Information quantity for secondary structure propensities of protein subsequences in the Protein Data Bank

Elucidating the principles of sequence–structure relationships of proteins is a long-standing issue in biology. The nature of a short segment of a protein is determined by both the subsequence of the segment itself and its environment. For example, a type of subsequence, the so-called chameleon sequences, can form different secondary structures depending on its environments. Chameleon sequences are considered to have a weak tendency to form a specific structure. Although many chameleon sequences have been identified, they are only a small part of all possible subsequences in the proteome. The strength of the tendency to take a specific structure for each subsequence has not been fully quantified. In this study, we comprehensively analyzed subsequences consisting of four to nine amino acid residues, or N-gram (4≤N≤9), observed in non-redundant sequences in the Protein Data Bank (PDB). Tendencies to form a specific structure in terms of the secondary structure and accessible surface area are quantified as information quantities for each N-gram. Although the majority of observed subsequences have low information quantity due to lack of samples in the current PDB, thousands of N-grams with strong tendencies, including known structural motifs, were found. In addition, machine learning partially predicted the tendency of unknown N-grams, and thus, this technique helps to extract knowledge from the limited number of samples in the PDB.

For all the polypeptide chains in the dataset, they were decomposed into N-gram segments by sliding the window of Nsuccessive residues along the sequence. The N value ranged from four to nine. The secondary structure and rASA of each N-gram segment were assigned based on the properties of the residue at the middle of the segment; when N is an even number, the (N/2)-th residue was chosen as a representative. We assessed the following properties of each segment: (i) the secondary structure, (ii) whether highly exposed to solvent or not based on the threshold of rASA ≥ 0.8, and (iii) whether exposed to solvent or not based on the threshold of rASA ≥ 0.2. For property (i), the secondary structure was defined by the DSSP software [29]. The eight classes of secondary structures produced by DSSP were reduced into three classes: helix ('H', 'G', and 'I' in the DSSP output), beta ('E' and 'B'), and coil (other symbols). For properties (ii) and (iii), the rASA was defined as the ASA value of the residue divided by the reference ASA value, which is defined by the ASA of the center residue in the Gly-X-Gly tripeptide.

Information Quantity of an N-gram Sequence
If many segments with the same N-gram sequence always exhibit the same properties, this N-gram sequence is considered to have a strong tendency to that property. Conversely, if the same N-gram sequence can yield diverse properties, such as the chameleon sequences, this N-gram sequence is considered to have a weak tendency. We quantified the strength of the tendency of each N-gram sequence in terms of the expected probability of obtaining an observed Figure 1 Method overview. The dataset was constructed by extracting entries meeting the criteria from the Protein Data Bank (PDB) with the elimination of redundancy. For each N-gram segment with 4 ≤ N ≤ 9 included in the dataset, five properties were assessed including the secondary structure and rASA. Information quantity to predict these properties from the sequence was evaluated in terms of the negative log value of the expected probability to observe the samples in the dataset. In addition, the predictability of the properties was also evaluated using a machine learning technique. structural data of proteins deposited in the Protein Data Bank (PDB) [4][5][6][7]. Statistical studies on the PDB have achieved success for a wide range of prediction tasks from protein sequences, such as predictions of secondary structures [8], solvent-accessible residues [9,10] and intrinsic disorder [11,12]. In the tasks to predict structural features for a short segment of protein sequence or subsequence, three levels of information hidden in the sequence are used. The first is the propensity of amino acids composing the target segments [13]. The second is context-dependent information, which means that information from the regions other than the target site [14,15]. The interactions with the other regions separated along the sequence from the target segment, including non-local interactions, create challenges for predictions [16][17][18]. The third is evolutionary information, such as the sequence profile generated from a multiple sequence alignment [19,20]. State-of-the-art prediction methods based on the sequence profile drastically improve the predictions [8]. Toward understanding the principles of sequence-structure relationships, the distinction of the contributions of the information to determine the structures is of importance. At the standpoint of Anfinsen's dogma, the first two levels of information are enough to predict the structures. Trials to maximize the prediction performances without evolutionary information provide indispensable insights [21]. In addition, differences in roles of the first and second types of the information should be dissected. Although the first type of information, i.e., a short segment of the amino acid sequence, has been studied for a long time, the principles have not been fully clarified. When we consider 20 types of standard amino acids, subsequences consisting of N residues have 20 N variations of the sequence. Different sequences should exhibit different physical properties. Characterizing the universe of N-residue subsequences, or N-gram, is the key to understanding the principles of sequence-structure relationships [22]. A prominent example of well-characterized classes of N-gram sequences is chameleon sequences, which are subsequences with the capability to form distinct secondary structures depending on the environment [23,24]. Chameleon sequences have a weak tendency to form a specific structure, and their structures are determined by their surroundings rather than by their sequence. On the contrary, non-chameleon sequences have a strong tendency to form specific structures. Previous studies have successfully identified many chameleon sequences by taking advantage of the wealth of data in the PDB based on the binary classification of subsequences into chameleon. However, the strength of the tendency to form a specific structure in each subsequence can be considered as a spectrum in principle. Classifying the subsequences into chameleon may overlook the quantitative features of the sequence-structure relationships.
In this study, we quantified the strength of the tendency to form a specific structure of each subsequence in terms of the information quantity calculated from the statistics on the PDB. The information quantity is the negative logarithm of the expected probability for a probability distribution. When segments with the same subsequence more frequently form the same secondary structure than expected, this subsequence is considered to have a high information quantity. Alternatively, when segments with the same subsequences take various secondary structures by chance, this subsequence has no information about its secondary structure.
In addition to the secondary structures, we also assessed the relative accessible surface area (rASA) of the residues. We aimed to characterize the N-grams observed in the PDB in terms of information quantity for their structures encoded in each N-gram sequence. We collected N-grams (4 ≤ N ≤ 9) from the PDB and evaluated the propensity for secondary structure (helical, beta, and coil structures) and rASA. We found thousands of N-grams with strong tendencies for these structural features. They are potential candidates for structural motifs of proteins. We also found that a machine learning technique can be partially applied to estimate the tendency of unknown N-grams.

Dataset Construction
The method overview is shown in Figure 1. We analyzed the same dataset as in our previous study [25]. This dataset is a subset of the PDB [26] snapshot in 2017 filtered by the following criteria: i) X-ray structures with a resolution better than or equal to 3.0 Å, ii) the number of atoms is less than one million, and iii) the redundancy in the dataset is eliminated by single-linkage clustering based on the sequence identity ≤ 40 % using CD-HIT software [27]. The data from the PDB were processed using in-house scripts via PDBML [28]. The entry consisting of the highest number of N-gram segments in each cluster was chosen as a representative. The dataset is shown in Supplementary Data S1.

N-gram Analyses
In this article, we define the two terms, the N-gram segment and N-gram sequence, as follows. The former indicates a region consisting of N-successive residues in a polypeptide chain contained in a PDB entry. The latter indicates the amino acid sequence of N-successive residues. An N-gram sequence is a property of each N-gram segment. N-gram segments with the same N-gram sequence were collected from the dataset to characterize an N-gram sequence, and their statistics were assessed.  For all the polypeptide chains in the dataset, they were decomposed into N-gram segments by sliding the window of Nsuccessive residues along the sequence. The N value ranged from four to nine. The secondary structure and rASA of each N-gram segment were assigned based on the properties of the residue at the middle of the segment; when N is an even number, the (N/2)-th residue was chosen as a representative. We assessed the following properties of each segment: (i) the secondary structure, (ii) whether highly exposed to solvent or not based on the threshold of rASA ≥ 0.8, and (iii) whether exposed to solvent or not based on the threshold of rASA ≥ 0.2. For property (i), the secondary structure was defined by the DSSP software [29]. The eight classes of secondary structures produced by DSSP were reduced into three classes: helix ('H', 'G', and 'I' in the DSSP output), beta ('E' and 'B'), and coil (other symbols). For properties (ii) and (iii), the rASA was defined as the ASA value of the residue divided by the reference ASA value, which is defined by the ASA of the center residue in the Gly-X-Gly tripeptide.
In this article, the symbols signifying these properties of N-grams are introduced. The three properties are signified as the symbol pc,

Information Quantity of an N-gram Sequence
If many segments with the same N-gram sequence always exhibit the same properties, this N-gram sequence is considered to have a strong tendency to that property. Conversely, if the same N-gram sequence can yield diverse properties, such as the chameleon sequences, this N-gram sequence is considered to have a weak tendency. We quantified the strength of the tendency of each N-gram sequence in terms of the expected probability of obtaining an observed Figure 1 Method overview. The dataset was constructed by extracting entries meeting the criteria from the Protein Data Bank (PDB) with the elimination of redundancy. For each N-gram segment with 4 ≤ N ≤ 9 included in the dataset, five properties were assessed including the secondary structure and rASA. Information quantity to predict these properties from the sequence was evaluated in terms of the negative log value of the expected probability to observe the samples in the dataset. In addition, the predictability of the properties was also evaluated using a machine learning technique. Figure S4. Cluster analysis for 2,205 tetra-grams with strong tendency to form helical   [11,12]. In the tasks to predict structural features for a short segment of protein sequence or subsequence, three levels of information hidden in the sequence are used. The first is the propensity of amino acids composing the target segments [13]. The second is context-dependent information, which means that information from the regions other than the target site [14,15]. The interactions with the other regions separated along the sequence from the target segment, including non-local interactions, create challenges for predictions [16][17][18]. The third is evolutionary information, such as the sequence profile generated from a multiple sequence alignment [19,20]. State-of-the-art prediction methods based on the sequence profile drastically improve the predictions [8]. Toward understanding the principles of sequence-structure relationships, the distinction of the contributions of the information to determine the structures is of importance. At the standpoint of Anfinsen's dogma, the first two levels of information are enough to predict the structures. Trials to maximize the prediction performances without evolutionary information provide indispensable insights [21]. In addition, differences in roles of the first and second types of the information should be dissected. Although the first type of information, i.e., a short segment of the amino acid sequence, has been studied for a long time, the principles have not been fully clarified. When we consider 20 types of standard amino acids, subsequences consisting of N residues have 20 N variations of the sequence. Different sequences should exhibit different physical properties. Characterizing the universe of N-residue subsequences, or N-gram, is the key to understanding the principles of sequence-structure relationships [22]. A prominent example of well-characterized classes of N-gram sequences is chameleon sequences, which are subsequences with the capability to form distinct secondary structures depending on the environment [23,24]. Chameleon sequences have a weak tendency to form a specific structure, and their structures are determined by their surroundings rather than by their sequence. On the contrary, non-chameleon sequences have a strong tendency to form specific structures. Previous studies have successfully identified many chameleon sequences by taking advantage of the wealth of data in the PDB based on the binary classification of subsequences into chameleon. However, the strength of the tendency to form a specific structure in each subsequence can be considered as a spectrum in principle. Classifying the subsequences into chameleon may overlook the quantitative features of the sequence-structure relationships.
In this study, we quantified the strength of the tendency to form a specific structure of each subsequence in terms of the information quantity calculated from the statistics on the PDB. The information quantity is the negative logarithm of the expected probability for a probability distribution. When segments with the same subsequence more frequently form the same secondary structure than expected, this subsequence is considered to have a high information quantity. Alternatively, when segments with the same subsequences take various secondary structures by chance, this subsequence has no information about its secondary structure.
In addition to the secondary structures, we also assessed the relative accessible surface area (rASA) of the residues. We aimed to characterize the N-grams observed in the PDB in terms of information quantity for their structures encoded in each N-gram sequence. We collected N-grams (4 ≤ N ≤ 9) from the PDB and evaluated the propensity for secondary structure (helical, beta, and coil structures) and rASA. We found thousands of N-grams with strong tendencies for these structural features. They are potential candidates for structural motifs of proteins. We also found that a machine learning technique can be partially applied to estimate the tendency of unknown N-grams.

Dataset Construction
The method overview is shown in Figure 1. We analyzed the same dataset as in our previous study [25]. This dataset is a subset of the PDB [26] snapshot in 2017 filtered by the following criteria: i) X-ray structures with a resolution better than or equal to 3.0 Å, ii) the number of atoms is less than one million, and iii) the redundancy in the dataset is eliminated by single-linkage clustering based on the sequence identity ≤ 40 % using CD-HIT software [27]. The data from the PDB were processed using in-house scripts via PDBML [28]. The entry consisting of the highest number of N-gram segments in each cluster was chosen as a representative. The dataset is shown in Supplementary Data S1.

N-gram Analyses
In this article, we define the two terms, the N-gram segment and N-gram sequence, as follows. The former indicates a region consisting of N-successive residues in a polypeptide chain contained in a PDB entry. The latter indicates the amino acid sequence of N-successive residues. An N-gram sequence is a property of each N-gram segment. N-gram segments with the same N-gram sequence were collected from the dataset to characterize an N-gram sequence, and their statistics were assessed. For all the polypeptide chains in the dataset, they were decomposed into N-gram segments by sliding the window of Nsuccessive residues along the sequence. The N value ranged from four to nine. The secondary structure and rASA of each N-gram segment were assigned based on the properties of the residue at the middle of the segment; when N is an even number, the (N/2)-th residue was chosen as a representative. We assessed the following properties of each segment: (i) the secondary structure, (ii) whether highly exposed to solvent or not based on the threshold of rASA ≥ 0.8, and (iii) whether exposed to solvent or not based on the threshold of rASA ≥ 0.2. For property (i), the secondary structure was defined by the DSSP software [29]. The eight classes of secondary structures produced by DSSP were reduced into three classes: helix ('H', 'G', and 'I' in the DSSP output), beta ('E' and 'B'), and coil (other symbols). For properties (ii) and (iii), the rASA was defined as the ASA value of the residue divided by the reference ASA value, which is defined by the ASA of the center residue in the Gly-X-Gly tripeptide.
In this article, the symbols signifying these properties of N-grams are introduced. The three properties are signified as the symbol pc, where ∈ { , 08 , 02}, and ∈ {ℎ , , }

Information Quantity of an N-gram Sequence
If many segments with the same N-gram sequence always exhibit the same properties, this N-gram sequence is considered to have a strong tendency to that property. Conversely, if the same N-gram sequence can yield diverse properties, such as the chameleon sequences, this N-gram sequence is considered to have a weak tendency. We quantified the strength of the tendency of each N-gram sequence in terms of the expected probability of obtaining an observed Figure 1 Method overview. The dataset was constructed by extracting entries meeting the criteria from the Protein Data Bank (PDB) with the elimination of redundancy. For each N-gram segment with 4 ≤ N ≤ 9 included in the dataset, five properties were assessed including the secondary structure and rASA. Information quantity to predict these properties from the sequence was evaluated in terms of the negative log value of the expected probability to observe the samples in the dataset. In addition, the predictability of the properties was also evaluated using a machine learning technique.   [11,12]. In the tasks to predict structural features for a short segment of protein sequence or subsequence, three levels of information hidden in the sequence are used. The first is the propensity of amino acids composing the target segments [13]. The second is context-dependent information, which means that information from the regions other than the target site [14,15]. The interactions with the other regions separated along the sequence from the target segment, including non-local interactions, create challenges for predictions [16][17][18]. The third is evolutionary information, such as the sequence profile generated from a multiple sequence alignment [19,20]. State-of-the-art prediction methods based on the sequence profile drastically improve the predictions [8]. Toward understanding the principles of sequence-structure relationships, the distinction of the contributions of the information to determine the structures is of importance. At the standpoint of Anfinsen's dogma, the first two levels of information are enough to predict the structures. Trials to maximize the prediction performances without evolutionary information provide indispensable insights [21]. In addition, differences in roles of the first and second types of the information should be dissected. Although the first type of information, i.e., a short segment of the amino acid sequence, has been studied for a long time, the principles have not been fully clarified. When we consider 20 types of standard amino acids, subsequences consisting of N residues have 20 N variations of the sequence. Different sequences should exhibit different physical properties. Characterizing the universe of N-residue subsequences, or N-gram, is the key to understanding the principles of sequence-structure relationships [22]. A prominent example of well-characterized classes of N-gram sequences is chameleon sequences, which are subsequences with the capability to form distinct secondary structures depending on the environment [23,24]. Chameleon sequences have a weak tendency to form a specific structure, and their structures are determined by their surroundings rather than by their sequence. On the contrary, non-chameleon sequences have a strong tendency to form specific structures. Previous studies have successfully identified many chameleon sequences by taking advantage of the wealth of data in the PDB based on the binary classification of subsequences into chameleon. However, the strength of the tendency to form a specific structure in each subsequence can be considered as a spectrum in principle. Classifying the subsequences into chameleon may overlook the quantitative features of the sequence-structure relationships.
In this study, we quantified the strength of the tendency to form a specific structure of each subsequence in terms of the information quantity calculated from the statistics on the PDB. The information quantity is the negative logarithm of the expected probability for a probability distribution. When segments with the same subsequence more frequently form the same secondary structure than expected, this subsequence is considered to have a high information quantity. Alternatively, when segments with the same subsequences take various secondary structures by chance, this subsequence has no information about its secondary structure.
In addition to the secondary structures, we also assessed the relative accessible surface area (rASA) of the residues. We aimed to characterize the N-grams observed in the PDB in terms of information quantity for their structures encoded in each N-gram sequence. We collected N-grams (4 ≤ N ≤ 9) from the PDB and evaluated the propensity for secondary structure (helical, beta, and coil structures) and rASA. We found thousands of N-grams with strong tendencies for these structural features. They are potential candidates for structural motifs of proteins. We also found that a machine learning technique can be partially applied to estimate the tendency of unknown N-grams.

Dataset Construction
The method overview is shown in Figure 1. We analyzed the same dataset as in our previous study [25]. This dataset is a subset of the PDB [26] snapshot in 2017 filtered by the following criteria: i) X-ray structures with a resolution better than or equal to 3.0 Å, ii) the number of atoms is less than one million, and iii) the redundancy in the dataset is eliminated by single-linkage clustering based on the sequence identity ≤ 40 % using CD-HIT software [27]. The data from the PDB were processed using in-house scripts via PDBML [28]. The entry consisting of the highest number of N-gram segments in each cluster was chosen as a representative. The dataset is shown in Supplementary Data S1.

N-gram Analyses
In this article, we define the two terms, the N-gram segment and N-gram sequence, as follows. The former indicates a region consisting of N-successive residues in a polypeptide chain contained in a PDB entry. The latter indicates the amino acid sequence of N-successive residues. An N-gram sequence is a property of each N-gram segment. N-gram segments with the same N-gram sequence were collected from the dataset to characterize an N-gram sequence, and their statistics were assessed. Figure S7. Cluster analysis for 703 tetra-grams with strong tendency to be exposed to solvent. For all the polypeptide chains in the dataset, they were decomposed into N-gram segments by sliding the window of Nsuccessive residues along the sequence. The N value ranged from four to nine. The secondary structure and rASA of each N-gram segment were assigned based on the properties of the residue at the middle of the segment; when N is an even number, the (N/2)-th residue was chosen as a representative. We assessed the following properties of each segment: (i) the secondary structure, (ii) whether highly exposed to solvent or not based on the threshold of rASA ≥ 0.8, and (iii) whether exposed to solvent or not based on the threshold of rASA ≥ 0.2. For property (i), the secondary structure was defined by the DSSP software [29]. The eight classes of secondary structures produced by DSSP were reduced into three classes: helix ('H', 'G', and 'I' in the DSSP output), beta ('E' and 'B'), and coil (other symbols). For properties (ii) and (iii), the rASA was defined as the ASA value of the residue divided by the reference ASA value, which is defined by the ASA of the center residue in the Gly-X-Gly tripeptide.
In this article, the symbols signifying these properties of N-grams are introduced. The three properties are signified as the symbol pc, where ∈ { , 08 , 02},

Information Quantity of an N-gram Sequence
If many segments with the same N-gram sequence always exhibit the same properties, this N-gram sequence is considered to have a strong tendency to that property. Conversely, if the same N-gram sequence can yield diverse properties, such as the chameleon sequences, this N-gram sequence is considered to have a weak tendency. We quantified the strength of the tendency of each N-gram sequence in terms of the expected probability of obtaining an observed Figure 1 Method overview. The dataset was constructed by extracting entries meeting the criteria from the Protein Data Bank (PDB) with the elimination of redundancy. For each N-gram segment with 4 ≤ N ≤ 9 included in the dataset, five properties were assessed including the secondary structure and rASA. Information quantity to predict these properties from the sequence was evaluated in terms of the negative log value of the expected probability to observe the samples in the dataset. In addition, the predictability of the properties was also evaluated using a machine learning technique. Figure S8. Cluster analysis for 586 tetra-grams with strong tendency to be highly exposed to   [11,12]. In the tasks to predict structural features for a short segment of protein sequence or subsequence, three levels of information hidden in the sequence are used. The first is the propensity of amino acids composing the target segments [13]. The second is context-dependent information, which means that information from the regions other than the target site [14,15]. The interactions with the other regions separated along the sequence from the target segment, including non-local interactions, create challenges for predictions [16][17][18]. The third is evolutionary information, such as the sequence profile generated from a multiple sequence alignment [19,20]. State-of-the-art prediction methods based on the sequence profile drastically improve the predictions [8]. Toward understanding the principles of sequence-structure relationships, the distinction of the contributions of the information to determine the structures is of importance. At the standpoint of Anfinsen's dogma, the first two levels of information are enough to predict the structures. Trials to maximize the prediction performances without evolutionary information provide indispensable insights [21]. In addition, differences in roles of the first and second types of the information should be dissected. Although the first type of information, i.e., a short segment of the amino acid sequence, has been studied for a long time, the principles have not been fully clarified. When we consider 20 types of standard amino acids, subsequences consisting of N residues have 20 N variations of the sequence. Different sequences should exhibit different physical properties. Characterizing the universe of N-residue subsequences, or N-gram, is the key to understanding the principles of sequence-structure relationships [22]. A prominent example of well-characterized classes of N-gram sequences is chameleon sequences, which are subsequences with the capability to form distinct secondary structures depending on the environment [23,24]. Chameleon sequences have a weak tendency to form a specific structure, and their structures are determined by their surroundings rather than by their sequence. On the contrary, non-chameleon sequences have a strong tendency to form specific structures. Previous studies have successfully identified many chameleon sequences by taking advantage of the wealth of data in the PDB based on the binary classification of subsequences into chameleon. However, the strength of the tendency to form a specific structure in each subsequence can be considered as a spectrum in principle. Classifying the subsequences into chameleon may overlook the quantitative features of the sequence-structure relationships.
In this study, we quantified the strength of the tendency to form a specific structure of each subsequence in terms of the information quantity calculated from the statistics on the PDB. The information quantity is the negative logarithm of the expected probability for a probability distribution. When segments with the same subsequence more frequently form the same secondary structure than expected, this subsequence is considered to have a high information quantity. Alternatively, when segments with the same subsequences take various secondary structures by chance, this subsequence has no information about its secondary structure.
In addition to the secondary structures, we also assessed the relative accessible surface area (rASA) of the residues. We aimed to characterize the N-grams observed in the PDB in terms of information quantity for their structures encoded in each N-gram sequence. We collected N-grams (4 ≤ N ≤ 9) from the PDB and evaluated the propensity for secondary structure (helical, beta, and coil structures) and rASA. We found thousands of N-grams with strong tendencies for these structural features. They are potential candidates for structural motifs of proteins. We also found that a machine learning technique can be partially applied to estimate the tendency of unknown N-grams.

Dataset Construction
The method overview is shown in Figure 1. We analyzed the same dataset as in our previous study [25]. This dataset is a subset of the PDB [26] snapshot in 2017 filtered by the following criteria: i) X-ray structures with a resolution better than or equal to 3.0 Å, ii) the number of atoms is less than one million, and iii) the redundancy in the dataset is eliminated by single-linkage clustering based on the sequence identity ≤ 40 % using CD-HIT software [27]. The data from the PDB were processed using in-house scripts via PDBML [28]. The entry consisting of the highest number of N-gram segments in each cluster was chosen as a representative. The dataset is shown in Supplementary Data S1.

N-gram Analyses
In this article, we define the two terms, the N-gram segment and N-gram sequence, as follows. The former indicates a region consisting of N-successive residues in a polypeptide chain contained in a PDB entry. The latter indicates the amino acid sequence of N-successive residues. An N-gram sequence is a property of each N-gram segment. N-gram segments with the same N-gram sequence were collected from the dataset to characterize an N-gram sequence, and their statistics were assessed. Figure S9. The amino acid propensities for N = 4, 5, 6, 7, 8, and 9. See the legend of Figure 5.