Chapter 16

Fractal Related Methods for Predicting Protein Structure Classes and Functions

ZU-GUO YU, VO ANH, JIAN-YI YANG, and SHAO-MING ZHU

16.1 Introduction

The molecular function of a protein can be inferred from the protein's structure information [1]. It is well known that an amino acid sequence partially determines the eventual three-dimensional structure of a protein, and the similarity at the protein sequence level implies similarity of function [2, 3]. The prediction of protein structure and function from amino acid sequences is one of the most important and challenge problems in molecular biology.

Protein secondary structure, which is a summary of the general conformation and hydrogen bonding pattern of the amino acid backbone [4, 5], provides some knowledge to further simplify the complicated 3D structure prediction problem. Hence an intermediate but useful step is to predict the protein secondary structure. Since the 1970s, many methods have been developed for predicting protein secondary structure (see the more recent references cited in Ref. [6]).

Four main classes of protein structures, based on the types and arrangement of their secondary structural elements, were recognized [7]: (1) the helices, (2) the strands, and (3) those with a mixture of and shapes denoted as and . This structural classification has been accepted and widely used in protein structure and function prediction. As a consequence, it has become an important problem and can help build protein database and predict protein function. In fact, Hou et al. [1] (see also a short report published in Science [8]) constructed a map of the protein structure space using the pairwise structural similarity of 1898 protein chains. They found that the space has a defining feature showing these four classes clustered together as four elongated arms emerging from a common center. A review by Chou [9] described the development of some existing methods for prediction of protein structural classes. Proteins in the same family usually have similar functions. Therefore family identification is an important problem in the study of proteins.

Because similarity in the protein sequence normally implies similarity of function and structure, the similarities of protein sequences can therefore be used to detect the biological function and interaction of proteins [10]. It is necessary to enrich the concept and context of similarity, because some proteins with low sequence identity may have similar tertiary structure and function [11].

Fractal geometry provides a mathematical formalism for describing complex spatial and dynamical structures [12, 13]. Fractal methods are known to be useful for detection of similarity. Traditional multifractal analysis is a useful way to characterize the spatial heterogeneity of both theoretical and experimental fractal patterns [14]. More recently it has been applied successfully in many different fields, including time series analysis and financial modeling [15]. Some applications of fractal methods to DNA sequences are provided in References [16–18] and references cited therein.

Wavelet analysis, recurrence quantification analysis (RQA), and empirical mode decomposition (EMD) are related to fractal methods in nonlinear sciences. Wavelet analysis is a useful tool in many applications such as noise reduction, image processing, information compression, synthesis of signals, and the study of biological data. Wavelets are mathematical functions that decompose data into different frequency components and then describe each component with a resolution matched to its scale [19, 20]. The recurrence plot (RP) is a purely graphical tool originally proposed by Eckmann et al. [21] to detect patterns of recurrence in the data. RQA is a relatively new nonlinear technique introduced by Zbilut and Webber [22, 23] that can quantify the information supplied by RP. The traditional EMD for data, which is a highly adaptive scheme serving as a complement to Fourier and wavelet transforms, was originally proposed by Huang et al. [24]. In EMD, a complicated dataset is decomposed into a finite, often small, number of components called intrinsic mode functions (IMFs). Lin et al. [25] presented a new approach to EMD. This method has been used successfully in many applications in analyzing a diverse range of datasets in biological and medical sciences, geophysics, astronomy, engineering, and other fields [26, 27].

Fractal methods have been used to study proteins. Their applications include fractal analysis of the proton exchange kinetics [28], chaos game representation of protein structures [29], sequences based on the detailed HP model [30, 31], fractal dimension of protein mass [32], fractal properties of protein chains [33], fractal dimensions of protein secondary structures [34], multifractal analysis of the solvent accessibility of proteins [35], and the measure representation of protein sequences [36]. The wavelet approach has been used for prediction of secondary structures of proteins [37–45]. Chen et al. [41] predicted the secondary structure of a protein by the continuous wavelet transform (CWT) and Chou–Fasman method. Marsolo et al. [42] combined the pairwise distance with wavelet decomposition to generate a set of coefficients and capture some features of proteins. Qiu et al. [43] used the continuous wavelet transform to extract the position of the helices and short peptides with specialscales. Rezaei et al. [44] applied wavelet analysis to membrane proteins. Pando et al. [45] used the discrete wavelet transform to detect protein secondary structures. Webber et al. [46] defined two independent variables to elucidate protein secondary structures based on the RQA of coordinates of -carbon atoms. The variables can describe the percentage of carbons that are composed of an helix and a sheet, respectively. The ability of RQA to deal with protein sequences was reviewed by Giuliani et al. [47]. The IMFs obtained by the EMD method were used to discover similarities of protein sequences, and the results showed that IMFs may reflect the functional identities of proteins [48, 49].

More recently, the authors used the fractal methods and related wavelet analysis, RQA and EMD methods to study the prediction of protein structure classes and functions [50–54]. In this chapter, we review the methods and results in these studies.

16.2 Methods

16.2.1 Measure Representation Based on the Detailed HP Model and Six-Letter Model

The detailed HP model was proposed by Yu et al. [30]. In this model, 20 different kinds of amino acids are divided into four classes: nonpolar, negative polar, uncharged polar, and positive polar. The nonpolar class consists of the eight residues ALA, ILE, LEU, MET, PHE, PRO, TRP, and VAL; the negative polar class consists of the two residues ASP and GLU; the uncharged polar class is made up of the seven residues ASN, CYS, GLN, GLY, SER, THR, and TYR; and the remaining three residues ARG, HIS, and LYS designate the positive polar class.

For a given protein sequence with length , where is one of the 20 kinds of amino acids for we define

16.1

This results in a sequence , where is a letter of the alphabet . The mapping (16.1) is called the detailed HP model [30].

According to Chou and Fasman [55], the 20 different kinds of amino acids are divided into six classes: strong , former (); former ();weak , former (); indifferent (); breaker (); and strong breaker (). The class consists of the three residues Met, Val, and IlE; the class consists of the seven residues Cys, Tyr, Phe, Gln, Leu, Thr, and Trp; the class consists of the residue Ala; the class consists of the three residues Arg, Gly, and Asp; the class is made up of the five residues Lys, Ser, His, Asn, and Pro; and the remaining residue Glu constitutes the class.

For a given protein sequence with length , where is one of the 20 kinds of amino acids for , we define

16.2

This results in a sequence , where is a letter of the alphabet . The mapping (16.2) is called the six-letter model [51].

Here we call any string made up of letters from the set (for the detailed HP model) or (for the six-letter model) a -string. For a given , there are in total or different strings. In order to count the number of strings in a sequence from a protein sequence , we need or counters. We divide the interval into or disjoint subintervals, and use each subinterval to represent a counter. For which is a substring with length , we define

16.3

For , , which is a substring with length , we define

16.4

We then use the subinterval to represent substring . Let be the number of times that a substring with length appears in the sequence (when we count these numbers, we open a reading frame with width and slide the frame one amino acid each time). We define

16.5

to be the frequency of substring . It follows that . We can now define a measure on by , where

16.6

when defined by (16.3). We define a measure on by , where

16.7

when defined by (16.4). We see that and . We call and the measure representation of the protein sequence corresponding to the given based on the detailed HP model and the six-letter model, respectively.

16.2.2 Measures and Time Series Based on the Physicochemical Features of Amino Acids

Measured in kcal/mol, the hydrophobic free energies of the 20 amino acids are , , , , , , , , , , , , , , , , , , , and [39].

The solvent accessibility (SA) values for solvent exposed area are , , , , , , , , , , , , , , , , , , , and [56].

The Schneider–Wrede scale (SWH) values of the 20 kinds of amino acids are , , , , , , , , , , , , , , , , , , , and [47]. Yang et al. [51] added a constant 12.30 to these values to make all the 20 values nonnegative, yielding the revised Schneider–Wrede scale hydrophobicity (RSWH).

Rose et al. [57] proposed different measures of hydrophobicity of proteins. They gave four kinds of values for surface area and hydrophobicity of each amino acid. We use , the stochastic standard state accessibility, that is, the solvent accessible surface area (SASA) of a residue in standard state. The SASA of the 20 kinds of amino acids are , , , , , , ,, , , , , , , , , , , , and .

Each amino acid can also be represented by the value of the volume of sidechains of amino acids [58]. These values are , , , , , , , , , , , , , , , , , , , and .

We convert each amino acid according to hydrophobic free energy, SA, RSWH, SASA, and volume of sidechains along the protein sequence to calculate five different numerical sequences, and view them as time series.

Let , , be the time series with length . First, we define as the frequency of . It follows that . Now we can define a measure on the interval by , where

16.8

We denote the interval by . It is seen that and . We call the measure for the time series.

16.2.3 -Curve Representation of Proteins

The concept of -curve representation of a DNA sequence was first proposed by Zhang and Zhang [59]. We propose a similar concept for proteins [50]. Once we get the sequence for a protein, where is a letter of the alphabet as in Section 16.2.1, we can define the Z-curve representation of this protein as follows. This curve consists of a series of nodes , whose coordinates are denoted by , and . These coordinates are defined as

16.9

where denote the number of occurrences of the symbols 0,1,2,3 in the prefix , respectively, and . The connection of nodes to one another by straight lines is defined as the Z-curve representation of this protein. We then define

16.10

where , , and can only have values 1 and .

16.2.4 Chaos Game Representation of Proteins and Related Time Series

Chaos game representation (CGR) of protein structures was first proposed by Fiser et al. [29]. We denote this CGR by 20-CGR as 20 kinds of letters are used to represent protein sequences. Later Basu et al. [60] and Yu et al. [31] proposed other kinds of CGRs for proteins, in which 12 and 4 kinds of letters were used for protein sequences, respectively. We denote them by 12-CGR and 4-CGR.

16.2.4.1 Reverse Encoding for Amino Acids

It is known that there are several kinds of coded methods for some amino acids. As a result, there should be many possible nucleotide sequences for one given protein sequence. We have used [52], the encoding method proposed by Deschavanne and Tufféry [61], which is listed in Table 1 of our study [52]. Deschavanne and Tufféry [61] explained that the rationale for the choice of this fixed code is to keep a balance in base composition so as to maximize the difference between the amino acid codes.

After one protein sequence is transformed into nucleotide sequences, we can use the CGR of nucleotide sequences [62] to analyze it; the CGR obtained is abbreviated AAD-CGR (amino acids to DNA CGR). The CGR for a nucleotide sequence is defined on the square , where the four vertices correspond to the four letters A, C, G, and T: the first point of the plot is placed half way between the center of the square and the vertex corresponding to the first letter of the nucleotide sequence; the th point of the plot is then placed half way between the th point and the vertex corresponding to the th letter. The plot is then called the CGR of the nucleotide sequence, or the AAD-CGR of the protein sequence.

We can decompose the AAD-CGR plot into two time series [52]. Any point in the AAD-CGR plot is determined by two coordinates: and coordinates. Because the AAD-CGR plot can be uniquely reconstructed from these two time series, all the information stored in the AAD-CGR plot is contained in the time series, and the information in the AAD-CGR plot comes from the primary sequence of proteins. Therefore, any analysis of the two time series is equivalent to an indirect analysis of the protein primary sequence. It is possible that such analysis provides better results than direct analysis of the protein primarysequences.

16.2.5 Time Series Based on 6-Letter Model, 12-Letter Model, and 20-Letter Model

According to the 6-letter model, the protein sequence can be represented by a numerical sequence ; here for each .

We now analogously define a 12-letter model and a 20-letter model. With the idea of chaos game representation based on a 12-sided regular polygon [60], we define the 12-letter model as , , , , , , , , , , , , , , , , , , , and according to the order of vertices on the 12-sided regular polygon to represent these amino acids.

The 20-letter model can be similarly defined as , , , , , , , , , , , , , , , , , , , and according to the dictionary order of the 3-letter representation of each amino acid listed in Brown's treatise [63].

These three models can be used to convert a protein sequence into three different numerical sequences (and also can be viewed as time series).

16.2.6 Iterated Function Systems Model

In order to simulate the measure representation of a protein sequence, we proposed use of the iterated function systems (IFS) model [30]. IFS is the acronym assigned by Barnsley and Demko [64] originally to a system of contractive maps . Let be a compact set in a compact metric space, and

Then is called the attractor of the IFS. The attractor is usually a fractal set and the IFS is a relatively general model to generate many well-known fractal sets such as the Cantor set and the Koch curve. Given a set of probabilities , we pick an and define the iteration sequence

where the indices are chosen randomly and independently from the set with probabilities . Then every orbit is dense in the attractor [64]. For sufficiently large, we can view the orbit as an approximation of . This process is called chaos game.

Let the characteristic function for the Borel subset , then, from the ergodic theorem for IFS [64], the limit

exists. The measure is the invariant measure of the attractor of the IFS. In other words, is the relative visitation frequency of during the chaos game. A histogram approximation of the invariant measure may then be obtained by counting the number of visits made to each pixel.

The coefficients in the contractive maps and the probabilities in the IFS model are the parameters to be estimated for a real measure that we want to simulate. A moment method [30, 65] can be used to perform this task.

From the measure representation of a protein sequence based on the detailed HP model, it is logical to choose and

in the IFS model. For a given measure representation of a protein sequence based on the detailed HP model, we obtain the estimated values of the probabilities by solving an optimization problem [65]. Using the estimated values of the probabilities, we can use the chaos game to generate a histogram approximation of the invariant measure of the IFS that can be compared with the real measure representation of the protein sequence.

16.2.7 Detrended Fluctuation Analysis

The exponent in a detrended fluctuation analysis can be used to characterize the correlation of a time series [17, 18]. We view , , and in the -curve representation of proteins as time series. We denotethis time series by . First, the time series is integrated as , where is the average over the whole time period. Next, the integrated time series is divided into boxes of equal length . In each box of length , a linear regression is fitted to the data by least squares, representing the trend in that box. We denote the coordinate of the straight-line segments by . We then detrend the integrated time series by subtracting the local trend in each box. The root-mean-square fluctuation of this integrated and detrended time series is computed as

16.11

Typically, increases with box size . A linear relationship on a log–log plot indicates the presence of scaling . Under such conditions, the fluctuations can be characterized by the scaling exponent , the slope of the line in the regression against . For uncorrelated data, the integrated time series corresponds to a random walk, and therefore, . A value of indicates the presence of long memory so that, for example, a large value is likely to be followed by large values. In contrast, the range indicates a different type of power-law correlation such that positive and negative values of the time series are more likely to alternate. We consider the exponents for the , , and of the -curve representation of protein sequences as candidates constructing parameter spaces for proteins in this chapter. These exponents are denoted by , , and respectively.

16.2.8 Ordinary Multifractal Analysis

The most common algorithms of multifractal analysis are the so-called fixed-size box counting algorithms [16]. In the one-dimensional case, for a given measure with support , we consider the partition sum

16.12

with , where the sum runs over all different nonempty boxes of a given side in a grid covering of the support , that is, . The exponent is defined by

16.13

and the generalized fractal dimensions of the measure are defined as

16.14

16.15

where . The generalized fractal dimensions are numerically estimated through a linear regression of against for , and similarly through a linear regression of against for . The value is called the information dimension and , the correlation dimension.

The concept of phase transition in multifractal spectra was introduced in studies of logistic maps, Julia sets, and other simple systems. By following the thermodynamic formulation of multifractal measures, Canessa [66] derived an expression for the analogous specific heat as

16.16

He showed that the form of resembles a classical phase transition at a critical point for financial time series.

The singularities of a measure are characterized by the Lipschitz–Hölder exponent which is related to by

16.17

Substitution of Equation (16.13) into Equation (16.17) yields

16.18

Again the exponent can be estimated through a linear regression of

against [35]. The multifractal spectrum versus can be calculated according to the relationship .

16.2.9 Analogous Multifractal Analysis

The analogous multifractal analysis (AMFA) is similar to multiaffinity analysis, and can be briefly sketched as follows [51]. We denote a time series as . First, the time series is integrated as

16.19

16.20

where is the average over the whole time period. Then two quantities and are defined as

16.21

16.22

where denotes the average over ; typically varies from 1 to for which the linear fit is good. From the – and – planes, one can find the following relations:

16.23

16.24

Linear regressions of and against will result in the exponents and , respectively.

16.2.10 Wavelet Spectrum

As the wavelet transform of a function can be considered as an approximation of the function, wavelet algorithms process data at different scales or components. At each scale, many coefficients can be obtained and the wavelet spectrum is calculated on the basis of these coefficients. Hence the wavelet spectrum provides useful information for analyzing data. Given a function , one defines its wavelet transform as [19]

16.25

where is the position and is the scale. The scale in wavelet transform means the th resolution of the data. Taking and , we get the wavelet spectrum as

where .

For simplicity, the scale can be selected as , which are more adjacent and can be used to capture more details of the data. The wavelet spectrum is calculated by summing the squares of the coefficients in each scale . The local wavelet spectrum is defined through the modulus maxima of the coefficients [67] as , where

The maximum of the wavelet spectrum and the maximum of the local wavelet spectrum were applied in the prediction of structural classes and families of proteins [53].

In our work, we chose the Daubechies wavelet, which is commonly used as a signal processing tool. These wavelet functions are compactly supported wavelets with extremal phase and highest number of vanishing moments for a given support width. They are orthogonal and biorthogonal functions. The Daubechies wavelets can improve the frequency domain characteristics of other wavelets [19].

16.2.11 Recurrence Quantification Analysis

The recurrence plot (RP) is a purely graphical tool originally proposed by Eckmann et al. [21] to detect patterns of recurrence in the data. For a time series with length , we can embed it into the space with embedding dimension and a time delay . We write , where . In this way we obtain vectors (points) in the embedding space . We gave some numerical explanations for the selection of and in our earlier paper [52].

From the points, we can calculate the distance matrix (DM), which is a square matrix. The elements of DM are the distances between all possible combinations of points and points. They are computed according to the norming function selected. Generally, the Euclidean norm is used [47]. DM can be rescaled by dividing each element in the DM by a certain value as this allows systems operating on different scales to be statistically compared. For such a value, the maximum distance of the entire matrix DM is the most commonly used (and recommended) rescaling option, which redefines the DM over the unit interval (0.0–100.0%).

Once the rescaled DM = is calculated, it can be transformed into a recurrence matrix (RM) of distance elements within a threshold (namely, radius). RM = and where is the Heaviside function

16.26

RP is simply a visualization of RM by plotting the points on the plane for those elements in RM with values equal to 1. If , we say that points recur with reference to points. For any , since , the RP has always a black main diagonal line. Furthermore, the RP is symmetric with respect to the main diagonal as . is a crucial parameter of RP. If is chosen too small, there may be almost no recurrence points and we will not be able to learn about the recurrence structure of the underlying system. On the other hand, if is chosen too large, almost every point is a neighbor of every other point, which leads to a large number of artifacts [68]. Selection of was discussed numerically in [52].

Recurrence quantification analysis (RQA) is a relatively new nonlinear technique proposed by Zbilut and Webber [22, 23] that can quantify the information supplied by RP. Eight recurrence variables are usually used to quantify RP [68]. It should be pointed out that the recurrence points in the following definitions consist only of those in the upper triangle in RP (excluding the main diagonal line). The first recurrence variable is %recurrence (%REC). %REC is ameasure of the density of recurrence points in the RP. This variable can range from 0% (no recurrent points) to 100% (all points are recurrent). The second recurrence variable is %determinism (%DET). %DET measures the proportion of recurrent points forming diagonal line structures. For this variable, we have to first decide at least how many adjacent recurrent points are needed to define a diagonal line segment. Obviously, the minimum number required (and commonly used) is 2. The third recurrence variable is linemax (), which is simply the length of the longest diagonal line segment in RP. This is a very important recurrence variable because it inversely scales with the largest positive Lyapunov exponent. The fourth recurrence variable is entropy (ENT), which is the Shannon information entropy of the distribution probability of the length of the diagonal lines. The fifth recurrence variable is trend (TND), which quantifies the degree of system stationarity. It is calculated as the slope of the least squares regression of %local recurrence as a function of the displacement from the main diagonal. It should be made clear that the so called %local recurrence is in fact the proportion of recurrent points on certain line parallel to the main diagonal over the length of this line. %recurrence is calculated on the whole upper triangle in RP while %local recurrence is computed on only certain lines in RP, so it is termed as local. Multiplying by 1000 increases the gain of the TND variable. The remaining three variables are defined on the basis of the vertical line structure. The sixth recurrence variable is %laminarity ( %LAM). %LAM is analogous to %DET but is calculated with recurrent points constituting vertical line structures. Similarly, we also select 2 as the minimum number of adjacent recurrent points to form a vertical line segment. The seventh variable, trapping time (TT), is the average length of vertical line structures. The eighth recurrence variable is maximal length of the vertical lines in RP (), which is similar to .

16.2.12 Empirical Mode Decomposition and Similarity of Proteins

Empirical mode decomposition (EMD) was originally designed for non-linear and nonstationary data analysis by Huang et al. [24]. The traditional EMD decomposes a time series into components called intrinsic mode functions (IMFs) to define meaningful frequencies of a signal.

Lin et al. [25] proposed a new algorithm for EMD. Instead of using the envelopes generated by spline, a lowpass filter is used to generate a moving average to replace the mean of the envelopes. The essence of the shifting algorithm remains. Let be a lowpass filter operator, for which represents a moving average of . We now define . In this approach, the lowpass filter is dependent on the data . For a given , we choose a lowpass filter accordingly and set , where is the identity operator. The first IMF in the new EMD is given by , and subsequently the th IMF is obtained first by selecting a lowpass filter according to the data and iterations , where . The process stops when has at most one local maximum or local minimum. Lin et al. [25] suggested using the filter given by . We selected the mask

in our work [53].

Let . The original signal can be expressed as , where the number can be chosen according to a standard deviation. In our work, the number of components in IMFs was set as 4 due to the short length of some amino acid sequences [53].

The similarity value of two proteins at each component (IMF) is obtained as the maximum absolute value of the correlation coefficient. In our work [53], a new cross-correlation coefficient is defined by

16.27

where is the length of the intersection of two signals with lag , is the length of signal , and is the length of signal . The maximum absolute value C of all the correlation coefficients of the components is considered as the similarity value for two proteins.

16.3 Results and conclusions

In our earlier paper [50], we selected the amino acid sequences of 43 large proteins from the RCSB Protein Data Bank (http://www.rcsb.org/pdb/index.html). These 43 proteins belong to four structural classes according to their secondary structures:

1. We converted the amino acid sequences of these proteins into their measure representations based on the detailed HP model with . We found that the IFS model corresponding to is a good model for simulating the measure representation of protein sequences, and the estimated value of the probability from the IFS model contains information useful for the secondary structural classification of proteins [30]. We performed an IFS simulation for the proteins selected and adopted the estimated parameter as one parameter to construct the parameter space for proteins.

2. We converted the amino acid sequences of these proteins to their Z curve representations and performed their detrended fluctuation analysis. The exponents were estimated and used as candidate parameters to construct the parameter space.

3. We computed the generalized fractal dimensions and the related spectra , multifractal spectra of hydrophobic free energy sequences and solvent accessibility sequences of all 43 proteins.

4. For a structural classification of proteins, we considered the following parameters: from the IFS estimations of the measure representations; the exponents from the detrended fluctuation analysis of the Z curve representations; the range of (i.e. the value – in our frame); the maximum value of (denoted ); the value of that corresponds to the maximum value of ; the maximum value of (denoted ), the minimum value of (denoted ) and (defined by ) from the multifracal analysis of the hydrophobic free-energy sequences and solvent accessibility sequences of proteins as candidates for constructing parameter spaces.

In a parameter space, one point represents a protein. We wanted to determine whether the proteins can be separated from four structural classifications in these parameter spaces. We found that we can propose a method which consists of three components to cluster proteins [50]. We used Fisher's linear discriminant algorithm to give a quantitative assessment of our clustering on the selected proteins. The discriminant accuracies are satisfactory. In particular, they reach 94.12% and 88.89% in separating proteins from proteins in a 3D space.

We [51], considered a set of 49 large proteins that included the 43 proteins studied earlier [50]. Given an amino acid sequence of one protein, we first converted it into its measure representation based on the six-letter model with length . Then we calculated , and for the measures of the 49 selected proteins. We then converted the amino acid sequences of proteins into their RSWH sequences according to the revised Schneider–Wrede hydrophobicity scale. We used such sequences to construct the measures . The ordinary multifractal analysis was then performed on these measures. The AMFA was also performed on the RSWH sequences. Then nine parameters from these analyses were selected as candidates for constructing parameter spaces. We proposed another three steps to cluster protein structures [51]. Fisher's linear discriminant algorithm was used to assess our clustering accuracy on the 49 selected large proteins. The discriminant accuracies are satisfactory. In particular, they reach and in separating the proteins from the proteins in a parameter space; and , in separating the proteins from the proteins in another parameter space; and and , in separating the proteins from the proteins in the last parameter space.

We [52], intended to predict protein structural classes (, , , or ) for low-similarity datasets. Two datasets were used widely: 1189 (containing 1092 proteins) and 25PDB (containing 1673 proteins) with sequence similarity values of 40% and 25%, respectively. We proposed decomposing the chaos game representation of proteins into two kinds of time series. Then we applied recurrence quantification analysis to analyze these time series. For a given protein sequence, a total of 16 characteristic parameters can be calculated with RQA, which are treated as feature representations of the protein. On the basis of such feature representation, the structural class for each protein was predicted with Fisher's linear discriminant algorithm. The overall accuracies with step-by-step procedure are 65.8% and 64.2% for 1189 and 25PDB datasets, respectively. With one-against-others procedure used widely, we compared our method with five other existing methods. In particular, the overall accuracies of our method are 6.3% and 4.1% higher for the two datasets, respectively. Furthermore, only 16 parameters were used in our method, which is less than that used by other methods.

Family identification is helpful in predicting protein functions. Since most protein sequences are relatively short, we first randomly linked the protein sequences from the same family or superfamily together to form 120 longer protein sequences [53], and each structural class contains 30 linked protein sequences. Then we used, the 6-letter model, 12-letter model, 20-letter model, the revised Schneider–Wrede scale hydrophobicity, solvent accessibility, and stochastic standard state accessibility values to convert linked protein sequences to numerical sequences. Then we calculated the generalized fractal dimensions , the related spectra , the multifractal spectra , and the curves of the six kinds of numerical sequences of all 120 linked proteins. The curves of , showed that the numerical sequences from linked proteins are multifractal-like and sufficiently smooth. The curves resemble the phase transition at a certain point, while the and curves indicate the multifractal scaling features of proteins. In wavelet analysis, the choice of a wavelet function should be carefully considered. Different wavelet functions represent a given function with different approximation components. We [53], chose the commonly used Daubechies wavelet db2 and computed the maximum of the wavelet spectrum and the maximum of the local wavelet spectrum for the six kinds of numerical sequences of all 120 linked proteins. The parameters from the multifractal and wavelet analyses were used to construct parameter spaces where each linked protein is represented by a point. The four classes of proteins were then distinguished in these parameter spaces. The discriminant accuracies obtained through Fisher's linear discriminant algorithm are satisfactory in separating these classes. We found that the linked proteins from the same family or superfamily tend to group together and can be separated from other linked proteins. The methods are also helpful to identify the family of an unknown protein.

Zhu et al. [54] applied component similarity analysis based on EMD and the new cross-correlation coefficient formula (16.27) to protein pairs. They then considered maximum absolute value of all the correlation coefficients of the components as the similarity value for two proteins. They also created the threshold of correlation [54]. Two signals are considered strongly correlated if the correlation coefficient exceeds and weakly correlated if the coefficient is between and . The results showed that the functional relationships of some proteins may be revealed by component analysis of their IMFs. Compared with those traditional alignment methods, component analysis can be evaluated and described easily. It illustrates that EMD and component analysis can complement traditional sequence similarity approaches that focus on the alignment of amino acids.

From our analyses, we found that the measure representation based on the detailed HP model and six-letter model, time series representation based on physicochemical features of amino acids, -curve representation, the chaos game representation of proteins can provide much information for predicting structure classes and functions of proteins. Fractal methods are useful to analyze protein sequences. Our methods may play a complementary role in the existing methods.

Acknowledgment

This work was supported by the Natural Science Foundation of China (Grant 11071282); the Chinese Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT) (Grant No. IRT1179); the Research Foundation of Education Commission of Hunan Province, China (Grant 11A122); the Lotus Scholars Program of Hunan Province of China; the Aid program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province of China; and the Australian Research Council (Grant DP0559807).

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