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Structural Organization of Molecules

Module by: Lydia E. Kavraki. E-mail the author

Summary: This module helps students to view the proteins as three-dimensional graphs. The graph representation of these macromolecules is crucial to understand the existence and the generation of different three dimensional structures of the same protein.

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Three-dimensional Structure

The spatial arrangement of aminoacids in a protein is known as a conformation. Every protein can have many conformations generated by rotation around bonds since bond length and angles are fairly invariant in the known protein structures. Among all these conformations a protein has a unique or nearly unique three-dimensional structure also known as a native conformation. The native conformation is the surviving conformation under specific biological conditions. The critical forces stabilizing conformations are noncovalent interactions. These are responsible for common structural patterns, through which we organize our understanding of protein architecture. The local structure of proteins is recognized by specific backbone torsion angles and specific main chain hydrogen bond pairings. Therefore, the key to protein folding lies in the torsion angles (also known as dihedrals) of the backbone. Since the dihedrals are mainly responsible for generating different conformations, they are also known as degrees of freedom (dofs). A protein that has nn rotatable dihedrals is also said to have nn dofs.

Bonds

The aminoacids in proteins are connected to one another through bonds. The bond lengths for different atom pairs are determined. By viewing the protein as a collection of atoms connected by bonds, one can picture the protein as an undirected graph where the atoms are the nodes and the bonds between the atoms are undirected edges. To ease the understanding of the protein as a graph, one can designate an atom to serve as the root of the graph. This atom can be chosen so that it is not part of a cycle. The root of the graph is named the anchor atom.

Figure 1: The root of the graph is a designated atom of the protein called the anchor atom.
Underlying Graph
 Underlying Graph  (protein_graph.jpg)

Bond Angles

Although the bonds are enough to give us information on how the protein is connected, this information is not enough. You will probably realize that the positions of atoms following the root are not fully determined by simply knowing the position of the anchor atom and the bond lengths of all different pairs. Two atoms can rotate with respect to a common atom they are connected to. This rotational degree of freedom is known as the bond angle. The bond angle is defined as the smallest angle between two consecutive bonds on the unique plane defined by them.

Dihedrals

Besides the bond angle, an extra rotational degree of freedom is given by the dihedral angles. A dihedral angle is defined by four consecutive atoms. Given four consecutive atoms connected by bonds b i 1 b i 1 , b i b i , and b i + 1 b i 1 , the dihedral angle is defined as the smallest angle between the projections of b i 1 b i 1 and b i + 1 b i 1 , on the plane perpendicular to bond b i b i . We say that the dihedral is on bond b i b i . The following figure illustrates this.

Figure 2
Dihedrals
 Dihedrals  (dihedral.jpg)

Protein Conformations

Biologists classify their understanding of dihedral angles into three categories:

  • φφ-dihedral on the bond between NN and C α C α
  • ψψ-dihedral on the bond between C α C α and C 1 C 1
  • ωω-dihedral on the bond between C 1 C 1 and NN
Figure 3: Three repeating torsion angles along the backbone
Three different dihedrals
 Three different dihedrals  (dihedral_aminoacid.jpg)
Now that we have classified the degrees of freedom of the protein as bonds, bond angles, and dihedral angles, it is not hard to imagine that one can generate different three dimensional structures of the same protein by varying the values of these varibles. However, in order to avoid unrealistic 3D structures one cannot stretch bonds too much. Many other constraints are added by the fact that the atoms are not inanimate nodes but in fact can attract each other or repel each other. φφ, ψψ, and ωω cannot take any values on the range 0 π 0 π . Certain conformations are not possible since atoms do not like to clash into one another, get too close or too far away from one another (remember that during folding the covalent bonds should not be broken). It has been biologically observed that omega tends to take two different values: 0 or 180. Conformations where omega is 0 are known as cis conformations, whereas conformations where oomega is 180 degres are known as trans conformations. Out of all the 32,539 angles in 154 X-ray structures, onnly 116 (0.36 %) are found to be cis (Stewart et al. 1990). Therefore, since trans is generally favored over cis, this leaves phi and psi to be mainly responsible for the folding of the chain. However, steric conflicts limit even these angles as well.

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