Dr' Eduardo Mendoza

1
Canonical Models of Metabolic Networks, Part 2

• Dr. Eduardo Mendoza
• Lecture 4 Jan-Feb 04
• 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
PLAS Software

• PLAS Power Law Analysis and Simulation
• available at http//correio.cc.fc.ul.pt/aenf/pl
  as.html

3
Canonical Modeling (Power Law Formalism) website
http//www.udl.es/usuaris/q3695988/PowerLaw/

• This technique has been applied both to
  theoretical and applied problems in fields as
• Immunology
• Genetics
• Biochemistry
• Physiology
• Statistics
• Risk Assessment
• Forestry
• Epidemiology and
• Applied Mathematics.

• The search of design principles in biochemistry
  and metabolism
• Analysis of gene circuits
• Growth models
• Modeling and optimization of biotechnological
  processes
• Recasting
• Statististical applications and S-distributions
• Applications
• Complete list of publications on the power-law
  formalism 
• Recommended papers
• The starting years
• 1975-80
• 1981-85
• 1986-89
• 1990 -2003
•       

4
Topics to be covered

• 4.0 Math justification for S-System approx
• 4.1 Examples
• 4.2 Reversible Pathways
• 4.3 Dynamics of S-Systems
• 4.4 Parameter estimation
• 4.5 Rate Laws and Michaelis-Menten
• 4.6 Estimation of rate constants
• 4.7 Estimating from dynamic data

5
4.1 Examples Branched pathways (cf. Voit 3.3)
X2
T
T
X4
X1
T
T
X3
6
Branched Pathways (2)

• Discussion
• S-System model
• GMA model
• Derivation of constraints
• Numerical simulation of S-System and GMA models

7
Branched Pathways (3)

• File BranchG.plc
• Branched Pathway as GMA model in Chapter 3
• X1' 0.8 X2g12 X3g13 X4.5 - 3 X10.5
  X2-.1 - 2 X10.75 X3-.2
• X2' 3 X10.5 X2-.1 - 1.5 X20.5
• X3' 2 X10.75 X3-.2 - 5 X30.5
• X1 0.5
• X2 0.5
• X3 1
• X4 0.25
• X5 0.5
• g12-1
• g13-1
• Fluxes, defined as "transformations"

8
Branched Pathways (4)

• File BranchS.plc
• Branched Pathway as S-system model in Chapter 3
• X1' 0.8 X2g12 X3g13 X4.5 -
  4.959813621 X10.615672498 X2-0.053731001
  X3-.092537998
• X2' 3 X10.5 X2-.1 - 1.5 X20.5
• X3' 2 X10.75 X3-.2 - 5 X30.5
• X1 0.5
• X2 0.5
• X3 1
• X4 0.25
• X5 0.5
• g12-1
• g13-1

9
S-System vs. GMA
10
Homework Exercise No. 1

11
Homework No. 2

• Exercise
• Variables (d, i)?
• S-System model?
• Any constraints?

Cascade example
12
Example 2 Bimolecular reaction
X4
X5

• Discussion
• Variables (d, i)?
• S-System model?
• Any constraints?

Constraint The production of X3 must equal
the degradation of X1 and X2 at the steady state
(at least)
Homework 3!
13
General derivation of constraints

• Constraint equation
• ai X1gi1 X2gi2 ....Xngin Vi (X1,X2,.... Xn)
• Then (near the operating point)
• ai V(X1,X2,.... Xn) / X1gi1 X2gi2 ....Xngin
• and
• gij (?Vi / ?Xi ) (Xj / Vi )

14
Formal definition of S-System
Mishra-Policriti, 2003
15
Further Derivation of Constraints

• Stoichiometric Constraints
• In certain reactions, there are more specific
  constraints due to stoichiometry
• Example Splitting reaction of fructose
  diphosphate X3
• Ignoring modulating effects
• V3- ß3 X3h33
• V4 a4 X4g43
• Normally g43 h33 and a4 ß3 , however due to
  stoichiometry,
• a4 2ß3 .

16
Conserved Masses

• Up to now constraints due to equivalence of
  fluxes
• Further constraints stem from conservation of
  mass, eg
• Pooling of closely related metabolites
• Enzyme present in bound or unbound form
• See Voit p. 91 for details

17
4.2 Reversible Pathways (1)
v12
v23
X1
X2
X3
v21
v32

• 2 sets of equations dX2/dt
• dX2/dt (v12 v21) (v23 v32)
• dX2/dt (v12 v32) (v21 v23)

18
Reversible Pathways (2)

• Is the second option better (it contains all 3
  variables)?
• Results
• For reactions close to thermodynamic equilibrium,
  this is correct (reversible strategy)
  (Sorribas-Savageau, 1989)
• Far from thermodynamic equilibrium, the first
  option (irreversible strategy) is better

19
Condensation of pools
X2
X3
X4
X5
X1
X5
X1
...
X1
X5
What happens to - steady state? - dynamics?
20
Closed systems, Open systems

• Closed system
• no material enters or leaves the system
• there may be signals affecting the system from
  outside
• Open system
• Material enters or leaves the system
• may remain constant in size if input output

21
4.3 Dynamics of S-Systems

• Transient Responses
• It takes infinitely long to reach the true
  steady-state point ( mathematical limit)
• Cf. Table option under Results menu
• Bolus Experiments
• System is at steady state
• Inject at t 2 a single bolus of 0.1 mM glucose
  1 phospate
• Persistent Changes
• Study effect of reducing the constant input of
  glucose by 50

22
Changes in System parameters

• Structural Changes
• Up to now model structure remained the same
  changing parameters affects the structure
• Example initiate G6P.plc with 90 reduced
  concentration of glucose-6-phospate (or X20.04)

23
Example Simple Pathway with Surprise

• Pathway diagram s. board
• Equations
• X1' X2-2 X3 - X10.5 X2
• X2' X10.5 X2 - X20.5
• X1 1
• X2 1
• X3 2
• t0 0
• tf 25
• hr .01
• Simulations with PLAS
• Vary X1 X2 X3 1.5
• Vary tf 50, 100
• Vary X12, X21.5, X3 0.75

24
Surprise _at_ X1X21, X3 0.75
25
Surprise _at_ X12, X21.5, X3 0.75
26
2D/3D Phase Plots
File Sine.plc Sine Function in
Chapter 4 Differential equations X1' 3 -
X2 X2' X1 - 3 Initial values X1 3 X2
4 Solution parameters t0 0 tf 20 hr 0.01

• For oscillatory results, it is often useful to
  plot a variable as function of another variable
• This is called a phase-plane plot

27
Phase Plots (2D/3D) at g13 - 16
28
Steady-State Analysis

• Example Simple Pathway with Surprise

• Steady state values ?
• Flux at steady state ?
• What happens if I double
• the rate constants?
• if X3 initial value is doubled?

T
X3
X1
X2

29
Thanks for your attention !

• Questions?