Bioprocess Engineering

Michael L Shuler

Chapter 3

Enzymes - all with Video Answers

Educators

Chapter Questions

Problem 1

Consider the following reaction sequence:
$$
\mathrm{S}+\mathrm{E} \underset{k_{2}}{\rightleftharpoons}(\mathrm{ES})_{1} \rightleftharpoons_{k_{4}}^{k_{3}}(\mathrm{ES})_{2} \stackrel{k_{g}}{\longrightarrow} \mathrm{P}+\mathrm{E}
$$
Develop a suitable rate expression for production formation $\left[v=k_{3}(\mathrm{ES})_{2}\right]$ by using (a) the equilibrium approach, and (b) the quasi-steady-state approach.

Dominador Tan

Dominador Tan

Numerade Educator

Problem 2

Consider the reversible product-formation reaction in an enzyme-catalyzed bioreaction:
$$
\mathrm{E}+\mathrm{S} \stackrel{k_{1}}{\rightleftharpoons}(\mathrm{ES}) \stackrel{k_{2}}{\stackrel{k_{3}}{\sum}} \mathrm{E}+\mathrm{P}
$$
Develop a rate expression for product-formation using the quasi-steady-state approximation and show that
$$
v=\frac{d[\mathrm{P}]}{d t}=\frac{\left(v_{s} / K_{m}\right)[\mathrm{S}]-\left(v_{p} / K_{p}\right)[\mathrm{P}]}{1+\frac{[\mathrm{S}]}{K_{m}}+\frac{[\mathrm{P}]}{K_{p}}}
$$ $$
\text { where } K_{m}=\frac{k_{-1}+k_{2}}{k_{1}} \text { and } K_{p}=\frac{k_{-1}+k_{2}}{k_{-2}} \text { and } V_{x}=k_{2}\left[\mathrm{E}_{0}\right], V_{p}=k_{-1}\left[\mathrm{E}_{0}\right] \text {. }
$$

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 3

$$
\text { where } K_{m}=\frac{k_{-1}+k_{2}}{k_{1}} \text { and } K_{p}=\frac{k_{-1}+k_{2}}{k_{-2}} \text { and } V_{x}=k_{2}\left[\mathrm{E}_{0}\right], V_{p}=k_{-1}\left[\mathrm{E}_{0}\right] \text {. }
$$

AG

Ankit Gupta

Numerade Educator

Problem 4

The hydration of $\mathrm{CO}_{2}$ is catalyzed by carbonic anhydrase as follows:
$$
\mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2} \rightleftharpoons \mathrm{HCO}_{3}^{-}+\mathrm{H}^{+}
$$
The following data were obtained for the forward and reverse reaction rates at pH $7.1$ and an enzyme concentration of $2.8 \times 10^{-9} M$.
\begin{tabular}{cccc}
\hline \multicolumn{2}{c}{ Hydration } & \multicolumn{2}{c}{ Dehydration } \\
\cline { 1 - 2 } \begin{tabular}{ccc}
$1 / v, M^{-1}$ & {$\left[\mathrm{CO}_{2}\right]$} & $1 / v, M^{-1}$ & {$\left[\mathrm{HCO}_{3}^{-1}\right]$} \\
$\left(\mathrm{s} \times 10^{-3}\right)$ & $\left(M \times 10^{3}\right)$ & $\left(\mathrm{s} \times 10^{-3}\right)$ & $\left(M \times 10^{3}\right)$ \\
\hline 36 & $1.25$ & 95 & 2 \\
20 & $2.5$ & 45 & 5 \\
12 & 5 & 29 & 10 \\
6 & 20 & 25 & 15 \\
\hline
\end{tabular}
\end{tabular}
$v$ is the initial reaction rate at the given substrate concentration. Calculate the forward and reverse catalytic and Michaelis constants.
[Courtesy of D. J. Kirwan from "Collected Coursework Problems in Biochemical Engineer-

Dominador Tan

Numerade Educator

Problem 5

An inhibitor (I) is added to the enzymatic reaction at a level of $1.0 \mathrm{~g} / \mathrm{l}$. The following data were obtained for $K_{w}=9.2 \mathrm{~g} \mathrm{~S} / \mathrm{l}$.
a. Is the inhibitor competitive or noncompetitive?
b. Find $K_{\mathrm{r}}$.

Dominador Tan

Numerade Educator

Problem 6

During a test of kinetics of an enzyme-catalyzed reaction, the following data were recorded:
a. Determine the Michaelis-Menten constant for the reaction with no inhibitor present at $30^{\circ} \mathrm{C}$ and at $49.6^{\circ} \mathrm{C}$.
b. Determine the maximum velocity of the uninhibited reaction at $30^{\circ} \mathrm{C}$ and an enzyme concentration of $1.6 \mathrm{~g} / \mathrm{l}$.
c. Determine the $K_{1}$ for the inhibitor at $30^{\circ} \mathrm{C}$ and decide what type of inhibitor is being used.

Dominador Tan

Numerade Educator

Problem 7

An enzyme ATPase has a molecular weight of $5 \times 10^{4}$ daltons, a $K_{M}$ value of $10^{-4} M$, and a $k_{2}$ value of $k_{2}=10^{4}$ molecules ATP/min molecule enzyme at $37^{\circ} \mathrm{C}$. The reaction catalyzed is the following:
$$
\mathrm{ATP} \stackrel{\text { ATPase }}{\longrightarrow} \mathrm{ADP}+\mathrm{P}_{i}
$$
which can also be represented as
$$
\mathrm{E}+\mathrm{S} \underset{k_{1}}{\stackrel{A_{1}}{\longrightarrow} \mathrm{ES} \stackrel{k_{2}}{\longrightarrow} \mathrm{E}+\mathrm{P}
$$
where $S$ is ATP. The enzyme at this temperature is unstable. The enzyme inactivation kinetics are first order:
$$
\mathrm{E}=\mathrm{E}_{0} e^{-k_{\alpha} r}
$$
where $\mathrm{E}_{0}$ is the initial enzyme concentration and $k_{d}=0.1 \mathrm{~min}^{-1}$. In an experiment with a partially pure enzyme preparation, $10 \mu \mathrm{g}$ of total crude protein (containing enzyme) is added to a $1 \mathrm{ml}$ reaction mixture containing $0.02 \mathrm{M}$ ATP and incubated at $37^{\circ} \mathrm{C}$. After 12 hours the reaction ends (i.e., $t \rightarrow \infty$ ) and the inorganic phosphate $\left(\mathrm{P}_{i}\right)$ concentration is found to be $0.002 M$, which was initially zero. What fraction of the crude protein preparation was the enzyme? Hint: Since $[\mathrm{S}]>>K_{m}$, the reaction rate can be represented by
$$
\frac{d(\mathrm{P})}{d t}=k_{2}[\mathrm{E}]
$$

Lottie Adams

Numerade Educator

Problem 8

Assume that for an enzyme immobilized on the surface of a nonporous support material the external mass transfer resistance for substrate is not negligible as compared to the reaction rate. The enzyme is subject to substrate inhibition (eq. 3.34).
a. Are multiple states possible? Why or why not?
b. Could the effectiveness factor be greater than one?

Christina Sorrentino

Christina Sorrentino

Numerade Educator

Problem 9

The following data were obtained for an enzyme-catalyzed reaction. Determine $\mathrm{V}_{\max }$ and $K_{m}$ by inspection. Plot the data using the Eadie-Hofstee method and determine these constants graphically. Explain the discrepancy in your two determinations. The initial rate data for the enzyme-catalyzed reaction are as follows:
\begin{tabular}{cc}
\hline${[\mathrm{S}] }$ $\mathrm{mol} / 1$ & $v$ $\mu \mathrm{mol} / \mathrm{min}$ \\
\hline $5.0 \times 10^{-4}$ & 125 \\
$2.0 \times 10^{-4}$ & 125 \\
$6.0 \times 10^{-5}$ & 121 \\
$4.0 \times 10^{-5}$ & 111 \\
$3.0 \times 10^{-5}$ & $96.5$ \\
$2.0 \times 10^{-5}$ & $62.5$ \\
$1.6 \times 10^{-5}$ & $42.7$ \\
$1.0 \times 10^{-5}$ & $13.9$ \\
$8.0 \times 10^{-6}$ & $7.50$ \\
\hline
\end{tabular}
Do these data fit into Michaelis-Menten kinetics? If not, what kind of rate expression would you suggest? Use graphical methods.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 10

a. H. H. Weetall and N. B. Havewala report the following data for the production of dextrose from corn starch using both soluble and immobilized (azo-glass beads) glucoamylase in a fully agitated CSTR system.
1. Soluble data: $T=60^{\circ} \mathrm{C},\left[\mathrm{S}_{0}\right]=168 \mathrm{mg}$ starch $/ \mathrm{ml},\left[\mathrm{E}_{0}\right]=11,600$ units, volume $=1000 \mathrm{ml}$.
2. Immobilized data: $T=60^{\circ} \mathrm{C},\left[\mathrm{S}_{0}\right]=336 \mathrm{mg} \operatorname{starch} / \mathrm{ml},\left[\mathrm{E}_{0}\right]=46,400$ units initially, immobilized, volume $=1000 \mathrm{ml}$.
\begin{tabular}{ccc}
\hline & \multicolumn{2}{c}{ Product concentration (mg dextrose/ml) } \\
\cline { 2 - 3 } Time (min) & Soluble & Immobilized \\
\hline 0 & $12.0$ & $18.4$ \\
15 & $40.0$ & 135 \\
30 & $76.5$ & 200 \\
45 & $94.3$ & 236 \\
60 & $120.0$ & 260 \\
75 & $135.5$ & 258 \\
90 & $151.2$ & 262 \\
105 & $150.4$ & 266 \\
120 & $155.7$ & 278 \\
135 & $160.1$ & 300 \\
150 & $164.9$ & 310 \\
165 & $170.0$ & 306 \\
225 & $-$ & 316 \\
415 & $-$ & 320 \\
\hline
\end{tabular}
Determine the maximum reaction velocity, $V_{m}(\mathrm{mg} / \mathrm{ml}-\mathrm{min}$ - unit of enzyme) and the saturation constant, $\mathrm{K}_{M}(\mathrm{mg} / \mathrm{ml})$.
b. The same authors studied the effect of temperature on the maximum rate of the hydrolysis of corn starch by glucoamylase. The results are tabulated next. Determine the activation energy ( $\Delta E \mathrm{cal} / \mathrm{g}$ mole) for the soluble and immobilized enzyme reaction.
\begin{tabular}{ccc}
\hline & \multicolumn{2}{c}{$V_{\max }\left(\mathrm{m} \mathrm{mol} / \mathrm{min} 10^{6}\right)$} \\
\cline { 2 - 3 } $\mathrm{T},{ }^{\circ} \mathrm{C}$ & Soluble & Azo-immobilized \\
\hline 25 & $0.62$ & $0.80$ \\
35 & $1.42$ & $1.40$ \\
45 & $3.60$ & $3.00$ \\
55 & $8.0$ & $6.2$ \\
65 & $16.0$ & $11.0$ \\
\hline
\end{tabular}
c. Using these results, determine if immobilized enzyme is diffusion limited. [Courtesy of A. E. Humphrey from "Collected Coursework Problems in Biochemical Engineering" compiled by H. W. Blanch for 1977 Am. Soc. Eng. Educ. Summer School.]

Shazia Naz

Numerade Educator

Problem 11

Michaelis-Menten kinetics are used to describe intracellular reactions. Yet $\left[\mathrm{E}_{0}\right] \approx\left[\mathrm{S}_{0}\right]$. In in vitro batch reactors, the quasi-steady-state hypothesis does not hold for $\left[\mathrm{E}_{0}\right] \approx\left[\mathrm{S}_{0}\right]$. The rapid equilibrium assumption also will not hold. Explain why Michaelis-Menten kinetics and the quasi-steady-state approximation are still reasonable descriptions of intracellular enzyme reactions.

Josee Pacheco

Numerade Educator

Problem 12

You are working for company A and you join a research group working on immobilized enzymes. Harry, the head of the lab, claims that immobilization improves the stability of the enzyme. His proof is that the enzyme has a half-life of 10 days in free solution, but under identical conditions of temperature, $\mathrm{pH}$, and medium composition, the measured half-life of a packed column is 30 days. The enzyme is immobilized in a porous sphere $5 \mathrm{~mm}$ in diameter. Is Harry's reasoning right? Do you agree with him? Why or why not?

Sam Limsuwannarot

Sam Limsuwannarot

Numerade Educator

Problem 13

The following data were obtained from enzymatic oxidation of phenol by phenol oxidase at different phenol concentrations.
$\begin{array}{lrccllllcrrr}\mathrm{S}(\mathrm{mg} / \mathrm{l}) & 10 & 20 & 30 & 50 & 60 & 80 & 90 & 110 & 130 & 140 & 150 \\ v(\mathrm{mg} / \mathrm{l}-\mathrm{h}) & 5 & 7.5 & 10 & 12.5 & 13.7 & 15 & 15 & 12.5 & 9.5 & 7.5 & 5.7\end{array}$
a. What type of inhibition is this ?
b. Determine the constants $V_{\mathrm{m}}, K_{m}$ and $K_{s i-}$
c. Determine the oxidation rate at $[\mathrm{S}]=70 \mathrm{mg} / 1$.

Sam Limsuwannarot

Sam Limsuwannarot

Numerade Educator

Problem 14

Uric acid is degraded by uricase enzyme immobilized in porous Ca-alginate beads. Experiments conducted with different bead sizes result in the following rate data:
$\begin{array}{lllllllll}\text { Bead Diameter, Dp }(\mathrm{cm}) & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 \\ \text { Rate, } v(\mathrm{mg} / \mathrm{l} . \mathrm{h}) & 200 & 198 & 180 & 140 & 100 & 70 & 50 & 30\end{array}$
a. Determine the effectiveness factor for particle sizes $D p=0.5 \mathrm{~cm}$ and $D p=0.7 \mathrm{~cm}$.
b. The following data were obtained for $\mathrm{Dp}=0.5 \mathrm{~cm}$ at different bulk uric acid concentrations. Assuming negligible liquid film resistance, calculate $V_{m}$ and $K_{x}$ for the enzyme. Assume no substrate or product inhibition.
$\begin{array}{llllrrr}\mathrm{S}_{0}(\mathrm{mg} \text { UA/1) } & 10 & 25 & 50 & 100 & 200 & 250 \\ v(\mathrm{mg} \text { UA } / \mathrm{L} . \mathrm{h}) & 10 & 20 & 30 & 40 & 45 & 46\end{array}$

Rashmi Sinha

Numerade Educator

Problem 15

The enzyme, urease, is immobilized in Ca-alginate beads $2 \mathrm{~mm}$ in diameter. When the urea concentration in the bulk liquid is $0.5 \mathrm{~m} M$ the rate of urea hydrolysis is $v=10 \mathrm{mmoles-1- \textrm {h } .}$ Diffusivity of urea in $\mathrm{Ca}$-alginate beads is $D_{e}=1.5 \times 10^{-5} \mathrm{~cm}^{2} / \mathrm{sec}$, and the Michaelis constant for the enzyme is $K_{m}^{\prime}=0.2 \mathrm{~m} M$. By neglecting the liquid film resistance on the beads (i.e., $\left.\left[\mathrm{S}_{0}\right]=\left[\mathrm{S}_{\mathrm{s}}\right]\right)$ determine the following:
a. Maximum rate of hydrolysis $V_{m}$, Thiele modulus $(\phi)$, and effectiveness factor $(\eta)$.
b. What would be the $V_{w}, \phi$, and $\eta$ values for a particle size of $\mathrm{Dp}=4 \mathrm{~mm}$ ?

Rashmi Sinha

Numerade Educator

Problem 16

Decarboxylation of glyoxalate (S) by mitochondria is inhibited by malonate (I). Using the following data obtained in batch experiments, determine the following:
a. What type of inhibition is this?
b. Determine the constants $V_{\ldots} K^{\prime}$. and $K_{\text {. }}$

Sam Limsuwannarot

Sam Limsuwannarot

Numerade Educator

Problem 17

Urea dissolved in aqueous solution is degraded to ammonia and $\mathrm{CO}_{2}$ by the enzyme urease immobilized on surfaces of nonporous polymeric beads. Conversion rate is controlled by transfer of urea to the surface of the beads through liquid film, and the conversion takes place on the surfaces of the beads. The following parameters are given for the system.
$k_{L}=0.2 \mathrm{~cm} / \mathrm{s} ; K_{m}=200 \mathrm{mg} / 1$
$V_{m}^{\prime}=0.1 \mathrm{mg}$ urea/cm $^{2}$ support surface $-\mathrm{s}$.
$S_{b}=1000 \mathrm{mg}$ urea/l
a. Determine the surface concentration of urea.
b. Determine the rate of urea degradation under mass transfer controlled conditions.

Rashmi Sinha

Numerade Educator

Problem 18

Two enzymes are both immobilized on the same flat, nonporous surface. For enzyme A the substrate is $S_{1}$. For enzyme B the substrate is $S_{2}$. The product of the first reaction is $S_{2}$. That is:
$$
S_{1} \underset{E_{A}}{\longrightarrow} S_{2} \underset{E_{A}}{\longrightarrow} P
$$
a. Figure 3.P1 depicts the rate of the first reaction on the surface as a function of local concentrations of $S_{1}$. If the bulk concentration of $S_{1}$ is $100 \mathrm{mg} / \mathrm{l}$ and the mass transfer coefficient is $4 \times 10^{-5} \mathrm{~cm} / \mathrm{s}$, what is the rate of consumption of $S_{1}$ for a $1 \mathrm{~cm}^{2}$ surface? What is the surface concentration of $S_{1}$ ?
b. The rate of the second reaction is:
$$
-d\left[S_{2}\right] / d t=d[P] / d t=\frac{V_{m}^{\prime \prime} S_{2 \text { saptace }}}{K_{m}+S_{2 \text { surface }}}
$$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 19

Consider the case of two enzymes immobilized on the same nonporous, planar surface. $S$ is a substrate used by both enzymes in the following reactions: $$
\begin{aligned}
&\mathrm{S}+\mathrm{E}_{1} \longrightarrow \mathrm{E}_{1}+\mathrm{P}_{1} \\
&\mathrm{~S}+\mathrm{E}_{2} \longrightarrow \mathrm{E}_{2}+\mathrm{P}_{2}
\end{aligned}
$$
The final product $P_{3}$ is formed by the spontaneous reaction of $P_{1}$ and $P_{2}$ :
$$
\mathrm{P}_{1}+\mathrm{P}_{2} \longrightarrow \mathrm{P}_{3}
$$
Reactions 1 and 2 occur only at the surface and reaction 3 is a homogeneous reaction occurring throughout the bulk liquid phase.

Figure 3.P2 gives the predicted reaction-rate dependence of reaction 1 (bottom curve) alone and reaction 2 (top curve) alone based on the measured amount of each enzyme immobilized and assuming the intrinsic reaction kinetics are not altered in the process of immobilization.
a. If $k_{L}=6 \times 10^{-5} \mathrm{~cm} / \mathrm{s}$ and the bulk concentration of substrate is $500 \mathrm{mg} / \mathrm{l}$, what is the total rate of substrate disappearance?
b. What is the overall effectiveness factor under the conditions of part a?
c. What will be the ratio of $P_{2}$ to $P_{1}$ under the conditions of part a?
d. If you want to produce equimolar amounts of $P_{1}$ and $P_{2}$ and if $k_{L}=6 \times 10^{-5} \mathrm{~cm} / \mathrm{s}$, what value of bulk substrate concentration must you pick?

Susan Hallstrom

Susan Hallstrom

Numerade Educator