Chapter 8 Applications

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Chapter 8 Applications

• In physics
• In biology
• In chemistry
• In engineering
• In political sciences
• In social sciences
• In business

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In chemistry chemical reaction

• Biochemical reactions
• Take place in all living organisms
• Most of them involve proteins called as enzymes
• Enzymes react selectively on compounds
  substrates
• Biological biochemical processes are
  complicated
• Develop a simplifying model to understand the
  phenomena
• Give qualitative understanding
• First step to develop more realistic
  complicated model

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Reaction kinetics

• Basic enzyme reaction Michaelis Menten (1913)
• A substrate S reacting with an enzyme E to form a
  complex SE
• The complex SE is converted into a product P and
  the enzyme
• The laws of mass action the rate of a reaction
  is proportional to the product of the
  concentrations of the reactants

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Reaction equations

• Concentrations of the reactants
• Nonlinear reaction equations
• Explain The first equation for s is simply the
  statement that the rate of change of the
  concentration S is made up of a loss rate
  proportional to S E and a gain rate
  proportional to SE.

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Model reduction

• Initial conditions
• The last equation is uncoupled
• Conservation of enzyme catalyst
• Reduced system

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Pseudo-steady state

• Pseudo-steady state solution The reaction of
  complex to product is much faster than that of
  substrate to complex, i.e. enzyme is almost at
  equilibrium
• The equation

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Pseudo-steady state solution

• Define Michaelis constant
• Pseudo-steady state solution
• Determine rate of reaction v
• Take a sequence of different initial values of
• Measure the corresponding variation of s with t,
• Rate of reaction
• Obtain for each experiment a measurement of the
  initial rate

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Lineweaver-Burk plot
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Michaelis-Menten rate
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Qualitative analysis

• Nondimensionalization
• Dimensionless equation
• Qualitative understanding
• Steady state uv0
• v increases from v0 until attains its maximum at
  vu/(uK) then decreases to v0
• u decreases monotonically from u1 to u0

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Qualitative analysis
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Michaelis-Menhten theory

• Pseudo-steady state hypothesis The remarkable
  catalytic effectiveness of enzymes is reflected
  in the small concentrations needed in their
  reactions as compared with the concentrations of
  the substrates.
• Approximate (asymptotic) solution
• Assume
• Substitute and equate powers of
• The O(1) equations

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Michaelis-Menten theory

• Solution (nonsingular or outer solution, valid
  for )
• Difficulty The second equation is algebraic
  does not satisfy the initial condition

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Michaelis-Menten theory

• The solution is not a uniformly valid
  approximation for all
• The original problem is a singular perturbation
  problem since the second equation is multiplied
  by a small parameter
• The assumption is not valid near
  
• Initial layer exists
• Introduce the transformations
• New equations

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Michaelis-Menten theory

• Assume
• O(1) equations
• The solutions (singular or inner solution, valid
  for )

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Michaelis-Menten theory

• Matching
• The limit of the outer solution when
• The limit of the inner solution when
• Initial (or boundary) layer

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Michaelis-Menten theory

• Singular perturbation, a systemic way
• Outer solution in the form of a regular series
  expansion
• Inner solution expansion

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Michaelis-Menten theory

• Initial conditions
• Thus the singular solutions are determined
  completely
• Outer solutions
• Matching of the inner and outer solutions

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Michaelis-Menten theory

• Uniformly expansion

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Michaelis-Menten theory
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Michaelis-Menten theory

• Uniformly matched asymptotic expansion
  innerouter-middle
• Explain
• Rapid change in substrate-enzyme takes place in
  dimensionless time
• Very short, in many experimental cases, singular
  solutions is not observed
• The reaction for the complex is essentially in a
  steady state
• The v-reaction is so fast it is more or less in
  equilibrium at all times
• This is Michaelis and Mentens pseudo-steady
  state hypothesis

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Other chemical reactions

• Cooperative phenomena
• Enzyme has more than one binding site for
  substrate molecules
• An enzyme a substrate is called as cooperative
  if a single enzyme molecule, after binding a
  substrate molecule at one site can then bind
  another substrate molecule at another site.
• Example enzyme molecule E binds a substrate
  molecule S to form a single bound
  substrate-enzyme complex C1. C1 can break to form
  a product P and enzyme E combine with another
  substrate molecule S to form a dual bound
  substrate-enzyme complex C2. C2 breaks down to
  form the product P and single bound complex C1.
• Autocatalysis, Activation Inhibition systems
  with feedback controls

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Other models

• Biological oscillators switches Feedback
  control
• Reaction diffusion, Chemotaxis