Kauffman Networks: Analysis and Applications

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Kauffman Networks Analysis and Applications

• Elena Dubrova,
• Maxim Teslenko,
• Andrés Martinelli
• IMIT/KTH
• Kista, Sweden

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Overview
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I. Introduction to Random Boolean Networks
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Random Boolean Network

• Create n vertices

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State of an RBN

• Vertices take values 0 or 1 (vertex state)
• Current state of the network is the vector of
  current values of its vertices
• Next state of a vertex v is determined by the
  current states on its parents u1,,uk
• xv fv (xu1,,xuk)
• An RBN is a synchronous Boolean automata

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State transition graph for an RBN with n4 and k2
xa xcxd xb x'a xc xbx'd xd 1
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II. RBNs in modeling of gene regulatory networks
modeling
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Background

• In a given organism, all types of cells have the
  same number of genes
• In a human
• ?25.000 genes
• 254 cell types
• Cell cycle time is the amount of time required
  for a cell to grow and divide itself into two
  daughter cells

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Jacob and Monod, 1964

• Any cell contains a number of regulatory genes,
  which act as tiny switches
• By exposing the cell to a certain hormone, they
  can be turned on or off
• Activated genes send signals to other genes
  which, in turn, get activated or repressed
• Signals propagate until the cell settles down
  into a stable pattern

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Cell Differentiation

• How can a single egg differentiate into many
  different cell types?

• Each cell type corresponds to a different
  pattern of activated genes

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Kauffman model

• In 1969, Kauffman proposed using RBNs for
  modeling of gene regulatory networks of living
  cells
• Each gene is represented by a vertex
• An edge from one vertex to another implies the
  former gene regulates the latter
• Boolean functions assigned to vertices represent
  rules of regulatory interactions between genes

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Kauffman model

• Kauffman has shown that if
• Each vertex has two parents, (k 2)
• Boolean functions are assigned independently and
  uniformly at random
• then the statistical features of RBNs match
  the characteristics of living cells
• number of attractors number of cell types
• length of attractors cell cycle time

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General model

• Suppose that Boolean functions are assigned so
  that they
• evaluate to 0 with probability p
• evaluate to 1 with probability 1-p
• For example, p0.5 means that Boolean functions
  are assigned independently and uniformly at
  random

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Dynamics of RBN, n??

• Number of attractors and their length depend on k
  and p
• kc 1/(2 p(1-p))

• k lt kc frozen phase
• ?(2n) attractors of length ?(1)

• k gt kc chaotic phase
• ?(1) attractors of length ?(2n)

• k kc self-organized critical behavior
• ?(?n) attractors of length ?(?n)

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III. Analysis of RBNs
redundancy removal
analysis
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Goal of analysis

• To compute attractors
• Cannot be done by full enumeration of states for
  large n
• n ? 25.000 (human genome)
• Number of states ?225.000

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Basic steps

• Remove redundant vertices
• Partition into components
• Compute attractors of components
• E. Dubrova, M. Teslenko, A. Martinelli, Kauffman
  Networks Analysis and Applications, ICCAD'05,
  Nov. 2005
• Compose attractors of the original RBN
• E. Dubrova, M. Teslenko, Compositional Properties
  of Random Boolean Networks, Physical Review E,
  71, May 2005, 056116

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Redundant vertices

• A vertex is redundant if its removal does not
  change the number and the length of attractors

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Redundant vertices

• A vertex is redundant if its removal does not
  change the number and the length of attractors

b
b
xa
xa
a
a
c
c
xcxd
xc
xbxd
xb
0
d
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Reduced STG
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Removing redundancy

• NP-complete problem
• In O(n) time we can remove some of the
  redundant vertices
• remove vertices with constant associated function
• remove vertices with no outputs

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Simulation results

• On average, the number of vertices is reduced
  from n to ?n
• State space is reduced from 2n to ?(2?n)
• 225.000 to ?(2158) for human genome

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Partitioning

• Two vertices belong to the same component if
  there is an undirected path between them
• Components can be computed in O(n) time

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Simulation results

• Number of components is of order of ?(log n)
• Size of the largest component in ?(n)
• "giant" component phenomenon in random graphs

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Computing Attractors Algorithm I

• Suppose we have a component with r vertices. Its
  state space is of size 2r
• If we pick up an arbitrary state and jump 2r
  steps forward, we are in some attractor

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Algorithm I

• Transition relation T2r (s,s) describes the set
  of next states s reachable from a current state
  s in 2r steps
• Can be computed by iterative squaring in
  r steps
• T, T2, T4, , T2r

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Algorithm I

• If we are in some attractor A and we jump up
  to 2r steps backward, then the set of reached
  states is the basin of attraction of A

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Algorithm II

• If we jump 2r steps forward from all states,
  then we get a set of states of all attractors
• A(s) ?s.T2r (s,s)

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Simulation results

• Number of attractors is of order of ?n
• Length of attractors in of order of ?n

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Summary

• Computation of attractors can be simplified by
  applying
• redundancy removal
• partitioning
• composition
• In this way, exact results for larger networks
  can be obtained

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IV. RBN-based Computing
RBN model
synthesis
implementation
redundancy addition
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Functions defined by RBNs

• Consider an RBN with r vertices and m attractors
• Basins of attraction partition the Boolean space
  0,1r into m parts
• The RBN defines the mapping
• f 0,1r ? 0,1,,m-1

logic AND
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Open problems
redundancy removal
analysis
modeling
RBN state space
Gene regulatory network
reduced RBN
RBN model
synthesis
implementation
redundancy addition
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Future work I Analysis

• Using Boolean circuits instead of BDDs to
  represent the set of states and the transition
  relation
• Better partitioning technique

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Future work II Synthesis

• Algorithms
• deriving a reduced RBN for a given STG
• redundancy addition
• The speculations that RBNs are advantageous (e.g.
  fault-tolerant, self-healing) need to be
  confirmed or refuted formally

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In a distant future

• When the level of understanding of gene
  regulatory networks matures, a functional
  nano-scale device operating on the principles of
  gene interactions might become a reality.

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Thank you!