Chaitali Mandal Æ Ravindra D. Gudi
G. K. Suraishkumar
Multi-objective optimization in Aspergillus niger fermentation
for selective product enhancement
Abstract A multi-objective optimization formulation that
reflects the multi-substrate optimization in a multi-product fermentation is proposed in this work. This formulation includes the application of e-constraint to generate
the trade-off solution for the enhancement of one selective
product in a multi-product fermentation, with simultaneous minimization of the other product within a
threshold limit. The formulation has been applied to the
fed-batch fermentation of Aspergillus niger that produces
a number of enzymes during the course of fermentation,
and of these, catalase and protease enzyme expression
have been chosen as the enzymes of interest. Also, this
proposed formulation has been applied in the environment of three control variables, i.e. the feed rates of sucrose, nitrogen source and oxygen and a set of trade-off
solutions have been generated to develop the paretooptimal curve. We have developed and experimentally
evaluated the optimal control profiles for multiple substrate feed additions in the fed-batch fermentation of A.
niger to maximize catalase expression along with protease
expression within a threshold limit and vice versa. An
increase of about 70% final catalase and 31% final protease compared to conventional fed-batch cultivation
were obtained. Novel methods of oxygen supply through
liquid-phase H2O2 addition have been used with a view to
overcome limitations of aeration due to high gas–liquid
transport resistance. The multi-objective optimization
problem involved linearly appearing control variables
and the decision space is constrained by state and end
point constraints. The proposed multi-objective optimization is solved by differential evolution algorithm, a
C. Mandal Æ R. D. Gudi
Department of Chemical Engineering, IIT Bombay, 400 076 Powai,
Mumbai, India
G. K. Suraishkumar
Biotechnology Department, IIT Madras, 600 036 Chennai, India
relatively superior population-based stochastic optimization strategy.
List of symbols
aP
protease yield with respect to cell (unit protease (kg cell mass)1)
bP
protease yield with respect to maintenance
(unit protease (kg cell mass)1 s1)
C
concentration of sucrose (kg m3)
Cfeed
sucrose concentration in feed (kg m3)
CR
crossover factor
d
hyphal diameter (m)
Ei
intracellular enzyme (catalase) (kmol (kgcell)1)
F
feed rate (m3 s1)
FC
sucrose feed rate (m3 s1)
FH
hydrogen peroxide feed rate (m3 s1)
FN
nitrogen source feed rate (m3 s1)
H
hydrogen peroxide concentration in the medium (kmol m3)
Hfeed
hydrogen peroxide concentration in feed
(kmol m3)
Hi
hydrogen peroxide concentration in the cell
(kmol m3)
K
rate constant for enzymatic reaction inside the
cell (kmol m3)1 s1)
KC
monod type constant for sucrose (kg m3)
Kd
rate constant for enzymatic reaction in the
medium (kmol m3)1 s1)
KH
hydrogen peroxide permeability across the cell
envelope (m s1)
K Hi
monod type constant for intracellular hydrogen peroxide (kmol m3)
Ki
inhibition constant for hydrogen peroxide in
the medium (kmol m3)
KN
monod type constant for nitrogen source
(kg m3)
N
nitrogen source concentration (kg m3)
Nd
number of discretizations in fermentation
time
150
Nfeed
NP
Pcat
Pprot
qP
V
Vf
X
YC/X
YC/peat
YC/Ppeat
YN/X
nitrogen source concentration in feed
(kg m3)
population size
catalase concentration (kmol m3)
protease concentration (U m3)
catalase yield with respect to cell (kmol catalase (kg cell mass)1)
reactor volume (m3)
maximum working volume of the reactor
(m3)
cell concentration (kg m3)
inverse of cell yield with respect to sucrose
inverse of catalase yield with respect to
sucrose
inverse of protease yield with respect to
sucrose
inverse of cell yield with respect to nitrogen
source
Greek letters
e
penalty factor
lm
maximum specific growth rate (s1)
q
density of the cell (kg m3)
Subscripts
f
final time
Introduction
Many industrial bioprocesses involve expression of more
than one product during the course of fermentation.
Optimization of such bioprocesses therefore needs to be
focussed towards obtaining more favourable product
expression rates, so that (i) the substrate is better utilized
towards the main product of interest and (ii) downstream processing towards separation of the desired
product, in the presence of other side products, is facilitated. Thus, a multi-objective optimization problem
results when one includes constraints on the expression
rates/final concentrations on the other products. The
application of multi-objective optimization to fermentation problems has not been extensively explored so far,
but has been useful in the context of the chemical process applications [1]. However, multi-objective optimization is quite relevant in fermentation that has a
number of different product expressions (for example, a
number of enzymes could be expressed). The design of
an optimal control strategy in a multi-product fed-batch
fermentation involves the development of a substrate
feeding strategy that maximizes a given performance
index and simultaneously accommodates other conflicting or non-conflicting objectives, subject to several
physical constraints arising out of the nature of the
problem. The optimization procedure requires a reliable
dynamic model of the system that typically includes the
mass balances on biomass growth, substrate consumption and product formation.
In most of the cases considered in the biochemical
engineering literature, the optimization of fed-batch
processes has involved a single operating variable.
However, to better optimize a fed-batch process, multiple operating variables can also be simultaneously
optimized [2]. Such optimizations result in optimal time
profiles for several variables say, the concentration of
carbon, nitrogen and oxygen (substrates). This necessitates the regulation of those substrate levels in the bioreactor during optimal operation. The optimization in a
multi-product fermentation typically includes the end
point constraint to reflect realistic bounds on the final
levels of product concentrations. Also, optimization of
fed-batch fermentations could typically include statepath and end point constraints. For example, a typical
end point constraint could be the level of substrate at the
end of fermentation; a typical state-path constraint
could include bounds on the substrate level at any instant in time. Thus, the overall problem of arriving at
optimal profiles of multiple substrates, while accommodating state-path and end point constraints, is fairly
complex and interesting.
Generally, multi-objective optimization deals with
more than one conflicting types of objective functions
and if both the objective functions are equally important
(in case of two objectives), then it is impossible to
completely satisfy the two objective functions simultaneously. Therefore, multi-objective optimization results
in a trade-off solution between the two objective functions. An evolutionary algorithm (EA) is one of the
popular means to solve the multi-objective optimization
problem. The solution of such optimization follows the
preference-based approach where a relative preference
vector is used to scalarize a multiple objective. In classical search, the optimization methods use a point-bypoint approach, where one solution in each iteration is
modified to a better solution and results in a single
optimized solution. Therefore, the use of classical search
would motivate the preference-based approach to generate a single optimized solution in a single run. Thus,
one approach and solution could be to convert the
multi-objective optimization problem to a single objective optimization and use classical search [3]. A number
of other approaches based on the concept of non-elitism
in evolutionary computation have also been proposed [3]
and applied to engineering problems [1].
One major difficulty in optimal control problem
arises when the control variable appears linearly in the
state equations as well as in the Hamiltonian; this results
in a singular control problem [4]. In the majority of the
available literature, the Pontryagin’s principle [5, 6] has
been used to solve the singular control problem. The
singular control problem is normally difficult to solve,
for a constraint-bound complex optimization problem
consists of four or more dynamic balance equations
describing the system. It is possible to transform a singular problem formulation to a relatively easier non-
151
singular problem formulation depending on the structure of the state equations [7–9]. But again, transformations to convert the problem from singular to nonsingular for a constraint-bound optimization problem
are not easy to find.
Evolutionary algorithms are therefore increasingly
becoming popular to solve complex optimization problems. Differential evolution (DE), as developed by Storn
and Price [10] in 1996, is one of the relatively superior
population-based evolutionary algorithms. It is a stochastic parallel direct search method that operates on a
population of potential solutions by applying the principle of the survival of the fittest to produce an optimal
solution. It is capable of solving problems involving
general constraints, as shown in Chiou and Wang [11].
However, the DE strategy has not been applied to solve
a multi-objective optimization problem involving
manipulation of addition rates for multiple substrates.
Further, the experimental evaluations of the optimization strategies are not widely reported.
This work is concerned with the development of
optimal control profiles for maximizing selected enzyme
expressions through appropriate manipulations of multiple substrate additions. The system of focus is the fedbatch fermentation involving the organism Aspergillus
niger which is an industrially important filamentous
fungi due to its versatile application for the production
of various enzymes and acids. We specifically focus on
the expressions rates for two of these enzymes, viz.,
catalase and protease, in this study. Catalase is a hemecontaining enzyme that is present in almost all aerobic
organisms and it protects the cell against the damaging
effects of H2O2 [12]. Protease is one of the most widely
used industrial enzymes [13] and is involved in the proteolytic degradation [14]. A liquid phase oxygen supply
strategy (LPOS) through need-based liquid phase H2O2
addition has been used for this system with a view to
overcome limitations of aeration due to high gas–liquid
transport resistance [15]. In this study as well, we consider this mode of oxygen supply and seek to optimize
the liquid phase peroxide additions. Other substrates
that are crucial to the fermentation are the sucrose and
nitrogen levels, which we also seek to optimize. The
problem is also typically (upper) bounded by the H2O2
level in the fermentation, which is treated in our work as
a path constraint. We additionally impose end point
constraints to minimize residual levels of substrates at
the end of the fermentation. High levels of protease and
catalase at the end of fermentation poses a problem in
the downstream separation of these enzymes [13, 16–18].
Therefore, we seek to optimize conditions in the
fermentation that maximize expressions of one of the
products (say, catalase) while keeping the other product
(say, protease) expressions within threshold limits. We
show that a multi-objective optimization problem thus
results and the objectives are conflicting. To solve this
complex multi-objective optimization problem, we propose the use of the DE algorithm. We demonstrate the
applicability of the proposed methodology through
simulations and experimental validations involving fedbatch fermentation of A. niger.
Process model
The production of catalase and protease enzyme by
A. niger, with liquid phase H2O2 addition as an alternative source of oxygen, has been chosen as the model
system. Of these, catalase is expressed as a growth-associated product, whereas the protease is expressed as a
non-growth-associated product. Since catalase is a
growth-associated product, its expression rate increases
with cell growth and then there is a decrease in catalase
production as the cell growth reaches its stationary phase.
Therefore, if catalase is chosen to be the enzyme of
interest, one of the objectives of this optimization problem is to maintain the cell growth at its exponential phase
in order to maximize the catalase production. On the
other hand, the protease production occurs both in
exponential and stationary growth phases with a significant production occurring in the stationary phase.
Therefore, to maximize protease, the other objective of
this optimization is to maintain a prolonged stationary
phase to maximize the protease production. However, for
a fixed time of fermentation, depending on which enzyme
expression is sought to be maximized, the optimization
algorithm would have to look at balancing these growth
phase or stationary phase related objectives, so as to
satisfy various constraints imposed on the problem.
Towards this end, it is necessary to first develop the
dynamic process model for the mold system, which can be
used for both catalase and protease maximization runs.
Since the growth model criteria are different in the maximization of these two different products, a simple growth
model was developed using the logarithmic phase only.
The mechanism of oxygen availability from H2O2 [19] has
also been considered while developing the model.
It has been observed from the experimental data that
increased H2O2 concentration (within toxic thresholds)
in the broth results in a 26% enhancement in the catalase production. Also, it is well known that the cell has a
natural tendency to produce catalase to avoid the peroxide toxicity in the broth [20, 21]. This is true in either
case of H2O2 concentration increase, viz., by external
addition or by the expression from the microorganisms
themselves. In contrary, the effect of increased H2O2
concentration on protease production is almost negligible. Accordingly, therefore, the catalase and the protease
models have been developed by taking care of the effects
of H2O2 on them.
In a submerged culture, the A. niger grows either as
a hyphal element (dispersed mycelia) or as a pelleted
form [22]. The hyphal morphology affects the rheological properties of the broth and imposed high viscosity. The mechanism of the filamentous growth is
complicated and different from that of unicellular
growth. All the cells within the mycelium take part in
the production of the protoplasm but the extension of
152
the hypha occurs only at the tips. The total length of
the hyphae may increase exponentially due to the formation of the new tips along the hyphae [23]. Therefore, the model is based on a simple assumption to
measure the cell surface area for H2O2 diffusion. It is
assumed that all the hyphal growth together formed an
equivalent single solid cylindrical hypha whose equivalent diameter, d, is constant and the hyphal length, l,
increases with time. Therefore, the area of the single
hypha can be written as
d 2 4XV pd 2
A ¼ pdl þ p ¼
þ
:
ð1Þ
dq
2
2
The dynamic mass balance equations for the fed-batch
fermentation involving the addition of sucrose and
nitrogen sources and H2O2 can be written as follows:
A mass balance for H2O2 in the medium, considering
that the flux of H2O2 in the cell is proportional to the
difference in H2O2 concentrations between the medium
and cell, and also considering the decomposition of
H2O2 by extracellular catalase, can be written as
dH FH Hfeed ðFH þ FC þ FN ÞH
¼
dt
V
V 4XV pd 2 ðH Hi Þ
þ
Kd Pcat H :
KH
dq
V
2
dC FC Cfeed ðFH þFC þFN ÞC
Ki
N
¼
YC=X lm X
dt
V
V
ðKi þH ÞðKN þN Þ
Ki
N
Hi
C
YC=peat qP lm X
ðKi þH ÞðKN þN ÞðKHi þHi ÞðKC þCÞ
Ki
N
C
þbP X
;
YC=Pprot aP lm X
ðKC þCÞ
ðKi þH ÞðKN þN Þ
ð7Þ
In Eq. 3, the last term on the right hand represents the
uptake of intracellular H2O2 due to growth.
The specific growth rate, l, is dependent on the H2O2
concentration and nitrogen levels in the medium. The
dependency on H2O2 is modelled assuming enzyme
inhibition kinetics in which nitrogen is considered as a
limiting growth factor. The dynamic equation describing
cell growth can be written as
ð4Þ
The dynamic equation for the catalase production, a
growth-associated product, is written as
dPcat
Ki
N
Hi
C
¼ qP lm X
dt
ðKi þ H Þ ðKN þ N Þ ðKHi þ Hi Þ ðKC þ CÞ
ðFH þ FC þ FN ÞPcat
;
V
The mass balances for sucrose and nitrogen concentrations in the medium were represented by the following
equations:
ð2Þ
A balance on the H2O2 concentration, considering the
cell as the system, yields the following dynamic balance
in terms of the intracellular hydrogen peroxide concentration, Hi,
dHi
4XV pd 2
q
þ
¼ KH
ðH Hi Þ
dq
XV
dt
2
ðFH þ FC þ FN ÞHi
KEi Hi q
V
Ki
N
:
ð3Þ
H i lm
ðKi þ H Þ ðKN þ N Þ
dX
Ki
N
ðFH þ FC þ FN ÞX
¼ lm X
:
dt
V
ðKi þ H Þ ðKN þ N Þ
The dynamic equation for the protease formation, a
non-growth-associated product, is developed from Luedeking–Piret model where sucrose is considered as the
limiting nutrient. The dynamic equation for the product
formation is written as
dPprot
Ki
N
C
¼ a P lm X
þ bP X
ðKC þ CÞ
dt
ðKi þ H Þ ðKN þ N Þ
ðFH þ FC þ FN ÞPprot
:
V
ð6Þ
ð5Þ
where intracellular H2O2 level and sucrose concentration
were considered as the limiting factors on catalase
product formation.
dN FN Nfeed ðFH þ FC þ FN ÞN
¼
dt
V
V
Ki
N
:
YN =X lm X
ðKi þ H Þ ðKN þ N Þ
ð8Þ
Finally, the variation of the broth volume as a function
of the addition rates of the three substrates can be
written as
dV
¼ ðFH þ FC þ FN Þ:
dt
ð9Þ
The model parameters in the above equations were
partly estimated from the batch experiment data and
partly by solving the parameter estimation problem
using the DE algorithm. Data from various batch and
fed-batch fermentations were used in this parameter
estimation, which was essentially based on minimization
of the prediction errors in a least squared sense. For
further details on the estimation problem and sensitivity
of the resulting parameters, the reader is referred to
reference [24]. For brevity, we present a single plot of the
comparison of the model predictions and experimental
data (for fed-batch fermentation with constant feed
addition of H2O2, sucrose and nitrogen source) in Fig. 1.
The values of the process parameter are given in
Table 1.
In the above problem, the objective is to maximize
the concentration of one product and simultaneously
keep the concentration of the other product at the end of
the fermentation within some threshold. The addition
rates of the three substrates, FC, FN and FH, are considered as the control variables. The search space of this
optimization problem is bounded by several constraints,
viz., end point constraints, constraint on the decision
153
Fig. 1 Protease concentration
profiles from experiment and
simulation
variables, constraint on feed rate and state-path
constraint.
Also, an end point constraint is considered on the final
working volume of the broth to bind the maximum
working volume of the fermenter as
End point constraint
V ðtÞ Vf 0;
ð13Þ
1
Towards obtaining minimal concentration of the substrate and a specific product at the end of the
fed-batch fermentation, end point constraints are imposed on the sucrose, nitrogen source and product
concentration as
C ðtf Þ Cf 0;
ð10Þ
N ðtf Þ Nf 0;
ð11Þ
P ðtf Þ Pf 0:
ð12Þ
where Cf, Nf and Vf are considered as 1.0 g l , 2.0 g l1
and 1.4 l, respectively, and the Pf is the minimum concentration of a specific product.
Constraint on decision variables
In this problem, the sucrose and nitrogen concentrations
in the feed are treated as decision variables. The lower
and upper bounds on the concentrations of sucrose and
nitrogen source in the feed are imposed as
Table 1 The values of the parameters used in the simulations
Parameter
Value of the parameter
Reference/basis
aP
bP
d
EI
KHi
Ki
KC
KN
KH
Kd
K
qP
Vf
YC/X
YN/X
YC/Protease
YC/Catalase
lm
q
41 unit protease (kg cell)1
7.99·104 unit protease (kg cell)1 s1
13.5·106 m
1.0·1013 kmol (kg cell)1
3·104 kmol m3
2·102 kmol m3
1.5·101 kg m3
2.4·101 kg m3
4·1010 m s1
1.0·103 (kmol m3)1 s1
7.5·106 (kmol m3)1 s1
1.28·108 kmol catalase (kg cell)1
1.4·103 m3
0.272
0.015
0.854
1.01·107
2.78·105 s1
1.0·103 kg m3
From parameter estimation by using DE
From parameter estimation by using DE
Average diameter measured in image analyzer (Olympus, Japan)
From parameter estimation by using DE
From parameter estimation by using DE
From parameter estimation by using DE
From parameter estimation by using DE
From parameter estimation by using DE
From parameter estimation by using DE
From parameter estimation by using DE
Bailey and Ollis [32]
From parameter estimation by using DE
Maximum working volume of the reactor
Estimated from batch fermentation
Estimated from batch fermentation
Estimated from batch fermentation
Estimated from batch fermentation
Calculated from experimental data
Based on 80% water in the cell
154
Cfeedmin Cfeed Cfeedmax ;
ð14Þ
Nfeedmin Nfeed Nfeedmax :
ð15Þ
In the above equations, the bounds are chosen to minimize the effect of dilution due to the nutrient additions.
Traditionally, in optimal control problems considered so
far in the fermentation literature, the initial conditions
on the state are assumed to be fixed or given. In the
proposed work, we consider the initial volume and the
feed concentrations to be decision variables. The lower
and upper bounds on the working volume need to be
included in the problem formulation and can be written
as
ðVmin Þ V ðVf Þ:
ð16Þ
Constraint on feed rate
Depending on the pump capacity, a lower and an upper
bound are imposed on the feed rate as
Fmin F ðFmax Þ:
ð17Þ
State-path constraint
Additionally, the variable path is also bounded by a
state-path constraint on the level of H2O2 in the broth, as
H
\0:005 mol g1 of cell.
X
ð18Þ
The value for H/X was chosen based on the change in
intracellular reduction state with extracellular H2O2
concentration as indicated by the NADH level [25].
Max
subject to
Multi-objective problem formulation
An optimization problem with more than one objective
function that has to be minimized or maximized is
known as a multi-objective optimization [3]. For catalase and protease production by A. niger fermentation,
we consider two objective functions, i.e. to maximize one
of the product expressions, while seeking to minimize or
keep within thresholds, the final concentration of the
other product. Thus, the optimization problem can be
generally posed as
Minimize/maximize
subject to
Pm ðxÞ
gj ðxÞ 0;
hk ðxÞ ¼ 0;
xLi xi xU
i ;
lower and upper limits on the decision variables. The
general problem deals with J inequality and K equality
constraints.
Generally, multi-objective optimization deals with
conflicting types of objective functions and if both the
objective functions are equally important, it is not possible to satisfy completely the two objective functions
simultaneously. Therefore, multi-objective optimization
generates a number of solutions as trade-offs between
the two objective functions, which are also called as
pareto-optimal solutions. This set of pareto-optimal
solutions can be generated by a number of methods. In
the simplest formulation, also called as the e-constraint
method, the multi-objective optimization is run as a
single objective problem with the other objectives being
posed as constraints. This single objective optimization
is run repeatedly for different values of the constraint
thresholds to generate the pareto-optimal set. In a relatively more complex formulation, Kasat et al. [1] used
the non-dominated genetic search algorithm to generate
the pareto-optimal set. This algorithm essentially reformulates the traditional genetic algorithm in a non-elitist
framework to generate the pareto-optimal set in a single
simulation. In our work, we prefer the use of the econstraint method to better understand the nature of the
trade-offs, and therefore formulate the multi-objective
optimization as a single-objective one with constraints.
At any step of the transformed problem, we accord
higher priority to one of the objectives (say the maximization of catalase) and accordingly the other objective
(say the minimization of protease) is posed as an end
point constraint on the final levels of protease. This
constraint is then accommodated in the objective function using the penalty function method. The resultant
problem can then be run for different values of the end
point to generate the trade-off (pareto-optimal) curve.
Therefore, the optimization problem in our work has
been formulated as
m ¼ 1; 2;
j ¼ 1; 2; . . . ; J ;
;
k ¼ 1; 2; . . . ; K;
i ¼ 1; 2; . . . ; n;
ð19Þ
where M represents the two objective functions. x is a
vector of n decision variables and xLi and xU
i are the
P1 ðtf Þ
; and other constraints
P2 ðtf Þ Pf 0
ð20Þ
where P1 and P2 are, respectively, the catalase and
protease concentrations at final time and Pf is the
threshold value which can be varied to understand the
trade-offs involved in the optimization.
The optimization problem is sought to be minimized
in the space of several decision variables. At first, the
total fermentation time, tf, is divided into small periods
of time and the three individual nutrient rates are
parameterized over the fermentation time, tf. It is assumed that each individual nutrient addition rate is to be
approximated by a piecewise constant value. Therefore,
the optimization problem is converted into a piecewise
optimal control problem. The other decision variables
are the initial batch volume, the feed concentration of
two individual nutrients and the fermentation time. If
Nd discretizations in the time duration [0,tf] are made,
155
each individual nutrient addition rate would give rise to
Nd decision variables, corresponding to its value in each
discrete time interval. Hence, the total number of decision variables for the above problem can be evaluated as
Nd·3 variables arising from each of sucrose, nitrogen
source and H2O2 additions, another two variables for
the feed concentration of the individual nutrients and
two other variables relating to the initial batch volume
and fermentation time.
In this work, we propose to use the DE algorithm [10]
to solve the multi-objective optimization problem in the
space of decision variables and constraints. The DE
method is a parallel direct search method that utilizes a
large number of populations of vectors, NP, for each
generation, W. Each vector carries the decision parameters of x dimensions. The value of NP is chosen so as to
reflect and obtain a diverse population.
The basic structure of DE consists of four main
operations, viz., initialization of the population, mutation, cross over and evaluation.
Initialization
The initialization involves the random selection of the
population, Zi (i=1, ..., NP). The mathematical representation of the initialization can be written as
Zi ¼ Zmin þ qi ðZmax Zmin Þ;
ð21Þ
where qi denotes a random number generated from a
uniform distribution. Zmax and Zmin are the upper and
lower limits, respectively, on the initialization value.
by a binomial distribution to perform crossover operation to generate an offspring. In the crossover operation,
the jth gene of the ith individual at the next generation is
produced from the perturbed individual ZiW ¼ ½Z1 ; . . . ;
W 1
ZNd W and current individual Z W 1 ¼ ½Z1 ; . . . ; ZNd as
i
i
Zji ¼ ZjiW 1
ZjiW
if a random number > CR
;
otherwise
ð24Þ
where j=1, ..., Nd, i=1, ..., NP and the crossover factor
CR2(0,1) is a user specified real number.
Evaluation
In DE, the evaluation function of an offspring competes
one to one with that of its parent (target individual in the
current generation) and the parent is replaced by its offspring in the next generation only if the objective function
value of the offspring is lower than that of its parent.
The DE algorithm was generally formulated for
unconstrained minimization problems. Therefore for
constrained problems as above, dynamic/adaptive penalty functions [27] are used to convert the constrained
problem to an unconstrained problem. In these methods, the penalty terms associated with constraints are
added to the objective function, i.e. penalty term reflects
the violation of the constraints and assigns higher costs
of the penalty function to the individuals that are far
from the feasible region. Therefore, the objective functional can be modified as
J ½uðtÞ ¼ P ðtf Þ þ
l
X
e2l ;
ð25Þ
l¼1
Mutation
Mutation involves the generation of the difference vector, Djk, between any two individuals of the NP vectors.
The weighted difference vector, Djk, is added to a third
randomly selected individual or best performing individual to have a perturbed individual, ZiW [10]. Among
the five types of mutation strategies proposed by Storn
[26], we use the following strategy for simplicity:
Djk ¼ ZjW 1 ZkW 1
ð22Þ
ðW 1Þ
ðW 1Þ
ZiW ¼ ZpW 1 þ f Zj
Zk
;
i
where e is the penalty function associated with violation
of a constraint.
Hence, any candidate individual that violates the
constraints would inherit a worse fitness value and will
find it difficult to survive in the next generation. The
steps of mutation, crossover and evaluation are repeated
for a number of generations (iterations) till a population
of individuals that maximize the above objective function is obtained.
Materials and methods
ð23Þ
where f2(0, 1.2) is a scaling factor introduced by Storn
and Price [10] to ensure convergence.
Crossover
Crossover is an interesting step to increase the diversity
of the new individuals in the next generation. The
resulting perturbed individual, ZiW ; from Eq. 23 and a
target individual in the current generation are selected
Culture, growth medium and cultivations
Aspergillus niger (A. niger—618, ATCC—10594, CMI—
15955, NCIM, Pune, India) was grown on potato
dextrose agar medium for sporulation. All the slants
were maintained at 4 C. The growth medium for the
seed culture consisted of 30 g l1 sucrose, 5 g l1 soluble
starch [28], 0.4 g l1 (NH4)2HPO4, 0.2 g l1 KH2PO4,
0.2 g l1 MgSO4Æ7H2O and 10 g l1 peptone in tap
water [29]. The medium was inoculated with a spore
suspension (transmittance adjusted to 10%) at 10% (v/
156
v) and the cells were grown in a shaker incubator at
30 C. The growth medium used for the bioreactor
consisted of 6.0 g l1 initial sucrose concentration,
0.2 g l1 KH2PO4, 0.2 g l1 MgSO4Æ7H2O and 1.5 g l1
initial nitrogen concentration.
The feed concentrations of H2O2 and sucrose were
0.3 M and 300 g l1, respectively, for both the catalase
and protease maximization runs. The feed concentration
of nitrogen source was 300 g l1 for protease maximization run and 268 g l1 for catalase maximization run.
(NH4)2HPO and peptone were used as the nitrogen
sources in the ratio of 1:14 [(NH4)2HPO4nitrogen:Peptonenitrogen]. The experiments were carried out in a 2 l
bioreactor with a constant pH of 3.8, at a constant
temperature of 30 C and at a constant rpm of 300. The
sucrose, nitrogen source and H2O2 were added in fedbatch mode and the feed rates of additions were regulated according to the control profiles derived by DE.
Experimental setup
A 2 l bioreactor with 1.4 l working volume (Vaspan
Industries, India) was used. The stirred tank bioreactor
was equipped with two Rushton turbine impellers with
temperature and RPM controls. A polarographic DO
probe (Bela Instruments, India) was used to measure the
dissolved oxygen level (DO) in the broth and data was
acquired with a data acquisition system (S.C.R. Elektroniks, India). The pH was controlled online by using a
pH controller (Bela Instruments, Mumbai, India, Model
672 P). The H2O2, sucrose and the nitrogen were added
intermittently through peristaltic pumps (Cole Parmar,
USA).
Analysis
Samples were taken at regular intervals, filtered and
dried to obtain dry cell concentrations. The filtrates
obtained were stored at 20 C for further analysis and
enzyme assays were performed immediately after the
completion of the cultivation. Standard methods were
used for enzyme assay: the azocasein degradation
method for protease [30] and the decomposition rate (of
H2O2) method for catalase [31].
Results and discussion
Simulation results
The DE-based optimization was run for over 10,000
iterations. The optimizer was run twice for the two
objective functions, viz., (i) catalase maximization with
threshold on final protease concentration and (ii) protease maximization with threshold on final catalase
concentration.
In all the simulations, the fed-batch fermentation was
assumed to start with an initial condition on cell mass
(0.75 g l1), nitrogen source (1.5 g l1), sucrose
(6.0 g l1), product (0.0 g l1) and H2O2 concentration
both in the medium and in the cell (0.0 mol l1). Although the fermentation time can be accommodated in
the formulation as a control variable, it was set as a
parameter to simplify the optimization task. The overall
bioreactor run was assumed to be of 30 h duration
(based on a priori knowledge) and the total run time was
divided into 15 intervals of 2 h duration each, i.e. the
value of Nd was assumed to be 15. Therefore, as discussed in an earlier section, each individual nutrient
addition gave rise to 15 decision variables. Thus, in this
problem a total number of 49 decision variables were
used consisting of 45 decision variables generated from
each of the sucrose, nitrogen source and H2O2 additions,
two variables for sucrose and nitrogen concentrations in
the feed and two variables for initial batch volume and
fermentation time. A population of 100 vectors of
decision variables in one generation was used to solve
the optimization problem, i.e. NP=100. Lower and upper bounds were imposed on the decision and control
variables as
Cfeedmin ð200 g l1 Þ Cfeed Cfeedmax ð300 g l1 Þ;
Nfeedmin ð200 g l1 Þ Nfeed Nfeedmax ð300 g l1 Þ:
A concentrated feed condition was considered to reduce the dilution effect due to fed-batch addition and to
meet the nutrient requirement.
Vmin ð700 mlÞ V Vf ð1; 400 mlÞ;
Fmin ð0:0 cm3 s1 Þ F Fmax ð1:5 102 cm3 s1 Þ:
The initial value of the decision and control variables
were determined by using Eq. 21. A crossover factor
(CR) of 0.5 was used and the program was run for 10,000
generations. In the following subsections, the results of
the overall optimization as well as the influence of each
of the individual constraints/decision variables on the
overall productivity are discussed.
Optimum run for catalase maximization
The first multi-objective optimization problem was run
for the maximization of catalase enzyme and simultaneously for the minimization of the protease enzyme.
The results of the optimization are shown in Fig. 2a–e.
The optimizer predicted that the fed-batch must start
from an optimum initial batch volume of 1,070 ml and
optimum nitrogen and sucrose concentrations in the
feed of 268 and 300 g l1, respectively. Figure 2a shows
the optimal feed rate profiles for the additions of the
nitrogen source, sucrose and H2O2.
From Fig. 2a, it is clear that the nitrogen source feed
addition occurs almost in one initial pulse, whereas
sucrose feed addition occurs over the entire duration of
time with the higher initial feed addition followed by a
157
Fig. 2 a Optimal feed rate profiles for sucrose, nitrogen and H2O2
in catalase maximization run. b Sucrose and nitrogen profiles for
catalase maximization. c Hydrogen peroxide concentration in the
medium (H) and inside the cell (Hi) for catalase maximization. d
Cell mass, catalase and protease profiles for catalase maximization.
e Volume profile for catalase maximization
decreasing profile. For H2O2, the additions occur over the
entire time duration mostly with an increasing profile. The
optimal sucrose feed additions are able to maintain the
sucrose concentration profile in an increasing nature almost up to the half of the fermentation duration to
accelerate the growth-associated catalase enzyme, followed by a decreasing trend in sucrose concentration to
maintain the end point constraint on the final residual
sucrose concentration, as shown in Fig. 2b. The increasing nature of H2O2 feed addition supports the fact that
enhanced H2O2 concentration favours the catalase
production. Again there is H2O2 inhibition on growth,
which the optimizer has to consider and therefore bring
down the H2O2 concentration in the broth so as to enhance the cell growth. Hence, there is a trade-off between
the inhibition and promoting nature of H2O2 on the
growth and on the catalase enzyme. Also, this optimal run
is for multi-objective optimization where the simultaneous minimization of protease enzyme lowers the final
cell concentration (enhances the inhibition term) as well as
the protease production to satisfy the end point constraint
on final protease concentration. Hence, a trade-off solu-
158
Optimum run for protease maximization
tion is obtained for the final cell concentration to satisfy
both the objectives. The maximum nutrient addition, to
enhance the product concentration, is limited by the final
batch volume. As is evident from the results, these profiles
are therefore predicted by the optimizer to balance the
three effects, viz., those resulting from constraints on final
batch volume, the end point residual protease and nutrient concentration as well as the nutrient (sucrose, nitrogen
and H2O2) requirements for growth and product
formation. A final cell and catalase concentrations of
7.23 g l1 and 0.45 lM, respectively, were obtained at
30 h. A final protease concentration in the catalase maximization run of 4.99 U l1 was obtained. The individual
profiles of the variables for this optimized case are shown
in Fig. 2b–e.
The second optimization problem was solved to maximize the protease enzyme and to minimize the catalase
enzyme. The results of the optimization are shown in
Fig. 3a–e. In this second case, the optimizer predicted
that the fed-batch must start from an optimum initial
batch volume of 1,250 ml and optimum nitrogen source
and sucrose concentrations in the feed of 300 g l1 each.
Figure 3a shows the optimal feed rate profiles for the
additions of the nitrogen source, sucrose and H2O2.
From Fig. 3a, it is clear that the nitrogen source feed
additions are almost the same as the catalase run: in one
initial pulse, whereas the sucrose feed additions are different from that of the catalase maximization run: an
Fig. 3 a Optimal feed rate profiles for sucrose, nitrogen source and
H2O2 in protease maximization run. b Sucrose and nitrogen profiles
for protease maximization. c Hydrogen peroxide concentration in
the medium (H) and inside the cell (Hi) for protease maximization.
d Cell mass, catalase and protease profiles for protease maximization. e Volume profile for protease maximization
159
increasing rate of sucrose feed addition up to half of the
fermentation followed by a decreasing feed addition
profile to maintain the end point constraint on residual
sucrose. The optimizer predicts a sucrose addition rate
that increases the sucrose concentration at a slow rate up
to 6 h of fermentation followed by a sharp increase in
the sucrose concentration up to 17 h of fermentation to
favour the expression of the protease enzyme. Then, the
optimizer maintains the sucrose concentration at its
maximum level to maximize the secondary metabolite,
protease, followed by a decrease in sucrose concentration to meet the end point constraint as seen in Fig. 3b.
For H2O2, the additions occur in small pulses up to half
of the time period, with a comparatively large addition
at the end. Since it is a multi-objective optimization with
the minimization of the final catalase concentration as
one of the objectives, the optimizer seeks to maintain a
low H2O2 concentration in the broth in order to minimize the catalase concentration. As discussed earlier,
protease expression rate does not get substantially affected with the H2O2 addition except at low H2O2 concentration when the inhibition effect on cell growth is
lowered, and hence the final cell concentration is increased. The higher final cell concentration enhanced the
protease production. A final cell concentration of
11.27 g l1 and a final protease of 7.50 U l1 at 30 h
were obtained. The final catalase concentration in this
protease maximization run was predicted to be
0.316·103 lM. The individual profiles of the variables
for this optimized case are shown in Fig. 3b–e.
Analysis of the optimum runs
The final concentration of the state variables obtained
from the two optimized runs is shown in Table 2.
The optimal feed addition profiles for sucrose and
H2O2 obtained for both catalase and protease maximization runs are entirely different in nature as discussed
earlier. The optimizer generally predicts higher final cell
mass concentration in the case of protease maximization, keeping in view the non-growth-associated nature
of the expression. Although protease is produced in both
growth and stationary phases, its non-growth nature
favours expression during the stationary phase, thus
requiring higher cell concentration towards the end of
the fermentation, to maximize the final protease concentration. In contrast, the catalase, being the growthassociated product, is produced during growth and
requires a suitable cell concentration to maximize the
catalase concentration. The optimum initial batch
volume is lower in case of catalase than protease. In
catalase run, there is a higher dilution due to the
continuous addition of H2O2, when compared with the
protease run where the pulse addition of H2O2 is observed. Therefore, it can be concluded that the optimizer
recognizes this requirement and predicts lower initial
batch volume for catalase to allow higher dilution in the
presence of constraints on the maximum batch volume.
In the case of protease run, the prediction of higher
batch volume does not violate the maximum limit on
final batch volume.
Analysis of other runs
Some other simulations have also been carried out for
both catalase and protease maximization runs, to
understand the nature of the trade-offs. Generally, catalase production is favoured by high growth rate which
essentially translates to keep extracellular peroxide levels
low and the nitrogen and intracellular peroxide levels
high (see Eq. 5). The optimizer would therefore seek
dilution conditions and feed concentrations which will
achieve levels of peroxide and nitrogen as a compromise
between these requirements. On the other hand, protease
production is favoured by a large cell concentration,
which translates to higher growth rates and lower dilution factors (see Eq. 6). In the light of the above relationships, the results obtained for various optimization
runs are discussed and presented below in a tabular form.
Different runs
1. Catalase max. without any end point constraint.
2. Catalase max. with end protease higher than the
optimal threshold value.
3. Catalase max. with end protease lower than the
optimal threshold value.
4. Catalase max. with end Hi higher than the optimal
threshold value.
5. Protease max. without end point constraint.
6. Protease max. with end catalase higher than the
optimal threshold value.
7. Protease max. with end catalase lower than the
optimal threshold value.
8. Protease max. with end Hi lower than the optimal
threshold value.
The final concentration of the state variables obtained
from the above runs is shown in Table 3.
Effect of end protease concentration in catalase
maximization
In catalase maximization, Run 2, the higher final concentration of protease than its threshold optimal value
Table 2 Final concentration obtained from optimal run
Run
H (M)
Hi (M)
X (g l1)
Catalase (lM)
Protease (U l1)
N (g l1)
C (g l1)
V (ml)
Catalase maximum
Protease maximum
1.77·103
1.12·106
2.56·104
1.48·107
7.23
11.27
0.45
0.316·103
4.99
7.50
1.99
1.99
0.94
0.87
1,070
1,250
160
obtained from the optimized catalase maximization run
decreases the final catalase concentration from its optimal value. This is because the enhancement of one
product in a multi-product fermentation simultaneously
decreases the concentration of the other product. Similarly in the same catalase maximization, Run 3, the decrease in protease concentration from its threshold
optimal value should increase the final optimal catalase
concentration. But, Run 3 shows a lower value of final
catalase than its optimal value. The reason is the end
point constraint to lower the final protease concentration from its optimal threshold value directed the optimizer towards the low cell mass production. In other
words, this also means lower growth rates. Since catalase
is a growth-associated product, its expression decreased
from its optimal value due to this low cell concentration.
Effect of end catalase concentration in protease
maximization
H2O2 concentration (H) indirectly and has an adverse
effect on the cell concentration as well as the catalase
indirectly. This is because of the inhibition effect of H on
the catalase levels (see Eq. 5).
Effect of end H2O2 concentration in protease
maximization
In the case of protease maximization, Run 8, the further
decrease in the intracellular H2O2 concentration, Hi,
practically has no effect on the final protease concentration. Still, there is a trend to decrease the final protease. The decrease in Hi concentration indirectly
decreases the final cell concentration through a decrease
in final catalase concentration, and hence a decrease in
the final protease concentration is observed.
End point constraints on the final residual nutrient
concentrations
In case of protease maximization, Run 7, the specification of a lower final catalase concentration from its
optimal threshold value obtained from the optimum
protease maximization run directed the optimizer towards a decrease in the cell concentration. As an effect of
this decrease, the final protease concentration also decreased from its optimal value, though the change in the
final concentration is almost negligible. On the other
hand, a specification of the end point constraint to be
higher than the final optimal catalase concentration
(Run 6) decreased the protease concentration from its
optimal value. Therefore, we can say that the enhancement in one product reduced the production of the other
product to maintain the mass balance in the competitive
substrate utilization system.
Effect of end H2O2 concentration in catalase
maximization
It is observed from Run 4 that the further enhancement
in the final intracellular H2O2 concentration, Hi, does
not increase the catalase concentration further. The increase in Hi concentration also increases the extracellular
One of the other interesting aspects of this work is the
reduction of the final residual nutrient concentrations by
including endpoint constraints in the problem formulation; this has been proposed here to ease separation
during downstream processing. The optimization problems without imposing any end point constraints on the
final residual nutrient concentration, Run 1 and Run 5,
yield a high residual nutrient concentration at the end of
the fermentation that could interfere with the downstream processing. It shows high residual nitrogen concentrations of 11.34 g l1 (Run 1) and 11.38 g l1 (Run
5) at the end of the fermentation when there is no end
point constraint imposed on residual nitrogen. On the
other hand, in the case of the optimum (Table 2), a very
low residual nitrogen concentration of 1.99 g l1 is obtained for both the runs. In Run 1 and Run 5 where
there is no end point constraint, the final concentration
of the product is about 13% higher in the case of catalase and 10% higher in the case of protease than those
with end point constraints. This indicates a trade-off
between high final concentrations of the product and the
residual nutrient levels. Relaxation of the end point
constraints could yield higher final product concentra-
Table 3 Final concentration obtained from other simulations
Run
H (M)
Hi (M)
X (g l1)
Catalase (lM)
Protease
(Ul1)
N (g l1)
C (g l1)
V (ml)
Optimal catalase maximum
1
2
3
4
Optimal protease maximum
5
6
7
8
1.77·103
2.22·103
1.08·103
5.88·103
3.0·103
1.12·106
3.58·107
7.52·107
6.91·107
5.25·107
2.56·104
3.09·104
1.6·104
8.0·104
4.13·104
1.48·107
4.78·108
1.01·107
8.8·108
6.62·108
7.23
8.4
8.8
4.28
7.2
11.27
12.56
11.17
11.18
11.22
0.45
0.51
0.422
0.337
0.45
0.316·103
0.171·103
0.411·103
0.288·103
0.14·103
4.99
5.7
6.0
3.0
4.96
7.50
8.25
7.43
7.44
7.47
1.99
11.34
2.0
1.99
1.99
1.99
11.38
1.99
2.0
2.0
0.94
1.7
0.96
0.9
1.0
0.87
1.13
0.76
0.87
0.99
1,070
1,000
1,170
897
1,060
1,250
1,080
1,266
1,266
1,267
161
tions but at the expense of higher residual nutrient levels
which translate to higher processing/downstream costs.
State-path constraints on nutrient additions
As stated in Eq. 18, a state-path constraint is used in this
optimization problem to bind the toxic level of H2O2
concentration on the cell growth. The results obtained
are presented in Figs. 4 and 5. It shows that the H/X
value was maintained below the threshold value of
0.005 mol g1 of cell over the entire duration of fermentation.
Experimental evaluation of the simulation results
specific growth rate and the catalase yield coefficient
were reduced by approximately 10% in the simulations,
there was a good agreement between the experimental
and simulated values.
Furthermore, when compared with conventional
methods of constant feed additions of sucrose, nitrogen
and hydrogen peroxide, experimental results demonstrated a 70% improvement in the final catalase levels,
when the additional profiles suggested based on the
simulation models were implemented. We would like to
submit that while better models and parameter estimates
can be generated to further improve the quality of predictions, even the existing models do provide enough
validation to demonstrate the efficacy of the methods
proposed.
Catalase maximization
Protease maximization
The final product and the cell concentration profiles
obtained from the simulation result have been validated
here through experiments. In the results presented in the
sequel, all the experiments have been done in duplicate
with an average error of 6–8%. For the biomass, the
experimental results match the simulation predictions
very closely. Figure 6 shows the cell concentration
profile obtained from the experiments and simulation results. A final cell concentration of 7.35 g l1 obtained from the experiments compares well with
7.23 g l1 from the simulation. Also, the catalase profile
obtained from the experiment (Fig. 7) is comparable
with that from the simulation.
The final product (catalase) predicted from the
simulations was approximately 0.45 lM whereas that
obtained from the experiment was 0.265 lM. The primary reason for this departure from the predicted values
is that the variance errors associated with the estimation
of the parameters are high; these can be improved by
using data from a larger number of experiments during
the parameter estimation step. In fact, when both the
The final product and the cell concentration profiles
obtained from the simulation result have been validated
here through experiments. The experimental results obtained for protease maximization (Figs. 8 and 9) show
the most deviations from the profiles predicted via
simulations. The reasons for this can be summarized as
follows: the optimal nutrient addition policies predicted
based on the simulation model, when implemented
during experiments, resulted in a batch-like growth due
to the pulse additions of peroxide and the resulting
limited oxygen environment for the cells. This took the
experiment into batch-like operating regions; however,
the parameters were estimated using experimental data
that had nutrient addition policies typically found in fedbatch fermentation. The mismatch between these
parameter estimates for the fed-batch case and the true
parameters for batch-like growth was large enough to
make the experimental profiles for protease deviate from
those predicted from the simulations. As seen in Fig. 9,
the simulation predicts an exponential rate of protease
expression, whereas the experiments actually realize a
Fig. 4 Profile of the constraint,
H/X for catalase maximization
162
Fig. 5 Profile of the constraint,
H/X for protease maximization
Fig. 6 Cell concentration
profile from experimental result
for catalase maximization
Fig. 7 Catalase concentration
profile from experimental result
for catalase maximization
163
Fig. 8 Cell concentration
profile from experimental result
for protease maximization
batch-like protease production in the stationary phase.
A final cell concentration of approximately 3.1 g l1 and
protease concentration of 3.7 U l1 are obtained from
the experiment.
Similar to the catalase maximization case, the protease production in the current fermentation employing
the three-feed multi-objective optimization was 31%
higher than the fed-batch cultivation employing constant feed addition of H2O2, sucrose and nitrogen
source. These experimental results demonstrate the
effectiveness of the proposed optimization scheme.
The experimental results for both the protease and
catalase maximization cases presented show a significant
deviation from the values predicted via simulations. This
can be attributed to the large variance errors in the
parameter estimates resulting from using data from
fewer experiments. Also, for the protease case, the
optimal nutrient addition policies took the operation of
the process into a batch-like region, whereas the
Fig. 9 Protease concentration
profile from experimental result
for protease maximization
parameters were estimated from fed-batch experiments.
Nevertheless, the experimental results for both the catalase and protease maximization cases did show the
efficacy of the proposed optimization scheme in terms of
higher enzyme yields when compared with the case of
constant nutrient additions. To fully realize the benefits
of such optimization schemes, the models must be
accurate; this can be achieved via a method that is based
on design of experiments and evolutionary learning used
to regularly update the parameters in the fed-batch
model using data from new, evolving batches. This is an
aspect of future research.
Conclusions
This work proposed the formulation of the multiobjective optimal control problem with three-feed
nutrient addition for fed-batch fermentations that
164
produce more than one product. The formulation
proposed the maximization of one product with
simultaneous minimization of the other product. It
included an end point constraint to pose the requirement of end product concentration more realistically.
Additionally, it also proposed an inclusion of decision
variables such as initial batch volume, substrate feed
concentrations and end point constraints such as the
minimization of the end residual nutrients, final batch
volume and also state-path constraint on permissible
oxygen levels. A robust stochastic optimization algorithm based on DE was found to solve the resulting
optimization problem. The results obtained from the
simulations were verified experimentally and showed an
increase of about 70% final catalase and 31% final
protease compared to conventional fed-batch
cultivation.
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