HOME
PAGE of Jack Quine
Charles W. McArthur Professor
of Mathematics,
Florida State University
Department
of
Mathematics
Director of Biomedical
Mathematics
Faculty Member of Institute
of Molecular
Biophysics, Florida State University
Faculty Member, NHMFL
(National High Magnetic Field
Laboratory
quine@math.fsu.edu
Research
Protein Structure and NMR
Crystallography and NMR (Nuclear Magnetic Resonance) are the two most
important experimental methods for finding the structure of
proteins. Analysis of crystallographic data requires finding the
coefficients of the Fourier series for the electron density
function. Analysis of NMR data relies on finding distances
between atoms (distance constraints) and the angles of bond directions
with the magnetic field direction (orientational
constraints). Distance constraints are used when the
molecules are in random motion, and orientational constraints are used
when the molecules are in a relatively fixed position in relation to
the magnetic field direction.
We research mathematics useful in analyzing orientational
constraints. One tool is the discrete Frenet frame (DFF)
which adapts the method of moving frames used to analyze continuous
curves to discrete curves consisting of a sequence of points (atoms) in
space. In analogy to the curvature and torsion in the
continuous case are the bond angles and torsion angles familiar to
chemists. Information about typical bond and torsion angle for
proteins is used together with orientational constraints to obtain a
structure.
Orientational constraints are obtained by using NMR data to find the
coordinates of the unit magnetic field direction B in molecular frames
rigidly attached to the molecule. The relationship of the
molecular frames to the frenet frames is known. It is
necessary to know the change of frame matrices between the different
frames used to interpret the NMR data. Here the DFF formalism is
useful in giving the change of frame in terms of bond and torsion
angles.
The DFF is also useful in modeling secondary structure in
proteins. The periodicity of the torsion angles gives
explicit formulas for the axis of helical structures such as the alpha
helix. This aids in finding the orientation of entire helical
segments and computing the relationship between them.
The analysis of orientational constraints is complicated by
degeneracies. These are of two types, chiral and quadratic.
Chiral degeneracies arise because the sign of cross products of vector
cannot be determined. Quadratic degeneracies arise because of
multiple solutions to quadratic equations used to solve for the
coordinates of B in a frame. This leads to multiple choices for
the correct structure.
FSU and NHMFL Collaborators:
Former Students
For a list of publications in this area
see my NHMFL
homepage