Transcript Kinetics of multi substrate enzyme catalysed reactions

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
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
Sequential bi-bi
BX
A
AX
E.AX
E-BX
E.AX.B
E.A.BX
E.A
AX
BX
A
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
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 AX B
v B
Km AX   KmAX B  AX B  K sAX KmB
•
Vmax : max vo when both AX and B are saturating
KmAX : [AX] which gives 1/2V when B is saturating
max
K mB : [B] which gives 1/2V when AX is saturating
max
KSAX : dissociation constant for E + AX
EAX
Steady state kinetics-2
Vmax AX B
v B
Km AX   KmAX B  AX B  K sAX KmB
At very large [B]:
Vmax AX 
v
AX   KmAX
At constant but non saturating [B]:

B
K1  B
K m  B 
Vmax K1 AX 
v
AX   K2
K sAX KmB  KmAX B
K2 
KmB  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
General rate equation of Dalziel
(1957)
[E ]
AX
B
AXB
 0 


v
[ AX ] [ B] [ AX ][B]
 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
King and Altman procedures
(1956)
Extremely useful when the mechanisms involved in calculating kinetic
constants are laborious
Example: Ordered bi-bi
[enzymeform] sum of kappaproductsof thatform

[ Etotal ]
sum of all kappaproducts
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
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
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
For varied [A]
For varied [B]
Compulsory ordered
A binding first
A
C
N
B
U
C
Compulsory ordered
B binding first
A
C
U
B
N
C
Random ordered
A
C
N
B
N
C
A
C
U
B
U
C
Ping-pong
Inhibitor pattern observed
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
Isotope exchange studies-2
• General group transfer reaction:
AX + B
A + BX
[ BX ][ A]
• Equilibrium constant: K eq 
[ AX ][B ]
• 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
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
E
E
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
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
Adair equation
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:
Y
Kb [ S ]
1  Kb [ S ]
Identical for a protein with
a single binding site...
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
A schematic
presentation of
positive cooperativity
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
Models of allosteric behavior
• Velocity equation for KNF model
Ks
aKs
S
+S
S
S
+S
• 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 ] [ S ]2 

Vmax 

2 
K S aKS 

v
2[ S ] [ S ]2
1

K S aKS2
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
Models of allosteric behavior
Velocity equation for MWC model
• Allosteric constant: L 
[T0 ]
[ R0 ]
• Assume an enzyme in which the T
state has no affinity at all for the
substrate and has h number of ligand
binding sites
v
[S ]  [S ] 
1 

Vmax
KS  KS 
 [S ] 

L  1 
 KS 
h
h 1
KS
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
MWC vs KNF
Models of allosteric behavior
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
Vmax[ S ]h
v
K D  [ S ]h
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
• n > h > 1.0
• h =n
• h < 1.0
no cooperativity (same
as the MM equation)
positive cooperativity
completely cooperative
negative cooperativity
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
v0
[ ES ]


Vmax [ E0 ]

v0

1   Vmax  v0
  
log Kb  n log[S ]  log

1 
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
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 nonheme 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
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
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
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
Hemoglobin
T-R Shift
Hemoglobin
Oxygen Binding Curve
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
Allosteric Enzymes
9-X 
3-X 
80-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 
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
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
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
Some Examples
Aspartate transcarbamoylase (ATCase)
T state
Key catalytic residues too far apart
R state
Key catalytic residues in proper positions
Some Examples
Phosphofructokinase
When a multi-subunit enzyme
is fully in the active form, it
approximates MichaelisMenten kinetics
(hyperbolic curve)