A New Theory of Cell Growth and Division

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.

See also: Two mathematical proofs of constant A/V cell growth and division

About Frank Davis

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33 Responses to A New Theory of Cell Growth and Division

  1. Professor Im in awe!

  2. magnetic01 says:

    Some more ammunition for HorecaClaim:

    Anti-smoking group Clean Air Nederland is taking the Dutch state to court on Friday in an effort to enforce a ban on smoking in all bars and cafes.

    One of the first acts of the current government in 2010 was to relax the ban so that bars smaller than 70m2 with no staff did not have to comply.

    However, the ban, brought in four years ago, is widely flouted in bigger bars, cafes and night clubs.

    Clean Air Nederland argues the current situation has led to unfair competition with bars which do keep the law.

    That’s an admission by an antismoking organization that bars that flout the law – that permit smoking – have an unfair advantage over those that observe the antismoking law. If that’s the case, then the antismoking claim that smoking bans are good for business cannot be true.

  3. Rose says:

    Frank, trying to understand that hurt my head.

    So I will update you on the bee crisis.

    Government to reconsider nerve agent pesticides
    31 March

    “The Government is to reconsider its refusal to ban neonicotinoid pesticides, the nerve-agent chemicals blamed for the collapse of bee colonies worldwide, the chief scientist at the Department of the Environment, Sir Robert Watson, told The Independent.

    Sir Robert, a former head of the UN climate panel, moved quickly to begin a comprehensive re-evaluation of the Government’s stance after two new scientific studies, from Britain and France, strongly linked neonicotinoid use to bee declines.

    He said the new studies, and others, would be closely analysed.

    The Government has refused previous requests to consider a precautionary suspension of the chemicals, which have been banned in France and Italy, despite mounting evidence that they are harmful to bees and other pollinating insects, even in minute doses.

    Bees’ role in pollinating crops is worth billions of pounds annually to global agriculture.

    Even on Thursday, after the new studies were published, a spokesman for Defra said the new research did not change the Government’s position, and that “the evidence shows that neonicotinoids do not pose an unacceptable risk to honey bees”.

    “Systemic pesticides are those that are transported in the sap of a plant from the seed up through the stem into the leaves and flowers. Here, they contaminate nectar and pollen and hence any insect that picks them up – including bees.

    Since then, imidacloprid and other neonicotinoid systemic pesticides, such as thiamethoxam, have been implicated in the worldwide collapse of honeybee colonies.

    As well as being systemic, they act as a neurotoxin attacking an insect’s nervous system on contact or ingestion and are designed to protect over 140 commercial crops, including cereals, oilseed rape, maize, cotton, sunflower and sugar beet.”

    Don’t forget that we are all eating these which rather puts yesterday’s nonsense into perspective.


    Anger at advisers’ biotech links
    July 2003

    “Dossier reveals Ministers’ worries over connections between science experts and leading drugs firms

    Dozens of the Government’s most influential advisers on critical health and environmental issues have close links to biotech and drug corporations, according to a dossier of Whitehall documents obtained by The Observer.

    Internal papers from the Department for the Environment, Farming and Rural Affairs (Defra) reveal for the first time the extent of the close connections between big business and scientists hired to give independent advice to Ministers. Many work as consultants for the firms, own shares in the companies or enjoy lucrative research grants from them.

    Confidential documents disclose that former Environment Minister Michael Meacher and Food and Farming Minister Lord Whitty, were deeply concerned that scientists with industry links were dominating committees on everything from food safety and air quality to the imminent arrival of GM crops. Both Meacher and Whitty were alarmed that the scientists’ commercial links jeopardised the independence of the advice they gave.”

    · A key member of the committee advising Ministers on the safety of GM products has received research funding from biotech giants Monsanto and Syngenta. Professor Phil Mullineaux also works for the John Innes Centre – the GM research centre funded by Science Minister Lord Sainsbury;

    · More than three-quarters of the members of the committee which advises Ministers on food safety have direct links to major food companies and drug giants including Novartis, Astra-Zeneca and Syngenta. Its chair, Professor Ieuan Hughes, has personal interests in Pharmacia – which in April was bought by Pfizer to create the biggest drugs company in the world – and owns shares in BP Amoco where his daughter works.

    · A former deputy chairman of the committee which examines the safety of pesticides, Professor Alan Boobis, received research funding from GlaxoSmithKline for his department at Imperial College but never declared it. Other members of this committee have links to agrochemical firms like Aventis, Astra Zeneca and Monsanto. The current head of the body, Professor David Coggon, was a close friend of Esso’s chief medical officer and received a gift from the oil giant.

    · The chair of a group examining air quality in Britain, Professor Stephen Holgate, is a consultant to drug giant Merck. His university department has received grants from Glaxo and Astra Zeneca. Others work for biotech and drug giants like Novartis and Schering-Plough.

    · Almost three out of four members of the committee advising Ministers on the cancer risks of chemicals in food and other consumer products either own shares in or work for major biotech and drug corporations;

    While the scientists openly declare their interests, Meacher was so exasperated by the structure of committees advising him that he personally intervened on a number of occasions in an attempt to get more environmentally friendly members on them.

    Last week it emerged that Whitty was so alarmed about the industry links on the committee advising him on the safety of farming chemicals that he broke official rules and hired a toxicologist, Dr Vyvyan Howard, who is known to be more sensitive to environmental issues.”

    “A Defra spokesman said the committees publish their members’ interests.

    He went on: ‘Defra has full confidence in the capability of independent advisory committees across the range of issues the department deals with to provide high-quality, well-informed advice and support.’

    Which might explain our current governments attitude, goodness knows what the previous lot signed.

    • Rose says:

      Now this worried me, it sounded like the British public were being offered as guinea pigs via the NHS.

      “Big push” for UK biotech – 2007

      “Now is the moment for a big push” in improving the environment for the UK biotech industry, argues Tony Blair, Prime Minister.
      “To allow the biotech industry to get some strength in research out of the NHS is something we need to look at in this country,”

      “The Prime Minister added that he believes the industry and Government have now curbed the actions of animal rights extremists, and underlined his support for stem cell research. The discussions covered the cost of clinical trials, strengthening links with universities and funding. The BioIndustry Association (BIA) described the meeting as, “incredibly positive from all around the table”, and that Tony Blair’s leadership on creating the right policy climate for biotech had “reverberated across the sector”.

      Which could explain why

      Dying bees ‘were not a priority’ – 2009

      • Rose says:

        From the days before colony collapse, when nicotine was a widely used pesticide.

        Production and Use of Nicotine from Crops in Peace and War- The Yearbook of Agriculture 1950-1951

        Here’s where it gets even more curious, look at the dates.

        “The use of neonicotinoid insecticides has grown considerably since their introduction in 1990s.

        They are used extensively for the control of agriculturally important crop pests and also in the control of cat and dog fleas. Imidacloprid exploited through an elaborated structural and substituent optimization of nithiazine was launched to market in 1990. The selectivity ofneonicotinoid compounds for insect species has been attributed to their binding on nicotinic acetylcholine receptors in which the negatively charged nitro- or cyano-groups of neonicotinoid compounds interact with a cationic subsite within insect nicotinic acetylcholine receptors. The first example of a pest evolving resistance to field use of neonicotinoids was Bemisia tabaci.

        Resistance to neonicotinoids can arise either through nAChR subtypes expression, detoxification mechanisms and/or structural alterations of target-site proteins. Consequently, a number of derivatives and analogues ofimidacloprid have been generated to date. In 1992, a new neonicotinoid using acetylcholine as the lead compound has been found. This was dinotefuran, which has a characteristic tetrahydro-3-furylmethyl group instead of the pyridine-like rings of others neonicotinoids.”

        Bearing in mind that these are synthetic and are supposed to mimic the action of nicotine on an insect,and therefore can be patented.
        I originally thought that the nicotine in nicotine patches must be synthetic so that Pharma could patent them, but apparently the patent is on the patch as a delivery method.

        Patents; A Nicotine Drug Patch to End Smoking – 1990

        “Transdermal drug patches, which dispense medication into the blood stream through the skin, may have a new use: helping cigarette smokers to quit.
        Researchers at the Schering-Plough Corporation, the pharmaceutical manufacturer in Madison, N.J., obtained a patent this week for a drug patch designed to reduce tobacco craving by dispensing nicotine in controlled doses 24 hours a day.”

        So at the very same time we were beginning to be bombarded by nicotine patch sellers, mostly based on hundred year old folklore as it turns out, the bees were also being attacked by synthetic nicotine based presumably on it’s previous successful but unpatentable use.

        Now that really does look like a concerted effort to take over a large part of the nicotine market, both of which are rapidly descending into catastrophe for both species, while still being staunchly defended by the manufacturers and their wrong footed political allies.

        That’s enough thinking for one morning, I am off to commune with my bees.

  4. garyk30 says:


    That is the only way they can multiply and the ones that did not multiply, long ago, ceased to exist.

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  6. garyk30 says:

    A new idiocy by your beloved Debbie:

    ‘Deborah Arnott, chief executive of Action on Smoking and Health (ASH), said: “There is no safe level of exposure to tobacco smoke and children are at risk of a range of diseases such as […] potentially fatal meningitis as a result of breathing in second-hand smoke in the home or car.”‘

    Total idiocy, there are NO viruses, bacteria, fungi, or parasites to be found in SHS.


    Meningitis is usually caused by infection from viruses or microorganisms. Most cases are due to infection with viruses, with bacteria, fungi, and parasites being the next most common causes.

    • beobrigitte says:

      A new idiocy by your beloved Debbie:

      ‘Deborah Arnott, chief executive of Action on Smoking and Health (ASH), said: “There is no safe level of exposure to tobacco smoke and children are at risk of a range of diseases such as […] potentially fatal meningitis as a result of breathing in second-hand smoke in the home or car.”‘

      Ah, the ‘poor chiiiildren’ are being abused by the anti-smoking zealots – AGAIN – for trying – AGAIN – to push for a smoking ban in cars.

      DEBBIE, IT’S MY CAR AND MY RULES. Get used to it.

      I seriously wonder how poor Debbie explains the survival of the generation of kids growing up happily and healthily with smoking adults in cars and at home, becoming healthy adults and by now are beginning to give governments the headache of having to fund their pension. May I suggest the monies wasted on Tobacco Control, especially ASH, and their useless campaigns would be of help here?

      Poor Debbie must be desperate (is the government axing the funding for ASH?) to even think about peddling such utter nonsense.

      Debbie will encounter potentially fatal third hand smoke by accepting every taxpayer’s money.

  7. Frank Davis says:

    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.

    Cell membranes are very stretchy, with Young’s Modulus of about 12 megaPascals. But this may be unimportant, because underneath the membrane there is an actin cytoskeleton. Actin has a Young’s Modulus of about 2 gigaPascals, and is much stiffer.

    “Actin filaments form a layer under the plasma membrane that provides mechanical strength”


    Click to access janmey.pdf


  8. junican says:

    Cells of the ‘epithelium’ (the ‘inner skin’ of the bronchial tubes) are plate-like. They are flat like fish scales or snake skin.

    • Frank Davis says:

      I know. But I reckon I could show how those grow and divide too.

      • Frank Davis says:

        In principle, there’s nothing to stop it. A 1 x 1 x 1 cubic cell has a volume of 1 cubic unit, and an area of 6 square units, and so an AV-ratio of 6. But a 4 x 4 x 0.4 cell has a volume of 6.4 and a surface area of 38.4 and so an AV-ratio of 6 also. So a thin flat cell can grow from a cubic one – the important thing being to preserve the same AV-ratio at all times.

        However, this particular flat cell is over 6x the volume of the cubic parent, and so would probably grow 6 times more slowly. And to the best of my knowledge epithelial cells grow and multiply fastest of all.

  9. Frank Davis says:

    Thoughts to self:

    The A/V ratio of cubes varies depending on what size they are. A 1 x 1 x 1 cube has a volume of 1 cubic unit, and an area of 6 square units, and so an A/V ratio of 6. But a 10 x 10 x 10 cube has a volume of 1000 cubic units and an area of 600 square units, and so an A/V ratio of 0.6.

    I think that the A/V ratio of a unit (1 x 1 x 1) cube has to be used as the basis: the cubic unit A/V ratio. The A/V ratio of a 10 x 10 x 10 cube is simply the cubic unit A/V ratio divided by side length, which works out as 6 / 10, which is 0.6 – and the correct value.

    Let’s put this into practice. Ordinary human cells (liver cells) are about 50 micrometres long. Assuming they’re cubical, then their A/V ratio will be 6 / 50 x 10-6, which is 1.2 x 10-5..

    When cells grow, they must maintain their A/V ratio. And they can grow to form a variety of shapes. Long thin cells ( e.g. nd x d x d ) have an A/V ratio that’s given by 2(n+2) / nd where n is the number of units, and d the length. Wide flat cells ( e.g. nd x nd x d ) have an A/V ratio that’s given by 2(2n+1) / nd. So if we know that human cells have an A/V ratio of 1.2 x 10-5, then for any long thin human cell we can write:

    2(n+2) / nd = 1.2 x 10-5

    and for wide flat cells we can write:

    2(2n+1) / nd = 1.2 x 10-5

    And so for any value of n there’ll be a corresponding value of d.

    The limit unit A/V ratio for rods ( n –> infinity, d = 1 ) is 4, and for plates is 2.

    While cells remain rods or plates, they won’t divide. In order to divide, they have to start to form a notch. Then, as the notch grows (and the A/V ratio continues to remain the same), the cell will divide.

    • junican says:

      I mentioned it because it seems to me that an elongated, flat cell might ‘split in two’ more easily than a cube or sphere.

      Entirely as a novice, I would think that cells do not magically split into two replicas. There must be an element of time, during which they acquire the materials, in advance, which are needed for each chromosome to replicate. In fact, it makes little sense to imagine anything other than that the chromosomes have already duplicated before the cell division. That is, a cell divides when it is, materially, ready to divide. However, For some reason or other, for a cell to divide, it needs space. The space is provided by cell ‘apoptosis’ – the death of old cells.

      It is a wonderful subject, about which little is known (apoptosis). It is at this level that epidemiology breaks down. It breaks down in the sense that not all the ‘confounders’ have been eliminated.

      • Frank Davis says:

        Cells certainly have duplicated their chromosomes prior to division. And they spend quite a long time growing before they divide. But I’ve never heard that cells only grow and divide if there’s space for them provided by the death of other cells.

        P.S. In mitotic division, the chromosomes have been duplicated, leastways. In meiotic division, they haven’t.

        • junican says:

          Perhaps I ought not to talk about this subject since I know so little!

          Clearly, a growing foetus needs cells to divide and divide in order to grow according to ‘the plan’ as does a child. But it is reasonable to ask why people do not go on getting bigger and bigger and bigger.

          I came across the subject of apoptosis some years ago. I don’t know how or why. It was research by someone or other who investigated the question of the ‘suicide’ of cells and he was the person who coined the word ‘apoptosis’. It was in that paper/article that the process of ‘making space’ by apoptosis for cells to divide was mentioned. I suppose that it could also be the other way round! Maybe cells that need to divide force the apoptosis of old cells.

        • Frank Davis says:

          in order to grow according to ‘the plan’ as does a child.

          Well, this is a most fundamental matter. Is there a ‘plan’?

          The current biological view seems to be that there is indeed a plan, and it’s buried somewhere in our genes/chromosomes. It seems to be inherent in human thinking to see plans in nature, to see design.

          But I’ve begun to doubt that there is any ‘plan’. There is instead simply an unfolding logic. And living things form like rivers form. There is no ‘plan’ for rivers. Nature doesn’t come along and drive stakes into the ground and say, “This is where I want the river to go, I want nice sweeping curves along this bit, and a tight corner here with an ox-bow lake, and little tributaries coming in here and here.” No. There’s nothing like that. That’s what humans do when they design and build rivers (or rather, canals). Natural rivers just happen. They grow out of the logic of rain and water and gravity and motion.

        • beobrigitte says:


          Thalassemia is a genetic disease that results in the production of an abnormal
          ratio of hemoglobin subunits.

          This is not quite correct. Thalassaemia, strictly speaking, is not a disease. It is the result of genetic adaptation to Malaria.
          There are mainly alpha, beta and delta thalassaemias, each of which refers to the amino acid subunits. (There are further variations in combination with other haemoglobinopathies)
          The various Malaria parasites cannot complete their reproduction cycle in these erythrocytes due to the affected cells “collapsing”. Unfortunately a difference in air pressure (e.g. a climb up a mountain) will do the same, so thalassaemia people are indeed very vulnerable, even though Malaria (which at some point was the greatest threat) will not kill them.

          Hope this is useful. Mutation is not necessarily a bad thing.

  10. Frank Davis says:

    Some of my theoretical cells are dumb-bell shaped. But it seems that quite a few cells in nature have this shape. Guard cells in some plants are dumb-bell shaped.

    And human red blood cells are dumb-bell shaped in cross-section. It will be interesting to find out if such cells can preserve a constant A/V ratio. I suspect that they can.

    It seems that red blood cells in other animals have some quite fantastical shapes.

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