Vincent
Jansen is
Professor of Mathematical Biology at Royal Holloway University of London.
He uses mathematical models to understand how things work in biology, evolution
and epidemiology. This is an
edited version of a contribution to The Scientists’ Scribe, the biology student led magazine of
Royal Holloway University of London.
hat does a mathematical biologist do? A colleague ends his emails with a quote from Asimov: "The most exciting phrase to hear in science, the one
that heralds the most discoveries, is not Eureka but That's funny..."
Spot on! That is science for me: I want to know -need to know- how things work,
even trivial things. Why are the nuts
always on the top when I open the muesli box? I want to know why. That is
what I do; that makes me a scientist. And I use mathematics to find out that
why.
A
while ago, when walking home I spot a pile of wood, ready to be taken to the
tip. The wood has strange markings on it. I pull up some of the dried bark;
under it are wriggly lines, making an intriguing and beautiful pattern. I see a
frazzled wheatsheaf, a crumpled fleur-de-lys and a confused looking double
headed eagle with rather ruffled feathers.
It looks as if the wood has been in the hands of a designer on
I-don’t-know-what who carved it with organic, semi-symmetric psychedelic
patterns; Laura Ashley on acid.
These
funky patterns are the galleries
of bark beetle larvae.
Mother bark beetles drill a tunnel between the bark and the wood. In this
tunnel they lay eggs, nicely spaced out. When the eggs hatch, the larvae feed
themselves on the bark and the wood. Whilst eating, they drill their own little
tunnels. Initially they move away from mother’s tunnel. As the larvae grow, the
tunnels get wider, and the tunnels seem to radiate out, going their own way,
but hardly ever cross other tunnels. Once the larvae are fully grown they
pupate, and after that they eat themselves through the bark to discover that
there really is light at the end of their tunnels.
But
why the crazy pattern? All these larvae munch their way independently through
the wood, all minding their own business. How can such an intricate and
beautiful pattern emerge? Why do the larvae’s paths hardly ever cross and why
are the galleries so recognisable the same (each species of bark beetle has
their own, typical
pattern)? If all the larvae would mind their own business, why don’t we get
one big fuzzball of a pattern?
It
must be that the larvae obey some rules, probably very simple rules that allows
them to create the pattern. The rules must be genetically determined, as mummy
beetle is not there to tell them. Now the modelling part of my brain kicks
in. What could these rules be? Can the
larvae smell each other and use that to keep distance? This seems unlikely in a
tunnel in a piece of wood. Can they hear each other? You can sometimes hear
woodborers at work in a forest, munching wood creates sound. All these little
larvae munching through wood would make quite a racket if you get close enough.
But
how do they hear? Do insect larvae have ears? It seems they don’t but it is not
impossible that they perceive sound. Even if they can’t hear sound, they might
pick up vibrations. I find it hard to imagine that insect larvae are able to
detect the direction that the sound is coming from without ears. But by simply
moving their heads from left to right they probably can detect which side is
louder, or they could do the same things by having simple sensors each side of
their heads. What would happen if they have a tendency to move away from the
loudest side? 1
Here
we go, this is mathematical biology. I work out the gradient of the sound in a
sound scape. It isn’t that hard really, basically A-level physics. I then write
a little computer programme in which I let virtual larvae move, a step at a
time. They can deviate a little to the left and to the right so that they don’t
just go in a straight line. But if they hear a sound that is louder on their
right they move a little more to the left and the other way around. I let the
programme run on my laptop.
First,
I try to see what happens if a larva has two sound sources ahead of it with a
gap in the middle. On my screen the virtual larva navigates beautifully in a
straight line straight down the middle. Once it has passed the sound sources it
starts to wander about aimlessly. It seems to work. If the larva is close to
the sound sources the path is very straight. I have learned something, with
clear sources the larvae go straight and their paths wouldn’t easily cross.
Then I
put a lot of virtual larvae together on my computer screen. I let a line of
larvae start off, evenly spaced, and facing in two different directions. They
all try to move away from the others and radiate out from their starting
points. It works! A regular pattern emerges as the larvae move away from the
place where they have hatched. The larva at the end of the line of eggs has
most free space and curves out very fast. The larva next to it follows, but
can’t quite curve as much so has a little bit of a straighter line, and so on.
Thus the wheat sheaf, fleur-de-lys and double headed eagle patterns emerge. So
from the simple rule that larva move forward, and turn away from the vibrations
emitted by the other larvae I can construct similar pretty patterns.
But
the patterns are very regular, and not quite like the ones on the logs. On logs
some of the paths curl around crazily; none of that happens in my simulations.
What I see is perhaps fit for the windows of Laura Ashley, but on acid it is
not. I fiddle with the parameters, and I can’t get the frazzled look. I give
up. I know that my simple rule sort of works and am happy to leave it there.
But
then I want to do one more thing: I place the eggs on a W shape and see what
happens. And to my surprise, the larvae now at first create a regular shape
but, particularly at the bottom of the W do all sorts of crazy curls, the paths
cross and I get the ruffled feathers. It seems that in the W shape the larvae
get sounds from all directions and don’t really go in one direction, but swirl
around in the acoustic confusion.
So I
have figured out how the pattern can emerge from a simple rule. By changing the
parameters in my programme I get slightly different patterns: so species could differ in the parameters
that are genetically programmed into them. It all makes some sense. Would
different types of wood give different parameters, and therefore different
patterns? Could you make an identification key to determine the species by
figuring out what the parameters are? If you wanted to go further you could
probably do experiments and build a whole research programme around it. But
this is not for me, I am content with the insight I have gained. This wasn’t a
very important problem, but it was fun and I have scratched my scientific itch.
I
varnish the log; it is beautiful and looks like a sculpture with esoteric
etchings on it. The barking W will make a nice initial for this blog.
Vincent Jansen, April 2019.
1.
I later find a paper that suggests the larvae might indeed be
guided by sound: de Jong
and Saarenmaa 1985, Ecological Modelling, 27 (1985) 109-138 109, who in
turn cite Schmitz, R.F. and Rudinsky, J.A., 1968. Res. Pap. 8, Forestry
Research Laboratory, School of Forestry, Oregon State University, as a source
for this idea.
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