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CMU BSC 03231 - Lecture

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1Biochemistry I Fall Term, 2004October 4, 2004Lecture 14: Enzyme KineticsAssigned reading in Campbell: Chapter 5.1-5.6Key Terms:First order reactionSecond order reactionTransition state theoryActive siteSteady-State KineticsMichaelis Menten EquationVmax & KMkcat & kcat/KMLinks:(I) Review Quiz on Lecture 14 concepts(I) Substrate Saturation Kinetics shows various graphs used to analyze kinetic data(S) Graphing Quiz: Michaelis- Menten Kinetics: Determine Vmax and KMCampbell's text is excellent on all of the assigned sections for today's lecture. These notesdevelop the same concepts in a slightly different order and add some material in the TransitionState Theory and Catalytic Efficiency and the Specificity sections below.For the simple enzyme-catalyzed reaction:S <==> PS is substrate; and P is product.The enzyme forms a complex with the substrate, in much the same way as a protein-ligandcomplex, and performs some chemical reaction/transformation of the bound substrate. Theresultant product is released.Important Features of Enzyme Catalysis.1. Enzymes increase the rate of reactions.2. Enzymes do not change the equilibrium point of reactions.3. Enzymes are specific for their substrates:Geometric complementarityEnergetic complementarity.4. Enzymes are regulated:2• at genetic level (transcription, translation);• by concentration of substrate and product;• allosterically.Background Kinetics1. Order of Reactions:A -> P rate ~ [A]1 1st orderA + A -> P rate ~ [A]2 2nd orderA + B -> P rate ~ [A][B] 2nd order2. Velocity of a reaction:v = -d[A]/dt = d[P]/dt= k[A][B] (for a second order reaction)3. Transition State Theory:Is there a physical limit to the rate of a chemical reaction? There is indeed such an upper limit;and transition state theory provides an estimate of the rate constant. At some point along thereaction progress curve, the chemical species reach a high energy configuration called thetransition state. It is represented by the symbol, X‡ (Campbell, Fig. 5.1).Transition state theory states that the rate of reaction is directly proportional to the concentrationof the transition state:v = k'[X‡]where k' is the rate at which the transition state will decay to products. The key assumptionmade in transition state theory is that the substrates and products are in equilibrium with thetransition state. For example, the equilibrium constant of a second order reaction in the forwardreaction is:K‡ = [X‡]/[A][B]Therefore the velocity of the reaction is: v = k'K‡[A][B]. Comparing this equation to that for asecond order reaction gives: k = k' K‡.An expression for k' was derived by H. Eyring who postulated that a reaction velocity could notexceed the vibrational frequencty of the bond that is broken. This maximum frequency can beexpressed using only fundamental physical constants:3ν = kBT/hwhere kB is Boltzmann's constant and h is Planck's constant.The vibrational frequency, ν = 6.3*1012 sec-1 at T = 300 K.Finally, we substitute the free energy equivalent of the equilibrium constant K‡, to get theexpression for a rate constant according to transition state theory:k = (kBT/h)exp(-∆G‡/RT)The pre-exponetial portion is the maximum frequency possible; the exponential represents thefraction of reactants in the transition state (at temperature, T). From the above equation it isclear how enzymes increase the rate of the reaction - they must do so by lowering the free energyof the transition state in the enzyme substrate complex. This is shown by the free energydiagrams below:Note that enzymes bind to both the substrates and products and stabilize the transition state. Thetransition state of the enzyme substrate complex is stabilized in two ways:• First, the enzyme transition-state complex is stabilized by direct interactions between theenzyme and the transition state. This reduces the free energy of the transition state.• Second, the enzyme may provide a number of functional groups to aid in catalysis. Sincethese groups are positioned in well defined locations, the entropy of bringing thesegroups into the catalytic site is substantially less than having these functional groupsdiffuse freely in solution.4Steady State Enzyme Kinetics:• Steady-state reactions are easy to measure because the rate of the reaction is constant forrelatively long periods of time.• Steady-state rates are those which are most relevant to metabolic levels.• The analysis of steady state kinetics can only provide limited information on the kineticmechanism.The simplest reaction scheme is:where, k1 is the forward rate constant for substrate binding k-1 is the reverse rate constant for substrate binding. k2 is the catalytic rate constant (containing terms related to the transition state).The ES complex is also called the "Michaelis complex".The velocity of the reaction is:v = d[P]/dt = k2[ES]The change in [ES] as a function of time is:d[ES]/dt = k1[E][S] - k-1[ES] - k2[ES]During the steady state: d[ES]/dt = 00 = k1[E][S] - k-1[ES] - k2[ES]The goal is to relate this equation to readily measurable experimental parameters, such as:• The total amount of enzyme: ET = [E] + [ES]• The concentration of substrate: [S]• The measured steady state velocity (v = k2 [ES])We do not have a suitable way to measure [E], but since we know the total enzymeconcentration, we can substitute as follows:[E] = ET - [ES][ES](k-1 + k-2) = k1[S](ET - [ES])5[ES](k-1 + k2) = k1 E[ES] - k1[ES][S][ES](k-1 + k2 + k[S]) = k1 E[S][ES] = k1E[S]/(k-1 + k2 + k1[S])The velocity of the reaction:v = k2[ES]= k1k2E[S]/{k-1 + k2 + k1[S]}= k2E[S]/{(k-1 + k2)/ k1+ [S]}Defining a few terms:The Vmax or maximal velocity: Vmax = k2ETThis is the highest reaction rate that can be attained because all (i.e. ET) of the enzyme issaturated with substrate.The KM or Michaelis constant: KM = (k-1 + k2)/k1This is the substrate concentration that gives a reaction velocity equal to 1/2 of Vmax. Notethat in the case of slow reaction kinetics (k2<<k-1), the KM is also the dissociation constantfor substrate binding.The use of these definitions gives the Michaelis-Menten EquationThe class handout, Substrate Saturation Kinetics shows a plot of reaction velocity versus [S].Also shown is a graph of the data in a double reciprocal plot, otherwise known as a "Lineweaver-Burk Plot".6Catalytic Efficiency and the Specificity of Enzyme Reactions: kcat/KMThe catalytic constant of an


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