Do energy costs affect the size of colour-changing animals? (Shadeshifting II)

This post is not the second post about shadeshifters that I mentioned in the first post on that subject. In case you missed that first one, it was about colour-changing animals; I reserved the word ’shadeshifters’ for animals that can change colour in seconds, minutes, or a few hours at most. The post asked the question whether shadeshifters elsewhere in the universe might be larger than the ones we have on Earth. The promised second post was to be about possible workarounds to allow large shadeshifters. Well, that will -probably- become the third post on this subject.

This new second post will be about the energy costs of colour changing, also called metachrosis. I had mentioned costs aspects without going into detail, for the simple reason that papers on the subject did not provide measurements of the energy bill of colour change.

What would a colour-changing system cost? If such a system on another world is similar to ones on Earth, four cost factors can be distinguished. The first factor is a ‘visual environment analysis centre’; this brain part makes use of an already existing good visual system and analyses environmental visual information in terms of colour, contrast, contours, etc. It then makes a sort of recipe that describes the visual environment. The second part, the ‘body mapper’ is also a brain centre, one that translates the recipe into a map fitting the animal precisely, holding instructions for all colour-producing skin cells (chromatophores). The third factor is the cost of firing neurones that  run from the ‘mapping centre’ to every chromatophore, turning it on or off. The fourth factor deals with the chromatophores themselves: spreading or concentrating pigment granules must also cost something. 

The first three factors have to do with neurons, and on Earth neurons are expensive (for example, the human brain requires some 20% of all blood pumped out by the heart while having a mass of only about 1.5% of the body). I do not know whether neurons on all planets will also be expensive, but let’s assume so. Mind you, if neurons would be really cheap somewhere, an organism would not be punished for throwing brain power at any problem; would that allow a runaway evolution of intelligence? In reverse, are we perhaps hampered by overly expensive neurones? If so, we are perhaps more stupid than the rest of the universe; no wonder they don’t come visiting...

Anyway, back to shadeshifting. Lacking facts, we can still think about whether animal size would affect these costs. Let’s assume that a shadeshifter needs four layers of chromatophores of different colours to produce a wide range of colour effects. I see no need for a large animal to thicken each of these four layers. Whether the colours are visible will depend on the dead tissue above, not on the thickness of the colour-producing layers. A skin area of 10 square centimeter would therefore contain the same amount of chromatophores for a small as for a large animal. In other words, the total number of chromatophores and the number of neurons to turn them on and off are linearly related to the skin area of the animal. If a large animal has three times the skin area of the small one,  it will have three times the number of chromatophores. Note that we just concluded that the third and fourth cost factors linearly depend on skin area. 

Click to enlarge; copyright Gert van Dijk

Aha! Now we are back on familiar ground, meaning ‘scaling’. ‘Scaling’ has been discussed on this blog many times (***). Say we have a chameleon with a length of 25 cm; in other words, L=25. We increase its length, width and height all five times, so the animal become 5 times longer, wider and higher; its length will be 5L=125 cm. Now, surface area scales as a square, so the animal’s skin area becomes larger by the square of 5, meaning it becomes 5^2=25 times larger. The third and fourth cost factors of our larger shadeshifter are therefore 25 times those of the small one.

By itself this does not mean that much. A large animal would in general obviously have higher energy costs than a small one. The key question is how the costs of shadeshifting relate to the animal’s overall energy budget. To answer that, we need to know how metabolism scales with size.

First, we need to realise that each kg of animal will need an amount of energy, so we need to think about the mass of the animal. Mass is related to volume, and volume is easy to work out; it scales with the third power. Our super-chameleon, 5 times the length, width and height of the small one, has 5^3=125 times the volume of the small one, and therefore also about 125 times the mass.

You might think that the energy needed to sustain one kg of animal is the same for a large and a small animal. If that were true, then the larger animal, 125 times the mass of the smaller one, would also have 125 times more energy to spend. That would be very nice for shadeshifting, because the costs of shadeshifting would only become 25 times larger. Brilliant!

But no. It doesn’t work that way.

It actually costs more to sustain one kg of rabbit than one kg of elephant. Small animals have relatively high energy costs, which means that small animals run more quickly out of energy than large ones (this is why mice need to eat often). Mind you, the effect is relative: in total, an elephant of course needs more energy than a mouse; it’s just per kg than the elephants needs less energy. The relationship of metabolism to body mass is well known in zoology:

        Minimal metabolic rate (MMR) = a M^0.75

‘Minimal metabolic rate’ means the animal does nothing besides being alive. The constant ‘a’ differs between groups of animals, and we can forget that for now. The exponent of 0.75 is less than one, which is another way of saying that large animals use relatively little energy. Can we now work out what scaling by length (L) does, rather than by mass (M)? Yes, we can:  mass relates to volume and volume was length to the third power:

        mass (M) ~ volume ~ L^3

We can now replace ‘M’ in the first equation with ‘L^3’, giving:

        Minimal metabolic rate (MMR) = a (L^3)^0.75 = a L^2.25

This is the equation we need. It tells us what happens to MMR if we make the length of an animal 5 times larger. MMR then becomes 5^2.25 times larger, which is 37.4 times larger. Remember that the skin area became 25 times larger. In other words, the energy budget increased 37 times, and the costs for colour changing 25 times. That’s a comfortable margin! In short, large animals can more easily afford shadeshifting than small ones. . 

But how about the first and second cost factors? I see no reason why analysing the visual environment should depend on animal size, so the first factor should not cost more. The second factor was making a map telling each chromatophore what to do. In its simplest form, the costs for such a map would also reflect the number of chromatophores, meaning skin area, which does not alter the reasoning above.

All in all, the relative costs of shadeshifting get smaller as size increases. That is good news for large shadeshifters! Of course, the problems posed by thicker dead skin layers are still there…

 

PS the video at the top of the video was made for the instagram channel I am trying out (j.gertvandijk)