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**Systems biology 2 – Reaction kinetics Edda Klipp Sommmersemester 2010**

Humboldt-Universität zu Berlin Institut für Biologie Theoretische Biophysik

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**Enzymes remain unchanged after reaction as catalyst**

Proteins, often complexed with cofactors Anorganic cofactors: metall ions Organic cofactors (coenzymes): vitamin-derived complex groups remain unchanged after reaction as catalyst have a catalytical centre are in general highly specific are often pH- and temperature dependent Turnover number: 1000 /sec (100 /sec million /sec) Acceleration (compared to non-catalyzed reaction) by 106 to fold Thermodynamics: Enzymes reduce the necessary activation energy for the reaction

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**Classification of enzymatic reactions**

irreversible reversible S P S P

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**Classification of enzymatic reactions**

Number of substrates (and products) uni S P S2 large (0.5) bi S1+S2 P S2 small (0) ter S1+S2 +S3 P 1

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**Classification of enzymatic reactions**

Type of kinetics . Linear Mass action v = k S Hyperbolic Michaelis-Menten Sigmoidal Hill kinetics, Monod, Koshland v “Hyperbolic” and “Sigmoidal” show saturation, “Linear” involves unlimited reaction rates.

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**Kinetics of Enzymatic Reactions**

Deterministic kinetic modeling of biochemical reactions Basic quantities: Concentration S : number of molecules per unit of volume Reaction rate v : concentration change per unit time Postulat: The reaction rate v at point r in space at time t can be expressed as a unique function of the concentrations of all substances at point r at 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))

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**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 to the concentration of reactants to the power of their molecularity. (Guldberg and Waage, 19. century) A+B 2 C Reaction rate Rate constants Equilibrium constant

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**Michaelis-Menten Kinetics**

Brown (1902): Mechanism for Invertase reaction (with sucrose), Which holds for one-substrate-systemes with backward reaction of effectors: E – catalyst S – substrate P – product ki – kinetic constant complex formation reversible complex degradation irreversible Michaelis, Menten (1913): rate equation under the assumption That second reaction will not influence the first equilibrium (Hypothesis of quasi-equilibrium) Briggs, Haldane (1925): more general derivation of Rate law under the assumption of a steady state for the enzyme-substrate-complex (where )

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**Michaelis-Menten Kinetics: derivation of rate law**

Non-linear ordinary differential equation system (1) The rate of product formation is equal to the reaction rate (2) The sum of equations (2) and (3) is a conservation relation for the enzyme (3) (4) The whole set of equations cannot be solved analytically. Using quasi-steady state assumption

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**Michaelis-Menten-Kinetics: The rate equation**

Reaction rate v Maximal velocity Vmax Vmax Michaelis constant Michaelis-Menten- Rate expression S

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**Integrated Form of MM rate law**

Reaction rate = Product increase or substrate decrease per unit time Integration from t0, S0 to t, S results in Henri-Michaelis- Menten-equation and for Wenn die Änderung der Reaktionsgeschwindigkeit nur von der Änderung der Substrat-sättigung des Enzyms und nicht von einer Hemmung durch das Produkt oder das Erreichen des Gleichgewichtes abhängt, ist This is a function or One can record a progress curve and estimate the kinetic constants using non-linear regression.

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**Estimation of Parameters Vmax and Km**

1. Measurement of initial rates Measure initial rates for different initial concentrations , i.e. measure initial change of S. 2. Interpretation Plot measurement results in (S,V)-Diagram; Compare with Michaelis-Menten rate law; Estimate parameters by non-lineare regression, for example least-squares methode

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**Linearizations of the MM rate law**

Lineweaver-Burk-Plot Eadie-Plot Hanes-Plot

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**Additional aspects Relation to thermodynamics**

Vmax is related to turnover number, kcat Condition: completely saturated enzymes, maximal rate: [1/(mol*s)] Dissoziation constante KS of the enzyme-substrate-complex: [mol]

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**Regulation of Enzyme Activity**

Important mechanism for the regulation of cellular processes 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

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Enzyme Inhibition Competitive inhibition: substrate and inhibitor compete for the binding place at the enzyme Equilibrium for inhibitor binding Conservation relation for the enzyme Rate equation

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**Examples Competitive Inhibition**

Bernsteinsäuredehydrogenase 1. Succinic acid dehydrogenase has as substrate succinic acid and is inhibited by Malonic acid. 2. Acetylcholin esterase has 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.

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**Enzyme Inhibition 2. Uncompetitive inhibition:**

Inhibitor binds only to the enzyme-substrate-complex 3. Non-competitive inhibition: Inhibitor binds to free and bound enzyme

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Enzyme Inhibition, 3 4. Irreversible inhibition : inhibitor binds the enzyme irreversibly, partial or complete loss of catalytic effectivity Example: Reaction of Iod acetate with –SH groups in cystein side chains of the reaction centre 1. Di-isopropyl-fluorophosphate (DFP) and other alkylphosphates bind covalently to acetylcholinesterase. This enzyme is responsible for Transmission of nerve stimuli. The organsims die of paralysis (Lähmung) of organ function. (used in military gases and insektizids)

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**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 coordinate Product Inhibition : Inhibition by the product due to allosteric inhibition (prevents excess production) Reduction of the net reaction rate, due to an accumulation of product which is substrate of the backward reaction.

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**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 assumptions Enzyme conservation Reaction rate Optimum Example: Succinic acid dehydrogenase

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**Enzyme Activation Activation Increase of the rate by**

- Change of substrate binding - Acceleration of product formation Example Substrate activation: Substrats S acts as activator A. Reaction rate = Product formation rate Enzyme conservation Quasi-equilibrium condition Reaction rate

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**Activation and Inhibition for Mass Action Kinetics**

+ S P compulsory additional I - S P

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**Ligand Binding and Cooperativity**

Ligand: compound that binds to enzyme / protein Here: Binding of ligands to monomeric und oligomeric proteins. several ligand binding sites at a protein: Possibility of interactions between these sites during binding This phenomenon is called cooperativity Positive/negative cooperativity: Binding of a ligand molecule increases/reduces the affinity of the protein for further ligands. Homotrope/heterotrope cooperativity : Binding of a ligand molecule affects binding of further molecules Of the same/ other ligands.

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**Fractional Saturation**

Case of 1 binding site: Binding of S (Ligand) to E (Protein) Binding constante Definition: Fractional Saturation Fractional saturation for 1 subunit Plot of Y versus S is hyperbolic

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**Hill-Kinetik Positive, homotrope cooperativity**

Simplest case: dimeric protein - two similar ligand binding sites - Binding of first ligand increases affinity to second ligand M = monomere Untereinheit, M2 = Dimer Assumption: Binding of S increases affinity M2S reacts with S as soon as it is formed Fractional saturation: Complete cooperativity (each subunit is either empty or completey saturated) Binding constante Fractional saturation:

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**Hill Kinetics For complete homotrope cooperativity**

of a protein with n subunits holds: This is a form of the Hill equation Y S Hemoglobin: sigmoid bindung curve of oxygen against oxygen partial pressure Hill (1909): Interaction between binding sites - positive cooperativity Known: hem binds oxygen molecules Unknown: number of subunits per protein Assumption: complete cooperativity - experimental Hill coefficient h=2.8 Hintergrund: experimentelle Befunde zur Bindung des Sauerstoffs ans Hämoglobin fanden Bohr und Mitarb., dass die Auftragung der fraktionellen Sättigung des Hämoglobin mit Sauerstoff gegen den Sauerstoffpartialdruck eine sigmoide Kurve ergab.Hill (1909) erklärte das auf der Grundlage von Wechselwirkungen zwischen den Bindungsstellen, die positive Kooperativität bewirken. Damals wußte man schon, dass jedes Häm ein Sauerstoffmolekül bindet, allerdings nicht aus wieviel UE ein oligomeres Protein besteht. Hill leitete seine Gleichung ab für Er nahm vollständige Kooperativität an und fand experimentellen Hillkoeff. von h=2.8. Heute weiß man, dass es vier Bindungsstellen an jedem Hämoglobinmolekül gibt, so dass keine voll- ständige Kooperativität vorliegt. Praktischer Nutzen der sigmoiden Binungscharakteristik: In der Lunge ist der Sauerstoffpartialdruck hoch, dort kann Hb den Sauerstoff gut binden; im Körper ist der Sauerstoffpartialdruck geringer und Hb kann O2 leicht abgeben Four Binding sites per hemoglobin molecule No complete cooperativity High oxygen partial pressure in lungs: good binding of oxygen to Hb Low oxygen partial pressure in body – easy delivery of O2

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**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 conformation Conformation change for all SU at the same time (concerted transition). R - active T - inactive Conformation equilibrium R – Conc. active conformation R0 – R- Conc. without bound substrate R1 – R- Conc. with 1 bound substrate T – Conc. of inactive conformation T0 – Conc. without bound substrates L Allosteric constant

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**Monod-Wyman-Changeux model**

n = 4 subunits KR Binding 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 S R2 - Conc. of R with 2 bound molecules of S R3 - Conc. of R with 3 bound molecules of S R4 - Conc. of R with 4 bound molecules of S 1 possibility 4 possibilities 6 possibilities. 4 possibilities 1 possibility General: Possibilities of substrate binding for Ri

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**Monod-Wyman-Changeux model**

It holds: General: Sum of all active states: with binomic Formula: Fractional saturation Replacement of R and Ri T exists only as T0

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**Monod-Wyman-Changeux model**

It follows Reaction rate Michaelis-Menten- Term "Regulatory Term"

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**Monod-Wyman-Changeux model**

For S∞ : Monod-Kinetics approaches Michaelis-Menten-Kinetics small S: regulatory term important depending on L L = 0: MM-Kinetics L >> 0: sigmoidal curve, shifted to right. 102 103 v activation 104 inhibition S Explanation of the action of activators and inhibitors: - Activators bind to active conformation - Inhibitors bind to inactive conformation -Shift of equilibrium to R or T Bindungskonstanten

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**Monod-Wyman-Changeux model**

Example: Phosphofructokinase: experimentaly well studied system Activators: Inhibitors: DPG, ATP Typical value for

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**Kinetics of Reversible Reactions**

Derivation of rate equation for steady state Relation between equilibrium constant q and kinetic constants of elementary steps Reaction rate

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**Kinetics of Reversible Reactions**

Relation to phenomenological quantities S very high, P=0 P very high, S=0 Half-maximal forward rate Half-maximal backward rate For S and P very small holds This resembles Mass action kinetics (Also called linear kinetics).

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**Several activated complexes**

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**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 Enzym 2. Relative concentration of each enzyme species is equal to ratio of two sums of terms, where every term Tij is the product of n-1 rate constants and 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:

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**King-Altman for 3-Step reaction mechanism**

Sk1 1. Conservation of total enzyme amount: : E ES k-1 k-2 Pk-3 k2 k3 EP 2., 3. Listing of all possibilities of n-1 = 2 lines leading to each enzyme species: k-1 k-1 For E k3 k-2 k3 k2 Sk1 Sk1 For ES k3 k-2 Pk-3 k-2 k-1 Sk1 For EP Pk-3 k2 Pk-3 k2 4. Reaction rate:

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**Further typical Mechanisms**

Ordered bi-bi-Mechanismus (Example: Kreatinkinase)

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**Further typical Mechanisms**

Ordered bi-bi-Mechanism (Example: Kreatinkinase) Ping-Pong-Mechanism (Example : Transaminase, Nukleosid-Diphosphokinase) Random bi-uni-Mechanism (Example : an Aldolase-Type)

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**Unbranched Reaction Chain**

EXn Apparent rate constants EXn-1 EX1 Apparent equilibrium constants EX2 EX2 General rate law Holds for all sequential reaction mechanisms

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Example

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**Convenience Kinetics (actually a generalized random kinetics….)**

Ordered Kinetics Ping-pong Kinetics Convenience Kinetics Convenience Kinetics Ordered Kinetics Ping-pong Kinetics r=0.946 r=0.975 r=0.983

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**Other types of kinetics: S-Systems**

Introduced by M. Savageau, 1976 („synergistic systems“) Xj4 Xj3 Xj2 Xj1 Xi Xj5 Vi+ Vi- For i = 1...n n independent variables m dependent variables g, h – positive or negative, usually no integers Steady state:

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**Other types of kinetics: Lin-Log Kinetics**

Sef Heijnen and others

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