Biochemistry: A Short Course Tymoczko • Berg • Stryer Biochemistry: A Short Course Second Edition CHAPTER 7 Kinetics and Regulation © 2013 W. H. Freeman and Company

Much of life is motion, whether at the macroscopic level of our daily life or at the molecular level of a cell. Capturing the idea of motion is what motivated French photographer Jacques Henri Lartigue when he photographed a fast-moving automobile in 1912. In biochemistry, kinetics (derived from the Greek kinesis, meaning “movement”) is used to capture the dynamics of enzyme activity.

Consider a simple reaction: The velocity or rate of the reaction is determined by measureing how much A disappears as a function of time or how much B appears as a function of time. Suppose that we can readily measure the disappearance of A. The velocity of the reaction is given by the formula below, where k is a proportionality constant. When the velocity of a reaction is directly proportional to reactant concentration, the reaction is called a first-order reaction and the proportionality constant has the units s-1.

Many important biochemical reactions are biomolecular or second-order reactions. The rate equations for these reactions are: and The proportionality constant for second-order reactions has the units M-1s-1.

A common means of investigating enzyme kinetics is to measure velocity as a function of substrate concentration with a fixed amount of enzyme. Consider a single-substrate reaction in which the enzyme E catalyzes the conversion of SP. We assume k-2 is very, very small, so… Under these conditions, the velocity is called the initial velocity or Vo.

This is a Michaelis-Menten curve; enzyme follows Michaelis-Menten kinetics Figure 7.1 Reaction velocity versus substrate concentration in an enzyme-catalyzed reaction. An enzyme-catalyzed reaction approaches a maximal velocity.

How do we study enzyme-catalyzed reactions? The initial velocity is determined by measuring product formation as a function of time, and then determining the velocity soon after the reaction has started. Experiment: Mix enzyme + substrate Record rate of substrate disappearance/product formation as a function of time (the velocity of reaction) Plot initial velocity versus substrate concentration. Change substrate concentration and repeat

Figure 7. 2 Determining initial velocity Figure 7.2 Determining initial velocity. The amount of product formed at different substrate concentrations is plotted as a function of time. The initial velocity (V0) for each substrate concentration is determined from the slope of the curve at the beginning of a reaction, when the reverse reaction is insignificant. The initial velocity is illustrated for substrate concentration [S]4.

Leonor Michaelis and Maud Menten derived an equation to describe the initial reaction velocity as a function of substrate concentration. o

s E s E s E s E s E s E s E s E s E s E s E s E s E s E s E s E s E s

When Vo = ½ Vmax, KM =[S]. Thus, KM is the substrate concentration that yields ½ Vmax. Figure 7.3 Michaelis–Menten kinetics. A plot of the reaction velocity, V0, as a function of the substrate concentration, [S], for an enzyme that obeys Michaelis–Menten kinetics shows that the maximal velocity, Vmax, is approached asymptotically. The Michaelis constant, KM, is the substrate concentration yielding a velocity of Vmax /2.

How to view kinetics at low [S] and high [S] This graph shows the kinetic parameters that define the limits of the curve at high and low [S]. At low [S], Km >> [S] and the [S] term in the denominator of the Michaelis-Menten equation (Eqn 6-9) becomes insignificant. The equation simplifies to V0 = Vmax[S]/Km and V0 exhibits a linear dependence on [S], as observed here. At high [S], where [S] >> Km, the Km term in the denominator of the Michaelis-Menten equation becomes insignificant and the equation simplifies to V0 = Vmax; this is consistent with the plateau observed at high [S]. The Michaelis-Menten equation is therefore consistent with the observed dependence of V0 on [S], and the shape of the curve is defined by the terms Vmax/Km at low [S] and Vmax at high [S].

Given… k1 k2 E + S ES E + P k-1 If rate of formation of ES ( given by k1) = rate of breakdown of ES (given by (k-1 + k2)), can derive the following relationship: The Michaelis constant, Km, is Often k2 << k-1, so Km = k-1/k1 = [reactants]/[product] = 1/Keq = 1/Kaffinity

KM is a [S] with units of moles/liter (remember this!). Note KM is a [S] with units of moles/liter (remember this!). When KM is a small number (concentration), the substrate affinity is high. When KM is a large number (concentration), the substrate affinity is low. Think of: E + S ↔ ES How much S does it take to drive rx in direction of ES complex formation? Important if E has more than one S (ribonucleotide reductase: Km values of 0.12 mM for CDP, 0.14 mM for ADP and 0.026 mM for GDP), or S more than one E (example, sensitivity to ethanol).

Two enzymes play a key role in the metabolism of alcohol.

Some people respond to alcohol consumption with facial flushing and rapid heart beat, symptoms caused by excessive amounts of acetaldehyde in the blood. There are two different acetaldehyde dehydrogenases in most people, one with a low KM and one with a high KM. The low KM enzyme is inactivated in susceptible individuals. The enzyme with the high KM cannot process all of the acetaldehyde, and so some acetaldehyde appears in the blood.

Determining KM and VMAX The Michaelis-Menten equation can be manipulated into one that yields a straight-line plot. In the form of y = m ● x + b This double-reciprocal equation is called the Lineweaver-Burk equation.

Figure 7. 5 A double-reciprocal, or Lineweaver-Burk, plot Figure 7.5 A double-reciprocal, or Lineweaver-Burk, plot. A double-reciprocal plot of enzyme kinetics is generated by plotting 1/V0 as a function 1/[S]. The slope is KM /Vmax, the intercept on the vertical axis is 1/Vmax, and the intercept on the horizontal axis is –1/KM.

KM values for enzymes vary widely and evidence suggests that the KM value is approximately the substrate concentration of the enzyme in vivo.

M

If the enzyme concentration, [E]T, is known, then and k2 is also called kcat or the turnover number of the enzyme. It gives the number of substrate molecules converted into product per second.

In living cells, [S]<<KM, so we can assume that free enzyme [E] ≈ [E]T. The Michaelis-Menten equation can be manipulated to yield: Under these conditions, is a measure of catalytic efficiency because it takes into account both the rate of catalysis (kcat) and nature of the enzyme substrate interaction (KM).

Multiple substrate reactions can be divided into two groups. Sequential reactions are characterized by formation of a ternary complex consisting of the two substrates and the enzyme. Double-displacement reactions are characterized by the formation of a substituted enzyme intermediate. Double-displacement reactions are also called ping-pong reactions.

Michaelis-Menten enzymes Enzyme activity not regulated in cell, but rather the rate of rx is determined by [E] and [S] (activity governed by law of mass action). hyperbolic Vo vs. [S] curve But, …

Figure 7. 7 Complex interactions require regulation Figure 7.7 Complex interactions require regulation. (A) A part of a street map of Paris. (B) A schematic representation of several interconnecting metabolic pathways in plants. [(A) ©2008 Google-Map data © 2008 Tele Atlas.]

Allosteric enzymes control the flux of biochemical reactions in metabolic pathways. are regulatory enzymes. are cellular information sensors. are often (not always) regulated by products of the pathways they control.

Assume in this example that the conversion of A to B is the committed step, because once this occurs B is committed to being converted into F. Allosteric enzymes catalyze the committed step of metabolic pathways, and catalyze essentially irreversible rxs. Michaelis-Menten enzymes catalyze unregulated rxs.

The amount of F synthesized can be regulated by feedback inhibition. The pathway product F inhibits enzyme e1 by binding to a regulatory site on the enzyme that is distinct from the active site.

The regulation of metabolic pathways can be quite complex. Allosteric enzymes may be inhibited or stimulated by multiple regulatory molecules.

Figure 7.8 Two pathways cooperate to form a single product.

The reaction velocity of allosteric enzymes displays a sigmoidal relationship to substrate concentration.

All allosteric enzymes display quaternary structure with multiple active sites and regulatory sites. Can be activated (by + effectors) or inhibited (by – effectors). Changes in activity result from changes in structure.

One model that explains the behavior of allosteric enzymes is the concerted model. Features of the concerted model: The enzyme exists into two different quaternary structures, designated T(tense) and R (relaxed). T and R are in equilibrium, with T being the more stable state. The R state is enzymatically more active than the T state. All active sites must be in the same state. The binding of substrate to one active site traps the other active sites in the R state and removes the substrate-bound enzyme from the T<->R equilibrium. This disruption of the T<->R equilibrium by the binding of substrate favors the conversion of more enzymes to the R state.

Figure 7. 10 The concerted model for allosteric enzymes Figure 7.10 The concerted model for allosteric enzymes. (A) [T] >>> [R], meaning that L0 is large. Consequently, it will be difficult for S to bind to an R form of the enzyme. (B) As the concentration of S increases, it will bind to one of the active sites on R, trapping all of the other active sites in the R state (by the symmetry rule.) (C) As more active sites are trapped in the R state, it becomes easier for S to bind to the R state. (D) The binding of S to R becomes easier yet as more of the enzyme is in the R form. In a velocity-versus-[S] curve, V0 will be seen to rise rapidly toward Vmax.

Allosteric regulators change the R<->T equilibrium (T/R ratio) when they bind to the enzyme’s allosteric, or regulatory, sites. Inhibitors (– effectors) stabilize the T state while activators (+ effectors) stabilize the R state.

Figure 7.11 The effect of regulators on the allosteric enzyme aspartate transcarbamoylase. (A) ATP is an allosteric activator of aspartate transcarbamoylase because it stabilizes the R state, making it easier for substrate to bind. As a result, the curve is shifted to the left, as shown in blue. (B) Cytidine triphosphate (CTP) stabilizes the T state of aspartate transcarbamoylase, making it more difficult for substrate binding to convert the enzyme into the R state. As a result, the curve is shifted to the right, as shown in red.

The sequential model for allosteric enzymes proposes that subunits undergo sequential changes in structure.

sequential model represents cooperativity Main Feature of Model: the binding of substrate induces a conformational change from the T form to the R form the change in conformation is induced by the fit of the substrate to the enzyme, i.e., through the induced-fit model of substrate binding sequential model represents cooperativity Figure 7.12 The sequential model. The binding of a substrate (S) changes the conformation of the subunit to which it binds. This conformational change induces changes in neighboring subunits of the allosteric enzyme that increase their affinity for the substrate. The K1, K2, etc., represent rate constants for the binding of substrate to the different states of the enzyme.

Phosphoribosylpyrophosphate synthetase (PRS) is an allosteric enzyme in the purine nucleotide synthesis pathway. A mutation leading to the loss of regulatory control without an effect on catalytic activity leads to the overproduction of purine nucleotides. The overproduction results in the painful disease gout.

Figure 7.13 A gout-inflamed joint. [Medical-on-Line/Alamy Images.]

Helpful Questions Quick quizzes 1 and 2; #3, 4, 7, 10, 11 (except for h and i), 15, 20, 21, 22, 23 Use graph paper, not notebook paper, for the L-B plots; scale axes appropriately; don’t try to get exactly same answer as text.