Polymath Tutorial: Solution to Classwork

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Polymath TutorialSolution to Classwork

• By Kelvin WONG
• CENG 364 _at_ HKUST
• Feb, 2009

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Objective

• To familiar how to use Polymath (to set up ODE
  problem).
• To analyze a problem and convert it into a
  mathematical model.

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Classwork Problem

• A reaction pathway is given.
• And reaction kinetics, etc. are given too.

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Problem Type

• First of all, we need to find out what kind of
  problem it is.
• We have some rate equations (e.g. ) that
  describe the change in intermediate
  concentrations (e.g. B, C, ...).
• We are going to find out those concentrations at
  a certain time, or concentration profiles over
  time.
• I.e. we are going to integrate those differential
  equations.
• Therefore, it is differential equation problem
  type in Polymath.

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Equation Type (1)

• Next, we need to identify what kind of equations
  we are going to input into Polymath.
• We have 4 type of variables

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Equation Type (1)

• However, there are two types of metabolites
• Boundary metabolites (A and H), and
• Intermediate metabolites.
• Where boundary metabolites are fixed at certain
  concentrations.
• Therefore, we need to further classify metabolite
  concentrations as

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Model Input (1)

• Explicit equations
• Boundary metabolite concentrations (A, H)
• Rate equations (v1, ..., v5)
• Kinetic parameters (Vmax1, ..., Keq6)

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Model Input (2)

• We are going to define differential equations.
• B is generated by reaction v1 and consumed by v2.
• Therefore

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Model Input (3)
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Inhibition

• Inhibition by the existence of E on reaction v1
  is already described by the kinetic equation v1

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Reversible Reactions (1)

• As stated in the classwork, there are two
  reversible reactions.
• We can break down a reversible reaction into
  two reactions in opposite directions
• It is also the same for the kinetic equation

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Reversible Reactions (2)

• The first term represents the consumption of E,
  and the second term represents the consumption of
  F
• If the net reaction rate (v4) is positive, i.e.
  net consumption of E
• If the net v4 is negative, i.e. net consumption
  of F.
• Therefore, v4 is defined as E ? F as forward and
  F ? E as backward.

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Reversible Reactions (3)

• Analyze the rate equation for reaction v6, we
  find that it is defined as C ? G as forward and G
  ? C as backward.
• If the net v6 is positive, i.e. net consumption
  of C
• If the net v6 is negative, i.e. net consumption
  of G.

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Model Input (4)