"Defining the Connnectivity of a Backbone Chain"

Figure 1 provides examples of some aminoacids, from long ones such as arginine to bulky ones such as proline, and to the smallest one, glycine. Note that as we have mentioned, while different in their sidechain atoms, all these aminoacids share a common group of atoms that includes N, CA, C, and O. The chain that goes through this shared core is known as the backbone. For the purpose of this module, you will consider as the backbone the chain that goes through N, CA, and C for every aminoacid. Note that there are two backbone dihedrals per aminoacid, with the number of sidechain dihedrals varying depending on the aminoacid. To simplify the rotation by dihedrals, in this assignment you will only manipulate protein structures from which the sidechain atoms have been removed. This means that you will only deal with a backbone chain that goes through the N, CA, and C of every aminoacid.

Figure 1: Sample Aminoacids

You should be familiar with how aminoacids connect to each other through the peptide bond. Figure 2 illustrates the peptide bond formed between the C of an aminoacid and the N of the following aminoacid. Consecutive applications of the peptide bond create the backbone chain. Recall that due to its double covalent nature, the peptide bond is planar (illustrated with the plane in Figure 2), and unrotatable. Therefore, in your manipulation of the backbone chain, you do not need to consider the peptide bond for rotation. You are left with two dihedral bonds per aminoacid.

Figure 2: Due to its double covalent nature, the peptide bond is rigid.

Figure 3 shows four consecutive aminoacids in a polypeptide chain. Note the peptide bonds that connect the aminoacids. You need to understand that rotation of a backbone dihedral will affect the location of all atoms following it due to the accumulation of rotations along the backbone. Therefore, it is necessary for you to define an orientation for the backbone. The easiest orientation is that used for the sequence of a protein, where the backbone is defined as the chain starting at the N atom of the first aminoacid of the sequence and ending at the C atom of the last aminoacid of the sequence. For your convenience, the native conformation supplied to you later in this assignment will list the coordinates of the atoms in the order N, CA, C for every aminoacid. So, reading in order from this file will give you the orientation of the backbone.

Figure 3: A series of applications of the peptide bond give rise to the polypeptide chain.

Before you proceed to the next section, make sure that you can answer these questions:

Dihedral Rotations

Now it is time to actually rotate dihedral bonds. Recall from our discussion of Forward Kinematics that one can define the transformation matrix/matrices to account for a dihedral rotation, depending on the representation method you choose (global or local coordinates, as discussed below). If you need to compute values such as bond lengths and bond angles, You can compute the bond length between two atoms as the Euclidean distance between them. You can also compute the angle between two bonds, the bond angle, as the the angle between two normalized vectors, which amounts to taking the dot product of the normalized vectors. If you need to identify one atom as your anchor atom, with the default backbone orientation, your anchor atom is the N of the very first aminoacid (the very first atom of the chain).

Now we will put our knowledge into practice. You will manipulate the structure in the pdb file backbone_native.pdb, which is the backbone of the 1COA structure of CI2. You may want to save this pdb file for later visualization in VMD.

     for i = 1:N
         for j = 1:3
             backbone_chain(i,j) = cartesian_coordinates((i-1) * 3 + j);
         end;
     end;
  

Conduct the following rotations:

If you wish to visualize the resulting conformations in VMD, you simply have to produce a file with the transformed coordinates. This file should be in ASCII (text) format and must have an empty/dummy first line, and all atom coordinates following (for example all in a single line, separated by spaces). You can save this file with ".crd" extension. Then, you can use the same "backbone_native.pdb" file, and use VMD's option "Load data into molecule" to load your new coordinates from the .crd file.

Ranking by Energy

Protein conformations are not only geometric objects but are characterized by energetic stability. You have seen that there are many empirical ways to compute the potential energy of a conformation. Functions such as CHARMM are very involved for the purpose of this assignment. Therefore, let us not worry about all the energetic terms to be considered from the atomic interactions. Consider only unfavorable interactions due to collisions between atoms, also referred to as steric clashes. Think about a simple energy function that checks for steric clashes. Your function should report high energies for conformations with collisions and low energies for collision-free conformations. You can model each atom as a sphere with a certain radius known as the Van der Waals (VDW) radius. Even though different atoms have different VDW radii, you can assume that all atoms have the same radius of 1.7 A. To assist you in devising a function that checks for collisions over all pairs of atoms, consider the following:

Figure 4: For all different atoms i and j, make sure that the distance between their centers C is more than twice their radiuses. In this simple example we are using the same radius R.

Again, as in dihedral rotations, you are encouraged to use Matlab. After you have the cartesian coordinates of the manipulated conformation, you can iterate over the atoms and check whether they are in collision with one another. You can do this with a double loop. Assume that the radius of every atom is the same, 1.7 A. In order to avoid reporting collisions for bonded atoms, you can consider only pairs of atoms that are 4 positions apart. That is, check atom i with atom i+4 for a possible steric clash.

For Submission

NOTE: remember that in order to open a .crd file in VMD (to attach it to a loaded structure) it has to contain a dummy first line. If your implementation writes out ASCII coordinates, remember to add an empty/dummy line at the beginning before opening with VMD. Please follow this list of deliverables closely: