Protein Folding Structures: Formation of Folding Structures Based on Probability Theory.

To the best of our knowledge, this is the first study that shows that the X-ray structures of proteins can be dissected into their continuous folding structure units. Each folding structure unit was designed such that both the terminal di- or tri-peptide sequences shared common sequences with the two adjacent folding structure units. To encode the folding structure information of proteins into their amino acid sequences, we proposed 44 kinds of folding elements, which covered all of the amino acids in the protein chains, and defined all folding structure units. The folding element was defined to mean a minimum structural piece, which covered the frame of the main chain of each amino acid in a protein chain. A folding structure unit of a local sequence could be fully characterized by the sequential combination of individual folding elements assigned to each amino acid. The folding structure information showed amino acid preferences in various positions in folding structure units. Folding structure formation proceeded on the basis of probability theory. Strikingly, relative formation ability analysis clearly indicated that we can decode the types and the chain length of folding structure units from the amino acid sequence of a protein.

Introduction

Anfinsen’s basic tenet of protein folding maintains that the information determining the native structure of a protein is encoded in its amino acid sequence.[1,2] Research conducted to understand protein folding has strongly supported his proposal.[3−14] In kinetic folding, we consider that not secondary structures but folding structures repeatedly unfold and refold. Secondary structure is a local part of a tertiary structure or, in other words, is the conformation of a local sequence. In aqueous solution of a globular protein, a nucleus can grow rapidly by addition of peptide chain segments to direct protein folding.[15] Globular proteins fold to create separately cooperative folding structures along folding pathways. However, they are inherently unstable, and noncovalent tertiary interactions among the folding structures are primarily responsible for the formation of secondary and tertiary structures.[15−27] One of the important objectives in this article is to define the folding structure units (Figure ), which can be verified by X-ray structures of proteins. Folding structures of proteins having X-ray structures could be statistically analyzed to yield the folding structure information. Here, we confirmed for the first time that folding structure formation can be derived from folding structure information, on the basis of probability theory. Folding structure, instead of secondary structure, enabled us to understand protein folding on an amino acid level. The reason why secondary structure is not suitable for decoding of secondary structure information is described below.

Figure 1

Amino acid sequence and the notation of a folding structure unit.

Nearly 50% of amino acids in globular proteins are in either α-helix or β-strand forms.[28,29] Namely, half of the sequences form simple secondary structures, and residual secondary structures are irregular structures. As three-dimensional structures of globular proteins are available from PDB,[30] their secondary structures, except for irregular structures, can be defined using dihedral angles (ϕ, ψ) of each amino acid. However, the secondary structure information has never been encoded in amino acid sequences precisely as many kinds of secondary structures still have no definition. Although N-cap and C-cap residues of α-helices have been defined,[31] those of other secondary structures are ambiguous. As a result, many kinds of secondary structures cannot be assigned to local sequences of proteins. Thus, the introduction of folding structures, which cover all of the protein chains, is indispensable for the dissection of folding structure information. Furthermore, the tertiary structure of a protein is derived from its secondary structures. Thus, the relationship between folding and secondary structures is critical for conformational analysis of tertiary structures of proteins.

Protein folding pathways must be correctly described in the amino acid sequences of proteins.[14] However, recent examples of protein fold switching indicate that an amino acid sequence of a polypeptide chain can encode a stable fold, while simultaneously hiding latent propensities for alternative states with novel functions.[32] Thus, detailed encoding and decoding of the folding structure information is the most important step to understand protein folding. The objective of this study is to develop statistical decoding of the folding structure information encoded in the 20 kinds of amino acids.

In 1988, Richardson reported the precise amino acid preferences for 17 individual positions relative to the α-helical ends.[33] Using Richardson’s analysis, we analyzed the amino acid preferences in type II β-turn and type I α-turn.[34,35] Although there is no mention of decoding of the folding structure information in these reports, these treatments involve a study of the normalized folding structure information encoded in the 20 kinds of amino acids. This is exactly the encoding and decoding of folding structure information that we realized in this article. Decoding of the folding structure information enables us to understand the folding structure formation of a local sequence on the basis of probability theory.

Detailed encoding of the folding structure information into the amino acid sequences of proteins requires precise definition of the folding structure units. Two-dimensional representations of native structures of proteins, derived from the backbone dihedral angles (ϕ, ψ) and the amino acid sequences, clearly displayed their three-dimensional structures.[34] The representations distinctly indicate that the precise definition of cap residues of all of the folding structure units is possible, using the relationship between dihedral angles and amino acid sequences. Furthermore, the precise definition of cap residues strongly suggests the existence of overlapping regions of folding structure units at both the terminal regions and develops the design concept of folding structure units.

Here, we propose 44 kinds of folding elements to encode the folding structure information into amino acid sequences of proteins. By use of folding elements, we designed the folding structure units on the basis of the design concept that protein folding can be derived from continuous folding structure units with overlapping regions. Subsequently, the folding structure information for the amino acid preferences in the 44 different folding elements was encoded and decoded statistically. Finally, on the basis of probability theory, folding structure formation was demonstrated by relative formation ability (RFA) analysis. Folding structure, instead of secondary structure, enabled us to evaluate the RFA value of a local sequence for any local structure. The decoding of the folding structure information appears to yield the initiation mechanism of protein folding.

Results and Discussion

Definition of Folding Structure Units

To clarify the position of each amino acid of the folding structure units, relative to their cap residues, we introduced symbols termed as folding elements. The folding element means a minimum structural piece, which covers the frame of the main chain of each amino acid in protein chains, and represents the position and single or multiple conformational regions of each amino acid in the folding structure units. Here, we proposed 44 kinds of folding elements and defined folding structure units that cover all of the protein chains. As an identical amino acid expresses folding elements of α-helices, β-strands, and irregular structures in protein chains, each of the 20 kinds of amino acids in protein sequences possessed characteristics described by our 44 kinds of folding elements. For the statistical treatment, the folding structure units were classified into four categories: α-helix (H), α-turn and β-turn (T), β-strand (S), and interconnecting folding structure units (HH, HS, SH, and SS).

Folding structure units can be fully characterized by local sequences, assigned with a single folding element for each amino acid (Figure ). As a result, the notation of a folding structure unit can be represented by a sequential combination of folding elements. The local sequence of a protein forms a variety of dynamic folding structures in the denatured state. The particular dynamic folding structure is fixed to the static folding structure in its native structure by tertiary interactions.

Design of α-Helix, Turn, and β-Strand Folding Structure Units

On the basis of two-dimensional representations of native structures of proteins, each of the folding structure units was designed such that both the terminal di- and tri-peptide sequences shared common sequences with two adjacent folding structure units. A common sequence forms a single irregular structure, which always functions as an overlapped folding structure. It is statistically treated as an overlapping sequence.

Primarily, we determined the cap residues of the folding structure units on the basis of the continuity of the backbone dihedral angles (ϕ, ψ) of the α- and β-regions.[34,36] First, N-cap and C-cap residues of H and T folding structure units were located at both ends of the sequence having continuous α-helix dihedral angles. Dihedral angles of the cap residues were out of the α-region, as shown in Figure , for the H19 folding structure unit of protein GB1, the immunoglobulin-binding B1 domain of protein G.[37] The sequence of protein GB1 is shown in Figure .

Figure 2

H19 (α-helix with 19 amino acids residues) structure unit.

Figure 9

Amino acid sequence of protein GB1, the notation of its continuous folding structure units, and RFA values of local sequences for respective folding structure units.

N-Cap and C-cap residues were assigned to folding elements and . For the terminal amino acids positioned outside the caps, the terminal folding elements of ′ and ′ were ascribed. The terminal amino acids were coincident with cap or internal residues of two adjacent folding structure units and conserved their characteristics. Both the terminal di- and tri-peptide sequences of folding structure units were always common sequences (Figure ), which are signals of termination and initiation of adjacent folding structure units; therefore, continuous folding structure units can be derived from the presence of these common sequences.

The notations of H folding structure units were as follows. For example, when seven continuous α-region residues were found in a protein chain, these residues were assigned to folding elements . In this example, the eleven residues were assigned to folding elements ′′ and denoted an H11 folding structure unit. The seven folding elements (, , , , , , and ) were in a single conformational region (α-region), and the four folding elements (′, , , and ′) were in any of the multiple conformational regions ( and are in a region other than α-region). When a sequence contained more than 12 amino acids, all of the central residues were assigned to folding element (Figure ). Figure shows the notation of an α-helix (H19) sequence of 19 amino acids V21–V39 in protein GB1 together with that of its secondary structure. The notations of secondary structures can be represented one-dimensionally using conformational regions, instead of folding elements. Folding element was introduced to designate H9 folding structure units, with folding elements ordered as ′′. The shortest H folding structure unit (H8) is formed with eight amino acids assigned to the folding elements, ′′.

Hexa- and hepta-peptide sequences formed T6 and T7 folding structure units. The notations of T6 and T7 units are shown in Figure . Folding elements and were assigned to cap residues in a T6 folding structure unit. Hexa- and hepta-peptide sequences were assigned to T6 (′′) and T7 (′′) folding structure units, respectively. Residues assigned to and were out of the α-region. T6 unit comprised type I and III β-turns,[38] and T7 unit comprised type I α-turn.[35] The three folding elements, , , and , were in the α-region, and the four folding elements, ′, , , and ′, were in any of the multiple conformational regions.

Figure 3

T7 (α-turn) and T6 (β-turn) structure units.

Second, the dihedral angles of cap residues in the S folding structure unit were still associated with the β-region. To avoid the complication of overlapped folding structures, which always grow at both terminal sequences of folding structure units (Figure ), a different definition of the cap residues of S folding structure units was introduced. The cap residues of S folding structure units were located at both ends of the continuous β-region residues, and the folding elements ′···′ were used for the longer S folding structure units, where the central residues in β-strand sequences longer than a heptapeptide sequence were assigned to folding element . N-Cap and C-cap residues were assigned to strand folding elements and , respectively. For the terminal amino acids positioned outside the caps, the terminal folding elements of ′ and ′ were ascribed. The terminal amino acids were coincident with cap or internal residues of two adjacent folding structure units and conserved their characteristics. The S5 folding structure units that consist of pentapeptide sequences were expressed by the sequence of folding elements ′′, where N-cap and C-cap residues were assigned to and (Figure ). The eight folding elements, , , , , , , , and ; were in the β-region; and the four folding elements, ′, ′, ′, and ′, were in any of the multiple conformational regions, other than the β-region.

Figure 4

S7 (β-strand with seven amino acid residues) and S5 (β-strand with five amino acid residues) structure units.

Design of Interconnecting Folding Structure Units

Interconnecting folding structure units were located between the two types (H (T) and S) of folding structure units, whose cap residues are determined on the basis of the continuity of the backbone dihedral angles (ϕ, ψ) of the α- and β-regions. Both the terminal dipeptide sequences (cap and terminal amino acids) of interconnecting folding structure units were always coincident with the terminal dipeptide sequences of two adjacent folding structure units. The interconnecting sequences were designed to conserve the characteristics of two adjacent folding structure units.

When both sides of a pentapeptide sequence were β-strands, the sequence was assigned to an SS5 folding structure unit. The SH5, HS5, and HH5 folding structure units were also assigned (Figure ). Symbol H of these structure units included H and T folding structure units. Cap folding elements 1–4 were ascribed depending on the adjacent folding structure unit. For the cap residues of the interconnecting folding structure unit, the cap folding elements of 1 (adjacent to and ), 2 (adjacent to and ), 3 (adjacent to and ), and 4 (adjacent to and ) were defined. For the amino acids positioned outside the caps, the terminal folding elements of 1′, 2′, 3′, and 4′ were ascribed. Only the folding elements 1′ and 3′ were in the β-region. For the amino acids positioned inside the caps, the internal folding elements of were ascribed. Figure shows the notations of interconnecting folding structure units formed by a pentapeptide sequence.

Figure 5

Interconnection structure unit (1).

A minimum interconnecting sequence was a tripeptide sequence, and both the terminal dipeptide sequences conserved the characteristics of the two adjacent folding structure units. In about 2% of the amino acids in proteins, single amino acid residues can be assigned to between cap elements of the adjacent folding structure units. In such cases, is replaced by 13 (between β-strand and β-strand), 14 (between β-strand and helix), 23 (between helix and β-strand), and 24 (between helix and helix) (Figure ). A helix in parenthesis consisted of an α-helix and turn folding structure units. The local sequence A20–D22 in protein GB1 (Figure ) was an example of this type of interconnecting sequence. When a tripeptide sequence was found between S and H folding structure units, the sequence was assigned as an SH3 folding structure unit. The SS3, HS3, and HH3 folding structure units were also assigned. All of the folding elements, except for 1′ and 3′, of the interconnecting folding structure units were in any of the multiple conformational regions.

Figure 6

Interconnection structure unit (2).

Contrast between Folding and Secondary Structures

The X-ray structures of globular proteins could be dissected into their continuous folding structure units on the basis of the precise definition of N-cap and C-cap amino acids of H, T, and S folding structure units. Fundamentally, all of the folding structure units of globular protein were determined on the basis of the continuity of the backbone dihedral angles (ϕ, ψ) of the α- and β-region. The continuity of folding structure units with overlapping regions at both the terminal sequences was clearly displayed in Figure using protein GB1 chain. The overlapping region shared a common sequence with the two adjacent folding structure units. The common sequences were statistically treated as overlapping sequences, which were assigned overlapped folding structures independently. Any of the folding structures of protein GB1, as well as globular proteins, could be represented by a local sequence of individual folding element assigned to each amino acid. All of the local sequences of globular proteins were covered with folding structures. Thus, the folding structure information of all of the local sequences for their respective folding structures could be analyzed.

The notations of all of the folding structure units were represented by sequential combinations of specific folding elements. Out of the 44 folding elements, 21 were in a single conformational region. They are as follows: , , , , , , , , , , , , , , , , , , , 1′, and 3′. However, the remaining 23 folding elements represented multiple conformational regions. The most significant feature of a folding structure, in relation to a secondary structure, is that the assignment of individual folding elements to each amino acid of a local sequence enables the determination of precisely simple secondary structures with continuous α- or β-region residues. Concurrently, we can specify the terminal sequences whose secondary structures cannot be determined. Most significantly, we may consider dynamic local structures in the denatured state using folding structures, as the folding structure information is expected to be derived from Richardson’s analysis.[33]

The tertiary structure of a protein is derived from its secondary structures. Thus, the relationship between folding and secondary structures is critical for conformational analysis of tertiary structures of proteins. The notation of secondary structure is represented by the sequential combination of conformational regions assigned to each amino acid, although the definition of conformational regions is generally ambiguous. To differentiate between folding and secondary structures, both the notations, assigned to the local sequence V21–V39 of protein GB1, are represented in Figure . Amino acids assigned to and were in any of the multiple conformational regions, other than the α-region.[36] Secondary structures of proteins were primarily classified into two categories: simple secondary structures, having continuous α- or β-region residues, and irregular, undefined structures. Simple secondary structures are internal folding structures, and most of the irregular secondary structures are terminal folding structures. Conformational regions could not be assigned to many portions of local sequences of proteins, as many kinds of secondary structures remain undefined.

Furthermore, a common sequence forms a single irregular secondary structure, which always functions as an overlapped folding structure. On the contrary, identical terminal folding structures of different proteins contain multiple secondary structures. The secondary structure information has never been decoded, as the relationship between secondary structures and amino acid sequences is complicated. Thus, we cannot discuss dynamic local structures in the denatured state by secondary structures.

Originally, the precise definition of all of the folding structure units enabled detailed sequence analysis for the formation of simple secondary structures and irregular structures. Most importantly, sequences forming simple secondary structures and irregular structures were characterized by assignment of the 44 kinds of folding elements. Folding structures of proteins included single or multiple secondary structures, for example, terminal folding structures of proteins composed of multiple irregular structures. Internal folding structures, except for interconnecting folding structure units, consisted of single simple secondary structures. The precise definition of cap residues enabled the determination of single simple secondary structures having continuous α- or β-region residues and sequences forming irregular structures, on the basis of the assignment of individual folding elements to each amino acid of a local sequence.

For terminal regions of H folding structure units, we could determine the terminal sequences precisely, but not the secondary structures. Any of the simple secondary structures of α-helices could be determined precisely as the H folding structure units are characterized by 12 kinds of folding elements. Similarly, simple secondary structures formed by central di- or tri-peptide sequences of T6 and T7 folding structure units could be determined precisely as the T folding structure information was encoded by 13 kinds of folding elements. For the S folding structure units, any of the simple secondary structures having continuous β-region residues could be determined precisely as the S folding structure information was encoded by 12 kinds of folding elements. The precise definition of interconnecting folding structure units enabled determination of the interconnecting sequences that form irregular structures, but not secondary structures. Statistical analysis of the relationship between interconnecting sequences and secondary structures was expected to provide the secondary structure information.

Encoding the Folding Structure Information of Proteins

By the assignment of individual folding elements of folding structure units to each amino acid of local sequences, the folding structure information could be encoded into the amino acid sequences of proteins. Folding structure units were determined on the basis of the continuity of the backbone dihedral angles (ϕ, ψ) of the α- and β-regions. The dihedral angles used for data set preparation were as follows: α-region: −130 ≤ ϕ ≤ −30, −80 ≤ ψ ≤ +30; β-region: −180 ≤ ϕ ≤ −45, +90 ≤ ψ ≤ +180. N-Cap and C-cap residues of the H, T, and S folding structure units were located at both ends of the sequence. The dihedral angles of the cap residues of H and T folding structure units were out of the α-region, whereas the dihedral angles of the cap residues in the S folding structure unit were in the β-region. Both the terminal di- and tri-peptide sequences of H, T, and S folding structure units always belonged to two adjacent folding structure units in a protein chain. Each of the amino acid in overlapping local sequences was independently assigned to each of the overlapped folding element. Interconnecting folding structure units were located between the α-region sequences and the β-region ones. The cap residues of the interconnecting folding structure units were defined depending on the adjacent folding structure unit. Both their terminal dipeptide sequences always shared common sequences with the adjacent folding structure units.

The folding structure information was encoded into protein GB1 chain as follows. Each amino acid in protein GB1 chain was assigned to one or more of folding elements pertaining to a folding structure unit. For example, the local sequence G41–D47 of protein GB1 formed S7 (G41–D47), and the terminal dipeptide sequence G41E42 and the terminal tripeptide sequence Y45–D47 were the common sequences in the adjacent folding structure units (Figure ). In Figure , all of the amino acid residues were assigned to 77 folding elements in total, along with the folding structure units. When a tripeptide sequence formed an interconnecting folding structure unit, the central amino acid residue expressed three folding elements (e.g., N8 and V21 in Figure ). The central amino acid was independently assigned to two additional terminal folding elements of the adjacent folding structure units. About 2% of the amino acids in the protein data set expressed three folding elements. These data provided important statistical values useful for analysis of the amino acid–folding element relationship.

Figure 7

Overlapped folding structure.

Decoding the Folding Structure Information

The protein data set prepared above includes 1 000 666 (T10) amino acid residues, which we assigned to the 44 folding elements. The common sequence was statistically treated as overlapping local sequences. The number of amino acids assigned to α-helix folding element in the data set was 41 752 (T0) in total. The number of Ala assigned to α-helix folding element was 1383 (TA).

The number of amino acid residues assigned to a particular folding element could be regarded as the number of the folding element. As the number of amino acid residues that were transformed into α-helix folding element in the protein chains used in this study was 41 752 (T0), the formation probability of all of the amino acid residues found in the protein data set for α-helix folding element was 41 752/1 000 666 (Q0 = T0/T10). This value was termed “folding element value” (Table ).

Table 1

Normalized Formation Probability (NFP) Value of the 20 Kinds of Amino Acids for the 44 Kinds of Folding Elements

	P1Xa	a′	a	b	c	d	e	f	g	h	i	i′	j	k	l′	l	m	n	o	p	p′	r′	r
A	8.1%	0.75	0.41	1.1	1.3	1.2	1.5	1.3	1.4	1.1	0.68	0.76	1.0	1.4	0.47	0.95	0.77	0.82	0.76	0.74	0.77	0.60	0.97
C	1.4%	1.0	1.1	0.56	0.51	0.67	0.91	0.77	0.70	1.1	1.2	0.81	0.54	1.2	0.52	1.2	1.2	1.3	1.3	1.4	0.75	0.63	1.2
D	5.8%	0.77	3.0	0.83	1.6	1.6	0.65	0.61	0.92	0.86	0.90	1.1	2.0	1.2	1.7	0.72	0.48	0.38	0.35	1.6	1.4	1.4	0.49
E	6.8%	0.72	0.56	1.2	2.5	2.1	1.1	1.2	1.6	1.0	0.51	0.87	2.3	1.3	0.73	0.96	0.78	0.68	0.69	0.71	0.82	0.71	0.94
F	4.1%	1.3	0.38	0.92	0.66	0.89	1.1	0.98	0.67	1.2	1.1	0.75	0.54	1.5	0.42	1.2	1.1	1.4	1.5	0.93	0.57	0.63	1.5
G	7.3%	1.1	1.6	0.46	0.76	0.65	0.42	0.30	0.39	0.37	3.5	1.1	0.66	0.46	4.1	0.45	0.29	0.40	0.38	0.44	2.6	3.6	0.44
H	2.3%	0.98	1.3	0.72	1.0	1.0	0.82	0.90	0.91	1.5	1.3	0.86	1.3	0.79	1.1	1.2	0.88	0.95	0.88	1.2	0.91	1.1	1.1
1	5.9%	1.3	0.18	0.86	0.47	0.69	1.2	1.2	0.61	0.57	0.76	0.87	0.31	0.86	0.29	1.1	1.5	1.7	1.8	0.96	0.54	0.43	1.3
K	5.9%	0.83	0.55	0.98	1.1	0.81	1.1	1.6	1.8	1.3	0.84	1.2	1.3	0.67	1.0	1.2	1.1	0.75	0.87	0.73	0.83	0.89	1.0
L	9.4%	1.2	0.28	1.0	0.58	0.88	1.5	1.6	1.1	1.4	0.86	0.77	0.42	1.1	0.39	0.86	0.96	1.2	1.2	0.70	0.66	0.59	1.1
M	1.9%	1.4	0.40	0.85	0.66	0.94	1.4	1.4	1.2	1.1	0.88	0.74	0.48	1.0	0.52	1.0	0.91	1.2	1.2	0.69	0.59	0.60	1.2
N	4.2%	0.86	2.4	0.53	0.88	0.73	0.71	0.70	0.91	1.5	1.5	0.91	1.3	0.76	2.0	0.76	0.58	0.50	0.47	1.3	1.6	1.8	0.68
P	4.6%	1.2	1.5	3.6	0.51	0.36	0.21	0.15	0.28	0.059	0.031	3.0	0.27	0.34	0.62	0.58	1.9	0.96	1.2	1.9	1.8	0.69	0.71
Q	3.7%	0.80	0.57	0.83	1.2	1.6	1.2	1.2	1.5	1.4	0.89	0.87	1.2	0.99	0.89	1.1	0.86	0.77	0.78	0.70	0.69	0.88	0.97
R	5.1%	0.84	0.63	0.92	0.94	0.78	1.2	1.4	1.6	1.2	0.84	0.98	1.0	0.84	0.88	1.1	1.0	0.87	0.93	0.74	0.74	0.83	1.0
S	5.9%	0.83	2.4	0.87	1.3	0.84	0.69	0.73	1.1	1.1	0.87	0.95	1.9	1.0	0.95	1.1	0.79	0.79	0.66	1.6	1.0	0.93	0.92
T	5.5%	1.0	1.8	0.79	0.92	1.0	0.73	0.55	0.64	1.2	0.64	0.97	1.0	0.88	0.86	1.3	1.2	1.2	1.0	1.4	1.0	0.90	1.1
V	7.3%	1.2	0.18	0.84	0.53	0.88	1.0	0.85	0.57	0.51	0.63	0.82	0.37	0.96	0.33	1.3	1.7	2.0	1.9	1.1	0.57	0.47	1.2
W	1.4%	1.2	0.39	1.3	1.2	0.92	1.1	1.0	0.72	0.88	0.75	0.71	0.99	1.8	0.45	1.2	1.4	1.1	1.3	0.86	0.65	0.65	1.6
Y	3.5%	1.2	0.50	0.85	0.75	0.88	1.0	0.93	0.69	1.4	0.99	0.76	0.71	1.2	0.51	1.3	1.2	1.4	1.4	0.96	0.59	0.70	1.4
Qx0b		0.042	0.042	0.041	0.032	0.027	0.20	0.026	0.031	0.040	0.040	0.042	0.0086	0.002	0.042	0.041	0.04	0.10	0.041	0.041	0.043	0.013	0.013

Composition of amino acid (X) in all of the amino acid residues found in the data set.

Folding element value Q0 ( = folding element) is the formation probability of all of the amino acid residues found ∑Q0 = 1.39.

As the terminal and cap residues always had two or three folding elements, the sum total of folding element values of the protein data set was not equal to 1.0, but to 1.39. This value means that about 40% of protein sequences were common sequences. As an example, the sum total of the folding element values of protein GB1 was calculated as follows. Each amino acid of protein GB1 is treated as a structural piece of a folding structure. Each amino acid was assigned to one or two folding elements, except for the N8 and V21 residues, which were assigned to three folding elements, as shown in Figure . Protein GB1 consists of 56 amino acids, which expressed 77 folding elements in the peptide sequence. As the common sequences were treated statistically as overlapping sequences, a total of 77 amino acids were assigned to 77 folding elements. The total folding element value for protein GB1 was 1.38 (77/56).

Using this approach, we could calculate the formation probability of each amino acid for a given folding element. The number of Ala (T1A) in our data set prepared above was 81 527. Among them, the number of Ala that expressed α-helix folding element was 1383 (TA). Thus, the formation probability of Ala for α-helix folding element was 1383/81 527 (QA = TA/T1A). The formation probability of Ala for (QaA) was then normalized by the formation probability of all of the amino acid residues for , folding element value of (Q0). The “NFP” (NFP = QA/Q0 in the case of Ala for ) expressed the formation preference of an amino acid in a folding element. In this way, NFP values of the 20 kinds of amino acids for the 44 folding elements were calculated (Table ). NFP values consisted of the normalized folding structure information of the amino acids for their respective folding elements. Each amino acid in a protein sequence expressed single or multiple folding elements through a dihedral angle transition, with a particular formation probability.

Table shows the NFP values of amino acids for all of the 44 folding elements. These results indicate that identical amino acids in a protein chain can express different folding elements to form different folding structures. As a result, the folding elements of an identical amino acid revealed the chameleon character of the native structure. NFP values of amino acids depended on the structure of their side chains. The amino acid pairs, such as D–N, E–Q, F–W–Y, I–V, K–R, L–M, and S–T, showed similar NFP values for the 44 kinds of folding elements. Figure shows the similarity between the NFP values of L and M. Other pairs of amino acids also showed high level of similarity between their NFP values. These results clearly indicate that amino acid preferences in positions of folding structure units strongly depend on the similarity of amino acid residues and that probability theory can be used to evaluate the folding structure information appropriately. The determined NFP values for folding elements of α-helices are in good agreement with amino acid preferences for specific locations at the ends of α-helices.[32] Statistical decoding of the folding structure information encoded in local sequences may disclose protein folding pathways.

Figure 8

Similarity on the NFP values of Leu and Met.

The probability of occurrence of an Ala residue at α-helix folding element (PA) was found to be 1383/41 752 (TA/T0 = PA). Comparison of this value with the occurrence probability of all of the Ala residues found in the data set, 81 527/1 000 666 (T1A/T10 = P1A), showed the preference of Ala at α-helix folding element . We used the “normalized occurrence probability” (NOP = PA/P1A in the case of Ala at ) to express the preference of an amino acid residue appearing in a particular folding element. The above statistical analysis used Richardson’s analysis as a basis for the position-specific amino acid preferences in α-helices. The NOP values corresponded to the position-specific amino acid preference values found by Richardson.[32] The NOP values in the α-helix region (′–′) were in good agreement with Richardson’s preference values. These values are the normalized folding structure information of α-helices encoded in the 20 kinds of amino acids. Small differences between NOP values and Richardson’s preference values are caused by the definition of the helix region.[32] From the discussion above, the NFP value can be assumed to be mathematically equivalent to the NOP value of an amino acid at a folding element (eq ). The NFP values of the 20 kinds of amino acids for the 44 folding elements were calculated for the peptide sequence design.

Statistical analysis of folding element and amino acid in our data set showed the occurrence number of amino acid assigned to each folding element. Amino acid preferences were normalized by the occurrence probability of an amino acid on the basis of the overall percentage of each amino acid found in our data set. The result gave NOP. We can compare these values in any sequence to discuss the folding structure. The NFP values in Table show the normalized folding structure information of each amino acid in the peptide sequences. They can be used for the evaluation of designed peptide sequences for folding structure formation.where NFP is the normalized formation probability; NOP is the normalized occurrence probability; is the folding element; X is the amino acid residue; TX is the number of the folding element assigned to amino acid X, equals to the number of the amino acid X at ; T1X is the number of amino acid X in the data set; T0 is the number of folding element in total, equals to the number of all of the amino acids at ; T10 is the number of total amino acid residues in the data set; QX is the formation probability of amino acid X for folding element ; Q0 is the formation probability of all of the amino acid residues in the data set for folding element (folding element value); PX is the occurrence probability of amino acid X at folding element ; and P1X is the occurrence probability of amino acid X in the data set.

RFA Analysis of the Continuous Folding Structure Units of Protein GB1 Sequence

The normalized folding structure information of a local sequence gave the RFA value for any folding structure. RFA values are the products of NFP values of each amino acid for a respective folding element. As all of the NOP values of the 20 kinds of amino acids in a peptide sequence were 1.0, the relative occurrence ability value of any of the local sequences in a peptide was 1.0, regardless of its chain length and amino acid sequence. As the NFP value of each amino acid in a peptide sequence was the relative folding structure information, the normalized folding structure information can be regarded as the relative folding structure information in the peptide sequences.

As a protein chain example for RFA analysis, we chose protein GB1 sequence (Figure ). The notations of the folding structure units of protein GB1 were assigned to protein GB1 sequence by referring to a two-dimensional representation of the native structure of protein GB1.[34]Figure shows all of the amino acid residues assigned to the 77 folding elements in total, along with the continuous folding structure units. In protein GB1 chain, both the terminal sequences of the folding structure units always overlapped each other in the continuous folding structure units. In protein GB1 sequence, all of the RFA values for folding structure units were more than 1.4.

The internal folding structures of H, T, and S folding structure units consisted of simple secondary structures, and interconnecting sequences expressed irregular structures. Sequences forming simple secondary structures and irregular structures could be determined precisely on the basis of the assignment of individual folding elements to each amino acid of protein GB1 sequence. For the H folding structure unit of protein GB1, we determined the terminal dipeptide sequences, V21D22 and G38V39, forming irregular structures and the 15-residue sequence, A23–N37, forming the simple secondary structure precisely. For T folding structure units, we determined common sequences, N8G9, L12K13, Y45–D47, and K50T51, forming irregular structures and central di- and tri-peptide sequences, K11T12 and D47–T49, forming simple secondary structures. For S folding structure units, hexa- and penta-peptide sequences forming simple secondary structures were determined precisely. The RFA analysis clearly indicated that any of simple secondary structures could be decoded from the protein GB1 sequence on the basis of the tertiary structure information. Precise decoding of simple secondary structures from protein GB1 sequence will appear elsewhere.

The folding structure units could be formed by simple accumulation of folding elements encoded in amino acids along the local sequences, on the basis of probability theory. The simple accumulation is the general solution for folding structure formation in the denatured state. The local structure formation of a protein using folding structure, instead of secondary structure, could lead to the first general solution for local structure formation, on the basis of probability theory. Toward fully understanding the continuous folding structure units, the most important issue is to dissect the continuity of folding structure units using RFA analysis and tertiary structure information.

Methods

Preparation of the Protein Data Set

The protein data bank of well-determined three-dimensional protein structures could provide statistically meaningful data for analysis of the relationship between the 20 kinds of amino acids and the 44 kinds of folding elements. First, we extracted protein chains in which the sequence similarity was minimal to enable statistical analysis of the data. Folding structure units were determined on the basis of the continuity of the backbone dihedral angles (ϕ and ψ) of the α- and β-regions. The dihedral angles used for data set preparation are as follows: α-region: −130 ≤ Φ ≤ −30, −80 ≤ Ψ ≤ +30, β-region: −180 ≤ Φ ≤ −45, +90 ≤ Ψ ≤ +180, ω-region: the regions other than α- and β-regions. The dihedral angle data used in this article were obtained from Database of Secondary Structure in Proteins (DSSP) provided by Kabsch and Sander.[29] N-Cap and C-cap residues of folding structure units are located at both ends of the sequence. The dihedral angles of the cap residues of the H and T folding structure units are out of the α-region, whereas the dihedral angles of the cap residues in the S folding structure unit are still in the β-region. This definition may avoid the complication of overlapped folding structures. Interconnection folding structure units are located between the α-region sequences and the β-region ones. Depending on the sequences in front and behind of the interconnection folding structure unit, the folding structure units are classified into HH (between α-region sequences), HS (between α-region and β-region sequences), SH (between β-region and α-region sequences), and SS (between β-region sequences).

We used the CATH classification database[39] to avoid sequence similarity. One protein chain was picked from each “Sequence Family” in the CATH classification, each containing domains characteristic of the different Sequence Families. Some protein chains contained repeated sequences. Chains with multiple repeated sequences longer than a heptapeptide were removed from the data set. A total of 4636 protein chains were selected for the data set, which included 1 000 666 amino acid residues. Information regarding the folding structure units, N-terminal and C-terminal folding structures, was extracted from all sequences of the selected protein chains.

Extraction of All of the Folding Structure Units

Folding structure units were determined on the basis of the continuity of the backbone dihedral angles (ϕ and ψ) of the α- and β-regions. The dihedral angles used for data set preparation were described above.

The common sequences were statistically treated as overlapping local sequences, and each of their amino acids was independently assigned to each of the overlapped folding elements. All of the folding structure units could be extracted from protein chains, and each of the 20 kinds of amino acids was assigned to their appropriate folding elements.

Conclusions

By use of folding elements, we could design the folding structure units based on the concept that protein folding should be derived from continuous folding structure units. Each of the folding structure units was designed so that both the terminal di- or tri-peptide sequences shared common sequences with the two adjacent folding structure units. The folding structure information showed amino acid preferences in positions of folding structure units. It is simply described as a one-dimensional sequence of folding elements assigned to an amino acid sequence. In protein GB1 sequence, all of the RFA values for folding structure units verified by the X-ray structure were more than 1.4, and the continuous folding structure units led to the native structure through tertiary interactions. In the protein chain of GB1, both the terminal sequences of the folding structure units always overlapped each other in the continuous folding structure units. The simple accumulation of folding elements encoded in amino acids along local sequences, on the basis of probability theory, could be the general solution for folding structure formation.