V11: Genetic networks

11. Lecture WS 2005/06

Bioinformatics III 1

V11: Genetic networks

Methods to describe genetic networks:

(1) boolean networks (today)

(2) clustering gene expression data

( Bioinformatics II lecture)

Clustering is a relatively easy way

to extract useful information out of

large-scale gene expression data sets.

However, it typically only tells us

which genes are co-regulated,

not what is regulating what.

Need to reverse engineer networks from their activity profiles!

JCell manual, U Tübingen



Intergenic interaction matrix M

Since the introduction detecting gene expression by microarrays,

a major problem has been the estimation of the intergenic interaction matrix M.

The matrix element mij of the interaction matrix M is

- positive if gene Gj activates gene Gi

- negative if gene Gj inhibits gene Gi

- equal to 0 if gene Gj and gene Gi have no interaction.

Gi = +1 if it is expressed, otherwise = 0.

Aracena & Demongeot, Acta Biotheoretica 52, 391 (2004)



simulating the dynamics of regulatory networks

Given the interaction matrix M, the change of state xi of gene Gi between t and t +1

obeys a threshold rule:


btMxHtx

btxmHtx

i

nkikiki

1

or1,1

where H is the Heavyside function

H(y) = 1 if y 0 and

H(y) = 0 if y < 0,

and the bi‘s are threshold values.

In the case of small regulatory genetic systems, the knowledge of such a matrix M

makes it possible to know all possible stationary behaviors of the organisms having

the corresponding genome.



Example

Mendoza, Alvarez-Buylla, JCB, 1998

In the genetic regulatory network which

rules Arabidopsis thaliana flower

morphogenesis (right), the interaction

matrix is a (11,11) matrix with only 22

non zero coefficient.

Below: A fixed configuration (attractor) of

its Boolean dynamics that is obtained

from propagating xi(t).



Interaction matrix - interaction graph

For each genetic regulatory network, we can define an interaction graph built from

the interaction matrix M by drawing an edge + (resp. -) between the vertices

representing the genes j and i, iff mij > 0 (resp. < 0).

To calculate the mij´s, we can either determine the s-directional correlation ij(s)

between the state vector {xj(t – s)}t C of gene j at time t – s and the state vector

{xi(t)}t C of gene i at time t , t varying during the cell cycle C of length K = | C | and

corresponding to the observation time of the bio-array images:


21

22 1where

1

Ct Ctjjj

ij

Ct Ct Ctijij

ij

stxKstxs

ss

txstxKtxstxs



interaction matrix

and then take


ijij

ijmsijij

mifm

mifsKsignm

0

,1,...,1

where is a de-correlation threshold.

Alternatively, one may identify the system with a Boolean neural network.

When it is impossible to obtain all the coefficients of M in this manner

(either from the literature or from such calculations),

it may be possible to complete M by appyling an heuristic approach.



estimation of interaction values

We may randomly choose the missing coefficients by considering

- the connectivity coefficient K(M) = I / N, the ratio between the number I of

interactions and the number N of genes, and

- the mean inhibition weight I(M) = R / I , the ratio between the number of inhibitions

R and I.

For many known operons and regulation networks, K(M) is between 1.5 and 3, and

I(M) between 1/3 and 2/3.

If M is structurally stable, then the random estimation of M can be used to obtain an

approximate estimation on the control mechanisms of the regulatory network.




Mathematical Aspects of the Inverse Problem

A network with two or more connected components, i.e. two or more sub-networks,

has as fixed configurations the combination (Cartesian product) of all fixed

configurations of each sub-network.

We say that the fixed configurations are factorizable.

Thus, the inverse problem consists of determining whether a fixed configurations

set is factorizable.

In this way, we can obtain some information on the connectivity of the network.




Factorization

Given S {0,1}n and a permutation function : {1,...,n} {1,...,n},

we denote by (S), or simply S the set {s(1)s(2) ... s(n) : s1s2...sn S }.

A set S {0,1}n is said to be factorizable if there exist sets of vectors

S1 {0,1}j(1) and S2 {0,1}j(2) and , ..., Sk {0,1}j(3) and a permutation function

: {1, ..., n} {1,...,n} such that S can be written as S = (S1 S2 ... Sk) ,

where the symbol „“ is the cartesian product between sets.

If S is a factorizable set, then j(1) + j(2) + ... + j(k) = n.

The set defined by F = {S1,S2, ...,Sk} is called a factorization of S and each

Sj F a factor of S.

F is called a maximal factorization if every factor Sj F is not factorizable.




Examples

i) S = {0100, 0111, 1000, 1011} = {01, 10} {00, 11}.

Here, the permutation function is the identity.

ii) S = {0010, 0111, 1000, 1101} =

({0100, 0111, 1000, 1011})(2,3) = ({01, 10} {00, 11})(2,3) ,

where (2,3) is the function which permutes the second and third coordinates.

Given the sets I {1, ..., n} and S (0,1)n, let PI(S) be the projection set defined by

PI(S) = {(sj(1),sj(2), ...,sj(I)): s S, j(k) I, k = 1, ..., | I |, and j(k) < j(l) for all k < l }.




Proposition 2

Proposition 2

If a set S {0,1}n is factorizable, then the maximal factorization of S is unique.

ProofLet F = {S1,S2, ...,Sk} and G = {T1,T2, ...,Tk} be two distinct maximal factorizations of S.

S = (S1 S2 ... Sk)1 = (T1 T2 ... Tk)2

Hence, the permutation = (1)-1 ○ 2 is such that

S1 S2 ... Sk = (T1 T2 ... Tk) Since F and G are maximal factorizations, there is a factor of F not included into G,

which is supposed to be S1 {0,1}q, q {1, ..., n}.


Let T = T1 T2 ... Tm , so S1 = P{1,...,q} (T)

Hence, if we denote by I(k) {1, ..., n} the set of indices such that

PI(k)(T) = T, for every k = 1, ...,m and by J = { j {1,...,m}: I(j) {(1),...,(p)} }

then there exists a permutation function ‘ such that

jsjJ

TPTPS pjsIpjI

,...,1

where',...,1,...,111

Therefore, S1 is factorizable, a contradiction.



Algorithm

Let : {0,1}n {0,1}n P({1,...,n}) be the function called the difference function

where P({1,...,n}) is the set of subsets of {1,...,n} and defined by

(x,y) = {i: xi yi}, where x,y {0,1}n.

Given S {0,1}n, the idea of the Factorization algorithm is first to construct a matrix

with all the values of (x,y) for every x,y S.

Next, for each row i of the matrix we construct a finite and undirected graph

Gi = (Vi,Ei), where the set of nodes Vi is equal to the set {1,...,n} and the set of arcs Ei

is determined by the values of each row of the matrix, according to the algorithm.

Finally, the connected components of the union of all graphs Gi determine the factors

of the maximal factorization of S.

In the case that S is not factorizable, the output of the algorithm will be a graph with a

unique connected component.




Algorithm




Theorem 3

Given a set S {0,1}n, if I = { I(1), I(2), ..., I(k) } is the output of the Factorization

algorithm with input S,

then F = { P(I)(S): I = 1, ..., k) is the maximal factorization of S

and the complexity of the algorithm is O(|S|3 + n2)




Example 2

Let S = { x1 = 000, x2 = 001, x3 = 100, x4 = 010, x5 = 011, x6 = 110}.

The difference matrix is

and the partial graphs and the

output graph of the algorithm are:


The output is I(1) = {1,3} and I(2) = {2}.

the maximal factorization of S is given

by

S = (PI(1)(S) PI(2) (S))(2,3)

= ({00,01,10} {0,1})(2,3)

where (2,3) is the permutation of the

second and third coordinates.



Example 3


The maximal factorization of S is given

by

S = (PI(1)(S) PI(2) (S))(4,5)

= ({0010,0001,1100) (000,111)}(4,5)

The following set of vectors corresponds to the observed fixed points of the A.thaliana

regulatory network, considering only genes whose activity is not constant.

Let S = { x1 = 0010000, x2 = 0011011, x3 = 0000100, x4 = 0001111, x5 = 1100000, x6 = 1101011}.

The difference matrix is

The graph G of the algorithm and the

connected components

I(1) = {1,2,3,5} and I(2) = {4,6,7} are:



Design principles of regulatory networks

Wiring diagrams of regulatory networks resemble somehow electrical circuits.

Try to break down networks into basic building blocks.

Search for „network motifs“ as patterns of interconnections that recur in many

different parts of a network at frequencies much higher than those found in

randomized networks.

Shen-Orr et al. Nature Gen. 31, 64 (2002)

Uri Alon

Weizman Institute



Detection of motifs

Represent transcriptional network as a connectivity matrix M

such that Mij = 1 if operon j encodes a TF that transcriptionally regulates operon i

and Mij = 0 otherwise.

Scan all n × n submatrices of M generated by choosing n nodes that lie in a

connected graph, for n = 3 and n = 4.

Submatrices were enumerated efficiently by recursively searching for nonzero

elements.

Compute a P value for submatrices representing each type of connected subgraph

by comparing # of times they appear in real network vs. in random network.

For n = 3, the only significant motif is the feedforward loop.

For n = 4, only the overlapping regulation motif is significant.

SIMs and multi-input modules were identified by searching for identical rows of M.




DOR detection

Consider all operons regulated by ≥ 2 TFs.

Define (nonmetric) distance measure between operons k and j, based on the # of

TFs regulating both operons:

d(k,j) = 1/ (1+n fnMk,n Mj,n)2)

Where fn = 0.5 for global TFs and fn = 1 otherwise.

Cluster operons with average-linkage algorithm.

DORs correspond to clusters with more than 10 connections

with a ratio of connections to TFs > 2.




Network motifs found in E.coli transcript-regul network

a, Feedforward loop: a TF X regulates a second TF

Y, and both jointly regulate one or more operons

Z1...Zn.b, Example of a feedforward loop (L-arabinose utilization).

c, SIM motif: a single TF, X, regulates a set of

operons Z1...Zn. X is usually autoregulatory. All

regulations are of the same sign. No other

transcription factor regulates the operons.

d, Example of a SIM system (arginine biosynthesis).

e, DOR motif: a set of operons Z1...Zm are each

regulated by a combination of a set of input

transcription factors, X1...Xn. DOR-algorithm detects

dense regions of connections, with a high ratio of

connections to transcription factors. f, Example of a DOR (stationary phase response).




Significance of motifs




Regulatory network


Each TF appears only in a single subgraph except for

global TFs that can appear in several subgraphs.



Structural organization of transcript-regul networks

Modules: observation that reg. Networks are highly interconnected, very few

modules can be entirely separated from the rest of the network.

Babu et al. Curr Opin Struct Biol. 14, 283 (2004)



Evolution of the gene regulatory network

Larger genomes tend to have more TFs per gene.




Cross-organism comparison

Many TF families are specific to

individual phylogenetic groups or

greatly expanded in some genomes.


In contrast to the high level of conservation of other regulatory and signalling

systems across the crown group eukaryotes,

some of the TF families are dramatically different in the various lineages.



Regulatory interactions across organisms

Are regulatory interactions conserved among organisms? Apparently yes.

Orthologous TFs regulate orthologous target genes.

As expected, the conservation of genes and interaction is related to the

phylogenetic difference between organisms.

Above: Many interactions of (a) can be mapped to pathogenetic Pseudomonas

aeruginosa that is related to E.coli (b).

Very few interactions can be mapped from (a) to (c).




Regulatory interactions across organisms

Observation: there is no bias towards conservation of network motifs.

Regulatory interactions in motifs are lost or retained at the same rate as the other

interactions in the network.

The transcriptional network appears to evolve in a step-wise manner, with loss

and gain of individual interactions probably playing a greater role than loss and

gain of whole motifs or modules.

Observation: TFs are less conserved than target genes, which suggests that

regulation of genes evolves faster than the genes themselves.




Most research on biological networks has been focused on static topological

properties, describing networks as collections of nodes and edges rather than as

dynamic structural entities.

Here this study focusses on the temporal aspects of networks, which allows us to

study the dynamics of protein complex assembly during the Saccharomyces

cerevisiae cell cycle.

The integrative approach combines protein-protein interactions with information on

the timing of the transcription of specific genes during the cell cycle, obtained from

DNA microarray time series shown before.

a quality-controlled set of 600

periodically expressed genes,

each assigned to the point in the

cell cycle where its expression peaks.

Analysis of complexome during cell cycle

Science 307, 724 (2005)

Ulrik Lichtenberg Peer Bork



Temporal protein interaction network in yeast cell cycle

Cell cycle proteins that are part

of complexes or other physical

interactions are shown within

the circle.

For the dynamic proteins, the

time of peak expression is

shown by the node color;

static proteins are represented

as white nodes.

Outside the circle, the dynamic

proteins without interactions

are positioned and colored

according to their peak time.

Science 307, 724 (2005)



Just-in-time synthesis vs. just-in-time-assembly

Transcription of cell cycle–regulated genes is generally thought to be turned on

when or just before their protein products are needed: often referred to as

just-in-time synthesis.

Contrary to the cell cycle in bacteria, however, just-in-time synthesis of entire

complexes is rarely observed in the network. The only large complex to be

synthesized in its entirety just in time is the nucleosome, all subunits of which are

expressed in S phase to produce nucleosomes during DNA replication.

Instead, the general design principle appears to be that only some subunits of

each complex are transcriptionally regulated in order to control the timing of

final assembly.

Science 307, 724 (2005)



Integrate transcriptional regulatory information and gene-expression data for

multiple conditions in Saccharomyces cerevisae.

5 conditions cell cycle

sporulation

diauxic shift

DNA damage

stress response

Something spectacular at the end

Luscombe, Babu, … Teichmann, Gerstein, Nature 431, 308 (2004)

Sarah Teichmann Mark Gerstein



SANDY: topological measures + network motifs

Luscombe et al. Nature 431, 308 (2004)

+ some post-analysis



Dynamic representation of transript. regul. network

c, Standard statistics (global topological measures and local network motifs) describing network structures. These vary between endogenous and exogenous conditions; those that are high compared with other conditions are shaded. (Note, the graph for the static state displays only sections that are active in at least one condition, but the table provides statistics for the entire network including inactive regions.)

Luscombe, Babu, … Teichmann, Gerstein, Nature 431, 308 (2004)

a, Schematics and summary of properties for the endogenous and exogenous sub-networks.

b, Graphs of the static and condition-specific networks. Transcription factors and target genes are shown as nodes in the upper and lower sections of each graph respectively, and regulatory interactions are drawn as edges; they are coloured by the number of conditions in which they are active. Different conditions use distinct sections of the network.




Interpretation

Half of the targets are uniquely expressed in only one condition; in contrast, most

TFs are used across multiple processes.

The active sub-networks maintain or rewire regulatory interactions, over half of

the active interactions are completely supplanted by new ones between conditions.

Only 66 interactions are retained across ≥ 4 conditions.

They are always „on“ and mostly regulate house-keeping functions.

The calculations divide the 5 condition-specific networks into 2 categories:

endogenous and exogenous.

Endogenous processes are multi-stage, operate with an internal transcriptional

program

Exogenous processes are binary events that react to external stimuli with a

rapid turnover of expressed genes.



Figure 2 Newly derived 'follow-on' statistics for network structures. a, TF hub usage in different cellular conditions. The cluster diagram shades cells by the normalized number of genes targeted by TF hubs in each condition. One cluster represents permanent hubs and the others condition-specific transient hubs. Genes are labelled with four-letter names when they have an obvious functional role in the condition, and seven-letter open reading frame names when there is no obvious role. Of the latter, gene names are red and italicised when functions are poorly characterized. Starred hubs show extreme interchange index values, I = 1. b, Interaction interchange (I) of TF between conditions. A histogram of I for all active TFs shows a uni-modal distribution with two extremes. Pie charts show five example TFs with different proportions of interchanged interactions. We list the main functions of the distinct target genes regulated by each example transcription factor. Note how the TFs' regulatory functions change between conditions. c, Overlap in TF usage between conditions. Venn diagrams show the numbers of individual TFs (large intersection) and pair-wise TF combinations (small intersection) that overlap between the two endogenous conditions.





Interpretation

Most hubs (78%) are transient = they are influential in one condition, but less

so in others.

Exogenous conditions have fewer transient hubs (different ).

„Transient hub“: capacity to change interactions between connections.



a, The 70 TFs active in the cell cycle. The

diagram shades each cell by the normalized

number of genes targeted by each TF in a

phase. Five clusters represent phase-specific

TFs and one cluster is for ubiquitously active

TFs. Both hub and non-hub TFs are included.

b, Serial inter-regulation between phase-

specific TFs. Network diagrams show TFs

that are active in one phase regulate TFs in

subsequent phases. In the late phases, TFs

apparently regulate those in the next cycle.

c, Parallel inter-regulation between phase-

specific and ubiquitous TFs in a two-tiered

hierarchy. Serial and parallel inter-regulation

operate in tandem to drive the cell cycle while

balancing it with basic house-keeping

processes. Luscombe et al. Nature 431, 308 (2004)

TF inter-regulation during the cell cycle time-course




Summary

Integrated analysis of transcriptional regulatory information and condition-specific

gene-expression data; post-analysis, e.g. - Identification of permanent and transient hubs- interchange index- overlap in TF usage across multiple conditions.

Large changes in underlying network architecture in response to diverse stimuli, TFs alter their interactions to varying degrees,

thereby rewiring the network some TFs serve as permanent hubs, most act transiently environmental responses facilitate fast signal propagation cell cycle and sporulation proceed via multiple stages

Many of these concepts may also apply to other biological networks.

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V11: Genetic networks