Algae

Aristid Lindenmayer introduces a system of substitutions that would later bear his name. He did this in a set of two articles; Mathematical Models for Cellular Interactions in Development. As the title implies it offers a mathematical theory to model growth in certain type of cells. Stripping away the mathematics his article described an example that modeled algae. A more modern description follows below.

We will study words, i.e. a sequence of symbols, over an alphabet. In our example we take our alphabet to be the symbols A, and B. We will examine a series of words. In the algae example we are starting with the word A, it is our axiom. With the current word in the series, we replace each symbol with a sequence of symbols and concatenating the sub-sequences in a new word. In our example we will replace each occurring symbol A with the sequence AB, and each occurring symbol B with the sequence A.

Below you find the first few iterations of this process.

  1. A, our starting word.
  2. AB, because the single A is replaced with AB.
  3. ABA, because the single A is replaced with AB and the B is replaced with A.
  4. ABAAB, because each A is replaced with AB and each B is replaced with A.

In the above model the symbol A is a model for a mature cell, ready to divide itself. The division however is asymmetric. It allows the original cell to comfortable remain in place, making place for a young cell B. In its stead, the young cell B first must mature to and become an A cell before it can start reproducing itself.

Although this is a simple model. A lot can be learned from this. For an few examples, see the exercises.

Exercises

  1. Extend the example with the a few iterations.
  2. Count the number of symbols in each word of the series. Guess what number comes next.
  3. Count the number of A's and the number of B's separately. What do you get.
  4. We will number the words in our series, \(W_{0}\) for our start word, \(W_{1}\), for the next, \(W_{2}\) for the one after that, etcetera. Pick any number, let's say 4. Notice how \(W_{4}\), i.e. ABAAB is the concatenation of ABA, which is \(W_{3}\), and AB, which is \(W_{2}\). In other words \(W_{4}=W_{3}W_{2}\).
  • Check if something similar holds for \(W_{3}\) and \(W_{5}\).
  • Does this property, i.e. \(W_{k} = W_{k-1}W_{k-2}\) always hold?
  1. The above properties remind us of the Fibonacci sequence. For the Fibonacci sequence there is a formula that instantly calculates the the value in the sequences. Robert Dimartino found out that a similar property holds for our words. Search for Fibonacci word for more information.