Over the past few months, starting on 1 April, I’ve been presenting the theory that when cells grow and divide, they maintain a constant ratio of surface area to volume. This is an unorthodox view.
I’ve managed to show this numerically, and I also now have the mathematical proof (not yet published) that it’s possible for a cube to double in volume and divide in half to produce two perfect replicas of the original cube (which may also be the 2500 year-old Delian Riddle). I’ve also been able to double a sphere in the same way, but I haven’t got a mathematical proof of that yet. [Update Nov 2012: Both proofs can be found here]
There are several virtues in having a constant A/V ratio. First of all, the cell never starves, because its surface area never grows too small relative to its volume, as happens if it just swells in all directions. Secondly, in order to grow and divide, the cell only has to make new internal material and external membrane in the same constant ratio, and not in some complex manner. And thirdly, this method of growth and division is (probably) a Least Action variant of all the numbers of ways in which a cell might grow and divide.
If it were possible (and it may be possible) I would build a model of of such a cell, using a polythene bag of water. But I can’t, because while it would be easy to add volume to the bag of water by pouring more water into it, I don’t know how to add surface area to such a bag by adding more polythene.
So instead, I’ve set out to construct a computer simulation model of a bag of water. And I started with an icosahedron, a regular solid which has 20 triangles on its surface. And I then sub-divided each of the triangles into 4 triangles, and pushed the vertices of these triangles radially outwards from the centre of the icosahedron until they lay on the surface of the sphere that encompasses the icosahedron, to create an 80-faced polyhedron. And then I repeated the process, to produce a 320-faced polyhedron.
This 320-faced polyhedron replicates Buckminster Fuller’s very first geodesic dome.
I’m able to show this in 3D colour because about 10 years ago I got interested in 3D graphics, and wrote my own code to do it, so I used that code to produce these images.
Because this is going to be a physical model, I then introduced point masses at each vertex of the polyhedron, and connected the point masses with ties with a known coefficient of elasticity. I’d written the code to model the dynamic behaviour of such structures nearly 20 years ago with my orbital siphon model, so I’ve used that old code to do it.
I now had a spherical polyhedron made up of point masses connected by ties. I wondered how it would behave when something happened to it. So I dropped it onto a floor from a height of about 4 times the diameter of the polyhedron. When it hit the floor, the underside of the polyhedron buckled, but the upper portion retained surprising integrity. And when I looked at the underside, I found that the buckled faces had produced a concavity, much like those you see on ping pong balls that have been hit too hard. This seemed very plausible. And if I strengthened the ties, and dropped it from a lesser height, the polyhedron even bounced a little. Which was also plausible.
Thinking that the multi-coloured polyhedron’s buckled surface was a bit hard to read, I tried to introduce simple shading of the faces, so that ones facing upwards were white, and ones facing downward were black, with shades of grey on the ones in between. But I didn’t get it quite right, somehow (see right). Why are two of the faces so bright? I don’t know. So for now I’m sticking to my multi-coloured polyhedra.
The next thing I want to do is fill the polyhedral cell with water, and work out the pressure on its surfaces, and hence the force acting on each one of the point masses on the polyhedron’s 320 vertices. This will be new physics for me, so I’ve been poring over my physics textbooks hydrostatics pages trying to figure out how to do it. I think I’ve more or less worked it out, although I’m not sure if I should have water outside the cell as well as inside. I also have to use the bulk elasticity of water to calculate the force it exerts when it’s squeezed.
And when I’ve got that, and also worked out the surface area and volume of the polyhedron, I’ll have a tiny little water-filled sphere.
Which I can then try to grow. I still haven’t quite figured out how I’m going to do this. But I’ve started out by stretching first stretching the equator of the polyhedron so that it became sausage-shaped, and then pinching the waist of the stretched bit. This was interesting to watch, because when the waist became very narrow, it was insufficiently strong to prevent the two halves folding apart (which is what would happen when this cell divides).
So it looks like I’m going to have no trouble producing cancer cells. I might even manage to produce a working model of the rather astonishing HeLa cells which spring apart when they divide.
What looks like it’s going to be hardest to do is to reproduce the growth and division of normal spherical cells. At least, I’m supposing that normal cells start out spherical.
Until this Wednesday, I was just thinking about cells as little bags of water. I’m not interested (yet) in what might be going on inside them. That’s to say I’m not interested in DNA or the nucleus or mitochondria of the endoplasmic reticulum and all the rest of the crazy things that are inside real cells. I’m not a biologist or a chemist or a geneticist, and so I don’t know what all these damn things are. As far as I’m concerned right now, my cells are just full of goo – or will be when I’ve filled them with water. I’m only interested in the geometry and physics of cells. But on Wednesday I got to thinking how the geometry might extend inside the cells, with a surprising result, which I’ll write something about sometime.
I’m also somewhere near the limits of my mathematical skills. I’m doing all this using my tried and tested trigonometry. But I’m beginning to think that I need to upgrade to vector algebra. I was actually once taught vector algebra, but I never quite ‘got it’. That’s how it is with a lot of my mathematics. I use a simple instruction set, a bit like a carpenter who’s only got a hammer and a screwdriver in his toolbox, and hasn’t got any of the fancy electric drills and power saws and stuff that most carpenters have. But you can do one hell of a lot with just a hammer and a screwdriver.
All this may seem a long way from smoking and smoking bans, but it’s not really. Because smoking, as everybody knows, causes cancer. And I’ve already got a little bit of a handle on cancer. And maybe even a better one than the ‘experts’ in CRUK. They’ve had 70 years of getting nowhere, and blaming it all on smoking. It’s time for somebody else to have a go at the problem.