1 Systems biology 2 – Reaction kinetics Edda Klipp Sommmersemester 2010 Humboldt-Universität zu BerlinInstitut für BiologieTheoretische Biophysik
2 Enzymes remain unchanged after reaction as catalyst Proteins, often complexed with cofactorsAnorganic cofactors: metall ionsOrganic cofactors (coenzymes): vitamin-derived complex groupsremain unchanged after reaction as catalysthave a catalytical centreare in general highly specificare often pH- and temperature dependentTurnover number: 1000 /sec (100 /sec million /sec)Acceleration (compared to non-catalyzed reaction) by 106 to foldThermodynamics:Enzymes reduce the necessaryactivation energy for the reaction
3 Classification of enzymatic reactions irreversible reversibleSPSP
4 Classification of enzymatic reactions Number of substrates (and products)uniSPS2 large (0.5)biS1+S2PS2 small (0)terS1+S2 +S3P1
5 Classification of enzymatic reactions Type of kinetics.LinearMass actionv = k SHyperbolicMichaelis-MentenSigmoidalHill kinetics,Monod, Koshlandv“Hyperbolic” and “Sigmoidal” show saturation,“Linear” involves unlimited reaction rates.
6 Kinetics of Enzymatic Reactions Deterministic kinetic modeling of biochemical reactionsBasic quantities:Concentration S : number of molecules per unit of volumeReaction rate v : concentration change per unit timePostulat:The reaction rate v at point r in space at time t can be expressed as aunique function of the concentrations of all substances at point rat time t :Simplifying assumptions:- spatial homogeneity (well-stirred)- autonomous systemes (not directly dependent on time)v(r,t) = v(S(r,t),t)v(t) = v(S(t))
7 The Mass Action Law A+B 2 C The reaction rate is proportional to the probability of collision of reactants,This is in turn proportional tothe concentration of reactants to the power of their molecularity.(Guldberg and Waage, 19. century)A+B2 CReaction rateRate constantsEquilibrium constant
8 Michaelis-Menten Kinetics Brown (1902): Mechanism for Invertase reaction (with sucrose),Which holds for one-substrate-systemes with backward reaction of effectors:E – catalystS – substrateP – productki – kinetic constantcomplexformationreversiblecomplexdegradationirreversibleMichaelis, Menten (1913): rate equation under the assumptionThat second reaction will not influence the first equilibrium(Hypothesis of quasi-equilibrium)Briggs, Haldane (1925): more general derivation ofRate law under the assumption of a steady statefor the enzyme-substrate-complex (where )
9 Michaelis-Menten Kinetics: derivation of rate law Non-linear ordinary differential equation system(1)The rate of product formation is equal to thereaction rate(2)The sum of equations (2) and (3) isa conservation relation for the enzyme(3)(4)The whole set of equations cannot be solvedanalytically.Using quasi-steady state assumption
10 Michaelis-Menten-Kinetics: The rate equation Reaction ratevMaximalvelocityVmaxVmaxMichaelis constantMichaelis-Menten-Rate expressionS
11 Integrated Form of MM rate law Reaction rate =Product increase or substrate decreaseper unit timeIntegration from t0, S0 to t, S results inHenri-Michaelis-Menten-equationand forWenn die Änderung der Reaktionsgeschwindigkeit nur von der Änderung der Substrat-sättigung des Enzyms und nicht von einer Hemmung durch dasProdukt oder das Erreichen des Gleichgewichtes abhängt, istThis is a function orOne can record a progress curveand estimate the kinetic constantsusing non-linear regression.
12 Estimation of Parameters Vmax and Km 1. Measurement of initial ratesMeasure initial rates for different initialconcentrations , i.e.measure initial change of S.2. InterpretationPlot measurement results in (S,V)-Diagram;Compare with Michaelis-Menten rate law;Estimate parameters bynon-lineare regression, for exampleleast-squares methode
13 Linearizations of the MM rate law Lineweaver-Burk-PlotEadie-PlotHanes-Plot
14 Additional aspects Relation to thermodynamics Vmax is related to turnover number, kcatCondition: completely saturated enzymes, maximal rate:[1/(mol*s)]Dissoziation constante KS of theenzyme-substrate-complex:[mol]
15 Regulation of Enzyme Activity Important mechanism for the regulation of cellularprocesses upon the adaptation to internal and external changes.Regulation of enzyme amount (Gene expression / proteine degradation)Action of effectors (inhibitors, activators)Composition of mediums (pH, ions)Regulation of protein activity by kinases / phosphatases / methylases....Here: the enzyme as target of effectors
16 Enzyme InhibitionCompetitive inhibition: substrate and inhibitor compete forthe binding place at the enzymeEquilibrium forinhibitor bindingConservation relationfor the enzymeRate equation
17 Examples Competitive Inhibition Bernsteinsäuredehydrogenase1. Succinic acid dehydrogenasehas as substrate succinic acid and is inhibited by Malonic acid.2. Acetylcholin esterasehas as substrate acetylcholin and is inhibited by Neostigmin. Note that obviously only the charged N(CH3)3+-group is active.3. Sulfonamide(antibiotica)block as competitive inhibitors the production of DNA,Since they are used by the enzyme instead of the vitamine precursor p-Aminobenzoesäure.
18 Enzyme Inhibition 2. Uncompetitive inhibition: Inhibitor binds only to theenzyme-substrate-complex3. Non-competitive inhibition:Inhibitor binds to free and boundenzyme
19 Enzyme Inhibition, 34. Irreversible inhibition : inhibitor binds the enzyme irreversibly, partial or complete loss of catalytic effectivityExample: Reaction of Iod acetate with –SH groupsin cystein side chains of the reaction centre1. Di-isopropyl-fluorophosphate (DFP)and other alkylphosphates bind covalently to acetylcholinesterase. This enzyme is responsible forTransmission of nerve stimuli. The organsims die of paralysis (Lähmung) of organ function.(used in military gases and insektizids)
20 Enzyme Inhibition , 5 Allosteric Inhibition: Product Inhibition : Inhibition by a molecule that does not bind to the reaction centre.conformation change of the enzyme,Change of reaction coordinateProduct Inhibition :Inhibition by the product due to allosteric inhibition(prevents excess production)Reduction of the net reaction rate, due to an accumulation of productwhich is substrate of the backward reaction.
21 Substrate Excess Inhibition Binding a further substrate molecule to ES-complex Enzyme-Substrate-Complex ESS,Which does not transforms to reaction products. Reversible inhibition, if one molecule dissociates.Equilibrium assumptionsEnzyme conservationReaction rateOptimumExample:Succinic acid dehydrogenase
22 Enzyme Activation Activation Increase of the rate by - Change of substrate binding- Acceleration of product formationExample Substrate activation:Substrats S acts as activator A.Reaction rate = Product formation rateEnzyme conservationQuasi-equilibrium conditionReaction rate
23 Activation and Inhibition for Mass Action Kinetics +SPcompulsoryadditionalI-SP
24 Ligand Binding and Cooperativity Ligand: compound that binds to enzyme / proteinHere: Binding of ligands to monomeric und oligomeric proteins.several ligand binding sites at a protein:Possibility of interactions between these sites during bindingThis phenomenon is called cooperativityPositive/negative cooperativity:Binding of a ligand molecule increases/reduces the affinity of the proteinfor further ligands.Homotrope/heterotrope cooperativity :Binding of a ligand molecule affects binding of further moleculesOf the same/ other ligands.
25 Fractional Saturation Case of 1 binding site:Binding of S (Ligand) to E (Protein)Binding constanteDefinition: Fractional SaturationFractional saturation for 1 subunitPlot of Y versus S is hyperbolic
26 Hill-Kinetik Positive, homotrope cooperativity Simplest case: dimeric protein- two similar ligand binding sites- Binding of first ligand increases affinity to second ligandM = monomere Untereinheit, M2 = DimerAssumption: Binding of S increases affinityM2S reacts with S as soon as it is formedFractional saturation:Complete cooperativity (each subunit iseither empty or completey saturated)Binding constanteFractional saturation:
27 Hill Kinetics For complete homotrope cooperativity of a protein with n subunits holds:This is a form of the Hill equationYSHemoglobin: sigmoid bindung curve of oxygen against oxygen partial pressureHill (1909): Interaction between binding sites - positive cooperativityKnown: hem binds oxygen moleculesUnknown: number of subunits per proteinAssumption: complete cooperativity - experimental Hill coefficient h=2.8Hintergrund: experimentelle Befunde zur Bindung des Sauerstoffs ans Hämoglobin fandenBohr und Mitarb., dass die Auftragung der fraktionellen Sättigung des Hämoglobin mit Sauerstoffgegen den Sauerstoffpartialdruck eine sigmoide Kurve ergab.Hill (1909) erklärte das auf derGrundlage von Wechselwirkungen zwischen den Bindungsstellen, die positive Kooperativitätbewirken. Damals wußte man schon, dass jedes Häm ein Sauerstoffmolekül bindet, allerdings nichtaus wieviel UE ein oligomeres Protein besteht. Hill leitete seine Gleichung ab fürEr nahm vollständige Kooperativität an und fand experimentellen Hillkoeff. von h=2.8. Heuteweiß man, dass es vier Bindungsstellen an jedem Hämoglobinmolekül gibt, so dass keine voll-ständige Kooperativität vorliegt. Praktischer Nutzen der sigmoiden Binungscharakteristik: In derLunge ist der Sauerstoffpartialdruck hoch, dort kann Hb den Sauerstoff gut binden; im Körper istder Sauerstoffpartialdruck geringer und Hb kann O2 leicht abgebenFour Binding sites per hemoglobin moleculeNo complete cooperativityHigh oxygen partial pressure in lungs: good binding of oxygen to HbLow oxygen partial pressure in body – easy delivery of O2
28 Monod-Wyman-Changeux model for enzymes with sigmoidal kinetics Model assumptions (J.Mol.Biol.(1965),12,88) :Enzyme consists of several identical subunits (SU)each SU can assume one of two conformations (active = R or inactive = T)all SU of an enzyme have the same conformationConformation change for all SU at the same time (concerted transition).R - activeT - inactiveConformation equilibriumR – Conc. active conformationR0 – R- Conc. without bound substrateR1 – R- Conc. with 1 bound substrateT – Conc. of inactive conformationT0 – Conc. without bound substratesLAllosteric constant
29 Monod-Wyman-Changeux model n = 4 subunitsKRBinding constante for substrate S to one SU: KR or KT(Assumption: Binding only to active form,S +For each enzyme there are the following possible bound states:R0 - Concentration of R without substrate binding,R1 - Conc. of R with 1 bound molecule of SR2 - Conc. of R with 2 bound molecules of SR3 - Conc. of R with 3 bound molecules of SR4 - Conc. of R with 4 bound molecules of S1 possibility4 possibilities6 possibilities.4 possibilities1 possibilityGeneral: Possibilities of substrate binding for Ri
30 Monod-Wyman-Changeux model It holds:General:Sum of allactive states:with binomicFormula:Fractional saturationReplacement of R and RiT exists only as T0
31 Monod-Wyman-Changeux model It followsReaction rateMichaelis-Menten-Term"Regulatory Term"
32 Monod-Wyman-Changeux model For S∞ :Monod-Kinetics approaches Michaelis-Menten-Kineticssmall S: regulatory term important depending on LL = 0: MM-KineticsL >> 0: sigmoidal curve, shifted to right.102103vactivation104inhibitionSExplanation of the action of activators and inhibitors:- Activators bind to active conformation- Inhibitors bind to inactive conformation-Shift of equilibrium to R or TBindungskonstanten
33 Monod-Wyman-Changeux model Example: Phosphofructokinase: experimentaly well studied systemActivators:Inhibitors: DPG, ATPTypical value for
34 Kinetics of Reversible Reactions Derivation of rate equationfor steady stateRelation between equilibrium constant qand kinetic constants of elementary stepsReaction rate
35 Kinetics of Reversible Reactions Relation to phenomenological quantitiesS very high, P=0P very high, S=0Half-maximal forward rateHalf-maximal backward rateFor S and P very small holdsThis resembles Mass action kinetics(Also called linear kinetics).
37 Methode of King and Altman Empirical methode to derive steady-state rate equations for reactions,Which are catalyzed by an enzyme (no interaction between enzymes!)1. Conservation of total enzyme amount:EXi - freies Enzym2. Relative concentration of each enzyme speciesis equal to ratio of two sums of terms,where every term Tij is the product of n-1 rate constantsand the related concentrations.3. Every term Tij contains the rate constants (times substrate conc.),which are associated with the steps leading individually or sequentially to EXi .The sum of all possible combinations (j) are the numerator,the sum of all numerators for all EXi is the denominator.4. The reaction is:
38 King-Altman for 3-Step reaction mechanism Sk11. Conservation of total enzyme amount::EESk-1k-2Pk-3k2k3EP2., 3. Listing of all possibilities of n-1 = 2 lines leading to eachenzyme species:k-1k-1For Ek3k-2k3k2Sk1Sk1For ESk3k-2Pk-3k-2k-1Sk1For EPPk-3k2Pk-3k24. Reaction rate:
39 Further typical Mechanisms Ordered bi-bi-Mechanismus (Example: Kreatinkinase)
40 Further typical Mechanisms Ordered bi-bi-Mechanism (Example: Kreatinkinase)Ping-Pong-Mechanism (Example : Transaminase, Nukleosid-Diphosphokinase)Random bi-uni-Mechanism (Example : an Aldolase-Type)
41 Unbranched Reaction Chain EXnApparent rate constantsEXn-1EX1Apparent equilibrium constantsEX2EX2General rate lawHolds for all sequential reaction mechanisms
43 Convenience Kinetics (actually a generalized random kinetics….) Ordered KineticsPing-pong KineticsConvenience KineticsConvenience KineticsOrdered KineticsPing-pong Kineticsr=0.946r=0.975r=0.983
44 Other types of kinetics: S-Systems Introduced by M. Savageau, 1976 („synergistic systems“)Xj4Xj3Xj2Xj1XiXj5Vi+Vi-For i = 1...nn independent variablesm dependent variablesg, h – positive or negative, usually no integersSteady state:
45 Other types of kinetics: Lin-Log Kinetics Sef Heijnen and others