5_torsion

5_torsion - 5. Torsion angles and pdb files In the study of...

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5. Torsion angles and pdb files In the study of space curves, the Frenet frame is used to define torsion and cur- vature, and these are used to describe the shape of the curve. A long molecule such as DNA or a protein can be thought of as a curve in space. Rather than being described by continuous functions, it is described by line segments which represent covalent bonds between atoms. The concept of curvature and torsion from differen- tiable curves can be adapted to study the structure of these molecules. Curvature corresponds to the angle between adjacent bonds, and torsion corresponds to the torsion angle discussed here. 5.1. Torsion Angles. In the study of molecular structure, torsion angles are fre- quently used to describe the shape of the molecule. In figure 1, we see four atoms p 1 , p 2 , p 3 , and p 4 . Think of the vectors p j as vectors giving the coordinates of Figure 1. Torsion angle φ = Tor ( p 1 , p 2 , p 3 , p 4 ). The angle is measured in the plane perpendicular to b = p 3 - p 2 . the centers of the atoms. Let a = p 2 - p 1 (1) b = p 3 - p 2 c = p 4 - p 3 . and let P a and P c be the projections of a and c respectively onto the plane per- pendicular to b . The angle, φ from - P a to P c , measured counterclockwise around b , is the torsion angle. Denote this angle as φ = Tor ( p 1 , p 2 , p 3 , p 4 ) . It is important to note that this angle is measured not between the two vectors - a and c , but between their projections onto the plane perpendicular to b . Since the torsion angle depends only on the vectors a , b , c also write φ = τ ( a , b , c ) . In this case the torsion angle is also called the dihedral angle . The angle is usually measured in degrees and chosen in the interval ( - 180 , 180]. 1
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2 The dihedral angle can be thought of as the angle between two planes (See figure 2). It is the angle counterclockwise from the normal vector a × b of the plane containing a and b to the normal vector b × c of the plane containing b and c . Both a × b and b × c are in the plane perpendicular to b MOLECULAR MODELING OF PROTEINS 411 α β p 1 p 2 p 3 φ p a b c F IG .3 . Bond vectors, bond angles, and the dihedral angle. specifying the position of the atom in space. If two atoms with labels j and k are joined by a chemical bond, we consider the corresponding bond vector r = x k - x j , with bond length k r k = p ( r, r ) , where ( p, q ):= p 1 q 1 + p 2 q 2 + p 3 q 3 is the standard inner product in R 3 .
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5_torsion - 5. Torsion angles and pdb files In the study of...

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