Mathematical Systems Biology

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Mathematical Systems Biology

• 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

• What is Biomathematics (or Mathematical Biology)?
• Relationship to Bioinformatics (or Computational
  Biology)
• Mathematical Modeling Techniques for Biological
  Networks
• Case Study 1 EFM-Analysis of E.Coli metabolic
  network
• Case Study 2 Boolean model of Drosophila segment
  polarity network
• Interpretation as polynomial model
• Case Study 3 ODE model of Dictyostelium TCA cycle

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What is Biomathematics?

• Synonymous with Mathematical Biology
• My favorite answer is an adaptation of the aims
  scope of the Journal of Mathematical Biology
  Biomath either
• provides biological insight as a result of
  mathematical analysis or
• identifies and opens up challenging new types of
  mathematical problems that derive from biological
  knowledge

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The context of Biomathematics (1)

• Related areas
• Mathematical Biosciences, Mathematical Life
  Sciences
• Theoretical Biology
• Theoretical Biophysics (Biological Physics)
• Biostatistics
• Computational Biology/Bioinformatics
• ...

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The context of Biomathematics (2)

• Biomath/Math Bio
• can look back to an (at least) 70 year-tradition
  with work of Fisher, Haldane, Lotka, Volterra in
  the 1930s ( but also work of Malthus in the
  1890s?)
• Focuses on Math Modelling (traditionally with
  differential equations) while
• Bioinformatics on algorithms and related
  computational aspects
• Biostatistics on experimental design and
  (available) data analysis

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Communities and organizations (1)

• Special journals
• Bulletin of Mathematical Biology (-gt SMB)
• Journal of Mathematical Biology (-gt ESMTB)
• Mathematical Biosciences
• Journal of Theoretical Biology
• IMA Journal of Mathematical Applications in
  Medicine and Biology
• Theoretical Population Biology
• Ecological Modelling
• ...

• Organizations (regional/national)
• Society of Mathematical Biology (mainly US-based)
• European Society for Mathematical and Theoretical
  Biology
• Japan Society for Mathematical Biology
• ....

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Communities and organizations (2)

• Main international event
• SMB Annual Conference (usually called
  International Conference on Mathematical
  Biology)
• Plenary talks at ICMB2003 (Dundee, UK, Aug 6-9)
• A. Fogelson (Utah) Computational Modelling of
  Blood Clotting
• P. Hogeweg (Utrecht) Evolution of Morphogenesis
  The interface between generic and informatic
  processes
• L. Keshet (UBC, Vancouver) Modelling Type-1
  Diabetes
• P. Maini (Oxford) A multiple scale model for
  tumor growth
• J. Murray (Oxford/UW Seattle) Modelling Marital
  Interaction divorce prediction and marital
  therapy
• A. Stevens (MPI Leipzig) Structure and
  Function Interacting Cell Systems
• J. Sherratt (Heriot-Watt U, UK) Spatiotemporal
  patterning of Cyclic Field Voles

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The context of Biomathematics (3)

• Highlights
• Math modeling key to coping with biosystems
  complexity gt 70 increase in NSF math funding in
  past 3 years, 3 new Math Inst this year
• Growing use of computational techniques gt coop
  opportunities with other sciences
• Modeling simulation essential for new drug
  discovery development

• Lowlights
• Currently a double-edged sword
• Mathematicians do not seem interested in Biomath
  results
• Biological predictions based on math are not even
  read by biologists
• Need for commitment to the biological problems at
  hand gt good understanding of biological details,
  results relevant to biologists

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Bioinformatics back to the roots?

• Synonymous with Computational Biology
• Bioinformatics broad term originally coined in
  mid-80s to encompass computer applications in
  the biological sciences
• Commandeered by different disciplines to mean
  rather different things, e.g. in the context of
  genome initiatives, it meant computational
  manipulation and analysis of biological sequence
  data
• Transitioning back to original meaning in the
  Systems Biology framework

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Basic Bioinformatics Problems ZIMM02

• shows merging of traditional
  Bioinformatics and Computational Systems Biology

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Modtech diversity to match bio-complexity

• Graphs (directed and undirected)
• Bayesian networks
• Boolean and generalized logical networks
• 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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Choosing the right Modtech
Source DEJO02
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Structural analysis of biochemical networks

• Main 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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Def example Stoichiometry Matrix
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Elementary Flux Mode (EFM) Example 1
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EFM Example 2 Adenine nucleotide metabolism
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EFM Example 2(contd)
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Flux Mode
Definition 1. A flux mode, M, is defined as the
set M V ? Rr V ?V, ? gt 0, where V is an
r-vector (unequal to the null vector, i.e.,
nonzero) fulfilling the following two
conditions (C1) Steady-state condition,
NV0 (C2) Sign restriction V contains a
subvector, Virr , that fulfills inequality Virr
gt 0, with the components of Virr corresponding
to the irreversible reactions.
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Elementary Flux Mode

• Definition 2. A flux mode M with a representative
  V is called an elementary flux mode if and only
  if, V fulfills the condition
• (C3) Non-decomposability. V cannot be
  represented as a positive linear combination,
• V ? ?1V ?2V ?1, ?2 gt 0, (1)
• of two nonzero flux vectors V and V that have
  the properties
• (i) V and V themselves obey restrictions (C1)
  and (C2),
• (ii) both V and V contain zero elements
  wherever V does, and they include at least one
  additional zero component each,
• S(V) c S(V), S(V) c S(V).

Note (1) Represents the property of generating
vector elements (basis) of a pointed cone, that
vectors in the cone cannot be decomposed into
vectors belonging to the cone.
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Convex Polyhedral Cone
From Yingyu Ye, Dept. of Management Science and
Engineering, Stanford University
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CPC Example
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Extension of the Convex Basis

• Set of elementary flux modes not only includes
  the elements of the convex basis but also
  includes flux modes that satisfy conditions of an
  elementary mode in the sense of Definition 2 in
  order to satisfy uniqueness.

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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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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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• Full E.Coli Metabolic Network
• 791 vertices (chem substrates)
• 744 edges (chem reactions)
• ltkgt 2.1

KARP01
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Elementary modes in E.Coli metabolism

• Elementary (flux) mode analysis introduced by
  Schuster Hilgetag (1994)
• References
• 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)
• A. Cornish-Bowden, M.L. Cardenas Metabolic
  balance sheets, Nature 420 (14/11/02)
• 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)

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Elementary modes in E.coli some results
analysis
Stelling et al Nature Nov 14 02
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Boolean Network Models
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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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Review Boolean Functions
NOT Gate
NOR Gate
AND Gate
NAND Gate
OR Gate
XNOR Gate
XOR Gate
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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 nodes
• 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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Example Kauffman network (N,k)

• Studied by Stuart Kauffman since late 60s

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Kauffman Networks (2)
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A basin of attraction (DDLab)
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(No Transcript)
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Introducing Drosophila M.

• Model organism
• Main Resource FlyBase
• http//flybase.bio.indiana.edu/
• ia database of genetic and molecular data for
  Drosophila. FlyBase includes data on all species
  from the family Drosophilidae the primary
  species represented is Drosophila melanogaster.
• FlyBase is produced by a consortium of
  researchers funded by the National Institutes of
  Health, U.S.A., and the Medical Research Council,
  London. This consortium includes both Drosophila
  biologists and computer scientists at Harvard
  University, University of Cambridge (UK), Indiana
  University, University of California, Berkeley,
  and the European Bioinformatics Institute.

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Example ALOT02 Drosophila segment polarity GRN
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Sample Functions
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Albert-Othmer Drosophila model
ALOT02
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Boolean Functions as Polynomials
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Finite dynamic networks

• We consider in the following only
• X1 X2 X3 .... Xn k finite field
• Update schedule is parallel, all nodes updated
  synchronously ? f (f1,f2,..., fn)
• ?
• A dynamic network on n nodes over a finite field
  is simply a function kn ? kn

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Fixed Points of Dynamic Networks

• Let N ( fi, Y) be a dynamic network, f global
  update, ie f (f1,... fn) kn ? kn
• A fixed point of f is a u ? kn with f(u)
  u or fi(u) ui.

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Mapping to polynomial functions

• Added 6 new variables for intercellular
  interactions
• (total of 21 variables)

LAST03B
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List of polynomial functions
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List of fixed points
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Introducing Dicty (www.dictybase.org)
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Dicty TCA Cycle (Function)

• TCA (tricarboxylic acid) cycle very efficiently
  produces ATP while decomposing pyruvate to water
  and CO2 via acetyl-CoA
• Nutrient-rich conditions cycle is fed from
  ingested proteins that are broken down to amino
  acids
• Starvation conditions cellular proteins are used
  up
• With reasonable simplication cycle involves 13
  dependent metabolites and 26 enzyme-catalized
  processes

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Dicty TCA Cycle (Diagram)
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Dicty TCA Cycle Model History

• Series of models produced by B.E. Wright et al
  (1968 1992)
• Initial model (6 equations) progressively refined
  to current model (50 variables)
• Key question addressed explore validity of using
  in vitro information for predictions in vivo
• Both Michaelis-Menten (Wright) and S-System
  (Shiraishi-Savageau, 1992-93) approaches used
• Interesting remark by Voit since most of these
  models were created by Wright and
  co-workers...consistent philosophy and
  nomenclature..makes it easy to compare the
  models

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Dicty TCA Cycle Model Node Equations
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Dicty TCA Cycle Model Structure (1)

• Model consists of
• 13 dependent concentration variables
• 31 independent variables, incl. enzyme and
  cofactor concentrations (X14 to X39 ) and (X46 to
  X48 ) resp. and fixed reservoirs for protein and
  CO2 (X49 , X50)
• X40 to X45 )reserved for extension

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Michaelis-Menten vs. S-System
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Dicty TCA Cycle Model Structure (2)
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Dicty TCA Cycle Model Structure (3)
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Dicty TCA Cycle Model Structure (4)
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Consistency and robustness analysis

• Model has the expected steady state with
  concentrations in previous table
• Eigenvalue analysis
• Real parts are all negative ? system is locally
  stable (returns to the steady state after
  perturbations)
• Real parts have substantial range of magnitudes ?
  perturbation responses at different time scales
  (and a warning sign)

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Example
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Further robustness test

• Response to changes in enzymes (malate
  dehydrogenase and malic enzyme)
• use PLAS to simulate concentration changes
  (especially pyruvate)
• Multiply X18 by 0.98 ? new steady state of X5?
• At 0.95, 0.93, 0.92, 0.90
• Note ease of analysis in PLAS (vs. Other
  modeling approaches which do not even allow
  direct computations of the steady state)

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Model modification (Shiraishi-Savageau 1993)

• Problems with pyruvate metabolism and flux
  distribution at malate branch point
• Possible cause constant supply of amino acids
  arising from protein catabolism reactions 34-39
• Promising modification allow for
  re-incorporation of amino acids in proteins or
  (in the pathway map) add arrows from amino acids
  to protein

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Model modification (2)
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Modified model analysis

• Same steady state values as before
• Eigenvalue analysis
• locally stable
• magnitude range much narrower (-gt less variation
  in response times)
• Analysis of gains and sensivities several orders
  of magnitude less sensitive than the previous
  model

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Concluding remarks

• Voit amazing model improvement thru small
  conceptual changes
• Example shows some advantages of BST(S-Systems)
  modeling to traditional Michaelis-Menten
• BST currently applied/feasible for the
  meso-level (50-100 variables)? subnetwork
  (modular) approach

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Thanks for your attention !
MSBF

• Questions?