Kinetic models

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Kinetic models

• Markus Heinonen

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Enzyme kinetics

• Enzymes catalyse biochemical reactions
• Enzyme complex protein
• 1 Enzyme catalyses 100 1000000 reactions /s
• Spatial homogeneity assumption (well-stirred)
• Not time dependent

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Enzyme kinetics

• Mass-action law reaction rate is proportional to
  probability of collision of reactants ?
  proportional to concentration of reactants
  (homogeneity)

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Reaction equilibrium

• Reaction is in equilibrium when
• Equilibrium constant marks the ratio of
  concentrations in reaction equilibrium

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Reaction thermodynamics

• Spontaneous reactions occur only if it (1)
  increases entropy and/or (2) lowers free energy
• Laws of thermodynamics
• Gibbs free energy
• Spontaneous reaction
• Equilibrium
• Need supplied energy
• Enzyme doesnt change ?G

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Reaction thermodynamics
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Definitions

• Steady state
• dx/dt 0 for all metabolites
• Reaction rates and metabolite concentrations not
  zero
• Equilibrium
• All reaction rates are zero
• Dead system
• Transient state
• System moving towards eg. Steady state
• Homeostasis
• Resistance to change

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Definitions

• Stoichiometric matrix A representing mass
  balances and metabolite connectivities through
  reactions
• Metabolite rate of change
• Reaction rate

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Moieties

• Self conserving metabolite sets
• Ie. AMPADPATP constant
• Enzyme levels constant (E ES constant)
• Algebraic contraints on the model

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Example
ODE system
2AB ? 3C ? D
2A
v1
v2
R1
3C
D
R2
1B
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Properties of metabolic kinetics

• Rate of reaction must be proportional to enzyme
  level
• Enzyme levels ltlt metabolite levels
• At high metabolite concentrations theres a
  downward concave behavior of rate vs
  concentration ? saturation
• Desirable properties of kinetic formats
• Low number of kinetic parameters
• Analytical solutions of steady-state balances

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Reaction rate determination at molecular level

• Reaction rate v is dependent on
• enzyme used, its activity
• regulation effectors, inhibitors,
  (dis)activators of enzyme
• metabolite concentrations
• enzyme concentration
• surrounding reactions and molecules
• pH, ion-balance, molecule-gradients, energy
  potentials
• Generally regarded as linear in enzyme level
  (exceptions), hyperbolic in metabolite
  concentrations

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Problems of metabolic kinetics

• Kinetics are problematic
• Obtained from test tube tests of purified enzymes
• Measurement doesnt apply on cell environment
• Obtained kinetics non-linear with lots of
  parameters
• Thus no analytical solutions to metabolic network
• Numerical integration assumed parameters
• Kinetic approximations allow analytical solving
  of the model

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Kinetic models of different levels

• Mass-action form (mechanistic level)
• Each reaction as elementary mechanistic steps
• Large models
• Rate-law form (molecular level)
• Reaction aggregated into single step
• Roughly equal to mass action form
• Eg. michaelis-menten model, Hill kinetics,
• Power law form
• All metabolite producing/consuming reactions
  aggregated together
• Reaction described as power law, eg.
• Thermokinetic

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Rate law
Eg. Michaelis-Menten (irreversible, S-gtP, form)

• Reaction rate law
• Reaction as an aggregated rate law

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Michaelis-menten

• More complex function for reversible reactions

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Inhibitors

• Inhibitor binds to free enzyme rendering it
  unusable

k1 k-1
k2 k-2
ES
EP
ES
k3 k-3
EIS
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Activation

• Activator binds to free enzyme forming complex
  which produces P
• Activation if
• ES gt EAP
• gt ES gt EP

k1 k-1
k2 k-2
ES
EP
ES
k3 k-3
k4 k-4
k5 k-5
EAP
EAS
EAS
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Approximations

• Different approximations are linearized versions
  of kinetics, eg. rate law
• Aim at analytical solutions of mass balances
• Approximations only applicable near reference
  state
• Gene modifications most often result in large
  changes in enzyme activities
• Homeostasis

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Reference state steady state

• Approximations are defined around a reference
  state
• Defined (as in MCA) as steady state
• Each reaction has a steady state enzyme level e0,
  metabolite levels xi0 and steady-state flux J0
• Elasticities
• Elasticity-matrix
• represents the elasticity (effect) of
  metabolite j on reaction rate i

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Reference state

• Elasticity of michaelis-menten
• Hyperbolic rate in relation to reference state

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Linearized approximation

• Linear both in enzyme level and metabolite
  concentrations
• Allows only very small changes

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Log-linear approximation

• Logarithmic-linear in enzyme level and
  metabolite concentrations
• Note for (0.76 lt y lt
  1.31) with relative error of lt 15
• Several fold changes in metabolite
  concentrations, less than two-fold for enzyme
  levels allowed

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Linlog kinetics

• linear sum of logarithms

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Linlog kinetics

• Rewritten as

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Linlog accuracy

• Perturbation tests on a small metabolite network
  whose detailed kinetic model is known
• Linlog parameters fitted from model

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Network

• Perturbations on level of S
• 1-gt5
• 1-gt20

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Results of (S 1-gt20)
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References

• Heijnen, J. Approximative Kinetic Formats Used
  in Metabolic Network Modeling, Biotechnology and
  Bioengineering (2005), 91(5), 534-545
• Visser, D. and Heijnen, J. Dynamic simulation
  and metabolic re-design of a branched pathway
  using linlog kinetics, Metabolic engineering
  (2003), 5, 164-176
• Hofmeyr, JH. and Snoep, J. and Westerhoff, H.
  Kinetics, Control and Regulation of Metabolic
  Systems, (2002), chapters 1-4
• Klipp, E. and Herwig, R. and Kowald, A. and
  Wierling, C. and Lehrach, H. Systems Biology in
  Practise Concepts, Implementation and
  Application, (2005), Wiley-VCH, chapter 5