Boolean Models for Biological Networks

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Boolean Models for Biological Networks

• Lecture 2 Jan-Feb 04
• Dr. Eduardo Mendoza
• Physics Department
• Mathematics Department Center for
  NanoScience
• University of the Philippines
  Ludwig-Maximilians-University
• Diliman Munich, Germany
• eduardom_at_math.upd.edu.ph
  Eduardo.Mendoza_at_physik.uni-muenchen.de

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Topics

• Course organization
• Boolean Network Basics
• Kauffman networks
• Boolean models in Biology
• The cellular Apoptosis network
• A Boolean model for Apoptosis
• Exercise from boolean to binary polynomial

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1. Course Organization

• Papers for presentation
• Group (Janice,Marrick , Feb 12/16)
• J. Heidel, J. Maloney, C. Farrow Finding Cycles
  in Synchronous Boolean Networks with Applications
  to Biochemical Systems (preprint, 37 pp)
• Group 2 (Romina,Emmy Feb 16/19)
• R. Laubenbacher, B. Pareigis Finite Dynamical
  Systems, Adv. In Applied Math. 26 (2001), 14 pp

• Need substitute time for
• February 5
• February 12
• February 19
• 8.30 10.00

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2. Boolean Network Basics
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Boolean Network - Definition

• Let F2 0,1
• A Boolean network consists of
• 1) n Boolean variables (xi0,1)
• 2) local update Boolean functions
• fi (F2) n ? F2
• 3) a directed graph G with n vertices vi and
  edges connecting vi to vj if xi appears in fj.

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Model properties
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Boolean network dynamics (1)

• Boolean networks have at most 2n states, where n
  is the number of genes
• therefore after at most 2n 1 iterations, a
  repeating state must be found
• repeating state may occur as single state (point
  attractor) or as a cycle of several states
  (dynamic attractor)

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Boolean network dynamics (2)
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3. Kauffman networks (N,k)

• Studied by Stuart Kauffman since late 60s, also
  called random boolean networks (RBN)

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Review A basin of attraction
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Review biological interpretation
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Kauffman Networks (2)
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Kauffman Networks (3)
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Kauffman Networks (4)
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Periods of Kauffman networks
Albert-Barabasi Conjecture and numerical evidence
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4. Boolean models in Biology
Diversity of Modeling Techniques...

• Graphs (directed and undirected)
• Bayesian networks
• Boolean, generalized logical networks, polynomial
  models
• Nonlinear ODEs (ordinary differential equations)
• Special cases S-Systems, GMA Systems, pieceweise
  linear, qualitative
• PDEs (partial differential equations) and other
  spatially distributed models
• Stochastic master equations
• Rule-based formalisms
• Petri nets, transformational grammars, process
  algebras,.

Modtech Modeling techniques
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Boolean Networks and Biology

• Need quantitative analysis to understand complex
  biological networks
• What mathematical framework is appropriate for
  analysis? Depends...
• Case 1 Detailed knowledge of biochemical
  mechanisms
• Case 2 Data imply connectivities, but molecular
  details unknown

Biochemical Mechanisms
Causes
Effects
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Where does Boolean fit in? (1) Case 1 Detailed
knowledge ofbiochemical mechanisms

• Model with system of differential equations
• 2 types of dynamics
• Analog ODE crucial to describe key features
• Discrete steady-states capture behavior ODE is
  sufficient but not necessary
• Can be abstracted to Boolean algebra, where new
  framework offers new insights while retaining
  analysis capabilities

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Where does Boolean fit in? (2) Case 2 Data
imply connectivities, but molecular details
unknown

• When data show only two steady-states, cause and
  effect relationships can be modeled with Boolean
  logic functions

A
B
A
B
C
C
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Caspase Cascade in Apoptosis
Extrinsic
Intrinsic

• Missing some mechanistic detail
• Data show 2 steady-states

Death
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Previous Caspase Modeling

• Details of underlying mechanisms and important
  parameters were unknown, but Bailey attempted to
  model the cascade with a set of differential
  equations coupled with specialized functions.
• Their goal was to obtain qualitative results in
  the form of identifying combinations of drug
  targets to inhibit apoptosis despite both
  intrinsic and extrinsic death signals.

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Previous Results
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Review Boolean Functions
NOT Gate
NOR Gate
AND Gate
NAND Gate
OR Gate
XNOR Gate
XOR Gate
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Updated Model (Boolean)
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Analysis

• Mathematical manipulation
• Extract how output depends on input
• Blake Canonical Form
• Caspase-Dependent Death
• External Death Signal AND not FLIPs AND not IAPs
• OR
• Cell Damage AND not ARC AND not IAPs

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Model with Drug Targets
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Thanks for your attention !

• Questions?