O. Roeva, K. Kosev
Institute of Biophysics and Biomedical Engineering – BAS
105 Acad. George Bonchev St., 1113 Sofia, Bulgaria
E-mails: [email protected], [email protected]
FED-BATCH FERMENTATION PROCESS
OF E. COLI MC4110
dX
S
F
 μmax
X X
dt
kS  S
V
dS
1
S
F

μmax
X   Sin  S 
dt
YXS
kS  S
V
dV
F
dt
The values of the model parameters used in simulations are:
max  0.52 h -1, kS  0.023 gl-1, YSX  0.5.
Initial conditions of the process variable are:
X(0) = 1.252 g·l-1; S(0) = 0.812 g·l-1; V(0) = 1.35 l; Sin = 100 g·l-1.
APPLICATION OF GENETIC ALGORITHMS
FOR FEED RATE PROFILES DESIGN
A pseudo code of a GA is presented as:
i=0
set generation number to zero
initpopulation P(0)
initialize a usually random population
of individuals
evaluate P(0)
evaluate fitness of all initial individuals
while (not done) do
test for termination criterion
(time, fitness, etc.)
begin
i=i+1
increase the generation number
select P(i)
from P(i – 1)
select a sub-population for offspring reproduction
recombine P(i)
recombine the genes of selected parents
mutate P(i)
perturb the mated population stochastically
evaluate P(i)
evaluate its new fitness
end
Initial population: The initialization is done randomly. A binary 20 bit encoding is
considered.
Reproduction: The best known selection mechanism, roulette wheel selection, is used
in the proposed GA.
Recombination: Here, double point crossover is employed.
Mutation: In accepted encoding here a bit inversion mutation is used.
The GA operators and parameters are summarized in Tables 1 and 2.
Table 2.
Table 1.
Operator
Type
Parameter
Value
encoding
binary
generation gap
0.97
crossover
double point
crossover rate
0.70
mutation
bit inversion
mutation rate
0.05
selection
roulette wheel
selection
precision of binary
representation
20
number of individuals
100
number of generations
150
fitness
function
linear ranking
Representation of chromosomes:
Representation of chromosomes is a critical part of GA application.
The profile is divided into equal intervals and the feed rate values at the breakpoints
are registered. The sequence of numbers obtained is considered a chromosome and
each gene represented the feed rate after definite time.
Three chromosomes representations are proposed:
1st: division into equal 30 intervals (30 genes);
2nd: division into equal 60 intervals (60 genes);
3rd: division into equal 100 intervals (100 genes).
Every gene is coded in range F = 0 - 0.05 l·h-1.
Evaluation:
After every generated population, the individuals of the population should be
evaluated to be able to distinguish between good and bad individuals.
Here linear ranking is used.
The objective function (JOF) utilized here is presented as:
JOF = f(XActual, XTheory, S) → min
The genetic algorithm syntheses feed rate profile based on minimization of the ration of the
substrate concentration (S) and the difference between actual cell concentration (XActual)
and theoretical maximum cell concentration (XTheory).
Feed Rate Profiles Design
Three problems (30, 60 and 100 genes) are running 50 executions with the GA.
All experiments reported were run on a PC with a Pentium IV 3.2 GHz processor in Matlab
environment.
The genetic algorithm produce the same results with more than 85% coincidence.
Gene
JOF
Xend, g·l-1
FTotal, l
30
0.0308
4.32
0.51
60
0.0295
5.26
1.38
100
0.0295
5.29
1.96
Table 3. Results from the
feed rate design
RESULTS
Cultivation of E. coli MC4110
Cultivation of E. coli MC4110
4.5
4
Cultivation of E. coli MC4110
0.9
0.05
0.8
0.045
0.04
0.7
3.5
0.035
3
2.5
Feed rate, [l/h]
Substrate, [g/l]
Biomass, [g/l]
0.6
0.5
0.4
0.3
0.2
1
6.5
7.5
8
8.5
9
9.5
Time, [h]
10
10.5
11
11.5
a) biomass concentration
0.02
0.01
0.1
7
0.025
0.015
2
1.5
0.03
0.005
0
6.5
7
7.5
8
8.5
9
9.5
Time, [h]
10
10.5
11
b) substrate concentration
11.5
0
6.5
7
7.5
8
8.5
9
9.5
Time, [h]
10
10.5
c) feed rate profile
Fig. 1. Resulting dynamics of biomass and substrate and feed rate profile
in case of 30 genes in chromosome
11
11.5
Cultivation of E. coli MC4110
Cultivation of E. coli MC4110
0.9
0.05
5
0.8
0.045
4.5
0.7
4
0.6
3.5
3
0.4
0.3
2
0.2
1.5
0.1
6
7
8
9
Time, [h]
10
11
12
a) biomass concentration
0.035
0.5
2.5
1
0.04
Feed rate, [l/h]
Substrate, [g/l]
Biomass, [g/l]
Cultivation of E. coli MC4110
5.5
0
0.03
0.025
0.02
0.015
0.01
0.005
6
7
8
9
Time, [h]
10
11
b) substrate concentration
12
0
6
7
8
9
Time, [h]
10
c) feed rate profile
Fig. 2. Resulting dynamics of biomass and substrate and feed rate profile
in case of 60 genes in chromosome
11
12
Cultivation of E. coli MC4110
Cultivation of E. coli MC4110
0.9
0.05
5
0.8
0.045
4.5
0.7
4
0.6
3.5
3
0.4
0.3
2
0.2
1.5
0.1
0
7
7.5
8
8.5
9
9.5
Time, [h]
10
10.5
11
11.5
a) biomass concentration
0.035
0.5
2.5
1
6.5
0.04
Feed rate, [l/h]
Substrate, [g/l]
Biomass, [g/l]
Cultivation of E. coli MC4110
5.5
0.03
0.025
0.02
0.015
0.01
0.005
6
7
8
9
Time, [h]
10
11
b) substrate concentration
12
0
6
7
8
9
Time, [h]
10
c) feed rate profile
Fig. 3. Resulting dynamics of biomass and substrate and feed rate profile
in case of 100 genes in chromosome
11
12
CONCLUSIONS
The proposed genetic algorithm is found to be an effective and efficient method for solving
the optimal feed rate profile problem.
The GA is capable of simultaneously optimizing feed rate profile for a given objective
function.
However, the results seem to indicate that the feed profile formed using chromosome with
60 genes is superior to the rest feeding trajectories. Based on obtained feed rate profile cell
concentration has an ideal increase for the complete fermentation period, achieving final cell
concentration of 5.26 g·l-1 using 1.38 l feeding solution. This is a satisfactory result for the
fermentation system due to the economical effect and process effectiveness.
ACKNOWLEDGEMENT
This work is partially supported by National Science Fund Grants DMU 02/4 “High quality
control of biotechnological processes with application of modified conventional and
metaheuristics methods” and DID-02-29 “Modelling Processes with Fixed Development
Rules”.