The importance of this problem to protein modeling and simulation should be clear: as stated earlier, the only internal degrees of freedom usually considered for a protein are its dihedral angles. Thus, moving a protein will be achieved by setting some of its dihedral angles to new values. For some applications, such as the rendering of an image of the protein and the computation of its Energy, however, the Cartesian (x,y,z) coordinates for each atom are needed. These are obtained by forward kinematics.
Mathematical background: matrices and transformations
The math involved in solving forward kinematics requires some background in linear algebra, specifically in the anatomy and application of transformation matrices. The links provided in this section should provide enough mathematical background to understand the rest of this module and eventually write a simple protein manipulation program.
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Transformation Matrices: The main transformations you will apply to polypeptide chains will be acombination of
translations and
rotations . Please see
introduction to translations and
introduction to two- and three-dimensional rotations . One special rotation matrix is the
Euler matrix . Please pay particular attention to the different conventions used for defining the Euler matrix. The one adopted for this module is the XYZ convention (there is also the ZXZ convention). Now that you know what an Euler matrix looks like, you need to get familiar with rotations about an arbitrary vector or line. Please read more on
rotations around an arbitrary vector .
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Homogeneous Transformations: The use of homogenous coordinates and transformations can simplify some of the calculations involved in using three-dimensional transformations. In particular, they allow
translation , which is not a linear operator in 3D, to become a linear operator in the 3D subspace (x,y,z,1) of a 4D space. The advantage of this representation is that translation becomes achievable by multiplying a vector by a matrix, and so becomes composable. A direct benefit from this is the ability to express, as a matrix, a rotation around an
arbitrary point , not just the origin as in the pure 3D case. See
homogenous transformations .
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Quaternions Quaternions are an efficient, robust method of representing three-dimensional rotations. In particular, they are not subject to the undesirable singularities and numerical instability of rotations represented by orthonormal matrices and Euler angles. Please visit this introduction to
quaternions to see how they relate to homogenous transformations. In this class quaternions will be used for the optimal structural alignment of two proteins and it is recommended that the reader familiarizes him/herself with the concept of quaternions as soon as possible.
A more detailed discussion of spatial descriptions and transformations can be found in chapter 2 of
. The most widely used transformations to manipulate protein chains are rotations. Several representations are possible for rotations:
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Euler angles : The orientation of an object is given as three rotations about set axes. For example, in the ZXZ convention, the angles specify a rotation about the global z-axis, followed by one about the global x-axis, and finally, one more about the global z-axis. The use of Euler angles is subject to an undesirable phenomenon called
gimbal lock , in which two of the rotational axes become aligned in such a way that a degree of freedom is lost.
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Cardan angles : The orientation is specified as a set of three rotations about axes defined by the object. The typical example is the pitch-roll-yaw set of rotations for an aircraft. Pitch corresponds to a rotation about the axis from wingtip to wingtip. Roll corresponds to a rotation about an axis from the nose to the tail, and yaw corresponds to rotation about a third "vertical" axis through the center of the plane, and roughly corresponds to a notion of horizontal heading. This method is also subject to gimbal lock.
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Axis-angle representation : It can be proven that any three-dimensional rotation can be represented as a single rotation about an axis, represented by a unit vector.
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Rotation matrices : A rotation matrix is an orthonormal matrix that represents a rotation. Rotation matrices are discussed later in the module. Applying the matrix to a vector yields the rotated vector. Given two rotations represented by matrices A and B, the result of applying both rotations in sequences is given by the matrix product AB.
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Unit quaternions : A rotation of angle theta about the axis represented by the unit vector v = [x, y, z] is represented by a unit quaternion. Quaternions are described in
this module .
