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Mathematical Systems Biology

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Title: Mathematical Systems Biology


1
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

2
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

3
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

4
The context of Biomathematics (1)
  • Related areas
  • Mathematical Biosciences, Mathematical Life
    Sciences
  • Theoretical Biology
  • Theoretical Biophysics (Biological Physics)
  • Biostatistics
  • Computational Biology/Bioinformatics
  • ...

5
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

6
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
  • ....

7
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

8
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

9
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

10
Basic Bioinformatics Problems ZIMM02
  • shows merging of traditional
    Bioinformatics and Computational Systems Biology

11
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
12
Choosing the right Modtech
Source DEJO02
13
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

14
Def example Stoichiometry Matrix
15
Elementary Flux Mode (EFM) Example 1
16
EFM Example 2 Adenine nucleotide metabolism
17
EFM Example 2(contd)
18
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.
19
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.
20
Convex Polyhedral Cone
From Yingyu Ye, Dept. of Management Science and
Engineering, Stanford University
21
CPC Example
22
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.

23
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.

24
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

25
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

26
  • Full E.Coli Metabolic Network
  • 791 vertices (chem substrates)
  • 744 edges (chem reactions)
  • ltkgt 2.1

KARP01
27
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)

28
Elementary modes in E.coli some results
analysis
Stelling et al Nature Nov 14 02
29
Boolean Network Models
30
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.

31
Review Boolean Functions
NOT Gate
NOR Gate
AND Gate
NAND Gate
OR Gate
XNOR Gate
XOR Gate
32
Model properties
33
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)

34
Boolean network dynamics (2)
35
Example Kauffman network (N,k)
  • Studied by Stuart Kauffman since late 60s

36
Kauffman Networks (2)
37
A basin of attraction (DDLab)
38
(No Transcript)
39
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.

40
Example ALOT02 Drosophila segment polarity GRN
41
Sample Functions
42
Albert-Othmer Drosophila model
ALOT02
43
Boolean Functions as Polynomials
44
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

45
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.

46
Mapping to polynomial functions
  • Added 6 new variables for intercellular
    interactions
  • (total of 21 variables)

LAST03B
47
List of polynomial functions
48
List of fixed points
49
Introducing Dicty (www.dictybase.org)
50
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

51
Dicty TCA Cycle (Diagram)
52
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

53
Dicty TCA Cycle Model Node Equations
54
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

55
Michaelis-Menten vs. S-System
56
Dicty TCA Cycle Model Structure (2)
57
Dicty TCA Cycle Model Structure (3)
58
Dicty TCA Cycle Model Structure (4)
59
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)

60
Example
61
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)

62
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

63
Model modification (2)
64
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

65
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

66
Thanks for your attention !
MSBF
  • Questions?
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