A total of 146 globular proteins classified by the SCOP database, ^[30] including 50 all-α , 49 all-β , 24 α + β , and 23 α /β proteins, were selected as the sample source for our present work. The amino acid sequence and secondary structure of each protein are obtained from the Protein Date Bank (PDB). Three types of amino acids, namely hydrophobic residues (H) consisting of Phe, Met, Leu, Ile, Trp, Cys, Tyr, and Val, neutral residues (N) consisting of Gly, His, Thr, and Ala, and polar residues (P) consisting of Asn, Lys, Asp, Ser, Glu, Gln, Arg, and Pro, are adopted by the MJ matrix and other literatures.^{[19, 20, 24, 25, 31]} Then, the protein is represented by a chain containing 3 types of beads. Each bead in a protein molecule is represented by the position of an α -carbon atom (C_α ). The PDB codes, chain length, and the components (i.e., the number of residues categorized by hydrophobic scale and secondary structure) for all protein samples used in this present article are given in Table  1.

Table 1.

Table 1.

Table 1. PDB codes and structural information of globular proteins classified as all-α , all-β , α + β , and α /β types.   NO. PDB code Chain length NH NN NP Nα − helix Nβ − sheet Others   1 1ALA 316 106 77 133 220 0 96   2 1BBL 37 10 12 15 23 0 14   3 1BP2 123 44 18 61 67 10 46   4 1CCR 111 30 30 51 53 0 58   5 1EA8 140 46 26 68 127 0 13   6 1ECA 136 45 41 50 116 0 20   7 1ECD 136 45 41 50 116 0 20   8 1ECO 136 45 41 50 116 0 20   9 1F63 154 49 43 62 133 0 21   10 1FCS 154 50 43 61 121 0 33   11 1FHA 172 57 34 81 125 0 47   12 1FIP 73 27 14 32 59 0 14   13 1H96 167 55 37 75 135 0 32   14 1HBG 147 44 56 47 120 0 27   15 1IFA 158 66 21 71 91 0 67   16 1LE4 139 45 25 69 126 0 13   17 1LH1 153 53 41 59 126 0 27   18 1LH2 153 53 41 59 126 0 27   19 1LPE 144 46 29 69 128 0 16   20 1MBA 146 45 43 58 112 0 34   21 1MBC 153 48 45 60 121 0 32   22 1MBD 153 48 45 60 121 0 32   23 1MBS 153 46 44 63 121 0 32   24 1PPA 121 44 23 54 66 4 51 all-α proteins 25 1RCB 129 41 31 57 80 9 40   26 1RRO 108 30 21 57 57 4 47   27 1UTG 70 25 12 33 53 0 17   28 1YCC 108 30 31 47 52 7 49   29 1YEA 112 28 32 52 54 0 58   30 2C2C 112 31 33 48 59 0 53   31 2CDV 107 24 37 46 19 6 82   32 2CTS 437 158 104 175 315 7 115   33 2CY3 118 27 41 50 26 4 88   34 2CYP 293 93 62 138 155 16 122   35 2END 137 49 30 58 61 0 76   36 2FAL 146 45 43 58 112 0 34   37 2HBG 147 44 56 47 120 0 27   38 2LHB 149 50 39 60 113 0 36   39 2MHR 118 42 24 52 76 0 42   40 2PDE 43 14 12 17 22 0 21   41 2WRP 104 35 20 49 78 0 26   42 3C2C 112 31 33 48 59 0 53   43 451C 82 23 23 36 42 0 40   44 4BP2 64 25 11 28 39 0 25   45 4CPV 108 30 34 44 70 0 38   46 4ICB 76 24 9 43 56 2 18   47 4MBN 153 48 45 60 121 0 32   48 5CPV 108 30 34 44 60 6 42   49 5CYT 103 30 29 44 52 0 51   50 5TNC 161 50 32 79 106 10 45   51 1A45 173 63 30 80 7 62 104   52 1ACX 108 26 45 37 0 49 59   53 1CA2 256 81 58 117 41 59 156   54 1CD8 114 42 22 50 5 53 56   55 1CID 177 57 29 91 4 99 74   56 1EST 240 86 67 87 19 119 102   57 1F3G 150 55 33 62 29 74 47   58 1F53 84 29 16 39 0 32 52   59 1GPR 158 55 33 70 17 53 88   60 1H6X 159 53 47 59 0 82 77   61 1HJC 118 38 32 48 0 50 68   62 1HOE 74 25 24 25 0 50 24   63 1IFC 131 44 29 58 16 77 38   64 1KL9 165 62 21 82 87 33 45   65 1MPP 357 122 91 144 50 150 157   66 1NN2 388 130 82 176 11 188 189   67 1PAZ 120 42 28 50 21 54 45   68 1PYP 280 87 62 131 52 105 123   69 1QNY 237 74 59 104 18 108 111   70 1RBP 174 61 34 79 16 81 77 all-β proteins 71 1SGT 223 68 79 76 28 71 124   72 1STP 121 31 54 36 13 67 41   73 1TEN 89 26 21 42 0 48 41   74 1TIE 166 58 31 77 11 61 94   75 1TLK 103 35 21 47 4 61 38   76 1TON 226 82 51 93 29 71 126   77 2ALP 198 57 75 66 19 83 96   78 2APR 325 106 99 120 40 150 135   79 2AYH 214 72 61 81 13 102 99   80 2BB2 181 56 33 92 22 80 79   81 2BUK 184 62 49 73 28 82 74   82 2CA2 256 81 58 117 41 59 156   83 2CAB 256 75 59 122 48 72 136   84 2CNA 237 73 60 104 4 121 112   85 2ER7 330 100 115 115 34 184 112   86 2GCH 131 44 37 50 0 61 70   87 2MCM 112 31 50 31 0 93 19   88 2PCY 99 33 21 45 5 61 33   89 2REN 320 120 85 115 32 134 154   90 2RHE 114 32 30 52 8 51 55   91 2SGA 181 50 75 56 14 108 59   92 2SNS 141 40 36 65 34 32 75   93 2SNV 151 45 46 60 5 77 69   94 3APP 323 96 97 130 44 160 119   95 3CNA 237 73 60 104 0 112 125   96 3EST 240 86 67 87 19 120 101   97 4FGF 124 43 24 57 17 71 36   98 4PEP 326 118 78 130 42 156 128   99 5PTP 222 77 52 93 23 70 129   100 102L 163 53 41 69 124 14 25   101 125L 162 53 39 70 107 14 41   102 190L 162 54 41 67 108 17 37   103 1BKF 107 35 27 45 16 41 50   104 1E3V 128 46 32 50 45 44 39   105 1EZM 298 96 84 118 136 35 127   106 1FDD 106 41 15 50 47 12 47   107 1FKB 107 34 28 45 7 41 59   108 1FKD 107 34 28 45 7 42 58   109 1FKF 107 34 28 45 7 41 59   110 1FRH 106 41 15 50 44 12 50   111 1GWD 129 42 32 55 64 8 57 α + β proteins 112 1I1Z 130 45 31 54 63 8 59   113 1IET 94 25 23 46 9 6 79   114 1IEU 94 25 23 46 10 0 84   115 1LHI 130 45 32 53 44 6 80   116 1LZ1 130 45 31 54 44 6 80   117 1POP 212 76 52 84 58 47 107   118 1PPN 212 76 52 84 72 16 124   119 2LYZ 129 42 32 55 35 20 74   120 2LZM 164 56 38 70 107 18 39   121 3SSI 108 36 37 35 18 36 54   122 9PAP 212 76 52 84 58 42 112   123 9RNT 104 31 28 45 17 36 51   124 1ABA 87 32 17 38 36 16 35   125 1AZL 147 48 42 57 58 48 41   126 1C4W 128 47 31 50 65 21 42   127 1DHR 236 76 72 88 100 65 71   128 1E49 206 74 60 72 109 35 62   129 1E6K 130 47 31 52 69 24 37   130 1EAF 243 87 63 93 85 63 95   131 1EX7 186 57 39 90 97 39 50   132 1FX1 147 47 43 57 55 32 60   133 1Q21 171 60 36 75 70 43 58   134 1RHD 293 100 69 124 109 43 141 α /β proteins 135 1TFD 294 101 61 132 89 50 155   136 2FCR 173 56 42 75 58 42 73   137 2FOX 138 52 25 61 52 27 59   138 2FX2 147 47 44 56 58 48 41   139 3ADK 194 64 43 87 124 25 45   140 3CHY 128 46 31 51 50 24 54   141 3CLA 213 87 33 93 66 61 86   142 3DFR 162 54 46 62 40 60 62   143 3PGM 230 74 43 113 78 23 129   144 4CLA 213 87 33 93 65 50 98   145 5P21 166 58 36 72 61 44 61   146 8DFR 186 67 31 88 49 54 83

Table 1. PDB codes and structural information of globular proteins classified as all-α , all-β , α + β , and α /β types.

To minimize the running time during simulation processes on numerous protein samples and show the general properties of globular proteins instead of a specific one, the modified off-lattice minimalist HNP model, in which a single protein chain is described as a sequence of three types of beads H, N, and P, is selected in this article. The beads interact via the non-bonded potentials, and bonded potentials including bond-stretching, bond-bending and bond-torsional potentials. The coarse-grained force field used here is the same as that in the literature.^{[24, 32]} The potential energy functions adopted in our work can be thought of as a simple version used in the study of the structure and dynamics of real proteins. The four potential functions of a protein are as follows.

(I) The non-bonded interaction involves beads that are at least four residues apart in the amino acid sequence. For two beads, the non-bonded interaction is given by the truncated 12– 6 Lennard– Jones potential

where r_{i j} is the distance between beads i and j, ε is the parameter that defines the energy scale, ε sets the strength of the nonlocal interaction and Λ sets the nonlocal interaction that is attractive or repulsive. Both of these two parameters depend upon the nature of the two beads involved (see Table  2); σ is the length parameter representing the equilibrium bond length in Eq.  (2).

Table 2.

Table 2.

Table 2. Parameters ε and Λ in the non-bonded interaction term of the potential function depending on the nature of the two residues involved. H N P H ε = ε h; Λ = 1 ε = 2/3ε h; Λ = 0 ε = 7/12ε h; Λ = 0 N ε = 2/3ε h; Λ = 0 ε = 1/3ε h; Λ = 0 ε = 1/4ε h; Λ = 0 P ε = 7/12ε h; Λ = 0 ε = 1/4ε h; Λ = 0 ε = 1/6ε h; Λ = − 1

Table 2. Parameters ε and Λ in the non-bonded interaction term of the potential function depending on the nature of the two residues involved.

(II) The bond-stretching potential energy models the covalent bond connecting adjacent residues as a spring with an equilibrium value of σ rather than as a rigid rod. It takes the form

where r_i is the bond length between beads i − 1 and i. The bond-stretching potential energy constant is expressed as k_b = 100ε _h/σ ².

(III) The bond-bending potential energy models the harmonic potential to constrain the bond angle at equilibrium angle θ ₀ between three successive beads

where the equilibrium bond angle θ ₀ = 1.8326  rad and θ _i is the bond angle between three beads i − 2, i − 1, and i. The bond-bending potential energy constant is k_θ = 13.33ε _h· rad^{− 2}.

(IV) The bond-torsional potential energy describes the rotation of the three bonds that connect four successive residues in the amino acid sequence

where ϕ _α = 1.0  rad, and ϕ _β = π   rad, ϕ _i is the dihedral angle of two planes consisting of three successive bonds. The parameters A and B vary with different secondary regions of the protein. For α -helices: A = 6.0ε _h, B = 5.6ε _h. For β -sheets: A = 5.6ε _h, B = 6.0ε _h. For other cases: A = B = 0.

The same as the previous work, the simulation results are presented in terms of reduced units.^[33] Here, the reduced temperature T* = K_BT/ε , the reduced energy E* = E/ε , and the reduced time . A single protein chain exposed to a vacuum is simulated by the MD method. MD simulations are executed from the random configuration of the protein. The Nose– Hoover method is applied to keep the constant temperature of the system.^{[34, 35]}

Firstly, the ability of the three types of residues to form secondary structures, such as α -helix or β -sheet, is investigated based on 146 protein samples. The statistic results on the relationship between the residue types H, N, and P, and secondary structures such as α -helix, β -sheet or others, are listed in a 3 × 3 matrix (see Table  3(a)). A total of 23992 residues are found in 146 protein samples. The number of H, N, and P residues is 7915, 6026, and 10051, respectively. Then, the frequency of occurrence is easily obtained, i.e., H (33%), N (25%), and P (42%). In the HNP model, 20 amino acids are replaced by three kinds of residues. Hydrophobic residues (H) and hydrophilic residues (P) both consist of 8 types of amino acids, whereas neutral residues (N) have the half number of residue types compared with H or P. Based on the theory of statistics, the frequency of the occurrence of H and P should be equal, which is double that of N residues. It shows that the occurrence of N residues, which consists of only 4 kinds of amino acids, is common among the proteins. In 146 protein samples, 8586 and 5780 residues participate in forming α -helices or β -sheets, respectively. Both numbers are smaller than 9629 that the number of residues excluded from the secondary structure is neither α -helix nor β -sheet. More detailed data are shown in Table  3(a) and Fig.  1. For example, α -helical structures consist of 8586 residues, including 2964 H, 1955 N, and 3667 P residues, respectively. Among 5780 residues, which as the fundamental elements of β -sheets, H type presented a maximum value of 2665. These results conclude that P residues are found more frequently in an α -helix, meanwhile, H ones present the higher tendency to construct β -sheet secondary-structural elements. Previously, several groups have derived a helix propensity scale based on studies of the protein stability by experimental or simulation methods.^{[36, 37]} In contrast, the propensity for β -sheet formation of 20 naturally occurring amino acids is also measured in the literature.^[38] According to the helix propensities of amino acids achieved by Pace and Scholtz, the average value of helix propensities of H residues is 0.464  kcal/mol, which is higher than that of P type residues, i.e., 0.443  kcal/mol.^[36] Thus, P residues are favorable to the occurrence of α -helix secondary structures.

Table 3.

Table 3.

Table 3. Residual occurrence in the secondary regions based on all protein samples (a), and four types of globular proteins (b), respectively. (a) all protein samples H N P α -helix 2964 1955 3667 β -sheet 2665 1278 1837 others 2286 2793 4547 (b) all-α proteins H N P α -helix 1639 1099 1988 β -sheet 40 12 33 others 525 637 907 all-β proteins H N P α -helix 311 199 460 β -sheet 1836 1021 1339 others 955 1246 2126 α + β proteins H N P α -helix 442 281 524 β -sheet 307 74 181 others 392 486 714 α /β proteins H N P α -helix 572 376 695 β -sheet 482 171 284 others 414 424 800

Table 3. Residual occurrence in the secondary regions based on all protein samples (a), and four types of globular proteins (b), respectively.

Fig.  1.

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	Fig.  1. Histogram of Residual occurrence. Twenty residues are simplified as three types of beads, i.e., H, N, and P according to the MJ matrix in Ref.  [26]. The occurrences of H, N, and P are obtained from the statistical results in 146 protein samples listed in Table  1. The sequence and secondary structures of each protein can be downloaded from http://www.pdb.org.

The data were also analyzed according to four structural classes of globular proteins, i.e., all-α , all-β , α + β , and α /β proteins, and the results are shown in Table  3(b). For 50 all-α proteins, α -helices dominate at its secondary-structural level with 4726 residues, which is 55.6 times larger than ones that construct β -sheets. For all-β proteins, 4196 residues participate in the β -sheet folds. Although nearly 4.33 times larger than the number of residues forming an α -helix, it is even less than the number of residues, which construct other secondary structures except α -helices and β -sheets. In four such protein categories, except for all-β proteins, the residues forming α -helices always exceed the ones forming β -sheets. In α + β and α /β proteins, the difference of residue number between constructing α -helices and β -sheets is within 2.2 times, which is smaller than the condition of all-α or all-β proteins. The all-α and all-β proteins consist almost exclusively of helical and β -sheet secondary structures, respectively, whereas the α + β and α /β classes contain varying degrees of both secondary-structural elements. Helics and sheets are spatially segregated in α + β proteins. In the case of α /β proteins, α -helices and β -sheets alternate typically, allowing the β -sheets to organize in a parallel fashion. In these four matrices arranged by four protein categories, the number of P residues is always larger than that of H residues, although both of them consist of eight different amino acids. The number of N type residues is the minimum, as it just includes four different amino acids. However, the proportion of occurrence remains higher than 4/20. These results verify that the residual composition of proteins is independent of protein category. It totally conforms to the discussion in a previous paragraph.

The temperature dependence of the potential energy of each protein sample is investigated. Figures  2(a)– 2(d) show the simulation results of PDB code 1FCS, 2MCM, 9RNT, and 3CHY proteins classified as all-α , all-β , α + β , and α /β , respectively. The chain length, residual composition, and secondary-structural information are listed in Table 1. These figures show that: (i) the values of bond-stretching/bending potential energies are always larger than 0 under any simulation temperatures. Both energies are linearly increasing with the temperature in the four protein samples belonging to different protein categories. The values of bond-stretching and bond-bending potential energy approximate each other, especially when T* = 0.3. When T* > 0.3, the bond-stretching energy is slightly higher than the value of bond-bending energy, regardless of protein categories. On the contrary, when T* < 0.3 the bond-stretching energy is slightly lower than the one of bond-bending energy. It indicates that the slope of bond-stretching potential energy as a function of the reduced temperature is larger than that of bond-bending energy. (ii) Bond-torsional potential energy is the lowest one among the four components of potential energy in any simulation case. According to Eq.  ( 4), bond-torsional potential energy is invariably lower than 0. Bond-torsional energies increase initially and then decrease with increasing temperature. The difference of bond-torsional potential energies shown in the four figures is the transition temperature shifting between T* = 0.2 and T* = 0.3. (iii) The trend of non-bonded interaction with the change of temperature is opposite to that of bond-torsional energy, i.e., non-bonded interaction evidently decreases and then increases to a stable value when the reduced temperature remains escalating. In the four figures, the non-bonded potential energies are always lower than 0. According to Eq.  (1) and parameter values in Table  2, we can easily conclude that only the interaction between H– H residues present a prospectively negative value. In the other cases the residue pair interactions are always repulsive with different strengths. Therefore, H– H interactions perform the main functions in achieving the equilibrium state of each protein.

Fig.  2.

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	Fig.  2. The total and components of reduced potential energy versus reduced temperature T* of four types of globular proteins. (a) 1FCS classified as all-α , (b) 2MCM classified as all-β , (c) 9RNT classified as α + β , and (d) 3CHY classified as α /β protein, respectively.

From the perspective of protein categories, these figures also show that: (i) four categories of protein indicate the same tendencies of bond- and non-bonded potential energy with variable temperature. (ii) The transition temperature of a non-bonded interaction is the same at T* = 0.2. (iii) The difference among all-α , all-β , α + β , and α /β proteins is the transition temperature of bond-torsional potential energy. For all-α protein 1FCS and α /β protein 3CHY, the transition temperatures of bond-torsional energy are the same at T* = 0.3. In addition, in the case of all-β protein 2MCM and α + β protein 9RNT, the transition temperatures of bond-torsional energy are both at T* = 0.2. (iv) The change in total potential energy as a function of temperature is related to four parts of bond- and non-bonded potentials. Taking 1FCS as an example, figure  2(a) shows that the non-bonded potential energy of 1FCS is the main contributor during energy transition when 0.1 < T* < 0.2, because the total potential energy presents the same tendency as that of the non-bonded energy. In the temperature scope 0.2 < T* < 0.6, the total potential energy increases evidently with increasing temperature. The increasing tendencies are dominated by bond-stretching, bond-bending, and non-bonded interaction. When 0.6 < T* < 1.0, the total potential energy slightly increases with the temperature, and the major parts in the function of total potential energy are attributed to bond-stretching and bond-bending energies. According to the discussion mentioned above, we can conclude that the change in total potential energy with the system temperature corresponds to the competition between bond- and non-bonded interactions.

To obtain the general rules on the energy transition of proteins, MD simulations on 146 protein samples have been performed under 0.1 < T* < 1.0. According to Eqs.  ( 2) and (3), the bond-stretching and bond-bending potential energies are both linearly proportional to the protein chain length, and independent of secondary structure or protein category. Based on the view of thermodynamics, an amplitude of the variation of bond length/angle is quantified under the equilibrium state, i.e., or is independent of the protein at a certain temperature. Therefore, the total bond-stretching/bending potential energy of the protein chain is the summation of all bonds. In the present study, the bond-stretching/bending energies of each bond at a certain temperature are calculated by averaging simulation results on 50 all-α and 49 all-β proteins (protein No. from 1 to 99 listed in Table  1), and the statistical results are shown in Fig.  3(a). The figure shows a strong linear relationship between the bond-stretching/bending potential energy per bond and the reduced temperature. Based on the theorem of equipartition of energy, the average energy associated with each independent degree of freedom is linearly proportional to the system temperature, which can explain the straight lines in Fig.  3(a). On the other hand, the slope of bond-stretching energy per bond as the function of temperature is 0.53, which is larger than 0.45, which is the slope of bond-bending energy. At T* = 0.3, the values of bond-stretching and bond-bending energy per bond are exactly the same. This result is in good agreement with the previous discussion in Section 3.2, in which the bond-stretching and bond-bending potential energies of 1FCS, 2MCM, 9RNT, and 3CHY proteins are all the same at T* = 0.3 in Fig.  2.

Fig.  3.

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Fig.  3. Statistical and predicted values of bond-stretching and bond-bending potential energy. (a) Bond-stretching (• ) and bond-bending (▲ ) potential energy of each bond versus reduced temperature T* based on simulation results from 99 all-α and all-β proteins, (b)  simulated and predicted values of bond-stretching potential energy of 47 other proteins at T* = 0.5, (c) simulated and predicted values on bond-bending potential energy of 47 other proteins at T* = 0.5.

The bond-stretching and bond-bending potential energies of each bond as a function of temperature are obtained by the statistical results on 99 protein samples, which are independent of residue type and protein category. Therefore, the values of bond-stretching/bending potential energy of any protein can be predicted with a certain chain length. Figures  3(b) and 3(c) show the bond-stretching and bond-bending potential energy of 24 α + β and 23 α /β protein samples (protein No. from 100 to 146 listed in Table  1), respectively, by comparing simulated and predicted results at T* = 0.5. The extremely close values of bond-stretching/bending potential energy obtained by simulation or the prediction method are shown in these two figures. The correlation coefficient R is higher than 0.999, which indicates the nearly perfect agreement is achieved.

Similar to the analysis of bond-stretching/bending potential energy, the statistical result of bond-torsional potential energy per bond as a function of temperature is shown in Fig.  4(a). As the expression of bond-torsional potential energy is not the square of the independent degree of freedom, then the curve of bond-torsional energy per bond shows a nonlinear relationship with the temperature. Equation  (4) also shows that the bond-torsional potential energy involves the addition to the residues in its secondary regions. The parameters in Eq.  (4) are different according to the different secondary structures which the residues participate in. Here, the statistical results in Fig.  4(a) are obtained from 38 all-α and 9 all-β protein samples that their secondary-structural regions exclusively contain α -helices and β -sheets, respectively. In Fig.  4(a), the change of two curves corresponding to the α -helix and β -sheet with variable temperature is similar, i.e., the value of bond-torsional potential energy of each bond gradually increases to the maximum and then decreases with the increasing temperature. The differences between the bond-torsional energy per bond of α -helix and β -sheet include two aspects. First, bond-torsional energy per bond of the α -helix is lower than that of the β -sheet at any temperature, indicating that the α -helical structure is more stable than the β -sheet. Second, in case of the α -helix, the transition temperature corresponding to the maximum value of bond-torsional potential energy occurs at T* = 0.3, which is higher than that of the β -sheet at T* = 0.2. According to the statistical results shown in Fig.  4(a), the value of bond-torsional potential energy of any protein that even contains varying degrees of both secondary-structural elements can be predicted. For example, when T* = 0.5, the statistical values of bond-torsional energy per bond of the α -helix and β -sheet are − 3.939 and − 3.372, respectively. Then, the bond-torsional potential energy of each protein chain at T* = 0.5 is calculated as − 3.939N_{α − helix} − 0.372N_{β − sheet}. The values of N_{α − helix} and N_{β − sheet} are the number of residues as the elements of the α -helix and β -sheet, respectively. Figure  4(b) shows the bond-torsional potential energy of 24 α + β and 23 α /β protein samples, which include both helices and sheets at its secondary structure level at T* = 0.5. The values of bond-torsional potential energy by simulated and predicted methods are closely approximated to each other, and the correlation coefficient is R = 0.995. This result also verified the prediction is reasonable and effective in the study of bond-potential energy of proteins. The correlation coefficient R increases with the increasing temperature, which indicates the relatively small error between simulated and predicated bond-potential energy at high temperature.

Fig.  4.

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	Fig.  4. Statistical and predicted values of bond-torsional potential energy. (a) Bond-torsional potential energy of each bond with the residues constructing α -helices (▲ ) or β -sheets (▼ ) versus reduced temperature T* based on simulation results from 99 all-α and all-β proteins, (b) simulated and predicted values of bond-torsional potential energy of other 47 proteins at T* = 0.5.

The non-bonded potential energy involves two body interactions that are at least four residues apart in the protein sequence. In the HNP model, 20 amino acids are classified as three types by its hydrophobic property. According to the discussion mentioned in Section 3.2, the interactions between H– H residues contribute significantly in stabilizing the configuration of a protein. Then, the non-bonded potential energy of each protein sample under variable temperature is simply divided by possible H– H pairwise number N_H(N_H − 1)/2, and the results are shown in Fig.  5(a). Here, N_H is the number of H type residues of the protein. The curve of non-bonded potential energy per H– H pair as a function of temperature shows the same tendency in Fig.  2. The non-bonded potential energy of each H– H pair decreases when T* increases from 0.1 to 0.2. In the medium temperature range 0.2 < T* < 0.6, the values increase rapidly to − 0.01 with the increasing temperature. When T* > 0.6, the non-bonded potential energy of each H– H pair reaches a plateau with almost constant value. Figure  5(a) also shows that the non-bonded potential energy per H– H pair of all-α proteins is lower than that of all-β proteins, especially under the low temperature, which illustrated that all-α proteins present higher structural stability than all-β proteins. The non-bonded potential energy of 24 α + β and 23 α /β protein chains with the number of H components listed in Table  1 are also evaluated at T* = 0.5. The statistical results of non-bonded potential energy per H– H pair of all-α and all-β proteins under T* = 0.5 are − 0.0312 and − 0.0262, respectively. The total non-bonded potential energy of each chain at T* = 0.5 is calculated as

The comparative results are shown in Fig.  5(b). The value of correlation coefficient, R = 0.757, is consequently obtained. Comparing Figs.  3(b), 3(c), and 4(b), the numerical gap of the non-bonded potential energies between simulated and predicted results in Fig.  5(b) is relatively large. The correlation coefficient R as a function of temperature is also shown in Fig.  5(b). A lower value of non-bonded potential energy corresponds to a higher value of R obtained. The reason for the comparatively low value of correlation coefficient R mainly lies in disregarding other residue– residue interactions, such as H– N, H– P, N– P, N– N, and P– P. Besides H– H pairwise interactions, interactions between other residue types also partly contribute to the non-bonded potential energy, especially with the large value of potential energy under high temperature. Therefore, the higher non-bonded potential energy leads to the relatively large error.

Fig.  5.

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Fig.  5. Statistical and predicted values of non-bonded potential energy. (a) Non-bonded potential energy of each H– H pair versus reduced temperature T* based on simulation results from 50 all-α proteins (■ ) and 49 all-β proteins (• ), (b) simulated and predicted values of non-bonded potential energy of 47 other proteins at T* = 0.5 and the correlation coefficient R versus reduced temperature T* in the inserted figure.