The polypeptide is free swivel chain and has levers which collide to water molecules. The collision in the protein folding in the cell follows the Brownian movement. It is much stronger. The calculation of the translational molecular kinetics is based on the translational enthalpy data. Thus, the kinetic energy of the water molecule is,
And the velocity of water molecule gives,
1 calory is 4.1868 J (1 cal = 4.1868J), so
Hence, velocity of water molecule is
This velocity is fast enough to rotate backbone’s residue with torques(Serway, R. A. and Jewett, Jr. J.W., 2003; F. H. Stillinger, 1975).
Torsional inertia in the swivels are different for each amino acid. In a backbone, only one residue rotates, the others do not. While the former has small inertia, the latter has larger one. The difference of torsional inertia comes from the type and number of atoms and the length of side chain, and the mass and length of the backbone on both sides of the residue. The middle residue has a small rotational inertia, because the rotational inertia is proportional to the square of the distance.
The following(Figure 1) represents only two residues among many in the backbone. Among these connected residues, only two residues with the least rotational inertia were shown. Between them, A has smaller rotational inertia than B. When water molecules collide, A rotates first. B is a residue that stops while residue A rotates, and then returns as soon as A stops.
Residue A, has rotational momentum Pa is,
If B rotates, the rotation momentum Pb of B is,
When A does not rotates, the rotation momentum Pb of B is
If A rotates, the rotation momentum Pb of B is
If A stops rotation, Pa becomes zero and Pb is not reduced. So B can rotate more easily. If A rotates, Pa becomes larger and Pb is strongly reduced. Therefore, B cannot rotate easily.
In figure 2, the angle between the axel with A residue and B1 is 104.5°. Assuming that B rotates while A also rotates, residue A must handle this rotational movement of residue B. However, if it is 90°, because the radius of rotation(R = 1×R0) is the longest, the rotational inertia of residue B is affected to residue A. Since, residue A can hardly turn without the strongest force because rotational inertia(I = R2×m) is the maximum.
To receive the half of the magnitude of the force mentioned above, it must be 135 °. The radius of B2 is,
And inertia I is,
The bond angle between the two residues A and B is 104.5°. The inertia of B2 at 104.5° is,
This is almost the same as the value of 90° with the most difficulty. In conclusion, residue B2 cannot rotate with B1 at the same time.
In short, all of the above are summarized as follows. If there are many residues of different sizes, they do not rotate at the same time. The residue with the smallest inertia rotates first. As soon as it stops rotating, the residue with the next smallest inertia turns. When stops rotating, the next residue with more inertia than this rotates. In the case of multiple residues, the rotational inertia is ordered from the smallest.
The coordinates of short polypeptide with important points above was set. And it was compared with the structure from solution NMR. The cotranslational folding and torsional movement of atoms which are different from others was proved. The most representative model of each NMR assay was compared with the structure from the new folding algorithm. Typical structure alignment algorithms including TM-align(Y. Zhang and J. Skolnick, 2005) were not used in the comparison. These algorithms moves frame with insertions and deletions even in the case of the comparison of identical amino acid sequence. The result of the comparison was represented by logPr and RamRMSD(S. Jung, et al., 2011). And it was illustrated in the graph of torsion angle along the residue number(Figure 3). The change of the potential energy during the initial folding and following optimization is also displayed in the graph(Figure 4).
1. Data set
A single asymmetric chain structure was adopted in the PDB archive on the condition that the length of the chain was 8, the sequence identity was 90% or less, and that there were no heteroatoms. Structures which only contains protein without nucleic acids of DNA, RNA, or DNA-RNA hybrid was selected. Two structures of 1n9v and 1oeh was set. And 1oeh was abandoned because it was fragmented. Finally, 1n9v which is the angiotensin peptide was used in this work. ‘DRVYIHPF’ is the amino acid sequence of 1n9v.
2. Cotranslation Folding of Initial Structure
Cotranslational folding was performed with ProtTorter using torsion angles. Whenever a new amino acid was added, the potential energy was calculated, considering every conformation following the change of ϕ and ψ angles. As peptide bonds revolve around the backbone, we supposed they move 1 degree by 1 degree. So, 360 cases were observed. Local minima and the global minimum was gotten. These were to predict the initial structure of angiotensin.
3. Iterative Optimizations
This simulation became accurate by the peptide collision with water molecules, residues received larger torques, and planar amino acids to rotate faster. The length of the side chain was estimated as the maximum of bonds from the Cα atom. The order of calculation of the pair of bonds of the residue from the sum of the priority from torsional property.
The 8 amino acids of 1n9v were arranged into ‘D(7)R(2)V(8)Y(1)I(5)H(4)P(6)F(3)’. The smaller number was calculated first and received higher priority.
Consequently, the 4th peptide bond between Tyr(Y) and Ile(I) was first calculated(1+5=6), and the bond between Tyr(Y) and Val(V)(1+8=9). The order of priority is ‘D-(3)-R-(4)-V-(2)-Y-(1)-I-(6)-H-(7)-P-(5)-F’. The numbers in the round brackets designate the order of the linkage among residues. The dihedral angles flanking peptide bond are φ and ψ angles. Between these two dihedral angles, we calculated the one close to the higher priority first, and the other close to the lower priority later.
In this order, we iteratively optimized the initial structure from cotranslational folding. This was performed until the convergence of potential energy. The structure in which potential energy converged in six times from the initial structure was taken.