A two-dimensional transition

 by Nandakishor Krishnan

I started my expat life in Budapest in early 2022; my first international visit. Being already nervous about what to expect in Europe, I hadn’t heard much about the city of Budapest before. The first thing that used to come to my mind was hearing the city name recurring in some of the Avengers movies (Hawkeye and Black Widow background story). Although having heard of the legends Judit Polgár, Ernő Rubik, Frigyes Riesz, Alfréd Rényi, and Pál Erdős, I probably did not know they were of Hungarian origin. Other than that, I wasn’t aware of any other aspects. At times I feel like I should have known a bit more about Hungarian culture, lifestyle, history, etc., even before arriving. However, based on my experiences, I must admit I have been fascinated by the realization of how typical Hungarians value family and traditions over anything, maybe because my own culture values one’s strong affinity with family. Nevertheless, my naivety also extended to the work domain, as simultaneously I had a change of research field too.

During my early life of reading graduate-level textbooks on applied mathematics, it was books on dynamical systems and their applications that intrigued me and introduced me to applications of mathematics in biology. For example, the mathematics textbook “Partial Differential Equations” by Lawrence C. Evans and the bible of present-day evolutionary game theory, “Evolutionary Games and Population Dynamics” by Josef Hofbauer and Karl Sigmund. Having a background in pure mathematics as an undergraduate, taking up research in theoretical biology was a “major (evolutionary?) transition” for me. Even preparing slides for presentations took a turn, as I have been using LaTeX typesetting compilers before. Including more biology into the slides required a bit more flexibility while preparing the slides, which, at least to my capabilities, is time-consuming in LaTeX. I felt differences in the way I was supposed to read, interpret, and write articles. In pure mathematical articles, one first establishes the notations, states the theorems, maybe some lemmas and definitions in between, and finally concludes with the proofs. As original work, pure mathematicians are often expected to create new theorems and proofs. But in theoretical biology what comes first with most concern is the “story”. One first gets motivated by biological principles and develops the hypotheses, which are later the basis for mathematical modeling. The key element here is the starting point, that is, the story in the hypothesis, and its biological relevance and convincibility.

The kind of mathematical concepts that I have been using for my work are mostly based on graduate-level textbooks. However, the novelty and the contributions are in developing models based on real-life biological scenarios, and the efficiency of the models in providing interesting biological inferences. The models can get complicated to analyze and often one must use brute-force approaches. For a simple example, it is easy to find the eigenvalues of a second order general matrix either analytically or computationally, but it can get complicated for matrices of order higher than five or six. There isn’t always much one can do about it. The more one tries to get closer to biological realism, the more the models get complicated and difficult to study analytically, and perhaps even computationally. Whereas general models may not have sufficient details to rely on for significant conclusions. I have been advised that mathematical modelers should always have this thought of tradeoff between biological accuracy and model analyzability at the back of their mind while developing models. Models that are novel in the sense of how accurately they describe biological scenarios usually get really complicated. Whereas models that are built on existing simpler models and that give new results can be easy to comprehend and usually receive better recognition. After all, it is a matter of subjectivity and there are researchers who are inclined to either side. This reminds me of the famous quote by British statistician George E. P. Box that all mathematical modelers are familiar with: “All models are wrong, but some are useful” (George E. P. Box, 1976). This is indeed the case because it is nearly impossible to model anything with perfect precision, and there are always some limitations. However, even the simplest of the models can be helpful in gaining insights and better understanding, at least to some extent. Where exactly the models “cut the corners” to make it feasible to analyze is often a tricky question to tackle.

I used to take things for granted, that is, I was under the impression that once what you want to convey is properly written mathematically, it is sufficient. But learning to state even the obvious and interpreting it from a different perspective made me realize that even the simplest of things might actually be useful, and it really opened my way of thinking. I remember my supervisor mentioning that each time a biologist reader sees a mathematical equation while reading an article, the likeliness of the person continuing reading declines by almost half. This might have been a joke, but I have realized that there is some fact in it. Pure biologists prefer to visualize everything to comprehend material, especially mathematical concepts like equations, inequalities, or functions. To someone with a mathematics background, equations, inequalities, functions, etc., give more intuitive information.  The importance of visualization to the reader is something that I have been ignorant about, except in cases where it is critical and inevitable. For example, phase portraits are often inevitable in dynamical system analysis, but region plots that divide an area based on inequality may not be necessary for a proper understanding of the point (at least, that’s what I thought). But I have started to think otherwise, being open, and including all information without any prejudice might be helpful while making biological inferences from the mathematical results. For a simple example, consider linear stability analysis of equilibrium points of a simple two-dimensional dynamical system. A mathematician would be happy just with finding the solutions, conditions for existence and stability of all the equilibrium points. But the very idea of existence and stability of an interior equilibrium in a population dynamics framework implies that the two populations represented as the two dimensions of the dynamical system can stably coexist. The example I have provided is a basic one, but I have come across much more relevant cases where I missed noticing some interesting inferences, while I was happy with the mathematical analysis. It is these simple but elegant observations that can often make a difference. Furthermore, most often, pure mathematicians are happy with proving the existence of solutions to problems rather than clear-cut methods of synthesis. This approach has often led me to underexplain certain things while anticipating that the audience or readers would be annoyed by stating the obvious. To me, the whole notion of “obvious” has evolved over time, I would now happily reconsider adding all the details if possible.

I conclude my words with a note on the importance of the interface between mathematics and biology, and with a caveat that my opinions might be a bit naive, but I believe at least a few people could align with my thoughts. I personally would like to think that mathematical modelling in biology is a tool that has the potential to open further avenues for different other fields of biology to proceed. For instance, mathematical models have been utilized in epidemiology to analyze and predict epidemic dynamics; thus, facilitating potential disease control measures. Similarly, experimentalists and statisticians could also benefit from taking inspiration from theoretical models. The transferable skills of mathematical modelers also enable them to work on diverse projects, perhaps even in different environments. For instance, a theoretical biologist with expertise in stochastic processes and data processing could possibly have the skills to suit a financial analyst. After all, mathematical language is universal and its utilization to solve real life problems has wide scope.

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