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.

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

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.

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

Biologists classify their understanding of dihedral angles into three categories:

Figure 3: Three repeating torsion angles along the backbone

Three different dihedrals