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Lectures 25 and 26

Active Intermediates / Free Radicals (Chapter 7)*

Mechanism:

(1)

(2)

(3)

Pseudo Steady State Hypothesis (PSSH)The PSSH assumes that the net rate of species A*

(in this case, NO3*) is zero.

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This is why the rate decreases

as temperature increases.

Enzymes (Chapter 7)

Michaelis-Menten Kinetics (Section 7.4.2)

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Bioreactors (Chapter 7)

Rate Laws

Stoichiometry

A.) Yield Coefficients

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B.) Maintenance

A Word of Caution

on

A.) Growth Phase

B.) Stationary Phase

Mass Balances

Cell:

Also, for most systems.

Substrate:

Polymath Setup

1.) d(Cc)/d(t) = - D*Cc + (rg - rd)

2.) d(Cs)/d(t) = D*(Cso - Cs) - Ysc*rg - m*Cc

3.) d(Cp)/d(t) = - D*Cp + Ypc*rg

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4.) rg = (((1 - (Cp/Cpstar))**0.52) * mumax*(Cs/(Ks + Cs))*Cc

5.) D = 0.2

6.) kd = 0.01

7.) rd = kd*CC

8.) Cso = 250

9.) Ypc = 5.6

10.) m = 0.3

11.) mumax = 0.33

12.) Ysc = 12.5

13.) Ks = 1.7

Polymath Screen Shots

Polymath Equations

Summary Table

Cc and Cp vs. Time

Cs vs. Time

Wash Out:

1.) Neglect Death Rate and Cell Maintenance

2.) Steady State

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*

All chapter references are for the 3rd Edition of the textElements of ChemicalReaction Engineering.

Back to thetop of the page.

Professional Reference Shelf

CD7.4 Multiple Enzyme and Substrate Systems

In Section CD7.2 we discussed how the addition of a second

substrate, I, to enzyme-catalyzed reactions could deactivatethe enzyme and greatly inhibit the reaction. In the present

section we look not only at systems in which the addition of a

second substrate is necessary to activate the enzyme, but also

other multiple-enzyme and multiple-substrate systems in

which cyclic regeneration of the activated enzyme occurs.

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CD7.4A Enzyme RegenerationAs a first example we shall consider the oxidation of glucose

(Sr) with the aid of the enzyme glucose oxidase [represented

as either G.O. or (Eo)] to give -gluconolactone (P):

In this reaction, the reduced form of glucose oxidase

(G.O.H2), which will be represented by Er , cannot catalyze

further reactions until it is oxidized back to Eo. This oxidation

is usually carried out by adding molecular oxygen to the

system so that glucose oxidase, Eo, is regenerated. Hydrogen

peroxide is also produced in this oxidation regeneration step.

Overall, the reaction is written

In biochemistry texts, reactions of this type involving

regeneration are usually written in the form

The reaction is believed to proceed by the following

sequence of elementary reactions:

We shall assume that reaction involving the dissociation

between reduced glucose oxidase and -lactone is rate

limiting. The rate of formation of -lactone (P1) is given by

the equation

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(CD7-

26)

After applying the pseudo-steady-state hypothesis to the rates

of formation of(Eo Sr ), (Er So), and (Er),

we can solve for the following concentrations of the active

intermediates in terms of the concentrations of glucose,

oxygen, and unbound oxidized enzyme.

(CD7-

27)

(CD7-28)

(CD7-

29)

After substituting Equation (CD7-28) into Equation (CD7-

26), the rate law is written as

(CD7-

30)

The total enzyme initially present is given by the sum

(Et ) = (Eo ) + (Eo Sr) + (Er So ) + (Er)(CD7-

31)

After using Equations (CD7-27), (CD7-28), and (CD7-29) to

substitute for the active intermediate in Equation (CD7-31),

one can then solve for the unbound oxidized enzyme

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concentration, Eo, and substitute it into Equation (CD7-30) to

obtain the form of the rate law

(CD7-32)

Example CD7-6

Construct a Lineweaver-Burk Plot for Different Oxygen

Concentrations

The reaction above illustrates how an enzyme can be

regenerated through the addition of another substrate, in this

case O2.

CD7.4B Enzyme Cofactors

In many enzymatic reactions, and in particular biological

reactions, a second substrate (i.e., species) must be

introduced to activate the enzyme. This substrate, which is

referred to as a cofactororcoenzyme even though it is not an

enzyme as such, attaches to the enzyme and is most often

either reduced or oxidized during the course of the reaction.

The enzyme-cofactor complex is referred to as aholoenzyme.

An example of the type of system in which a cofactor is used

is the formation of ethanol from acetaldehyde in the presence

of the enzyme alcohol dehydrogenase (ADH) and thecofactor nicotinamide adenine dinucleotide (NAD). After the

enzyme is activated by combination with the cofactor in its

reduced state, NADH,

the holoenzyme (ADH NADH) reacts with acetaldehyde in

acid solution to produce ethanol and the oxidized form of theenzyme-cofactor coupling

(ADH NAD+):

The inactive form of the enzyme-cofactor complex for a

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specific reaction and reaction direction is called

anapoenzyme. This reaction is followed by dissociation of the

apoenzyme (ADH NAD+), which is usually relatively slow.

The values of the specific rates are3:

Typical initial concentrations for a small laboratory batch

reactor experiment might be [acetaldehyde]0-1mol/L, [ADH]0=

10-7 g mol/L, and [NADH]0 = 10-4 g mol/L. The

overall reaction is often written in the form

alcohol dehydrogenase

The ADH enzyme molecule produced by the dissociation of

(ADH NAD+) can participate in subsequent reactions

involving the formation of ethanol, while the nicotinamide

adenine dinucleotide from the dissociation cannot participate

until it is reduced back to NADH. Since the initial

concentration of NADH is usually several orders of

magnitude greater than the initial concentration of enzyme,

the consumption of NADH will not limit the overall rate of

formation of ethanol nearly so much as the slow dissociation

of the (ADH NAD+) complex. This apoenzyme essentially

ties up the enzyme, preventing it from becoming free

(unbound) to combine with NADH to form the holoenzyme,

which reacts with acetaldehyde to produce ethanol. We note

that the reaction rate might be increased considerably if wehad a way of going directly from (ADH NAD+) to (ADH

ADH); that is,

rather than having the enzyme ADH go through the steps of

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dissociating from

and then combining with NADH:

Example CD7-7

Derive an Initial Rate Law for Alcohol Dehydrogenates

Equation (CD7-33) is of a form that is often used in the

interpretation of initial rate data for enzymatic reactions

involving two substrates. The parameters K12,K1,K2,andVmax in Equation (CD7-33), which was first developed by

Dalziel,4may be evaluated through a series of Lineweaver-

Burk plots. After substituting the numerical values

forK1,K2,K12 and recalling the initial concentrations specified

(S1,0 = 0.1 g mol/L, S2,0 = 10-4 g mol/L), we see that we can

neglectK12 andK2(S2) with respect to the other terms in the

denominator, in which case Equation (G25-7) becomes

The initial rate is rp0 = 3.7 x 10-6 g mol/L s. In the next

section we compare the rate above with one in which a third

substrate is added to the system.

CD7.4C Multiple-Substrate Systems

In the preceding section we stated that the rate of formation

of ethanol might be increased if the (ADH NAD1) complex

could be converted by some means directly to the enzyme-cofactor complex (ADH NADH) without having the

enzyme ADH go through a series of reactions. This can be

achieved by the addition of a third substrate, S3, (e.g.,

propanediol), which during reaction (to form DL-

lactaldehyde) will also regenerate the cofactor (NADH). The

overall reaction sequence for this case is

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As a first approximation, this sequence of reactions could be

represented by the following elementary steps:

Example CD7-8Derive a Rate Law for a Multiple Substrate System

This same reaction has been carried in the reverse

directionby Gupta and Robinson.5They measured the initial

rate of conversion of DL-lactaldehyde to propanediol in the

presence of NAD 1 and ADH. The rate of dissociation of the

enzyme-cofactor complex (ADH NADH) is believed to be

rate limiting. This is con- firmed by the fact that whenethanol was added to the system, the reaction rate increased

100-fold by having the ethanol convert the (ADH NADH)

directly back to (ADH NAD+).

Example CD7-9

Calculate the Initial Rate of Formation of Ethanol in the

Presence of Propanediol

In analyzing multiple reactions in this manner, one shouldalways question the validity of the application of the PSSH to

the various active intermediates.

Since the nicotinamide adenine dinucleotide is continually

regenerated and the total concentration of the cofactor (in

itsoxidized, reduced, bound, and unboundforms) remains

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constant throughout the course of the reaction, it might be

desirable to replace S2 in the rate law in terms of the total

cofactor concentration, St . Neglecting any unbound, the total

(initial) cofactor concentration is

The total concentration of enzyme is

Subtracting Equation (CD7-34) from Equation (CD7-35), one

obtains

Equation (G25-7) can be rewritten in the form

where

After adding Equations (CD7-36) and (CD7-37) and

rearranging, we obtain

which is solved for the unbound enzyme concentration in

terms of S1, S3, P2, Et , and St .

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We can substitute Equation (CD7-39) into (CD7-34) and

rearrange to determine the concentration of the unbound

cofactor in its reduced form, that is,

The rate law was given by

One could substitute Equations (CD7-39) and (CD7-40) for

S2 and E into Equation (CD7-41) to arrive at a reasonably

complicated rate law involving S1, St, S 3, P2, and Et. However,

a computer solution would be used in most reaction

sequences that are this involved algebraically, in which case

further substitution would not be necessary and one could use

Equations (CD7-39), (CD7-40), and (CD7-41) directly.

CD7.4D Multiple Enzymes Systems

We shall again consider the production of ethanol from

acetaldehyde which uses the cofactor NADH. However, the

regeneration of NAD+ to NADH is brought about in a

reaction catalyzed by acetaldehyde dehydrogenease (E2),

which produces acetic acid from acetaldehyde:

This sequence can be written in abstract notation as

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One could apply the PSSH to E1 S2, E1 E2, and E2 S

and represent the total enzyme concentrations as

in deriving the rate law for this system. We shall not carry

through the necessary algebraic manipulation to obtain the

rate law here, as all the principles for determining the rate

law have been presented and there would be little more to be

accomplished by doing this.

The sections above on enzymatic reactions were meant to

serve as a brief, yet somewhat encompassing discussion of

enzyme kinetics. Further discussion can be found in theSupplementary Reading for Chapter 7.

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