11. Appendix 1: A Very Brief Linear ALgebra Review
Introduction. Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics. Very often in this course we study the shapes
and the symmetries of molecules. Motion of
10. HP model for Protein Folding
Protein structures can be investigated using energy functions and trying to nd
the native structure by looking for the conguration with lowest energy. Often
ignored in these computations are the facts that proteins obey la
9. Protein Folding
The structure of proteins is described on several scales. The primary structure
is the amino acid sequence which is coded in the gene for the protein. Secondary
structure describes the chain forming alpha helices, beta sheets and loops.
8. Discrete Frenet Frame
We saw in the previous section how distance geometry is useful in nding protein
structures from NMR distance data. When the sample can be held rigid in relation
to the magnetic eld, another method is used based on orientational co
7. Nuclear Magnetic Resonance
Nuclear Magnetic Resonance (NMR) is another method besides crystallography
that can be used to nd structures of proteins. NMR spectroscopy is the observation
of spins of atoms and electrons in a molecule that is placed in a m
6. X-ray Crystallography and Fourier Series
Most of the information that we have on protein structure comes from x-ray
crystallography. The basic steps in nding a protein structure using this method
are:
a high quality crystal is formed from a sample of
5. Torsion angles and pdb files
In the study of space curves, the Frenet frame is used to dene torsion and curvature, 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. Ra
4. Orthogonal transformations and Rotations
A matrix is dened to be orthogonal if the entries are real and
(1)
A A = I.
Condition (1) says that the gram matrix of the sequence of vectors formed by the
columns of A is the identity, so the columns are an or
3. Frames
In 3D space, a sequence of 3 linearly independent vectors v1 , v2 , v3 is called a
frame, since it gives a coordinate system (a frame of reference). Any vector v can
be written as a linear combination v = xv1 + y v2 + z v3 of vectors in the fram
2. Molecular Genetics: Proteins
2.1. Amino Acids. Proteins are long molecules composed of a string of amino
acids. There are 20 commonly seen amino acids. These are given in table 1 with
their full names and with one and three letter abbreviations for the
1. Molecular Genetics: DNA
This lecture reviews a few basic facts about genetics and DNA. There is a lot
of information available from the Department of Energy. They have a primer on
molecular genetics available on the web. This is a short course in some
Math Biophysics, Fall 2011
Homework 1
(1) Read about nding eigenvalues using Maple. Let
4 3 3 3
3
2 3 3
C=
3 4 3
3
6
6 6 5
Use Maple to nd a non-zero vector X such that CX = 2X .
(2) Use Maple to nd the inverses of the matrices
123
2
2i
M = 3 4 5 and N
Math Biophysics, Fall 2011
Homework 2
(1) The face centered cubic (fcc) lattice is generated by the basis vectors
(0, 1, 1), (1, 1, 0), (1, 0, 1),
which means it is the set of all vectors of the form
a(0, 1, 1) + b(1, 1, 0) + c(1, 0, 1)
where a, b, and c
Math Biophysics, Fall 2011
Homework 3
(1) Recall the delta function for the integer lattice Z can be approximated by
N
e2ikx .
k=N
Show the above sum is equal to
sin 2 N +
sin[x]
1
2
x
.
for x = 0. Hint: Use Eulers formula and the identity
2N
z k = (z 2N
Math Biophysics, Fall 2011
Homework 4
(1) Suppose G and M are real matrices with G = M M .
(a) Show that all the eigenvalues of G are non-negative.
(b) If M is a 3 n matrix show there can be at most 3 non-zero eigenvectors. (Hint: What is the dimension of