Dr. Eduardo Mendoza

1
Canonical Models of Metabolic Networks, Part 1

• 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
Papers for reports (1)

• Group 1 (4 members, Feb 11)
• S. Schuster et al Reaction routes in
  biochemical reaction systems Algebraic
  properties, validated calculation procedure, and
  example from nucleotide metabolism, Journal of
  Math. Biology 45 (2002)
• Group 2 (4 members, Feb 13)
• R. Albert, H. Othmer The topology of the
  regulatory interactions predicts the expression
  pattern of the segment polarity genes in
  Drosophila melanogaster, Journal of Theor.
  Biology 223 (2003),pp. 1-18

3
Papers for reports (2)

• Group 3 (3 members, Feb 18) N. Torres et al
  Metabolic Modeling and Optimization of
  Biochemical Systems. Application to Citric Acid
  Production in Aspergillus niger, 2003 (14 pgs)
• Group 4 (3 members, Feb 20)
• A. Salvador Synergism analysis of biochemical
  systems I. Conceptual framework, Math Biosciences
  (2000), pp 105-129
• Group 5 (3 members, Feb 25)
• D. Irvine, M. Savageau Efficient solution of
  nonlinear ordinary differential equations
  expressed in S-System Canonical Form, SIAM Rev.
  Num.Anal. 27 (1990), pp 704-735

4
Topics to be covered

• 3.1 Biochemical maps
• 3.2 From maps to equations
• 3.3 Canonical Modeling Background
• 3.4 Parameters
• 3.5 Generalized Mass Action (GMA) Systems
• 3.6 Canonical Modeling examples

5
References

• VOIT00 E.O. Voit Computational Analysis of
  Biochemical Systems, Cambridge University Press,
  2000
• VOSA01 E.O. Voit, Savageau, M Introduction to
  the Analysis of Biochemical and Genetic Systems,
  Lectures 2001
• BOBO01 J.M.Bower, H. Bolouri Computational
  Modeling of Genetic and Biochemical Networks, MIT
  Press, 2001
• MISH02 B. Mishra, Topics in Computational
  Biology, NYU Lectures, Spring 2002

6
3.1 Biochemical maps

• How do chemists describe a biochemical system?
• words (linear, 1-dimensional)
• chemical structures (2-dimensional)
• Biochemical map
• Simplify thru abbreviation to highlight global
  features, to leave out details (eg chemical
  structures not relevant)
• Key elements of a biochemical map
• System components or pools of components
• Arrows that indicate the flow of material (heavy
  arrows)
• Arrows that indicate the flow of information or
  signals (light, dashed, differently colored)

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Purine Synthesis Map
8
Graphical Representation (1)
Source Mishra
9
Graphical Representation (2)
The reaction between X1 and X2 requires coenzyme
X3 which is converted to X4
Source Mishra
10
Example Glycolysis
Glycogen
P_i
Glucose-1-P
Glucose
Phosphorylase a
Phosphoglucomutase
Glucokinase
Glucose-6-P
Phosphoglucose isomerase
Fructose-6-P
Phosphofructokinase
Source Mishra
11
Review Examples of Ambiguity

• Failure to account for removal (dilution)
• Failure to distinguish types of reactants
• Failure to account for molecularity
• Confusion between material and information flow
• Confusion of states, processes, and logical
  implication
• Unknown variables and interactions

VOSA01
12
Failure to Distinguish Types of Multireactants
VOSA01
13
Failure to Account for Molecularity
(Stoichiometry)
VOSA01
14
Confusion Between Material and Information Flow
VOSA01
15
Confusion of States, Processes, and Logical
Implication
VOSA01
16
3.2 From maps to equations (Voit, pp 41-49)

• variable Xi
• describes status of variable at time t,
  explicitly expressed as Xi (t)
• often difficult to measure directly (in
  experiments) ? changes measured
• Example kinetic laws related reaction rates to
  concentrations
• In almost all cases info about changes in
  concentration is sufficient to deduce the
  dynamics of a biochemical system

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Spatial Simplifications

• Abundant in natural systems
• Compartmentation is common in eukaryotes (e.g.
  mitochondria)
• Specificity of enzymes limits interactions
• Multi-enzyme complexes, channels, scaffolds,
  reactions on surfaces
• Implies ordinary rather than partial differential
  equations

VOSA01
18
Temporal Simplifications

• Vast differences in relaxation times
• Evolutionary -- generations
• Developmental -- lifetime
• Biochemical -- minutes
• Biomolecular -- milliseconds
• Simplifications
• Fast processes in steady state
• Slow processes essentially constant

VOSA01
19
Functional Simplifications

• Feedback control provides a good example
• Some pools become effectively constants
• Rate laws are simplified
• Best shown graphically

VOSA01
20
Criteria of a Good Approximation

• Capture essence of system under realistic
  conditions
• Be qualitatively and quantitatively consistent
  with key observations
• In principle, allow arbitrary system size
• Be generally applicable in area of interest
• Be characterized by measurable quantities
• Facilitate correspondence between model and
  reality
• Have mathematically/computationally tractable form

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In real life....a balance of features
1
7
2
6
5
3
4
Remember all models are wrong, but some models
are...
22
Systems of Differential Equations

• dXi/dt (instantaneous) rate of change in Xi at
  time t Function of substrate concentrations,
  enzymes, factors and products
• dXi/dt f(S1, S2, , E1, E2, , F1, F2,, P1,
  P2,)
• E.g. Michaelis-Menten model for substrate S
  product P
• dS/dt - Vmax S/(KM S)
• dP/dt Vmax S/(KM S)

Source Mishra
23
Terminology

• Dependent Variable
• Variable that is affected by the system
  typically changes in value over time
• Independent Variable
• Variable that is not affected by the system
  typically is constant in value over time
• Parameter
• constant system property e.g., rate constant

VOSA01
24
3.3 Canonical Modeling Background (1)

• Founded by M. Savageau in the late 60s to model
  (sets of) biochemical reactions in vivo
• Also called Biochemical Systems Theory or BSA
  (AAnalysis)
• Comes in two forms S-Systems (S synergistic)
  and GMA (Generalized Mass Action)

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Canonical Modeling Background (2)

• Applied in various fields
• Metabolic Networks
• Genetic (Regulatory) Networks
• Protein (Signal Transduction) Networks
• But also
• Forestry, Fisheries Management,...
• Excellent Textbook VOIT00

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10 (Mostly Easy) Steps to a Canonical Model
VOIT02

• Identify the components to be included in the
  model.
• If they change over time, assign to them variable
  names Xi, if not the components might still be
  variable by name or they might be absorbed in
  some of the model parameters.
• Identify the flow of materials between variables.
• Identify the regulatory signals, such as feedback
  loops.
• Create a diagram (e.g. a biochemical map)
• For each variable that changes over time, define
  an equation that relates its change over time(
  derivative w. resp to time d Xi /dt) to influxes
  and efluxes
• change in Xi influxes in Xi - efluxes in
  Xi.

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10 (Mostly Easy) Steps...

• S-System Variant
• 7. Collect all influxes into Xi into one function
  Vi , all efluxes into another function Vi- .
• 8. Approximate Vi , Vi- with power law
  functions, eg
• 9. Estimate numerical values for all
• parameters from measurements
• or literature information.
• 10. Analyze dynamics, steady states,
• robustness, responses under
• different scenarios

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Example Metabolic Pathway

• dX1/dt V1(X3, X4) V1(X1)
• dX2/dt V2(X1) V2(X1, X2)
• dX3/dt V3(X1, X2) V3(X3)
• No equation for independent variable X4

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3.4 Parameters Rate Constants

• In the following equation
• dXi/dt ai Õj1nm Xjgij - bi Õj1nm Xjhij
• ais and bis are rate constants in the
  production and the depletion terms respectively.
• These terms are positive or zero, but cannot be
  negative.
• At any point, which term (production or
  depletion) dominates depends on the
• rate constants ai and bi
• other parameters gij and hij, and
• the current concentration of all the metabolites
  that are involved in Vi and Vi-.

Source Mishra
30
Indices Kinetic Order

• The roles of the kinetic order parameters gij
  and hij
• i the first index of the kinetic order and
• j second index of the kinetic order.
• gij represents how the production of Xi is
  influenced by the variable Xj
• hij represents how the degradation of Xi is
  influenced by the variable Xj
• The kinetic orders need not be integers, they can
  be any real number !!!
• Positive kinetic orders indicate activating
  influences and negative kinetic orders express
  inhibition.
• If the kinetic order is zero, then it indicates
  independence from the metabolite.

Source Mishra
31
Comparison

• In traditional chemical kinetics (also called
  conventional mass action)
• kinetic orders are interpreted as the number of
  molecules involved in each chemical reaction
• kinetic orders are 0,1,2,....
• Recent biochemical studies show
• Many enzyme catalized reactions in vivo have
  non-integer kinetic orders eg.

SAVA03
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Example Branched Pathway

• Pathway diagram s. board
• Equations
• X1' 10 X3g13 X5 - 5 X10.5
• X2' 5 X10.5 - 10 X20.5
• X3' 2 X20.5 - 1.25 X30.5
• X4' 8 X20.5 - 5 X40.5
• g13 0

• X1 1.1
• X2 0.5
• X3 0.9
• X4 0.75
• X5 0.5
• t0 0
• hr 0.1
• tf 10
• Simulation with PLAS
• g13 -1, -2, -4, -16

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Varying the kinetic order
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Parameter estimation is hard

• Analytic solutions usually limited to small
  systems
• Parameter estimation is a major computational
  task
• A light of hope
• E. Voit claims generally more approaches for
  this task in biochemical systems than for other
  biological phenomena

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Which results do we expect in biological systems?

• Kinetic orders are closely related to stability
  (s. later), hence values 100 or more would make
  a system extremely sensitive or unreliable (Note
  1)
• Rules of thumb for kinetic orders (based on
  analysis of over 1000 recently modeled
  biochemical reactions)
• Flow of material and activations between 0 and 1
• Unmodulated between 0.5 (hyperbolic) and
  1(linear)
• Modulated between 0 and 0.5
• Inhibitory effects between 0 and -0.5
• Mostly around -0.1
• Useful when info on system is scarce, for initial
  estimates and interest only for orders of
  magnitude

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Biochemical justification

• Justification for S-Systems approach
• Basis biochemical finding that it is often more
  appropriate to study relative effects in response
  to relative changes in metabolites rather than to
  study absolute effects
• Relative change is dimensionless and hence
  automatically allows for widely diverse
  concentrations of metabolites, enzymes and
  effectors typical in vivo and even in vitro

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3.5 GMA Systems

• Do not aggregate the influxes and efluxes
  approximate each influx and each eflux with a
  power law function

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Relationships (Shiraishi-Savageau, 1992)
Kinetic orders weighted averages of more
elementary kos (Alves-Savageau, 2000)

Homogeneous 3D reactions -gt pos. integers
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Steady State for S-Systems

• If all equations are balanced (i.e., production
  is balanced by depletion), then dXi/dt 0,
  i1,,n.
• Thus the steady-state is achieved at
• 0 ai Õj1nm Xjgij - bi Õj1nm Xjhij
• or
• ai Õj1nm Xjgij bi Õj1nm Xjhij
• A steady state is characterized by the condition
  that no metabolite is changing (i.e., that dXi/dt
  0) and they remain constant..

Source Mishra
40
S-Systems and GMA Approaches

• S-System approximate V , V with power laws
  (quasi-monomials, fractional values allowed )
• GMA decompose V into component fluxes and
  approximate each with a quasi-monomial
• GMA equivalent to Generalized Lotka-Volterra
  (GLV)
• Results in HBFA97 for recasting GLV in
  S-System form

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3.6 Canonical Modeling Examples
Linear Pathways

• Consider the following simple case (linear
  pathway)
• Set up the S-System equations
• How can we simplify?

X3
X4
Voit pp 78-80 PLAS Linear1.plc
42
X4

• Exercise Set up the S-System equations

Voit pp 80-81 PLAS Linear2.plc
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Thanks for your attention !

• Questions?