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Kinetic models: Kinetic models Markus Heinonen
Enzyme kinetics: Enzyme kinetics Enzymes catalyse biochemical reactions
Enzyme = complex protein
1 Enzyme catalyses 100 – 1000000 reactions /s
Spatial homogeneity assumption (’well-stirred’)
Not time dependent
Enzyme kinetics: Enzyme kinetics Mass-action law: reaction rate is proportional to probability of collision of reactants  proportional to concentration of reactants (homogeneity)
Reaction equilibrium: Reaction equilibrium Reaction is in equilibrium when
Equilibrium constant marks the ratio of concentrations in reaction equilibrium:
Reaction thermodynamics: 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 doesn’t change ΔG
Reaction thermodynamics: Reaction thermodynamics
Definitions: 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
Definitions: Definitions Stoichiometric matrix A representing mass balances and metabolite connectivities through reactions
Metabolite rate of change
Reaction rate
Moieties: Moieties Self conserving metabolite sets
Ie. AMP+ADP+ATP = constant
Enzyme levels = constant (E + ES = constant)
Algebraic contraints on the model
Example: Example R1 2A 1B 3C v1 2A+B  3C  D D R2 v2 ODE system
Properties of metabolic kinetics: Properties of metabolic kinetics Rate of reaction must be proportional to enzyme level
Enzyme levels andlt;andlt; metabolite levels
At high metabolite concentrations there’s 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
Reaction rate determination at molecular level: Reaction rate determination at molecular level Reaction rate v is dependent on
enzyme used, it’s 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
Problems of metabolic kinetics: Problems of metabolic kinetics Kinetics are problematic
Obtained from test tube tests of purified enzymes
Measurement doesn’t 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
Kinetic models of different levels: 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
Rate law: Rate law Reaction rate law
Reaction as an aggregated rate law
Eg. Michaelis-Menten (irreversible, S-andgt;P, form)
Michaelis-menten: Michaelis-menten More complex function for reversible reactions
Inhibitors: Inhibitors Inhibitor binds to free enzyme rendering it unusable E+S ES E+P k1
k-1 k2
k-2 EI+S k3 k-3
Activation: Activation Activator binds to free enzyme forming complex which produces P
Activation if
’E+S’ =andgt; ’EA+P’
andgt; ’E+S’ =andgt; ’E+P’ E+S ES E+P k1
k-1 k2
k-2 EA+S k3 k-3 EAS EA+P k4
k-4 k5
k-5
Approximations: 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
Reference state – steady state: 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
Reference state: Reference state Elasticity of michaelis-menten
Hyperbolic rate in relation to reference state:
Linearized approximation: Linearized approximation Linear both in enzyme level and metabolite concentrations
Allows only very small changes
Log-linear approximation: Log-linear approximation Logarithmic-linear in enzyme level and metabolite concentrations
Note for (0.76 andlt; y andlt; 1.31) with relative error of andlt; 15%
Several fold changes in metabolite concentrations, less than two-fold for enzyme levels allowed
Linlog kinetics: Linlog kinetics linear sum of logarithms
Linlog kinetics: Linlog kinetics Rewritten as
Linlog accuracy: Linlog accuracy Perturbation tests on a small metabolite network whose detailed kinetic model is known
Linlog parameters fitted from model
Network: Network Perturbations on level of S
1-andgt;5
1-andgt;20
Results of (S 1->20): Results of (S 1-andgt;20)
References: 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