Boogie Woogie

The Buffalo AKG Electric Op exhibit reminded me of an old folly.

Piet Mondrian

A Piet Mondrian exhibit at MoMA or the National Gallery in 1994-1995 helped me to understand a little bit about modern art; there’s a catalog available. The exhibit was arranged chronologically, so one could see Mondrian start with pictorial representations (e.g., The Weavers’ House, Winterswijk, 1899; p. 96 of the catalog), through impressionist (Mollen; Mill in Sunlight, 1908, p. 106) and cubist (Still Life with Gingerpot II, 1912, p. 130; Tableau No. 2; Composition No. VII, 1913, p. 146) renderings before landing on the geometrical representations (Composition A; Composition with Black, Red, Gray, Yellow, and Blue, 1920, p. 197) that define most of the last 20 years of Mondrian’s life. One of the last works in the exhibit was Broadway Boogie Woogie (1942-1943, p. 297) where Mondrian seems to relax his tight structure to introduce an almost dynamic and playful component; apparently the image is inspired by traffic and lights on Broadway in New York City, enlivened by the music of the blues.

The main theme I took from the exhibit was Mondrian’s desire to identify universal values that transcend reality. Thus simplifying representation to primary colors and geometric shapes, and then to assemble these colors and shapes into aesthetic meaning.

Population Genetics

Simplified representations are often used in mathematical models, and in particular in the ‘infinite alleles’ model describing genetic drift in population genetics. The infinite alleles model imagines a population of individuals, represented by the alleles at a particular genetic locus. Alleles experience mutation to a novel form. The fate of the allele (whether and how quickly it is lost from the population, for instance) is determined by purely stochastic processes.

There is a kind of dynamic equilibrium between mutation and drift. Often a single allele dominates, with a small number of recently introduced mutations at low frequency. Sometimes the dominant allele is replaced by a new allele. I imagined representing a population by a grid of squares, each square colored to indicate the allele. Genetic drift is visualized by updating the grid to reflect new mutation and sampling between generations. The dynamics of new alleles reminded me of flickering lights, and of Mondrian’s final paintings.

I explored simulations of genetic drift while teaching in the 90’s, and eventually included the art history and simulation in a single memorable (for me, anyway) lecture in a graduate Population Genetics course at Washington State University.

Boogie Woogie

It’s easy to simulate the infinite alleles model (literally two lines of R code, for instance), making it a fun use case for exploring new programming languages.

Several of the Electric Op works used the ‘Processing’ programming language for computer-based visualization. A modern implementation is in the p5 javascript library. A perfect way for me to learn a bit more javascript (my use of p5 is very rudimentary)! Here is the result:

The population is represented by a 100 x 50 grid of alleles, so 5000 haploid ‘individuals’. Initially all alleles are the same (random) color. Each generation introduces new mutations (new colors) at rate μ, and then sampling with replacement the entire population. The display is updated every 50 generations, at a maximum rate of 12 frames (50 x 12 = 600 generations) per second. Progress is reported in number of generations scaled by population size (e.g., t / N = 1 indicates 5000 generations). ‘Segregating’ alleles are just the number of alleles in the population at each generation. ‘Replacements’ summarize how many times a common allele has been replaced by a new allele.

There is a lot of activity in the display. New alleles flicker on and off fairly rapidly, perhaps existing for only a few frames. The display becomes quite animated when a couple of alleles become more equally frequent. But replacements are pretty rare, maybe once every 10 time units, and with a lot of variability. It’s tempting to feel that a new mutation that has become frequent will ‘win’, but actually the chance of winning is just the frequency of the allele in the population, independent of whether the allele has recently increased in frequency or is new, etc. Hopefully this provides some entertainment and maybe a bit of education.

Notes

See the companion Boogie Woogie Bubble post.

In addition to p5, I use the stdlib javascript library (to binomial random deviates, and to sample the population). I also needed to learn how to integrate a p5 canvas into the jekyll static site generator (basically, adding an ‘assets’ folder to the root of the site, with javascript scripts and libraries, and inserting <script> and <div> tags into the markdown for this page). Here’s the Boogie Woogie javascript.

Population genetics and WSU remind me of Richard Gomulkiewicz, to whom I dedicate this post.