12 Feb 2008 Enzyme Kinetics and Inhibition Andy Howard Introductory Biochemistry, Spring February 2008

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 2 of 65 What’s the relationship between concentration and reaction rate? In the presence of an enzyme, there is a definable relationship between substrate concentration and reaction rate.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 3 of 65 Plans for Today Enzyme kinetics Michaelis-Menten kinetics: overview Kinetic Constants Kinetic Mechanisms Induced Fit Measurements and calculational tools Multisubstrate reactions Enzyme Inhibition Why study it? The concept Types of inhibitors Kinetics of inhibition Pharmaceutical inhibitors What makes an inhibitor a useful drug?

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 4 of 65 More complex cases More complicated than this if >1 reactant involved or if a catalyst whose concentration influences the production of species B is present. If >1 reactant required for making B, then usually the reaction will be linear in the concentration of the scarcest reactant and nearly independent of the concentration of the more plentiful reactants.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 5 of 65 Bimolecular reaction If in the reaction A + D  B the initial concentrations of [A] and [D] are comparable, then the reaction rate will be linear in both [A] and [D]: d[B]/dt = v = k[A][D] = k[A] 1 [D] 1 I.e. the reaction is first-order in both A and D, and it’s second-order overall

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 6 of 65 Forward and backward Rate of reverse reaction may not be the same as the rate at which the forward reaction occurs. If the forward reaction rate of reaction 1 is designated as k 1, the backward rate typically designated as k -1.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 7 of 65 Multi-step reactions In complex reactions, we may need to keep track of rates in the forward and reverse directions of multiple reactions. Thus in the conversion A  B  C we can write rate constants k 1, k -1, k 2, and k -2 as the rate constants associated with converting A to B, converting B to A, converting B to C, and converting C to B.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 8 of 65 Michaelis-Menten kinetics A very common situation is one in which for some portion of the time in which a reaction is being monitored, the concentration of the enzyme-substrate complex is nearly constant. Thus in the general reaction E + S  ES  E + P where E is the enzyme, S is the substrate, ES is the enzyme-substrate complex (or "enzyme- intermediate complex"), and P is the product We find that [ES] is nearly constant for a considerable stretch of time. [ES] t

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 9 of 65 Michaelis-Menten rates Rate at which new ES molecules are being produced in the first forward reaction is equal to the rate at which ES molecules are being converted to (E and P) and (E and S). Rate of formation of ES from left = v f = k 1 ([E] tot - [ES])[S] because the enzyme that is already substrate-bound is unavailable!

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 10 of 65 Equating the rates Rate of disappearance of ES on right and left is v d = k -1 [ES] + k 2 [ES] = (k -1 + k 2 )[ES] This rate of disappearance should be equal to the rate of appearance Under these conditions v f = v d.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 11 of 65 Derivation, continued Thus since v f = v d. k 1 ([E] tot - [ES])[S] = (k -1 + k 2 )[ES] K m  (k -1 + k 2 )/k 1 = ([E] tot - [ES])[S] / [ES] [ES] = [E] tot [S] / (K m + [S]) But the rate-limiting reaction is the formation of product: v 0 = k 2 [ES] Thus v 0 = k 2 [E] tot [S] / (K m + [S])

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 12 of 65 Maximum velocity What conditions would produce the maximum velocity? Answer: very high substrate concentration ([S] >> [E] tot ), for which all the enzyme would be bound up with substrate. Thus under those conditions we get V max = v 0 = k 2 [ES] = k 2 [E] tot

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 13 of 65 Using V max in M-M kinetics Thus since V max = k 2 [E] tot, v 0 = V max [S] / (K m +[S]) That’s the famous Michaelis-Menten equation

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 14 of 65 Graphical interpretation

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 15 of 65 Physical meaning of K m As we can see from the plot, the velocity is half-maximal when [S] = K m Trivially derivable: if [S] = K m, then v 0 = V max [S] / ([S]+[S]) = V max /2 We can turn that around and say that the K m is defined as the concentration resulting in half-maximal velocity K m is a property associated with binding of S to E, not a property of turnover

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 16 of 65 Physical meaning of V max V max is the initial reaction velocity when the substrate concentration is high compared to K m : v 0 = V max when [S] >> K m. This result follows directly from Michaelis-Menten equation: if [S] >> K m, then [S] + K m ~ [S], so [S]/(K m +[S]) = 1, and v 0 = V max.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 17 of 65 k cat A quantity we often want is the maximum velocity independent of how much enzyme we originally dumped in That would be k cat = V max / [E] tot Oh wait: that’s just the rate of our rate-limiting step, i.e. k cat = k 2

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 18 of 65 What do V max and k cat tell us? These are properties associated with the enzyme’s ability to turn over substrate, not its ability to bind substrate. So if we influence an enzyme’s substrate affinity without altering its ability to convert substrate to product, we won’t change V max or k cat.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 19 of 65 Physical meaning of k cat Describes turnover of substrate to product: Number of product molecules produced per sec per molecule of enzyme More complex reactions may not have k cat = k 2, but we can often approximate them that way anyway Some enzymes very efficient: k cat > 10 6 s -1

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 20 of 65 Specificity constant, k cat /K m k cat /K m measures affinity of enzyme for a specific substrate: we call it the specificity constant for the enzyme for that particular substrate Useful in comparing primary substrate to other substrates (e.g. ethanol vs. propanol in alcohol dehydrogenase)

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 21 of 65 Kinetic Mechanisms If a reaction involves >1 reactant or >1 product, there may be variations in kinetics that occur as a result of the order in which substrates are bound or products are released. Examine figure 5.7 in Horton, which depicts bisubstrate reactions of various sorts. As you can see, the possibilities enumerated include sequential, random, and ping-pong mechanisms.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 22 of 65 Historical thought Biochemists, examined effect on reaction rates of changing [reactants] and [enzymes], and deducing the mechanistic realities from kinetic data. In recent years other tools have become available for deriving the same information, including static and dynamic structural studies that provide us with slide-shows or even movies of reaction sequences. But things like fig. 5.7 still help!

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 23 of 65 Sequential, ordered reactions Substrates, products must bind in specific order for reaction to complete A B P Q _____________________________ E EA (EAB) (EPQ) EQ E W.W.Cleland

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 24 of 65 Sequential, random reactions Substrates can come in in either order, and products can be released in either order A B P Q EA EQ __ E (EAB)(EPQ) E EB EP B A Q P

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 25 of 65 Ping-pong mechanism First substrate enters, is altered, is released, with change in enzyme Then second substrate reacts with altered enzyme, is altered, is released Enzyme restored to original state A P B Q E EA FA F FB FQ E

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 26 of 65 Historical thought Biochemists, examined effect on reaction rates of changing [reactants] and [enzymes], and deducing the mechanistic realities from kinetic data. In recent years other tools have become available for deriving the same information, including static and dynamic structural studies that provide us with slide-shows or even movies of reaction sequences. But things like fig. 5.7 still help!

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 27 of 65 Induced fit Conformations of enzymes don't change enormously when they bind substrates, but they do change to some extent. An instance where the changes are fairly substantial is the binding of substrates to kinases. Danger:enzyme will catalyze the unproductive hydrolysis of ATP in the same site where the kinase reaction might occur Daniel Koshland

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 28 of 65 Kinase reactions unwanted reaction ATP + H-O-H ⇒ ADP + P i will compete with the desired reaction ATP + R-O-H ⇒ ADP + R-O- P Kinases minimize the likelihood of this unproductive activity by changing conformation upon binding substrate so that hydrolysis of ATP cannot occur until the binding happens. Illustrates the importance of the order in which things happen in enzyme function

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 29 of 65 Measurements and calculations The standard Michaelis-Menten formulation is v 0 =f([S]), but it’s not linear in [S]. We seek linearizations of the equation so that we can find K m and k cat, and so that we can understand how various changes affect the reaction.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 30 of 65 Lineweaver-Burk Simple linearization of Michaelis-Menten: v 0 = V max [S]/(K m +[S]). Take reciprocals: 1/v 0 = (K m +[S])/(V max [S]) = (K m /(V max [S])) + [S]/(V max [S])) = (K m /V max )*1/[S] + 1/V max Thus a plot of 1/[S] as the independent variable vs. 1/v 0 as the dependent variable will be linear with Y-intercept = 1/V max and slope K m /V max Dean Burk Hans Lineweaver

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 31 of 65 How to use this Y-intercept is useful directly: computeV max = 1/(Y-intercept) We can get K m /V max from slope and then use our knowledge of V max to get K m ; or X intercept = -1/ K m … that gets it for us directly!

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 32 of 65 Demonstration that the X-intercept is at -1/K m X-intercept means Y = 0 In Lineweaver-Burk plot, 0 = (K m /V max )*1/[S] + 1/V max For nonzero 1/V max we divide through: 0 = K m /[S] + 1, -1 = K m /[S], [S] = -K m. But the axis is for 1/[S], so the intercept is at 1/[S] = -1/ K m.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 33 of 65 Graphical form of L-B 1/[S], M -1 1/v 0, sLmol -1 1/V max, sLmol -1 -1/K m, Lmol -1 Slope=K m /V max

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 34 of 65 Are those values to the left of 1/[S] = 0 physical? No. It doesn’t make sense to talk about negative substrate concentrations or infinite substrate concentrations. But if we can curve-fit, we can still use these extrapolations to derive the kinetic parameters.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 35 of 65 Advantages and disadvantages of L-B plots Easy conceptual reading of K m and V max (but remember to take the reciprocals!) Suboptimal error analysis [S] and v 0 values have errors Error propagation can lead to significant uncertainty in K m (and V max ) Other linearizations available (see homework) Better ways of getting K m and V max available

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 36 of 65 Don’t fall into the trap! When you’re calculating K m and V max from Lineweaver-Burk plots, remember that you need the reciprocal of the values at the intercepts If the X-intercept is M -1, then K m = -1/(X-intercept) =(-)(-1/5000 M -1 ) = 2*10 -4 M Sanity check: typically M < K m < M Typically k cat ~ 10 to 10 7 s-1 (table 5.1), so for typical [E]=10 -7 M, V max = [E]k cat = Ms -1 to 1 Ms -1

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 37 of 65 iClicker quiz 1. The kinase reaction just described probably operates according to a (a) sequential, random mechanism (b) sequential, ordered mechanism ( c) ping-pong mechanism (d) none of the above.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 38 of 65 iClicker quiz 2. If we alter the kinetics of a reaction by increasing K m but leaving V max alone, how will the L-B plot change? (a) X intercept will move toward the origin and the Y intercept will remain as it was (b) X intercept will move away from origin and Y intercept will remain as it was ( c) Y intercept will move away from origin and X intercept will remain as it was (d) Y intercept will move toward origin and X intercept will remin as it was (e) None of the above

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 39 of 65 L-B plots for ordered sequential reactions kinetics/Chapter_4/chapter4_3.html kinetics/Chapter_4/chapter4_3.html Plot 1/v 0 vs. 1/[A] for various [B] values; flatter slopes correspond to larger [B] Lines a point in between X intercept and Y intercept

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 40 of 65 L-B plots for ping- pong reactions Again we plot 1/v vs 1/[A] for various [B] Parallel lines (same k cat /K m ); lower lines correspond to larger [B] Chapter_4/chapter4_3_2.html Chapter_4/chapter4_3_2.html

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 41 of 65 Why study inhibition? Let’s look at how enzymes get inhibited. At least two reasons to do this: We can use inhibition as a probe for understanding the kinetics and properties of enzymes in their uninhibited state; Many—perhaps most—drugs are inhibitors of specific enzymes. We'll see these two reasons for understanding inhibition as we work our way through this topic.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 42 of 65 The concept of inhibition An enzyme is a biological catalyst, i.e. a substance that alters the rate of a reaction without itself becoming permanently altered by its participation in the reaction. The ability of an enzyme (particularly a proteinaceous enzyme) to catalyze a reaction can be altered by binding small molecules to it sometimes at its active site sometimes at a site distant from the active site.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 43 of 65 Inhibitors and Accelerators Usually these alterations involve a reduction in the enzyme's ability to accelerate the reaction; less commonly, they give rise to an increase in the enzyme's ability to accelerate a reaction.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 44 of 65 Why more inhibitors than accelerators? Natural selection: if there were small molecules that can facilitate the enzyme's propensity to speed up a reaction, nature probably would have found a way to incorporate those facilitators into the enzyme over the billions of years that the enzyme has been available. Most enzymes are already fairly close to optimal in their properties; we can readily mess them up with effectors, but it's more of a challenge to find ways to make enzymes better at their jobs.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 45 of 65 Types of inhibitors Irreversible Inhibitor binds without possibility of release Usually covalent Each inhibition event effectively removes a molecule of enzyme from availability Reversible Usually noncovalent (ionic or van der Waals) Several kinds Classifications somewhat superseded by detailed structure-based knowledge of mechanisms, but not entirely

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 46 of 65 Types of reversible inhibition Competitive Inhibitor binds at active site Prevents binding of substrate Noncompetitive Inhibitor binds distant from active site Interferes with turnover Uncompetitive (rare?) Inhibitor binds to ES complex Removes ES, interferes with turnover Mixed (usually Competitive + Noncompetitive)

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 47 of 65 How to tell them apart Reversible vs irreversible dialyze an enzyme-inhibitor complex against a buffer free of inhibitor if turnover or binding still suffers, it’s irreversible Competitive vs. other reversible: Structural studies if feasible Kinetics

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 48 of 65 Competitive inhibition Put in a lot of substrate: ability of the inhibitor to get in the way of the binding is hindered: out-competed by sheer #s of substrate molecules. How many substrate molecules it will take to overwhelm the inhibitor depends on how strongly the enzyme is attracted to the substrate as compared to the attraction of the enzyme for the inhibitor. However, once the substrate is bound, the inhibitor does not influence how quickly the enzyme turns the substrate over.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 49 of 65 Kinetics of competition Competitive inhibitor hinders binding of substrate but not reaction velocity: Affects the K m of the enzyme, not V max. Which way does it affect it? K m = amount of substrate that needs to be present to run the reaction velocity up to half its saturation velocity. Competitive inhibitor requires us to shove more substrate into the reaction in order to achieve that half-maximal velocity. So: competitive inhibitor increases K m,

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 50 of 65 L-B: competitive inhibitor K m goes up so -1/ K m moves toward origin V max unchanged so Y intercept unchanged

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 51 of 65 Competitive inhibitor: Quantitation of K i Define inhibition constant K i to be the concentration of inhibitor that increases K m by a factor of two. K m,obs = K m (1+[ I c ]/K i ) So [ I c ] that moves K m halfway to the origin is K i. If K i = 100 nM and [ I c ] = 1 µM, then we’ll increase K m,obs elevenfold!

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 52 of 65 Noncompetitive inhibition Noncompetitive inhibitor has no influence on how available the binding site for substrate is, so it does not affect K m at all However, it has a profound inhibitory influence on the speed of the reaction, i.e. turnover. So it reduces V max and has no influence on K m. S I

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 53 of 65 L-B for non-competitives Decrease in V max  1/V max is larger X-intercept unaffected

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 54 of 65 K i for noncompetitives K i defined as concentration of inhibitor that cuts V max in half V max,obs = V max /(1 + [ I n ]/K i ) In previous figure the “high” concentration of inhibitor is K i

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 55 of 65 Uncompetitive inhibition Inhibitor binds only if ES has already formed It creates a ternary ESI complex This removes ES, so by LeChatlier’s Principle it actually drives the original reaction (E + S  ES) to the right; so it decreases K m But it interferes with turnover so V max goes down If K m and V max decrease at the same rate, then it’s classical uncompetitive inhibition.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 56 of 65 L-B for uncompetitives K m moves toward origin V max moves away from the origin Slope (  K m /V max ) is unchanged

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 57 of 65 K i for uncompetitives Defined as inhibitor concentration that cuts V max or K m in half Easiest to read from V max value I u labeled “high” is K i in this plot

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 58 of 65 iClicker quiz! 3. Treatment of enzyme E with compound Y doubles K m and leaves V max unchanged. Compound Y is: (a) an accelerator of the reaction (b) a competitive inhibitor (c) a non-competitive inhibitor (d) an uncompetitive inhibitor

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 59 of 65 iClicker quiz, question 4 4. Treatment of enzyme E with compound X doubles V max and leaves K m unchanged. Compound X is: (a) an accelerator of the reaction (b) a competitive inhibitor (c) a non-competitive inhibitor (d) an uncompetitive inhibitor

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 60 of 65 Most pharmaceuticals are enzyme inhibitors Inhibitors of enzymes that are necessary for functioning of pathogens Others are inhibitors of some protein whose inappropriate expression in a human causes a disease. Others are targeted at enzymes that are produced more energetically by tumors than they are by normal tissues.

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 61 of 65 Characteristics of Pharmaceutical Inhibitors Usually competitive, i.e. they raise K m without affecting V max Some are mixed, i.e. K m up, V max down Iterative design work will decrease K i from millimolar down to nanomolar Sometimes design work is purely blind HTS; other times, it’s structure-based

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 62 of 65 Amprenavir Competitive inhibitor of HIV protease, K i = 0.6 nM for HIV-1 No longer sold: mutual interference with rifabutin, which is an antibiotic used against a common HIV secondary bacterial infection, Mycobacterium avium

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 63 of 65 When is a good inhibitor a good drug? It needs to be bioavailable and nontoxic Beautiful 20nM inhibitor is often neither Modest sacrifices of K i in improving bioavailability and non-toxicity are okay if K i is low enough when you start sacrificing

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 64 of 65 How do we lessen toxicity and improve bioavailability? Increase solubility… that often increases K i because the van der Waals interactions diminish Solubility makes it easier to get the compound to travel through the bloodstream Toxicity is often associated with fat storage, which is more likely with insoluble compounds

12 Feb 2008 Biochemistry: Enzyme Kinetics & inhibition P. 65 of 65 Drug-design timeline 2 years of research, 8 years of trials log K i Time, Yrs Cost/yr, 10 6 $ Improving affinity Toxicity and bioavailability Research Clinical Trials Preliminary toxicity testing Stage I clinical trials Stage II clinical trials