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Theory
ANM CGdeformation and Allatom Reconstruction
One of the most used function of ENM (ANM in this case) is to predict the possible conformations from experimentally solved structures along the slowest ANM modes. The deformed structures (R) can be created by adding sU_{k} [Eq. 8 of the ANM theory] to the atom coordinates (R_{o}) of xraycrystallography or NMRsolved proteins, where s = RMSD•N^{1/2} is a scaling factor defined by userchosen root mean square deviation (RMSD) value, the size of the deformation, and U_{k} [Eq. 8 of the ANM theory] is the kth ANM modes. R, U_{k }and R_{o} are all arranged as 3Nelement vectors where N is the number of CGnodes (e.g. C_{a} atoms in proteins) in the system.
Such ANMguided deformation sometimes can entail, especially when the size of deformation is big, overstretched (or compressed) pseudobonds (connecting two consecutive CGnodes in the primary sequence) in the deformers. The following algorithm is designed to (A) resolve the overstretched bond lengths between consecutive CGnodes and (B) reconstruct the allatom structures from the CGdeformers for the following docking or MD simulation purposes.
The reconstructed finegrained (allatom) structures from normalmode deformed coarsegrained (CG) structures (CGdeformers) and restoration of bond lengths in the equilibrium state from overstretched or compressed bond lengths between nodes in the CGdeformers take three steps –
(1) Detailed below, general (internal) coordinates are derived from the Cartesian coordinates of the CGdeformers, which are the dihedral angles described by four consecutive CGnodes, bending angles formed by three consecutive CGnodes and bond lengths between two consecutive CGnodes in the primary sequence. Let’s denote the derived set of general coordinates for R_{o} as F{R_{o}}. To resolve the overstretching issue, an attachment of the socalled “tip effect”(1), we use the original bond lengths in F{R_{o}} to replace the bond lengths in F{R} while preserving the dihedral and bond angles in F{R} (the CGdeformer), and then we convert the bondlengthadjusted F{R} back to its corresponding Cartesian coordinates to obtain R’ (Fig. 1). The conversion between Cartesian coordinate system and general coordinate system is described toward the end of the document, according to Flory’s formulation(2).
Figure 1. Scheme of the ANM deformation with fixed bond lengths. A.The input allatom structure. The blue, red and green balls represent the representative nodes (C_{a}) for a residue, sidechain and backbone atoms, respectively. B. The coarsegrained structure with CGnodes only. C. The nodes are deformed along the deformation vector sU_{k} with the scaling factor s. The red arrows indicate the direction and scaling of the deformation of nodes. Some of the bond lengths could be overstretched (C_{a } C_{a} distance between neighboring residues > 3.8 Å) in the deformed structure. D. The dihedral angles and bending angles are taken from the newly deformed structure while the bond lengths are restored using the original bond lengths of the input structure.
(2) The backbone atoms between 2 consecutive C_{a}'s (C_{i1} O_{i1} N_{i}) were rebuilt by superimposing the original structure to the ANM deformed structure with the fixed bond lengths. Every superimposition only includes 3 consecutive C_{a}'s. For every 2 consecutive C_{a}'s (C_{a,i1}, C_{a,}_{}), there are 2 ways perform this superimposition, C_{a,i2}, C_{a,i1}, C_{a,i} or C_{a,i1}, C_{a,i}, C_{a,i+1}. The superimposition with the lowest RMSD was selected. The translation vector and rotation matrix used to do the superimposition were then applied to the backbone atoms between 2 of the C_{a}'s (C_{a,i1}, C_{a,i}). (Fig. 2).
Figure 2. Reconstruction of the backbone atoms for the ANMdeformed coarsegrained structure. A. The backbone assignment for the deformed coarsegrained model. B. The ANM deformed structure with backbone atoms reconstructed.
(3) Finally, side chains are reconstructed by psfgen plugin in VMD (3) and an energy minimization is performed using NAMD (4) with CHARMM36 force field (5) to resolve the possible clash between spatially close atoms to render the final allatom structure.
Figure 3. The final allatom structure with reconstructed side chains.
Convert from Cartesian to General coordinates
Let’s define the bond length vectors l_{i}, pointing from node i1 to node i. The following equations are used to transform Cartesian into General coordinates. Bond lengths, bond angles and dihedral angles (see Fig. 4) can be obtained as
(1)
(2)
(3)
where n_{k} is the unit normal vector, the normal of the plane spanned by l_{k} and l_{k+1}, which can be obtained as n_{k} =(l_{k} ´ l_{k+1})/l_{k} ´ l_{k+1}; “´” is cross product while “” is inner product. Sign[x] is the sign (+ or –) of x.
Figure 4. Notations for a chain segment of four bonds used in the general coordinate system. i1, i, i+1, i+2 are 4 consecutive CGnodes.
Convert from General to Cartesian coordinates
Using the transformation matrix between frames i+1 and i per Flory’s convention(2), the Cartesian coordinates of the ith (r_{i}) node at frame i1 can be expressed as
r_{i }= T_{i1}(l_{i}, 0, 0)’; where ‘ is the transpose, and the transformation matrix T is
(4)
Hence the Cartesian coordinates of the r_{i} at frame 2 (the first frame), following a recursive relation, can be expressed as
r_{i }= T_{2}…T_{i2}T_{i1}(l_{i}, 0, 0)’ (5)
where frame 2 is spanned by x2 and y2 shown in Figure 5; In this frame, the first node has a coordinate (0, 0, 0) and the second node is at (l_{2}, 0, 0).
We can thus obtain the Cartesian coordinates of every CGnode at frame 2 (the first frame).
Figure 5. Transformation matrix to express node i +1 in the coordinate frame of node I with examples in the upperright corner. Coordinates of node 5, r_{5} = [x_{5}, y_{5}, z_{5}] = [l_{5}, 0, 0] in its own coordinate system; where l_{5} is the bond length.
References
1. Lu,M. andMa,J. (2011) Normal mode analysis with molecular geometry restraints: Bridging molecular mechanics and elastic models. Arch. Biochem. Biophys., 508, 64–71.
2. Flory,P.J. (1989) Appendix B. In Statistical Mechanics of Chain Molecules. Hanser Gardner Pubns.
3. Humphrey,W., Dalke,A. andSchulten,K. (1996) VMD : Visual Molecular Dynamics. 7855, 33–38.
4. Phillips,J.C., Braun,R., Wang,W., Gumbart,J., Tajkhorshid,E., Villa,E., Chipot,C., Skeel,R.D., Kalé,L. andSchulten,K. (2005) Scalable molecular dynamics with NAMD. J. Comput. Chem., 26, 1781–1802.
5. Vanommeslaeghe,K., Hatcher,E., Acharya,C., Kundu,S., Zhong,S., Shim,J., Darian,E., Guvench,O., Lopes,P., Vorobyov,I., et al. (2010) CHARMM general force field: A force field for druglike molecules compatible with the CHARMM allatom additive biological force fields. J. Comput. Chem., 31, 671–690.