1. Kinetics of multi substrate enzyme catalysed reactions
Cleland Nomenclature for Enzymes
Cleland has devised a standardized way of referring to bisubstrate (Bi-Bi) enzymatic reactions, which make up 60% of all enzymatic transformations. The substrates, products and stable enzyme forms are denoted as follows:
Substrates are lettered A, B, C and D, in the order that they are added to the enzyme
Products are lettered P, Q, R and S, in the order that they leave the surface of the enzyme
Stable enzyme forms are lettered E, F and G, in the order that they occur
The number of reactants in the reaction are designated by the terms Uni, Bi, Ter and Quad
These are transfer reactions so can be presented as
AX + B BX + A

2. Sequential bi-bi
The first important type of bi-bi reaction is known as sequential, which means that all substrates must add to the enzyme before any reaction takes place
The sequential bi-bi can be
random, any substrate can bind first to the enzyme and any product can leave first
ordered, meaning that the substrates add to and products leave the enzyme in a specific order
A ternary complex (E + both substrates) is formed in both cases

3. Sequential bi-bi
E.A AX
E.AX.B
E.A.BX
E.AX A BX
E-BX

4. Ping-pong bi-bi (double-displacement)
One substrate bind first to the enzyme followed by product P release
Typically, product P is a fragment of the original substrate A
The rest of the substrate is covalently attached to the enzyme E, which we now designate as F
Now the second reactant, B, binds and reacts with the enzyme to form a covalent adduct with the covalent fragment of A still attached to the enzyme to form product Q
This is now released and the enzyme is restored to its initial form, E

5. Steady state kinetics-1
The general rate equation of Alberty (1953)
Many two-substrate reactions obey the MM equation with respect to one substate at constant concentration
Vmax : max vo when both AX and B are saturating
: [AX] which gives 1/2Vmax when B is saturating
: [B] which gives 1/2Vmax when AX is saturating
: dissociation constant for E + AX EAX

6. Steady state kinetics-2
At very large [B]:
At constant but non saturating [B]:
It works well for reactions using 1 or 2 substrate and producing 1 or 2 products but for more complex reactions, other approaches are used

7. General rate equation of Dalziel (1957)
 terms: kinetic coefficients found from primary and secondary plots
Primary plots of [E]/v versus 1/[AX] at constant [B] are drawn for series of different [B]
Secondary plots:
Slope vs 1/[B]  intercept: AX, slope: AXB
Intercepts vs 1/[B]  intercept: 0, slope: B

8. King and Altman procedures (1956)
Extremely useful when the mechanisms involved in calculating kinetic constants are laborious
Example: Ordered bi-bi

9. The use of primary plots
Lineweaver-Burk plots can be plotted at varying [A] and different fixed values of [B]
Ordered and random-sequential mechanisms can be distinguished from ping-pong mechanism, BUT NOT from each other by the use of primary plots
Competitive inhibitors or isotope exchange studies are used to differentiate these two mechanisms

10. Use of inhibitors
If there is an available competitive inhibitor for one of the substrates, addition of this compound will slow the overall forward rate and can allow the determination of the exact mechanism
Inhibitor can be a dead-end inhibitor or the product inhibitor
For an enzyme that requires 2 substrates, a competitive inhibitor of one of the substrate binding sites will display the behavior of competitive, noncompetitive and even uncompetitive inhibitor, depending on:
which substrate is varied
if inhibitor is reversible dead-end or product inhibitor
the mechanism of substrate interaction with enzyme

11. Dead-end inhibition patterns for a bi-bi reaction
Cleland has formulated a series of rules which enable the inhibition patterns for a particular mechanism to be predicted
Mechanism
Competitive inh. for substrate
Inhibitor pattern observed
For varied [A]
For varied [B]
Compulsory ordered
A binding first A C N B U
B binding first
Random ordered
Ping-pong

12. Isotope exchange studies-1
Rate of exchange between a radiolabeled substrate and a product under equilibrium conditions
First simple test: if exchange occurs between a substrate and a product when enzyme (+) but second substrate (-)  ping-pong mechanism...
e.g. Sucrose phosphorylase
Isotope exchange btw sucrose (S1) and fructose (P1) (no S2 and P2)
Sucrose fructose Pi G-1-P
E E
E.sucrose E.glucose.fructose E-glucose E.glucose-1-P

13. Isotope exchange studies-2
General group transfer reaction:
AX + B A + BX
Equilibrium constant:
Procedure:
Add a small amount of radioactively labelled B
Measure rate of formation of BX
Increase the concentrations of A and AX, keeping [A]/[AX] constant
The equilibrium concentrations of B and BX will remain unchanged but the rate of isotope exchange will be affected

14. Isotope exchange studies-3
B AX A BX
E E
EB E.B.AX E.BX.A EBX
Slight increase in [A] & [AX] may increase the rate of isotope exchange but substantial increase will force the formation of E.B.AX and E.BX.A, making it more difficult for B to dissociate from EB and BX from EBX  exchange rate will reduce
AX B BX A
EAX E.AX.B E.A.BX EA
Free enzyme forced to EAX and EA forms: EAX reacts with B and EA does not affect the initial velocity of liberation of BX from E.A.BX  exchange rate will increase

15. Binding of ligands to proteins
Binding of more than one ligand to an oligomeric receptor (or enzyme in our case) may occur sequentially with binding constants that may not be equal
The fractional saturation of such binding sites is described by the Adair equation (Gilbert Adair, 1924)
ES denotes enzyme-substrate complex and numbers (0, 1, 2, etc) are the number of substrate attached
Such a description don't say how it happens, i.e. why the first binding constant is weak and the second is strong
Thus, the Adair equation provides no physical insight as to why various microscopic dissociation differ from each other

16. Adair equation

17. Adair equation No interaction between binding sites:
Dimer: M2 + S  M2S (binding constant Kb1)
Protomer: M + S  MS (binding constant Kb)
Forward: dimer has two binding sites so ligand is 2 times more likely to bind
Reverse: in both cases, there is only one site that S can dissociate
 for overall reaction: Kb1=2Kb
M2S + S  M2S2 (Kb2)
Forward: both dimer and protomer have 1 free binding site
Reverse: two S can dissociate from dimer, only 1 from protomer
 for overall reaction: Kb2=1/2 Kb
If we substitute these to the general equation:
Identical for a protein with a single binding site...

18. Cooperativity
If there are more than one binding sites, there is a possibility of interaction btw the binding sites  cooperativity
Positive cooperativity
Negative cooperativity
Homotropic cooperativity
Heterotropic cooperativity
Allosteric inhibition: negative heterotropic
Allosteric activation: positive heterotropic

19. A schematic presentation of positive cooperativity

20. Models of allosteric behavior
Sequential model
(Koshland, Némethyl & Filmer (KNF), 1966)
Subunit interface is changed. Binding of substrate to one active site causes T to R transition and affinity of other subunit for substrate is increased due to substrate interface being altered

21. Models of allosteric behavior
Velocity equation for KNF model
Ks: dissociation constant
For positive cooperativity  a < 1.0
The model can be extended for a tetrameric enzyme and in this case, the effect of second and third substrate binding to KS is given by:
abKS & abcKS S Ks +S
aKs

22. Models of allosteric behavior
Concerted transition or symmetrical model
(Monod, Wyman & Changeux (MWC), 1965)
In a protein, all of the protomers are in the same conformational state: all must be in R- or T-form, no hybrids...
Two conformational forms are in equilibrium in favor of T-form in the absence of the ligand
Binding of the ligand shifts the equilibrium

23. Models of allosteric behavior
Velocity equation for MWC model
Allosteric constant:
Assume an enzyme in which the T state has no affinity at all for the substrate and has h number of ligand binding sites KS

24. Models of allosteric behavior
The currently accepted model for allosteric inhibitors and activators is based on the concerted model (MWC)
Inhibitors lock all subunits in the T- form
Activators lock all subunits in the R-form
The MWC model is useful to understand positive homotrophic cooperativity BUT can not explain the negative homotrophic cooperativity. KNF sequential model is usually used for that

25. MWC vs KNF

26. Models of allosteric behavior

27. A more general, simple equation
In the case of high cooperativity....
Y (or θ) = fractional saturation
h is the Hill constant and n is total number of substrate binding sites or

28. Hill Plot
When the cooperativity is moderate (that is in reality), the experimental data often still well modelled by this equation BUT h will no longer be equal to the number of binding sites (h may not be an integer)
Possibilities:
h =1.0 no cooperativity (same as the MM equation)
n > h > 1.0 positive cooperativity
h =n completely cooperative
h < 1.0 negative cooperativity

29. Hill Plot
At θ <0.1 and >0.9  slope approaches to 1 (i.e. no cooperativity)
Hill coefficient is calculated from the central linear portion of the graph
In case MM assumptions valid:
v0 is proportional to [ES] and

30. Binding of oxygen to Haemoglobin
In 1904, Bohr and coworkers
Fractional saturation of Hb with O2 vs pO2  sigmoidal curve
In 1909, Hill explained this on the basis of interaction of binding sites
He assumed complete cooperativity and derived Hill equation
He found h=2.8 for HB
In 1925, Adair developped more general equation for ligand binding
In 1960, X-ray data: binding sites are quite apart  cooperativity should be the result of interaction of subunits

31. Some Facts Oxygen has low solubility in blood (0.1mM)
Whole blood, which contains 150 g Hb/L, can carry up to 10 mM oxygen
Invertebrates can have alternative proteins for oxygen binding, including hemocyanin, which contains Cu and hemerythrin, a non-heme protein
On binding oxygen, solutions of Hb change color to bright red
Solutions of hemocyanin (most molluscs, and some arthropods) and hemerythrin change to blue and pink-violet colored, respectively
Some Antarctic fish don't require Hb since oxygen is more soluble at low temperature 

32. Hemoglobin and Myoglobin Structure
Hemoglobin of higher vertebrates is made up of two types of chains, referred to as  and β
The hemoglobin molecule is a 2-2 tetramer
Their primary structures are compared with that of myoglobin
The  and β sequences have considerable similarity to one another and some similarity to that of myoglobin
The myoglobin and hemoglobin chains have very similar tertiary structures
Protein structure affects ligand binding  CO binds 20,000x better than O2 to heme but only 200x better to Mb

33. Hemoglobin and Myoglobin Relation to Hill Plot
In the deoxy conformation  the binding initially occurs along the line corresponding to the weak-binding state
But partial oxygenation favors transition to the strong-binding oxy state
As oxygen is bound, more and more of the remaining available sites are in hemoglobin molecules that have this conformation
The binding curve passes over to that for the strong-binding state

34. Effect of O2 Binding
Heme is distorted into a dome shape and the axis of His is tilted by about 8°
When oxygen binds, it flattens the heme
Both the His and Val are too close to the heme...
His changes its orientation toward the perpendicular. This movement distorts and weakens the whole complex of H bonds and salt bridges
In the simplest terms, the binding of O2 pulls the iron a fraction of a nanometer into the heme, producing a lever effect which results in a much larger shift in the surrounding structure, particularly at the critical interfaces

35. Hemoglobin T-R Shift

36. Hemoglobin Oxygen Binding Curve

37. Allosteric Enzymes
If you examine the Michaelis Menten equation you will find that an increase in v from 0.1 to 0.9 Vmax requires an 81-fold change in substrate concentration. In other words the velocity is rather insensitive to substrate concentration
In allosteric enzymes, however, a small change in one parameter, e.g. substrate, inhibitor, activator concentration, brings about a large change in velocity
A consequence of a cooperative system is that the v vs. S plot is no longer hyperbolic
Most allosteric enzymes are oligomeric
They are generally located at or near branch points in metabolic pathways, where they are influential in directing substrates along one or another of the available metabolic paths

38. Allosteric Enzymes 9-X  80-X  3-X 
When h = 1, need ~80-fold  in [S] to  V0 by 9-fold
When h = 4, only need 3-fold  in [S]
When h < 1, don’t reach 90% of Vmax even at 1000-fold 

39. Some Examples Threonine deaminase in E.coli (Abelson, 1954)
Addition of isoleucin inhibited the formation of itself
Substrate: thereonine and inhibitor: isoleucine bind to different sites

40. Some Examples Aspartate transcarbamoylase (ATCase)
Aspartate carbamoyl transferase from E.coli is the first enzyme in which the active and regulatory sites were shown to be clearly separated
They are even in different subunits... Most allosteric enzymes are oligomers consisting of identical subunits
Aspartate + carbamoyl phosphate  N-carbamoyl aspartate + Pi
First reaction leading the biosynthesis of pyrimidine nucleotides (UMP, UDP, UTP and CTP)
Repressors: Uracil and CTP (metabolic end-product)
Activator: ATP

41. Some Examples Aspartate transcarbamoylase (ATCase)
1962: CTP  and ATP  sigmoidal behavior of the curve
1965: catalytic (c) and regulatory (r) sites are in different subunits
1968: sequence analysis and X-ray  c3(r2)3c3 and structural integrity maintained by Zn2+ ions
ATP and CTP are competing for the same binding site
1990: site-directed mutagenesis and X-ray  each catalytic subunit has separate domains for aspartate and carbamoyl phosphate binding
In T-form, binding sites are further apart
In T-form, two catalytic trimers are close to each other, hindering the access to active sites
T to R change is in concerted (symmetrical) fashion

42. Some Examples Aspartate transcarbamoylase (ATCase)
T state
Key catalytic residues too far apart
Key catalytic residues in proper positions
R state

43. Some Examples Phosphofructokinase
When a multi-subunit enzyme is fully in the active form, it approximates Michaelis-Menten kinetics
(hyperbolic curve)