0.1 Representing proteins in silico and protein forward kinematics  (Page 2/7)

The internal degrees of freedom of a protein

The degrees of freedom of a system are a set of parameters that may be varied independently to define the state of the system. For example, the location of a point in the Cartesian 2D plane may be defined as a displacement along the x-axis and a displacement along the y-axis, given as a (x,y) pair. It may also be given as a rotation about the origin by θ degrees and a distance r from the origin, given as a (r,θ) pair. In either case, a point moving freely in a plane has exactly two degrees of freedom.

As mentioned before, the spatial arrangement of the atoms in a protein constitute its conformation. In the PDB coordinate file above, we can see that one obvious way to define a protein conformation is by giving x, y, and z coordinates for each atom, relative to some arbitrary origin. These are not independent degrees of freedom, however, because atoms within a molecule are not allowed to leave the vicinity of their neighboring atoms (if no chemical reaction takes place). Pairs of atoms bonded to each other, for example, are constrained to remain close, so moving one atom causes others connected to it to move in a dependent fashion. In the kinematics terminology, this means that the true, effective or independent number of degrees of freedom is much less than the input space parameters -an (x,y,z) tuple for each atom-. The remainder of this section defines a set of independent degrees of freedom that more readily model how proteins and other organic molecules can actually move.

Bonds and bond length

The atoms in proteins are connected to one another through covalent bonds. Each pair of bonded atoms has a preferred separation distance called the bond length . The bond length can vary slightly with a spring-like vibration, and is thus a degree of freedom, but realistic variations in bond length are so small that most simulations assume it is fixed for any pair of atoms. This is a very common assumption in the literature and reduces the effective degrees of freedom of a protein; the remainder of this module makes this assumption.

Although bond lengths will not be allowed to vary in this work, the presence of bonds is still important because it allows us to represent the connectivity of the protein as an undirected graph data structure, where the atoms are the nodes and the bonds between them are undirected edges. In some cases, it is helpful to artificially break any cycles in the graph, and choose an atom from the interior as an anchor atom. The graph can then be treated as a tree data structure, with the anchor atom as the root.

Bond angles

Bond length is an independent degree of freedom given two connected atoms. A set of three atoms bonded in sequence defines another degree of freedom: the angle between the two adjacent bonds. This is, appropriately, referred to as the bond angle . The bond angle can be calculated as the angle between the two vectors corresponding to the bonds from the central atom to each of its neighbors. As a reminder, the angle between two vectors is the inverse cosine of the ratio of the dot product of the vectors to the product of their lengths. Like bond lengths, bond angles tend to be characteristic of the atom types involved, and, with few exceptions, vary little. Thus, like bond lengths, this module considers all bond angles as fixed (again, this is a common assumption).