Is the whole more, or less than the sum of its parts?

If you have ever heard about emergence, you have likely come across the phrase “the whole is more than the sum of its parts”. The idea is simple. If a collection of small parts give rise to something with more complex traits, the system does more than one could have expected from any of the smaller parts. Here I discuss some simple computer models to illustrate how one could also argue for the opposite view: the whole is less than the sum of its parts.

Emergent mesoscale patterns

Biology knows many scales. A single nucleotide is approximately 0.6 nm wide, making the physical length of the average human gene (16,200 nucleotides) roughly 10 µm. It is however the information encoded in DNA that gives rise to our bodies, which according to this back-of-the-envelope calculation are seven orders of magnitude larger than the genes that give rise to them. How do such small “rules” scale up to create bigger patterns?

Here I discuss a few classical Cellular Automata (CAs), which I graciously borrowed from the lectures by Paulien Hogeweg at Utrecht University, to illustrate the concepts of self-organisation, emergence, and mesoscale patterns. I do this to illustrate how these patterns can be though of as constraints, resulting in a whole that is, in fact, less than the sum of its parts.

A classical example of emergent behaviour

Without getting technical, CA are a basically grids with different states, which change over time. Think, for example, of different coloured pixels, all having their own 8 (or 4) neighbouring grid points. Every grid point changes its state, based on these 8 neighbouring points. The most famous CA is Conway’s Game of Life (see animation below). The rules are simple:

  • If a grid point is dead (white), and 3 neighbours are alive (black), the grid point also comes to life.
  • If a grid point is alive, and it has 2 or 3 living neighbours, it stays alive.
  • If a grid point has less than 2, or more than 3 living neighbours, it dies.

Conway’s Game of Life, 1970

Looking at the behaviour of Game of Life, we can notice a few interesting phenomena. Most importantly, patterns and behaviour emerge that one did not explicitly define in the rules of single grid points. In the image above we see the classical glider gun, which produces bullets that traverse the grid. We do however not have a rule that describes the bullets. Instead, these mesoscale patterns emerge, and have dynamics of their own. These emergent rules and dynamics are why one might say that the whole more than the sum of its parts.

Majority voting

While Game of Life is a beautiful prototype for emergence (and very enjoyable to watch and play with!), it is hard to see this system as a model of something. Furthermore, even the slightest variation in the rules described above changes the behaviour strongly. To illustrate a CA which is much more robust to the details of the rules, let’s consider the following majority rules:

  • Count the number of black pixels around the grid point, plus the grid point itself (maximally 8+1)
  • If this is a majority (>4), the grid point becomes black
  • If this is a minority (<=4) the grid point becomes white

Below, on the left-hand side, it is shown what happens with these rules after a short amount of time. Locally, patches of black and white appear, and then everything stays that way. On the right-hand side, we show what happens on the long run if we slightly modify the rules (e.g. a small amount of noise, or making 0, 1, 2, 3 and 5 go to the white state, and 4, 6, 7, 8, and 9 go to the black state). Despite having clearly bigger patterns, the behaviour also is consistent with that on the left: we see the emergence of “blobs that agree”.

Colourful spiral waves

Of course, one is not restricted to merely black and white. CAs truly are a prototype system for simple local information processing leading to complex “emergent” behaviour. This is once more illustrated in the image below. In another simple CA where red replaces blue, green replaces red, and blue replaces green, the system organises into beautiful spiral waves.

Just.. Wow!

Spiral waves in a simple rock-paper-scissors CA, “kiwibonga” on gamedev.net, 2011

Spiral waves in Hypercycles, Boerlijst & Hogeweg 1991

More or less than just a “wow”-effect?

So, what does this all mean? Why am I showing you all these pretty pictures? Consider the relatively small system above, where 9 colours exhibit a spiral waves pattern. This study was done on a 300×300 grid (90.000 points). Given that every grid point can be 9 different colours, the system as a whole can adopt 990.000 possible states. However, the system self-organises in a very particular state. In fact, this hypercycle system by Boerlijst & Hogeweg (1991) adapts this particular pattern in a wide variety of conditions. The aforementioned majority vote also clearly adopts a limited subset of all possible patterns, and the same is true for Game of Life. In other words: these systems can be in many states, but emergence / self-organisation makes it so that only a distinct subset of these states are actualised. Therefore, we can learn to understand that the whole is less than the sum of its parts.

When reading into this topic I came upon a book about meta-materials, which deliberately caricatured the two opposing views of “more” and “less”.

From the table above, one could conclude that the view that the whole is less than the sum of its parts seems rather empty and bleak. Words like reductionist, classical, reduced, linear, and simple, hardly seem to spark the imagination. I have come to learn however that biology, with all its awe, can be best inspected through the lens of local interactions and these “constrained” emergent patterns, allowing us to understand it so much more.

Magic does it for you.


References:

  • Mitchel, Campbell Reece. Biology Concept and Connections. California, 1997.
  • Conway, John. “The game of life.” Scientific American 223.4 (1970): 4.
  • Hogeweg, P., Stoker, D., 2018. An overview of the lectures on bioinformatic processes
  • Boerlijst, Maarten C., and Paulien Hogeweg. “Spiral wave structure in pre-biotic evolution: hypercycles stable against parasites.” Physica D: Nonlinear Phenomena 48.1 (1991): 17-28.
  • https://www.gamedev.net/blogs/entry/2249737-another-cellular-automaton-video/
  • Zouhdi, S., Sihvola, A. and Arsalane, M. eds., 2012. Advances in electromagnetics of complex media and metamaterials (Vol. 89). Springer Science & Business Media.

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