Research Topics in Computational and Mathematical Biology

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• Research Topics in Computational and Mathematical
  Biology
• Lecture 2B
• Dr. Eduardo Mendoza
• Mathematics Department Physics
  Department
• 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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Basic Bioinformatics Problems ZIMM02

• illustrates movement of traditional
  Bioinformatics to Systems Biology

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Topics to be covered

• Structural ( stoichiometric) analysis of
  biochemical networks
• Elementary (flux) mode analysis
• Application to E.coli metabolism
• Possible research areas

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Main References

• A. Cornish-Bowden, M.L. Cardenas Metabolic
  balance sheets, Nature 420 (14/11/02)
• R.Heinrich, S. Schuster The modelling of
  metabolic systems, BioSystems 47 (1998)
• S.Schuster, C. Hilgetag, J.H.Woods, D.A.Fell
  Reaction routes in biochemical systems Algebraic
  properties, validated calculation procedure and
  example from nucleotide metabolism Journal of
  Mathematical Biology 45 (2002)
• J. Stelling, S. Klamt, K. Bettenbrock, S.
  Schuster E.D. Gilles Metabolic network
  structure determines key aspects of functionality
  regulation, Nature 420 (14/11/02)

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1. Structural analysis of biochemical networks

• Traditional focus of math modelling kinetic
  models
• Aim predict system dynamics based on knowledge
  of network topology and kinetic parameters
• Methods
• solve algebraic equations for steady states and
• solve systems of differential equations for time
  dependent states
• Complementary methods developed recently to
  address limitations

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Issues in Kinetic Modelling SCHI02

• Kinetic properties (rate constants, etc) are not
  completely known
• Discrepancies exist between in vitro and in vivo
  behavior
• Enzyme activities in vivo are subject to frequent
  changes due to inhibition or activation. In
  contrast the structure..in terms of how
  substances are connected can be considered
  constant, unless evolutionary time scales are
  studied

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Non-kinetic approaches HESC98

• Structural Analysis aimed at elucidating
  relevant relationships between systems variables
  on the basis of network stoichiometry without
  reference to kinetic properties
• Metabolic Control Analysis serves to quantify,
  in terms of control coefficients, the extent to
  which different enzymes limit the flux under
  particular conditions
• Evolutionary Optimization analyzes systems
  parameters on the basis of evolutionary
  optimization principles (eg max reaction rates,
  min intermed. concentrations, min transient
  times)

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Def example Stoichiometry Matrix
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Structural Analysis aspects

• Conservation relations
• Eg certain linear combinations of several
  substances concentrations are constant over time
• Imply that some rows of stoichiometrx matrix are
  linearly dependent
• Can represent conservation of chemical units in
  this case, all vector coefficients must be
  non-negative
• Maximal conserved moieties
• Defined as largest molecular assemblies conserved
  in a given reaction system
• Frequent situation stoichiometry matrix given
  and conserved-moiety structure is sought
  (Schuster Hilgetag algorithm 1995)
• Steady states (cf. Elementary mode analysis)

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2. Elementary (flux) modes

• Elementary (flux) mode analysis introduced by
  Schuster Hilgetag (1994)

• Example 1

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Elementary (flux) mode example 2(1)
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Elementary (flux) mode example 2(2)
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Elementary modes mathematical definition and
properties

• Definitions
• index set of zero coordinates S(v)
• vectors in Ker N (called the null space V) with
  sign restriction
• (flux) mode M
• elementary (flux) mode (also called direct
  reaction route)
• Concepts related to elementary modes direct
  mechanism (chemistry), minimal T-invariant
  (Petri nets)

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Steady state, null space
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Definition Flux Mode
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Flux mode (2)
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Key fact about EFMs

• Any real flux can be represented as a
  superposition of elementary modes, in fact it is
  a linear combination with positive coefficients
• I.e. Any stationary state can be decomposed with
  respect to the flux values, in elementary modes
  which are realizable stoichiometrically and
  thermodynamically.

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Basic idea of EFM-algorithm

• Strategy extend a basis of convex flux cone to
  include all EFMs
• Algorithm for elementary modes is
  extension/adaptation of convex basis algorithm of
  Nozicka (1974)
• Start
• Tableau with stoich. Matrix N identity matrix,
  with N decomposed into rev irr reactions
• All metabolites are external ?each reaction is
  EFM on ist own
• In each step
• Preliminary elementary modes have to be linearly
  combined to give new preliminary elem modes in
  the next tableau
• Final result
• Final tableau contains a submatrix whose rows
  represent the elementary modes

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3. Elementary modes in E.Coli metabolism

• Introducing E.coli

• Key facts
• Cell length 1-3 microns
• DNA length 4500 kb
• 4400 genes
• 1 Chromosome
• 2500 active proteins
• 50-70 sensors on the membrane

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Elementary modes in E.coli some results
analysis
Stelling et al Nature Nov 14 02
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Understanding experiments(Cornish-Bowden,
Cardenas Nature Nov 14 2002)
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Elementary modes in E.coli prediction of gene
expression

• Direct correlation between metabolic fluxes and
  transcriptome/proteome patterns not yet observed
• Concept of a reactions control-effective flux
  introduced to measure flexibility-efficiency
  trade-off (average flux weighted by mode
  efficiency)
• Ratio of control-effective fluxes ( theoretical
  transciption rates) for E.coli growth on
  different substrates compared with DNA microarray
  data

(Stelling et al Nature Nov 14 02)
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Prediction details 1
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Prediction details 2
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Prediction details 3
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Prediction vs. experiment
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Elementary mode analysis prediction potential

• Examples
• (Cornish-Bowden, Cardenas Nature Nov 14 2002)
• Tolerable combination of mutations
• Genetic modifications enabling new properties
• Improvement of yield
• Robustness behavior

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Possible Bio-Network topics

• Presentation of EFM-Algorithm in detail proofs
  of underlying theorems
• Ref S.Schuster, C. Hilgetag, J.H.Woods,
  D.A.Fell Reaction routes in biochemical systems
  Algebraic properties, validated calculation
  procedure and example from nucleotide metabolism
  Journal of Mathematical Biology 45 (2002)
• 2. Specification of functional units in metabolic
  networks example
• Ref A. Kremling, K. Jahreis, J. Lengeler, E.D.
  Gilles The Organization of Metabolic Reaction
  Networks A Signal-Oriented Approach to Cellular
  Models, Metabolic Engineering 2 (2000)

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Bio-Networks topics (2)

• 3. Analysis and demo of BioSketchpad Modeling
  Tool (from BioSPICE)
• 4. Analysis and demo of JigCell Modeling Tool
  (from BioSPICE)
• 5. Dynamics of cell cycle regulation (paper by J.
  Tyson et al, BioEssays 24 (Dec 2002)
• 6. Model Building and Model Checking for
  Biochemical Processes
• (paper by Marco Antoniotti, Alberto
  Policriti, Nadia Ugel, Bud Mishra-preprint)