Lecture 27. Transition States and Enzyme Catalysis

Transcription

1 Lecture 27 Transition States and Enzyme Catalysis Reading for Today: Chapter 15 sections B and C Chapter 16 next two lectures 4/8/16 1

2 Pop Question 9 Binding data for your thesis protein (YTP), binding to its ligand, Hoopes. You are hoping that these data provide some clue as to its possible oligomeric state, especially because you have already determined that there is one Hoopes molecule bound per molecule of YTP. Slope = 1 Slope = 1 What quantitative value(s) can you extract about Hoopes binding to YTP? 0 < n H <1 (n H ~ 0.2) thus negative cooperativity K D for the high-affinity state ~ K D for the low-affinity state ~ K D (high) ~ Slope = n H <1 K D (low) ~ What can you determine about the YTP oligomeric state? Because there is cooperativity, YTP is an oligomer there has to be more than one binding site per functional unit to observe allostery. However, the Hill coefficient value does not provide direct information about the number of protomers in an oligomer. Note: If it is not a simple binding equilibrium (i.e. if the trend is not linear with a slope of 1), then the measure of [L] at f=1/2 is not a true K D. 4/8/16 2

3 Today s Goals We re continuing to explore reaction inetics: Reactions at steady state where the concentrations of reactants and products are constant Equilibrium reactions additional constraints on the rate constants Kinetics of ligand interactions Transition state theory what determines the rate constant Role of enzymes as catalysts Introducing Michaelis-Menten inetics of enzymes 4/8/16 3

4 Reactions at steady state Most biological reactions are not at equilibrium, but are at steady state 4/8/16 4

5 Reactions at steady state Most biological reactions are not at equilibrium, but are at steady state Formation and utilization of metabolites are equal Concentrations of the relevant reactants and products do not change with time (or only slowly) Case of: 1 A 2 B C Where [A] = [A] 0 at all times d[b] dt 1 [A] 0 2 [B] 4/8/16 5

6 d[b] dt 1 [A] Can be integrated to: 2 [ B] [A] 0(1 e 2 Reaction at steady state 0 1 t 2 [B] ) 1 A 2 B C 1 < 2 As t becomes large: [B] ss 1 [A] > 2 [B] ss is set by the ratio of rate constants and the concentration of reactant 4/8/16 6

7 Conditions of a true steady state Concentrations are constant There must be a flow through the system Formation and utilization of intermediates is equal Requirement for reaching steady state: the rate at each step beyond the 1 st step must depend on [reactant] Necessary so that if the [reactant] increases, the rate increases accordingly Total flux is determined by the slowest step Concentrations of intermediates before the slowest step are higher than those after the slowest step 4/8/16 7

8 If the change in free energy is small, the reaction is reversible When the DG is not much larger than RT, then the reverse reaction needs to be considered Applicable rates: To integrate the equations, we also now that: A The sum of [A] and [B] is constant f r B d[a] f [A] r [B] dt d[b] f [A] r [B] dt At equilibrium, f [A] eq = r [B] eq 4/8/16 8

9 Reversible reactions [A] ([A] ( f r ) t 0 [A] eq ) e [A] eq When t is large, [A]~[A] eq Similar equation can be developed for [B]: [B] [B] eq Equations correspond to an exponential approach to final concentrations ([B] 0 [B] eq ) e ( f r ) t The effective rate constant is the sum of the forward and reverse rate constants Figure from The Molecules of Life ( Garland Science 2008) 4/8/16 9

10 Rate constants and equilibrium constants At equilibrium, and Therefore d[a] dt Forward and bacward rates are equal: d[a] dt A 0 f r f [A] eq d[b] 0 dt r B [B] eq 0 [ A] [B] f eq r eq And f r [B] [A] eq eq K eq Although thermodynamics cannot be used to predict the inetics, it does constrain the ratio of the rate constants 4/8/16 10

11 Binding inetics of imatinib (Gleevec) Binding of inase inhibitor to inase leads to a decrease in fluorescence of Trp residues in the inase P + L on off (i.e. f ) (i.e. r ) PL Figure from The Molecules of Life ( Garland Science 2013) 4/8/16 11

12 Kinetics of a ligand binding equilibrium P + L on off PL Under conditions where [L]>>[P] on' on [ L] d [P] Rates: dt d[pl] dt Integrates to: [P][ L ] on off [P][ L ] on off [PL] [PL] on '[P] on off '[P] [PL] off [PL] [P] ([P] ( on ' off ) t 0 [P] eq) e [P] eq obs on ' off 4/8/16 12

13 Binding inetics of imatinib (Gleevec) From and obs on Linear dependence on [L]: ' ' on[l] off obs on on [L] off Binding of inase inhibitor to inase leads to a decrease in fluorescence of Trp residues in the inase Figure from The Molecules of Life ( Garland Science 2008) 4/8/16 13

14 Ligand binding inetics obs [L] on off Plot obs vs [L] Slope is on Note: [L] >> [P] for all experiments Extrapolate to [L] = 0 to get off at the y-intercept Figure from The Molecules of Life ( Garland Science 2008) 4/8/16 14

15 Ligand binding inetics Measured rate constants for Abl inase binding to its inhibitor imatinib (Gleevec), vs. Src, a non-specific target on on off off (Abl) (Src) (Abl) s (Src) s M M -1-1 s s -1-1 K K D D (Abl) (Src) P + L off on off on on off PL s M s M s s M M Figure from The Molecules of Life ( Garland Science 2008) [Imatinib] Ratio of /8/16 15

16 What determines the rate constant? A B C rate d[a] dt [A][ B] What determines the value of? Rate of collisions In gases, ~6x10 10 M -1 s -1 = collision In liquids, it depends on diffusion: collision 8RT / 3h h is the viscosity of the solution collision ~ M -1 s -1 for a typical aqueous solution 4/8/16 16

17 Diffusion-limited reactions vs. orientationdependent reactions 2I I 2 The delocalized electrons in the outermost orbitals come together to form the bond, and this can happen during ~any collision In this case, the collision has to happen in a specific orientation only a fraction of the collisions are effective Even 10 6 M -1 s -1 are considered fast rate constants ~1 in 10 4 collisions are productive Define factor A = collision x f p, where f p is the fraction of collisions at the correct angle Figure from The Molecules of Life ( Garland Science 2008) 4/8/16 17

18 Activation energy A second factor is the energy required to reach the transition state activation energy, E a Here the methyl hydrogens have to be pushed away from their low energy tetrahedral geometry to a planar arrangement Figure from The Molecules of Life ( Garland Science 2008) 4/8/16 18

19 Source of the activation energy The inetic energy of molecules is converted to potential energy as the reactants approach in the appropriate orientation The fraction of molecules that have sufficient energy is given by e -Ea/RT, related to the Boltzmann distribution The rate constant is a product of the collision factor, A, and the energetic factor, e -Ea/RT : Ae E a / RT A is called the pre-exponential factor = collision x f 4/8/16 19

20 Proline peptide bond isomerization The activation energy of a trans-to-cis isomerization of proline is 54 J/mol Figure from The Molecules of Life ( Garland Science 2008) 4/8/16 20

21 Arrhenius plot provides the activation energy value Ae E a / RT ln E 1 a R T ln A The slope of the Arrhenius plot gives E a /R 54 J mol -1 for proline isomerization Figure from The Molecules of Life ( Garland Science 2008) 4/8/16 21

22 Deviations from linearity What ind of reactions might show a plot lie this one? ln E 1 a R T ln A Figure from The Molecules of Life ( Garland Science 2008) 4/8/16 22

23 Transition state theory The process of getting to the transition state is an equilibrium: K eq A + B A B C + D K eq [AB ] [A][B] d[ C] [AB ] dt K eq [A][B] K eq e DG RT e DH RT e DS R ~ BT h Reflects the relevant vibration frequency leading the reaction forward Where h is Planc s constant 4/8/16 23

24 Another equation for the rate constant d[c] dt d[ C] K dt eq DS BT h e [A][B] R e DH RT [A][B] Analogous to the angle restriction for the collisions Analogous to the activation energy term Provides clear connections to thermodynamic concepts Useful to predict effects on reaction rates 4/8/16 24

25 Catalysts alter the rate constant rate (concentration factors) Ae E a / RT A collision f p A catalyst can modify both the pre-exponential and the exponential factors Favorable interaction with the transition state Increase the rate of collisions Favor the productive orientation Figure from The Molecules of Life ( Garland Science 2008) 4/8/16 25

26 Electrostatic potential to accelerate reactant encounters The substrate of acetylcholine esterase, acetylcholine (a neurotransmitter), is positively charged The negatively charged active site provides an electrostatic force pulling the substrate into the site Figure from The Molecules of Life ( Garland Science 2008) 4/8/16 26

27 Catalysts alter mechanism but not thermodynamics of the reaction equilibrium Single step: Two steps: A K eq B C [C] [A][B] K 1 A cat E B C [E] [A][cat] K 2 E cat [cat][c] [E][B] K1 cat A B E B D G eq DG 1 DG 2 K 2 C cat K eq K 1 K 2 4/8/16 27

28 Catalysts alter mechanism but not thermodynamics of the reaction equilibrium Single step: Two steps: A K eq B C [C] [A][B] Catalyst is not consumed during the reaction, but affects the mechanism and therefore the inetics. That s the role played by enzymes and ribozymes in life. K 1 K eq K eq A cat E B C [E] [A][cat] K 1K2 [C] [A][B] K 2 E cat [E] [A][cat] [cat][c] [E][B] [C][cat] [E][B] 4/8/16 28

29 Enzyme as catalyst an example Chorismate mutase isomerizes chorismate to prephenate step in the aromatic amino acid synthesis pathway Activation energy in water ~20 J mol -1 Activation energy with enzyme ~12 J mol -1 ~8 J mol -1 difference corresponds to the DG of the chorismate conformational change ~2x10 6 -fold increase in rate Figure from The Molecules of Life ( Garland Science 2008) 4/8/16 29

30 Transition state analogs Transition state analogs bind with high affinity because the enzymes have highest affinity for the transition state Figure from The Molecules of Life ( Garland Science 2008) 4/8/16 30

31 Enzyme-catalyzed reaction inetics S E P E rate Prediction: linear dependence on the concentration of both substrate and enzyme d[p] dt [S][E] Observations: Indeed ~linear at low [S] Hyperbolic function maximum = V max Michaelis constant K M = [S] where v = V max /2 in Implies a more complex inetic scheme Figure from The Molecules of Life ( Garland Science 2013) 4/8/16 31

32 Enzyme-catalyzed reaction inetics S E P E rate d[p] dt [S][E] Rate at which substrate will react to form [ES] complex depends on [S]: at low [S] it is first order at intermediate [S] it is between first and zero order due to partial saturation of the enzyme and at high [S] it is zero order At high [S], the enzyme is saturated, therefore the concentration of the enzyme is rate-limiting and the [S] does not affect the rate Figure from The Molecules of Life ( Garland Science 2013) in 4/8/16 32

33 Some concepts to remember Reversible reactions have additional constraints on the ratio of the rate constants The rate constant is determined by: Frequency of collisions Fraction of collisions that are in the favorable orientation Activation energy thermodynamics of the transition state Catalysts affect reaction rates but not equilibrium Positioning the reactants in favorable orientation Increasing the effective concentration Stabilizing the transition state to lower the activation energy Michaelis-Menten inetics of enzymes show hyperbolic function of rate vs. [S] 4/8/16 33