Modeling Genetic Network: Boolean Network

Yongyeol Ahn2004.08.18. KAIST

Genetic Network

• Genes interact with each other via proteins, RNAs and themselves.

Main Objectives

• To infer genetic network from biological data

• To explain and predict the behaviors of genetic regulatory network

Modeling Genetic Networks

• Statistics rules!Bayesian network

Hmm.. We need some dynamics.

• Let’s be realistic! Differential equation approach

• Simple is the best!Boolean network

Bayesian Network : Information Theory

• Shannon entropy:

• Joint probability: Pr(x,y)• Conditional probability:

Pr(y|x) = Pr(x,y)/Pr(x)• Mutual information:

Bayesian Network

• Find a directed acyclic graph which shows the relationships of nodes well.

Xi : Expression level

Differential Equation Approach

Gene A Gene R

AA

1 AA

1

50 0.01

A50

R 5

C

+

2

100.5

500 50

50 100

10.2

1‘The Clock’

Differential Equation Approach

30 40 50 60

0.2

0.4

0.6

0.8

Expressedgenes

30 40 50 60

20

40

60

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mRNAsR

A

30 40 50 60

500

1000

1500

2000

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R

250 500 750 1000 1250 1500 1750

500

1000

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Why Boolean network?

• It tells about the dynamics (vs. bayesian network)

• ‘Gene switch’ : There are attempts to make a ‘genetic computer’ using genetic ‘logic gates’. Binary state approximation is fine.

• In many cases, the exact timing may not be important.

• Simple, general, easy to implement, …

The Boolean Network

• Nodes, Directed links• Synchronous dynamics• Binary states: ‘on’ or ‘off’

• A node’s state is determined by states of other nodes which have a link to the node(by assigned boolean functions).

Example

0

1

2

1 20 0 10 1 01 0 11 1 1

Node 0 Node 2

1 0 11 0

0 1 2

0 0 01 0 10 0 10 0 1

t

Boolean Network Variations

• Multi-Valued model• Different updating scheme

(asynchronous, …)• Probabilistic model

Classification of boolean networks

(Gershenson2004)

Tools

• DDLab – http://www.ddlab.com

• RBNLab– http://rbn.sourceforge.net

• BN/PBN toolbox: – http://www2.mdanderson.org/app/ilya/PBN/PBN.htm

(Classical) Random Boolean Network

• Parameter: N, K, p– N: number of nodes– K: average in-degree– p: probability of ‘1’ in each boolean function

• Large ensemble ,state space(2^N) So big! Very high standard deviations

Phase Transition

• Stable (K<=2)• Critical• Chaotic (K>=3)

• Visualization method– Active nodes: ‘green’– Frozen nodes: ‘red’

Phase Transition

• Islands– Chaotic: green sea percolate & red islands– Stable: frozen red sea & green islands

• Robustness– Chaotic: damage spreads– Stable: robust

• Convergence and divergence of traj. (Lyapunov exponent)– Chaotic: similar states tend to diverge– Stable: tend to converge

Loops Trees

• For active dynamics, network needs Loops.

• Loops activate other parts (trees).• Active wave propagates from loops

to trees.

G – Density

• G-density : Garden of Eden states density

• Ordered: very high G-density, high in-degree frequency

• Critical: power-law in-degree distribution

• Chaotic: lower G-density,

Analytical Result of Phase Transition

• Derrida’s annealed approx. : Assuming connections and boolean functions are randomly reshuffled at each time step.

• Define overlap = 1 – Normalized Hamming distance between two states

• What will happen at tinf ?

Analytical Result of Phase Transition

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For a network with in-degree k,

Transforming with Hamming distance and consider bias

]))(1(1)[1(2)1( ktdpptd

Derrida Curves

)1( td

)(td

K=2

K=5

1)]1(2[ ppkCritical connectivity

Phase diagram

(In practice, the size of the network can play a role in the phase transitions)

Topology of boolean network

• In reality, genetic networks have very inhomogeneous degree distribution

• Using Derrida’s annealed approximation, the phase diagram for scale-free network can be obtained.

Derrida’s Annealed Approx. For Power-law Degree Distribution

• By the assumption, x(t) obeys the equation

Where,

Contd.

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Topology of boolean network: Scale-free boolean network

Attractors

The Number of Attractors

• The number of attractors grows faster than any power law with system size. (Samuelsson2003)

The Length of Attractor

• For K=1, root(N/2)• At critical phase, it is long believed

that the length proportional to root N ( Kauffman argued that this is related to the number of cell types )

• But it is linear

Applications

• Reverse Engineering• Morphogenesis model• Segment polarity development• Yeast transcriptional network

Reverse engineering: REVEAL

• REVerse Engineering Algorithm• It finds a minimal solution for a

boolean network given any set of time-series.

• Use entropy, mutual information

Neutral Mutation and Punctuated Equilibrium (Bornholdt1998)

• The model evolves under robustness principle (look for silent mutations)

• Threshold networks (restricted set of the boolean networks)

Weight = ± 1, 0

Evolutionary Rule

• Create a daughter network by ‘adding’, ‘removing’, ‘adding and removing a weight in the coupling matrix’ at random. (each p = 1/3)

• With a random initial state, if mother & daughter reach the same attractor, replace the mother with the daughter. In other case, keep the mother network.

Punctuated Equilibrium

• The evolution shows punctuated network connectivity (lifetime ~ 1/t^2)

• Evolved networks have much shorter attractors, large frozen components

Model for Morphogenesis(Sole2003)

• Modeling an organism with one dimensional cell array.

• Each cells have the same set of genes and hormones.

• Genes interact within the cell.• Hormones communicate with

neighboring cells.• Threshold model.

Morphogenesis Model

Development

Adaptive Walks

• ‘Toward more complex organism’• Complexity measure: the number

of cell types• Rule

– Evolve many organism in parallel– Addition, Removal, randomization of

link, link’s weight (each p=1/3)– Check the complexity

Logarithmic Increase of the Number of Patterns

• Consistent with Kauffman’s ‘rugged landscape’ explanation of Cambrian Explosion

Segment Polarity Network in Fly (Albert2003)

• The genetic network of Fly development

• This network is simulated with ODE(Dassow, 2002) and has shown a good result.

Boolean Network Construction Rules

• The effect of transcriptional activators and inhibitors is never additive, but rather, inhibitors are dominant.

• Transcription and translation are ON/OFF functions of the state

• If transcription/translation is ON, mRNAs/proteins are synthesized in one time step

• mRNAs decay in one time step if not transcribed• Transcription factors and proteins undergoing

post-translational modification decay in one time step if their mRNA is not present.

Constructed Boolean Network

Results

• Stable state is same as the real fly.• Essential gene deletion results also

agree with real data.• There are six distinct steady states

in the model. Three of these are well-known experimentally

Yeast transcriptional network(Kauffman2003)

• For a given network structure, they generate boolean network ensembles with nested canalyzing functions.

• The ensemble of networks are very stable.

Canalyzing Function

• Canalyzing boolean function has at least one input which the output value is fixed.

• In most cases, genetic networks consist of canalyzing functions.

Yeast Transcriptional Network

• Find topological transition point (to determine the confidence p value)

• Remove genes that have no output to other genes

Yeast Transcriptional Network

• For a given network structure..– Functions with null

hypothesis– Functions based

on literature (canalyzing)

Nested Canalyzing Function

• Inputs im, outputs Om, in degree k

• Assume i1 is canalyzing input, then we can define a new rule without i1 (with indegree k-1)

• In most cases, this new rule is also canalyzing.

Conclusion

• Boolean network model is simple, abstract, general.

• But, it’s powerful.