enzyme catalysis & kinetics · enzyme catalysis & kinetics wk5 - enzymes as individuals...

Enzyme Catalysis & Kinetics

Wk5 - Enzymes as Individuals (Wk6 - Enzymes in Complex Systems)

Susan Miller,

[email protected] GH S512B

Themes

• What do enzymes do… …from an energetic perspective

• How do they accomplish that… …from a structural perspective

• How do we discover and quantify… …from an experimental design perspective

• What does it all mean… …from a cellular perspective

Outline • Recap context • Catalysis - driving force (ΔGo) - reaction path(s) - ground states & transition states (ΔG‡) - specificity • Quantifying – Kinetics - elementary steps – microscopic level - 2nd order, 1st order, approach to equilibrium - overall flux - macroscopic behavior, kcat, KM, kcat/KM

• Vignettes – structures & free energy profiles • Design – testing understanding – Friday discussion papers

Biological/cellular context, three ideas…

For cells to thrive:

• Specific sets of molecules must interact at the right times and for the right lengths of time

• Specific sets of molecules must change into other sets of molecules at the right times and at the right rates

• Conformational dynamics at many time scales is important for molecular function

Three ideas, four concepts… • Specific sets of molecules must interact –

Thermodynamics ✔

• Specific sets of molecules must change into other sets of molecules – Chemical reactions – this section

• at the right times; right rates; right lengths of time – Kinetics – more depth

• Conformational dynamics – combination of thermodynamic stability and kinetics – JDG

Outline • Recap context • Catalysis - driving force (ΔGo) - reaction path(s) - ground states & transition states (ΔG‡) - specificity • Quantifying – Kinetics - elementary steps – microscopic level - 2nd order, 1st order, approach to equilibrium - overall flux - macroscopic behavior, kcat, KM, kcat/KM • Vignettes – structures & free energy profiles • Design – testing understanding

Free energy determines fate: Emphasis thus far on equilibrium “binding”, i.e. formation of noncovalent complexes…

ΔGfo = RT lnKdissoc

(std. state 1 M, T = 298 °K) ΔGf

o = 1.4 logKdissoc (kcal/mol) ΔGf

o = ΔHfo - TΔSf

o

? For noncovalent complexes in a cell, who can be “L”?? ? What “forces” contribute to ΔH and ΔS??

Free energy determines fate: Applies to changes in chemical/electronic structure, if reaction path available…

A couple of simple but important examples:

when “L” = e-, K = [PL]/[P] = [red]/[ox] and ΔGo = – nFEo = – RT lnK; n = # electrons F = Faraday constant Eo = std. reduction potential

when “L” = H+, – ln Kdissoc = pKa Ka = acid dissociation constant

Free energy determines fate: Applies changes in chemical/electronic structure, IF reaction path available…

ΔGo = – RT lnK ΔGo = – 1.4 logK (kcal/mol) ΔGo (std. states: 1 M, 298 °K) “thermodynamic driving force” ΔGo = ΔHo – TΔSo

Changes in covalent bond enthalpies, internal entropy, molecularity, solvation

Reaction Paths – Breaking Bonds - Transition States

For a path involving a single elementary step:

Transition State – covalent bond(s) are partially broken/made Recall: Covalent bond enthalpies are large, e.g.,

C-H ~99 kcal/mol C-C ~83 kcal/mol C-O ~86 kcal/mol P-O ~90 kcal/mol

∴ ΔG1

‡, ΔG-1‡ can be large


How large are ΔG‡ values for uncatalyzed biological reactions?

ΔG‡ values can be calculated from experimental rate constants using Eyring transition state rate theory.

If Q is removed rapidly, A -> Q becomes “irreversible” and the rate of the reaction, v =

rate = v = = k1 [A] – d[A]

dt

(more on measuring k1 later)

rate constant (experimentally measured)

Reaction Paths – Breaking Bonds - Transition States For a path involving a single elementary step:

Eyring transition state rate theory

K1‡ is a quasi-equilibrium constant

(quasi, because the lifetime of TS‡ < 10-12 s

ΔG1‡ = – RT lnK1

‡ K1

‡ = exp{–ΔG1‡/RT}

[TS‡] = [A] exp{–ΔG1‡/RT}

v = ν [TS‡] = [A] kBT/h exp{–ΔG1‡/RT}

v = –d[A]/dt = k1 [A] = ν [TS‡]

ν  ~ vibrational bond frequency kBT/h E = kBT = hν

kB – Boltzman’s const., h – Planck’s const.

= k1


For a path involving a single elementary step:

Eyring transition state rate theory

k1 = kBT/h exp{–ΔG1‡/RT}

Likewise, for the reverse reaction:

k-1 = kBT/h exp{–ΔG-1‡/RT}

~ 6.2 x 1012 s-1 @ 298 °K

Reaction Paths – Breaking Bonds - Transition States How large are ΔG‡ values for uncatalyzed biological reactions?

k1 = kBT/h exp{–ΔG1‡/RT}

Radzicka & Wolfenden 1995 Science Vol. 267

ΔGnon‡

(kcal/mol)

39.7

33.3

29.3

25.4

24.3 19.1

Reaction Paths – Breaking Bonds - Transition States Enzymes catalyze specific reactions by providing a lower energy path - increased rate: larger k1 & k-1, lower ΔG1

‡ & ΔG-1‡

- while ΔGo remains unchanged

noncatalyzed catalyzed

Profile drawn for standard state [A] = [Q] = 1 M

Reaction Paths – Breaking Bonds - Transition States How large are ΔΔG‡ values for biological reactions (ΔGnon

‡ - ΔGcat‡)?

Rate constants from A. Radzicka & R. Wolfenden 1995 Science Vol. 267

Enzyme knon (s-1) kcat (s-1) ΔG‡

non (kcal/mol)

ΔG‡cat (kcal/mol)

ΔΔG‡ (kcal/mol)

OMP decarboxylase 2.8 x 10-16 39 39.7 15.7 24.0

Staphylococcal nuclease 1.7 x 10-13 95 35.8 15.1 20.6

Adenosine deaminase 1.8 x 10-10 370 31.6 14.3 17.2

AMP nucleosidase 1.0 x 10-11 60 33.3 15.4 17.9

Cytidine deaminase 3.2 x 10-10 299 31.2 14.4 16.8

Phosphotriesterase 7.5 x 10-9 2,100 29.3 13.3 16.0

Carboxypeptidase A 3.0 x 10-9 578 29.8 14.0 15.8

Ketosteroid isomerase 1.7 x 10-7 66,000 27.4 11.2 16.2 Triosephosphate isomerase 4.3 x 10-6 4,300 25.4 12.8 12.6

Chorismate mutase 2.8 x 10-5 50 24.3 15.5 8.8

Carbonic anhydrase 2.8 x 10-1 1 x 106 19.1 9.5 9.6

Cyclophilin 2.8 x 10-2 13,000 20.1 12.1 7.9

Reaction Paths – Breaking Bonds - Transition States What features of enzyme structures contribute to providing a lower energy reaction path for specific sets of substrates?



Haldane (1930)- enzymes push and pull, i.e., distort or strain molecules

C Nu: X

What features of enzyme structures contribute to providing a lower energy reaction path for specific sets of substrates?

Eyring (1930’s) Transition State Rate Theory

A Q

[TS‡] K‡

TSf = [TS‡]

[A]

ΔG‡

ΔG‡ = -RT(lnK ‡TSf )

kf = κνK‡TSf = κνe-ΔG‡/RT

kf ∝ K‡TSf

lifetime ~0.1 ps “pseudostructure”


“push and pull” ~ both GS and TS effects

A Q (E +) (+ E)

[TS‡]

EA*

[E•TS‡]

[E•TS‡]

EA

EA*

[E•TS‡]

C Nu: X GS steric “push”

TS + charge “pull”

Pauling (1946)- enzymes “bind” TS more tightly than GS

A Q (E +) (+ E)

[TS‡]

E+A EA

EQ E+Q

|ΔGbind[TS]|>|ΔGbindGS|


60’s & 70’s - Jencks,Wolfenden, others - emphasize idea that catalysis occurs by binding TS more than GS

A Q

[TS‡]

E+A EA

EQ E+Q

KdA K‡

[ETS‡]

A + E

EA

Q

[ETS‡ ]

“KdTS”

K‡[TS‡]

[TS‡ ]

+ E

knon∝ K‡[TS‡] & kcat∝ K‡

[ETS‡]

kcat KdA

knon “KdTS”

=

“KdTS” = knonKdA kcat

∝ KdA kcat

60’s & 70’s - Jencks, - TS binding leads to specificity

E+A A Q

[TS‡]

EA EQ E+Q

“KdTS” ∝ KdA kcat

Two effects we will look at: -  Interactions directly with reaction center lead to major rate acceleration (and major inhibition)

- Optimized binding of other parts of substrate in TS provides specificity through further rate enhancement (and enhanced inhibition)

Not just ∝ KdA

“push and pull” ~ both GS and TS effects

A Q (E +) (+ E)

[TS‡]

EA*

[E•TS‡]

[E•TS‡]

EA

EA*

[E•TS‡]

GS – If A binding is favorable, how can A be “destabilized”?


Jencks, Fersht and many others envision…

E+A A Q

[TS‡]

EA EQ E+Q obs ΔGbind

additional favorable ΔGbind felt here and…

…in the TS

unfavorable ΔGdestab felt only in GS

Environmental changes that make “A” more reactive than in solution

Reaction Paths – Breaking Bonds - Transition States What features of enzyme structures contribute to providing a lower energy reaction path for specific sets of substrates? First – consider kinetic impact of introducing binding steps…



Reaction Paths – Breaking Bonds - Transition States First – consider kinetic impact of introducing binding steps…

For the noncatalyzed rxn, if Qinit = 0 and Q removed… -  single elementary step -  unimolecular – 1st order

For the catalyzed rxn, If Qinit = 0 and Q removed… -  3 elementary steps -  bimolecular binding – 2nd order -  chemical step – 1st order -  Q dissociation – 1st order

rate = v = = k1 [A] – d[A]

dt

M/s s-1

rate = ? Varies with [E] & [A] and depends on relative magnitudes of ΔG‡ for each of the 3 steps…

2nd order 1st order

Reaction Paths – Breaking Bonds - Transition States First – consider kinetic impact of introducing binding steps…


Profile with TS‡bind & TS‡

off [A] ≤ 1 M, [Q] = 0

Outline • Recap context • Catalysis - driving force (ΔGo) - reaction path(s) - ground states & transition states (ΔG‡) - specificity – (come back to this) • Quantifying – Kinetics - elementary steps – microscopic level – “single turnover” - 2nd order, 1st order, approach to equilibrium - overall flux - macroscopic behavior, kcat, KM, kcat/KM

• Vignettes – structures & free energy profiles • Design – testing understanding

…What do we mean by “single turnover” kinetics?

• In “single turnover” kinetics, enzyme is used as a chemical reagent to react with substrate (or other ligand) only once

- [E] comparable to [S] (may be > or <)

- monitor property(ies) of E, S or both to identify complexes or intermediates in reaction

- For example, if S is fluorescent and E used in xs…

ESE + S EI EP E + P

Binding could give ΔFl due to change in solvation of chromophore

If chemistry involves changes in electronic structure of chromophore, these steps should have ΔAbs &/or ΔFl

Here again change in solvation could give ΔAbs &/or ΔFl

• A few other examples of “Single turnover” kinetic situations

-  kinetics of ligand binding where L is unreactive

E + L EL kon

koff

- kinetics of covalent inactivation E + I EI kon

koff EI*

kinact

-  trapping/identification of intermediates E*S1 E*S1S2 E + *P1 + P2

xs S1+S2

- kinetics of half-reactions E + S1 ES1 E* + P1

E* + S2 E*S2 E + P2

Here * indicates label in S1. If E*S1 is kinetically competent, a burst of labeled *P1 appears. If E*S1 is not competent, the label will be diluted by xs S1 resulting in no burst of labeled *P1.

•Each kinetic phase or transient detected in a “single turnover” reaction indicates the presence of a “kinetically significant” elementary step, i.e., one that leads to accumulation of a new species.

• monophasic, biphasic, etc.

ESE + S EI EP E + P

•Each phase is defined by a time constant (τobs) or macroscopic rate constant (kobs = 1/τobs) that is typically a complex expression of the microscopic rate constants for each step.

e.g., this model may “fit” a four phase reaction curve

Let’s consider…

Single elementary step processes – exhibit monophasic, i.e., single exponential behavior defined by a macroscopic rate constant kobs (s -1):

EPkES

E + L ELk

EPESkfkr

• What are the distinguishing features of each of these? • How does kobs relate to the microscopic rate constants (k, kf , kr , kon, koff) in each case?

E + L ELkonkoff

Irreversible 1st order reactions – unimolecular steps

forward steps 2nd order overall, 1st order in both [E] and [L]

Approach to equilibrium

1st order irreversible process:

d[ES]dt

- = k[ES] =d[EP]dt=rate

1st order rate constant units = s-1 1st order in [ES] & overall = unimolecular

EPkES

∫ d[ES][ES] = ∫ -kt

ln[ES] =

0

t

ln[ES0] - kt

Half life: time pt when [ES] = [ES0]/2 t1/2 = (ln2)/k constant over full reaction

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

[ES

]/[E

S 0]

time (s)

k = 1 s-1

t1/2

= 0.693 s

τ = 1 s

t1/2

3*t1/2

2*t1/2

reaction ~finished in 10*t

1/2

amplitude

Time constant: τ = 1/k

M s =

1st order irrev. process (con’t): EPkES

[ES] = [ES0] e-kt

[EP] = [ES0](1 - e-kt)

Product formation, of course, occurs with the same rate constant and amplitude and has an endpoint of [ES0]

Alternative graph of kinetic in log(time) visualizes full time course, note reaction ~complete in 10*t1/2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

[EP

]/[E

S 0]

time (s)

0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1 10

[ES

]/[E

S 0]

time (s)t1/2

10*t1/2

2nd order irrev. process: E + L ELk

d[E]dt

- = k[E][L]=rate =d[EL]dt

2nd order rate constant units = M-1 s-1

Second order overall, first order each in [E] and [L]

In the limiting case where [E0] = [L0], the integrated rate equation is:

1[E]

=1[E0]

+ kt0

1

2

3

4

5

0 1 2 3 4 5

1/[E

]

time (s)

slope =

k (M-1 s-1)

intercept = 1/[E0]

Plotted in the “normal” sense of [E] vs time…

2nd order irrev. process: E + L ELk

However, as [L0] increases over [E0], t1/2 approaches a constant value and the reaction becomes pseudo-first order…

…we see that t1/2 is not constant, but increases with time

0.0 100

2.0 10-6

4.0 10-6

6.0 10-6

8.0 10-6

1.0 10-5

0 0.01 0.02 0.03[E

] (M

)time (s)

[L0] = 2*[E

0]

[L0] = 5*[E

0]

0.0 100

2.0 10-6

4.0 10-6

6.0 10-6

8.0 10-6

1.0 10-5

0 0.04 0.08 0.12 0.16 0.2

[E] (

M)

time (s)

t1/2

t3/4

≠ 2*t1/2

t7/8

≠ 3*t1/2

[E0] = [L

0]

k = 107 M-1 s-1

2nd order process under pseudo-first order conditions:

E + L ELk

d[E]dt

- = k[E][L]=rate

either… [L0] ≥ 10*[E0] or… [E0] ≥ 10*[L0]

For [L0] ≥ 10*[E0]… since [L0] only decreases by 10% over the whole reaction, kobs = k[L0] is ~ constant and…

The process behaves as single exp. but kobs is linearly dependent on [L0]…

d[E]dt

- = kobs[E]=rate

[E] = [E0] e-(kobs)t

0

5

10

15

20

0 2 4 6 8 10

k obs

(s-1

)[L0] (µM)

slope =

k (µM-1 s-1)

Approach to equilibrium: EPESkfkr

d[ES]dt

- = kf[ES] -kr[EP] =d[EP]dt=rate

but [EP] = [ES0] - [ES], so: d[ES]dt

- = (kf + kr)[ES] - kr[ES0]

d[ES]dt + (kf + kr)[ES] = kr[ES0]which rearranges to:

and integrates to:

A single exponential process with kobs = kf + kr i.e. monophasic 1st order process

amplitude equilibrium end point at t = ∞

[ES] = [ES0] e-(kf + kr)t[ES0]kr

(kf + kr)kf

(kf + kr)+

Approach to equilibrium (con’t): EPESkfkr

Comparing eqns for [ES] and [EP]:

[ES] = [ES0] e-(kf + kr)t[ES0]kr

(kf + kr)kf

(kf + kr)+

[EP] = [ES0] e-(kf + kr)t[ES0]kf

(kf + kr)kf

(kf + kr)-

kobs and amplitudes same but…

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10[E

S] o

r [E

P]

time (s)

kobs

= 1 s-1

kr = 0.4 s-1

kf = 0.6 s-1[ES]eq= [ES0]

kr(kf + kr)

[EP]eq= [ES0]kf

(kf + kr)

To determine kf, kr and Keq = kf /kr, ES and EP must have a measurable property that differs, e.g. NMR chemical shift

Eq. endpoints reflect Keq = kf /kr = [EP]eq/[ES]eq

amplitudes

Approach to equilibrium in a binding reaction:

Under pseudo-first order conditions, e.g. [L0] ≥ 10*[E0], the eqns are as above but with kf’ = kon[L0] in place of kf

Now, both kobs and the equil. end points vary with [L0]

E + L ELkonkoff

[E] = [E0] e-(kf' + koff)t[E0]

koff(kf' + koff)

kf'(kf' + koff)+

[EL] = [E0] e-(kf' + koff)t[E0]

kf'(kf' + koff)

kf'(kf' + koff)-

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6 7 8

[E] (

nM)

time (s)

10-8 M

3*10-8 M

10-7 M

3*10-7 M

10-6 M

[L0]=k

on = 107 M-1 s-1 k

off = 1 s-1

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6 7 8

[EL]

(nM

)

time (s)

kon

= 107 M-1 s-1 koff

= 1 s-1[L

0]=

10-6 M

3*10-7 M

10-7 M

3*10-8 M

10-8 M

kobs

Approach to equilibrium in a binding reaction (con’t):

Since kobs = kf’ + koff and kf’ = kon[L0]…

kobs shows a linear dependence on [L0], i.e.,kobs = kon[L0] + koff

…and the dissociation constant Kd = koff /kon = s-1/(M-1 s-1) = M

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0 0.2 0.4 0.6 0.8 1 1.2

k obs (s

-1)

[L0] (µM)

slope =

kon

(µM-1 s-1)

intercept = koff

(s-1)

E + L ELkonkoff

Limiting Values of 2nd order binding rate constants: kon

In some instances, the ΔG‡ for association is ≈ 0. In this case, the process is “diffusion limited”, i.e. it is simply limited by how fast the two molecules can diffuse together.

Models developed by Alberty, Hammes & Eigen (1958) and later by Chou (1970’s) – see G. Zhou & W. Zhong (1982) Eur. J. Biochem. 128, 383 which compares the models and lists relevant references

E + L ELkonkoff

kon, diff lim = (M-1 s-1)

4 π (rE + rL)(DE + DL) No

1000 cm3

rE, rL are radii of E & L in cm

No is Avogadro’s #

DE, DL are the Stokes-Einstein diffusion coefficients for E & L in cm2/s

η  = viscosity in dyn-s/cm2 = poise

kBT 6 π η rE

DE = kon, diff lim ≈ 1010 M-1 s-1

Test for diffusion limited: measure rate as f(η)

Biphasic reactions: 2nd order + 1st order

0.0

2.0

4.0

6.0

8.0

0 0.004 0.008

k obs (s

-1)

[S0] (M)

kmax

K1/2

ESk1E + S EP

k3k2

A 2-step process like this can exhibit 2 exponential phases IF there is a measureable signal for each phase.

kobs =k3k1S0

k2 + k3 + k1S0=

kmaxS0K1/2 + S0

kmax = k3 K1/2 = (k2 + k3)/k1When both steps are reversible, two new diagnostics arise…

kobs for appearance of EP

However, in many cases, the only measureable signal occurs in the 2nd step for ES -> EP. If pseudofirst order conditions are used, i.e., [So] ≥ 10*Eo, then kobs shows a hyperbolic dependence on [So]…

E decays biphasically

0.0 100

2.0 10-6

4.0 10-6

6.0 10-6

8.0 10-6

1.0 10-5

0 0.1 0.2 0.3 0.4 0.5time (s)

[spe

cies

] (M

)

EESEP

Biphasic reactions: 2nd order + 1st order

0.0

5.0

10.0

0 0.004 0.008[S

0] (M)

k obs (s

-1)

K1/2

kmax

kmax

= k3 + k

4

y-intercept = k4

K1/2

= (k2 + k

3)/k

1

k4ES

k1E + S EPk3

k2

appearance of EP

• all species reach equilbrium end pts • kobs for formation of [EP] exhibits a hyperbolic dependence on [S0] but with a positive y-intercept

An example of single turnover studies in tyrosyl-tRNA synthetase – Fersht, et al,1975 Biochemistry 14, 13-18.

Overall reaction occurs in two half-reactions:

Reaction 1 – “activation” of Tyrosyl carboxyl group by simple SN2 reaction



Reaction 2 – formation of Tyr-t-RNA ester



Reaction 1 – “activation” of Tyrosyl carboxyl group



binding steps chemistry PPi dissoc.

ATP binds too weakly to evaluate top pathway

First, evaluate equilibrium binding of Tyr: E + Tyr E•Tyr


First, evaluate equilibrium binding of Tyr: E + Tyr E•Tyr

Discover enzyme Tryptophan fluorescence decreases upon binding of Tyr - sensitive enough to measure rates

kon = 2.4 x 106 M-1 s-1

koff ≈ 24 s-1

KdTyr = koff/kon ≈ 10 µM

Compare with KdTyr ≈ 11 µM from equilibrium dialysis




2nd, eval. binding and rxn of Ad: E•Tyr + Ad E•Tyr•Ad E•Tyr-Ad


Trp fluorescence decreases further in reaction with Ad

Single exponential decay Gives kobs values which showed hyperbolic dependence on [Ad] with k3 ≈ 18 s-1 and K’dAd ≈ 3.9 mM (plot not shown)

2nd, eval. binding and rxn of Ad: E•Tyr + Ad E•Tyr•Ad E•Tyr-Ad




3rd, eval. reverse rxn of PPi: E•Tyr-Ad + PPi E•Tyr•Ad•PPi E


Trp fluorescence increases in Rxn of PPi with E•Tyr-Ad Verify formation of ATP using rapid chemical quench

Single exp fluor increase gives kobs values which showed hyperbolic dependence on [PPi] with k-3 ≈ 14 s-1and KdPPi ≈ 0.74 mM (plot not shown)

3rd, eval. reverse rxn of PPi: E•Tyr-Ad + PPi E•Tyr•Ad•PPi E

At left: Fluor kobs ≈ 0.39 s-

quench kobs ≈ 0.37 s-

Verifying that Δfluor is due to full reversal of the reaction (formation of ATP)

Outline • Recap context • Catalysis - driving force (ΔGo) - reaction path(s) - ground states & transition states (ΔG‡) - specificity – (come back to this) • Quantifying – Kinetics - elementary steps – microscopic level – “single turnover” - 2nd order, 1st order, approach to equilibrium - overall flux - macroscopic behavior, kcat, KM, kcat/KM

• Vignettes – structures & free energy profiles • Design – testing understanding

Reaction Paths – Breaking Bonds - Transition States First – consider kinetic impact of introduction of binding step…

Profile with TS‡bind & TS‡

off [A] ≤ 1 M, [Q] = 0 Recall, as [A] varies, the free

energy of A varies, i.e. at [A] < Kd, the binding equilibrium favors free E. A priori, cannot predict whether catalysis will have a higher energy TS than binding or dissociation, thus ΔG‡

rev and ΔG‡off may be

<, =, or > ΔG‡1

Quantifying – Overall flux or “classic” steady-state kinetics

- each enzyme molecule catalyzes multiple cycles of reaction

- [E] << [A], typically [E] too low to directly monitor its properties so only observe overall Rxn

-  Rates monitored as loss of A (-dA/dt) or appearance of Q (dQ/dt) using a chemical or spectroscopic property of A or Q

- Rates as f([A]) yield familiar steady-state “macroscopic” parameters, Vmax (kcat) and KM

• How are these defined and what can and cannot be learned from them?

A Q enzyme as catalyst

•Common experimental design:

- setting [ET] << [AT] and [Qinit] = 0 - measuring “initial velocity”, vi = dQ/dt = –dA/dt, i.e. <10% conversion where reaction remains ~ linear

•Steady-state assumption: No change in concentration of any enzyme species during time of measurement

•Accomplished by: [A]

[Q]

time

vi

Quantifying – Overall flux or “classic” steady-state kinetics

A Q enzyme as catalyst

So for:

= d[E] dt

d[EA] dt =

d[EQ] dt = 0 ET = E + EA + EQ

Hyperbolic dependence on [A] indicates minimal 2-step mechanism just as in 2-step “single turnover”…

• *Most common* behavior of vi vs [AT] & [ET]…

Vm

ax (M

/s)

[ET] (M)

kcat (s-1)

k1

k2

kcat E + A EA E + Q

kobs = vi /ET = kcat AT

KM + AT

vi = kcat ET AT

KM + AT

Vmax AT

KM + AT =

at const. [ET] << [AT]

2nd order overall 1st order in E, 1st order in A

1st order in EA

• Key Parameters k1

k2

kcat E + A EA E + Q

kobs = vi /ET = kcat AT

KM + AT vi =

kcat ET AT

KM + AT

Vmax AT

KM + AT =

at const. [ET] << [AT]

AT -> ∞ AT >> KM

vi = Vmax - maximal velocity

Vmax = kcat ET

kcat - macroscopic 1st order rate constant (s -1)

AT -> 0 AT << KM

vi = (Vmax/KM) AT = (kcat/KM)ETAT

Vmax/KM - pseudofirst order rate const

kcat/KM - macroscopic 2nd order rate constant (M-1 s -1)

AT = KM vi = Vmax /2 KM = Michaelis constant

[E] = ET/2 Σ[Ebound] = ET/2

• Key Parameters are Macroscopic Constants

k1

k2

kcat E + A EA E + Q

Applying the steady-state (ss) assumption: dE dEA dt dt = = 0

…to this minimal mechanism

rate = vi = d[Q] dt

= kcat [EA]

For initial velocity measurements, i.e., < 10% conversion, typically substitute [A] ≈ AT

d[EA] dt

= k1[E][A] – (k2 + kcat)[EA] = 0

(k2 + kcat)[EA] k1[A]

[ET] = [E] + [EA]

[ET] = [EA] [E] = 1 + (k2 + kcat) k1[A]

[EA] unknown, but ET and AT known

…combine [E] from ss assumption... …with [ET] from conservation of mass

vi = kcat[EA] = kcat ET AT

k2 + kcat k1

+ AT

• Key Parameters are Macroscopic Constants

k1

k2

kcat E + A EA E + Q

kcat appears to be a microscopic 1st order rate constant (s -1)

Applying the steady-state assumption: dE dEA dt dt = = 0

…to this minimal mechanism

vi /ET = kcat AT

KM + AT =

kcat AT

k2 + kcat + AT

k1

kcat/KM = k1kcat

k2 + kcat KM =

k2 + kcat

k1

…yields

…where at face value

…but both KM and kcat/KM are complex macroscopic constants

But further…

…the behavior of vi vs AT is macroscopic and consistent with many microscopic mechanisms, where kcat is also macroscopic, e.g.

k1

k2

k3 E + A EA EQ E + Q

k5

k1

k2

k3 E + A EA EQ

k4

E + Q k7 k5

EI k6

k1

k2

k3 E + A EA E*Q E* + Q

k5 k7

k8

E

kcat

kcat

kcat

kcat/KM

k1

k2

k3 E + A EA EQ

k4

E + Q k5

kcat

kcat/KM

kcat/KM

kcat/KM

•The expressions for kcat and kcat/KM include microscopic constants for all steps within the brackets in each case.

•kcat/KM expressions include all steps from binding of A through the 1st irreversible step

•kcat expressions include all first order steps including chemistry, conformational changes, product dissociation

•KM expressions can be derived from ratio of kcat/(kcat/KM) and are very complex

k1

k2

k3 E + A EA EQ E + Q

k5

kcat

kcat/KM

Without additional information, one cannot draw conclusions as to what types of processes (chemistry or binding, dissociation, conformational changes) limit the magnitudes of kcat and kcat/KM (i.e., are rate limiting) or how individual processes are altered by changes in protein structure (e.g., by mutation)

• Key point:

Consider the following energetic scenarios…

• What processes do kcat/KM and KM reflect? k1

k2

k3 E + A EA E + Q

kcat = k3

kcat/KM

ΔG

Rxn

E+A EA

E+Q

k = kBT/h exp{-ΔG‡/RT}, i.e. ln(1/k) ∝ ΔG‡

KM = k2 + k3

k1

ΔG

Rxn

ΔG

Rxn

E+Q E+Q EA EA E+A E+A

k2 >> k3

kcat/KM = k1 k3

k2 + k3

TS1,2

TS3

kcat/KM = k3

k2/k1 =

k3

KD

k2 ~ k3 k2 << k3

KM = KD

both kcat & kcat/KM limited by TS3, i.e., chemistry

kcat/KM = k1 k3

k2 + k3

KM = k2 + k3

k1 = 2KD

kcat/KM only partially limited by TS3 (chemistry)

kcat/KM = k1

KM = k3

k1 =

kcat

k1

kcat/KM limited only by TS1,2, i.e, diffusion

> KD

k1

k2

k3 E + A EA EQ

k4

E + Q k5

kcat

kcat/KM

…and what happens if EP accumulates?

kcat/KM = k1 k3k5

k2(k4 + k5) + k3 k5

Full expressions:

kcat = k3k5

k3 + k4 + k5 KM =

k2(k4 + k5) + k3 k5

k1(k3 + k4 + k5 )

kcat = k3

k3 + k4

k5 ( ) fraction of bound enzyme that accumulates as EQ

kcat reflects TS5 , i.e., product dissoc.

kcat/KM ~ k5

k2k4/k1 k3

kcat /KM also reflects TS5 for product dissoc.

EQ forms in equil with E & EA

KM = k2(k4)

k1(k3 + k4) = KD

k4 k3 + k4 ( )

In this case, KM < KD

ΔG

Rxn

E+A EA EQ

E+Q

TS1,2

TS3,4

TS5 k5 << k3, k4

Summary of Steady-State Points

• kcat, kcat/KM and KM are all macroscopic constants

• kcat may be limited or partially limited by any 1st order process including chemistry, conformational changes, product dissociation - chemistry often not fully rate limiting

• kcat/KM may be limited by any step reversibly connected to substrate binding. Diffusion limited means chemistry is faster than substrate dissociation

•KM is a Kinetic constant = [A] that gives half maximal velocity. It may be equal to KD, the dissociation constant for A, but often is not and does not a priori reflect the binding affinity of A for E

Σ (net flux from EA to E: back + fwd) KM =

kon

Outline • Recap context • Catalysis - driving force (ΔGo) - reaction path(s) - ground states & transition states (ΔG‡) - specificity • Quantifying – Kinetics - overall flux - macroscopic behavior, kcat, KM, kcat/KM

- elementary steps – microscopic level - 2nd order, 1st order, approach to equilibrium • Vignettes – structures & free energy profiles • Design – testing understanding

• Enzymes evolve to solve a chemical need for the cell, so specificity and rates are coupled – the lowest energy reaction path (with highest rates) is “fine-tuned” to the set of substrates encountered by the cell

• The degree of specificity is typically not absolute, but - depends on whether similar molecules are present

during the evolution that compete with the “important substrates”

- depends on functional role • Two aspects to specificity:

• reactant specificity observed in different rates • reaction specificity observed by analyzing products to determine if as expected!

A few thoughts about specificity…

From A. Fersht, Structure and Mechanism in Protein Science, Freeman, 1999.

• Enzymes evolve to solve a chemical need for the cell, so specificity and rates are coupled – the lowest energy reaction path (with highest rates) is “fine-tuned” to the set of substrates encountered by the cell

kcat & kcat/KM are experimental macroscopic steady-state rate constants described below


• The degree of specificity is typically not absolute, but - depends on whether similar molecules are present during the

evolution that compete with the “important substrates” - depends on functional role


Two P-450s… …Cyp21A2 – involved in steroid biosynthesis – exquisitely specific

…Cyp3A4 – major metabolizer of xenobiotics (drugs) – exquisitely nonspecific

• An example of reaction and reactant specificity

• reaction specificity observed by analyzing products -Rxn shown is fully coupled, i.e. stoichiometric -with some alternative substrates and in some mutants: *Rxn is uncoupled: NADPH and O2 consumed, but H2O2 produced instead of expected hydroxylated product*

• reactant specificity observed in different rates -PHBH exhibits highest rates with substrates shown -can use NADH as reductant at much lower rate -will tolerate p-SH or p-NH2 at similar or lower rates to expected product* -and some o-substitutions usually with lower rates to expected product*

(FAD cofactor)

N

O

NH2

RCO2

OH

O2 + +

N

O

NH2

RCO2

OH

H2O + +

HO

p-hydroxybenzoatehydroxylase (PHBH)(monooxygenase)

p-hydroxybenzoate NADPH 3,4-dihydroxybenzoate NADP+

H

• To observe changes in Reaction Specificity…

Easy assays in this system: -loss of Abs at 340 nm as NADPH -> NADP+ -consumption of O2 using oxygen electrode

(FAD cofactor)

…the assay must be carefully chosen

Problem: uncoupled reaction which produces H2O2 also consumes NADPH and O2

Solution: To observe change in reaction specificity - must explicitly monitor the main substrate/product

N

O

NH2

RCO2

OH

O2 + +

N

O

NH2

RCO2

OH

H2O + +

HO

p-hydroxybenzoatehydroxylase (PHBH)(monooxygenase)

p-hydroxybenzoate NADPH 3,4-dihydroxybenzoate NADP+

H

Outline • Recap context • Catalysis - driving force (ΔGo) - reaction path(s) - ground states & transition states (ΔG‡) - specificity • Quantifying – Kinetics - overall flux - macroscopic behavior, kcat, KM, kcat/KM

- elementary steps – microscopic level - 2nd order, 1st order, approach to equilibrium • Vignettes – structures & free energy profiles • Design – testing understanding

Catalytic “needs” are diverse… 1 substrate

1 substrate + H2O

3 substrates

TIM – chemistry requirements & specificity determinants – hard steps - breaking C-H bonds in DHAP & G3P – need a base (B:) – pulling e- density from π bond in C=O toward the O will lower pKa of C-H

– need B-H/B: to protonate/deprotonate C=O/OH – 2 catalytic steps with intermediate that must not escape – must prevent elimination of Pi in conversion of intermediate -> G3P – Pi provides strong binding energy for specificity – stereochemistry at C1 & C2 also a specificity element – need to accommodate changes in geometry at C1 and C2

J. R. Knowles 1970s, ‘80s, ’90s, many others ‘90s - current

enzyme catalysis & kinetics · enzyme catalysis & kinetics wk5 - enzymes as individuals...

Documents