A cybernetic view of microbial growth: Modeling of cells as optimal

A Cybernetic View of Microbial Growth:
Modeling of Cells as Optimal Strategists
P. Dhurjati, D. Ramkrishna, M. C. Flickinger, and G. T. Tsao
School of Chemical Engineering, Purdue University,
West Lafayette, Indiana 47907
Accepted for Publication February 27, 1984
A cybernetic framework is presented which views microbial response in multiple substrate environments as a
judicious investment of cellular resources in synthesizing different key proteins according to an optimal regulatory strategy. A mathematical model is developed
within the cybernetic framework for the diauxic growth
of Klebsiella pneurnoniae on a mixture of D-glucoseand
D-xylose. The “bang-bang” optimal policy describes
well the experimental observations obtained using a fermenter coupled to an Apple I1 microcomputer. Striking
variations in respiratory levels are observed experimentally during the switching of the cell’s adaptive machinery for the utilization of the less preferred substrate.
INTRODUCTION
The present work is concerned with modeling the
growth of microorganisms in multiple substrate environments. The problem of cell growth in multiple substrate
media is of importance in the fermentation of hemicellulose hydrolysates and in such processes as waste water
treatment. In such multisubstrate media, the cells exhibit a varied set of.metabolic activities, the most striking
of which is manifested in the phenomena of diauxic
growth. Here, the organism displays a marked preference
among two functionally equivalent substrates by deferring
utilization of one of the substrates until complete exhaustion of the other. Such calculated microbial responses to
their surroundings provide a considerable challenge to the
“chemical-kinetic framework” and it is doubtful whether
a proliferation of the number of lumped biochemical species, per se, could account for the broad spectrum of possible microbial responses.
In our approach, referred to as the cybernetic [the word
“cybernetic” is derived from the Greek “ ~ v & p n p ” which
means steersman] approach, we adopt a viewpoint at variance from modeling based purely on kinetic considerations. We consider the cells to be optimal control systems
involved in the maximization of a performance index
(goal). The main advantage of such an approach is that we
have a relatively simpler description of the microorganism
and the complex regulatory processes are reflected in
terms of the cell’s accomplishment of its optimal control
objectives. Having inferred some performance criteria for
Biotechnology and Bioengineering, Vol. XXVII, Pp. 1-9 (1985)
01985 John Wiley & Sons, Inc.
the cells based on our biological knowledge, we then explore the extent to which observed regulatory phenomena
can be modeled without having to detail all the rudimentary physicochemical interactions of the numerous intracellular constituents. Thus, what is being done is the interpretation of an elaborate facility acquired through
evolution in terms of what an optimal system would accomplish. Theviewpoint of treating a living entity (such as
a bacterium) as an optimal steering system is not alien to
biology. Bremerman2 states that “Most biological systems are goal oriented. . . A goal seeking system is called
cybernetic if it pursues its goal while interacting with an
environment.”
A mathematical model is developed within the cybernetic framework for the specific case of the diauxic growth
of Klebsiellapneumoniae. The model is solved using Pontryagin’s Maximum Principle and interpreted in the light
of experimental results.
CYBERNETIC FRAMEWORK
The cybernetic view of the cell presumes the existence of
certain characteristic features in the “model” microorganism. Thus, we consider the cell to possess an “adaptive
machinery” where specific metabolic changes occur in response to variations in the environment and also to have a
“permanent machinery” that does not participate “directly” in the adaptive response. The “regulator” which is
the third component of the model cell, is an idealization
that embodies the control information in the DNA and
guides the nature of the adaptive response. A schematic of
the cell as a cybernetic system is shown in Figure 1. A
detailed description of the various components of the
model cell follows.
Adaptive Machinery
The response of microorganisms to variations in a multiple substrate environment is very specific and usually
limited to the biosynthetic pathways of a few key proteins
necessary for incorporating the substrate into the cell’s
metabolism. The “adaptive machinery” is assumed to be
involved in the synthesis of these key proteins and includes
CCC 0006-3592/85/010001-09$04.00
r --
ENVIRONMENT
SUBSTRATES
Figure 1.
Schematic of cell as a cybernetic system.
a resource facility consisting of the precursor biomolecules required for the key proteins. From a modeling viewpoint, we may consider only a single critical resource such
as a particular amino acid to be the resource required for
the reaction networks concerned with the biosynthesis of
the key proteins.
The heart of the cybernetic approach proposed herein
lies in the “optimal” allocation of this resource among
competing reaction networks for the synthesis of key proteins for the various substrates (Fig. 1). Of course, the
most compelling reason for an optimal allocation of the
resource is their limited availability. Since metabolic processes must occur in time, we express the constraint on the
availability of resources by requiring that the “rate of
availability” of the critical resource be no greater than a
stipulated maximum (k,,,,,).
physiological realization of the regulator within the cell
would most likely be the control regions of the operon such
as the promoter and operator sites. Apter3 presents an
interesting discussion on how the operon could be considered to be a computing system or an automaton.
The “program” for utilizing the substrates has associated with it two important features: (1) an Optimization
criteria or a guiding strategy for the optimal allocation of
resources, and (2) the period for which such an optimal
strategy is to be established. There are other examples in
the biological literature where optimization criteria have
been used in the modeling of biological system^^.^ and detailed discussion on the appropriateness of such assumptions are presented in refs. 3 and 6-8.
From the viewpoint of the theory of evolution, an objectivefunction that suggests itself is the maximization of cell
mass productivity during a certain “period” of optimization. The term “period” here refers to the time interval for
which the optimization criteria are to be established. The
implementation of control decisions such as synthesis of
proteins involves thousands of reactions and long periods
of time (up to a couple of generations) and, consequently,
one could assume that such control decisions have associated with it a long-term projection of the future. Thus, one
could argue that the metabolic inertia inherent in a microbial system calls upon an efficient organism to prepare
“ahead.” The long-term policy is assumed to be initiated
as well as reevaluated when “external” changes occur in
the concentrations of substrates in the environment that
are brought about by other than the growth of the cells
themselves. From a biological viewpoint, the evolutionary
history (of environmental variations) embodied in the
genotype of the specific organism would dictate the nature
of its control strategy. In the present endeavor, we wish to
“infer” the appropriate combination of optimization and
constraints based on our biochemical knowledge so as to
best describe (for engineering purposes) the regulatory behavior of the microorganism.
Permanent Machinery
Once the adaptive machinery responds to the input of
environmental conditions with an output of certain key
proteins, the permanent machinery carries out the thousands of reactions necessary for the replication of cellular
material. The permanent machinery thus consists of
metabolic reactions which always occur and it is in this
sense that they are “permanent” features of the cell. The
nature of reaction networks that are included in the permanent machinery is largely dependent on the regulatory
problem under consideration just as the adaptive machinery is determined by the nature of the adaptation.
Regulator
The regulator is the “decision-making” apparatus of
the model cell that senses the environment and decides on
a “program” of its utilization by optimally allocating resources to the key protein synthesis systems. The possible
2
THE MATHEMATICAL MODEL
The various state differential equations for the mathematical model presented here have been formulated
within the context of the cybernetic framework discussed
in the previous section.
Consider the batch aerobic growth of a microorganism
in a multiple substrate environment where the carbon
source is the only growth-limiting nutrient and is provided
through “n” different carbon substrates. We denote by Si
the concentration of the ith-limiting carbon sources expressed as grams per liter of fermenter liquid volume. The
concentration of the microorganisms in the fermenter,
X , , refers to the dry mass of cells in grams per liter. The
physiological state of the cells, represented in terms of the
various key proteinsp;, is the concentration of the ith key
protein involved in the incorporation of the substrate Si.
We can then define an environmental vector given by
BIOTECHNOLOGY AND BIOENGINEERING, VOL. 27, JANUARY 1985
Parameter VSiis equivalent to (pmaxi/Ysj)
where pmax;
is
the maximum specific growth rate on the ith substrate.
The production of cell mass is the cumulative effect of
the utilization of the various substrates, and is given by
and a physiological vector given by
p = [PI, pz, . . .> P,ll
The state vector X provides information regarding the
fermenter in terms of the state of the environment and the
physiological state of the cells:
X
= [Xc,S1,
* - . , S ~ , P ~ , ***,Pnl
PZ~
We represent by Ri the allocation rate of the critical
resource to the ith key protein. The fractional allocation
given byRj/Rmaxare then denoted as ui and we can define
an allocation vector given by
u = [UI,u2,
* .
., u,l
i=l
The maximum possible key protein concentration of
the ith protein Pi during exponential growth on S; alone is
represented by Pi,,, . The nondimensionalized key protein concentration for the ith key protein, denoted by ej is
defined by ei = (P,./Pjm,,).
The rate of synthesis of ei is considered to be dependent
on the availability of resources, uikm,, , in the following
manner:
dt
i=l
Y.;
~
dt
where Ysj
is considered to be the yield of cell mass per unit
of substrate Sj for the case where cells are grown on S;
alone.
As discussed earlier in the section on the cybernetic
framework, the regulator has associated with it an optimization criterion. One possible optimization period could
be the maximization of cell mass productivity given by
i-
tr
max J O
~
r 0d t
where tf is the optimization period defined by the following stopping condition:
cuj=l
-dei
_
-c
dt
In the present case, we consider R,,, to be a constant.
Mandelstam9 has shown that the turnover of protein
could indeed lead to a constant pool of amino acids even
when no external substrates are consumed. Thus, the constraint on the total availability of resources can be expressed as
n
dS;
n
dXC
--
- a j f ( u j k m a x)
biei
In the above, a j is a constant dictating the rate of synthesis, f gives the filnctioiial dependence of the synthesis
rate on the fractional allocations u j ,and the last term represents the degradation of proteins (or protein turnover)
with bi being a degradation constant. For our present
problem, we assume a linear functional dependence and
include R,,, in the constant ai.
Once the adaptive machinery synthesizes the appropriate key proteins, the permanent reactions for the growth
proceed in an invariant manner and can be represented as
si + Pj + x, x,
+
The above is a macroscopic representation of the cell
growth process and does not reflect the actual stoichiometry. The rate of incorporation of substrate S; by key protein Pi can be described by simple Monod-type kinetics:
1 dSj - VSieiSi
X , dt
kSi Sj
+
Here, Vsi and k,,are growth and saturation constants, respectively, of Monod type and the negative sign is present
because substrate Si is being consumed by the cells X , .
n
C Sio
i= 1
+
se
S;dt = E
0
where Sjois the amount of substrate Sj at time t = 0.
OPTIMAL SOLUTION FOR A SPECIFIC CASE
The solution of the optimization problem discussed in
the previous section requires the use of calculus of variation. Specifically, we solve the problem by using Pontryagin’s maximum principlelO,llwhich involves the definition of a Hamiltonian and a set of adjoint differential
equations corresponding to the state variables. The numerical techniques for the solution of these optimization
problems are discussed by Denn’’ and Bryson and Ho.I2
One first guesses an initial optimal u, solves the state
equation forward in time, and then utilizes the state solution to solve the adjoint differential backward in time. The
value of u is then updated using the steep descent method
until no further improvement in the performance index is
obtained.
The values of the parameters for the specific case of
Klebsiellapneurnoniae growing aerobically in a batch culture consisting of two limiting carbon sources (D-glucose
and D-xylose) are summarized in Table I. The initial glucose and xylose concentrations were 1 and 2 g/L, respectively. The values of growth rates and yields were those
obtained from single substrate experiments while values
for other parameters such as a; and bi in the key protein
equation were assumed. For this problem, the Hamiltonian is linear in u1 and can be represented as
H=uq+y
DHURJATI ET AL.: CYBERNETIC VIEW OF MICROBIAL GROWTH
3
where a is referred to as the switching function, and depending on its value, we have the following optimal solution
u1 =
0 if u > 0
=lifu<O
When a = 0, we have a singular solution and the optimal u1 can be obtained by setting the successive derivatives of the switching function, with respect to time, equal
to zero. For the present problem, we find that we need not
be concerned with a singular solution and the optimal solution turns out to be what is referred to as a “bang-bang’’
optimal policy. Thus, as shown in Figure 2, the optimal u ,
Table I. Table of values.
Parameters
Pmaxl
Pmax2
YS!
ys2 a
yo2 a
Go,
yco,
ks, ks,
1
Values
Parameters
1.07 h-’
0.51 h-’
0.525
0.36
1.754
1.10
1.35
0.89
10 mg/L
SI(0)
Sz(0)
el@)
ez(O)
01 7
b,
a 2 . b,
x,(0)
kLac*
V
aD-glUCOSe.
bD-xylose.
Values
1.0 g/L
2.0 g/L
1 .o
0.002
10 h-’
1.0h-’
0.0175 g/L
0.023 g mol/L h
1.5 L
takes on a value equal to 1 followed by a switch in policy to
a value u , = 0 at switching time t,, . The optimal policy
can be substituted in the state differential equations to
yield the solution for the various states as a function of
time. Log (cell mass) and substrate concentrations are
plotted as function of time in Figure 3. We can see that
growth takes place in two distinct stages. The first stage
corresponds to the optimal policy u 1 = 1 and consists in
the exclusive utilization of glucose with exponential
growth of cells. This is followed by a switch in policy and
the consumption of xylose. At the point where a discontinuous change exists in the control variable, the WierstrassErdmann corner conditions need to be satisfied, as discussed by Citron.I3
The optimal strategy discussed previously could be detected by experimentally monitoring uptake of oxygen by
the cells. In this connection, it is useful to derive relationships between the dissolved oxygen tension (DOT) and the
adaptive machinery of the cell represented in terms of the
key proteins. This would enable us to utilize DOT to detect alterations in the adaptive machinery of the cell
caused by policy changes. An oxygen balance in the culture gives
dc/dt = kla(c* - c) - rx
where r, is the instantaneous oxygen consumption rate of
the cells, kLa is the mass transfer coefficient, c is the concentration of the oxygen in the liquid phase, and c* is the
CELL MASS AND STRATEGY
VS
TIME
1.m
----.l4bDEL
-EXP
‘“1
.€m2-
L7
c
w
E .mLo
n
a
-i.m-
5
m
Figure2. Optimal strategy and state solution for cell mass comparison with experimental
values for Klebsiellapneumoniae growing on 1 g/L D-glUCOSe and 2 g/L D-XyIOSe at 37°C.
4
BIOTECHNOLOGY AND BIOENGINEERING, VOL. 27, JANUARY 1985
LCIG[CELL MASS1 flND SUBSTRATES
Figw3.
TIME
Plot of substrate concentrations and natural logarithm of cell mass versus time.
saturation oxygen in the liquid phase. DOT expressed as
percent of air saturation is given by lOO(c/c*). The oxygen consumption rate r, may in general be
n
r, =
VS
C rxi
i= 1
where the index i refers to the ith substrate and rXiis given
by
5dSi)ei
-XC
Yo,(i)
where YO2(!)
is the yield of cells from oxygen for the ith
substrate (in g/g) and is assumed to be a constant.
One can adopt a pseudo-steady-state assumption (dc/
dt = 0) in the oxygen balance equation above and obtain
an expression for the dissolved oxygen tension as follows:
At the instant corresponding to the change in strategy,
one could expect the cell massXc and the value of the mass
transfer parameter kLac* to be constant. Thus, any
changes in DOT would be reflective of changes within the
cell which affect the intrinsic growth related parameters
or the physiological vector represented in terms of the
nondimensionalized key protein level e i . Thus, a switch in
cellular strategy should be more readily observed in the
dissolved oxygen response in contrast to cell mass which
cannot be expected to provide instantaneous indications
of the physiological changes. As discussed in the next section, the dissolved oxygen response is indeed a very sensitive experimental tool to diagnose switches in cellular policy.
RESULTS AND DISCUSSION
The experimental materials and methods are discussed
in the Appendix. The experimental plot of cell mass (g/L)
as afunction of time is shown in Figure 2. The initial levels
of D-glucose and D-xylose in this experiment were 1 and 2
g/L, respectively. As can be seen from this plot, growth
takes place in two distinct exponential phases and we have
the growth pattern referred to by Monod14 as "diauxie."
Also superposed on this plot is the state solution for cell
mass resulting from the bang-bang optimal policy. It can
be seen that the bang-bang optimal strategy can describe
fairly well the diauxic growth of Klebsiellapneumoniae on
a glucose-xylose mixture. From the point of the optimal
formulation, the cells first allocate resources to the key
proteins involved in the exclusive utilization of glucose.
This is followed by a switch in policy and an allocation of
cellular resources to key proteins for xylose, thus enabling
the utilization of the less preferred sugar. While the cell
mass measurement does indicate the policy switch, it is
not a satisfactory measurement from the cybernetic perspective. This is because corresponding to the change in
optimal strategy, the cell mass level does not give us a
sensitive indication of the nature of strategy involved. In
DHURJATI ET AL.: CYBERNETIC VIEW OF MICROBIAL GROWTH
5
this connection, the dissolved oxygen level as measured by
a galvanic oxygen probe turns out to be a sensitive monitor
of the cellular strategy.
Figure 4 consists of a plot of dissolved oxygen (as percent of air saturation) versus time (in h). The model values
were obtained by using the bang-bang strategy and solving for the differential equation describing the change in
oxygen level in the fermenter. The first stage of the dissolved oxygen up to just over three hours is quite similar to
the DOT curve for the single substrate. But corresponding
to the “bang-bang’’ switch in policy, there is a dramatic
and sudden variation in the dissolved oxygen which
changes from close to 60% to almost saturation levels.
According to the model, the switch corresponds to allocation of resources to the key proteins for the less preferred
substrate which results in the growth of cells and a consequent decrease in dissolved oxygen level. The dissolved
oxygen level drops again until complete exhaustion of the
less preferred xylose. At this time, both sugars have been
exhausted and the dissolved oxygen levels out at near saturation levels. As can be seen from the plot, the experimentally dissolved oxygen level does not “rise back” as fast as
predicted. This is probably due to the production of certain intermediates which are slowly consumed at the end.
In such a case, this phenomena could actually be a triple
substrate phenomena with the third substrate being an
intermediate fermentation product which is subsequently
consumed.
DUT
However, the most pertinent conclusion from the DOT
profile is that dissolved oxygen can indeed be a very sensitive tool to diagnose cellular strategies.
In addition to the cell mass and dissolved oxygen measurements, the exit gases from the fermenter were also
analyzed. The infrared analysis of C 0 2and the paramagnetic measurement of O2 can be utilized along with the
instantaneously measured inlet air flow rate to perform
on-line material balances on the gases flowing through the
fermenter as discussed by C o ~ n e y . These
’~
material balances yield the experimental values for the carbon dioxide
evolution rate (CER) and the oxygen uptake rate (OUR)
as shown in Figures 5 and 6 , respectively. The carbon dioxide evolution takes place in two stages corresponding to
the utilization of glucose and xylose with the switch occurring around 3 h. The oxygen uptake rate shows a similar
behavior with cells taking up oxygen in two different
stages corresponding to each of the policy periods. The
model values for OUR were obtained using the expression
for r, derived in the previous section. An equation similar
to the oxygen uptake rate equation was written for CER
replacing Yo2 with Yco2(,).
These stoicfiometric coefficients were estimated from
single substrate experiments and are listed in Table I.
These expressions, along with the bang-bang strategy,
are able to describe in general terms the experimental
CER and OUR profiles. The preciseness of these fits can
of course be improved with a more elaborate mathemati-
VS
TIME
150.0
__..
PODEL
-EXP
125.0
120.0
la5.0
-
90.0
w
(
I
I
t
x
75.0
>
v)
D
E
0 60.0
Y5.0
30 .o
15.0
0
.m
I
1.m
2.m
I
I
3.mTlMEY,.ZuRS)
5.m
1
I
I
6.m
7.m
8.mO
m
CER
TIME
VS
~ _
MODEL
_ .
-EXP
m
OUR
VS
TIME
-...FWDEL
-EXP
I
.m
1.m
I
2.m
I
3 . m TIME~i&RSI
s.m
r
e.mm
7 .t m
8 .I m
n
n
Figure 6. 0,uptake rate versus time for growth on glucose and xylose.
DHURJATI ET AL.: CYBERNETIC VIEW OF MICROBIAL GROWTH
7
cal description of the microbial system, but our main focus is on the conceptual aspects of the model.
LIMITATIONS AND ALTERNATIVES
The present approach to modeling has endeavored to
circumvent microscopic details (such as those occurring at
the level of the operon16) by postulating the microorganisms to be cybernetic systems capable of optimally steering themselves toward certain control objectives. The cybernetic framework, however, has the flexibility of
accounting for significantly more biological detail in its
state differential equation constraints than has been considered in the specific mathematic model. Such elaborations on the model would be most fruitful to the extent
that actual experimental measurements can be made
which relate to the kind of microscopic variables which are
included. The measurement of several of these microscopic variables in fermentation processes is in many
cases difficult or impossible, and as such, their inclusion
in the model may not be warranted at the present stage.
The mathematical model considers only a single intracellular resource to be limiting. A more complete model
within the cybernetic framework could account for a diversity of precursor biomolecules. The model makes an
assumption of a constant pool of resources available at
any time. It is more likely that this resource pool would be
influenced by the energy content of cells or by the growth
rate of cells. There is a possibility of mutations occurring
during the course of the fermentation leading to changes
in the genotype of microorganisms. One of the present
limitations of the model is that it is not possible to account
for such mutative changes because of the assumption of a
4
Cart
n
Amplifier
Boards
1
Analog to Digital
Converters
Apple II Computer
Figure 7.
8
Schematic of experimental setup.
constant genotype. By including the genotype itself as a
variable, it is conceivable that the cybernetic framework
can account for such mutative changes.
Some other limitations of the model are in regard to the
nature of the mathematical relations adopted. It is possible to explore a wide range of other objective functions
with different time scales of optimization within the cybernetic framework. Alternate state differential constraints
could also be considered. For example, the state equation
for key protein production could be postulated to have a
nonlinear dependence on the control variable. This leads
to a nonlinear dependence of the Hamiltonian in the control variable which could allow for simultaneous utilization of both sugars. A limitation of the present model is
that such simultaneous utilization is permitted only in the
event of a singular solution because of the postulated linear dependence on the control variable.
CONCLUSIONS
We have illustrated here that a cybernetic model is capable of describing regulatory phenomena such as those observed in diauxie. The switching “on” and “off” of
operons as described by the operon theory16of Jacob and
Monod is interpreted in our model to be simple “bangbang” type of optimal strategy for the cellular system. The
experimental results for the batch growth of Klebsiella
pneumoniae on D-glUCOSe and D-xylose agree well with a
cybernetic description of the intracellular control system.
This work was supported in part by National Science Foundation
Grant No. ENG 78-20964. The experimental portion of this work was
carried out at the Laboratory of Renewable Resources Engineering, Purdue University, Lafayette, IN.
APPENDIX: DESCRIPTION OF EXPERIMENTAL
MATERIALS AND METHODS
All experiments were performed with the microorganism Klebsiella
pneumoniae ATCC 8724 obtained from the American Type Culture Collection (Rockville, MD). The bacterium was maintained on agar slants
with xylose as the sole carbon source. The medium utilized for growing
the organism is listed in Table A I . All innocula were prepared in 250-mL
Erhlenmeyer flasks with a culture volume of 50 mL at 370°C using glucose as the carbon source. A 2-L New Brunswick fermenter (culture volume = 1.5 L) with a magnetic coupled drive was used in all experiments.
The power for agitation was provided by 1/8-hp motor (Model 42140
Dayton Mfg. Co., IL) and the agitation rate was 900 rpm. Dissolved
oxygen tension (DOT) was measured using a galvanic oxygen electrode
fitted with a 1 -mil Teflon membrane. The probe was constructed according to the specifications of Borkowskii7 and had a response time of less
than a minute. In our entire discussion, “cell mass” referred to the dry
mass of cells in grams per liter. Cell mass was monitored using optical
density measurements at a wavelength of 540 nm by means of a Bausch &
Lamb spectrophotometer (Spec 20) adapted with a special flow dilution
device.
The inlet air flow was measured using an air flow transducer (model
8141, Matheson, NJ). The CO, was measured using an infrared analyzer
(model IR-702, Infrared Industries, Inc., CA) while oxygen was measured with a paramagnetic analyzer (model 755, Beckman, NJ). More
details of the off-gas analysis equipment are presented by Flickinger.”
‘*
BIOTECHNOLOGY AND BIOENGINEERING, VOL. 27, JANUARY 1985
X
X,
state vector
concentration of cells (g/L)
stoichiometric coefficient for substrate S, (g cells/g C 0 2 )
yco2c,
stoichiometric coefficient for substrate S, (g cells/g 0 2 )
YOZCd
growth yield constant for substrate S, (g cells/g S, )
Ys,
Table AI. Composition of the medium.a
Component
K2HPO4 3HzO
K2HP04
(NH4 )2. so4
NaCl
CaC12.2H20
FeSO,. 7H20
MnS04. 4 H 2 0
ZnS04.7H20
EDTA
MgS04 7H2O
Concentration
(g/L)
11.4
1.5
3.0
0.1
0.014
0.01
0.0028
0.0075
0.4
0.24
Greek
switching function
specific growth rate as a function of S,
pmaxr maximum specific growth rate for S, (h-' )
Ll
p,(S,)
Superscripts and subscripts
differentiation with respect to time
1
refers to the ith substrate
aExcludingthe carbon source, it is the same
as the medium of Anderson.22
The recalibration of the off-gas equipment as well as continuous monitoring of all data was done by means of an Apple I1 Plus computer as
discussed by Forrest2' and DhurjatL2'
NOMENCLATURE
constant for key protein i synthesis rate (h-')
constant for key protein i breakdown rate (h-l)
concentration of dissolved O2 in liquid (g/L)
saturation concentration of dissolved 0,in liquid (g/L)
dissolved oxygen tension (7' of air saturation)
fraction of maximum possible key protein P,
Hamiltonian
oxygen solution rate constant (h-')
saturation constant for substrate S, (g/L)
number of limiting substrates
structural vector of key protein levels
key protein for substrates S, (g/g biomass)
maximum level of key protein P,
rate of allocation of critical resource R for synthesis of key
protein P,
maximum rate of availability of critical resource
cumulative oxygen uptake rate of the culture (g/L h)
oxygen uptake rate due to utilization of substrate S,
environmental vector of limiting substrate concentrations
concentration of limiting substrate i (g/L)
initial concentration of limiting substrate i (g/L)
time (h)
time period of optimization (h)
switching time (h)
decision vector
fraction allocation of resources to P,
maximum uptake rate of substrate S, (h-l)
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