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Unformatted text preview: 8. Discrete Frenet Frame We saw in the previous section how distance geometry is useful in finding protein structures from NMR distance data. When the sample can be held rigid in relation to the magnetic field, another method is used based on orientational constraints . This method uses the NMR spectrum to find the coordinates of the unit magnetic field direction in frames rigidly attached to the protein. A tool often used in calcu- lating protein structures from this information is the Discrete Frenet Frame (DFF). The DFF is also useful in the study of robotics and kinematics by transforming a rigid transformation into a sequence of simpler ones. 8.1. Discrete Frenet Frame. In calculus, a curve in 3D space is given by a vector function of a variable t . Using derivatives of this function the curvature and torsion can be computed in terms of a Frenet frame, a moving frame along the curve. Organic chemistry includes the study of long molecules such as a proteins and DNA. The backbone of a protein can be thought of as a sequence of points at the center of atoms rather than as a continuous function of t . Using differences rather than derivatives a Frenet frame can be defined which is useful in analyzing the shape of the protein, and in finding protein structures using NMR orientational constraints. 8.2. Definition of the discrete Frenet frame. A sequence of points v ,..., v n is called a discrete curve. To picture the curve, consecutive points are joined with line segments, which in chemical applications are thought of as chemical bonds. If no three consecutive points of a discrete curve are collinear, a sequence of orthonormal, right-handed frames F k = ( t k , n k , b k ) k = 1 ,...,n where t k = v k +1- v k | v k +1- v k | b k = t k- 1 t k | t k- 1 t k | n k = b k t k is defined called a discrete Frenet frame (DFF) for the curve. The unit vectors t , n and b are analogous to the tangent, normal, and binormal for continuous curves. The tangent vector t k is in the direction from v k to the next point v k +1 . The binormal vector b k is perpendicular to the plane containing t k and t k- 1 . Using the cross product formula a ( b c ) = ( a c ) b- ( a b ) c the normal vector can be written as n k =- t k- 1 + ( t k- 1 t k ) t k | - t k- 1 + ( t k- 1 t k ) t k | so it is in the plane containing t k and t k- 1 and perpendicular to t k ....
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- Fall '11