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The following information is taken from the 4th edition of Elements of Chemical Reaction Engineering, so the equation numbers correspond to those found in that book


R9.6.1: Microbial Growth on Multiple Substrates

Microbial Growth on Multiple Substrates


R9.6.2 Enzyme Regeneration


As 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 IMAGE 07eq52a.gif-gluconolactone (P):

IMAGE 07eq52.gif


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.


IMAGE 07eq53.gif

    Overall, the reaction is written  

IMAGE 07eq54.gif


In biochemistry texts, reactions of this type involving regeneration are usually written in the form


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    The reaction is believed to proceed by the following sequence of elementary reactions:  

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    We shall assume that reaction involving the dissociation between reduced glucose oxidase andIMAGE 07eq52a.gif-lactone is rate limiting. The rate of formation ofIMAGE 07eq52a.gif-lactone (P1 ) is given by the equation  

IMAGE 07eq57.gif

    After applying the pseudo-steady-state hypothesis to the rates of formation of
(Eomiddot Sr ), (Ermiddot So ), and (Er ),

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    we can solve for the following concentrations of the active intermediates in terms of the concentrations of glucose, oxygen, and unbound oxidized enzyme.  

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    After substituting Equation (R7.4-3) into Equation (R7.4-1), the rate law is written as  

IMAGE 07eq60.gif

    The total enzyme initially present is given by the sum  

(Et ) = (Eo ) + (EomiddotSr ) + (ErmidotSo ) + (Er )

    After using Equations (R7.4-2), (R7.4-3), and (R7.4-4) to substitute for the active intermediate in Equation (R7.4-6), one can then solve for the unbound oxidized enzyme concentration, Eo , and substitute it into Equation (R7.4-5) to obtain the form of the rate law  

IMAGE 07eq61.gif


Example R9.6-1
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 .  

R9.6.3 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 cofactor or coenzyme 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 a holoenzyme. 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 the cofactor nicotinamide adenine dinucleotide (NAD). After the enzyme is activated by combination with the cofactor in its reduced state, NADH,  

IMAGE 07eq67.gif

    the holoenzyme (ADHmiddotNADH) reacts with acetaldehyde in acid solution to produce ethanol and the oxidized form of the enzyme-cofactor coupling
(ADHmiddotNAD+ ):

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    The inactive form of the enzyme-cofactor complex for a specific reaction and reaction direction is called an apoenzyme. This reaction is followed by dissociation of the apoenzyme (ADHmiddotNAD+ ), which is usually relatively slow.  

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    The values of the specific rates are3 :  

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Typical initial concentrations for a small laboratory batch reactor experiment might be
[acetaldehyde]0 = 10-1 mol/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


IMAGE 07eq71.gif

    The ADH enzyme molecule produced by the dissociation of (ADHmiddotNAD+ ) 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 (ADHmiddotNAD+ ) 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 we had a way of going directly from (ADHmiddotNAD+ ) to (ADHmiddotNADH); that is,  

IMAGE /07eq72.gif

    rather than having the enzyme ADH go through the steps of dissociating from  

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    and then combining with NADH:  

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Example R9.6-2
Derive an Initial Rate Law for Alcohol Dehydrogenase

    Equation (RE7.4-1) 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 , and Vmax in Equation (RE7.4-1), which was first developed by Dalziel,4 may be evaluated through a series of Lineweaver-Burk plots. After substituting the numerical values for K1 , 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 neglect K12 and K2 (S2 ) with respect to the other terms in the denominator, in which case Equation (G25-7) becomes  

IMAGE 07eq89.gif

    The initial rate is rp 0 = 3.7 x 10-6 g mol/Lmiddots. In the next section we compare the preceding rate with one in which a third substrate is added to the system.  

R9.6.4 Multiple-Substrate Systems

In the preceding section we stated that the rate of formation of ethanol might be increased if the (ADHNAD1 ) complex could be converted by some means directly to the enzyme-cofactor complex (ADHNADH) 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


IMAGE 07eq90.gif

    As a first approximation, this sequence of reactions could be represented by the following elementary steps:  

IMAGE 07eq91.gif


Example R9.6-3
Derive a Rate Law for a Multiple Substrate System

This same reaction has been carried in the reverse direction by Gupta and Robinson.5 They 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 (ADHNADH) is believed to be rate limiting. This is con- firmed by the fact that when ethanol was added to the system, the reaction rate increased 100-fold by having the ethanol convert the (ADHNADH) directly back to (ADHmiddotNAD+ ).

Example R9.6-4
Calculate the Initial Rate of Formation of Ethanol in the Presence of Propanediol

    In analyzing multiple reactions in this manner, one should always 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 its oxidized, reduced, bound, and unbound forms) remains 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, S t . Neglecting any unbound, the total (initial) cofactor concentration is

IMAGE 07eq108.gif

    The total concentration of enzyme is  

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    Subtracting Equation (R7.4-9) from Equation (R7.4-10), we obtain  

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    Equation (R7.4-11) can be rewritten in the form  


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    After adding Equations (R7.4-11) and (R7.4-12) and rearranging, we obtain  

IMAGE 07eq113.gif

    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 (R7.4-14) into (R7.4-9) and rearrange to determine the concentration of the unbound cofactor in its reduced form, that is,  

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    The rate law was given by  

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    One could substitute Equations (R7.4-14) and (R7.4-15) for S2 and E into Equation (R7.4-16) 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 (R7.4-14), (R7.4-15), and (R7.4-16) directly.

R9.6.5 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 dehydrogenase (E2 ), which produces acetic acid from acetaldehyde:

Image 07eq117.gif


This sequence can be written in abstract notation as


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    We could apply the PSSH to E1middot S2 , E1middot E2 , and E2middot S and represent the total enzyme concentrations as  

IMAGE 07eq119.gif

    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 on enzymatic reactions were meant to serve as a brief, yet somewhat encompassing discussion of enzyme kinetics. Further discussion can be found in the Supplementary Reading for Chapter 9.