# Mathematical support for insect colonies as superorganisms

The mighty insect colonies of ants, termites and bees have been described as superorganisms. Through the concerted action of many bodies working towards a common goal, they can achieve great feats of architecture, agriculture and warfare that individual insects cannot.

That’s more than just an evocative metaphor. Chen Hou from Arizona State University has found that the same mathematical principles govern the lives of insect colonies and individual animals. You could predict how quickly an individual insect grows or burn food, how much effort it puts into reproduction and how long it lives by plugging its body weight into a simple formula.  That same formula works for insect colonies too, if you treat their members as a collective whole.

Life is fundamentally about the use of energy, about effectively harvesting it from food and channelling it into existence and offspring. As animals get bigger, their changing use of energy ripples across all aspects of their lives. Because of economies of scale, larger and more complex animals need less energy for each individual cell. They grow and reproduce more slowly and they live longer.

The astounding thing is that this variety can be captured by a deceptively simple equation. An animal’s metabolic rate is proportional to its mass to the power of three-quarters (0.75). So a cat that is 100 times heavier than a mouse would have a metabolic rate that was around 32 times greater, and a human that is 10 times heavier than a cat would have a metabolic rate around 6 times greater. This beautiful three-quarters “power law” links all animals from mice to elephants.

Hou showed that it applies to insect colonies. He gathered data on over 168 species of social insects and noted the total mass of all their members. They ranged from species of fire ants whose colonies weigh little more than 2 milligrams, to African termite colonies that tip the scales at around 4kg.

Hou showed that the metabolic rate of these species is proportional to their mass to the power of 0.81. It’s close enough to the 0.75 figure as to be statistically indistinguishable. You can see on the graph below that the trend for the colonies is just an extension of the trend for loners.

The same number applies to how quickly the colony grows, and how much effort it puts into reproduction. For individual animals, the same 0.75 power law links the mass of their sex organs with the mass of the rest of their bodies. For colonies, the queens are essentially the sex organs and the workers are the “rest of the bodies” – plug in their masses and the magic number is 0.65, lower than expected but at least within the right ballpark.

Finally, Hou also showed that the life spans of both colonies and individual insects are also related to their mass, this time to the power of a quarter. However, if you take a colony and an individual of the same mass, the chances are the colony will live for around 4-5 times longer. It’s not clear why this happens but it certainly suggested that for insects, a social life is a longer one.

In any case, the striking parallels in Hou’s study shows that the life of a superorganism is lorded over by the same numbers that governs the life of a solitary insect. This could help to explain why these insect societies are so successful. They are incredibly efficient engines of existence, with lesser energy demands per head and longer lives.

Reference: Hou et al. 2010. Energetic basis of colonial living in social insects. PNAS http://dx.doi.org/10.1073/pnas.0908071107

More on social insects:

1. #1 Kathy Orlinsky
January 18, 2010

The formula for energy usage makes a lot of sense to me. I’m not so sure about the reproductive organ/total body ratio. For example, in bee species, are the queens of large colonies that much bigger than solitary queens?

Reproductive organ size seems to depend a lot more on reproductive strategies (how much sperm competition there is, or how many eggs must be made) than on body size.

2. #2 Janne
January 18, 2010

I’m reading “Zou no jikan, nezumi no jikan”: “Elephant’s time, mouse’s time” by Tatsuo Motokawa right now. It’s a fun, engaging overview of this field, how the size of animals determine their pace of life, metabolic rate, size of their organs and so on.

I don’t think it has been translated to English, but someone should; it’s fairly short and to the point, and it became quite popular in Japan when released.

3. #3 Guillaume
January 19, 2010

Had anybody been able to retreive the original article? I had no luck with the DOI nor with PNAS (early edition or not).

4. #4 Ed Yong
January 19, 2010

You’ll have to wait. There’s up to a week delay between embargo lifts on PNAS papers and the actual publication.

5. #5 Emory Kimbrough
January 19, 2010

Very interesting post – thanks for writing this up.

One nitpick: “…fire ants whose colonies weigh little more than 2 milligrams…”

2 milligrams sounds like the weight of an individual ant, not the weight of an ant colony, even the lightest. Indeed the graph shows the lightest colonies at roughly a gram, not roughly a milligram.

6. #6 Ed Yong
January 19, 2010

From the methods:

with colonies ranging in mass from ~0.0017 g for the ant Solenopsis morphospecies ditt…

0.0017g = 1.7mg right?

7. #7 Scilogue
January 20, 2010

Very interesting indeed, this makes me wonder about mega cities and efficiency of human life. Could we, for instance, form social institutions that function as sepcific “organs” of the mega organism?

8. #8 Brent
February 1, 2010

Er… 14 days later and the article is still not apparently published by PNAS… any clues?

9. #9 Ed Yong
February 1, 2010

Yes. PNAS is the clue. ðŸ˜‰

10. #10 Brent
February 1, 2010

Sorry – I’m too stupid to figure that clue out. If we are talking Proceedings of the National Academy of Sciences of the USA (which I assume we are), the DOI still does not resolve, and the paper does not appear on the website, either as a published article, or in the early edition (my previous experience is that it appears there at the time of embargo lifting… just gets its page numbers later).

11. #11 Ed Yong
February 1, 2010

To elaborate further, PNAS often has substantial gaps between embargo lift and publication. Can be less than a day. Can be more than a fortnight.

12. #12 Dallas Krentzel
February 7, 2010

You may be interested to know that there was a recent paper that came out that questions the validity of the 3/4’s law: http://www.sciencedaily.com/releases/2010/02/100203101124.htm

13. #13 Jeremy Lent
February 11, 2010

This is an interesting and powerful validation of the evolutionary theory of Alfred J. Lotka who wrote in 1922 that “in the struggle for existence, the advantage must go to those organisms whose energy-capturing devices are most efficient in directing available energy into channels favorable to the preservation of the species.”

Lotka called this the “fourth law of thermodynamics” and went on to apply the theory to human cultural evolution, describing the role of culture “to accelerate the circulation of matter through the life cycle, both by ‘enlarging the wheel,’ and by causing it to ‘spin faster.’