ENZYME KINETICS A MODERN APPROACH – PART 6 pptx

112 MULTISITE AND COOPERATIVE ENZYMESrearrangement results in the following rate equation for a two-protomerallosteric enzyme: where Vmax = nkcat[ET], n is the number of protomers per en

110 MULTISITE AND COOPERATIVE ENZYMES

4 The binding affinity of a specific ligand depends on the conformation

of the enzyme (R or T), and not on neighboring site occupancy

Based on equilibrium arguments, a general expression for the velocity

of a cooperative enzyme-catalyzed reaction can be derived The rium macroscopic (KT, KR) and microscopic (kT, kR) dissociation con-stants for the different enzyme–substrate species present in a two-protomerenzyme are

CONCERTED TRANSITION OR SYMMETRY MODEL 111

To simplify the mathematical treatment, further assumptions have to bemade (see Fig 8.6):

1 Substrate can only bind to the R state of the protomer; substratedoes not bind to the T state of the protomer (c = 0).

2 The R state of the protomer is catalytically active and the T state iscatalytically inactive

3 The values of kR, kT, and L are the same for all ES n species

Thus, the rate equation for the formation of product and the mass ance for the enzyme are given by

Figure 8.6 Simplified version of the concerted transition model for a two-site cooperative

enzyme In this case the T state of the enzyme is assumed not to bind substrate.

112 MULTISITE AND COOPERATIVE ENZYMES

rearrangement results in the following rate equation for a two-protomerallosteric enzyme:

where Vmax = nkcat[ET], n is the number of protomers per enzyme, kR

is the intrinsic enzyme –substrate dissociation constant for the R-stateenzyme, and L is the allosteric constant for the R  T transition of the

native enzyme (L = [T0]/[R0])

One could envision how an allosteric effector would alter the balancebetween the R and T states, thus affectingL The presence of an activator

would lead to a decrease inL, while the presence of an inhibitor would

lead to an increase in L An activator is believed to bind preferentially

to, and therefore stabilize, the R state of an enzyme, while an inhibitor isbelieved to bind preferentially to, and stabilize, the T state of an enzyme

An activator would therefore decrease the sigmoidicity of thev versus [S]

curve, while an inhibitor would increase it

The effect of activators and inhibitors on the value of the tional equilibrium constantL can be determined from

assumed that activators bind exclusively to the R state of the protomers,while inhibitors bind exclusively to the T state of the protomers If onlyactivators or inhibitors are present, [I] or [A], correspondingly, would

be set to zero This expression could be included into Eq (8.23) This

CONCERTED TRANSITION OR SYMMETRY MODEL 113

is, however, not recommended, due to the complexity of the resultingequation and its effects on curve-fitting performance

Simulations of v versus [S] behavior using Eq (8.22) are shown in

Fig 8.7 Surprisingly, neithern nor L affect the sigmoidicity of the curve.

It is only the steepness of the curve that is affected by these parameters

As can be appreciated in Fig 8.7(a), the curve is very sensitive to the

value of n Small changes in n result in large changes in the observed

v versus [S] behavior As for the Hill model, the greater the value of n,

the more pronounced the steepness of the curve Increases in the value

of the allosteric constant L, on the other hand, lead to increases in the

steepness of thev versus [S] curve (Fig 8.7b) Thus, from a topological

perspective, the shape of the sigmoidal curve can be described by thesetwo parameters In the limit where the steepness of the curve is extreme,the sigmoidicity of the curve will not be apparent

Figure 8.7 (a) Simulation of the effects of varying the effective number of active sites

in an enzyme (n) on the shape of the initial velocity versus substrate concentration curve

for a cooperative enzyme (b) Simulation of the effects of varying the allosteric

con-stant (L) on the shape of the initial velocity versus substrate concentration curve for a

cooperative enzyme.

114 MULTISITE AND COOPERATIVE ENZYMES

It is of interest to assess the ability of these two models to describe the

v versus [S] behavior of an enzyme Figure 8.8a corresponds to a curve

fit using the Hill equation, while Fig 8.8(b) corresponds to a curve fit

using the simplified CT model The absolute sum of squares for the fit

of the Hill equation to the data set is 1.38 × 10−17 M2 min−2, while for

the CT model is 1.88 × 10−17 M2 min−2 In this case, there is no need

to carry out an F -test to decide which model fits the data best Since the

Hill equation has fewer parameters and the absolute sum of squares forthe fit of the model to the data is lower, one can safely conclude that theHill equation fits the data statistically better than does the CT model

Figure 8.8 Analysis of the initial velocity versus substrate concentration data for a

coop-erative enzyme using (a) the Hill model and (b) the MWC model.

REALITY CHECK 115

An advantage of the CT model, however, is the fact that it is possible

to estimate the magnitude of the enzyme–substrate dissociation constant

of the enzyme This is not possible with the Hill equation As describedbefore, the Hill constant is a complex term that is related but is notequivalent to, the enzyme–substrate dissociation constant By using the

CT model, it is also possible to obtain estimates of the allosteric stant, L This may prove useful in the study of allosteric modulators of

con-enzyme activity

One of the major problems with the use of any of these models, andparticularly more complex models of cooperativity and allosterism, is theinability independently to check the accuracy of the estimated catalyticparameters Even for the simple models discussed above, the experimentaldetermination of these catalytic parameters remains a daunting task In theabsence of independent experimental confirmation, estimates ofk
,n, kR,and L are nothing more than parameters obtained from curve fits of an

equation to data

In this simple treatment of cooperativity and allosterism, one should bereluctant to entertain more complex models It is our belief that an overre-ductionist approach inevitably leads to the development of extremelycomplex equations of limited analytical practicality This is due primarily

to both the excessive number of parameters to be estimated simultaneouslyand the inability ever to be able to check their accuracy independently

CHAPTER 9

IMMOBILIZED ENZYMES

The catalytic properties of an immobilized enzyme can be characterizedusing the Michaelis –Menten model The exact form of the modelwill depend on the type of enzyme reactor used In general,whenever non-steady-state conditions prevail, the integrated form of theMichaelis –Menten model is used:

K m
ln[S0][S] + [S0− S] = Vmaxt = kcat[ET]t (9.1)

where K m
is the apparent Michaelis constant for the enzyme, [ET] responds to total enzyme concentration, [S0] and [S] are, respectively,substrate concentration at time zero and timet, kcat is the zero-order rateconstant for the enzymatic reaction under conditions of substrate satura-tion, and t is the reaction time.

cor-The three main types of immobilized enzyme reactors used are batch(Fig 9.1), plug-flow (Fig 9.2), and continuous-stirred (Fig 9.3) In bothbatch and plug-flow reactors, non-steady-state reaction conditions pre-vail, while in continuous-stirred reactors, steady-state reaction conditionsare prevalent

For the case of a batch reactor, Eq (9.1) is modified to accountexplicitly for the volume of the reactor (V r) To do this, the total

116

BATCH REACTORS 117

ne/ Vr

Figure 9.1 Diagrammatic representation of a batch reactor.

[So] Q

(1 −X)[So ] Q

ne

Figure 9.2 Diagrammatic representation of a plug-flow reactor.

ne/ Vr

[So] Q

(1 −X)[So ] Q

Figure 9.3 Diagrammatic representation of a continuous-stirred reactor.

enzyme concentration term ([ET]) is substituted by n e /V r, thus yieldingthe expression

K m
ln[S0][S] + [S0− S] = kcatn e t

In this model,X is not an explicit function of time This can represent a

problem since most commercially available curve-fitting programs cannot

fit implicit functions to experimental data Thus, to be able to use thisimplicit function in the determination of kcat and K m
, it is necessary tomodify its form and transform the experimental data accordingly Dividingboth sides by t and K m
and rearranging results in the expression

ln(1 − X)

t = X[S0]

K m
t − kcatn e

A plot of ln(1 − X)/t versus X/t yields a straight line with slope =

[S0]/K m
, thex-intercept = kcatn e /V r[S0], and they-intercept = −kcatn e /

K m
V r (Fig 9.4a) The values of the slope and intercepts can readily be

obtained using linear regression Thus, from a single progress curve (i.e.,

a single X –t data set) it is possible to obtain estimates of K m
and kcat

For the case of a plug-flow reactor, the quantity V r /t in Eq (9.2) can

be substituted for by the flow rate (Q) through the packed bed, since

Q = V r /t Equation (9.2) then becomes

A plot of Q ln(1 − X) versus XQ yields a straight line with slope =

[S0]/K m
, thex-intercept = kcatn e /[S0], and they-intercept = −kcatn e /K m

CONTINUOUS-STIRRED REACTORS 119

X/t (a)

Figure 9.4 Linear plots used in determination of the catalytic parameters of immobilized

enzymes for the case of (a) batch, (b) plug-flow, and (c) continuous-stirred reactors.

(Fig 9.4b) Thus, by determining X as a function of different Q, it is

possible to obtain estimates of K m
and kcat

In a continuous-stirred reactor, steady-state reaction conditions prevail.Therefore, the model used is different from the one used for batch andplug-flow reactors For the case of a continuous-stirred reactor, the reac-tion velocity (v) equals the product of the flow rate (Q) through a reactor

120 IMMOBILIZED ENZYMES

of volumeV r times the difference between inflowing and outflowing strate concentrations, which itself equals the Michaelis –Menten modelexpression:

A plot of QX/(1 − X) versus XQ yields a straight line with slope =

−[S0]/K m
, thex-intercept = kcatn e /[S0], and they-intercept = kcatn e /K m

(Fig 9.4c) Thus, by determining X as a function of different Q, it is

possible to obtain estimates of K m
and kcat

CHAPTER 10

INTERFACIAL ENZYMES

Interfacial enzymes act on insoluble substrates Phospholipases and lipasesare two important examples from this group of enzymes Lipases, forexample, hydrolyze the ester bond of triacylglycerols, which are insol-uble in aqueous media During the digestion of lipids, triacylglycerolsare emulsified by surfactants such as bile salts, forming large emulsiondroplets Thus, to hydrolyze triacylglycerols, lipases must first bind tothe oil droplets The kinetics of this binding process are described by arate constant of adsorption and a rate constant of desorption (Fig 10.1).Upon binding to the interface, the enzyme will usually undergo a structuralchange and adopt an interfacial conformation (Fig 10.1) Once bound, theenzyme is effectively sitting on the substrate that it must act on— at theinterface between oil and water The concept of substrate concentration israther difficult to define in this case More relevant to the case of interfacialcatalysis is the concept of concentration of interfacial area or the amount

of interfacial area per unit volume ([As]) As depicted in Fig 10.2, for agiven amount of substrate, the smaller the substrate droplets, the greaterthe amount of interfacial area per unit volume Thus, for a given amount

of substrate, an interfacial enzyme would “see” a higher effective substrateconcentration in case 1 versus case 2 The use of volumetric substrate con-centrations in the treatment of interfacial enzyme kinetics is therefore notrecommended The amount of available interfacial area per unit volumeeffectively becomes the substrate concentration in this treatment

In determination of the catalytic parameters of an enzyme-catalyzedinterfacial reaction, increasing amounts of substrate are added to a solution

121

122 INTERFACIAL ENZYMES

kon

koff

Substrate Droplet

Interface

Figure 10.1 Binding of an interfacial enzyme to a substrate interface Upon binding, the

enzyme adopts an interfacial conformation The kinetics of binding is described by the rate constants of binding (kon ) and dissociation (koff ).

[As

2 ] [As

1 ] > [As

2 ]

Figure 10.2 Decreases in the amount of interfacial area per unit volume on increases in

the size of the globules at a fixed substrate concentration.

containing a fixed amount of enzyme The velocity of the enzymaticreaction is then determined at each substrate concentration As before, thisvelocity versus substrate concentration curve is used in the determination

of the apparent catalytic parameters Increasing substrate concentrationrefers to the increase in the number of substrate droplets present in thesystem This effectively results in an increase in the amount of interfacialarea per unit volume, which translates into a higher reaction velocity

10.1.1 Interfacial Binding

In this treatment we consider the binding of an interfacial enzyme to asubstrate interface to be accurately described by the Langmuir adsorption

The change in interfacial enzyme coverage (θ) as a function of time

can be expressed as

d θ

d t = kon[E](Emax∗ − E∗)[A s]− koff(E∗)[A s] (10.2)

wherekonis the rate constant for the adsorption, or binding, of enzyme tothe interface, koff is the rate constant for the desorption, or dissociation,

of enzyme from the interface, [E] represents the concentration of freeenzyme in solution (mol L−1), and [A

s] corresponds to the amount ofsurface area per unit volume in the system (m2 L−1).

At equilibrium, d θ/dt = 0, and Eq (10.2) can be rearranged to

θ = (E

∗) (Emax∗ ) = [E]

In this model it is assumed that the rate-limiting (slow) step in the tion is still the breakdown of substrate to product We also treat enzymeinterfacial binding as an equilibrium process that can be described by

reac-an equilibrium dissociation constreac-ant (K d∗) We also assume that once the

124 INTERFACIAL ENZYMES

enzyme has partitioned toward the interface, it will rapidly bind substrate.Thus, interfacial binding and substrate binding are grouped as a singlestep in this treatment This assumption was made because of difficulties

in defining substrate concentration at the interface, since the enzyme isbound to an interface composed of substrate More appropriate perhapswould be a treatment that considers the extraction of a substrate moleculefrom the interface to the enzyme’s active site This possibility, however,was not explored further in this treatment An important consideration

in enzyme interfacial catalysis is the loss of activity of the enzyme atthe interface Enzyme inactivation will happen at the interface, both uponinitial binding and in time In this treatment velocity measurements takeplace in the initial region where time-dependent enzyme inactivation isminimal For the instantaneous (initial) component of enzyme inactiva-tion, if a constant proportion of enzyme is inactive during measurements

of enzyme activity, this will translate into a decrease in the specific ity of the enzyme This may lead to an underestimation of the values

activ-of Vmax and kcat, without affecting estimates of K d∗ The effects of thisconstant amount of inactive enzyme can be factored out by determining(Emax∗ ) properly, as described below

As discussed previously, the rate equation for the formation of product,the dissociation constants for enzyme–interface and enzyme–substratecomplexes, and the enzyme mass balance are, respectively,

Thus, a velocity versus “substrate concentration” (α) plot is still a

rectan-gular hyperbola (Fig 10.3) It is informative to explore the effects of the