Matlab Functions:

pickCA.m
Retrives CA coordinates from a PDB file
pickN.m
Retrives N coordinates from a PDB file
make_contmap.m
Given a coordinate matrix, plot its contact map
trans_origin.m
Translate a coordinate matrix to origin

The HW2 Matlab Script:

%%%%%%%% Plot the coordinates and contact matrix  %%%%%%%%

coorCA = pickCA('1LAP.pdb');                    % Extract the CA coordinates
%plot3(coorCA(:,1), coorCA(:,2), coorCA(:,3));  % Plot the CA coordinates
make_contmap(coorCA);                           % Create and plot the contact map


%%%%%%%% Analyze distances between representations %%%%%%%

coorN = pickN('1LAP.pdb');          % Retrieve the N coordinates 
dist1 = norm(coorCA - coorN)        % Compute and output distance between coorCA and coorN

origCA = trans_origin(coorCA);      % Re-center the CA coordinates at 0.
origN  = trans_origin(coorN);       % Re-center the N  coordinates at 0.
dist2  = norm(origCA - origN)       % Output distance between re-centered matrices

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The Graphs:

Figure1
The 3D plot of the CA coordinates
Figure 2
The 3D plot of the CA coordinates
Figure 3a, Figure 3b, Figure 3c
Several zoomed-in plots identifying secondary-structure elements

Secondary Structure Analysis:

We can use the plot of the contact matrix to identify elements of secondary structure. Most of the information about the structural elements is contained near or along the main diagonal. However, because the protein we are considering is quite long, we cannot very well see what goes on along the main diagonal without zooming in on it (see Figure 2). Fortunately, Matlab has tools to do just that.

Once we have zoomed in on a region of the main diagonal, we can proceed to identify the structural elements. Alpha helices are compact structures, where several neighboring amino acids are packed close together. Therefore, in an alpha helix, we expect the alpha carbon i to have at least four contacts: i + 1, ..., i + 4. If we see a region along the main diagonal where position i has 4 or more contacts for several consequtive values of i, then that region indicates an alpha-helix. For example, see Figure 3a and Figure 3c.

One the other hand, Beta-Sheet strands are fairly spread-out structures, and so we don't expect their alpha-carbons to be packed closely. In a typical strand, each amino acid will have one or two contacts. In addition, when strands run parallel to one another in a beta-sheet structure, amino acids can wind up having contacts in remote positions, as illustrated below.


Therefore, regions along diagonal the main diagonal of the contact map where each position has 1 or 2 contacts are indicative of strands. In addition, continuous regions above the main diagonal, running either parallel (or anti-parallel) to it, are indicative of several paraller (or anti-parallel) strands of a Beta sheet. See Figure 3b for an example.

Distances between CA and N representations:

By running the matlab script shown above, we obtain the following results: 
dist1 =

   19.5119

dist2 =

   19.5069
As we can see, the distance between the CA and N representation has not decreased significantly, even though we translated both coordinate matrices to the origin. In order to decrease the distance further, we would need to rotate on the coordinate matrices until it overlaps the other.