For the past 20 years, on and off, I’ve been building computer simulation models of growing populations of strange imaginary living things, many of which were very like biological cells. I was still doing it last year.
But as I grew countless billions of these proto-cells in my computer, I was always wondering: How does one cell turn into two cells? How does one cube or sphere grow and divide into two cubes or spheres? How might one bag of water grow and divide into two bags of water? I wanted a nice, neat, simple way for this to happen. But for the life of me I just couldn’t see how it might be done.
And then, late last March, I picked up the old puzzle again, for what must have been the hundredth or two-hundredth time, and I had an idea. I noted that a sphere or a cube that has just grown or divided in two has doubled its volume and also doubled its surface area, and both have the same ratio to each other with which they started. And my idea was: what if cells always maintain the same ratio of surface area to volume as they grow? It seemed implausible, but I began to construct notched cubes to see if I could find some geometrical configuration that maintained the ratio of their surface area to their volume as a constant. Within a few days, to my surprise, I’d managed to do it. I had solved the old puzzle. I’d found a nice, easy way for cells to grow and divide.
But my puzzle had always been an abstract, geometrical one. I was thinking about cubes and spheres, not real biological cells. But as soon as I had solved the geometrical problem, and provided mathematical proofs of it, I began to seriously wonder whether real, biological cells grew and divided in the same nice, easy way.
Yet it seemed to me at the outset that, while this insight might be able to say something about the gross characteristics of cells – their shape and surface area and volume -, it could never say very much about the amazing complexity of what went on inside them.
But then I had another idea: When a cell grew and divided, it clearly didn’t all happen at once. The cell had to be adding new volume and surface area bit by bit, much like a house is built brick by brick. Perhaps when cells grew they added little bits of cell, each of which had a volume and surface area in the correct proportions. What might such cellular ‘bricks’ look like?
With a circular or spherical cell, the simplest way of drawing such things would be as wedges or slices radiating from the centre, much like the way cakes or pies are cut. The volume V of each slice would be the apple sauce inside the pie, and the surface area A would be the pastry crust around the edge. And however thick or thin such slices were, they would always have the same ratio of volume to surface area – or of sauce to crust. And cells would be composed of many such slices, all bound together.
But these slices wouldn’t necessarily be neat shapes. They could also be very irregular shapes. All that mattered was that they had the same ratio of surface area to volume. And because these were strange, abstract, theoretical entities, which didn’t actually seem to exist in real cells, I called them cunei (which is Latin for wedges). Each cell would be composed of tens or hundreds or thousands of such cunei. And when a cell grew and divided, the cunei doubled in numbers, with each cuneus being exactly replicated.
I then began to wonder how a cell composed of such cunei might actually grow and divide. Where in the cell would the new cunei grow? Might they just appear anywhere? Or, since each new cuneus would replicate an existing cuneus, would it appear next to it? I wondered if a daughter cuneus would grow side by side with its parent. But if it did that, when the cell divided in two, both parents and daughters would most likely end up in one cell or the other, not in both. More likely, it seemed that daughters would grow end to end with their parents, in radially opposite directions. That way, it was much more likely that parents and daughters would end up in equal numbers in the two cells, and the two daughter cells would have the right numbers of cunei.
If the cunei replicated end to end, then if a cell split along one of its axes, and the growing cunei appeared in the gap between the two halves, as these cunei grew they would gradually push the two halves of the cell apart. So let’s look at the complete process using this simple model:
A circular or polygonal cell contains 20 cunei, each one of which is the mirror image of the cuneus radially opposite in the cell, so that there are 10 pairs of cunei arranged around a central hub, each one a different colour.
This cell splits down the centre to form two separate halves with two separate hubs. And then each cuneus gradually grows a daughter cuneus radially opposite to it, so that the two halves of the cell are slowly pushed apart by the growing cunei, which act like twin opposed pistons.
As each cuneus grows in volume, it also grows surface material in the correct proportion (this is more clearly seen in the enlargement). The result is that when each new cuneus is a complete and perfect replica of its parent cuneus, it emerges at the cell perimeter with the exact amount of surface membrane to cover it. And when all the cunei have been replicated, the daughter cells have the exact same number of cunei, arranged in same order.
And if n is the number of unique cunei (i.e. ones with the same colour), and there are 2n cunei in a cell, then when the cell starts dividing, there are 4n cunei in the dividing cell, and 2n cunei in each of the two final daughter cells.
But the cuneus is a theoretical construct which cell biologists wouldn’t recognize. So let’s fade out all the cunei, and leave only their outlines. And we get something like:
And, oddly enough, this looks very like a real cell. Here’s a photo and a drawing by Boveri in 1901 of a cell growing and dividing:
Here’s the Wikipedia description of cell division (mitosis):
In real cells, the two hubs are called ‘asters’ or ‘centrosomes’ which separate, and radiate ‘microtubules’ to form a ‘spindle’. Along these microtubules, the DNA-carrying ‘chromosomes’ or ‘chromatids’ travel. And as the cell enlarges, it forms a ‘cleavage furrow’.
My theoretical cells don’t have chromosomes, but the following passage from Molecular Cell Biology (3rd edition, Scientific American 1995) may provide a clue as to how to introduce them:
In a diploid cell before DNA replication there are two morphologic chromosomes of each type, and the cell is said to be 2n. In G2, after DNA replication, the cell is 4n.
My cells have 2n cunei, which multiply to 4n cunei during cell division. So most likely each cuneus has its own chromosome. And one possible link between a chromosome and its cuneus could be that each chromosome contains the instructions for making a cuneus. When a cell starts to divide, the chromosomes in each cuneus are duplicated, and the duplicate is then used to start building the daughter cuneus from the hub (aster/centrosome) outwards. As the cuneus is built, the chromosome rises up it, much like a crane on top a rising skyscraper. And this is why the chromosomes from each half of the cell remain in close proximity at the partition (metaphase plate) between the two halves. When cuneus construction is complete, the chromosomes return to the hub (much like cranes returning to ground level) to re-form the cell’s nucleus.
This is one possible way in which chromosomes might be included in this simple cell model.
So this simple cell model seems to be doing fairly well when it’s compared with the observed reality of cell growth and division. I’ve maybe managed to get inside cells a bit. A further development might be to build a physical model of this cell, to explore its dynamic behaviour, and perhaps also look for variant forms of growth and division.