Chapter 14. Enzyme Kinetics Chemical kinetics • Elementary reactions A → P (Overall stoichiometry) I1 → I2 (Intermediates) • Rate equations aA+ bB+ … +zZ → P Rate = k[A]a[B]b…[Z]z k: rate constant • The order of the reaction (a+b+…+z): Molecularity of the reaction – Unimolecular (first order) reactions: A → P – Bimolecular (second order) reactions: 2A → P or A + B → P – Termolecular (third order) reactions Rates of reactions A → P (First-order reaction) d[P] d[A] v = = - = k[A] dt dt 2A → P (Second-order reaction) d[P] d[A] v = = - = k[A]2 dt dt A + B → P (Second-order reaction) d[P] d[A] d[B] v = = -= - = k[A][B] dt dt dt Rate constant for the first-order reaction d[A] v = − = k[A] dt d[A] = −kdt [A] [A] d[A] t = −k dt ∫∫[A]0 [A] 0 ln[A]− ln[A]0 = −kt ln[A] = ln[A]0 − kt −kt [A] = [A]0 e (t) The reactant concentration decreases exponentially with time Half-life is constant for a first-order reaction ln[A]− ln[A]0 = −kt [A] ln = −kt [A]0 1 ln = −kt1 / 2 2 ln 2 0.693 t1 / 2 = = k k For the first-order reaction, half-life is independent of the initial reactant concentration Second-order reaction with one reactant 2A → P d[A] v = − = k[A]2 dt d[A] 1 Slope= k = −kdt [A]2 [A] [A] d[A] t 2 = −k dt ∫∫[A]0 [A] 0 1 1 1 = + kt [A]0 [A] [A]0 Time (t) 2 1 1 kt1 / 2 = − = [A]0 [A]0 [A]0 1 t1 / 2 = Half-life is dependent on the initial reactant concentration k[A]0 Pseudo first-order reactions v = k[A][B] When [B] >> [A], v = k′[A] where k′ = k[B]0 e.g. B is water (55.5M): k'= 55.5k Arrhenius equation k = Ae-Ea/RT = Activation energy (Ea) Multistep reactions have rate-determining steps k1 > k2 k1 < k2 k 1 k2 Rate-determining step: the slow step (= the higher activation energy) Catalysts reduce the activation energy ΔEa (the reduction in Ea by the catalyst) ΔEa/RT Rate enhancement = kcat/kuncat = e e.g. When ΔEa = 5.7 kJ/mol, the reaction rate increases 10 fold 6 When ΔEa = 34 kJ/mol, the reaction rate increases 10 fold Michaelis-Menten equation Vmax[S] v0 = KM +[S] • v0 = the initial rate • Vmax = the maximum rate • KM = the Michaelis constant • [S] = the substrate concentration Steady state approximation k1 k2 E + S ES P + E k-1 Derivatization of Michaelis-Menten equation k1 k2 E + S ES P + E k-1 d[P] v = = k 2[ES] dt d[ES] = k 1[E][S]− k − 1[ES]− k 2[ES] = 0 (Steady state approximation) dt [E]T = [E]+[ES] k 1([E]T −[ES])[S] = (k − 1 + k 2)[ES] (k − 1 + k 2 + k 1[S])[ES] = k 1[E]T[S] k − 1 + k 2 ( +[S])[ES] = [E]T[S] k 1 [E]T[S] [ES] = k − 1 + k 2 +[S] k 1 [E]T[S] k − 1 + k 2 [ES] = where KM = KM +[S] k 1 k 2[E]T[S] The initial velocity v0 = k 2[ES] = KM +[S] The maximum velocity (Vmax) occurs at high substrate concentration when the enzyme is entirely in the [ES] form : Vmax = k 2[E]T Vmax[S] v0 = KM +[S] Michaelis constant KM KM = [S] at which v0 = Vmax/2 • If an enzyme has a small value of KM, it achieves maximal catalytic efficiency at low substrate concentrations • Measure of the enzyme’s binding affinity for the substrate (The lower KM, the higher affinity) Lineweaver-Burke plot Vmax[S] v0 = K M +[S] 1 K +[S] = M v0 Vmax[S] 1 ⎛ KM ⎞ 1 1 = ⎜ ⎟ + v0 ⎝Vmax ⎠ [S] Vmax kcat/KM is a measure of catalytic efficiency Catalytic constant or turnover number of an enzyme: Vmax kcat = [E]T When KM >> [S] , very little ES is formed and consequently [E] ≈ [E]T Vmax[S] Vmax[S] kcat[E]T[S] ⎛ kcat ⎞ v0 = ≈ = ≈ ⎜ ⎟[E][S] KM + [S] KM KM ⎝ KM ⎠ Catalytic perfection kcat k 2 k 1k 2 = = KM KM k − 1 + k 2 The ratio is maximal when k 2 >> k − 1, kcat = k 1 KM (Diffusion-controlled limit: 108 to 109 M-1s-1) Inhibitors • Substances that reduce an enzyme’s activity – Study of enzymatic mechanism – Therapeutic agents • Reversible or irreversible inhibitors - O CO2 - O CO2 - O N CO2 H - NH2 N CO2 N H HN N N N N H H2N N N CH3 H N N N H 2 Dihydrofolate Methotrexate (Dihydrofolate reductase substrate) (Dihydrofolate reductase inhibitor, anticancer drug ) Modes of the reversible inhibition • Competitive inhibitors – Binds to the substrate binding site • Uncompetitive inhibitors – Binds to enzyme- substrate complex • Non-competitive inhibitors – Binds to a site different from the substrate binding site • Mixed inhibitors – Binds to the substrate- binding site and the enzyme-substrate Competitive inhibition [E][I] KI = [EI] d[ES] = k 1[E][S]− k − 1[ES]− k 2[ES] = 0 (Steady state approximation) dt ⎛ k − 1 + k 2 ⎞ [ES] KM[ES] [E] = ⎜ ⎟ = ⎝ k 1 ⎠ [S] [S] [E][I] KM[ES][I] [EI] = = KI [S]KI [E]T = [E]+[EI]+[ES] KM[ES] KM[ES][I] ⎧KM ⎛ [I]⎞ ⎫ [E]T = + +[ES] = [ES]⎨ ⎜1+ ⎟ +1⎬ [S] [S]KI ⎩[S] ⎝ KI ⎠ ⎭ [E]T[S] [ES] = ⎛ [I]⎞ KM ⎜1+ ⎟ +[S] ⎝ KI ⎠ k 2[E]T[S] v0 = k 2[ES] = ⎛ [I]⎞ KM ⎜1+ ⎟ +[S] ⎝ KI ⎠ Since Vmax = k 2[E]T, Vmax[S] [I] v0 = where α =1+ αKM +[S] KI Competitive inhibitors affect KM Vmax[S] v0 = αKM +[S] [S] → ∞; v0 →Vmax regardless of α Determination of KI of the competitive inhibitor 1 ⎛αKM ⎞ 1 1 = ⎜ ⎟ + v0 ⎝ V max ⎠ [S] V max Uncompetitive inhibition [ES][I] KI'= [ESI] KM[ES] [E] = [S] [ES][I] [ESI] = KI' [E]T = [E]+[ES]+[ESI] KM[ES] [ES][I] ⎛ KM [I] ⎞ [E]T = +[ES]+ = [ES]⎜ +1+ ⎟ [S] KI' ⎝ [S] KI' ⎠ [E]T[S] [ES] = ⎛ [I] ⎞ KM + ⎜1+ ⎟[S] ⎝ KI' ⎠ k 2[E]T[S] v0 = k 2[ES] = ⎛ [I] ⎞ KM + ⎜1+ ⎟[S] ⎝ KI' ⎠ Since Vmax = k 2[E]T, V max [S] Vmax[S] α' [I] v0 = = where α'=1+ K +α'[S] K KI' M M +[S] α' Uncompetitive inhibitors decrease both Vmax and KM V max [S] v = α' 0 K M +[S] α' V [S] → ∞; v → max 0 α' Vmax (Effects of an uncompetitive inhibitor KM >> [S]; v0 → [S] K M become negligible) Vmax no inhibitor (α'=1) Increasing [I] I Vmax v0 Vmax/2 [I] I Vmax /2 α'=1+ KI' I KM KM [S] Determination of KI’ of the uncompetitive inhibitor 1 ⎛ KM ⎞ 1 α' = ⎜ ⎟ + v0 ⎝V max ⎠ [S] V max Mixed inhibition [E][I] [ES][I] KI = KI'= [EI] [ESI] [E][I] [EI] = KI [ES][I] [ESI] = KI' [E]T = [E]+[EI]+[ES]+[ESI] ⎛ [I]⎞ ⎛ [I] ⎞ [E]T = [E]⎜1+ ⎟ +[ES]⎜1+ ⎟ ⎝ KI ⎠ ⎝ KI' ⎠ [E]T = [E]α +[ES]α' KM[ES] ⎛αKM ⎞ [E]T = α +[ES]α'= [ES]⎜ +α'⎟ [S] ⎝ [S] ⎠ [E]T[S] [ES] = αKM +α'[S] k 2[E]T[S] v0 = k 2[ES] = αKM +α'[S] Since Vmax = k 2[E]T, Vmax[S] [I] [I] v0 = where α =1+ and α'=1+ αKM +α'[S] KI KI' Lineweaver-Burk plot of a mixed inhibition 1 ⎛αKM ⎞ 1 α' = ⎜ ⎟ + v0 ⎝ V max ⎠ [S] V max Noncompetitive inhibition A special mixed inhibition when KI = KI’ Vmax[S] [I] [I] v0 = where α =1+ and α'=1+ αKM +α'[S] KI KI' When KI = KI', α = α' V max [S] Vmax[S] α v0 = = α(K M +[S]) KM +[S] Noncompetitive inhibitors affect not KM but Vmax V max [S] α [I] v0 = where α =1+ K M +[S] KI V [S] → ∞; v → max 0 α Vmax no inhibitor (α=1) I Increasing [I] Vmax v0 Vmax/2 V I/2 [I] max α =1+ KI [S] KM Determination of KI of the noncompetitive inhibitor 1 ⎛αKM ⎞ 1 α = ⎜ ⎟ + v0 ⎝ V max ⎠ [S] V max 1/v0 α = 2 Increasing α= 1.5 [I] α= 1 (no inhibitor) Slope=αKM/Vmax [I] α =1+ α/Vmax KI 1 − 0 1/[S] KM Effects of inhibitors on Vmax and KM of the Michaelis-Menten equation app Vmax [S] v0 = app K M +[S] Effects of pH ' Vmax[S] • Binding of substrate to enzyme v0 = ' KM +[S] • Catalytic activity of enzyme ' ' Vmax = Vmax / f 2 and KM = KM ( f 1 / f 2) • Ionization of substrate where + • Variation of protein structure [H ] KE2 f 1 = +1+ + (only at extreme pHs) KE1 [H ] + [H ] KES2 f 2 = +1+ + KES1 [H ] - E ES- K H+ + E2 KES2 H k1 k2 EH + S ESH P + EH k + -1 + KE1 H KES1 H + + EH2 ESH2 Approximate identity of catalytic amino acid residues pKa ~4 → Catalytic Asp or Glu residue pKa ~6 → Catalytic His residue pKa ~10 → Catalytic Lys residue Caution should be taken because pKa of amino acid residues are environmentally sensitive Bisubstrate reactions • 60% of biochemical reactions involve two substrates and two products • Transfer reactions and oxidation-reduction reactions Cleland’s nomenclature system for the enzymatic reactions • Substrates: A, B, C, D… in the order that they add to the enzyme • Products: P, Q, R, S… in the order that they leave the enzyme • Inhibitors: I, J, K, L… • Stable enzyme complexes: E, F, G, H… with E being the free enzyme • Numbers of reactants and products: Uni (one), Bi (two), Ter (three), and Quad (four) e.g. Bi Bi reaction: a reaction that requires two substrates and yields two products Types of Bi Bi Reactions • Sequential reactions (single displacement reactions): all substrates bind before chemical event Ordered mechanism Random mechanism • Ping pong reactions (double displacement reactions): chemistry occurs prior to binding of all substrates Differentiating bisubstrate mechanisms • Measure rates • Change concentration of substrates and products • Lineweaver-Burk plot – Intercept (1/Vmax): the velocity at saturated substrate concentration → It changes when the substrate A binds to a different enzyme form with the substrate B – Slope (KM/Vmax): the rate at low substrate concentration → It changes when both A and B reversibly bind to an enzyme form Ping Pong Bi Bi Mechanism • Intercept changes because A and B bind to the different enzyme forms E and F, respectively • Slope remains same because the binding of A and B is irreversible due to the release of the product (P) Sequential Bi Bi Mechanism or Ordered Random • Intercept changes because A and B binds to the different enzyme forms (E or EB) and (EA or E), respectively • Slope changes because the binding of A and B is reversible Differentiating Bi Bi mechanisms by product inhibition Competitive inhibition → Substrate and inhibitor competitively bind to the same site of the enzyme A vs Q: Competitive B vs P: Competitive Ping Pong A vs P: Noncompetitive B vs Q: Noncompetitive A vs Q: Competitive B vs P: Noncompetitive Ordered A vs P: Noncompetitive sequential B vs Q: Noncompetitive Under assumption of dead-end complex formation (A is similar with Q and B is similar