Microsoft Word - Zhang et al.pdf


African Journal of Biotechnology Vol. 10(32), pp. 6162-6171, 4 July, 2011     
Available online at http://www.academicjournals.org/AJB 
DOI: 10.5897/AJB10.2086 
ISSN 1684–5315 © 2011 Academic Journals  

 
 
 
 

Full Length Research Paper 
 

Comparison of ν-support vector regression and logistic 
equation for descriptive modeling of Lactobacillus 

plantarum growth 
 

Lijing Zhang
1,3

, Zhijian Song
2,3

, Xiaodong Pan
3
, Mingguang Feng

1
 and Zhihua Jin

3
* 

 
1
Institute of Microbiology, Zhejiang University, Hangzhou, 310058, China. 

2
Institute of Biomedical Informatics/Zhejiang Provincial Key Laboratory of Medical Genetics, Wenzhou Medical College, 

Wenzhou 325000, China. 
3
Ningbo Institute of Technology, Zhejiang University, Ningbo 315104, China. 

 
Accepted 15 April, 2011 

 

Due to the complexity and high non-linearity of bioprocess, most simple mathematical models fail to 
describe the exact behavior of biochemistry systems. As a novel type of learning method, support 
vector regression (SVR) owns the powerful capability to characterize problems via small sample, non-
linearity, high dimension and local minima. In this paper, we developed a ν-SVR model with genetic 
algorithms (GA) in the pre-estimate in Lactobacillus plantarum fermentation by comparing the 
predicting capability of logistic model and SVR model. 5-fold cross validation technique was applied in 
the SVR train to avoid over-fitting. The information of SVR parameters were obtained in the generation 
of 150 and the optimal parameters were C= 235.8935, σ= 8.3608, ν=0.7587. Correspondingly, the logistic 

model parameters maxµ  and  maxx  were estimated as 0.4791 and 0.3498, respectively. The 
experimental results demonstrated that, SVR model excelled the logistic model based on the 
normalized mean square error (NMSE), mean absolute percentage error (MAPE) and the Pearson 
correlation coefficient R. We found that the ν-SVR model optimized by genetic algorithms could be a 
potential monitoring method for prediction of biomass. 
 
Key words: Support vector regression, genetic algorithm, logistic model, prediction of biomass. 

 
 

INTRODUCTION 
 
Because microorganism fermentation process is complex 
and non-linear, many parameters such as the 
concentration of bacteria, substrates and products are 
difficult to measure on-line (Wang et al., 2006). In order 
to optimize bioprocess and put the advanced algorithms 
of control into practice, it is necessary to monitor and 
diagnose the bioprocess parameters.  

Kinetic models are generally experimentally derived 
mathematical formulas that can agree well with the 
cultivation data (Bessaou and Siarry, 2001). Kinetic 
models can be linear or nonlinear,  such  as  ordinary  (or  
 

 
 
*
Corresponding author. E-mail: zlj@nit.zju.edu.cn. Tel: +86 574 

88229516. Fax: +86 574 88229516.  

partial) differential equations, cellular automata (Xiao et 
al., 2005; Xiao et al., 2006) and logistic model. The 
logistic model firstly, proposed by Verhulst in the 
eighteenth century(Liu et al., 2002), has been 
demonstrated to be the most illustrative model of 
microbial growth dynamics in a habitat of finite resources 
(Fujikawa et al., 2002; Peleg et al., 2007; Schepers et al., 
2000). Logistic equation is based on the notion that the 
momentary growth rate of a given population inoculated 
into a virgin environment is proportional to the momentary 
population’s size.  

Recently, a statistical learning theory based on 
formalism known as the support vector machines (SVMs) 
emerged as a novel powerful tool for data classification 
and regression, which are called support vector 
classification (SVC) and support vector regression (SVR). 



 
 
 
 
SVMs could solve the problems resulted from small 
samples, nonlinearity, high dimension and local minimum 
(Vapnik, 1995; Borin et al., 2006; Cai et al., 2003; Moreira 
et al., 2007). Therefore, SVR is gradually accepted in the 
data-driven nonlinear modeling applications (Desai et al., 
2006). 

Unfortunately, for the generalization performance of 
SVR which depends on SVR parameters, how to set 
these parameters by a given set of data is the main issue 
for the application of SVR. Cherkassky (2004) described 
ε-support vector regression (ε-SVR) for setting meta-
parameters. However, as the parameter ε influences the 
solution indirectly and lacks intuitionistic means, it is 
difficult to give appropriate value for ε. To overcome this 
problem, a new parameter ν was introduced to control the 
fitness and predication accuracy instead of parameter C 
in SVR (Schepers et al., 2000). Chalimourda et al. (2004) 
studied optimal ν in SVR for different noise models using 
the data from sine function and Boston housing problem. 
Recently, the performance of SVR in bioprocess was also 
researched (Li et al., 2006a; Li and Yuan, 2006b). 
However, ν-SVR coupled with genetic algorithm has not 
been investigated in fermentation process.  

In this study, ν-SVR was applied to determine the 
concentration of cell via the data from fermentation 
process. We aimed to find a suitable method for the 
simulation of microbial growth by comparing different 
modeling strategies in process simulation and 
fermentation control. The organism used in the model 
was Lactobacillus plantarum.  

 
 
MODEL STRUCTURE 

 
Logistic model 

 
As unstructured kinetic model, logistic model is the most 
frequently employed for modeling microbial systems, 
because it is simple and good enough for technical 
purposes ((Kiviharju et al., 2006; Mu et al., 2006). The 
logistic model can describe the kinetics of the microbial 
growth as follows: 
 

m

1
max

ax

dx x
x

dt x
µ

 
= − 

 
              (1) 

 
Where, x (g l

−1
) is the cell concentration, t (h) is the 

fermentation time, maxµ (h
-1

) is the maximum specific 

growth rate and m axx  (g l
−1

) is the maximum cell 

concentration. Using x  = 
0

x  (t = 0), integration of 

equation 1 gives the following equation for microbial 
biomass: 

Zhang et al.         6163 
 
 
 

m

0 m

m 0 0

axm

ax

ax

ax

t

t

x x e
x

x x x e

µ

µ
=

− +          (2) 

 
 
Support vector regression 
 
Support vector regression 
 
The ν-support vector regression (ν-SVR) is a promising 
method for solving non-linear regression problem, which 
depends on the theory of support vector machines 
(SVMs). In this section, a brief introduction of SVR is 
given.  
 
Given a set of data points: 

l

i i i=1
T={ (x , y )}

N
R R∈ × , where 

m

i
x R∈  is the 

input vector, iy R∈ is the desired value and l is the 
total number of data points. SVR estimation aims to seek 
for a function using the following formula: 
  

( )
T

i
y w x b= ⋅ Φ +                        (3) 

 

Where, w  is the weight, b is the coefficient, and ( )ixΦ
denotes the data in high dimensional feature space 
mapped by applying nonlinear kernel function from the 

input space
i

x . 

The coefficients (both w and b) could be optimized only 
by minimizing the regularized risk function: 
        

2

1

1 1
( )

2

l

i ik
i

w C
l

ζ ζ ∗

=

+ +∑        (4) 
 

Where, 
21

2 k
w is called the regularized term. 

1

1
( )

l

i i

il
ζ ζ ∗

=

+∑  is the empirical risk measured by the 
ε − insensitive loss function. The slack variable

i
ζ , is 

called the upper training error and iζ
∗

represents the 

lower training error subject to the " ε -radius-tube” where

(( ( ) )
i i

y x bω ε− ⋅ Φ + ≤ . C is the regularization factor 
which has the ability of determining the trade-off between 
the empirical risk and the regularized term. 

As   a   consequence,   the   ε-SVR   problem   can   be  



6164         Afr. J. Biotechnol. 
 
 
 
described as the following quadratic optimization 
problem: 

 

2

, ,
1

1 1
min ( )

2

l

i ikb
i

w C
lω ζ

ζ ζ ∗

=

 
+ ⋅ + 

 
∑

       (5) 
 

(( ( ) ) ,

(( ( ) ) ,

, 0, 1, 2, 3

i i i

i i i

i i

subject to x b y

y x b

i l

ω ε ζ

ω ε ζ

ζ ζ

∗

∗

⋅ Φ + − ≤ +

− ⋅ Φ + ≤ +

≥ = L

     

  
However, the varied value of the ε could be selected 

from all values in real number and would deeply influence 
the solution of quadratic optimization problem. 
Fortunately, Schölkopf et al. (2000) cited a weight 
parameter (ν) multiplied by ε in the ε-SVR problem to 
control the tube size, which avoids exhausting search of 
the ε parameter in the real number range. After this 
transition, the optimization problem is as follows: 
 

2

, ,
1

1 1
m in ( )

2

l

i ikb
i

w C
lω ζ

υ ε ζ ζ ∗

=

  
+ ⋅ + +  

  
∑       

                                                    (6) 
 

(( ( ) ) ,

(( ( ) ) ,

, 0, 1, 2, 3

i i i

i i i

i i

su b jec t to x b y

y x b

i l

ω ε ζ

ω ε ζ

ζ ζ

∗

∗

⋅ Φ + − ≤ +

− ⋅ Φ + ≤ +

≥ = L

      

   
Usually, the problem stated earlier could be 

successfully solved in its dual form by exploiting the 
Lagrange multipliers approach (Vapnik, 1995; Belousov 
et al., 2004). Through the Karush-Kuhn-Tucker (KKT) 
conditions of solving quadratic programming problem, the 
dual Equation 5 leads to the solution by maximizing the 

,
i i

α α∗ , which are Lagrange multipliers 
. 

*
,

, 1 1

1
max ( )( ) ( , ) ( )

2

l l

i i j j i j i i i

i j i

K x x y
α α

α α α α α α∗ ∗ ∗

= =

− − + −∑ ∑  

                                        (7) 
 

1

1

( ) 0

0 , , 1, 2, 3 ;

( )

l

i i

i

i i

l

i i

i

su bject to

C
i l

l

C

α α

α α

α α υ

∗

=

∗

∗

=

− =

≤ ≤ ∀ =

+ ≤ ⋅

∑

∑

L             

 
 
 
 
They satisfy the condition of 

0, 0, 0
i i i i

xα α α∗ ∗× = ≥ ≥ . Where 1, 2...i l= . 
 
 

Parameters determination of the ν-SVR model 
 
To construct an efficient ν-SVR model, SVR’s parameters 
are crucial and should be set carefully (Fujikawa et al., 
2004). These parameters include: 
  
 

Kernel function: The term ( , )i jk x x in Equation 7 is 

defined as kernel function. Generally speaking, as the 
key in the SVM theory, kernel function is used to map the 

input feature data ix into a higher-dimensional feature 
space via non-linear mapping (Bessaou and Siarry, 
2001). Three typical examples of kernel functions are 
polynomial, radial basis function (RBF) and linear kernel. 
As a kind of RBF kernel functions, Gaussian function 
yielding better prediction performance was used in this 
study (Samanta et al., 2003). 
 
 
Penalty parameter C: As mentioned earlier, C 
determines the trade off between margin maximization 
and error minimization. 

 
 

Bandwidth of the kernel function ( 2σ ): In this paper, 

sigma square ( 2σ ) denotes the variance of the Gaussian 

kernel function. 
 
 
Parameter ν: ν represents an upper bound on the 
fraction of errors and a lower bound on the fraction of 
parameters to build the regression function and the range 
is from 0 to 1. 

In general, the accurate estimation of ν-SVR model is 
based on the hyper-parameters (C, σ , ν), these 
parameters need to be selected in ν-SVR. But now it is 
still a complex problem and hard to obtain the optimized 
parameters value in an effective algorithm. Heuristic 
search techniques are usually used to find the optimal 
SVR parameter settings, such as genetic algorithm (GA) 
or simulation annealing algorithm (SA). Thereinto, GA is 
better for robust optimization in a complex search space 
(Potocnik and Grabec, 1999; Ustun et al., 2005). In this 
study, the coupled-model of GA-SVR was proposed to 
find the optimal values of ν-SVR parameters. 
 
 

GA-SVR optimization procedure 
 
Genetic algorithm (GA), which was formerly introduced 
by Holland (1975), employs the  biological  techniques  of  



 
 
 
 

 
 
Figure 1. GA-ν-SVR model. 

 
 
 
mutation and crossover to search the local optimal 
solutions. Figure 1 illustrates the algorithm of the GA-
SVR model. Our proposed GA- SVR is described as 
follows in detail: 

 
 

Step 1. Population initialization: In this process, the 
three ν-SVR parameters (C, σ , ν) were converted 
decimal code into a binary format. The chromosome X 
was represented as X = p1, p2, p3, where p1, p2 and p3 
denote the regularization parameters C, σ and ν, 
respectively. One population contains n sizes of such 
kind chromosomes and is randomly generated. In this 
study, we used 100 sizes of such chromosomes and 
when the chromosomes were put into SVR to train, the 
binary format was translated into decimal once again. 
 

 
Step 2. Evaluate the fitness function: The fitness of 
training data set is easy to be calculated, but also, it is 
prone to be over-fitting. The 5-fold cross validation 
technique was applied in the SVR train to avoid over-
fitting and to make the result generalized and reasonable 
(Holland, 1975; Browne, 2000). The fitness function of 
the parameter set is measured as the criteria of 
minimizing the  MAPE  (mean absolute percentage error): 

Zhang et al.         6165 
 
 
 

MAPE= 
1 100%

/
×

n

i i ii
a p a

n

=
−∑

 

 

Where, ip denotes the predicted value in SVR model, 

i
a  represents the actual experimental observation and n 
is the number of experimental observations. The smaller 
the values of MAPE in the regression, the better the 
prediction performance would be. 
 
 

Step 3. Genetic operation: The genetic operators 
(Figure 2) which contain selection, crossover and 
mutation are mainly illustrated as follows: 
 

Selection: A spin of a weighted roulette wheel is applied 
to choose chromosomes in the current population with 
the possibility of selection frequency rate. 
 
 

Crossover: Crossover is a typical genetic operation in 
genetic algorithm which allows new population to be 
created in the search space. This mechanism operation 
aims to create a crossover point randomly in a pair of 
parental chromosomes and then, exchange genes 
between these two chromosomes. This situation occurs 
between two chromosomes and the crossover rate could 
be described with the frequency of crossover.  
 
 

Mutation: Mutation process is the same genetic operator 
of crossover which also alters the solution of parental 
chromosome. However, it happened in one chromosome 
and the genes may occasionally be altered. In binary 
code, one mutation means genes changing code from 0 
to 1 at one random point. 

After these operators, if the minimum fitness value of 
the new population is smaller than that of the old 
population, the offspring chromosome could replace the 
old chromosome and create a new population. 
 
 

Step 4. Termination criteria: The process was repeated 
from the genetic operation and fitness function evaluation 
until the number of generations was up to termination. 
After that, the best chromosomes would be presented as 
a solution. Of course, the information of parameters in the 
chromosomes could be acquired and the best MAPE 
could be represented. In this study, we defined 150 
generation. 
 
 
MATERIALS AND METHODS 

 
Strain 

 
L. plantarum C263, originally isolated from diary food were  



6166         Afr. J. Biotechnol. 
 
 
 

 
 
Figure 2. Genetic operation. 

 
 
 

rendered by China Center of Industrial Culture Collection (CCICC) 
and cultured in MRS broth (Difco). Culture stocks were inoculated 
into MRS medium at 5% and incubated for 24 h at 37°C. This 
culture was propagated twice under the same conditions. 

 
 
Fermentation experiment 

 
The optimized fermentation medium consisted of (g l

-1
): glucose, 2; 

peptone, 1; beef extract, 1; yeast extract, 1; Na3C6H5O7 ·2H2O, 0.1; 
(NH4)2HPO4, 1; K2SO4, 0.2. The seed culture was inoculated at 5% 
into 250 ml Erlenmeyer flasks containing 150 ml MRS medium 
cultured at 37°C. For batch cultures in a bioreactor, 150 ml seed 
culture was added to a KF-5l fermentor (KBT, Korea) with 3 L 
production medium at 37°C anaerobically. The samples were 
fetched at intervals and the pH was recorded.  
The microbial growth was measured by recording the optical 
density (OD) at 660 nm. The glucose concentration in the 
fermentation broth was determined by the DNS method (Zhang et 
al., 1997).  

 
 
Data scaling 

 
Because original data may disturb each other during the training 
process, which could mislead the predicting result, data re-scaling 
plays an important role to improve the predicting accuracy. Another 
advantage for data scaling is to avoid numerical difficulties during 
the calculation. All the data sets are scaled within the range of 0 
and 1 in this formation shown as follows: 

 

min
,

max min

i

N i

x x
x

x x

−
=

−  

 

Where, i
x  is the feature vector which influences the L. plantarum 

growth in the batch fermentation, ,N i
x  is the normalized value of 

the ith  process variable after the process of scaling, maxx  is the 

maximized value of the feature vector i
x , minx  is the minimized 

value of the feature vector ix  and N is the number of experimental 
observations. 

  
 
RESULTS AND DISCUSSION 
 
Variables selection 
 
Variables selection is necessary for the v-SVR model 
with a good prediction. In this paper, we selected the 
most important features to describe the fermentation 
process, which included temperature, pH value, glucose 
concentration and fermentation time. We set the four 
vectors as the input variables and the cell concentration 
as output variables. 
 
 
Performance criteria 
 
In the statistical prediction, the following three cross-
validation methods are often used to examine a 
prediction model for its effectiveness in practical 
application; independent dataset test, subsampling (K-
fold cross-validation) test and jackknife test. However, 
among the three cross-validation methods, the jackknife 
test is deemed the most objective which can always yield 
a unique result for a given benchmark dataset and thus, 
has been increasingly used and widely recognized by 
investigators to examine the performance quality of 
various predictors (Chou, 2009; Lin et al., 2009; Wu et 
al., 2010; Xiao et al., 2008a; Xiao et al., 2008b; Xiao et 
al., 2009a; Xiao et al., 2009b; Xiao et al., 2009c; Xiao et 
al., 2010). In this work, we used 5-fold cross-validation to 
examine the performance quality of the two models and 
chose some statistical metrics such as normalized root 
mean square error (NRMSE), MAPE and R. Table 1 
shows these performance metrics and their calculations. 
NRMSE, MAPE were used to measure the deviation 
between the measured and predicted values. The smaller



 Zhang et al.         6167 
 
 
 

Table 1. Performance metrics and their calculations. 
 

Metric Calculation 

MAPE MAPE= 
1

100%
/

×

n

i i ii
a p a

n

=
−∑

 

NRMSE 
2

1

2

1

( )
NRMSE=

n

i ii

n

ii

a p

a

=

=

−∑
∑

 

R 
1

2 2

1 1

( )( )

( ) ( )

n

i ii

n n

i ii i

a a p a
R

a a p a

− −

=

− −

= =

− −
=

− −

∑

∑ ∑

 

 
 
 
the values of NRMSE and MAPE were the closer the 
predicted values were to the measured values. The 
Pearson correlation coefficient R was adopted to 
measure the correlation of the experimental and 
predicted values. 

 
 
Growth models obtained by v-SVR  

 
The GA was used for searching the optimal parameter 
sets when the MAPE was at its minimum. The searching 
process of optimal parameters was operated with 150 
generations in total. The genetic process was recorded. 
Figure 3 shows that the tendency of the optimal and 
average fitness of population varied during genetic 
process. We found that when the generation of 
population rose to 60, the optimal fitness value (MAPE) 
became steady. The information of the best parameters 
were acquired in the generation of 150 and the 
corresponding optimal ones were C = 235.8935, σ = 
8.3608 and ν = 0.7587. The mean error applied to 
describe the average MAPE in the total chromosome 
population was 0.068969. 

After the GA method was applied to search for the 
optimal parameter of ν-SVR, the predicting model for the 
batch fermentation with L. plantarum was constructed. 
The graphical comparison of the measured and predicted 
value modeling by GA-ν-SVR is shown in Figure 4. 

 
 
Growth models obtained by logistic model 

 
The logistic model is an approximation of the microbial 
growth curve for the batch experiments in this study. 
Figure 5 showed the relationship between the cell 
concentration and cultivation time, including both 
experimental data and predicted values obtained based 
on the logistic model. Correspondingly, the logistic model 

parameters maxµ and maxx were estimated as 0.4791 
and 0.3498. 

Performance comparison of SVR and logistic model 
 
As shown in Table 2, the values of MAPE and NRMSE 
generated from GA-ν-SVR model were smaller than 
those from logistic model, which indicated smaller 
deviations between the measured and predicted values. 
Moreover, the Pearson correlation coefficient R was 
higher.  

To illustrate the better performance of the SVR model, 
we compared the measured value with the predicted one. 
The plots between the measured value and the predicted 
value are shown in Figures 6 and 7. The predicting 
simulation was performed with the testing data. From 
Figures 6 and 7, the proposed GA-ν-SVR model qualified 
this particular data set very well (Pearson correlation 
coefficient was 0.9966), while two particular points lacked 
accuracy in the logistic model (Pearson correlation 
coefficient was 0.9828). Therefore, the performance of 
GA-ν-SVR model was better than that of the logistic 
equation. 

 
 
Conclusions  
 
Fermentation process is very complex and it is very 
difficult to obtain a complete picture of what is actually 
going on in a particular fermentation. In order to estimate 
the biomass online, the models presented in this work 
can be use to predict fermentation process of microbes.  
In this paper, a relatively easy strategy was given based 
on the investigation of ν-SVR parameters, which 
determined the value of ν beforehand and then, selected 
other two parameters(C and σ). From the results of 
MAPE, NRMSE and Pearson correlation coefficient, we 
concluded that, the approximation ability and 
generalization of v-SVR model were better than the 
logistic model for the on-line monitoring of biomass. SVR 
models could be considered as intracellular metabolic 
pathways, which are described by four variables 
(temperature, pH value, glucose concentration and 
fermentation time), while the logistic model has only one 
variable  to  describe  the  biomass.  In  conclusion, GA v- 



6168         Afr. J. Biotechnol. 
 
 
 

 
 
Figure 3. The fitness alternation during the evolutionary process. 

 
 
 

Table 2. Comparison of the predicted results from GA-ν-SVR and logistic models. 
 

Model R MAPE NRMSE 

GA-ν-SVR 0.9966 0.0185 0.0209 

Logistic 0.9828 0.0375 0.0465 
 
 
 

 
 
Figure 4. Measured and estimated cell concentration by ν-SVR (○-measured value 

▲-predicted value). 

0 10 20 30 40 50

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

T
he

 O
D

66
0 v

al
ue

 o
f L

. p
la
nt
ar
um

Fermentation time (h)



Zhang et al.         6169 
 
 
 

 
 
Figure 5. Measured and estimated cell concentration by logistic model (○-measured value ▲-
predicted value). 

 
 
 

 
 
Figure 6. The plots of measured cell concentration vs. predicted value by the ν-SVR model. 

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

SV
R

 p
re

di
ct

ed
 v

al
ue

 (
O

D
66

0)

The measured value (OD
660

)



6170         Afr. J. Biotechnol. 
 
 
 

 
 
Figure 7. The plots of the measured cell concentration vs. predicted value by logistic 
model. 

 
 
 

SVR has better predicting capacity on simulating bacteria 
biomass than logistic model and is a promising method 
for estimating the microbial growth.  
 
 
Acknowledgements 
 
This work was supported by the National Natural Science 
Foundation of China (Grant No. 20976161 and Natural 
Science Foundation of Ningbo City, China (Grant No. 
2009A610149). 
 
 
REFERENCES 
 
Belousov AI, Verzakov SA, Frese JV (2004). A flexible classification 

approach with optimal generalisation performance: support vector 
machines. Chemom. Intell. Lab. Syst. 64: 15–25.  

Bessaou M, Siarry P (2001). A genetic algorithm with real-value coding 
to optimize multimodal continuous functions. Struct. Multidisc. Optim. 
23: 63–74. 

Borin A, Ferrao MF, Mello C, Maretto DA, Poppi RJ (2006). Least-
squares support vector machines and near infrared spectroscopy for 
quantification of common adulterants in powdered milk. Anal. Chim. 
Acta. 579: 25–32. 

Browne MW  (2000). Cross-validation methods. J. Math. Psychol. 44: 
108-132. 

Cai CZ, W ang W L, Sun LZ, Chen YZ (2003). Protein function 
classification via support vector machine approach. Math. Biosci. 
185: 111-122. 

Chalimourda A, Schölkopf B, Smola AJ (2004). Experimentally optimal ν 
in support vector regression for different noise models and parameter 
settings. Neural Netw. 17: 127-141.  

Cherkassky V, Ma YQ (2004).  Practical  selection  of  SVM  parameters 

and noise estimation for SVM regression. Neural Netw. 17: 113-126. 
Cho KC (2009). Pseudo amino acid composition and its applications in 

bioinformatics, proteomics and system biology. Current Proteomics 6, 
262-274. 

Desai K, Badhe Y, Tambe SS, Kulkarni BD (2006). Soft-sensor 
development for fed-batch bioreactors using support vector 
regression. Biochem. Eng. J. 27: 225–239. 

Fujikawa H, Kai A, Morozumi SA (2004). New logistic model for 
Escherichia coli growth at constant and dynamic temperatures. Food 

Microbiol. 21: 501–509. 
Holland JH (1975). Adaptation in Natural and Artificial systems, 

University of Michigan Press. 
Kiviharju K, Salonen K, Leisola M, Eerikainen T (2006). Modeling and 

simulation of Streptomyces peucetius var. caesius N47 cultivation 
and ε-rhodomycinone production with kinetic equations and neural 
networks. J. Biotechnol. 126: 365–373 

Li YF, W ang ZF, Yuan JQ (2006a). On-line fault detection using SVM-
based dynamic MPLS for batch process. Chinese J. Chem. Eng. 14: 
754-758. 

Li YF, Yuan JQ (2006b). Prediction of key state variables using support 
vector machines in bioprocess. Chem. Eng. Technol. 29: 313-319.  

Lin, W .Z., Xiao, X., and Chou, K.C., 2009. GPCR-GIA: a web-server for 
identifying G-protein coupled receptors and their families with grey 
incidence analysis. Protein Eng Des Sel 22, 699-705. 

Liu JZ, W eng LP, Zhang QL, Xu H, Ji LN (2002). A mathematical model 
for gluconic acid fermentation by Aspergillus niger. Biochem. Eng. J. 
14: 137-141. 

Moreira GA, Micheloud GA, Beccaria AJ, Goicoechea HC (2007). 
Optimization of the Bacillus thuringiensis var. kurstaki HD-1 δ-
endotoxins production by using experimental mixture design and 
artificial neural networks. Biochem. Eng. J. 35: 48–55. 

Mu Y, W ang G, Yu HQ (2006). Kinetic modeling of batch hydrogen 
production process by mixed anaerobic cultures. Bioresource. 
Technol. 97: 1302–1307. 

Peleg M, Corradini MG, Normand MD (2007). The logistic (Verhulst) 
model for sigmoid microbial growth curves revisited. Food Res. Int. 
40: 808–818. 



 
 
 
 

Potocnik P, Grabec I (1999). Empirical modeling of antibiotic 
fermentation process using neural networks and genetic algorithms. 
Math. Comput. Simul. 49: 363-379. 

Samanta B, Al-Balushi KR, Al-Araimi SA (2003). Artificial neural 
networks and support vector machines with genetic algorithm for 
bearing fault detection. Eng. Appl. Artif. Intell. 16: 657–665. 

Schepers AW , Thibault J, Lacroix C (2000). Comparison of simple 
neural networks and nonlinear regression models for descriptive  

modeling of Lactobacillus helveticus growth in pH-controlled batch 

cultures. Enzyme. Microb. Tech. 26: 431-445. 
Schölkopf B, Smola AJ, W illiamson PC, Bartlett PL (2000). New 

Support Vector Algorithms. Neural Comput. 12: 1207-1245. 
Ustun B, Melssen W J, Oudenhuijzen M, Buydens LMC (2005). 

Determination of optimal support vector regression parameters by 
genetic algorithms and simplex optimization. Anal. Chim. Acta. 544: 
292–305. 

Vapnik V (1995). The Nature of Statistical Learning Theory, Springer, 
New York. 

W ang JL, Yu T, Jin CY (2006). On-line estimation of biomass in 
fermentation process using support vector machine. Chinese J. 
Chem. Eng. 14: 383-388. 

Wu ZC, Xiao X, Chou KC (2010). 2D-MH: A web-server for generating 
graphic representation of protein sequences based on the 
physicochemical properties of their constituent amino acids. J. 
Theoret. Bio. 267(1): 29-34. 

Xiao X, Lin W  Z (2009c). Application of protein grey incidence degree 
measure to predict protein quaternary structural types. Amino Acids. 

37(4) ：741-749. 
Xiao X, Lin W Z, Chou, KC (2008a). Using grey dynamic modeling and 

pseudo amino acid composition to predict protein structural classes. 
J. Comput. Chem. 29, 2018-2024. 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 

Zhang et al.         6171 
 
 
 
Xiao X, W ang P, Chou KC ( 2008b). Predicting protein structural 

classes with pseudo amino acid composition: an approach using 
geometric moments of cellular automaton image. J. Theor. Biol. 254, 
691-696. 

Xiao X, W ang P, Chou KC (2009a). GPCR-CA: A cellular automaton 
image approach for predicting G-protein-coupled receptor functional 
classes. J. Comput. Chem. 30, 1414-1423. 

Xiao X, W ang P, Chou, KC (2010). Quat-2L: a web-server for predicting 
protein quaternary structural attributes. Molecular Diversity, DOI 
10.1007/s11030-010-9227-8. 

Xiao X, W ang P, Chou KC (2009b). Predicting the quaternary structure 
attribute of a protein by hybridizing functional domain composition 
and pseudo amino acid composition. J. Applied Crystallography, 42: 
169-173. 

Zhang LX, Zhang TF, Li LY (1997). Biochemical experimental method 
and technology. Higher Education Press, Beijing.