enzyme catalysis & kinetics · enzyme catalysis & kinetics wk5 - enzymes as individuals...
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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
Reaction Paths – Breaking Bonds - Transition States
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
Reaction Paths – Breaking Bonds - Transition States
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?
noncatalyzed catalyzed
Profile drawn for standard state [A] = [Q] = 1 M
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”
What features of enzyme structures contribute to providing a lower energy reaction path for specific sets of substrates?
“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|
What features of enzyme structures contribute to providing a lower energy reaction path for specific sets of substrates?
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”?
What features of enzyme structures contribute to providing a lower energy reaction path for specific sets of substrates?
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…
noncatalyzed catalyzed
Profile drawn for standard state [A] = [Q] = 1 M
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 drawn for standard state [A] = [Q] = 1 M
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
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 2 – formation of Tyr-t-RNA ester
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
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:
binding steps chemistry PPi dissoc.
ATP binds too weakly to evaluate top pathway
First, evaluate equilibrium binding of Tyr: E + Tyr E•Tyr
An example of single turnover studies in tyrosyl-tRNA synthetase – Fersht, et al,1975 Biochemistry 14, 13-18.
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
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:
binding steps chemistry PPi dissoc.
2nd, eval. binding and rxn of Ad: E•Tyr + Ad E•Tyr•Ad E•Tyr-Ad
An example of single turnover studies in tyrosyl-tRNA synthetase – Fersht, et al,1975 Biochemistry 14, 13-18.
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
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:
binding steps chemistry PPi dissoc.
3rd, eval. reverse rxn of PPi: E•Tyr-Ad + PPi E•Tyr•Ad•PPi E
An example of single turnover studies in tyrosyl-tRNA synthetase – Fersht, et al,1975 Biochemistry 14, 13-18.
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
A few thoughts about specificity…
• 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
A few thoughts about specificity…
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