abstract tile assembly model

The DNA-based computer that can calculate π

Vasudevan Mukunth

— 3 min read

I’m not fond of biology. Of late, however, it’s been harder to avoid encountering it because the frontiers of many fields of research are becoming increasingly multidisciplinary. Biological processes are meshing with physics and statistics, and undergoing the kind of epistemic reimagination that geometry experienced in the 19th and 20th centuries. Now, scientists are able to manipulate biology to do wondrous things.

Consider the work of a team from the Dhirubhai Ambani Institute of Information and Communication Technology, Gujarat, India, which has figured out a way to compute the value of π using self-assembling strands of DNA. Their work derives from previous successful attempts to perform simple mathematical calculations by nudging these molecules to bind to each other in specific ways, a technique called tile assembly.

It was first formulated as a tiling problem by Chinese philosopher Hao Wang in 1961. Wang wanted to know if a set of square tiles could cover a plane in a periodic pattern if each tile had four different colored edges and only edges of the same color could abut each other. The answer was that they could cover a plane but only with an aperiodic pattern.

In a DNA tile assembly model (TAM), each tile represents a section of the DNA molecule, called a monomer. When adjacent tiles’ abutting sides line up with the same color, then the two monomers attach themselves across the abutting sides according to a strength corresponding to that color. This way, given a tile to start with – called the seed tile – and a sequence of tiles coming up next, the DNA monomers can link up to form diverse patterns.

By controlling the sequence of colors and their strengths, scientists can thus use TAM to control the values of variables moving through the resultant grid. Connections of monomers between tiles can be made become stronger or weaker, and to different extents, in ways mimicking how the voltage between different electronic components in a computer’s circuit allow it to perform mathematical calculations.

So, Shalin Shah, Parth Dave and Manish Gupta from the Institute used four new variations of TAM that they’d developed to calculate the value of π. Each of these variations performs a specific function, much like the logic gates inside an information processor.

  1. The compare tile system decides which number is greater between two numbers, or if they’re equal
  2. The shift tile system shifts the bits of a number by one bit to the right, and adds a 0 to the leftmost bit. For example, 11001 becomes 01100.
  3. The subtract and shift tile system subtracts one binary number from the other, then right-shifts its bits by one bit to the right, and finally adds a padding 0 to the leftmost bit
  4. The insert bit tile system inserts a bit in a number

Using a combination of these systems – all with the TAM at their hearts – the trio has been able to compute the value of π like below:

The gray tiles are input tiles, green are addition/subtraction tiles, yellow are copy/duplicate tiles, orange tiles are shift tiles, and blue tiles indicate the remainders of the corresponding division process. Image: Computing Real Numbers using DNA Self-Assembly, Shah et al, Laboratory of Natural Information Processing, DAIICT.
The gray tiles are input tiles, green are addition/subtraction tiles, yellow are copy/duplicate tiles, orange tiles are shift tiles, and blue tiles indicate the remainders of the corresponding division process. The calculation is growing upward and toward the right. Image: Computing Real Numbers using DNA Self-Assembly, Shah et al, Laboratory of Natural Information Processing, DAIICT.

You can see that the calculation is an ongoing infinite series – specifically, the Leibniz series, which estimates π as an infinitely alternating sequence of additions and subtractions between smaller and smaller fractions. Because it is infinite, the trio’s calculator’s ability to find a more precise value of π depends only on how many tiles are available. Second, because the calculator can compute infinite series, any number or problem that can be reduced to the solution of an infinite series is now solvable using this calculator.

This would merely be a curious yet tedious way to calculate if not for its potential to exploit the biological properties of DNA to enhance the calculator’s abilities. Although this hasn’t been elaborately outlined in the trio’s pre-print paper on arXiv, it is plausible that such calculators could be used to guide the development of complex and evermore intricate DNA structures with minimal human intervention, or to fashion molecular logic circuits commoving microscopic robots delivering drugs within our bloodstreams. Studies in the past have already shown that DNA self-assembly is Turing-universal, which means it can perform any calculation that is known to be calculable.

The DNA molecule is itself a wondrous device, existing in nature to store genetic data over tens of thousands of years only for a future inheritor to slowly retrieve information essential for its survival. Scientists have found the molecule can hold 5.5 petabits of data per cubic millimeter, without letting any of it become corrupted for 1 million years if stored at -18 degrees Celsius.

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