V11 Modelling genetic networks by boolean networks

11. Lecture WS 2007/08

Bioinformatics III 1

V11 Modelling genetic networks by boolean 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



Boolean networks

Boolean networks allow dynamic modelling of synchronous interactions between

vertices in a network. They belong to the simplest models that possess some of the

biological and systemic properties of real gene networks.

In Boolean logic, a Boolean variable x is a variable that can assume only two

values. The values are denoted usually as 0 and 1, and correspond to the logical

values true and false.

The logic operators and, or, and not are defined to correspond to the intuitive

notion of truthfulness and composition of those operators.

A Boolean function is a function of Boolean variables connected by logic

operators.



Boolean networks

A Boolean network is a directed graph G(X,E), where the vertices, xi X, are

Boolean variables.

To each vertex, xi, is associated a Boolean function, b(xi1, xi2, …, xil) , l n, xij X,

where the arguments xj are limited to the parent vertices of xi in G.

Together, at any given time, the values of all vertices represent the state of the

network, given by the vector

S(t) =(xi1(t), xi2(t), …, xilt)).

For gene networks, the vertex variables correspond to levels of gene expression,

discretized to either 0 or 1.

The Boolean functions at the vertices model the aggregated regulation effect of all

their parent vertices. The states of all nodes are updated at the same time

according to their respective Boolean functions: txtxtxbtx

iliiii,...,,1

21



Boolean networks

The transitions of all states together correspond to a state transition of the

network from S(t) to the new network state, S(t+1).

A series of state transitions is called a trajectory.

Since there is a finite number of network states, all trajectories are periodic.

This simply follows from the fact that as soon as one state is visited a second time,

the trajectory will take exactly the same path as for the first time.

The repeating parts of the trajectories are called attractors, and can be one or

more states long.

All the states leading to the same attractor are the basin of attraction for this

attractor.



Reverse engineering Boolean Networks

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 which gene is regulating what other gene(s).

The goal in reverse engineering Boolean networks is to infer both the underlying

topology (i.e. the edges in the graph) and the Boolean functions at the vertices from

observed gene expression data.

The actual observed data can come either from gene expression experiments

conducted at different time intervals or when the expression of various genes is

perturbed.

For time-course data, measurements of the gene expressions at two consecutive

time points simply correspond to two consecutive states of the network, S(i) and

S(i+1).



Intergenic interaction matrix M

Perturbation data come in pairs, which can be thought of as the input/output states

of the network, Ii/Oi, where the input state is the one before the perturbation and the

output the one after it.

Given the observations of the states of a Boolean network, in general many

networks may be constructed that are consistent with that data. Hence the solution

network is ambiguous.

There are several variants of the reverse engineering problem:

(a) finding one network consistent with the data,

(b) finding all networks consistent with the data, and

(c) finding the 'best' network consistent with the data (according to some pre-

specified criteria).

The first task is the simplest one and efficient algorithms exist.



Reverse engineering of boolean networks

The reverse engineering problems are intimately connected to the amount of

empirical data available.

Obviously, the inferred network will be less ambiguous the more data points are

available. The amount of data needed to completely determine a unique network is

known as the data requirement problem in network inference.

The amount of data required depends on the sparseness of the underlying topology

and the type of Boolean functions allowed. This can be understood intuitively. A

network with few connections may be defined with few data points.

In the worst case, the deterministic inference algorithms need on the order of

m = 2n transition pairs of data (experimental data points) to infer a densely

connected Boolean Network with general Boolean functions at the n vertices.

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



Intergenic interaction matrix M

Since introducing the detection of gene expression by microarrays,

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

In V10, the entries of M were either 0 or 1.

Today, we allow the matrix element mij of the interaction matrix M to be

- 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.



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).



Reverse engineering of the interaction matrix

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.




Monod and Jacob (1961) first proposed that complex networks of gene interactions regulate cell differentiation. The first formal models of genetic regulation of cell differentiation anticipated that real biological genetic networks would be too complex to be analyzed without the use of formal mathematical and/or computational tools. However, because the early models made some assumptions that were biologically unrealistic, experimentalists largely ignored them. Relatively complete genetic descriptions of developmental programs are now available in several model organisms, providing the necessary inputs for developing biologically realistic dynamic models of gene regulatory networks in cell differentiation. Such models should aid at building a formal framework for studies of developmental mechanisms and their evolution.

Espinosa-Soto et al. The Plant Cell 16, 2923 (2004)



methods

The model is discrete. N is the number of genes involved in the network,

Xn is a vector with expression state for each gene in a space of N dimensions,

representing the network state after n iterations:


where xi(n) represents the state of expression of the gene i at the iteration n.

We then write:

indicating that the state at the iteration (n + 1) is determined by the state at the

previous iteration.

Each node, except SEP (redundant SEP1, SEP2, and SEP3 genes), stands for the

activity of a single gene involved in floral organ fate determination.

Most nodes could assume three levels of expression (on, 1 or 2, and off, 0) to

enable different activation thresholds when experimental data was available.



functions of genes inferred from experiments


FLOWERING LOCUS T Double embryonic flower1 (emf1) flowering locus t (ft)

mutants do not develop embryonic flowers typical of emf1 single mutants,

suggesting that the lack of FT activity suppresses the emf1 phenotype because

EMF1 represses FT.

LFY

Double mutants of the weak emf1-1 allele and lfy-1 bear lfy-like flowers, suggesting

that, for this trait, lfy is epistatic. These genes have antagonistic activities; hence,

we infer that LFY is repressed by EMF1.

TFL1

In emf1-2 tfl1 double mutants, the emf1-2 mutation is epistatic. As these two genes

do not have opposite functions, this result suggests that EMF1 protein is needed for

TFL1 activity in wild-type Arabidopsis.

....



logical rules




logical rules




logical rules




network architecture


Figure 4. Gene Network Architecture for

the Arabidopsis Floral Organ Fate

Determination. Network nodes represent

active proteins of corresponding genes,

and

the edges represent the regulatory

interactions between node pairs (arrows

are positive, and blunt-end lines are

negative). Dashed lines are hypothetical

interactions for which there is no

experimental support (see logical rules).

The network includes F-box proteins

(UFO), membrane bound signaling

molecules (TFL1 and FT), cofactors

involved in transcriptional regulation (EMF1

and LUG), chromatin remodeling proteins

(CLF), and transcription factors (all others).

Interactions have been confirmed to be

direct transcriptional regulations in a few

cases (LFY on AG, LFY on AP1), and the

rest can either be direct or indirect and can

be transcriptional or other.



states of the system


The system has a finite number of possible initial conditions equal to 139,968,

and each one is represented by a vector of dimension 15 in which each column

corresponds to the expression state of each network node at initial conditions in the

following order:

FT EMF1 TFL1 LFY FUL AP1 AP3 PI AG UFO WUS AP2 SEP LUG CLF.

The vector of 15 entries that keeps track of the activity level of each node describes

the system at each time point. We updated the state of each node synchronously.

Starting on each initial condition the network is iterated until it reached an attractor.

The 139,968 initial states converge to only 10 stable attractors. All of them are fixed

point attractors in which the activity level of all genes remains the same as in the

previous iteration, see table 1.



simulated gene expression levels


These steady gene states (Table 1) predicted by the model coincide with the gene

expression profiles that have been documented experimentally in cells of wild-type

Arabidopsis inflorescence meristems and floral organ primordia.

For example, in the Infl steady states, floral meristem identity genes (LFY, AP1, and

AP2) and floral organ identity genes (AP1, AP2, AP3, PI, SEP, and AG) are off,

whereas the inflorescence identity genes (EMF1 and TFL1) are on.



attractors


The size of the basins of attraction may indicate how stable each morphogenetic response is and which genes are critical to attain each cell fate



petunia gene network


Deciphering network architectures underlying cell differentiation is a first step toward understanding the mechanisms that rule conservation and variation in morphological traits. Although most flowering species have an overall conserved plan of floral organ determination, mutations have revealed some important variations. In the case of petunia, the overall network of cell fate determination during flower organ development seems to be conserved with respect to Arabidopsis. However, mutant analyses have suggested some differences. Our simulation results suggest that an architecture like the one proposed here for the Arabidopsis network that includes a duplicated AP3 would yield the gene expression patterns observed for the wild type and a single AP3-like gene mutant and provides a prediction for the double loss-of-function mutations of AP3-like genes in petunia.



comparison arabidopsis - petunia


Figure 5. Arabidopsis and Petunia Wild Type (left), ap3 Mutant (right) Flowers, and Corresponding Network Models. Single petunia mutant for PhDEF is shown in the top part of (B) and a scheme of the predicted double mutant for PhDEF and PhTM6 is shown below. Arabidopsis is shown in (A). The networks indicate which nodes were turned off (yellow) to simulate mutants. The Arabidopsis orthologs of the cloned petunia genes are as follows: FLORAL BINDING PROTEIN26 (FBP26) is an AP1 ortholog, PhDEF (formerly known as GREEN PETALS) is an AP3 and DEFICIENS (DEF; from A. majus) ortholog, PhGLO1 (FBP1) and PhGLO2 (PETUNIA MADS BOX GENE2; pMADS2) are PI and GLOBOSA (GLO; from A. majus) orthologs, pMADS3 is an AG ortholog, PhAP2A is an AP2 ortholog, FBP2 and FBP5 are SEP orthologs, PhCLF1 and PhCLF2 are CLF orthologs, and PETUNIA HYBRIDA TM6 (PhTM6) is a paleoAP3 gene.



comparison arabidopsis - petunia




Control of Gene Expression

A bacterial cell lives in direct contact with its environment.

Its chemical composition may dramatically change from one moment to the other.

Consider bacteria growing either on lactose or tryptophan.

Fig. 2.16 Lactose: di-saccharide from glucose + galactose

oxidation provides cells with metabolic intermediates

and energy.

First step of lactose degradation (catabolism):

hydrolysis of the bond joining the 2 sugars by

-galactosidase

[Karp] Cell & Mol. Biol.



Transfer from minimal medium to lactose medium

When bacterial cells are grown in

a minimal medium, they don‘t

need -galactosidase and

contain < 5 copies and only 1

copy of its mRNA.

What happens when the cells are

transferred to a lactose medium?




lac Operon: an inducible operon

Inducible operon:

presence of substance (lactose) induces

transcription of the structural genes.

lac operon contains 3 tandem structural

genes:

z gene: encodes -galactosidase

y gene: encodes galactoside permease,

a protein that promotes entry of lactose

into the cell

a gene: encodes thiogalactoside

acetyltransferase




positive control by cyclic AMP

Repressors, such as those of the lac and trp operons, exert their influence by

negative control.

lac operon is also under positive control, the „glucose effect“.

If bacterial cells are supplied with glucose (as well as with other substances such as lactose

or galactose), the cells catabolize the glucose and ignore the other compounds.

glucose in the medium suppresses the production of various catabolic enzymes, such as

-galactosidase, needed to degrade the other substrates.

In 1965, cAMP was deteced in E.coli. The higher the glucose concentration, the lower the

cAMP concentration. When adding cAMP to the medium in the presence of glucose, the

catabolic enzymes that were normally absent were suddenly synthesized by the cell.

cAMP binds to CRP. The cAMP-CRP complex recognizes and binds to a specific site in the

lac control region. The presence of bound CRP changes the DNA conformation and allows

RNA polymerase to transcribe the lac operon.




positive control by cyclic AMP

Fig. 12.27




Growth on Trp medium

Trp is required for protein synthesis.

If no Trp is available in the medium, the bacterium must expend energy

synthesizing this amino acid cells contain enzymes and corresponding mRNA

of Trp-synthesis pathway.

If Trp becomes available in the medium, the cells no longer have to synthesize

their own Trp. Within a few minutes, the production of the enzymes of the Trp

pathway stops. In the presence of Trp, the genes encoding these enzymes are

repressed.




trp operon

In a repressible operon, the repressor

is unable to bind to the operator DNA

itself.




eukaryotic gene expression: PEPCK

model case: gene that codes for phosphoenolpyruvate carboxykinase (PEPCK).

This enzyme is one of the key enzymes of gluconeogenesis, the metabolic pathway that

converts pyruvate to glucose.

The enzyme is synthesized in the liver when glucose levels are low, e.g. when considerable

time has passed since your last meal. Synthesis drops sharply after carbohydrate-rich meal.


Level of synthesis of PEPCK mRNA is controlled by a variety of transcription factors,

including several hormone receptors that are involved in regulating carbohydrate

metabolism.

To understand the regulation of PEPCK gene expression, we need to

(1) unravel the functions of the numerous DNA regulatory sequences that residue upstream

from the gene itself

(2) identify the transcription factors that bind these sequences, and

(3) identify the signalling pathways that activate the machinery responsible for selective

gene expression.



eukaryotic gene expression: PEPCK

Fig. 12.32

TATA box followed by core promoter: site of assembly of a pre-initiation complex consisting

of RNA polymerase II and a number of general TFs

CAAT + GC boxes: bind global TFs such as NF1 and SP1; both are typically located 100 –

150 bp upstream proximal promoter elements [Karp] Cell & Mol. Biol.



Responsive elements from PEPCK gene

various hormones affect the expression of

PEPCK gene: insulin, thyroid hormone,

glucagon, glucocorticoid.

All of the act by means of specific TFs

that bind to the DNA.

Fig. shows responsive elements.




Activation of transcription

For example, let us focus on glucocorticoids, a group of steriod hormones (e.g. cortisol) that

are synthesized in response to stress.

Fig. 12.34




Conservation of regulatory elements?

Nature 423, 241 (2003)



Comparative genome analysis

Compare sequences of Saccharomyces paradoxus, S. mikatae, S. bayanus, with

S. cerevisae.

The three new yeast species have

sufficient sequence similarity to

S. cerevisiae to allow orthologous

regions to be aligned reliably, but

sufficient sequence divergence

to allow many functional elements

to be recognized by their greater

degree of conservation

by a four-way species comparison.

Assemble with Arachne program

Align 4 genomes.

Nature 423, 241 (2003)



Conservation of the Gal4-binding site

We first studied the binding site for one of the best-studied transcription factors,

Gal4, whose sequence motif is CGGn(11)CCG (which contains 11 unspecified

bases).

Gal4 regulates genes involved in galactose metabolism, including the GAL1 and

GAL10 genes, which are divergently transcribed from a common intergenic

region.

The Gal4 motif occurs three times in this intergenic region, and all three instances

show perfect conservation across the four species.

In addition, there is a fourth experimentally validated binding site for Gal4 that

differs from the consensus by one nucleotide in S. cerevisiae.

This variant site is also perfectly preserved across the species.

Nature 423, 241 (2003)



Conservation of the Gal4-binding site

We then examined the frequency and conservation of Gal4-binding sites across the aligned

genomes. In S. cerevisiae, the Gal4 motif occurs 96 times in intergenic regions and 415

times in genic (protein-coding) regions.

The motif displays certain marked conservation properties:

(1) occurrences of the Gal4 motif in intergenic regions have a conservation rate (proportion

conserved across all four species) that is about fivefold higher than for equivalent random

motifs.

(2) intergenic occurrences of the Gal4 motif are more frequently conserved than genic

occurrences. By contrast, random motifs are less frequently conserved in intergenic regions

than in genic regions, reflecting the lower overall level of conservation in intergenic regions.

Thus, the relative conservation rate in intergenic compared with genic regions is about 11-

fold higher for Gal4 than for random motifs.

(3) the Gal4 motif shows a higher conservation rate in divergent compared with convergent

intergenic regions (those that lie upstream compared with downstream of both flanking

genes); no such preferences are seen for control motifs. These three observations suggest

various ways to discover motifs based on their conservation properties.

Nature 423, 241 (2003)





Assign function

Assign candidate functions to these discovered motifs by the genes adjacent to conserved

occurrences of the motif with known gene categories.

Test for Gal4 motif. Given the biological role of Gal4, we considered the set of genes

annotated to be involved in carbohydrate metabolism (126 genes according to the Gene

Ontology classification) with the set of genes that have a Gal4-binding motif upstream. The

intergenic regions adjacent to carbohydrate metabolism genes comprise only 2% of all

intergenic regions, but 7% of the occurrences of the Gal4 motif in S. cerevisiae and 29% of

the conserved occurrences across the four species.

suggests that a function of the Gal4 motif could be inferred from the function of the genes

adjacent to its conserved occurrences. Such putative functional assignments can be useful

in directing experimentation for understanding the precise function of a motif.

Such considerations indicate that it should be possible to use comparative analysis, such

as explored here for yeast, to identify directly many functional elements in the human

genome that are common to mammals. More generally, comparative analysis offers a

powerful and precise initial tool for interpreting genomes.

Nature 423, 241 (2003)

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V11 Modelling genetic networks by boolean networks