Daniel B. Dix (Ph.D. in Mathematics, University of Chicago, 1988)
Analysis and Applied Math.
- Daniel B. Dix
Department of Mathematics
University of South Carolina
Columbia, SC 29208
- (803) 777-4320
- (803) 777-3783
Research interests include:
- mathematical chemistry, molecular quantum mechanics,
molecular geometry, molecular dynamics;
- protein dynamics and folding, statistical mechanics.
- mathematical cellular biology, systems biology.
- initial value problems for partial
differential equations governing the evolution of nonlinear waves,
asymptotic behavior of solutions, solutions with special symmetry,
completely integrable equations, and solitons;
Some papers, proposals, and preprints:
- Movie of neural tube folding using a simple model.avi
- Sharp Large-Time Asymptotics of the Correlation Integral
in the Suspended Flow over a Hyperbolic Toral Automorphism. pdf
- Polyspherical Coordinate Systems on Orbit Spaces
with Applications to Biomolecular Shape. pdf
- Biomolecular Modeling.pdf and
- Large-time Behavior of Functions Transported
along Chaotic Contact Flows.pdf
- Some Problems in Mathematical
Biochemistry.pdf and References.pdf
- Application of Iterated Line Graphs to Biomolecular
Conformation.pdf or Same thing.ps
- Mathematical Models of Protein Folding.pdf
- Large-time behavior of solutions of Burgers'
- Nonabelian Relative Cohomology and Bundles.pdf
IMIMOL: A Computer Program for three dimensional Z-system
(i.e. molecule) building, editing, manipulation, and
visualization, and for molecular geometry computations. IMIMOL allows
the user to visually organize a molecular system on a two-dimensional
canvas in a way which also allows its three dimensional geometry to be
fully specified. Other programs such as RASMOL or VMD should be used
to visualize the molecular system in three dimensions based on files
exported by IMIMOL.
IMIMOL was written by Scott Johnson at the Industrial Mathematics
Institute, Department of Mathematics, University of South Carolina. All
Berry Phase in the Molecular System H3. Master's
Thesis of Jialiang Wu. This thesis presents a
comprehensive treatment of the mathematical
foundations of the quantum mechanics of molecules, assuming the
Born-Oppenheimer approximation. In particular it
treats the molecular system H3, which consists of three protons and
three electrons. The mathematical formalism of quantum mechanics is
developed from first principles, and elementary examples of quantum
systems, such as spin systems, the hydrogen atom and molecule are
worked out as illustrations of the formalism. Molecular symmetry
groups and their
representation theory are also treated. The coordinate systems in the
shape space of three nuclei are studied. Vector bundles and connections
are developed and applied to the Berry-Simon connection in the complex
vector bundle over the shape space whose fibres are the energy ground
state eigenspaces. ``Berry phase'' is then interpreted as holonomy
associated to parallel translation in the
usual sense of differential geometry.
Building Geometric Models of Biological Molecules.
Master's Thesis of Haruna Katayama.
This thesis applies Z-system theory to a large biomolecular system,
namely the Light Harvesting Complex II (LH2) in the photosynthetic bacterium
Rhodobacter Sphaeroides. An elementary mathematical account of
Z-system theory is given and applied first to a simple molecular
system of sodium bicarbonate dissolved in water. The molecular biology
and the chemistry of the light harvesting complex are discussed from
first principles. Also, since there is no X-ray crystal structure of
LH2 in Rhodobacter Sphaeroides, the thesis discusses the process of
homology modelling whereby molecular structures can be transferred
between species sharing a high degree of amino acid sequence
similarity. The process of building a labelled Z-system for LH2 is
described in detail and the resulting structure is evaluated in terms of the
presence or absence of steric conflicts.
Stochastic Models of Gene Expression. Master's
Thesis of Lanjia Lin. This thesis presents a
detailed account of the mathematical theory of the stochastic processes
which are used to study gene expression in cells. The molecular biology
of transcription of genes and translation of mRNAs into proteins is reviewed.
A bipartite graph formalism, called a labeled Petri Net, is used to
describe any system of interreacting chemicals. This can be used to define
the corresponding State Graph, a directed graph with no loops and whose
edges are labeled with positive numbers. A continuous time Markov process
is defined which amounts to a random walk on the State Graph, where the
times between transitions are also random. The time dependent probability
distribution of this process satisfies a system of ordinary differential
equations called the Master Equations. The theory of the Master equations
is studied at some depth. A particular model of a self-regulating gene
circuit is formulated and some simple simulations are performed using Jarnac.
Classifying Three-Fold Symmetric Hexagons. Master's
Thesis of Amanda Gantt. This thesis studies the shapes
that a hexagon (not necessarily planar) can assume in three dimensional
space if all six side lengths and all six angles are specified. Actually
this thesis only does this if the six lenths are l0, l1, l0, l1, l0, l1
as one cycles around the hexagon, and the six angles are theta0, theta1,
theta0, theta1, theta0, theta1 as one cycles around the hexagon. This is
what is meant by a three-fold symmetric hexagon. Polynomial equations are
derived for the most general possible solution and explicit families of
solutions are found. One family forms a trefoil knot which has an axis
of three-fold rotational symmetry. The equations are derived and solved
using Maple worksheets, which accompany the thesis, and allow the reader
to view each type of hexagon in a rotatable three dimensional form.
Graduate Students Wanted! Some proposed research projects
are as follows.
- Mass flux equations associated to a Petri Net and the steady state solutions
of the associated Master equations, with applications to the control of metabolic
networks in living cells.
- Ph.D. Project: Singularities in the Born-Oppenheimer potential energy
function for the molecular system H3 near the Li atom limit.
- Ph.D. Project: The classical limit of the quantum dynamics of
nuclei subject to an electronic potential energy surface and a gauge
potential arising from geometric phase effects.
- Ph.D. Project: Coordinates systems in the shape space of flexible molecular
with applications to the dynamics of loop regions in proteins.
- Ph.D. Project: Large Motions in Protein Machines: Geodesics of the kinetic
energy Riemannian metric on shape space within the Rammachandran set.