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