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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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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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 Fall '11
 staff

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