Question: can be obtained the rate of the elementary reactions from steady-state kinetics? No: there are more unknown parameters than equations!
vi =
k2[E]t[S] k-1 + k2 + [S] k1
More informations from pre-steady-state kinetics, but the techniques are complicated Some important information may be quite easily be obtained by recent development of steady-state methods (for example, what is the nature of the rate-limiting step?)
Time domains of various techniques Laser scatter Dielectric relaxation and electric dichroism Fl polarization
Pressure jump
EPR and NMR Ultrasound absorption and electric field jump Spectroscopic methods
10-10
Stopped flow and continuous flow
Flash and T jump
10-5 seconds
Hand mixing
100
102
How can we study such a fast process? Flow methods: involves a continuous flow of enzyme and substrate through a mixing chamber in a long tube. Observation at a certain distance gives a population of molecule of a fixed "age" from the instant of mixing. This allows reactions with a halftime of H milliseconds to be studied with equipments requiring several o seconds for each measurement. w
Stopped flow methods: similar, but with a mechanical syringe stopping the flow almost instantaneously. Or involving the c addition a of a quenching (stopping) agent {e.g. trichloroacetic acid n that destroys the enzyme) or bringing the mixture to a very low temperature with liquid nitrogen, so as to "freeze" the situation at w a given instant. e Perturbation methods: for extremely quick reactions the mixture s at steady state is given a temperature-jump of +10 °C already t with an electric current. The steady-state rate will not be same at u the higher temperature and the "relaxation" to the new d equilibrium is observed. y
For fast reactions, rapid mixing and quench of the reaction is needed
We analize the product mixture
Mechanical mixing-spectroscopic observation
Usual deadtime ~ 1 ms; time resolution is less than 1 ms
s08c Dénaturation/renaturation des protéines. Extraire l’information quantitative
Gibbs environ 1960
From 1995, home-made by several laboratories
A few enzymes have been studied in great detail We will discuss now the dihydrofolate reductase
Dihydrofolate reductase
Dihydrofolate Reductase
A great deal of detail are known with this enzyme since both substrates/products absorbe light and some are fluorescent This is enough data for 2 PhD theses!
Complete reaction scheme of dihydrofolate reductase Insights into enzyme function from studies on mutants of dihydrofolate reductase. Benkovic SJ, Fierke CA, Naylor AM. Science. 1988 ;239(4844):1105-10. Review.
Complete reaction scheme of dihydrofolate reductase
Activation parameter of dihydrofolate reductase
3.7 ± 0.24 kcal/mol
3
k
cat
[s
-1
]
2
1 0.9 0.8 0.7 0.6 0.5 0.4
11.9 ± 0.47 kcal/mol 39.5°C I
12.7°C I
0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 -1
1/T [K ]
A discontinuity in the Arrhenius plot means that the rate-limiting step changes
The nature of the rate limiting step can be (quite) easily obtained by studying the dependence of kcat and kcat/Km in the presence of viscogenic agents. There are two kind of viscogenic agents (also called OSMOLYTES): a. Microviscogens (small molecules): sucrose, glycerol decrease the diffusion rate b. Macroviscogens (macromolecules): proteins, PEG decrease the accessible volume, do not decrease the diffusion rate PEG, polyethylene glycol
HO-(CH2-CH2-O)n-H
In a simple biomolecular reaction, the rate of collision (or dissociation) is inversely proportional to the viscosity of the medium (Kramers, 1940). A first-order reaction is NOT affected by the viscosity. By altering the viscosity of the enzyme assay solution, one can discern the effects on the apparent first- (kcat) and second-order (kcat/Km) rate constants. The maximal theoretical value for the ratios of maximal rates is predicted to be 1.0 .
Consider a simple one substrate kinetics.
The kcat in the presence of of a relative viscosity would be k1η = k1/η k-1η = k-1 /η k3η = k3,/η
k-3η = k-3 /η
But k2η = k2, k-2η = k-2 not diffusion-dependent reaction
since they are
Exemplu efect (micro)viscosity on reaction rate
*
*
*
*
Kinetic mechanism of NDP kinases: a ping pong reaction - parallel lines in double reciprocal plot - a phosphorylated intermediate, which is chemically and kinetically competent k2 E·ATP k 1·[ATP]
EP·ADP k3
k -1 k -1
E
EP k -4
k6
k -5 E·NTP
EP·NDP
k 4·[NDP]
k5
[Et] 1 k − 2 +k3 k2k3 +k − 1(k − 2 +k3) 1 k − 5 +k6 k5k6 +k − 4(k − 5 +k6) = + + + + + v k3 k2k3 k6 k5k6 k1k2k3[ATP] k4k5k6[NDP]
k2 E·ATP k1·[ATP]
EP·ADP k3
k-1
[Et] 1 k − 2 +k3 k2k3 +k − 1(k − 2 +k3) 1 k − 5 +k6 k5k6 +k − 4(k − 5 +k6) = + + + + + v k3 k2k3 k6 k5k6 k1k2k3[ ATP] k4k5k6[NDP]
k-1
E
EP k-4
k6
k-5 E·NTP
k4·[NDP]
EP·NDP k5
E·ATP
k1· [ATP] k-1 E
k2
EP·ADP
kcat/Km conditions [S]Km:
EP·ADP k3
k-1
k-1
E
EP k-4
k6
k-5 E·NTP
EP·NDP k5
k4·[NDP]
1 = 1 + k − 2 + k3 + 1 + k − 5 + k6 kcat k 3 k 2k 3 k6 k 5k 6
kcat = k 4 k 5k 6 Km, NDP k 5 k 6 + k − 4 ( k − 5 + k 6)
kxη = kx/η kyη = ky
x= 4, -4, 6 y = 5, -5
If: If:
>
What are the interesting conclusions? With TDP (a good substrate) the rate-limiting step is sensitive to viscosity: substrate binding or product dissociation) Rate is not sensitive to PEG TDP as substrate 1.2 1
kcat/Km
PEG as macroviscogen
0.8 0.6
Sucrose as microviscogen
0.4 0.2 0 1
1.5
2 2.5 3 3.5 Relative viscosity
4
What are the interesting conclusions? With AZT-DP (a poor substrate) the rate-limiting step is not sensitive to viscosity: the chemical step Rate is not sensitive to PEG AZT-DP as substrate 1.2
PEG as macroviscogen
1
Sucrose as microviscogen
0.8
kcat/Km
0.6 0.4 0.2 0 1
1.5
2 2.5 3 3.5 Relative viscosity
4
CONCLUSION: the rate limiting step is substrate binding with a good substrate and is the chemical step with a poor substrate kcat (s-1) TDP 1120 AZT diphosphate 5.5 TDP as substrate 1.2
Km (mM) 0.095 5.0
kcat/Km (M-1 s-1) 11 700 000 1100 AZT-DP as substrate 1.2
PEG as macroviscogen
1
1
kcat/Km0.8
PEG as macroviscogen
0.8
Sucrose as microviscogen
kcat/Km0.6
0.6 0.4
Sucrose as microviscogen
0.2
0.4 0.2
0
0
1
1.5
2 2.5 3 3.5 Relative viscosity
4
1
1.5
2 2.5 3 3.5 Relative viscosity
4
The activation energy is lower for diffusion-controlled reactions than for reaction controlled by the chemical transformation TDP AZT diphosphate
kcat (s-1) 1120 5.5
Km (mM) 0.095 5.0
kcat/Km (M-1 s-1) 11 700 000 1100
TDP
AZT-DP
The isotope effect
C—H or C—D interatomic distance The energy changes during the transfer of hydrogen or deuterium from carbon. The energy of the transition state is the same for both (subject to the provisos in the text), but the hydrogen is at a higher energy in the starting materials because of its higher zero-point energy. The activation energy for the transfer of hydrogen is therefore less than that for deuterium.
G. Kinetic isotope effects Information about the extent and nature of the bond making and breaking steps in the transition state may sometimes be obtained by studying the effects of isotopic substitution on the reaction rates. The effects may be divided into two classes, depending on the position of the substitution. 1. Primary isotope effects A primary isotope effect results from the cleavage of a bond to the substituted atom. For example, it is often found that the cleavage of a C—D bond is several times slower than that of a C — H bond. Smaller decreases in rate, up to a few percent, are sometimes found on the substitution of 15N for 14N, or of 18O for 16O. The magnitude of the change in rate gives some idea of the extent of the breaking of the bond in the transition state. A simple way of analyzing isotope effects is to compare an enzymatic reac-tion with a simple chemical model whose chemistry has been established by other procedures. In this section, we are interested primarily in the empirical re-sults of the model experiments, but the following oversimplified account of the theoretical origins of the effects is helpful in understanding their nature. The plot of the energy of a carbon-hydrogen bond against interatomic distance gives the characteristic curve shown in Figure 2.10. The carbon-deuterium bond gives an
Positional isotope exchange None of the experiments just discussed gives information on whether or not PICS the reactions are associative or dissociative, since both mechanisms predict the same configurational changes. This aspect of the mechanism may be probed, however, by a different approach which is based on the procedure of molecular isotope exchange. This may be illustrated by reactions of ATP. The dissociative mechanism is stepwise with prior formation of the metaphosphate intermediate. Thus, a kinase that catalyzes the transfer of the terminal phosphoryl group of ATP to an acceptor may, in the absence of the acceptor molecule, generate metaphosphate ion and ADP in a rapid and reversible reaction.
Positional isotope exchange PICS If the /3, γ-bridge oxygen is tagged as in equation 8.43, then because of the torsional symmetry of the phosphate group, the bridge oxygen may become scrambled.44 There are several "may"s in this argument, so what is the status of data from such positional isotope exchange experiments? The rules of proof discussed at the beginning of Chapter 7 for the detection of intermediates apply equally well here. If scrambling occurs at a rate that is consistent with the value of kCM for the transfer of the phosphoryl group to the acceptor in the steady state, this is very good evidence for the importance of the dissociative route. However, for the following reasons, the absence of scrambling does not necessarily rule out the dissociative mechanism. It is possible that:
Positional isotope exchange PICS 1. There is no scrambling in the absence of the acceptor because it is required to cause a conformational change in the enzyme, or to exert an electrostatic effect on the reaction (i.e., the acceptor is a promoter). 2. The lifetime of the intermediate is very short compared with that for rotation of the phosphate, so that the intermediate reverts to substrate faster than it can rotate. 3. Rotation of the intermediate may be restricted because an oxyanion is bound by a group on the enzyme (e.g., via a bridging metal ion). Positional isotope exchange experiments conducted on creatine kinase,45 hexokinase,46 and pyruvate kinase47 have provided no evidence in favor of a dissociative mechanism. In each case, the data are consistent with a straightforward nu-cleophilic attack on phosphorus to generate a trigonal bipyramidal intermediate.47
The stereochemistry of enzymatic phosphoryl transfers So far, all evidence is consistent with the interpretation that enzymatic reactions at phosphorus proceed with inversion by an in-line associative mechanism. There has been no need to invoke adjacent mechanisms, metaphosphate intermediates, or pseudorotation. Results are summarized in Table 8.1. Phosphokinases proceed with inversion at phosphorus, and it is thus usually assumed that these reactions involve direct transfer between the two substrates. This has been challenged, however, for acetate kinase: it has been argued that the inversion results from a triple-displacement mechanism with two phosphorylen-zyme intermediates and hence three transfers.
18
O
O
P 16
O S
Most phosphodiesterases catalyze transfer of the phosphoryl group to water with retention of configuration. Again, in keeping with other evidence for phosphorylenzyme intermediates, these probably represent doubledisplacement reactions. One exception occurs with the 3'—*■ 5' exonuclease of T4 DNA polymerase (Chapter 14); its reactions proceed with inversion. Mutases are, in effect, "internal kinases," in that they transfer a phosphoryl group from one hydroxyl to another in the molecule. Unlike kinases, however, their reactions proceed with retention. Since there is generally good corroborative evidence for a phosphorylenzyme in mutase reaction pathways, double-displacement reactions are most likely.
Testing whether diffusion limited at physiological [S] • W/ [S] reduced kcat – Repeat at [S] >> KM, where v0 = kcat[E] & substrate binding/viscosity do not affect rate
Stickiness • A substrate bound can either – React or Dissociate
• Stickiness SR = # reacting / # binding – Depends on relative rates of chemistry and dissociation
• If turnover < dissociation… – rapid pre-equilibrium binding occurs – chemistry must be limiting – thus can not be a diffusion-limited perfect enzyme
Measuring Stickiness by Isotope Partitioning • • • •
Start reaction w/ radiolabeled substrate Later - dilute w/ excess cold substrate Incubate for various short times Quench & measure distribution of label in substrate & product • Multimolecular: need to test different concentrations of other substrates • Variation: change viscosity to affect diffusion rate • These are very difficult experiments
