A little mathematical excursion.
I have both a theoretical and practical interest in biology. The little tobacco plants growing on my window sill represent the practical side of the interest. But persistent readers will be aware that I also grow a variety of theoretical plants, and occasionally even theoretical candles. In recent months I’ve returned to a question that has puzzled me since I first started thinking about living things over 20 years ago: Why do cells divide?
One explanation current in biological circles is that, as cells grow, the ratio of their surface area to their volume decreases, and since it is through their surfaces that they gain sugar and amino acids and oxygen, and lose heat and carbon dioxide and water, they have increasing problems with all of these, as the following little educational video explains:
Now I was thinking about the process of cell growth and division, when I noticed that when a cell grows and divides into two daughter cells, both the volume and the surface area of the original cell double. If the parent cell has surface area a and volume v, then the two equal sized daughter cells will have a combined surface area of 2a, and a combined volume of 2v. And given that cells begin and end their lives with the same ratio, I wondered whether cells might increase their surface area and volume in the same constant ratio throughout the cell cycle. That is to say that cells always produce additional external surface membrane and internal cell volume at the same relative rates, a bit like a fish and chip shop that serves 35 chips with every portion of cod, no more and no less.
However, if they did this, they couldn’t remain spherical or cubical in shape, because the surface area of these solids would decrease in proportion to their volume, as already described. But was there another geometric configuration that cells might adopt as they grew, that would preserve a strict ratio of their surface area to their volume?
I began to tinker around with cubical cells, wondering how such a ratio might be preserved, and remembered that cells form cleavage furrows as they grow. And so I began to explore ways in which cleavages in a cubical cell might form. One of simplest was a cell which grew in the middle forming a double truncated pyramid (see below), with the halves of the original cell on either side. While keeping a constant, and varying b, c, and h, I found I was able to produce growing cells which maintained the same surface area to volume ratio (av-ratio) of 0.6 as the parent cubical cell.
I then wrote a little computer program to calculate (by successive approximation) b and cell volume and area and av-ratio for a variety of values of c and h. At right a complete worked cell cycle is shown (click for enlargement), with a=10.0. The table below shows the results of one run. Under each (scaled) picture of a cell, complete with notch, is shown b (in black, and in tenths of a unit), cell volume (red), and cell surface area (blue). The top row in the table shows a number of cells in which h = 0 (no cleavage) with side width a=2c ranging from 0 to 10. Rows further down show the results for different values of c and h. Cells that have an av-ratio of 0.6 are shown in blue. Cells that have an av-ratio outside this range, or in which b = 0, are shown in red. The 10x10x10 ‘mother’ cell is coloured green, and twin 10x10x10 daughter cells are shown in orange.
Now this is a very interesting table. For it shows that in growing from the mother cell to produce two equal-sized daughter cells, the growing cell actually is able to maintain a constant av-ratio of 0.6. So it is actually possible for a cell to grow and divide if it is restricted to this geometry. In fact, the cell must divide. And it divides when b falls to zero, and the two daughter cells are only joined at the very apex of a pyramid, and the cell has no alternative but to break in two.
But the table shows far more than this. For the blue shaded area shows a large number of possibilities for cell growth. Going from top to bottom of the table, cells grow larger and longer. They become very long thin cells.
The blue area might be regarded as a ‘sea’ over which cells may navigate as they grow (and also as they shrink) after leaving the mother port. And the red areas may be seen as ‘land’ onto which they cannot move. The ‘north’ and ‘west’ coasts are merely obstructions which prevent cells going any further west or north. But the entire ‘east’ coast is one where cells divide in two should they become shipwrecked on it. And the daughter cells that appear at cell division along the east coast are not all the same size. The northernmost daughter cells are the same size as the mother, but further south along the coast, they are slightly smaller, and further south still they are larger. Which actually means that there are two places on the east coast where daughter cells are the same size as the mother. Where the south coast lies (if there is a south coast at all) is well off the above map. In principle there is nothing to stop southbound cells just getting bigger and bigger, and longer and longer.
Given such a range of possible cell transformations into slightly smaller and much larger daughters, it may seem unlikely that daughter cells will usually be the same size as their parental cells. But there may be a very simple explanation why this is the most likely outcome. And this is that it requires a lot more work to be done by cells to grow to a large size, and by the operation of a least action principle most cell growth will end up hugging the north coast as it moves east. Relatively few cells will be sufficiently energetic to embark on a long voyage south, and most of these will end up dividing to produce either slightly smaller or slightly larger cells than the mother.
In addition, because it requires more work to construct large cells, these cells will reproduce slowly. And the smallest of cells will reproduce the most rapidly.
Attempting to translate from theoretical cells to real cells, the great mass of cells that are around about the same size as their parental cells might be regarded as the ordinary cells which make up 99.99% of the mass of multicellular life. The very long cells might correspond to nerve cells, some of which can grow to be several metres long in some animals. And brain cells might be cells of intermediate length.
In addition, real cells usually aren’t cubes, but are much more like spheres. But there is most likely a spherical geometry in which the av-ratio remains constant. In fact there might well be a great many such geometries. And there may be geometries which allow not only mitotic cell division of cells into 2 daughter cells, but also meiotic division of cells into 4 daughter cells.
Also, this theory of cell division supposes that both cell interior material and external cell membrane are almost completely inelastic, and cell membranes can’t be stretched to cover unusual internal cell geometries. If cell membranes were highly elastic, cells might form any number of configurations.
And that’s about as far as I’ve got with the new theory. And it does seem to be new: I’ve not read of anything quite like it. And I like it a lot: it’s a purely geometrical explanation of cell growth and division. There’s no spindles and chromosomes and DNA. I don’t know why I didn’t think of it 20 years ago – except that I’m very stupid. It makes cell growth and division seem much simpler than it once did, to me at least. But at the same time it opens up an entire new and mysterious world (with its own sea and rocky shoreline), and a complete new set of strange questions. For example, I haven’t yet looked to see whether large and small daughter cells can grow and divide using the same geometrical constraints. I think they can, but I don’t know whether they revert back to the size of the parental line, or keep growing or shrinking. I should have the answer some time next week. I have lots of my own questions about these strange cells.
Anyway, I’m sure you lot will have a good laugh. “You’re pulling our legs, Frank!” you’ll say. “You must be. After all, today’s the first of April!” And cell biologists will probably say the same: this not how they think about cells at all.