Protein complexes and their shared components

10. Lecture WS 2004/05

Bioinformatics III 1

Protein complexes and their shared components

- Most cellular processes result from a cascade of events mediated by proteins

that act in a cooperative manner.-Protein complexes can share components: proteins can be reused and

participate to several complexes.

Methods for analyzing high-throughput protein interaction data have mainly used

clustering techniques.

They have been applied to assign protein function by inference from the biological

context as given by their interactors, and to identify complexes as dense regions

of the network (see V9).

The logical organization into shared and specific components, and its

representation remains elusive.

Gagneur et al. Genome Biology 5, R57 (2004)



shared components

Shared components = proteins or groups of proteins occurring in different

complexes are fairly common:

A shared component may be a small part of many complexes, acting as a unit that

is constantly reused for ist function.

Also, it may be the main part of the complex e.g. in a family of variant complexes

that differ from each other by distinct proteins that provide functional specificity.

Aim: identify and properly represent the modularity of protein-protein interaction

networks by identifying the shared components and the way they are arranged to

generate complexes.

Gagneur et al. Genome Biology 5, R57 (2004)Georg Casari, Cellzome (Heidelberg)



Modules

A graph and its modules.

Nodes connected by a link are called

neighbors.

In graph theory, a module is a set of

nodes that have the same neighbors

outside the module.

In addition to the trivial modules {a},

{b},...,{g} and {a,b,c,..,g}, this graph

contains the modules {a,b,c}, {a,b},

{a,c},{b,c} and {e,f}.




Quotient

Elements of a module have exactly the same neighbors outside the module

one can substitute all of them for a representative node.

In a quotient, all elements of the module are replaced by the representative node,

and the edges with the neighbors are replaced by edges to the representative.

Quotients can be iterated until the entire graph is merged into a final

representative node.

Iterated quotients can be captured in a tree, where each node represents a

module, which is a subset of ist parent and the set of its descendant leaves.




Modular decomposition

Modular decomposition of the

example graph shown before.

Modular decomposition gives a

labeled tree that represents iterations

of particular quotients, here the

successive quotients on the modules

{a,b,c} and {e,f}.

The modular decomposition is a

unique, canonical tree of iterated

quotients

(formal proof exists Möhring 1985).




Modular decomposition

The nodes of the modular decomposition

are labeled in 3 ways:

As series when the direct descendants

are all neighbors of each other,

as parallel when the direct descendants

are all non-neighbors of each other,

and by the structure of the module

otherwise (prime module case).


Series are labeled by an asterisk within a circle, parallel by two parallel lines within a circle,

and prime by a P within a circle. The prime is advantageously labeled by its structure.

The graph can be retrieved from the tree on the right by recursively expanding the modules

using the information in the labels. Therefore, the labeled tree can be seen as an exact

alternative representation of the graph.



Results from protein complex purifications (PCP), e.g. TAP

Different types of data:- Y2H: detects direct physical interactions between proteins

- PCP by tandem affinity purification with mass-spectrometric identification of the

protein components identifies multi-protein complexes

Molecular decomposition will have a different meaning due to different semantics

of such graphs.

Here, focus analysis on PCP content.

PCP experiment: select bait protein where TAP-label is attached Co-purify

protein with those proteins that co-occur in at least one complex with the bait

protein.

In future, integrated view combining both types of data would be preferred.




Clique and maximal clique

A clique is a fully connected sub-graph, that is, a set

of nodes that are all neighbors of each other.

In this example, the whole graph is a clique and

consequently any subset of it is also a clique, for

example {a,c,d,e} or {b,e}. A maximal clique is a

clique that is not contained in any larger clique. Here

only {a,b,c,d,e} is a maximal clique.


Assuming complete datasets and ideal results, a permanent complex will appear

as a clique.

The opposite is not true: not every clique in the network necessarily derives from

an existing complex. E.g. 3 connected proteins can be the outcome of a single

trimer, 3 heterodimers or combinations thereof.



Results from protein complex purifications (PCP), e.g. TAP

Interpretation of graph and module labels

for systematic PCP experiments.

(a) Two neighbors in the network are

proteins occurring in a same complex.

(b) Several potential sets of complexes

can be the origin of the same observed

network. Restricting interpretation to the

simplest model (top right), the series

module reads as a logical AND between

its members.

(c) A module labeled ´parallel´

corresponds to proteins or modules

working as strict alternatives with respect

to their common neighbors.

(d) The ´prime´ case is a structure where

none of the two previous cases occurs. Gagneur et al. Genome Biology 5, R57 (2004)



Obtain maximal cliques

Modular decomposition provides an instruction set to deliver all maximal cliques

of a graph.

In particular, when the decomposition has only series and parallels, the maximal

cliques are straightforwardly retrieved by traversing the tree recursively from top

to bottom.

A series module acts as a product: the maximal cliques are all the combinations

made up of one maximal clique from each „child“ node.

A parallel module acts as a sum: the set of maximal cliques is the union of all

maximal cliques from the „child“ nodes.




Modular decomposition of graphs

The notion of module naturally arises from different combinatorial structures and

has been introduced several times in different fields:

Modules have been called Decompositions have also been called- Autonomous sets - substitution decomposition- Closed sets - ordinal sum- Stable sets - X-join- Clumps- Committees- Externally related sets- Nonsimplifiable subnetworks- Partitive sets

…

Muller&Spinrad, J. Ass. Comp Mach 36, 1 (1989)



Consider undirected graph G=(V,E) with n =|V| vertices and m=|E| edges.

The complement of a graph G is denoted by G.

If X is a subset of vertices, then G[X] is the subgraph of G induced by X.

Let x be an arbitrary vertex, then N(x) and N(x) stand respectively for the

neighborhood of x and its non-neighborhood.

A vertex x distinguishes two vertices u and v if (x,u) E and (x,v) E.

A module M of a graph G is a set of vertices that is not distinguished by any

vertex.



A simple linear algorithm for modular decomposition

The modules of a graph are a potentially exponentially-sized family

However, the sub-family of strong modules, the modules that overlap no other

modules, has size O(n).

A overlaps B if A B , A \ B and B \ A

The inclusion order of this family defines the previously explained

modular tree decomposition, which is enough to store the module family of a

graph.

The root of this tree is the trivial module V and its n leaves are the trivial modules

{x}, xV.

Habib, de Montgolfier, Paul (2004)



Aim: a simple linear algorithm for modular decomposition

Any graph G with at least 3 vertices is either not connected

or its complement G is not connected

or G and G are both connected.

In the last case, the maximal modules define a partition of the vertex-set and are

said to be a prime composition.

The modular decomposition tree can be recursively built by a top-down approach.

At each step, the algorithm recurses on graphs induced by the maximal strong

modules. This technique gives an O(n4) complexity.

Here, derive a linear-time algorithm that computes a modular factorizing

permutation without computing the underlying decomposition tree.

This tree may be derived in a second step.Habib, de Montgolfier, Paul (2004)



Modular decomposition of protein interaction graphs

A graph and its modular tree decomposition. The set {1,2} is a strong module.

The module {7,8} is weak: it is overlapped by the module {8,9}.

The permutation = (1,2,3,4,5,6,7,8,9) is a modular factorizing permutation.




Module-factorizing orders

Let G=(V,E) be a graph and let O be a partial order on V.

For two comparable elements x and y where x <O y we state x precedes y and y

follows x.

Two subsets A and B cross if a,a‘ A and b,b‘ B such that a <O b and a‘ >O

b‘. A linear extension of a partially ordered set (‚poset‘) is a completion of the poset

into a total order.

Definition 1. A partial order O is a Module-Factorizing Partial Order (MFPO) of

V(G) if any pair of non-intersecting strong modules of G do not cross.

The modular factorizing permutations are exactly the module-factorizing total orders.

Proposition 1. A partial order O is an MFPO if and only if it can be completed into a

factorizing permutation.




Module-factorizing orders

Definition 2. An ordered partition is a collection {P1, ..., Pk} of pairwise disjoint

parts, with and an order O such that for all

x Pi and y Pj, x <O y if i < j.

Start with trivial partition (a single part equal to the vertex set) and iteratively

extend (or refine) it until every part is a singleton.

A center vertex c V is distinguished and two refining rules, preserving the MFPO

property, are used. They are defined in Lemma 1:




Defining rules

Lemma 1.

1. Center Rule: For any vertex c, the ordered partition

is module-factorizing.


The center rule picks a center and breaks a trivial partition to start the

algorithm.

Once launched, the process goes on based on the pivot rule, that splits each

part Pi (except the part Pi that contains the pivot), according to the neighborhood

of the pivot.



Lemma 1 continued.

2. Pivot Rule: Let be an ordered partition with

center c and p Pi such that Pj, ij, overlaps N(p) .

If O is an MFPO, then the following refinements preserve the module-

factorizing property:

Defining rules: pivot rule




Preliminary algorithm

Partition refinement scheme that outputs a partition of V into the maximal

modules not containing c.


When this algorithm ends, every part is a module. To obtain a factorizing

permutation it has to be recursively relaunched on the non-singleton parts.




Execution example of algorithm

The resulting factorizing permutation is (a, s, v, w, u, y, x, z, t).



Ordered chain partition yields linear-time algorithm

Definition 3. An ordered chain partition (OCP) is a partial order such that each

vertex belongs to one and only one chain, and one chain belongs to one and

only one part. The vertices of the same chain are totally ordered, the chains

of the same part are uncomparable, and the parts of totally ordered.


A trivial chain contains only 1 vertex, and a monochain part contains only one

chain. The OCPs generalize the Ordered Partitions since the latter ones contain

only trivial chains.




C(x) denotes the chain containing x while P(x) denotes the part of the partition

containing x.

Each chain C has a representative vertex r(C) C.

During the algorithm, the chains will behave as their representative vertices.

Chains are possibly merged. Then, the representative of the new chain is one of

the former representatives. But chains will never be split.

The algorithm still uses the center and pivot rules.

The chains are moved by these 2 rules, according to the adjacency between

their representative vertex and the center of the pivot.

But there is a third rule, the chaining rule (line 9 of algorithm).




Defining rule 3: Chaining rule

There is a third rule, the chaining rule

Unlike the two first ones, the third rule removes comparisons from the order.

It first concatenates a sequence of monochain parts, that occur consecutively in

O, into one chain. Then this new chain is inserted into one of the two parts,

say P, neighboring the chain.

Chaining rule, chaining the black vertices into P.


The comparisons between the chain and P are lost.

But since the number of chains strictly decreases during the algorithm,

the process is guaranteed to end.




Use each vertex a constant number of times as a pivot.









Modular decomposition of protein interaction graphs

Finally the following invariant is satisfied:

Invariant 1. The ordered chain partition O is an MFPO of V(G) and no chain is

overlapped by a strong module.

Position of a strong module M (black vertices) in O (Invariant 1).


To use any vertex O(1) times as a pivot, the algorithm picks only one vertex per

part to extend the OCP. A chain C is úsed´ if its representative vertex r(C) has

already been used as pivot by some extension rule. Similarly, a part is úsed´ if it

contains a used chain.

Pivots may be chosen from ùnused` parts only ensuring each vertex neighbor-

hood is used O(1) times. The non-trivial (multichain) parts are not necessarily

modules. The algorithm chooses a new center and recurses (line 12).



Choice of the new center.

The center plays an important role (see Lemma 1).

Invariant 2 Let M be a strong module and c be the center. Then either c belongs

to M, or M consists in consecutive monochain parts, or M is included in a single

part P.

The new center cn must fulfil Invariant 2 as the old center c did.

If all the strong modules containing c but not cn are included in P(c) then

Invariant 2 holds.




Choice of the new center.

Let PL (resp. PR) be the rightmost (resp. leftmost) multichain part that precedes

(resp. follows) c.

As both parts are ùsed`, their pivots pL and pR are defined. One of them is

chosen for the recursive call, and its pivot becomes the new center.

Only one pivot among pL and pR distinguishes the other from the center c.

The rule chooses that pivot as new center.

This can be implemented by an adjacency test.


… and so-forth … see original article



Implementation of an Ordered Chain Partition







Summary:- simple, linear-time

algorithm now available

for modular decomposition

of graphs.

What is the meaning of

such modules when

applied to real data?



In the modular decomposition tree, the leaves are proteins,

the root represents the whole network.

In between, each node is a module that is a sub-part of ist parent.

The label of a node gives the nature of the relationship between ist direct children.

Proteins or modules in a parallel module can be be seen as

alternatives. If A is neighbor of B and C, which are not neighbors

of each other, then A can belong to a complex together with

either B or C, but not with both at the same time.

B and C define a parallel module and thus are alternative

partners in a complex with their common neighbor A.

This situation corresponds to a logical „exclusive OR“

between B and C.

Interpretation for PCP protein interaction networks




Proteins or modules in a series module can be

seen as potentially combined in any way.

If A is neighbor of B and C, and B and C are

also neighbors, the A can belong to a complex

together with B or C, or with both at the same

time.

This corresponds to a logical „OR“ between B

and C.

A series module can be seen as a unit: a set of

proteins (modules) that function together.

A ‚prime‘ is a graph where neither of these cases

occurs.

Interpretation for PCP protein interaction networks




Three examples of modular

decomposition of protein-protein

interaction networks. In each case

from top to bottom: schema of

complexes, the corresponding

protein-protein interaction network as

determined from PCP experiments,

and its modular decomposition

(MOD).

(a) Protein phosphatase 2A. Parallel

modules group proteins that do not

interact but are functionally

equivalent. Here these are the

catalytic Pph21 and Pph22 (module

2) and the regulatory Cdc55 and

Rts1 (module 3).

Back to the real world …





RNA polymerases I, II and III

A good layout of the corresponding network

gives an intuitive idea of what the constitutive

units of the complexes are. Modular

decomposition extracts them and makes their

logical combinations explicit.




Transcriptional regulator complexes

Modular decomposition condenses the network to

its backbone prime structure (root of the tree) and

identifies its constitutive units.



NFB variants

Modular decomposition of NFκB members relB, c-rel, p50 and p52 delivers the

potential NFκB dimers and tetramers.

All combinations are possible (series) except those including both relB and c-rel

(parallel), and those including both p50 and p52.





Modular decomposition of the network of

NFκB members and their partners in

resting cells. The network is composed of

the NFκB members and their interactors. In

this step, interactions among the

interactors are disregarded. Baits are

outlined in green. Modular decomposition

organizes the interactors into modules. The

root is a prime whose structure is shown in

the encircled network.

Module 1 and module 2, respectively,

group the new interactors into activators

and inhibitors of NFκB.



Summary


Ongoing: need for modular description of molecular biology.

What are suitable modules?

Spirin&Mirny, Barabasi et al. : identify dense parts of the network

Alon and co-workers: identify (small) repeated motifs

Here: apply established method of modular graph decomposition

to protein interaction networks. Can (and has been) applied to other networks.

What is the biological relevance of modules at different levels?

Integrate with gene ontology?

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Protein complexes and their shared components

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