1. Reaction kinetics: 1st order reactions•
[A] t
Decay reactions, like radio-activity;
SN1 reactions
Rate: -
Rewriting: -
Integration gives:
So: ln[A]t – ln[A]0 = -kt or:
-kt

2. The time for half of the reactant initially present to decompose, its half-time or half-life, t1/2 , is a constant and hence independent of the initial concentration of reactant.
By substituting the relationship [A] = [A0] / 2 when t = t1/2 into ln [A]=ln [A]0 - kt
and rearranging:
t 1/2 = ln2/k = 0.693/k

3. Second-order reactionP
The half-time for a second order reaction is expressed t 1/2 = 1/k [A] and therefore, in contrast to a
first order reaction depends on the initial reactant concentration.
A+B P
Here, the reaction is said to be first order in A and first order in B.
Unimolecular and bimolecular reactions are common. Termolecular reactions are unusual because the simultaneous collision of three molecules is a rare event. Fourth and higher order reactions are unknown.

4. Enzyme Kinetics ß-fructofuranosidase: Sucrose + H2O glucose + fructose
When [S] » [E] : the rate is zero order with respect to sucrose

5. The Michaelis-Menten Equation
This equation cannot be explicitly integrated, however, without simplifying assumptions, two possibilities are
1. Assumption of equilibrium. Leonor Michaelis and Maud Menten, building on the work of Victor Henri, assumed that k-1 » k2, so that the first step of the reaction reaches equilibrium.
Ks is the dissociation constant of the first step in the enzymatic reaction

6. The Michaelis-Menten Equation
Assumption of steady-state. Figure illustrates the progress curves of the various participants in reaction
under the physiologically common conditions that substrate is in great excess over
Enzyme ([S] » [E]).
ES maintains a steady state and [ES] can be treated as having a constant value:
The so called steady state assumption, a more general condition than that of equilibrium, was first proposed in 1925 by G. E. Briggs and B. S. Haldane

7. The Michaelis-Menten Equation
Letting [E] = [E]T - [ES] and rearranging yields
The Michaelis constant, KM , is defined as
Solving for [ES],

8. The Michaelis-Menten Equation
The expression of the initial velocity (v0) of the reaction, the velocity
at t=0, thereby becomes
The maximal velocity of a reaction, Vmax occurs at high substrate concentrations when the enzyme is saturated, that is, when it is entirely in the ES form
Therefore, combining the last two equations, we obtain:
This expression, the Michaelis-Menten equation, is the basic equation of enzyme kinetic.

9. Significance of the Michaelis Constant
The Michaelis constant, KM, has a simple operational definition. At the substrate concentration at which [S] = KM, this equation
yields v0 = Vmax/2 so that
KM is the substrate concentration at which the reaction velocity is half maximal

10. Analysis of Kinetic Data
Lineweaver-Burk or double-reciprocal plot
Analysis of Kinetic Data

12. S >> Km
vi=Vmax
Vmax= k2Et

15. k2= 10/5 = 2 moles/mg seg Vmax= 10 M/seg Km=10 x10-5 M
Si en el ensayo se usaron 5mg/L de preparación enzimática, entonces:
v= Vmax = k2 ET
k2= 10/5 = 2 moles/mg seg
¿Qué predicciones podemos hacer a partir de esta información?

16. Significance of the Michaelis Constant
The magnitude of KM varies widely with the identity of the enzyme and the nature of the substrate. It is also a function of temperature and pH. The Michaelis constant can be expressed as
Since Ks is the dissociation constant of the Michaelis complex, as Ks decreases, the enzyme’s affinity for substrate increases. KM in therefore also a measure of the affinity of the enzyme for its substrate, provided k2/k1 is small compared to Ks, that is k2 ‹ k-1 so that the ES P reaction proceeds more slowly than ES reverts to E + S
kcat/KM Is a Measure of Catalytic Efficiency
We can define the catalytic constant, kcat, of an enzyme as
This quantity is also known as the turnover number of an enzyme because it is the number of reaction processes (turnovers) that each active site catalyzes per unit time.

17. Turn Over Numbers of Enzymes
Substrate
kcat (s-1)
Catalase H2O2
40,000,000
Carbonic anhydrase HCO3-
400,000
Acetylcholinesterase Acetylcholine
140,000
b-Lactamase Benzylpenicillin
2,000
Fumarase Fumarate
800
常數 k3 控制的是生成物的產生速率，因此也可看作一個酵素最後轉換出生成物的速率，特稱之為 turn over number (t.o.n)，也是一個重要指標。 注意其單位為 s-1，亦即每秒鐘能夠轉換得到生成物的分子數目；例如上面 RecA 的 t.o.n. 為 0.4，表示此酵素要 2.5 秒才能使用一分子 ATP 生成 ADP (DNA 重組時需要此酵素)。
RecA protein (ATPase) ATP
0.4
The number of product transformed from substrate
by one enzyme molecule in one second
Adapted from Nelson & Cox (2000) Lehninger Principles of Biochemistry (3e) p.263

18. kcat/KM Is a Measure of Catalytic Efficiency
When [S] « KM, very little ES is formed. Consequently, [E] ≈ [E]T, so
reduces to a second-order rate equation:
The quantity kcat/KM is a measure of an enzyme’s catalytic efficiency.
There is an upper limit to the value of kcat/KM : It can be not greater than k1; that is, the decomposition of ES to E + P can occur no more frequently than E and S come together to form ES. The most efficient enzymes have kcat/KM values near to the diffusion-controlled limit of 108 to 109 M-1.s-1

20. Chymotrypsin Has Distinct kcat / Km to Different Substrates
O R O
H3C–C–N–C–C–O–CH3
H H = –
kcat / Km
R =
Glycine –H
1.3 ╳ 10-1
–CH2–CH2–CH3
Norvaline
3.6 ╳ 102
–CH2–CH2–CH2–CH3
Norleucine
3.0 ╳ 103
用 kcat/Km 可以同時兼顧前半與後半反應，也就是說同時監視酵素對基質的親和力，以及該酵素的 Vmax (kcat)，是一個比較理想的酵素行為指標； 此一比值越大者，有越好的催化力。由上例可以看出，酵素對不同基質會有不同的表現，chymotrypsin 顯然偏好較大的胺基酸基團，且最好有芳香基團。
酵素的成功催化首先需與基質碰撞，而兩者的碰撞率決定在細胞內的擴散率，但是細胞內的擴散率有其極限；有人由此算出若在最高的擴散速率下，且每次碰撞都完成催化，則一個酵素最高的催化極限，其 kcat/Km 將在 10 8~10 9 (M-1s-1) 之間；的確最有效的酵素催化也都趨近此一範圍，目前最高的記錄是 triose phosphate isomerase 的 2.4×10 8。
–CH2–
Phenylalanine
1.0 ╳ 105
(M-1 s-1)
Adapted from Mathews et al (2000) Biochemistry (3e) p.379

21. - dS/dt = vi = So dX/dt Al iniciar: t = 0, S = So A cualquier tiempo:
T = t S = S
X = (So-S)/So
- dS/dt = vi = So dX/dt

23. Temperature Dependence of Enzymes
As is the case with most reactions, an increase in temperature will result in an increase in kcat for an enzymatic reaction.
From general principles, it can be determined that the rate of any reaction will typically double for every 10°C increase in temperature.
Many enzymes display maximum temperatures around 40°C, which is relatively close to body temperature.
There are enzymes that are isolated from thermophilic organisms that display maxima around 100°C, and some that are isolated from psychrophilic organisms that display maxima around 10°C.

24. Enzyme Inhibition (Mechanism)
Competitive
Non-competitive
Uncompetitive E
Substrate E X
Cartoon Guide
Compete for
active site
Inhibitor
Different site
E + S → ES → E + P + I ↓ EI ← ↑
E + S → ES → E + P
I I
↓ ↓
EI + S →EIS ← ↑
E + S → ES → E + P + I ↓
EIS ← ↑
Equation and Description
酵素 的抑制劑有不同的抑制機制，通常依照抑制劑對酵素的結合方式，可分成兩大類。其一為競爭同一活性區 (competitive)，可以用提高基質濃度的方法來競爭；另一則是結合在活性區之外的地方，又可分成 non-competitive 及 uncompetitive 兩種。後面兩種抑制方式大致相同，因此有些課本也就不再細分，其差別在於基質的結合，會不會影響抑制劑的結合。雖然這幾種抑制方式，都是可逆反應，但只有 competitive 可以用提高基質的方式來對抗抑制。
[I] binds to free [E] only,
and competes with [S];
increasing [S] overcomes
Inhibition by [I].
[I] binds to free [E] or [ES]
complex; Increasing [S] can
not overcome [I] inhibition.
[I] binds to [ES] complex
only, increasing [S] favors
the inhibition by [I].
Juang RH (2004) BCbasics

25. Competitive Inhibition
Product
Substrate
Competitive Inhibitor
Succinate
Glutarate
Malonate
Oxalate
C-OO-
C-H
C-OO-
H-C-H
C-OO-
H-C-H
C-OO-
H-C-H
C-OO-
競爭性 抑制劑通常都與正常的基質相像，可以與酵素結合，但無法繼續反應，產生生成物；因為都是競爭同一活性區，因此可提高基質來對抗抑制。
Succinate Dehydrogenase
Adapted from Kleinsmith & Kish (1995) Principles of Cell and Molecular Biology (2e) p.49

26. Sulfa Drug Is Competitive Inhibitor
Domagk (1939)
Para-aminobenzoic acid (PABA)
Bacteria needs PABA for
the biosynthesis of folic acid
-COOH
H2N-
Folic
acid
Tetrahydro-
folic acid
Precursor
-SONH2
H2N-
Sulfa drugs has similar
structure with PABA, and
inhibit bacteria growth.
磺胺藥 就是消炎藥，因為其構造類似細菌生長細胞壁所需之 PABA，會競爭性地抑制利用 PABA 的酵素，因而阻礙細菌的生長，但無法完全殺菌。
Sulfanilamide
Sulfa drug (anti-inflammation)
Adapted from Bohinski (1987) Modern Concepts in Biochemistry (5e) p.197

27. Enzyme Inhibition Competitive Inhibition
Many substances alter the activity of an enzyme by reversibly combining with it in a way what influence the binding of substrate and/or its turnover number. Substances that reduce an enzyme’s activity in this way are known as inhibitors
Competitive Inhibition
A substance that competes directly with a normal substrate for an enzyme’s substrate-binding site is known as a competitive inhibitor.
Here it is assumed that I, the inhibitor, bind reversibly to the enzyme and is in a rapid equilibrium with it so that
And EI, the enzyme-inhibitor complex, is catalytically inactive. A competitive inhibitor therefore reduces the concentration of free enzyme available for substrate binding.

28. Enzyme Inhibition Competitive Inhibition
This is the Michaelis-Menten equation that has been modified by a factor, , which is defined as
 Is a function of the inhibitor’s concentration and its affinity for the enzyme. It cannot be less than 1.

29. Enzyme Inhibition Competitive Inhibition
Recasting in the double-reciprocal form yields
A plot of this equation is linear and has a slope of KM/Vmax, a 1/[S] intercept of -1/ KM, and a 1/v0 intercept of 1/ Vmax

30. Enzyme Inhibition Uncompetitive Inhibition
In uncompetitive inhibition, the inhibitor binds directly to the enzyme-substrate complex but not to the free enzyme
In this case, the inhibitor binding step has the dissociation constant
The uncompetitive inhibitor, which need not resemble the substrate, presumably distorts the active site, thereby rendering the enzyme catalytically inactive.

31. Enzyme Inhibition Uncompetitive Inhibition
The double-reciprocal plot consists of a family of parallel lines with slope KM/Vmax, 1/v0 intercepts of ’/Vmax and 1/[S] intercept of -’/KM

32. Enzyme Inhibition Mixed Inhibition (noncompetitive inhibition)
A mixed inhibitor binds to enzyme sites that participate in both substrate binding and catalysis. The two dissociation constants for inhibitor binding
Double-reciprocal plots consist of lines that have the slope  KM/Vmax, with a 1/v0 intercept of ’/Vmax and 1/[S] intercept of -’/  KM

33. Enzyme Inhibition (Plots)
Competitive
Non-competitive
Uncompetitive
Vmax
Vmax Km
[S], mM
Vmax
[S], mM vo vo
Direct Plots
Double Reciprocal
Vmax’ I
Km’
Vmax’ I I Km Km
Km’
[S], mM
= Km’
Vmax unchanged
Km increased
Vmax decreased
Km unchanged
Both Vmax & Km decreased
1/[S]
1/Km
1/vo
1/ Vmax I
1/vo
1/ Vmax
1/[S]
1/Km
1/[S]
1/Km
1/ Vmax
1/vo I
Intersect
at X axis I
Two parallel
lines
這些 抑制機制都可以用酵素動力學來描述，使用雙倒數作圖更可明顯地指出是屬於何種抑制方式。不過，以上三種作圖都是屬於最典型者，很多時候實驗所得到的作圖結果，可能會有混合型態出現，則是較為複雜的抑制機制，或者有其他的干擾因子在內。
Intersect
at Y axis
Juang RH (2004) BCbasics

35. Bisubstrate Reactions
Almost all of these so called bisubstrate reactions are either transferase reactions in which enzyme catalyzed the transfer of a specific functional group, X, from one of the substrates to the other:
or oxidation-reduction reactions in which reducing equivalents are transferred between two substrates.
Sequential Reactions
Reactions in which all substrates must combine with the enzyme before a reaction can occur and products be released are known as Sequential reactions

36. Bisubstrate Reactions
Sequential Reactions
Ordered bisubstrate reaction
A and B : substrates in order that they add to the enzyme
P and Q : products in order that they leave the enzyme
Random bisubstrate reaction
Ping Pong Reactions
Group-transfer reactions in which one or more products are released before all substrates have been added are known as Ping Pong reactions