Here in MATH WORLD,
our focus will be pointed towards the mathematics which will ultimately help us have some fun at the level of very small things. The very small electromagnetic waves which we will need to deal with will require some Fourier math and the small atoms and molecules will need to be described in 3D coordinate sets.
We will plot them with the Python programming language;
Ultimately, we'd like to be able to show you the beauty and importance of structural biology by exploring the math and physics that underlie it all!
Quite a task!? Well, we have to start somewhere. With your help, we can contribute to the elucidation of protein chemistry whether it's benefitting the physician or the philosopher!
I am learning this in an out-of-school setting and will provide the resources that I have used.
def(A, B, C) = +/- volume of parallelepiped.
This section will attempt to explain the relationship between a circle and the sine and cosine functions.
If you look carefully at the code for the circle, the function xy() returns the sine and cosine of each value from 0 to 2 pi.
If you look carefully at the code for the SPHERE, the FILLER STUFF HAPPENSZZ
ALL ABOUT THE DOT PRODUCTTTTTTT
If vector A = (a1, a2, a3) and vector B = (b1, b2, b3) then A (dot) B = (a1b1 + a2b2 + a3b3)
The geometric interpretation of the dot product allows us to find angles between vectors.
We need to rearrange the equation A (dot) B = |A| x |B| x cos(theta)
to isolate the angle, giving us theta = the inverse cosine of [(dot product of A and B) / |A| x |B|]
A determinant is a matrix representation of one or more vectors. We use determinants
to add/subtract and/or multiply/divide vectors.
For example, in order to represent a magnetic field (which is a consequence of a moving charge and), we need to find the cross product of the velocity vector of the moving charge and the position vector between that charge and the point with which we are concerned (the point where the vector representing the magnetic field will be drawn). The cross product is calculated using determinants and is discussed in the CROSS PRODUCT tab.
def(A, B, C) = +/- volume of parallelepiped.
Exploring Structural Biology Techniques
Physics. The human body has in it a beautiful
instrument for detecting light. Our eyes, in conjunction
with our nervous system, have been evolving for 451 million years, and have been crafted into
a system that detects what you and I know as light. Light, the Almighty Light, which physicists have identified as
electromagnetic waves pervades and powers the world around us
and we, as humans, have molecules (see Box 1.0) in our eyes that respond to the waves with wavelengths of
anywhere between 400 and 700 nanometers.(Figure 1.0)
Since our visual systems are designed to interpret only wavelengths of this size, we can say that this is our
resolution. The wavelength of the light that is interacting with the observed object determines what resolution
is achievable. """We can see the chairs and trees around us just fine because the lightwaves
which we use is of a smaller length than tree."""
In order to see the atoms that make up proteins, we need wavelengths that are orders of magnitude shorter than visible light. (10,000 times shorter OR 5 orders of magnitude shorter OR 1x10^5 OR 1E5 shorter). Let's say that an atom is about .1 nanometers. When we gather data about atoms' positions in proteins we must use X-rays, whose waves are .01 to 10 nanometers in length.
Figure 1.0. Electromagnetic Spectrum
Box 1.0 Three-dimensional structure of bovine rhodopsin. The seven transmembrane domains are shown in varying colors. The chromophore is shown in red. Biology. The chromophore absorbs a photon of light resulting in a conformational change in the protein. This, through a series of interactions with subsequent proteins ultimately opens a channel in the membrane, allowing the influx of Na+ and Ca2+, which depolarizes the cell causing an action potential which is the means for communicating with the subsequent neuron on the signals path back to the visual cortex in the back of the brain, before it gets processed into what we call vision.
Our goal is to understand the math behind the electron density maps (Figure 1.1),
Figure 1. Representative part of the electron density replacement solution after one round of XYZ, B refinement using the simulated annealing protocol in Phenix. The contour level is 1 sigma. Rahman, MM
Please watch this great series from Dr. Paul Heiney at University of Pennsylvania!
Video 1. Dr. Paul Heiney explains sines, cosines, and the fourier transform. Please see supplemental interactive graph below.
Click !!!Here!!!to take a look at a cosine of 4x multiplied by a cosine of 5x.
Rahman MM, Germantsis DP, Machuca MA, Ud-Din AIS, Roujeinikova A. Crystallisation and Preliminary Crystallographic Analysis of Helicobacter pylori Periplasmic Binding Protein YckK. Crystals. 2017; 7(11):330.
Roland Deschain - Palczewski K, et al ("Crystal structure of rhodopsin: A G protein-coupled receptor." Science. 2000 Aug 4;289(5480):739-45
Video 2.. Some Fourier Transforms in One Dimension
Let's get to it!
Video 3. Fourier Transforms and Bragg's Law
Video 4. Crystal Structure and the Lattice
Video 5. The Reciprocal Lattice