Mathematical Biology – Bringing numbers to life.
Updated: Jul 24, 2021
B. Harshvardhan,
Mathematics Faculty,
Narayana Group of Institutions,
Hyderabad, Telangana, India
Q. Why do biologists not like maths? – Because it is so dry and lifeless. Q. Why do mathematicians not like biology? – Because in biology, a division is the same as multiplication.
Mathematics and Biology have a love-hate relationship. Tending more often towards hate than love. There is in fact, much love to be had when the two meet. People think mathematics is only about numbers. At its core, Mathematics is about patterns and ‘Biology’ is nothing if not a world of patterns. Living, breathing patterns. The patterns exist not only in the shape and structure of living things but also in their behaviour. Whether as an individual or in a group, whether at the level of molecules or communities, life forms exhibit patterns of breath-taking variety. Mathematicians, therefore, have had their work cut out for them, just waiting to be discovered. Let’s see where and how does Mathematics appear in Biology.
Have you heard of the Fibonacci series? It is a simple series –
1, 1, 2, 3, 5, 8, 13, 21, …
where every term is created by adding the two previous terms. Although not visibly remarkable, this series turns up at unexpected places (Fig. 1). If you take the ratio of successive terms, you get a new series
1, 2, 1.5, 1.67, 1.6, 1.63, 1.62…
If you were to keep going, you will reach a certain ratio that remains fixed – 1.618… This ratio dubbed as the “Golden Ratio” has baffled mathematicians and naturalists alike. Not only does it have the simple property of being one unit more than its’ reciprocal (as in half is the reciprocal of two), it also turns up at completely surprising places – from body proportions to ideal building shapes, from sunflower spirals to shape of a nautilus shell.
Fig. 1: Fibonacci series in most commonly seen petal numbers.
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Fig. 2: Example of Golden ratio and Fibonacci series in the human body: The ratio of the forearm to the hand is close to the Golden ratio and lengths of digits follow the Fibonacci series.
The Fibonacci series and the Golden ratio offer a small glimpse into the interdependence of Biology and Mathematics. In the last half-century, the development of computational methods has propelled mathematics into being an inevitable tool to understand biology at a systems scale and provide insights like never before.
Here is a glimpse into studies where Mathematical tools helped unravel the mysteries of Biology and beyond:
1. Fractals: Fractals are patterns that are self-similar across multiple levels of observation. They can be seen everywhere in nature, for example, in ferns, trees, clouds, waves, snowflakes, etc.
Fig. 3: Self-similarity in a fern. Observe that the shape of the entire blade is reproduced at each pinna and pinnule.
Fig. 4: A tree as a fractal - Observe that the branching pattern is repeated at the main stem, then larger branches, then smaller branches too.
Fig. 5: Fractal Pattern of bronchial branching.
Fractals are seen in the human body too – branching pattern of blood vessels, bronchioles, even neuronal dendrites. One of the most important applications of fractal mathematics is the measurement of irregular shapes. Fractal geometry arose from the paradox that the measured length of the coast of Britain depended upon (i.e. changed along with) unit of measurement used. In 1967 in a seminal paper published in Science (1), the ‘father of fractal mathematics’ – Benoit Mandelbrot used fractal geometry to find this length with precision. This approach was considered groundbreaking because previously known mathematics assumed shapes to be linear or regular – smooth shapes of lines, curves, etc. Fractal geometry by its very nature assumes shapes to be irregular. Mandelbrot established the same and then showed how fractals can be used to measure almost any irregular shape found in nature.
In the ‘Golden age’ of cell biology – the 1960s to 1990s – biologists also looked at fractal geometry and realized that it could be used to make measurements of cell organelles to previously unknown levels of precision. As Cell Biologist G.A. Losa has described (2), the precise measurement of organelles offered a new tool to characterize cell growth and morphology. Fractal measurements of nuclear chromatin have helped quantify the degree of malignancy in human breast cancer cells. Fractal analysis of the nuclear periphery has been used for the early detection of ovarian cancer.
In the same article, Dr Losa describes in particular how fractal geometry is essential to understanding the complex structure of the brain – including the folds at the macro scale and the dendritic pattern at a micro-scale. The fractal analysis allowed researchers and doctors to quantify the alterations in the diseased brain with “epilepsy, schizophrenia, stroke, multiple sclerosis, cerebellar degeneration” etc. A mathematical quantity called ‘Fractal Dimension’ takes into account the complexity of any pattern. It has been shown that the quantitative evaluation of surface fractal dimension of neurons (in particular axons, dendrites) allows modelling of not only the complex geometrical architecture of the brain but may also provide a tool for early detection of tumour vasculature.
2. Game Theory, Evolution, and development of cooperation:
Have you watched the movie “A Beautiful Mind”? It is the story of a Nobel Laureate afflicted with Schizophrenia - Prof John Nash. He is credited to have founded the field of Game theory. It is a theoretical framework underlying games of strategy. By providing such a framework, Prof. Nash’s work became the basis of everything from Stock market prediction to quantifying sociality among animal groups.
One of the major questions in the field of evolutionary ecology is the evolution of cooperation and altruism. On a superficial level if animals are motivated purely by survival instinct then every interaction would be a competition for resources leaving no space for cooperation. Moreover, in ecological terms, altruistic behaviour is the behaviour shown by an individual that reduces the fitness of the actor while increasing the fitness of another individual. Treating every individual as a selfish being, out there looking out for itself only, doesn’t allow for altruistic behaviour. Yet both cooperation and altruistic behaviours are widely seen in nature. From hunting in a pack (cooperation) to warning calls (the bird that provides the warning call increases its own risk of predation but lowers the same for the group).
At the most basic level, the Game theory deals with understanding multi-individual interactions and prediction of the outcomes of those interactions. The payoff (reward or punishment) to each individual depends not only on the strategy of the individual but also on the strategies of other individuals (3). The payoffs are represented in a matrix form and multiple interactions can be evaluated using basic matrix multiplication.
Often such interactions can be simplified into cooperation vs competition choice for both individuals. Game theory shows that if the interactions between the individuals are few, then competition is indeed the best strategy to maximize fitness (probability of survival).
In nature though, the interaction is repeated over thousands of generations. Such evolutionary time scales can be modelled by a simple ‘tit-for-tat’ strategy. Here individuals repeat the same response to their partner in the previous encounter. If the partner cooperated in the previous encounter, the individual would do the same now and vice versa. It is easy to see how over hundreds of iterations, cooperation would evolve as an ‘evolutionarily stable strategy’.
Fig. 6: (a)The payoff matrix here is of the “Prisoner’s Dilemma” – whether to confess to the crime and implicate their partner (defect) or to remain silent (Cooperate). The matrix shown here is from the perspective of Prisoner 1. Even though in the short term, defecting always gives a higher net benefit (b), a longer-term tit-for-tat strategy (c) can lead to the evolution of cooperation between the players. Source (3)
Game theory-based models of ecological interactions have evolved over the years (3). They now take into consideration mixed type strategies where the player strategy is different for different individuals based on traits such as size etc.; complex interactions where there is risk asymmetry between players (eg. Animals fighting for territory – the occupiers vs intruder – the occupier will often be more aggressive than intruder); even modification of payoffs that reward individuals that cooperate while punishing those that don’t. This framework even explains why many animal species engage in ritualistic fights (red deer exhibiting their superior antlers, fiddler crabs displaying their claws) rather than engage in physical combat.
Prof Vishwesha Guttal from the Indian Institute of Science, Bangalore uses Game theory, Network Dynamics, and Statistics in combination with field data to study swarm intelligence, predator-prey behaviour, migration in the animal kingdom among other things. A 2018 study on heterospecific sociality (cooperation between individuals across species) gave a conceptual framework to explain why in nature animals often choose partners from a species different from the animal’s species (4). Studies such as these help explain why species behave the way they do, why some interactions are more common than others, and in general why things are the way they are. While molecular biology and field data often answer the ‘how’ and ‘what’ of natural systems, mathematics often helps to answer the ‘why’ of these behaviours.
This was just a sneak peek into the world of Mathematical Biology. Mathematics has been used to create immunization strategies (5); analyze the spread of behaviour in social networks (6); even predict the violent clashes that happened in the US in 2020, back in 2012 (7). In developmental biology mathematical modelling has provided a window into processes by which animals like fish develop their complicated patterns (8). Mathematical modelling often helps researchers guesstimate exactly which molecular process might lead to some specific pattern seen at the systems level. It has even helped predict the existence of specific types of cells in the brain before they were discovered (9). With the advent of Machine Learning and Artificial Intelligence, computers have begun to delve into patterns that the human mind would ignore or be unable to comprehend (10). As expected it is revolutionizing the field of Medical research. The wall between Mathematics and Biology is an artificial creation that has long outlived its usefulness. Research doesn’t follow the same divisions, neither does knowledge itself. Life is what matters and how we choose to explore it is entirely up to us.
References:
1. Mandelbrot B.B. (1967) How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 155: 636–640.
2. Losa G.A. (2012) Fractals and their contribution to biology and medicine. Medicographia 34: 365–374.
3. Cowden C.C. (2012) Game Theory, Evolutionary Stable Strategies, and the evolution of Biological Interactions. Nature Education Knowledge 3(10): 6.
4. Sridhar H. and Guttal V. (2018) Friendship across species borders: factors that facilitate and constrain heterospecific sociality. Phil. Trans. Royal Society of London B 373: 20170014.
doi: https://doi.org/10.1098/rstb.2017.0014
5. Shen, Dongqin & Cao, Shanshan, (2018) An efficient immunization strategy based on transmission limit in weighted complex networks. Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 1–7.
6. Damon Centola, et al. (2010) The Spread of Behaviour in an Online Social Network Experiment. Science 329, 1194. doi: 10.1126/science.1185231
7. Turchin P. (2012) Dynamics of Political Instability in the United States, 1780–2009. Journal of Peace Research 4: 577–591.
8. Morelli L.G. et al. (2012) Computational Approaches to Developmental Patterning. Science 336: 187. doi: 10.1126/science.1215478
9. Soman K., Chakravarthy S. and Yartsev M.M. (2018) A hierarchical anti-Hebbian network model for the formation of spatial cells in three-dimensional space. Nat Commun 9: 4046. https://doi.org/10.1038/s41467-018-06441-5
10. Sarah Webb (2018) Deep learning for biology. Nature 554: 555–557. doi: https://doi.org/10.1038/d41586-018-02174-z