1. Enzymes: Kinetics & Inhibition
Andy Howard Biochemistry Lectures, Spring 2019 Thursday 14 February 2019

2. Enzyme Kinetics & Inhibition
After we finish describing Michaelis-Menten kinetics, we’ll introduce the notion of enzyme inhibition and its significance to pharmaceutical projects.
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Enzyme Kinetics & Inhibition

3. Enzyme Kinetics & Inhibition
Topics for today
M-M kinetics
kcat & meanings
Characteristic values
Bisubstrate reactions
Calculations
Inhibition
Concept
Irreversible
Reversible …
Kinetics
Pharmaceuticals
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Enzyme Kinetics & Inhibition

4. Enzyme Kinetics & Inhibition
kcat
A quantity we often want is the maximum velocity independent of how much enzyme we originally dumped in
That would be kcat = Vmax / [E]tot
Oh wait: that’s just the rate of our rate-limiting step, i.e. kcat = k2
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5. Physical meaning of kcat
Describes turnover of substrate to product: Number of product molecules produced per sec per molecule of enzyme
More complex reactions may not have kcat = k2, but we can often approximate them that way anyway
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6. Enzymes with large kcat values
Some enzymes very efficient: kcat > 106 s-1 (catalase: 4*107 s-1: 2H2O2 -> 2H2O + O2)
High values of kcat indicate rapid turnover; small values indicate slow turnover
Micrococcus catalase 58 kDa monomer EC PDB 1GWE, 0.88Å
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7. Specificity constant, kcat/Km
kcat/Km measures affinity of enzyme for a specific substrate: we call it the specificity constant or the molecular activity for the enzyme for that particular substrate
Useful in comparing primary substrate to other substrates (e.g. ethanol vs. propanol in alcohol dehydrogenase)
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8. Enzyme Kinetics & Inhibition
Meaning of kcat/Km
High values of kcat/Km indicate strong affinity for the substrate; low values indicate weak affinity.
High values mean near-diffusion-limited conditions
TIM kcat=4300 s-1, Km=20 µM; kcat / Km = 2.4*108 s-1M-1
Leishmania triosphosphate isomerase 56 kDa dimer; monomer shown EC PDB 2VXN, 0.82Å
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9. Dimensions of Km and Vmax
Km must have dimensions of concentration (remember it corresponds to the concentration of substrate that produces half-maximal velocity)
Vmax must have dimensions of concentration over time (d[P]/dt)
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10. Dimensions: kcat and kcat/Km
kcat must have dimensions of inverse time
kcat / Km must have dimensions of inverse time divided by concentration, i.e. inverse time * inverse concentration
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11. Typical units for kinetic parameters
Remember the distinction between dimensions and units!
Km typically measured in mM or µM
Vmax typically measured in mM s-1 or µM s-1
kcat typically measured in s-1
kcat / Km typically measured in (mM )-1 s-1
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12. Characteristic values of Km and kcat
A good way to ensure that you’re doing kinetics problems correctly is to make sure that the values you get correspond to physical realities
Km for the biologically relevant substrate is typically between 1 µM and 10 mM
kcat is typically s-1
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13. Enzyme Kinetics & Inhibition
Sanity checks
kcat/Km is therefore typically ~ M-1s-1
Vmax = kcat*[E]tot is typically ~ 10-4 Ms-1
If you get answers outside that range, you’ve probably done something wrong!
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14. Bisubstrate reactions
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.
The possibilities enumerated include sequential, random, and ping-pong mechanisms.
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15. Enzyme Kinetics & Inhibition
Historical thought
Biochemists, examined effect on reaction rates of changing [reactant] and [enzyme], and deducing the mechanistic realities from kinetic data.
Now other tools are 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 diagrams like these still help!
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16. Sequential, ordered reactions
W.W.Cleland
Substrates, products must bind in specific order for reaction to complete A B P Q _____________________________ E EA (EAB) (EPQ) EQ E
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17. Sequential, random reactions
Substrates can come in in either order, and products can be released in either order
Example: creatine kinase
Human B-type CK 86kDa dimer EC PDB 3B6R, 2Å
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18. Enzyme Kinetics & Inhibition
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 FP F FB FQ F E
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19. Enzyme Kinetics & Inhibition
Induced fit
Daniel Koshland
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.
Cartoon from textbookofbacteriology.net
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20. Enzyme Kinetics & Inhibition
Kinase reactions
unwanted reaction ATP + H-O-H ⇒ ADP + Pi
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
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21. Hexokinase conformational changes
G&G Fig
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22. Measurements and calculations
The standard Michaelis-Menten formulation is v0=f([S]), but it’s not linear in [S]. We seek linearizations of the equation so that we can find Km and kcat, and so that we can understand how various changes affect the reaction.
Enzyme Kinetics & Inhibition
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23. Enzyme Kinetics & Inhibition
Lineweaver-Burk
Simple linearization of Michaelis-Menten:
v0 = Vmax[S]/(Km+[S]). Take reciprocals:
1/v0 = (Km +[S])/(Vmax[S]) = (Km /(Vmax[S])) + [S]/(Vmax[S])) = (Km/Vmax)*1/[S] + 1/Vmax
Thus: plot of 1/[S] as independent variable vs. 1/v0 as dependent variable is linear: Y-intercept = 1/Vmax and slope Km/Vmax
Dean Burk
Hans Lineweaver
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24. Enzyme Kinetics & Inhibition
How to use this
Y-intercept is useful directly: computeVmax = 1/(Y-intercept)
We can get Km/Vmax from slope and then use our knowledge of Vmax to get Km; or
X intercept = -1/ Km … that gets it for us directly!
Enzyme Kinetics & Inhibition
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25. Demonstration that the X-intercept is at -1/Km
X-intercept means Y = 0
In Lineweaver-Burk plot,
0 = (Km/Vmax)*1/[S] + 1/Vmax
For nonzero 1/Vmax we divide through:
0 = Km /[S] + 1, -1 = Km/[S], [S] = -Km.
But the axis is for 1/[S], so the intercept is at 1/[S] = -1/ Km.
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26. Enzyme Kinetics & Inhibition
Graphical form of L-B
1/v0, s L mol-1
1/Vmax, s L mol-1
Slope=Km/Vmax
1/[S], M-1
-1/Km, L mol-1
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27. 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.
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28. Advantages and disadvantages of L-B plots
Easy conceptual reading of Km and Vmax (but remember to take the reciprocals!)
Suboptimal error analysis
[S] and v0 values have errors
Error propagation can lead to significant uncertainty in Km (and Vmax)
Other linearizations available (see homework)
Better ways of getting Km and Vmax available
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29. Don’t fall into the trap!
When you’re calculating Km and Vmax from Lineweaver-Burk plots, remember that you need the reciprocal of the values at the intercepts
If the X-intercept is M-1, then Km = -1/(X-intercept) =(-)(-1/5000 M-1) = 2*10-4M
Similarly, if the Y-intercept is 2000 sM-1, then Vmax = 5*10-4 Ms-1 and if [E]tot=10-7M, then kcat = 5*103 s-1
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30. Sanity checks, revisited
Sanity check #1: typically 10-7M < Km < 10-2M (cf. table 5.2)
Typically kcat ~ 0.5 to 107 s-1 (table 5.1), so for typical [E]=10-7M, Vmax = [E]totkcat = 10-6 Ms-1 to 1 Ms-1
If you get Vmax or Km values outside of these ranges, you’ve probably done something wrong
… and please don’t try to tell me that Km < 0!
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31. iClicker quiz question #1
1. If we alter the kinetics of a reaction by increasing Km but leaving Vmax alone, how will the L-B plot change?
Answer
X-intercept
Y-intercept a
Moves toward origin
Unchanged b
Moves away from origin c d
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32. Enzyme Kinetics & Inhibition
iClicker question 2
2. Enzyme E has a tenfold stronger affinity for substrate A than for substrate B. Which of the following is true?
(a) Km(A) = 10 * Km(B)
(b) Km(A) = 0.1 * Km(B)
(c) Vmax(A) = 10 * Vmax(B)
(d) Vmax(A) = 0.1 * Vmax(B)
(e) None of the above.
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33. Another physical significance of Km
Years of experience have led biochemists to a general conclusion:
For its preferred substrate, the Km value of an enzyme is usually within a factor of 50 of the steady-state concentration of that substrate.
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34. Enzyme Kinetics & Inhibition
What this means
So if we find that Km = 0.2 mM for the primary substrate of an enzyme, then we expect that the steady-state concentration of that substrate is between 4 µM and 10 mM.
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35. Example: hexokinase isozymes
Mutant human type I hexokinase
PDB 1DGK, 2.8Å 110 kDa monomer
Hexokinase catalyzes hexose + ATP  hexose-6-P + ADP
Most isozymes of hexokinase prefer glucose; some also work okay mannose and fructose
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36. Muscle vs. liver isozymes
Muscle hexokinases have Km ~ 0.1mM so they work efficiently in blood, where [glucose] ~ 4 mM
Liver glucokinase has Km = 10 mM, which is around the liver [glucose] and can respond to fluctuations in liver [glucose]
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37. L-B plots for ordered sequential reactions
1/v = (1/Vmax)(KmA+KsAKmB/[B])(1/[A])
+ (1/Vmax)(1+KmB/[B])
L-B plots for ordered sequential reactions
Plot 1/v0 vs. 1/[A] for various [B] values; flatter slopes correspond to larger [B]
Lines a point in between X intercept and Y intercept (= -1/KSA)
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38. L-B plots for ping-pong reactions
1/v = (KmA/Vmax)(1/[A]) + (1+KmB/[B])/(1/Vmax)
L-B plots for ping-pong reactions
Again we plot 1/v vs 1/[A] for various [B]
Parallel lines (same kcat/Km); lower (rightmore) lines correspond to larger [B]
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39. Enzyme Kinetics & Inhibition
Why study inhibition?
Let’s look at how enzymes get inhibited.
At least two reasons to do this:
Use inhibition as a probe for understanding kinetics & properties of enzymes themselves
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.
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40. 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.
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41. How to inhibit an enzyme
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.
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42. 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.
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43. 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.
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44. Enzyme Kinetics & Inhibition
Additional reason
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.
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45. Irreversible inhibitors
Inhibitor binds without possibility of release
Usually covalent
Each inhibition event effectively removes a molecule of enzyme from availability
Example: diisopropylfluorophosphate for serine proteases
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46. Reversible inhibitors
Usually noncovalent (ionic or van der Waals)
Several kinds
Classifications somewhat superseded by detailed structure-based knowledge of mechanisms, but not entirely
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47. Types of reversible inhibition: I
Competitive
Inhibitor binds at active site
Prevents binding of substrate
Noncompetitive
Inhibitor binds distant from active site
Interferes with turnover
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48. Types of reversible inhibition: II
Uncompetitive (rare?)
Inhibitor binds to ES complex
Removes ES, interferes with turnover
Mixed (usually Competitive + Noncompetitive)
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49. Putting that all together…
Ligands that influence enzyme activity
Accelerators
Inhibitors
(Usually allosteric)
Irreversible
Reversible
(Usually covalent)
Noncompetitive (often allosteric)
Competitive
Mixed
Uncompetitive
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50. Enzyme Kinetics & Inhibition
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
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51. 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.
This kind of inhibition manifests itself as interference with binding, i.e. with an increase of Km
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52. Competitive inhibitors don’t affect turnover
Within the active site of any given molecule of the enzyme, one of three states are possible:
The inhibitor is present
The substrate is present
Nothing is present
Therefore rate of turnover isn’t affected by the inhibitor: just the availability of binding sites.
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53. Kinetics of competition
Competitive inhibitor hinders binding of substrate but not reaction velocity:
Affects the Km of the enzyme, not Vmax.
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54. Effect of competitive inhibitor
Which way does it affect it?
Km = 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 Km
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55. L-B: competitive inhibitor
Km goes up so -1/ Km moves toward origin
Vmax unchanged so Y intercept unchanged
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56. Competitive inhibitor: Quantitation of KI
Define inhibition constant KI to be the concentration of inhibitor that increases Km by a factor of two.
Then Km,obs = Km{1+([Ic]/KI)}
So [Ic] that moves Km halfway to the origin is KI.
That corresponds to saying that KI is the value of [Ic] that doubles the Km.
If Ki = 100 nM and [Ic] = 1 µM, then we’ll increase Km,obs elevenfold!
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57. Enzyme Kinetics & Inhibition
Sample problem
Suppose a competitive inhibitor with KI = 5 µM is competing with a substrate for which Km = 100 µM, then the effective Km is Km, obs = Km {1 + ([Ic] / KI)}
Thus if [Ic] = 20 µM, Km, obs = Km{1 + ([Ic] / KI)} = 100 µM {1 + (20 µM/5µM)} = 100 µM (1 + 4) = 100 µM (5) = 500 µM
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58. Noncompetitive inhibitionS
Noncompetitive inhibitor has no influence on how available the binding site for substrate is, so it doesn’t affect Km at all
However, it has a profound inhibitory influence on the speed of the reaction, i.e. turnover. So it reduces Vmax and has no influence on Km.
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59. L-B for non-competitives
Decrease in Vmax  1/Vmax is larger
X-intercept unaffected
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60. KI for noncompetitives
KI defined as concentration of inhibitor that cuts Vmax in half
Vmax,obs = Vmax/{1 + ([In]/KI)}
In previous figure the “high” concentration of inhibitor is KI
If KI = KI’, this is pure noncompetitive inhibition
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61. Uncompetitive inhibition
Inhibitor binds only if ES has already formed
It creates a ternary ESI complex
This removes ES, so by LeChatelier’s Principle it actually drives the original reaction (E + S  ES) to the right; so it decreases Km
But it interferes with turnover so Vmax goes down
If Km and Vmax decrease at the same rate, then it’s classical uncompetitive inhibition.
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62. L-B for uncompetitives
-1/Km moves away from the origin
1/Vmax moves away from the origin
Slope ( Km/Vmax) is unchanged
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63. Enzyme Kinetics & Inhibition
Ki for uncompetitives
Defined as inhibitor concentration that cuts Vmax or Km in half
Easiest to read from Vmax value
Same equation as for noncompetitves: Vmax,obs = Vmax / {1 + ([Iu]/KI)}
Iu labeled “high” is equal to Ki in this plot
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64. Enzyme Kinetics & Inhibition
Mixed inhibition
Usually involves interference with both binding and catalysis
Km goes up, Vmax goes down
Easy to imagine the mechanism:
Binding of inhibitor alters the active-site configuration to interfere with binding, but it also alters turnover
Same picture as with pure noncompetitive inhibition, but with Ki ≠ Ki’
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65. Warning: this terminology is becoming challenged
Some biochemists say pure noncompetitive or pure uncompetitive inhibition are rare
Pure noncompetitive inhibition implies that the binding of the inhibitor has no influence on binding of substrate; these people say that it’s more common for KI  KI’.
I don’t agree, and in any case the approach we’ve taken helps you visualize realities.
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66. Enzyme Kinetics & Inhibition
iClicker question 3
3. If we compare the Lineweaver-Burk plot of an uninhibited enzyme to that of a noncompetitively inhibited enzyme, what will change?
(a) The X-intercept
(b) The Y-intercept
(c) The slope
(d) both (a) and (c)
(e) both (b) and (c)
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67. Most pharmaceuticals are enzyme inhibitors
Some are inhibitors of enzymes that are necessary for functioning of pathogens
Others are inhibitors of a protein whose inappropriate expression in a human causes disease.
Others are targeted at enzymes that are produced more energetically by tumors than they are by normal tissues.
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68. Characteristics of Pharmaceutical Inhibitors
Usually competitive, i.e. they raise Km without affecting Vmax
Some are mixed, i.e. Km up, Vmax down
A few are allosteric and noncompetitive
Iterative design work will decrease Ki from millimolar down to nanomolar
Sometimes design work is purely blind HTS; other times, it’s structure-based
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69. Enzyme Kinetics & Inhibition
Amprenavir
Competitive inhibitor of HIV protease, Ki = 0.6 nM for HIV-1
HIV-2 protease with amprenavir bound 23 kDa dimer EC PDB 3S45, 1.5 Å
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70. Collision with rifabutin
No longer sold: mutual interference with rifabutin, which is an antibiotic used against a common HIV secondary bacterial infection, Mycobacterium avium
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71. Pharmaceutical examples
Trade Generic Enzyme Action Name Name Inhibited Viagra Sildenafil PDE5 cGMP remains active Aspirin COX 1&2 Prostaglandins slowed Lipitor Atorvastatin HMGCoAR Cholesterol slowed Crestor Rosuvastatin HMGCoAR Cholesterol slowed Nexium Esomeprazole ATPase Stomach acid slowed Januvia Sitagliptin DPP-4 Increases insulin Atripla Efavirenz HIV RT HIV can’t replicate Celebrex Celecoxib COX-2 Prostaglandins slowed
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72. Biggest-selling drugs!
adalimumab
Biggest-selling drugs!
Five of the top 12 sellers worldwide are monoclonal antibodies
Another biologic (i.e. macromolecule) is a TNF inhibitor used for rheumatoid arthritis
Two interfere with replication in hepatitis C virus by incorporation into the genome
Some have complex or poorly understood mechanisms of action
Net sales correlate poorly with social importance
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73. When is a good inhibitor a good drug?
It needs to be bioavailable and nontoxic
Beautiful 20nM inhibitor is often neither
Modest sacrifices of Ki in improving bioavailability and non-toxicity are okay if Ki is low enough when you start sacrificing
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74. How do we lessen toxicity and improve bioavailability?
Increase solubility… that often increases Ki 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
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75. Enzyme Kinetics & Inhibition
2 years of research, 8 years of trials
Drug-design timeline
100 -3
Improving affinity
Toxicity and bioavailability
Stage I clinical trials
Stage II clinical trials
Cost/yr, 106 $
Preliminary toxicity testing
log Ki -8 10
Research
Clinical Trials 2
Time, Yrs 10
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