Cube-Sphere Intersection Volume

This week's math problem was particularly difficult. I wanted to know the intersection volume between a cube and a sphere. Imagine a sphere contained in a cube. The sphere will first touch the cube when its radius is equal to half the width of the cube. Until that point, its volume is just the formula for the volume of a sphere. Once the sphere's radius is 2 sqrt(3) times the width of the cube, then it completely engulfs the cube and the intersection of the 2 objects is just the volume of a cube. In between, I've found it looks like this:

Bacteria in Blender

I made this while drinking my morning coffee. I just stretched a UV Sphere, stuck two layers of hair on it (flagella, cilia) and added a few modifiers to get it bumpy. The texture is just some glass, Voronoi textures, and diffuse shaders. Meh, it's OK.

Generating n-Dimensional Lattice Coordinates

It comes up sometimes in my research that I need to generate coordinates along a lattice. For example, when generating an initial structure for a molecular dynamics simulation. I thought I share my code for this. The code works by recursively enumerating each dimension in a lattice and calling a given function each time it reaches a point. So, it could be used to execute any function along a lattice. Here's what it looks like for a 2D and 3D example. In the 3D example I didn't choose a cubic number of points on the lattice, so that it sort of stops in the middle.

Category Colors in R

I thought I'd write down the color palette I use for categories. I've seen many discussions about gradient color choices, but category colors are less often discussed. I have a function given below which generates colors where there is not meant to be an ordering to the color. It is important when data points are near, that they are easy to distinguish with the eye. It makes more sense to see it:

ggplot Style in Classic R Plots

ggplot is a highly acclaimed R package for plotting. I have had very little experience with the library because I've mostly memorized all the quirks of normal R plots. I like the ggplot default style though, so I thought write down how to replicate it in R plots. The distinctive feature of ggplot is the gray rectangle and white grid lines. This may be replicated like so:



Here's an example

DNA-Polymer Nanoparticle

DNA-Polymer nanoparticle are used to deliver DNA to cell nuclei for curing diseases at the genetic level. In my research group, we work with carboxy-betaine acrylamide polymers. I modeled an exceptionally large DNA-polymer nanoparticle below. The polymer is 80 kDa and the DNA is a single strand of 65 base pairs. After condensing it (in blender, not a true physical simulation), the nanoparticle comes out to about 15nm in diameter. 

The Accuracy of Ghose-Crippen ALogP for Peptides

I was using the Ghose-Crippen ALogP predictor implemented in CDK as a QSAR for looking at peptide solubilities. Unfortunately, this partition coefficient predictor is fit on small organic molecules and does a pretty bad job at predicting peptide partition coefficients. Here's what it looks like on single amino acids compared with experimental data.

Visualizing Size in Biochemical Systems - Update

After some helpful suggestions, I've revised the previous image I made. This time I've used the atom representation for the proteins and peptides, so atoms are used everywhere for visualization and they have the correct radii. Doesn't look as cool, but it is more consistent.

Visualizing Size in Biochemical Systems

I decided to waste a few hours and make a Blender image based on my post on the relative size of common elements in biochemical systems. Using 1 blender unit = 1 nm, I modeled a gold nanoparticle, an antibody, albumin, an iron oxide nanoparticle, a 10 amino acid length peptide, and salicylic acid.

Relative Size of Proteins, Nanoparticles, Antibodies, Peptides, and Drugs

Needing to know the relative size of these elements in biochemical systems often comes up in my research, so I've decided to write them.

Component	Size [nm]	Reference
Gold Nanoparticle	~20	Can vary, most common is 20. Frens (1973)
Iron Oxide Nanoparticle	5-15	Can be much larger with emulsion methods. This size is most common when magentic properties are desired. Called partial reduction coprecipitation method.
Antibody	15.5 x 11	PDB: 1igt measured in VMD
Human Serum Albumin	7.6 x 7	PDB: 1e7i measured in VMD
10 Amino Acid Peptide	~1.5 (coiled globule sphere)	Paper I wrote on Peptide Conformations
Salicylic Acid (drug)	0.6 x 0.5	PubChem structure measured in VMD

Approximate Restricted Integer Partition With Exact Part Number

I'm interested in a certain type of mathematical structure this week: integer partitions. An integer partition, p(n), is the set of sets of integers that add up to n. For example,

• p(3) = 1 + 1 + 1, 1 + 2, 3
• p(4) = 1 + 1 + 1 + 1, 1 + 1 + 2, 2 + 2, ...

Speeding up R

I've been programming in R for a while now. Whenever I've had performance problems, it almost always is due to data.frame usage. To check what's slowing down your R code, just use the Rprof command like so:

Changing Image Size in Tachyon in VMD

I always forget how to change the image size for rending molecular images in VMD using the (external) Tachyon rendering engine, so I thought I'd write down the steps here:

In the TCL console, get the aspect ratio correct for the image. So, for example to make a nice 1920x1080 image, we'll make the view at half of that:

 
display resize 960 540

Now, add simply add -res 1920 1080 to the end of the render command in the file render controls window. Here's an example of one of my render commands:

 
"$HOME/VMD/tachyon_LINUXAMD64" -aasamples 12 %s -format TARGA -o %s.tga  -res 1920 1080

Here's the result of one my renders:

Making nanoparticles in Blender

Today I made some nanoparticles in Blender. I used the node compositor to create the fluorophore effect. Making science in Blender is fun!