Enzyme Kinetics: The study of reaction rates. For the one-way st -order reaction: S the rate of reaction (V) is: V P [ P] moles / L t sec For each very short segment dt of the reaction: d[ P] d[ S] V dt dt P t n a st -order reaction: V k [S] k is a rate constant with units /s = s - ; V has units of moles per litre per second (mol L - s - ). n a reversible st order reaction, S k P k - the net reaction rate V is V k [S] k [P] At equilibrium, V = 0 and, k [S] eq k [P] eq or [P] eq [S] eq k k K eq
n a one-way 2 nd order reaction, A + B C the rate of reaction is V k[ A][ B] The rate constant k has the units litres moles sec moles litre sec litre moles sec moles litre moles litre n a 2 nd order reversible reaction, A + B k C k - the net rate of reaction is V k [A][B] k [C] The rate constant k has the units litres moles sec The backward rate constant k has the units s - 2
Factors affecting Enzyme-Catalyzed Reactions E + S ES P + E. Enzyme Concentration Usually [E] << [S], [S] can be considered a constant at the start of a reaction At time = 0, it is a one way reaction: V o = k [E][S]= k [E] f [S] is the same in each assay, a graph of V o vs. [E] is a straight line: Vo (M/s) [E] (M) 2. Substrate Concentration: The curve is hyperbolic. (M/s) [S] (M) At low [S], most of the E is unbound. As [S] increases, more and more E. S forms and increases. At high [S], all the E is present as E. S so adding S cannot increase. The E is saturated with S at. 3
Note: will be different for different [E]. We will now present an equation to describe this curve. t is based on the following mechanism: k E + S E. S E + P k - k - 2 k, k - etc. are rate constants. k 2 We define the Michaelis Constant k 2[ES][S] [S] k k 2 k k k 2 k When the Enzyme is saturated, [E. S] = [E tot ] and k 2 [E tot ] Therefore, [S] [S] This is the Michaelis-Menton Equation. t describes the relationship between and [S]; and are characteristics of particular enzymes. is the maximum rate of reaction at a particular [E tot ]. 4
When V then and 2 max 2 [S] [S] 2 V [S] max [S] [S] and! So is numerically equal to the [S] at which 2 V o V max 2 [S] Recall that. Quite often k 2 << k K - m k k 2 So k k k K S [E][S] [E S] which is the equilibrium Enzyme-Substrate dissociation constant, a measure of the affinity of S for E. k - E. S E + S k Big K S or means weak binding. Small K S or means strong binding. One way to determine and is by non-linear least-squares fitting of the Michaelis-Menton equation to measured data. Another way is a reciprocal plot of the MM equation. 5
The inverse of [S] [S] is [S] Y = m X + b Slope Y int ercept X int ercept V o (s/m) - / [S] (M - ) Recall that depends on the [E] and when the E is saturated with S V and V max k 2 [E tot ] max k 2 k cat E tot 6
So k cat is an E-parameter that is independent of [E] and reports how fast an E works. e.g. At ph 7, 35 o C carbonic anhydrase hydrates 4x0 5 moles of CO 2 yielding 4x0 5 moles of HCO 3 - per mole of enzyme per second k 2 is a st -order rate constant. k cat can be used for non-m-m enzymes as well. Since: k cat [E tot ][S] [S] when [S] is << k cat [E tot ][S] k cat The units are is a 2 nd order rate constant that measures how fast E and S react. litres moles sec The fastest that two molecules can diffuse together is about 0 9-0 M -. s - so this puts a limit on how fast the reaction can go. k cat ranges from 0.36 to 2.4 X 0 8 M -. s -. 7
3. Many enzymes exhibit an optimum ph at which maximum activity occurs. Pepsin =.5 Carboxypeptidase A = 7.5 Arginase = 9.7 log V o Why?.5 7.5 9.7 ph i. ph may change the ionization states of AA in the active site. ii. ph may change the ionization state of S. iii. ph may denature the E. 4. Temperature dependence is determined by: G RT ln e ( k h k B T ) k k B T h e ( G R T ) and by denaturation at high and low T. 00 % Activity 0 40 T o C 8
5. nhibitors i. Some are foreign, e.g. poisons, antibiotics, pesticides, herbicides. ii. Some are naturally present for regulation of enzymatic activity. Competitive nhibitors () is similar in structure to S and binds competitively to the same site on the E as S. E S E e.g. malonic acid, HOOCCH 2 COOH, is an inhibitor of succinate dehydrogenase: Succinic acid = HOOCCH 2 CH 2 COOH How does affect the V o? At high [S] the is displaced from E so is unchanged. E + S + ES + E V o 2 [S] 9
But is apparently increased because it takes much more S to reach. Recall that is the apparent affinity of E for S. V o -/ -/ / [S] Non-Competitive nhibitors binds at a site distinct from the substrate site, usually an allosteric site. Allos -Greek- other Stereos - shape. E + S ES E + P S + + E K K E + S ES X t may bind to free E or to ES. Once bound, it will prevent P formation. f the binding affinities of to E and ES are identical, there will be no effect on. 0
V o V max V max 2 But since the decreases active [E tot ] then must decrease [S] (= k cat [E tot ] ). / / -/ / [S] So graphical analysis can give important information about the nature of the inhibitor and its mechanism of action. Regulatory Enzymes A metabolic pathway is one in which the product of the st Enzyme is a substrate for the 2 nd Enzyme etc. L-Thr E E 2 E 3 E 4 E 5 A B C D L-le Often regulation occurs at the beginning of the path to prevent waste. E is Threonine dehydratase. t is a key regulatory enzyme in the pathway and is inhibited by L-le.
End-Product nhibition / Feedback nhibition The regulation shown above is called heterotropic because the enzyme is regulated by molecules that are not S or P of the enzyme being regulated. f S or P modulate the activity of their own enzyme the regulation is called homotropic. Directed Overflow Regulation CTP is synthesized from Asp in 7 steps. CTP feedback inhibits its own synthesis by inhibiting the first step of the pathway. However, when demand is low, CTP can still build up and it is eliminated by degradation to uracil followed by secretion. Allosteric Enzymes The vs [S] plot is Sigmoidal whereas MM plots are Hyperbolic. The enzyme will be very sensitive to [S] over a narrow range and behaves like an on-off switch. Hyperbolic Sigmoidal [S] 2
/ vs /[S] curves are not linear. t is not proper to speak of so S 0.5 is used for [S] needed to reach /2.. Allosteric enzymes are usually oligomeric, multi-subunit complexes. i.e. They have quaternary structure. 2. They often exhibit cooperativity. The binding of S to one active site of the enzyme makes the binding of subsequent S easier. How? 3. Each subunit can exist in 2 conformations: Low affinity - T-state (taut) High affinity - R-state (relaxed) Without S, the equilibrium favours T and the affinity of S for E is low. T R Lots Little When small amounts of S are added they don t bind because the affinity of T is too low. As S begins to bind it reveals more high affinity sites permitting more binding. 3
Binding of S stabilizes the R-state, pulling the equilibrium to the high affinity form simply by Le Chatelier s Principle: T R + S ncreasing the R-state, reveals more high-affinity sites and results in more S binding which continues until the E is saturated. S This can be shown to produce sigmoidal vs. [S] graphs. etc S nhibitors stabilize the T-state, increasing S 0.5, and making the enzyme less sensitive to substrate. e.g. ATP and citrate inhibit phosphofructokinase. Activators stabilize the R-state, decreasing S 0.5, and making the enzyme more sensitive to substrate. E.g. AMP increases PFK activity. V o +AMP +ATP [Fructose-6 phosphate] 4
Such inhibitors and activators do not bind at the substrate binding site. They bind at other sites and change the conformation of the protein. Hemoglobin binds O 2 cooperatively. This allows it to respond to changes in O 2 demand by different tissues. O 2 binding is also inhibited allosterically by 2,3- bisphosphoglycerate. Because fetal hemoglobin has a lower affinity for BPG it has a higher affinity for O 2. This permits the fetus to extract O 2 from the mother s blood. Covalent Regulation e.g. Glycogen phosphorylase is activated by phosphorylation of Ser. This is reversible by a phosphatase that removes the phosphate. Other possibilities: AMP, UMP, methyls, ADP ribose, sugars etc. can be added and removed. 5
n cancer cells, addition of N-acetyl-Glucose to Ser-529 of phosphofructokinase- inhibits the enzyme slowing down glycolysis and speeding up the pentose phosphate pathway. This permits cells to make more biosynthetic precursors and NADPH allowing faster growth. These changes permit much faster metabolic changes than are possible by gene regulation. rreversible nhibitors Usually they form a covalent bond with an active site AA. e.g. Penicillin binds to the active site Ser in transpeptidase, an enzyme involved in bacterial cell wall synthesis. e.g. Diisopropylfluorophosphate binds to Ser-95 in chymotrypsin. Zymogens: Enzyme activation by proteolytic cleavage. e.g. The pancreas produces inactive trypsinogen, chymotrypsinogen, proelastase, and procarboxypeptidase to prevent digestion of the pancreas. n the gut, a duodenal enzyme enteropeptidase activates trypsin by removing AA -6 of trypsinogen producing active trypsin. Then Trypsin 245 Chymotrypsinogen 5 6 245 -Chymotrypsin 3 6 46 49 245 -Chymotrypsin 6