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Bioprocess Engineering

Michael L Shuler

Chapter 6

How Cells Grow - all with Video Answers

Educators

Chapter Questions

01:24

Problem 1

A simple, batch fermentation of an aerobic bacterium growing on methanol gave the results shown in the table. Calculate:
a. Maximum growth rate $\left(\mu_{\max }\right)$
b. Yield on substrate $\left(Y_{X / S}\right)$
c. Mass doubling time $\left(t_{d}\right)$
d. Saturation constant $\left(K_{s}\right)$
e. Specific growth rate $\left(\mu_{n e t}\right)$ at $t=10 \mathrm{~h}$
$$
\begin{array}{cll}
\hline \text { Time (h) } & X(\mathrm{~g} / \mathrm{l}) & S(\mathrm{~g} / \mathrm{l}) \\
\hline 0 & 0.2 & 9.23 \\
2 & 0.211 & 9.21 \\
4 & 0.305 & 9.07 \\
8 & 0.98 & 8.03 \\
10 & 1.77 & 6.8 \\
12 & 3.2 & 4.6 \\
14 & 5.6 & 0.92 \\
16 & 6.15 & 0.077 \\
18 & 6.2 & 0 \\
\hline
\end{array}
$$

Dominador Tan
Dominador Tan
Numerade Educator
05:44

Problem 2

The growth of a microbial population is a function of pH and is given by the following equation:
$$
\mu_{g}=\frac{1}{X} \frac{d X}{d t}=\frac{\mu_{m} S}{K_{s}\left(1+\frac{H^{+}}{k_{1}}\right)+S}
$$
a. With a given set of experimental data ( $X$ and $S$ versus $t$ ), describe how you would determine the constants $\mu_{m}, K_{s}$, and $k_{1}$ -
b. How would the double-reciprocal plot $1 / \mu_{g}$ versus $1 / S$ change with $\mathrm{pH}\left(\right.$ or $\left.\mathrm{H}^{+}\right)$concentration?

Kader Bayar
Kader Bayar
Numerade Educator
06:31

Problem 3

The following data were obtained for the effect of temperature on the fermentative production of lactic acid by a strain of Lactobacillus delbrueckii. From these data, calculate the value of the activation energy for this process. Is the value of the activation energy typical of this sort of biological conversion? (See Chapter 3.)
$$
\begin{array}{cc}
\hline \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & \text { Rate constant (mol/l-h) } \\
\hline 40.4 & 0.0140 \\
36.8 & 0.0112 \\
33.1 & 0.0074 \\
30.0 & 0.0051 \\
25.1 & 0.0036 \\
\hline
\end{array}
$$
[Courtesy of A. E. Humphrey from "Collected Coursework Problems in Biochemical Engineering," compiled by H. W. Blanch for $1977 \mathrm{Am}$. Soc. Eng. Educ. Summer School.]

Mukesh Devi
Mukesh Devi
Numerade Educator
01:26

Problem 4

It is desired to model the growth of an individual bacterium. The cell transports $S_{1}$ into the cell enzymatically, and the permease is subject to product inhibition. $S_{1}$ is converted into precursors, $P$, that are converted finally into the macromolecular portion of the cell, $M$. The catalyst of all reactions is $M$.
(1) $S_{1}^{*} \longrightarrow S_{1}$ (per unit surface area) where $S^{*}=$ outside concentration of $S$
(2) $S_{1} \longrightarrow P$
(3a)
$\left.\begin{array}{l}\text { Energy }+P \longrightarrow M \\ S_{1} \longrightarrow{M} \rightarrow \text { energy }\end{array}\right\}$ coupled reaction
or
(3b) $S_{1}+P \longrightarrow M$
The dry weight of the cell is $T$ and is equal to
(4) $T=S_{1}+P+M_{1}=\rho V$
where $\rho=$ cell density and $V=$ cell volume
Write the equations and define all symbols necessary to describe the changes in $S_{1}, P, M$, and $T$ within the cell. Remember that the cell volume is always changing.

Dominador Tan
Dominador Tan
Numerade Educator
05:16

Problem 5

A biochemical engineer has determined in her lab that the optimal productivity of a valuable antibiotic is achieved when the carbon nutrient, in this case molasses, is metered into the fermenter at a rate proportional to the growth rate. However, she cannot implement her discovery in the antibiotic plant, since there is no reliable way to measure the growth rate $(d X / d t)$ or biomass concentration $(X)$ during the course of the fermentation. It is suggested that an oxygen analyzer be installed on the plant fermenters so that the OUR (oxygen uptake rate, $\mathrm{g} / \mathrm{l}-\mathrm{h}$ ) may be measured.
a. Derive expressions that may be used to estimate $X$ and $d X / d t$ from OUR and time data, assuming that a simple yield and maintenance model may be used to describe the rate of oxygen consumption by the culture.
b. Calculate values for the yield $\left(Y_{\mathrm{X} \mathrm{O}_{2}}\right)$ and maintenance $\left(m_{\mathrm{O}_{2}}\right)$ parameters from the following data:
$$
\begin{array}{rcc}
\hline & \text { OUR } & X \\
\text { Time } & (\mathrm{g} / \mathrm{h}) & (\mathrm{g} / 1) \\
\hline 0 & 0.011 & 0.60 \\
1 & 0.008 & 0.63 \\
2 & 0.084 & 0.63 \\
3 & 0.153 & 0.76 \\
4 & 0.198 & 1.06 \\
5 & 0.273 & 1.56 \\
6 & 0.393 & 2.23 \\
7 & 0.493 & 2.85 \\
8 & 0.642 & 4.15 \\
9 & 0.915 & 5.37 \\
10 & 1.031 & 7.59 \\
11 & 1.12 & 9.40 \\
12 & 1.37 & 11.40 \\
13 & 1.58 & 12.22 \\
14 & 1.26 & 13.00 \\
15 & 1.58 & 13.37 \\
16 & 1.26 & 14.47 \\
17 & 1.12 & 15.37 \\
18 & 1.20 & 16.12 \\
19 & 0.99 & 16.18 \\
20 & 0.86 & 16.67 \\
21 & 0.90 & 17.01 \\
\hline
\end{array}
$$
[Courtesy of D. Zabriskie from "Collected Coursework Problems in Biochemical Engineering," compiled by H. W. Blanch for $1977 \mathrm{Am}$. Soc. Eng. Educ. Summer School.]

Sana Riaz
Sana Riaz
Numerade Educator
02:25

Problem 6

Pseudomonas $\mathrm{sp}$. has a mass doubling time of $2.4 \mathrm{~h}$ when grown on acetate. The saturation constant using this substrate is $1.3 \mathrm{~g} / 1$ (which is unusually high), and cell yield on acetate is $0.46 \mathrm{~g}$ cell $/ \mathrm{g}$ acetate. If we operate a chemostat on a feed stream containing $38 \mathrm{~g} / 1$ acetate, find the following:
a. Cell concentration when the dilution rate is one-half of the maximum
b. Substrate concentration when the dilution rate is $0.8 D_{\max }$
c. Maximum dilution rate
d. Cell productivity at $0.8 D_{\max }$
[Courtesy of E. Dunlop from "Collected Coursework Problems in Biochemical Engineering," compiled by H. W. Blanch for 1977 Am. Soc. Eng. Educ. Summer School.]

AG
Ankit Gupta
Numerade Educator
01:24

Problem 7

The following data were obtained in a chemostat for the growth of $E$. aerogenes on a glycerol-limited growth medium.
For this system, estimate the values of:
a. $K_{s}, \mathrm{mg}$ glycerol $/ \mathrm{ml}$
b. $\mu_{m}, \mathrm{~h}^{-1}$
c. $Y_{X / S}, \mathrm{mg}$ cells/mg glycerol
d. $m_{s}, \mathrm{mg}$ glycerol $/ \mathrm{mg}$ cell- $\mathrm{h}$
[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.]

Dominador Tan
Dominador Tan
Numerade Educator
01:26

Problem 8

The kinetics of microbial growth, substrate consumption, and mixed-growth-associated product formation for a chemostat culture are given by the following equations:
$$
\begin{aligned}
\frac{d X}{d t} &=\frac{\mu_{m} S}{K_{s}+S} X \\
\frac{d S}{d t} &=\frac{\mu_{m} S}{\left(K_{s}+S\right) Y_{X / S}} X \\
\frac{d P}{d t} &=\alpha \frac{d X}{d t}+\beta X=\left(\alpha \mu_{g}+\beta\right) X
\end{aligned}
$$
The kinetic parameter values are $\mu_{m}=0.7 \mathrm{~h}^{-1}, K_{s}=20 \mathrm{mg} / 1, Y_{X / S}=0.5 \mathrm{~g}$ dw/g substrate, $Y_{P / X}$ $=0.15 \mathrm{gP} / \mathrm{g} \cdot \mathrm{dw}, \alpha=0.1, \beta=0.02 \mathrm{~h}^{-1}$, and $S_{0}=1 \mathrm{~g} / \mathrm{l}$.
a. Determine the optimal dilution rate maximizing the productivity of product formation $(P D)$.
b. Determine the optimal dilution rate maximizing the productivity of cell (biomass) formation $(D S)$.
[Problem adapted from one suggested by L. Erickson.]

Dominador Tan
Dominador Tan
Numerade Educator
08:07

Problem 9

Ethanol is to be used as a substrate for single-cell protein production in a chemostat. The available equipment can achieve an oxygen transfer rate of $10 \mathrm{~g} \mathrm{O}_{2} / 1$ of liquid per hour. Assume the kinetics of cell growth on ethanol is of the Monod type, with $\mu_{m}=0.5 \mathrm{~h}^{-1}, K_{s}=30$ $\mathrm{mg} / 1, Y_{X S}=0.5 \mathrm{cells} / \mathrm{g}$ ethanol, and $Y_{\mathrm{O}_{2} / \mathrm{S}}=2 \mathrm{~g} \mathrm{O}_{2} / \mathrm{g}$ EtOH. We wish to operate the chemostat with an ethanol concentration in the feed of $22 \mathrm{~g} / \mathrm{L}$. We also wish to maximize the biomass productivity and minimize the loss of unused ethanol in the effluent. Determine the required dilution rate and whether sufficient oxygen can be provided.

Christopher Nilsen
Christopher Nilsen
Numerade Educator
04:36

Problem 10

Plot the response of a culture to diauxic growth on glucose and lactose based on the following: $\mu_{\mathrm{g} \text { lucose }}=1.0 \mathrm{~h}^{-1} ; \mu_{\text {lactose }}=0.6 \mathrm{~h}^{-1} ; Y_{\text {glucose }}=Y_{\text {lactose }}=0.5 ;$ enzyme induction requires 30 min to complete. Plot cell mass, glucose, and lactose concentrations, assuming initial values of $2 \mathrm{~g} / \mathrm{l}$ glucose, $3 \mathrm{~g} / \mathrm{l}$ lactose, and $0.10 \mathrm{~g} / \mathrm{l}$ cells.

Dennis Howard
Dennis Howard
Numerade Educator
01:00

Problem 11

The following data are obtained in oxidation of pesticides present in wastewater by a mixed culture of microorganisms in a continuously operating aeration tank.
$$
\begin{array}{lcc}
\hline & S \text { (Pesticides), } & \\
D\left(\mathrm{~h}^{-1}\right) & \mathrm{mg} / \mathrm{l} & X(\mathrm{mg} / \mathrm{l}) \\
\hline 0.05 & 15 & 162 \\
0.11 & 25 & 210 \\
0.24 & 50 & 250 \\
0.39 & 100 & 235 \\
0.52 & 140 & 220 \\
0.7 & 180 & 205 \\
0.82 & 240 & 170 \\
\hline
\end{array}
$$
Assuming the pesticide concentration in the feed wastewater stream as $S_{0}=500 \mathrm{mg} / \mathrm{l}$, determine $Y_{X=5}^{M}, k_{d}, \mu_{m}$, and $K_{s}$.

David Collins
David Collins
Numerade Educator
01:26

Problem 12

In a chemostat you know that if a culture obeys the Monod equation, the residual substrate is independent of the feed substrate concentration. You observe that in your chemostat an increase in $S_{0}$ causes an increase in the residual substrate concentration. Your friend suggests that you consider whether the Contois equation may describe the situation better. The Contois equation (eq. $6.36$ ) is:
$$
\mu=\frac{\mu_{m} S}{K_{x} X+S}
$$
a. Derive an expression for $S$ in terms of $D, \mu_{m}, K_{s x}$, and $X$ for a steady-state CFSTR (chemostat).
b. Derive an equation for $S$ as a function of $S_{0}, D, K_{s x}, Y_{X / S}^{M}$, and $\mu_{m}$.
c. If $S_{0}$ increases twofold, by how much will $S$ increase?

Dominador Tan
Dominador Tan
Numerade Educator
01:26

Problem 13

Pseudomonas putida with $\mu_{m}=0.5 \mathrm{~h}^{-1}$ is cultivated in a continuous culture under aerobic conditions where $D=0.28 \mathrm{~h}^{-1}$. The carbon and energy source in the feed is lactose with a concentration of $S_{0}=2 \mathrm{~g} / \mathrm{l}$. The effluent lactose concentration is desired to be $S=0.1 \mathrm{~g} / \mathrm{l}$. If the growth rate is limited by oxygen transfer, by using the following information:
$$
Y_{X I S}^{M}=0.45 \mathrm{~g} X / \mathrm{gS}, \quad Y_{X I O}^{M S}=0.25 \mathrm{~g} X / \mathrm{g} \mathrm{O}_{2} \text { and } C^{*}=8 \mathrm{mg} / 1
$$
a. Determine the steady-state biomass concentration $(X)$ and the specific rate of oxygen consumption $\left(q_{\mathrm{O}_{2}}\right)$.
b. What should be the oxygen-transfer coefficient $\left(k_{L} a\right)$ in order to overcome oxygentransfer limitation (i.e., $\left.C_{L}=2 \mathrm{mg} / \mathrm{l}\right)$ ?

Dominador Tan
Dominador Tan
Numerade Educator
01:24

Problem 14

The maximum growth yield coefficient for Bacillus subtilis growing on methanol is $0.4 \mathrm{~g} X / \mathrm{g}$ $S$. The heat of combustion of cells is $21 \mathrm{~kJ} / \mathrm{g}$ cells and for substrate it is $7.3 \mathrm{kcal} / \mathrm{g}$. Determine the metabolic heat generated by the cells per unit mass of methanol consumption.

Dominador Tan
Dominador Tan
Numerade Educator
01:26

Problem 15

Calculate the productivity (i.e., DP) of a chemostat under the following conditions:
1. Assume Monod kinetics apply. Assume that negligible amounts of biomass must be converted to product $(<1 \%)$.
2. Assume the Luedeking-Piert equation for product formation (eq. 6.18) applies.
3. Assume steady state:
$$
\begin{array}{ll}
D=0.8 \mu_{m} & Y_{X J s}^{M}=0.5 \mathrm{~g} X / \mathrm{g} S \\
\mu_{m}=1.0 \mathrm{~h}^{-1} & S_{0}=1000 \mathrm{mg} / \mathrm{l} \\
K_{s}=10 \mathrm{mg} / \mathrm{l} & \beta=0.5 \mathrm{~h}^{-1} \mathrm{mg} P / \mathrm{g} X \\
\alpha=0.4 \mathrm{mg} P / \mathrm{g} X &
\end{array}
$$

Dominador Tan
Dominador Tan
Numerade Educator
04:16

Problem 16

Consider a chemostat. You wish to know the number of cells in the reactor and the fraction of the cells that are viable (i.e., alive as determined by ability to divide).
a. Write an equation for viable cell number $\left(n_{v}\right)$. Assume that
$$
\mu_{\text {met,rep. }}=\frac{\mu_{\mathrm{m}, \mathrm{rep}} S}{K_{\text {s,rep }}+S}-k_{d}^{\prime}
$$
where $\mu_{\text {net,rep }}=$ net specific replication rate, $\mu_{m, r e p}=$ maximum specific replication rate, and $K_{d}=$ death rate. $K_{s, \text { rep }}$ is the saturation parameter.
b. Derive an expression for the value of $S$ at steady state.
c. Write the number balance in the chemostat on dead cells $\left(n_{d}\right)$.
d. Derive an expression for the fraction of the total population which are dead cells.

Km Neeraj
Km Neeraj
Numerade Educator
01:52

Problem 17

E. coli is cultivated in continuous culture under aerobic conditions with a glucose limitation. When the system is operated at $D=0.2 \mathrm{~h}^{-1}$, determine the effluent glucose and biomass concentrations by using the following equations $\left(S_{0}=5 \mathrm{~g} / \mathrm{l}\right)$ :
a. Monod equation: $\mu_{m}=0.25 \mathrm{~h}^{-1}, K_{s}=100 \mathrm{mg} / \mathrm{l}$.
b. Tessier equation: $\mu_{m}=0.25 \mathrm{~h}^{-1}, K=0.005(\mathrm{mg} / 1)^{-1}$.
c. Moser equation: $\mu_{m}=0.25 \mathrm{~h}^{-1}, K_{s}=100 \mathrm{mg} / 1, n=1.5$.
d. Contois equation: $\mu_{m}=0.25 \mathrm{~h}^{-1}, K_{s x}=0.04, Y_{X S}^{M}=0.4 \mathrm{~g} X / \mathrm{g} S .$
$$
S_{0}=5 \mathrm{~g} / 1
$$
Compare and comment on the results.

Carson Merrill
Carson Merrill
Numerade Educator
01:26

Problem 18

Consider steady-state operation of a chemostat. Assume that growth is substrate inhibited and that endogeneous metabolism can be ignored such that:
$$
\mu_{\text {met }}=\frac{\mu_{m} S}{K_{S}+S+S^{2} / K_{1}}
$$
a. Derive an expression for the residual substrate concentration (i.e., $S$ ) as a function of dilution rate and the kinetic parameters $\left(\mu_{m}, K_{S}, K_{1}\right)$.
b. What are the implications for operation of a chemostat when the organism is subjected to substrate inhibition?

Dominador Tan
Dominador Tan
Numerade Educator
06:31

Problem 19

Formation of lactic acid from glucose is realized in a continuous culture by Streptococcus lactis. The following information was obtained from experimental studies.
$$
\begin{aligned}
&S_{0}=5 \mathrm{~g} / 1, \mu_{m}=0.2 \mathrm{~h}^{-1}, K_{S}=200 \mathrm{mg} / 1, k_{d}=0.002 \mathrm{~h}^{-1}, Y_{X S}^{M}=0.4 \mathrm{~g} X / \mathrm{g} S, Y_{P / S}=0.2 \mathrm{~g} P / \mathrm{g} S, \\
&q_{P}=0.1 \mathrm{~g} P / \mathrm{g} X-\mathrm{h} .
\end{aligned}
$$
a. Plot the variations of $S, X, P, \mathrm{DX}$, and DP with dilution rate.
b. Determine (graphically) the optimum dilution rate maximizing the productivities of biomass (DX) and the product (DP).

Mukesh Devi
Mukesh Devi
Numerade Educator