Mathematical Models and Experimental Verification of a CellDeath Process

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Mathematical Models and Experimental Verification
of a Cell-Death Process
Nadia Ugel NYU Bioinformatics Group (Joint work
with Shih-Chieh Lin,Yuri Lazebnik Bud Mishra)
July 25, 2004. International Conference for
Mathematics in Biology and Medicine.
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Reasoning and Experimentation
Model Simulation
Model Construction
Comparison
Hypotheses
Revision
Experimental Results
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How much of reasoning about biology can be
automated?

• We claim that, by drawing upon mathematical
  approaches developed in the context of dynamical
  systems, kinetic analysis, computational theory
  and logic, it is possible to create powerful
  simulation, analysis and reasoning tools for
  working biologists to be used in deciphering
  existing data, devising new experiments and
  ultimately, understanding functional properties
  of genomes, proteomes, cells, organs and
  organisms.

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Part I Mathematics

• Automated Translation from Graphical to
  Mathematical Model
• Automated Verification of Properties

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Different Representations
Lee, Salic, Krüger, Heinrich, Kirschner
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Who are the players?

• For each reaction we can identify one or more
• Reactant
• its concentration affects and is affected (-)
  by the reaction rate
• Product
• its concentration is affected () by the
  reaction rate
• Modifier
• its concentration affects the reaction rate

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What are the rules?

• To each reaction we can associate

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Simple Example
Pathway
ODE
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Canonical Form
for n dependent and m independent species

• Characteristics
• Predefined Modular Structure
• Computational Manipulation
• Scalability

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Temporal Logic and Model Checking

• Set of Queries (through Model Checking)
• summarize the numerical traces into an automaton
  with distinguishable biological states and a
  deterministic set of rules of transition from
  state to state
• check the automaton model for its ability to
  satisfy various temporal logic statements

• Set of Differential Equations

• Set of Traces

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Temporal Logic

• Next Time X P
• the property P holds in the second state of the
  path
• Eventually F P
• the property P will hold at some state on the
  path (in the future)
• Always G P
• the property P holds at every state on the path
  (globally)
• Until P U Q
• the property Q holds at some state on the path
  and property P holds at all preceding states

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Model Checking

• is an automatic technique for verifying
  correctness of temporal properties
• is efficient (linear on the number of states)
• will terminate with an answer indicating that the
  model satisfies the formula or show a
  counter-example in case it does not

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Automaton

• Simple Approach
• The values of the variables uniquely characterize
  the state of the system.
• The succession of the integration steps describes
  the possible transitions.
• More Sophisticated Approach
• Grouping several time instants according to some
  simple rules.

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Bisimulation

• Definition of projection operation
• (restriction to a subset of interesting
  variables)
• to generate reduced automata satisfying the same
  formulae as the initial ones.

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Part II Biology

• Apoptosis is a form of programmed cell death.
• Receptor Mediated (external)
• Receptor Independent (internal)

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Caspases
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Mitochondria
DNA
nucleus
DEVD-Afc
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Results
Rodriguez and Lazebnik (1999)
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Part III Model
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Technology - Framework
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Technology - Application
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Series of Models (I)
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Series of Models (II)
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To be continued (Apoptosis)

• Reversible Reactions
• Include different (cleaved) form of caspase 9
• Varying initial concentrations (of APAF1)
• Other reactants involved (e.g. XIAP)

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Possible Models
Reversible
Caspase 3 expanded
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Ongoing Research (Simpathica)

• Continuous vs Stochastic Models
• Spatial Models
• Numerical Integration
• Hybrid Systems
• Hierarchical Models

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Acknowledgements

• Bud Mishra Members of NYU Bioinformatics Group
• Shih-Chieh LIN, Yuri Lazebnik and Members of
  Yuris lab at CSHL
• Web-site http//www.bioinformatics.nyu.edu/
• Simpathica is supported by DARPA-BioSpice