Mathematical beauty (100 years part I)

Exactly a century ago, D’Arcy Thompson published his book On Growth and Form.

I’ve spoken about Mr. D’Arcy before, but as it is the 100-year anniversary of his masterwork, I feel it fitting to revisit the topic. Since mentioning him last, I have finished reading his book, and have also started to write up my thesis. I bring up my thesis because my work is related to D’Arcy’s work in the sense that I have been trying to bridge the gap between biology and physics, and I predict that some of my reading will inspire me to write more (hopefully more of my thesis but presumably also more on the general topic of bio-meets-phys).

Mr. D’Arcy wrote On Growth and Form to collect his observations on the mathematical principles of nature. He explains how biological phenomena of form and growth closely resemble physical and mathematical principles. Especially for some of the more simple examples (e.g. the shape and size of single or doublets of cells) the similarities between biology and physics (e.g. single and doublets of bubbles) are almost uncanny. These simple systems can easily be described using simple formulas, and he suggests that even more complex systems can be explained in a similar way (though he remarks that it will take a lot of formulas and paper space, luckily we have computers now). In 1917, this idea was pioneering, to say the least. Bio-mathematics and biophysics were nowhere near being the hot topics they are today.

One of the events organised for the anniversary of On Growth and Form, was the exhibition A Sketch of the Universe at the  City Art Gallery in Edinburgh, showcasing works of art that were inspired by the book, or by the idea that mathematics and biology are closely intertwined. The exhibition is closed now, but this weekend I did get the chance to go visit it.

So, I present to you, some of the highlights that I found interesting or cool-looking:

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Trifolium repens L – top view – No 10 by Macoto Murayama (2016) – 2D rendering of a 3D model depicting the structure of a white clover flower.

“I know that in the study of material things number, order and position are threefold clue to the exact knowledge; and that these three, in the mathematician’s hands, furnish the first outlines for a sketch of the Universe.” (D’Arcy Thompson, On Growth and Form)

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Aggregation 24/27 by Andy Lomas (2005) – These prints were “grown” using computer algorithms that simulated the paths of millions of particles flowing in a field of forces.
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Untitled by Gavin Rutherford (2010) – Prints citing D’Arcy Thompson.

“For the harmony of the world is made manifest in Form and Number, and the heart and the soul and all the poetry of Natural Philosophy are emodied in the concept of mathematical beauty.” (D’Arcy Thompson, On Growth and Form)

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Radiolarians by Amy Barber (2010) – The single-celled Radiolardia aggregate into complex and very diverse shapes. Their skeletons are incredibly delicate and look pretty neat.

“The waves of the sea, the little ripples of the shore, the sweeping curve of the sandy bay between the headlands, the outline of the hills, the shape of the clouds, all these are so many riddles of form, so many problems of morphology.” (D’Arcy Thompson, On Growth and Form)

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Scarus, Pomacanthus by Darran McFarlane (2012) – This painting was created by subjecting an existing portrait of D’Arcy to mathematical transformations.

You can read more about D’Arcy Thompson, On Growth and Form and some of the events being organised this year in honour of the 100 year anniversary, here. And presumably in the near future on this very blog.

[Note: Again, I realise that D’Arcy Thompson’s last name is “Thompson” so “Mr. Thompson” would be a more appropriate title (or Prof. Thompson) but I just cannot resist his practically Austenesque first name.]

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Physics of Cancer (2)

Two weeks ago, I told you that physics and cancer are, perhaps counterintuitively, intermingled and that this relationship has biological and clinical implications. I outlined how mechanical forces act on cells and tissue, and perhaps are responsible for one of the many ways of cancer progression.

In this post, I’d like to tell you about how being able to detect mechanical properties of tissue can help with diagnosing diseases. So while the previous post was more about how physics can influence the biology of a tissue, this time I’d like to focus on how biology can dictate physical properties of a tissue.

A very important issue to point out, before going into the differences between healthy tissue/cells and cancer, is the size scale we are considering. Depending on whether we are talking about cells (µm size range) or tissues (100s of µm to mm), we can make quite opposite conclusions: several studies have shown that tumour cells are softer than healthy cells (of the same tissue type), while tumour tissue is stiffer than healthy tissue.

First the cells. Experiments such as Atomic Force Microscopy (which I mention because I have used it myself) show that especially metastatic tumour cells are softer than healthy cells. If we consider what cells do during metastasis, this actually makes sense. (Metastasis is the process where cells migrate away from the initial tumour and spread to other parts of the body.) A softer cell is able to squeeze through other cells, and through the wall of a blood vessel, allowing it to travel to elsewhere in the body. This different mechanical property allows it to behave in its particular way. Knowing this property allows us to predict the aggressiveness (or invasiveness) of a certain cancer. If the tumour has cells that are much softer than other, it is usually a more aggressive type of cancer.

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That one time that being less flexible is actually healthier!

This difference in mechanical properties not only makes sense if we consider the behaviour of the cells, it can also help make prognoses and decide on what type of treatments to use.

Next, on a larger scale, tumours are stiffer than healthy tissue. This is exactly what we feel when we are “looking for lumps”. A bit of tissue feels different, namely stiffer, than what it should be. The reason tumours are stiffer is not actually due to the (softer) cells it contains, but due to what sits in between the cells: the extracellular matrix. The extracellular matrix is a very structured meshwork of structural proteins that acts as a scaffold for the cells: it provides the tissue with structural integrity, cell organisation and mechanical strength. For example, there are a lot of extracellular matrix proteins in our skin, which is why it is, well, our skin (hurray for circular reasoning): a sturdy barrier between the outside and the inside of our body. In healthy tissue, the extracellular matrix is usually very well organised. The fibers making up the matrix are regularly cross-linked and have and neatly organised. However, the matrix in tumourous tissue is chaotic. In essence, was “built” too quickly. In this fast-growing bit of tissue, the scaffolding had to be assembled fast to support the rapidly dividing cells. As a result, the fibres are not well organised and the crosslinking is random. It is as if the scaffolding of a building was built too quickly, so rather than nicely structured, there are random beams sticking out in all directions. As a result, pushing down on the matrix does not compress it as much, and it feels stiffer.

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Neatly structured, easily compressed.
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Just ugh. (Also, majestic artist skills, right?)

At a tissue level, these differences in mechanical properties are very useful for diagnosing cancer. Because of different mechanical properties, we can feel lumps, but we can also image it using techniques such as ultrasound, MRI and other imaging techniques. Due to different physical properties, the cancerous tissues interacts differently with whatever wave (light, sound, …) we are using to try and detect it. Thank you physics!

To wrap this up: the physics of cancer is important, and useful, and interesting, and cool and definitely worth researching. And this is why interdisciplinary research is not only a fancy buzzword, it can also increase our understanding certain phenomena and come up with better diagnoses and treatments by approaching the problem from a completely different perspective.

This subject was the topic of my second FameLab performance (Scottish regionals), which ended in a little song (to the tune of “What a Wonderful World” by Sam Cooke), in which I wanted to highlight the importance of interdisciplinary research and how studying diseases from a physics perspective can only be productive:

You know, cancer’s about biology
And perhaps a bit of chemistry
But I’m telling you there’s physics too
There’s physics happening inside you

With one subject you can never be sure
Put them together, and we might find a cure
And what a wonderful world that would be…


Some references I used to verify that my thoughts on this subject were not completely unsubstantiated:

Baker EL, Lu J, Yu D, Bonnecaze RT, Zaman MH. Cancer Cell Stiffness: Integrated Roles of Three-Dimensional Matrix Stiffness and Transforming Potential. Biophysical Journal. 2010;99(7):2048-2057. doi:10.1016/j.bpj.2010.07.051.

Suresh S. Biomechanics and biophysics of cancer cells. Acta biomaterialia. 2007;3(4):413-438. doi:10.1016/j.actbio.2007.04.002.

Kumar S, Weaver VM. Mechanics, malignancy, and metastasis: The force journey of a tumor cell. Cancer metastasis reviews. 2009;28(1-2):113-127. doi:10.1007/s10555-008-9173-4.

Plodinec M, Loparic M, Monnier CA, Oberman EC, Zanetti-Dallenback R, Oertle P, Hyotyla JT, Aebi U, Bentires-Alj M, Lim RYH, Schoenenberger C-A. The nanomechanical signature of breast cancer. Nature Nanotechnology. 2012; (7):7 57–765. doi:10.1038/nnano.2012.167