J. Biomedical Science and Engineering, 2009, 2, 136-143
Published Online June 2009 in SciRes. http://www.scirp.org/journal/jbise
Prediction of protein folding rates from primary
sequence by fusing multiple sequential features
Hong-Bin Shen1,3,*, Jiang-Ning Song2, Kuo-Chen Chou1,3
1Institute of Image Processing & Pattern Recognition, Shanghai Jiaotong University, 800 Dongchuan Road, Shanghai, 200240, China;
2Bioinformatics Center, Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan; 3Gordon Life Sci-
ence Institute, 13784 Torrey Del Mar Drive, San Diego, California 92130, USA.
*Corresponding author: firstname.lastname@example.org
Received 20 May 2009; revised 23 May 2009; accepted 1 June 2009.
We have developed a web-server for predicting
the folding rate of a protein based on its amino
acid sequence information alone. The web-
server is called Pred-PFR (Predicting Protein
Folding Rate). Pred-PFR is featured by fusing
multiple individual predictors, each of which is
established based on one special feat ure derived
from the protein sequence. The ensemble pre-
dictor thus formed is superior to the individual
ones, as demonstrated by achieving higher
correlation coefficient and lower root mean
square deviation between the predicted and
observed results when examined by the jack-
knife cross-validation on a benchmark dataset
constructed recently. As a user-friendly web-
server, Pred-PFR is freely accessible to the
public at www.csbio.sjtu.edu.cn/bioinf/Folding
Keywords: Protein Folding Rate; Ensemble Predictor;
Fusion Approach; Web-Server; Pred-PFR
Knowledge of protein three-dimensional (3D) structures
plays an indispensable role in molecular biology, cell
biology, biomedicine, and drug design . However,
each protein begins as a polypeptide, translated from a
sequence of mRNA as a linear chain of amino acids. A
protein can function properly only if it is folded into a
correct shape or conformation . Failure to fold into
the intended 3D structure usually produces inactive
proteins with different properties. Although many efforts
have been made trying to understand the mechanism of
protein folding (see, e.g., [3,4,5,6]), it still remains one
of the most challenging problems in molecular biology.
In addition to understanding how a protein chain is
folded, it is also important to find the folding rates of
proteins from their primary sequences. Protein chains
can fold into the functional 3D structures with quite dif-
ferent rates, varying from several microseconds to even
an hour [7,8].
Experimentally determining the three dimensional
structure of a protein is often very difficult and
expensive. However the sequence of that protein is
easily known. Therefore, for quite a long time, scientists
have tried to use the “least free energy principle” [2,9] to
predict the 3D structures of proteins. Unfortunately,
owing to the notorious local energy minimum problem,
so far it can only be successfully used to address very
limited structural characters, such as the handedness
tendency and packing arrangement in proteins (see, e.g.,
[10,11,12]). In the past two decades, various statistical
methods have been developed for predicting the struc-
tural classes of proteins and their folding patterns ac-
cording to the sequence information alone (see, e.g.,
[13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] and a
review ). Encouraged by the results obtained via
these statistical approaches, various methods were de-
veloped for predicting the folding rates of proteins be-
cause the information thus acquired would be very use-
ful for understanding the protein folding mechanism and
the sequence-structure-function relationship [8,30]. In
this regard, the approaches can be generally categorized
into two groups: (1) the prediction of protein folding
rates is based on the protein structure information; and
(2) the prediction is based on the primary sequence in-
For the first group, the features of proteins are ex-
tracted from their 3D structural information and hence
the predictions are feasible only after the structures have
been determined. Most of the methods in this group tried
to derive the statistical significance of the correlation
between the protein folding rate and the corresponding
structural topological parameters, such as contact order
(CO) , absolute contact order (Abs_CO) , total
contact distance (TCD) , long-range order (LRO)
, the fraction of local contact (FLC) , the chain
H. B. Shen et al. / J. Biomedical Science and Engineering 2 (2009) 136-143 137
SciRes Copyright © 2009 JBiSE
topology parameter (CTP)  and the most recent
geometric contact number (Nα) .
For the second group, the features of proteins are
mainly extracted from their primary amino acid sequences,
such as the amino acid biochemical properties  and the
effective folding length (Leff)  derived from the se-
quence-predicted secondary structure. The approaches in
the second group are particularly useful when the 3D
structural information of the protein concerned is not
Although the aforementioned methods in predicting
folding rates of proteins each have their own merits, they
were all established by focusing on one (or a few) spe-
cific feature(s). As is well known, a protein folding sys-
tem is very complicated that involves many physical and
chemical factors. For this kind of complicated biological
system, it would be particularly effective to treat it by
assembling many individual predictors with each oper-
ated based on its own special feature [37,38]. In view of
this, the present study was devoted to develop a novel
ensemble predictor for predicting the folding rate of a
protein chain by incorporating its many different fea-
tures through an optimal fusion process.
2. MATERIALS AND METHODS
To develop a powerful statistical predictor, the first im-
portant thing is to obtain an effective benchmark dataset
. To realize this and also for facilitating comparison
with the existing prediction methods, we use the bench-
mark dataset as described below.
2.1. Benchmark Dataset
The large dataset recently constructed by Ouyang and
Liang  was used in the current study. It contains 80
proteins whose folding rates have been experimentally
determined. Of the 80 proteins, 45 belong to the two-
state folding behaviors without the visible intermediates
while the other 35 belong to the three-state or multi-state
folding kinetics that exhibit the obvious intermediate
state during the folding process under the experimental
conditions. If classified according to their structural
classes,18 are all- proteins, 32 all-, and the remain-
ing 30 are proteins (where means the mix of
and α ). The folding rates of the 80 pro-
teins range fromf tof, spanning
more than eight orders of magnitude of f
6.9 12 .9
. For users’
convenience, the benchmark dataset, denoted as
is given in the Online Supporting Information A, which
can also be downloaded from the web-site at
www.csbio.sjtu.edu.cn/bioinf/FoldingRate/. It is instruc-
tive to point out that f
ench is actually an ap-
parent folding rate constant (see Appendix A). Therefore,
to develop a statistical method for predicting
protein according to its sequence information alone,
there is no need to discriminate whether the protein is
two-state or multi-state folding.
2.2. Sequence Feature Extraction
As mentioned above, although the features extracted
from the 3D structures of proteins are very useful for
predicting their folding rates, they can be used only
when the corresponding PDB codes are available. Owing
to such a limit, in this study we will focus on those fea-
tures that can be derived from the amino acid sequential
information alone, either directly or indirectly.
(a) Amino acid properties. Protein is composed of
different amino acids, which show different physical,
chemical, and conformational properties and hence may
have correlations with the folding rates. In this study, the
following four amino acid properties were used: c, the
propensity to be at the C-terminal of -helix ; S,
the propensity to form β-strand ; , the com-
pressibility ; and SA , the solvent accessible
surface area in an unfolding protein chain . Suppose
a protein P is expressed by
where 1 represents the 1st residue of the protein ,
2 the 2nd residue, and so forth. Thus, the protein’s
scores in the aforementioned four amino acid properties
can be formulated as
where represents the protein length, and
(1,2,3,4; 1,2, , 20)
(1, 2,3, 4i
) respectively represent the
original , , , and SA for the
SA -t hj
native amino acid, and their values can
be obtained from [41,42,43]; 0
ing the maximum one among …, ,
nMi the corresponding minimum one. For
reader’s convenience, the values thus obtained for ,ij
( 1,2,3,4;ij1,2, , 20)
(cf. Eq.3) are given in
(b) Protein size effect. Many studies have indi-
cated that the protein chain length and its fractional
powers (,, or ) or logarithm have a
good correlation with the folding rates, suggesting that
138 H. B. Shen et al. / J. Biomedical Science and Engineering 2 (2009) 136-143
SciRes Copyright © 2009 JBiSE
and its various expressions forms could be useful
features for predicting protein folding rates [8,30]. In the
present study, was adopted.
(c) Information derived from secondary
structure prediction. Given a protein sequence, its
secondary structure can be predicted by means of vari-
ous secondary structure prediction tools. In the present
study, based on the information thus obtained by using
PSIPRED , we have the secondary structure content
ratios for the protein , as formulated by
where , and are the ratios of the -helix,
-sheet, and coiled-coil residues for the protein .
Note that although the secondary structure content con-
tains three components (,
,), they were treated
as one feature because of the normalized condition im-
posed by Eq.4. Moreover, based on the secondary struc-
ture prediction results, the effective protein folding chain
length can be derived, as given by :
where is the total number of amino acids for the
entire protein chain;
L the number of predicted heli-
cal conformation residues;
N the number of predicted
helices; and the number of an
-helix turn ( is
generally ; for a standard -helix, ). In
the current study, was set at 3, and used
as the feature input.
2.3. Prediction Algorithm
According to the above section, we have a set of seven
different kinds of specific features, as can be summa-
rized by the following equation:
To study the folding rate of a protein chain, the key is
, the so-called folding rate constant.
For reader’s convenience, a brief discussion about the
role of f
(or its logarithm f
) on the protein
folding rate is provided in Appendix A. According to
Eq.6, we can construct the following seven linear re-
gression models for predicting the protein folding rate
ab L (7.5)
f66,1α6,2 β6,3 C
ln Kab bb
is the protein folding rate
constant predicted based on the specific feature i
(cf. Eq.6), while i and i are the corresponding pa-
rameters determined by using the regression analysis on
a training dataset such as
ench . For the details of how
to use the regression procedures to determine i and
, refer to . Note that f
of Eq.7 .6 is involved
with more parameters because the 6-th feature 6
contains three sub-features (cf. Eq.6).
All the above seven formulae (Eqs. 7.1–7.7) can be used
to predict the protein folding rates but they each reflect the
effect (s) of only one (or one kind) of specific feature (s).
To incorporate the effects from all the seven kinds of fea-
tures, let us consider the following formulation:
where is the weight that reflects the impact of the
on the protein folding rate. If
the impacts of the seven features were the same, we
should have 1/
. Since they are
actually not the same, it would be rational to introduce
some statistical criterion to reflect their different impacts,
as formulated below.
Given a statistical system consisting of samples,
the Pearson Correlation Coefficient (ACC) is defined by
and are, respectively, the observed and
predicted results for the sample, while
the corresponding mean values for the samples.
Since reflects the correlation of the predicted
results with the actual ones, its value can be used to
H. B. Shen et al. / J. Biomedical Science and Engineering 2 (2009) 136-143 139
SciRes Copyright © 2009 JBiSE
measure the quality of a prediction method. If all the
predicted results are exactly the same as the observed
ones, we have the perfect correlation of . For
different prediction algorithms, Eq.9 will yield different
values of . Therefore, the weight in
be formulated as
PCC( ) (1,2,
where is the Pearson Correlation Coeffi-
cient (Eq.9) obtained with the folding rate pre-
dicting formula in Eq.7 on the benchmark dataset
by the jackknife cross-validation.
The prediction method by fusing the seven individual
methods as formulated by Eq.7 is called the Pred-PFR
(Predictor of Protein Folding Rate).
3. RESULTS AND DICSUSSIONS
In statistical prediction, the following three
cross-validation methods are often used to examine a
predictor for its effectiveness in practical application:
independent dataset test, subsampling test, and jackknife
test . However, as elucidated in  and demon-
strated by Eq.5 of , among the three cross- valida-
tion methods, the jackknife test is deemed the most ob-
jective that can always yield a unique result for a given
benchmark dataset, and hence has been increasingly and
widely used by investigators to examine the accuracy of
various predictors (see, e.g., [46,47,48,49,50,51,52,53,
54]). To demonstrate the quality of Pred-PFR, here let
us also use the jackknife cross-validation on the bench-
(see the Online Supporting Infor-
Now, let us use f
to represent the Pearson
Correlation Coefficient (Eq.9) obtained with Pred-PFR
(Eq.8) on the benchmark dataset
ch by the jack-
knife cross-validation. For facilitate comparison of the
ensemble predictor with the individual predictors, the
values of f)PCC(
and those of
are given in Table 2.
Furthermore, to show the accuracy about the predic-
tion in a more intuitive manner, let us introduce the
RMSD oot Mean Square Deviation) as defined by
, and have the same meanings as
Eq.9. Obviously, the smaller the value of , the
more accurate the prediction. If all the predicted results
are identical to the corresponding observed ones, we
Similar to the case of , let us use
represent the value of obtained with the ensem-
ble predictor Pred-PFR (Eq.8) on the benchmark dataset
by the jackknife cross-validation, and
that by the formula
of Eq.7. All these values are also given in Table
As we can see from the table, the overall value
yielded by the ensemble prediction formula (Eq.8) is 0.88,
which is the closest to 1 in comparison with those by the
individual prediction formulae (Eqs 7.1-7.7). Such an
overall value is even higher than that by the pre-
diction method using the 3D structural information 
on the same benchmark dataset. Moreover, it can be seen
from Ta b le 2 that the overall RMSD value generated by
the ensemble prediction formula is the lowest one in
comparison with those by the seven individual prediction
formulae. The highest correlation and lowest deviation
results indicate that the Pred-PFR ensemble predictor
formed by the fusing approach is indeed more powerful
than the individual predictors.
Pred-PFR is developed for predicting the folding rate of
a protein based on its sequence information alone. It is
an ensemble predictor formed by fusing multiple indi-
vidual predictors with each based on one special feature.
As expected, the ensemble predictor is superior to the
individual predictors. The web-server for Pred-PFR is
freely accessible to the public at www.csbio.sjtu.edu.
This work was supported by the National Natural Science
Foundation of China (Grant no. 60704047), the Science and
Technology Commission of Shanghai Municipality (Grant no.
08ZR1410600, 08JC1410600), and sponsored by Shanghai
APPENDIX A. THE PROTEIN FOLDING
RATE CONSTANT Kf
For a given protein, its folding rate is generally re-
flected by the apparent rate constant f
by the following differential equation
140 H. B. Shen et al. / J. Biomedical Science and Engineering 2 (2009) 136-143
SciRes Copyright © 2009
Table 1. The values of the four amino acid properties that have been normalized according to the Max-Min normalization procedure
of Eq.3. For more explanation about the four amino acid properties, see the relevant text.
Amino acid code c
β τ SASA
Single letter Numerical index
A 1 0.58 0.82 0.34 0.21
C 2 0.20 0.25 0.61 0.56
D 3 0.96 0.23 0.12 0.20
E 4 0.90 0.00 0.00 0.29
F 5 0.34 0.12 0.75 0.84
G 6 0.12 0.70 0.28 0.00
H 7 0.09 0.33 0.37 0.51
I 8 0.16 0.33 0.92 0.79
K 9 0.11 0.29 0.27 0.35
L 10 0.10 0.33 0.69 0.69
M 11 0.18 0.38 0.51 0.83
N 12 0.30 0.40 0.39 0.24
P 13 1.00 1.00 0.13 0.23
Q 14 0.45 0.27 0.54 0.39
R 15 0.00 0.73 0.42 0.58
S 16 0.23 0.48 0.28 0.15
T 17 0.47 0.38 0.61 0.27
V 18 0.13 0.42 1.00 0.57
W 19 0.56 0.45 0.75 1.00
Y 20 0.18 0.08 0.82 0.82
Table 2. The jackknife test results by using different formulae on the benchmark dataset bench (see the Online Supporting Informa-
tion A). aNote that PCC may also have negative value (see Eq.9). However, the correlation strength of the predicted results with the
observed ones is generally measured by its absolute value.
Prediction formula PCC a
(cf. Eq.9) RMSD (cf. Eq.12)
(see Eq.7.1) -0.68 3.16
(see Eq.7.2) 0.27 4.17
(see Eq.7.3) -0.52 3.71
(see Eq.7.4) -0.39 3.99
(see Eq.7.5) 0.79 2.67
(see Eq.7.6) 0.29 4.14
(see Eq.7.7) 0.85 2.23
(see Eq.8) 0.88 2.03
where and represent the concentrations
of its unfolded state and folded state, respectively. Suppose
the total protein concentration is , and initially only the
unfolded protein is present; i.e., and
when . Subse-quently, the protein sys-
tem is subjected to a sudden change in temperature, solvent,
or any other factor that causes the protein to fold. Obvi-
ously, the solution for Eq.A1 is
It can be seen from the above equation that the larger
, the faster the folding rate will be. However, the
actual process is much more complicated than the one as
described by Eq.A1 even if the system concerned con-
sists of only two states. The reason is the folded state
may reverse back to the unfolded state, as described by
the following equation
where is the forward rate constant for con-
verting to folded , and 21 is the corresponding reverse
rate constant. Thus we have the following kinetic equation
21 folded12 unfold
dP( )P() P(
dP( )P() P(
H. B. Shen et al. / J. Biomedical Science and Engineering 2 (2009) 136-143 141
SciRes Copyright © 2009 JBiSE
Eqs. A3 and A4 can be expressed by an intuitive graph
called directed graph or digraph [55,56] as shown in
Fig.1a. To reflect the variation of the concentrations of
unfolded and folded proteins with time, the digraph is
further transformed to the phase digraph as shown in
Fig.1b, where is an interim parameter associated with
the following Laplace transform
where unfold and folded are the phase concentrations of
and , respectively [55,56]. Thus, using the
graphic rule 4 [55,56], also called “Chou’s graphic rule
for non-steady-state enzyme kinetics” , we can imme-
diately obtain the solutions of Eq.A4, as given by
21 012 0
12 012 0
12 2112 21
Accordingly, it follows
kk kkkt t
Comparing Eq.A7 with Eq.A1, we obtain the following
meaning: the apparent folding rate constant f
function of not only the detailed rate constants, but also
is actually not a constant but will
change with time. Only when and k,
Figure 1. (a) The directed graph or digraph [55,56] for the
two-state protein folding mechanism as schematically ex-
pressed in Eq.A3 and formulated in Eq.A4. (b) The phase di-
graph obtained from of panel (a) according to the
graphic rule 4 [55,56], which is also called “Chou’s graphic
rule for non- steady-state enzyme kinetics” in the literature (see,
e.g., ). The symbol in panel (b) is an interim parameter
(see Eq.A5) and the related text for further explanation).
can Eq.A8 be reduced to 12f
k and Eq.A6 to
dP( )P() P(
be treated as a constant.
It can be imagined that for a three-state or multi-state
folding system, f
will be much more complicated.
We can also see from the above derivation that using
graphic analysis to deal with kinetic systems is quite
efficient and intuitive, particularly in dealing compli-
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