Multiple Enzyme-Substrate System

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		R7.4.1: Microbial Growth on Multiple Substrates Microbial Growth on Multiple Substrates

		R7.4.2 Enzyme Regeneration
		As a first example we shall consider the oxidation of glucose (S_r ) with the aid of the enzyme glucose oxidase [represented as either G.O. or (E_o )] to give -gluconolactone (P):
		In this reaction, the reduced form of glucose oxidase (G.O.H₂ ), which will be represented by E_r , cannot catalyze further reactions until it is oxidized back to E_o . This oxidation is usually carried out by adding molecular oxygen to the system so that glucose oxidase, E_o , 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

			(R7.4-1)

		After applying the pseudo-steady-state hypothesis to the rates of formation of (E_o S_r ), (E_r S_o ), and (E_r ),



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

			(R7.4-2) (R7.4-3) (R7.4-4)

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

			(R7.4-5)

		The total enzyme initially present is given by the sum

		(E_t ) = (E_o ) + (E_oS_r) + (E_rS_o ) + (E_r )	(R7.4-6)

		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, E_o , and substitute it into Equation (R7.4-5) to obtain the form of the rate law

			(R7.4-7)

		Example R7.4-1 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 .

		R7.4.3 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,



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



		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 (ADHNAD⁺ ), 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 = 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


		The ADH enzyme molecule produced by the dissociation of (ADHNAD⁺ ) 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 (ADHNAD⁺ ) 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 (ADHNAD⁺ ) to (ADHNADH); that is,



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



		and then combining with NADH:



		Example R7.4-2 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 V_max 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 (S_1,0 = 0.1 g mol/L, S_2,0 = 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

			(R7.4-8)

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

		R7.4.4 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



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



		Example R7.4-3 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 NAD¹ 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 (ADHNAD⁺ ). Example R7.4-4 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

			(R7.4-9)

		The total concentration of enzyme is

			(R7.4-10)

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

			(R7.4-11)

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

			(R7.4-11)
		where

			(R7.4-12)

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

			(R7.4-13)

		which is solved for the unbound enzyme concentration in terms of S₁ , S₃ , P₂ , E_t , and S_t .

			(R7.4-14)

		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,

			(R7.4-15)

		The rate law was given by

			(R7.4-16)

		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 , S_t , S₃ , P₂ , and E_t . 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.


		R7.4.5 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:



		This sequence can be written in abstract notation as



		We 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 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 7.

Professional Reference Shelf

R7.4.1: Microbial Growth on Multiple Substrates

R7.4.2 Enzyme Regeneration

R7.4.3 Enzyme Cofactors

R7.4.4 Multiple-Substrate Systems

R7.4.5 Multiple Enzymes Systems