**Theory**

** **

**ANM CG-deformation and All-atom Re-construction**

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 *s***U**_{k}** **[Eq.
8 of the ANM theory] to the atom coordinates (**R**_{o})
of x-ray-crystallography- or NMR-solved proteins, where *s* = RMSD•*N*^{1/2}
is a scaling factor defined by user-chosen root mean square deviation (RMSD)
value, the size of the deformation, and **U**_{k}** **[Eq. 8 of the ANM theory] is the *k*-th ANM modes. **R,**
**U**_{k }and **R**_{o} are all arranged
as 3*N*-element vectors where *N* is the number of CG-nodes (e.g.
C_{a} atoms in proteins) in the system.

Such ANM-guided deformation sometimes can entail, especially when the size of deformation is big, over-stretched (or compressed) pseudo-bonds (connecting two consecutive CG-nodes in the primary sequence) in the deformers. The following algorithm is designed to (A) resolve the overstretched bond lengths between consecutive CG-nodes and (B) reconstruct the all-atom structures from the CG-deformers for the following docking or MD simulation purposes.

The reconstructed fine-grained (all-atom) structures from normal-mode deformed coarse-grained (CG-) structures (CG-deformers) and restoration of bond lengths in the equilibrium state from overstretched or compressed bond lengths between nodes in the CG-deformers take three steps –

(1) Detailed
below, general (internal) coordinates are derived from the Cartesian
coordinates of the CG-deformers, which are the dihedral angles described by four
consecutive CG-nodes, bending angles formed by three consecutive CG-nodes and
bond lengths between two consecutive CG-nodes in the primary sequence. Let’s
denote the derived set of general coordinates for **R _{o}** as

**Figure ****1****. Scheme
of the ANM deformation with fixed bond lengths.** **A.**The input all-atom structure. The *blue*, *red* and *green* balls
represent the representative nodes (C_{a}) for a residue, sidechain and backbone atoms,
respectively. **B.** The coarse-grained structure with CG-nodes only. **C.**
The nodes are deformed along the deformation vector *s***U**_{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_{i-1} O_{i-1} 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,i-1}, C_{a,}_{}), there are 2 ways
perform this superimposition, C_{a,i-2},
C_{a,i-1}, C_{a,i} or C_{a,i-1}, 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,i-1}, C_{a,i}). (**Fig. 2**).

**Figure 2****. Reconstruction of the backbone atoms for the ANM-deformed
coarse-grained structure.** **A.** The
backbone assignment for the deformed coarse-grained 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 all-atom structure.

**Figure 3****.** The final
all-atom structure with reconstructed side chains.

**Convert
from Cartesian to General coordinates**

Let’s
define the bond length vectors **l**_{i}, pointing from node i-1 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. i-1,
i, i+1, i+2 are 4 consecutive CG-nodes.

**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 i-th (**r _{i}**) node at frame i-1
can be expressed as

**r _{i
}**=

(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
}**=

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 CG-node 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 upper-right 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 drug-like molecules compatible
with the CHARMM all-atom additive biological force fields. *J. Comput. Chem.*,
**31**, 671–690.