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![Page 1: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/1.jpg)
Gene duplication models Gene duplication models and reconstruction of gene and reconstruction of gene
regulatory network regulatory network evolution from network evolution from network
structurestructure
Juris Viksna, David GilbertJuris Viksna, David Gilbert
Riga, IMCS, 10.02.2006Riga, IMCS, 10.02.2006
![Page 2: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/2.jpg)
Gene regulatory networks
[J.Rung,T.Schlitt,A.Brazma,K.Freivalds,J.Vilo Bioinformatics 18 S2 (ECCB), 202-210 ]
Yeast network:
![Page 3: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/3.jpg)
Gene regulatory networks
• Directed graph
• Graph vertices correspond to genes
• An edge from gene A to B means that gene B is (directly) regulated by gene A
![Page 4: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/4.jpg)
Properties of gene networks (1)
• Believed to be scale-free (vertex degrees satisfy so-called power law):
N(k) – number of vertices with degree k
N(k) k
![Page 5: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/5.jpg)
Properties of gene networks (1)
N(k) k
[F.Chung,L.Lu,T.Dewey,D.Gallas JCB 10, 677-687]
![Page 6: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/6.jpg)
Properties of gene networks (2)
• Believed to have a noticeable modularity
i - vertexki - number of neighbours for vertex iki - number of direct links between these
ki neighbours
Clustering coefficient (for vertex i):
Ci = 2ni/ki(ki1)
![Page 7: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/7.jpg)
Properties of gene networks (2)
Clustering coefficient (for vertex i):
Ci = 2ni/ki(ki1)
[E.Ravasz,A.Somera,D.Mongru,Z.Oltvai,A.Barabasi Science 297, 1551-1555]
![Page 8: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/8.jpg)
Network evolution models (1)
[A.Barabasi, R.Albert Science 286, 509-512]
(i) networks expand continuously by the addition of new vertices,
(ii) new vertices attach preferentially to sites that are already well connected.
A model based on these two ingredients reproduces the observed stationary scale-free distributions.
![Page 9: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/9.jpg)
Network evolution models (2)
"Hierarchical" model
[E.Ravasz,A.Somera,D.Mongru,Z.Oltvai,A.Barabasi Science 297, 1551-1555]
Sample hierarchical networks (scale-free and modular)
![Page 10: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/10.jpg)
Network evolution models (3)
"Duplication" model
Scale-free with < 2 for ½ < p < 1
[F.Chung,L.Lu,T.Dewey,D.Gallas JCB 10, 677-687]
![Page 11: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/11.jpg)
Network evolution models (4)
![Page 12: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/12.jpg)
Network evolution models (M1)
M1
![Page 13: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/13.jpg)
M1, p = 0.1, 5000 vertices
4.5
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M1, p = 0.01, 5000 vertices
3
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M1, p=0.05, d=0.2, 5000 vertices
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M1, p=0.05, d=0.2, 5000 vertices
2.5
![Page 17: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/17.jpg)
Network evolution models (M1)
M1
V E
20 4050 200100 700500
150001000
500005000
800000
![Page 18: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/18.jpg)
Network evolution models (M2)
A
X'X
A
X'X
genome evolution
![Page 19: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/19.jpg)
Network evolution models (M2)
A
X'X
genome evolution
A
X'X
A
X'X
or
![Page 20: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/20.jpg)
Network evolution models (M2)
M2
![Page 21: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/21.jpg)
M2, p = 0.1, 20000 vertices
![Page 22: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/22.jpg)
M2, p = 0.1, 20000 vertices
1
![Page 23: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/23.jpg)
Network evolution models (M2)
M2
V E
20 4050 80100 150500 7001000 15005000 7000
![Page 24: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/24.jpg)
Evolution graphs
k+2 vertices
two types of edges:
- for swappable events (black)- for dependent events (grey)
![Page 25: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/25.jpg)
Evolution graphs
![Page 26: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/26.jpg)
Evolution graphs
Initial graph G
Graph G' obtained from G afterk (in this example k=6) evolutionsteps
Intermediate graphs between G and G' correspond to cuts of evolution graph (G and G' can also be obtained in this way)
Numbered vertices correspondto evolution steps and are markedby the vertices duplicated in thecorresponding steps
![Page 27: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/27.jpg)
Evolution graphs – some questions
EquivalenceDecide whether 2 given evolution graphs are equivalent
Irreducible networks – networks that can’t be obtainedfrom simpler networks by evolution graph
Uniqueness of evolutionIs it possible that D(G1,E1)= D(G2,E2) for two different irreducible networks G1 and G2?
![Page 28: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/28.jpg)
"Reverse engineering" problems
Given: Reconstruct:
G'
G
E
![Page 29: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/29.jpg)
"Reverse engineering" problem (1)
(Assuming either model M1 or M2.)
Reconstruction of evolution graph
For a given network N’ find an irreducible network N, the sequence of duplication events D1,...,Dm and the corresponding evolution tree, such that N’=D(N,E).
![Page 30: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/30.jpg)
"Reverse engineering" problem (2)
(Assuming either model M1 or M2.)
Reconstruction of duplication event
For a given network N’ find a network N and a duplication event D, such that N’=D(N).
![Page 31: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/31.jpg)
"Reverse engineering" problem (3)
(Assuming either model M1 or M2.)
Reconstruction of the largest duplication event
For a given network N’ find a network N with the smallest possible number of genes and a duplication event D, such that N’=D(N).
![Page 32: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/32.jpg)
"Reverse engineering" - complexity
For a given network N’ find a network N with the smallest possible number of genes and a duplication event D, such that N’=D(N).
• at least as hard as graph isomorphism problem
• likely NP-hard (maximum clique for reconstruction graphs)
• reconstruction graphs are much smaller than networks
• still might be practically solvable for random graphs of reasonable size (few tens of thousands of vertices).
![Page 33: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/33.jpg)
Algorithm – stage 1
Partition G' vertices into orbits
Can be done e.g. with nauty package
One can try to use some property p which is more simple to compute than automorphisms and is such that p(G1)=p(G2) for isomorphic graphs G1 and G2.
![Page 34: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/34.jpg)
Reconstruction graphs
Vertices correspond to non-singleton orbits
Two types of edges: - (1) have to participate in the same duplication event (solid) - (2) can not participate in the same duplication event (dotted)
![Page 35: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/35.jpg)
Algorithm – stage 2
Find reconstruction graph
![Page 36: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/36.jpg)
Algorithm – stage 3
Find the largest independent set (according to type 2 edges)in reconstruction graph
![Page 37: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/37.jpg)
Algorithm – stage 4
- if all selected orbits contain just 2 nodes, we are practicallydone
- otherwise we have to find a pair of (largest) sets of vertices from selected orbits, which correspond to duplication event[currently exhaustive search]
![Page 38: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/38.jpg)
Algorithm
Evolution graph can be reconstructed by repeated use ofLargest duplication event
![Page 39: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/39.jpg)
Algorithm - efficiency
- using nauty we can deal with networks with < 200 genes
- for larger graphs one can use heuristics to computeorbits
- vertex/edge counts at different DFS levels seems to workquite well
- likely to find a large part of duplication event
- for <200 vertices often gives the exact result
![Page 40: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/40.jpg)
Algorithm – Model 2
General case – check automorphisms for all k-tuplesof vertices
A serious problem even for k=2
However, large components are duplicated not that often
Previous algorithm could be used to find "large" partof duplicated genes
Still an open problem
Also, a question about good heuristics
![Page 41: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/41.jpg)
Model 2 – Component sizesModel M2
550 vertices132 duplications
![Page 42: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/42.jpg)
Model 2 – Component sizes
Constructing random network with 20000 genes:
Component sizes #of events1 1770082 3423 974 495 376 187 138 1010,11,14 49,12,13,15,27 316,24 217,18,21,22,31,27 1
![Page 43: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/43.jpg)
Experiments with yeast network
6270 genes106 regulators
![Page 44: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/44.jpg)
Experiments with yeast network
p=0.0001
E=106
V=216
![Page 45: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/45.jpg)
Experiments with yeast network
277 pairs of duplication candidates were discovered
Few "real": COS5 and COS8, YLR460C and YNL134L
All 5962 genes were compared all-v-all using SW
Normalized compression score: ssearch_score(P1,P2)/min{length(P1),length(P2)}
Scores for the found duplication pairs were compared withaverage values
![Page 46: Gene duplication models and reconstruction of gene regulatory network evolution from network structure Juris Viksna, David Gilbert Riga, IMCS, 10.02.2006.](https://reader034.fdocuments.net/reader034/viewer/2022051618/56649c7b5503460f9492e822/html5/thumbnails/46.jpg)
Experiments with yeast network
Observed distances vs average, all non-adjacent gene pairs