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The Puzzling Problem Of Proportionate Growth

Biologists have long wondered how our organs all grow at the same rate. Now theoretical physicists think they’ve found a clue in the special way sandpiles grow

Humans increase their body weight by a factor of 30 or so as they grow from babies to adults. For elephants the factor is closer to 100. 

But this raises a problem for biologists. They know that internal organs all grow at almost exactly the same rate, a phenomenon known as proportionate growth. But how does the body organise this? 

At one level, the answer is clear. The growth is controlled by chemical regulators–hormones, promoters, inhibitors and so on. These in turn are controlled by various genes. 

But this isn’t an entirely satisfactory explanation. The reaction rates associated with these chemicals can vary hugely from cell to cell because only a relatively small number of molecules are involved. 

If these variations were independent, they would cause much greater variation in growth throughout the body than is observed. So some other organising principle must be at work.

Today, Tridib Sadhu at the Weizmann Institute of Science in Israel and Deepak Dhar at the Tata Institute of Fundamental Research in Mumbai, India, put forward an interesting idea.

They point out that proportionate growth may be an example of self organisation, a phenomenon that occurs in many natural systems. They go on to show that a particular type of self organisation that occurs when sandpiles grow has exactly the property needed to explain proportionate growth.

Sadhu and Dhar study a particular type of sandpile growth known as the abelian sandpile model. This consists of a square grid of ‘sandpile’ sites that can each hold up to three grains. Adding a fourth grain causes an avalanche in which the four grains are distributed to the four neighbouring sites.  

The model proceeds in time steps during which a single grain is added to a specific site and the resultant avalanches allowed to run their course. 

What’s remarkable about this model is that after several thousand time steps, complex symmetric patterns emerge. The image above shows just such a pattern after 50,000, 200,000 and 4000,000 time steps. 

The exact shape of the pattern and its symmetry depend on the distribution of grains at the beginning but all the patterns have the same remarkable property. “The patterns are composed of large distinguishable structures with sharp boundaries, all of which grow at the same rate, keeping their overall shapes unchanged,” say Sadhu and Dhar. That’s proportionate growth, exactly the behaviour that biologists want to understand.

Could it be that sandpiles and organs both tap into the same underlying principles of self organisation as they grow?

Sadhu and Dhar say that it’s easy to imagine how organs might grow in the same way, since cell division is ultimately governed by external resources such as the food and energy supply. “Our model takes into account the basic phenomenology that the cell-division process operates under some threshold conditions: it does not happen until adequate resources are available,” they say.

In that sense, sandpile and organ growth are similar but there is another interesting similarity too. 

One important feature of proportionate growth in biology is that it is remarkably robust to external noise.  Sadhu and Dhar add various types of noise to the sandpile model and say it is remarkably robust to small random variations in the site where grains are added and to noise in the initial distribution of grains. However, it is not robust to variations in the rules that govern how the grains are redistributed at each time step. 

Another interesting feature of sandpile models is the symmetry that emerges. Perhaps one direction for future work might be to link this to the emergence of bilateral symmetry in biology. When mammals grow, they preserve this symmetry to within just a few percent. Perhaps sandpile might models explain this too.

But the model has serious limitations too. Sadhu and Dhar certainly have a promising idea on their hands but they will have to work harder to persuade biologists that it is worth pursuing. The fact that one process has superficial similarities to another does not constitute proof or even a theory of anything.

The process of science requires testable predictions that allow a theory to be falsified. What these guys need to do now is develop a way of testing their model in the field against real data. Until then, it is little more than a curiosity.

Ref:  Modelling Proportionate Growth

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