1. Biochemistry 412
Enzyme Kinetics I
March 29th, 2005

2. Reading:
Mathews & van Holde, Biochemistry, Benjamin/Cummings Publishing Co., Redwood City, CA, pp in 1990 edition (or equivalent pages in a later edition)
Other (optional) resources:
>>> And special thanks for this lecture goes to Dr. Gabriel Fenteany, Department of Chemistry, University of Illinois at Chicago ( whose slides I liberally borrowed!

3. Enzymes Are Uniquely Powerful Catalysts
Enzymes are proteins that can accelerate biochemical reactions often by factors of 106 to 1012! This is much higher than chemical catalysts.
Enzymes can be extremely specific in terms of reaction substrates and products.
Enzymes catalyze reactions under mild conditions (e.g. pH 7.4, 37ºC).
The catalytic activities of many enzymes can be regulated by allosteric effectors.

4. For example:
Triose Phosphate Isomerase

5. And…

6. Chemical Kinetics

7. Irreversible First-Order Reactionsk
A  B
v = d[B]/dt = -d[A]/dt = k[A]
(k = first-order rate constant (s-1))
Rearrange:
d[A]/[A] = dln[A] = -kdt
Integrate and express [A] as a function of time (t):
[A]/[A]o = e –kt or
[A]= [A]o e –kt
([A]o = initial concentration)

8. Reversible First-Order Reactionsk1
A B
v = -d[A]/dt = k1[A] - k-1[B]
At equilibrium: k1[A]eq - k-1[B]eq = 0
[B]eq/[A]eq = k1/k-1 = Keq  
k-1

9. Second-Order Reactionsk
2A  P
v = -d[A]/dt = k[A]2
A + B  P
v = -d[A]/dt = -d[B]/dt = k[A][B]
(k = second-order rate constant (M-1s-1))
Change in [A] with time:
1/[A] = 1/[A]o + kt k
Note: third-order reactions rare, fourth- and higher-order reactions unknown.

10. Free Energy Diagrams K = e –∆Gº/RT For A  A‡ [A]‡/[A]o = e –∆Gº‡/RT
K = equilibrium constant
‡ = transition state
[A]‡ = concentration of molecules having the activation energy
[A]o = total concentration
–∆Gº‡ = standard free energy change of activation (activation energy)

11. Relationship of Reaction Rate Constant to Activation Energy and Temperature: The Arrhenius Equation
Reaction rate constant (k) determined by activation energy (∆Gº‡) and temperature (T) and proportional to frequency of forming product (Q = kBT/h, where kB = Boltzmann’s constant, h = Planck’s constant):
k = Q e -G°‡/RT =
(kBT/h) e -G°‡/RT
(G = H - T S)
k = Q e S°‡/R  e - H°‡/RT
k = Q´e -H°‡/RT
(Q´ = Q e -S°‡/R)
ln k = ln Q´ - H°‡/RT
L-malate  fumarate + H20
ln k

12. The Transition State Energy Barrier Opposes the Reaction in Both Directions
K = k1/k-1
K = (Q e -G1°‡/RT)/(Q e -G-1°‡/RT)
K = e -(G1°‡ - G-1°‡)/RT
∆Gº = G1º‡ - G-1º‡
K = e –∆Gº/RT
Equilibrium constant K says nothing about rate of reaction, only free energy difference between final and initial states.

13. Effect of a Catalyst on Activation Energy
Catalysts do not affect GA (initial) or GB (final) and so do not affect overall free energy change (∆Gº = GB - GA) or equilibrium constant K.
Equilibrium concentrations of A and B still determined solely by overall free energy change.
Catalysts only affect ∆Gº‡, lowering the activation energy.
They accelerate both the forward and reverse reaction (increase kinetic rate constants k1 and k-1).

14. Intermediate States in Multi-step Reactions

15. Q: How do enzymes work?
A: A number of ways: “propinquity”, catalytic groups at active site, catalytic metals at active site, etc. (see assigned reading).
>>> however, primitive enzymes may have behaved like catalytic antibodies, which can accelerate reactions merely by binding to and increasing the relative concentration of the transition state ([A]‡ = [A]o e –∆Gº‡/RT effect).

16. Enzyme Kinetics

17. Types of Enzymes
Oxidoreductases catalyze oxidation-reduction reactions.
Transferases catalyze transfer of functional groups from one molecule to another.
Hydrolases catalyze hydrolytic cleavage.
Lyases catalyze removal of a group from or addition of a group to a double bond, or other cleavages involving electron rearrangement.
Isomerases catalyze intramolecular rearrangement.
Ligases catalyze reactions in which two molecules are joined.

18. Two Models for Enzyme-Substrate Interaction

19. Induced Conformational Change in Hexokinase

20. Triose Phosphate Isomerase
Mutational analysis to study enzyme mechanism:
Site-directed mutagenesis - substitution mutation at specific position in sequence
Deletion mutation

21. Free Energy Barrier to the Glyceraldehyde-3-Phosphate <-> Dihydroxyacetone Phosphate Reaction
Mutant in flexible loop that closes over active site.

22. OK, but now let’s talk about kinetics….

23. The Effect of Substrate Concentration on Reaction Velocity
Q: for a fixed amount of enzyme, what happens if you keep adding more and more substrate?

24. The Steady State in Enzyme Kinetics

25. Michaelis-Menten Kinetics (1)
v = k2[ES], if this is the rate-limiting step*
[Enzyme]total = [E]t = [E] + [ES]
How to solve for [ES]?
1. Assume equilibrium, if k-1 >> k2:
KS = k-1/k1 = [E][S]/[ES] or
2. Assume steady state:
d[ES]/dt = 0
[*Note: v is always measured as an initial rate!]
E = free enzyme, S = substrate
ES = enzyme-substrate complex
P = product
(Michaelis and Menten, 1913)
(Briggs and Haldane, 1925)

26. Michaelis-Menten Kinetics Continued (2)
Because of steady state assumption:
d[ES]/dt = k-1[ES] + k2[ES] - k1[E][S] = 0
So: k1[E][S] = k-1[ES] + k2[ES]
Rearranging: [ES] = (k1/(k-1 + k2))[E][S]
Substituting (the “M” constant* = KM = (k-1 + k2)/k1):
[ES] = ([E][S])/KM
So: KM[ES] = [E][S]
*Note: Briggs & Haldane came up with this,
but they lost out when it came time to name things!

27. Michaelis-Menten Kinetics Continued (3)
Substituting ([E] = [E]t - [ES]):
KM[ES] = [E]t[S] - [ES][S]
Rearranging: [ES](KM + [S]) = [E]t[S]
So: [ES] = [E]t[S]/(KM + [S])
Now we can substitute for [ES] in the rate equation
v = k2[ES], so…

28. The Michaelis-Menten Equation (4)
v = k2[E]t[S]/(KM + [S]) or
v = Vmax[S]/(KM + [S])
(since Vmax = k2[E]t)

29. At [S] « Km, Vo is proportional to [S] At [S] » Km, Vo = Vmax

30. A Lineweaver-Burk Plot

31. An Eadie-Hofstee Plot

32. Multi-step Reactions E + S ES  EP  E + P  k1
k k3
E + S ES  EP  E + P
v = kcat[E]t[S]/(KM + [S])  
k-1
(kcat = general rate constant that incorporates k2 and k3)

36. kcat, KM, and kcat/Km: Catalytic Efficiency
=> “Perfect enzyme”
Diffusion-controlled limit: M-1s-1
Substrate preferences for chymotrypsin