**This article was originally published in The Oxford Scientist Hilary Term 2022 edition, Regeneration.**

One of the most stunning qualities about mathematics is its universality. Besides describing properties of numbers and shapes, the application of maths to modern science underpins many significant discoveries and technological developments.

Quantum mechanics and general relativity, for example, are both understood with the language of mathematics and have each impacted the design of GPS, spacecraft, and computers—to name but a few inventions.

At the centre of this language lie structures known as mathematical models: sets of equations that translate our understanding of a given natural phenomenon into abstract “words” supplied by calculus, probability, geometry, or some other mathematical field.

Although these models are inherently theoretical and, consequently, do not capture all aspects of reality, they nevertheless can be strikingly accurate.

Newton’s second law (force equals mass times acceleration), for instance, can be reformulated in terms of a differential equation—a structure from calculus that describes how the position and velocity of a given object change. By solving the equation, we can predict with extremely high precision where the object will be at any future time.

While mathematical models have been extraordinary at describing the laws of physics and chemistry, their application in biology has proven to be much more challenging.

This difficulty arises from multiple sources. The behaviour of cells, for example, results not only from their interactions with the environment, but also from unique internal properties, such as the set of genes that they express. Additionally, most biological systems tend to have many interacting components that cannot all be feasibly tracked.

Although these challenges make mathematical models for biology usually worse predictors than their counterparts for physics and chemistry, the models can nevertheless be useful to experimentalists. Indeed, they can help biologists explore different hypotheses in a tractable and cost-effective way, draw attention to potentially significant mechanisms, and guide experimental design.

These benefits can be readily seen in the field of developmental biology, where mathematical modelling has advanced our understanding of stem cell behaviour.

Stem cells are a crucial biological system to investigate because their descendants form the basis of practically all adult organs, and they normally help to replenish tissues in the body. Consequently, they are key for developing practical and useful tools and treatments in regenerative medicine.

Although mathematics has been applied to investigate multiple aspects of stem cell behaviours, I will focus on how the subject has helped discover new mechanisms that guide the migration of neural crest cells, which are a particular stem cell population.

Neural crest cells, which are common to vertebrates, emerge from a structure that eventually becomes the spinal cord. They migrate to specific areas throughout the organism, where, depending on their final location, they differentiate into skin cells, bone cells, or even peripheral neurons.

It is no surprise, then, that disruptions to neural crest cell migration can have serious consequences on normal development, including death or disorders such as cranial malformation. What is surprising, however, is that such complications are rare.

This quality of nature, known as robustness, suggests that cell movement is not entirely random but instead can be explained by their interactions with other cells and their environment—exactly the kind of behaviours that can be represented using mathematics.

Mathematical modelling has drawn from, and added to, experimental studies to suggest plausible mechanisms that confer robustness to neural crest cell migration.

One specific hypothesis, developed after observing that cells contact each other frequently during migration, suggested that migration could arise from a process in which cells repel each other. Such cell-cell repulsion, known as contact inhibition of locomotion in this context, explained why cells avoided becoming densely packed in the early stages of migration, and why they moved towards less populated regions.

Mathematical models developed with this hypothesis, however, did not always yield realistic results. Cells would spread out in simulations and colonise regions not normally traversed by neural crest cells.

This discrepancy between the theoretical model and the true biology led investigators to refine their hypothesis by reasoning that the cells should also attract each other over large distances. This “short-range repulsion, long-range attraction” scenario was able to produce realistic migrating collectives in simulations, inspiring experimentalists to search for molecules that attract, and are secreted by, neural crest cells.

This prediction was eventually verified in frog embryos by the discovery of molecules such as stromal cell-derived factor 1 (Sdf1), which cells move towards. These experimental observations confirmed that neural crest cells could indeed attract each other by secreting Sdf1, and thereby demonstrated the power of mathematics in making experimentally useful predictions.

As the example above illustrate, mathematical models are not set in stone but instead are continually updated based on experimental results.

As more data become available on neural crest cells, the mathematical models used to describe them will become better at anticipating their behaviour over larger time-scales. This in turn may aid the design of new technologies in stem cell engineering and regenerative medicine, as one must first have a reliable and accurate model to make an informed prediction before attempting to control where and when stem cells migrate and differentiate.

In this way, by harnessing mathematics to accurately guide stem cell behaviours, researchers will draw closer to establishing regenerative medicine as a grounded, practical field with reliable tools and treatments.

## References:

Giniunaite, R., McLennan, R., McKinney, M. C., Baker, R. E., Kulesa, P. M., & Maini, P. K. (2019). An interdisciplinary approach to investigate collective cell migration in neural crest. Dev. Dyn., 1–11.

Kulesa, P.M., Kasemeier-Kulesa, J C., Morrison, J.A., McLennan, R., McKinney, M. C., & Bailey, C. (2021). Modelling cell invasion: A review of what JD Murray and the embryo can teach us. Bull. Math. Biol., 83(4), 1–13.

Szabó, A., Melchionda, M., Nastasi, G., Woods, M. L., Campo, S., Perris, R., & Mayor, R. (2016). In vivo confinement promotes collective migration of neural crest cells. J. Cell Biol., 213(5), 543–555.

Colombi, A., Scianna, M., Painter, K. J., & Preziosi, L. (2020). Modelling chase-and-run migration in heterogeneous populations. J. Math. Biol., 80(1–2), 423–456.