46
SLIDE 46

Random graphs:N=8

Figure: Precision as a noise function, U — Umeyama algorithm results, LP — linear

programming algorithm, QCV — convex function approach (F0), PATH — path minimization algorithm, OPT — an exhaustive search (the global minimum).

JP Vert (ParisTech) Global alignment of PPI networks 28 / 47

slide-47
SLIDE 47

Random graphs:N=20

Figure: Precision as a noise function, U — Umeyama algorithm results, LP — linear

programming algorithm, QCV — convex function approach (F0), PATH — path minimization algorithm.

JP Vert (ParisTech) Global alignment of PPI networks 29 / 47

slide-48
SLIDE 48

Random graphs:N=100

Figure: Precision as a noise function, U — Umeyama algorithm results, QCV — convex

function approach (F0), PATH — path minimization algorithm.

JP Vert (ParisTech) Global alignment of PPI networks 30 / 47

slide-49
SLIDE 49

Algorithm complexity

Figure: Timing of U, LP, QCV and PATH algorithms as a function of graph size. Noise level is

0.3. Slope: tanLP = 6.67,tanU = tanQCV = tanPATH = 3.3

JP Vert (ParisTech) Global alignment of PPI networks 31 / 47

slide-50
SLIDE 50

Experiment results for QAPLIB benchmark

QAP MIN PATH QPB GRAD U chr12c 11156 18048 20306 19014 40370 chr15a 9896 19086 26132 30370 60986 chr15c 9504 16206 29862 23686 76318 chr20b 2298 5560 6674 6290 10022 chr22b 6194 8500 9942 9658 13118 esc16b 292 300 296 298 306 rou12 235528 256320 278834 273438 295752 rou15 354210 391270 381016 457908 480352 rou20 725522 778284 804676 840120 905246 tai10a 135028 152534 165364 168096 189852 tai15a 388214 419224 455778 451164 483596 tai17a 491812 530978 550852 589814 620964 tai20a 703482 753712 799790 871480 915144 tai30a 1818146 1903872 1996442 2077958 2213846 tai35a 2422002 2555110 2720986 2803456 2925390

JP Vert (ParisTech) Global alignment of PPI networks 32 / 47

slide-51
SLIDE 51

Eye vessels image processing

JP Vert (ParisTech) Global alignment of PPI networks 33 / 47

slide-52
SLIDE 52

Eye vessels image processing: Shape context

−100 −50 50 100 −80 −60 −40 −20 20 40 60 80 shape context only − 1 on 2 −100 −50 50 100 −100 −80 −60 −40 −20 20 40 60 80 100 shape context only − 2 on 1

JP Vert (ParisTech) Global alignment of PPI networks 34 / 47

slide-53
SLIDE 53

Combination of shape context and structural information

−150 −100 −50 50 100 150 −250 −200 −150 −100 −50 50 100 Linear combination of shape context and graph structure − 1 on 2 −100 −50 50 100 −100 −50 50 100 150 200 Linear combination of shape context and graph structure − 2 on 1

JP Vert (ParisTech) Global alignment of PPI networks 35 / 47

slide-54
SLIDE 54

Recognition of chinese characters

character 1 character 2 character 3

Figure: Chinese characters from the ETL9B dataset.

JP Vert (ParisTech) Global alignment of PPI networks 36 / 47

slide-55
SLIDE 55

Recognition of chinese characters

Table: Classification of chinese characters. (CV , STD)—mean and standard deviation of test error over cross-validation runs (five folds, 50 repetitions)

Method CV STD Linear SVM 0.377 ± 0.090 SVM with gaussian kernel 0.359 ± 0.076 KNN (PATH) (α=1): shape context 0.399 ± 0.081 KNN (PATH) (α=0.4) 0.248 ± 0.075 KNN (PATH) (α=0): pure graph matching 0.607 ± 0.072 KNN (U) (α=0.9): α best choice 0.382 ± 0.077 KNN (QCV) (α=0.3): α best choice 0.295 ± 0.061

JP Vert (ParisTech) Global alignment of PPI networks 37 / 47

slide-56
SLIDE 56

Alignment of PPI networks: Fly vs. Yeast

PPI networks and all-vs-all BLAST / Inparanoid clusters for D. melanogaster (fly) vs. S. cerevisiae (yeast) Data provided by Bandyopadhyay et al. (MSB 2006) Fly (7k nodes, 20k edges) Yeast (4k nodes,15k edges)

JP Vert (ParisTech) Global alignment of PPI networks 38 / 47

slide-57
SLIDE 57

Experiments: Constrained Alignment

There are Inparanoid 2244 clusters: 1552 clusters with only two proteins 692 ambiguous clusters

JP Vert (ParisTech) Global alignment of PPI networks 39 / 47

slide-58
SLIDE 58

Experiments: Constrained Alignment

There are 2244 clusters: 1552 clusters with only two proteins 692 ambiguous clusters There is no cycles in the graph of clusters!

JP Vert (ParisTech) Global alignment of PPI networks 40 / 47

slide-59
SLIDE 59

Experiments: Constrained Alignment

InParanoid clusters: 2244 clusters (1552 clusters with only two proteins + 692 ambiguous clusters) Message Passing Algorithm (MP) provides the optimal solution MRF (Bandyopadhyay et al., 2006), IsoRank (Singh et al., 2008), PATH and GA methods may be used as well Measure the number of conserved interactions Validation: count the number of Homologene pairs (gold standard for functional orthologs?) Algorithm MP GA PATH MRF IsoRank #cons. interactions 238 238 238 233 228 #HomoloG pairs 41 41 41 36 39 Timing(sec) 1-2 1-2 80 10 1-2

JP Vert (ParisTech) Global alignment of PPI networks 41 / 47

slide-60
SLIDE 60

Differences MP vs MRF: example

Solid red: interaction conserved by MP; Dotted black: interactions conserved by MRF.

JP Vert (ParisTech) Global alignment of PPI networks 42 / 47

slide-61
SLIDE 61

Differences MP vs MRF: example

Solid red: interaction conserved by MP; Dotted black: interactions conserved by MRF.

JP Vert (ParisTech) Global alignment of PPI networks 43 / 47

slide-62
SLIDE 62

Experiments: Balanced Alignment

Maximize: (1 − λ)J + λS

Number of conserved interaction J versus sequence similarity S.

JP Vert (ParisTech) Global alignment of PPI networks 44 / 47

slide-63
SLIDE 63

Outline

1 Identification of functional orthologs 2 Algorithm for constrained global network alignment 3 Algorithms for balanced global network alignment 4 Experiments 5 Conclusion

JP Vert (ParisTech) Global alignment of PPI networks 45 / 47

slide-64
SLIDE 64

Conclusion

What we did

Formulation of biological network alignment as a graph matching problem Message passing algorithm: exact solution for the constrained alignment problem Graph matching algorithms: good performance in the case of balanced alignment.

Future work

Interactions of a higher order (see paper) Synchronized alignment of several networks Many-to-Many graph matching

JP Vert (ParisTech) Global alignment of PPI networks 46 / 47

slide-65
SLIDE 65

Acknowledgements

Misha Zaslavskiy Francis Bach This presentation is supported by a JSPS Invitation Fellowship Program for Research in Japan, hosted by Tatsuya Akutsu (Kyoto University)

JP Vert (ParisTech) Global alignment of PPI networks 47 / 47