Preface to the special issue BIOMATH 2019

Both biology and mathematics have existed as well-established branches of science for hundreds of years, and both, maybe not in a well-defined way, have been with the humankind for a couple of thousands of years. Though any civilisation developed some mathematical concepts, possibly the Greeks were the first who studied mathematics for its own sake, as a collection of abstract objects and relations between them. It is quite remarkable that most of such abstract objects and relations, created by human mind, have turned out to be a good reflection of natural phenomena and processes. This has resulted in several centuries of the successful interplay between mathematics, physics and engineering where, on the one hand, the need to precisely describe the physical world pushed the frontiers of mathematics as a necessary tool for the task and, on the other, mathematical theories gave rise to discoveries in the physical and engineering sciences. On the other hand, for most of its history, biology was being developed using a colloquial, everyday language to describe its observations and formulate its laws, with mathematics present at best in an auxiliary capacity. However, over the last few centuries, five revolutions, namely, the microscope, the Linnaean classification, the theory of evolution, genetics and the discovery of the DNA structure, have brought biology closer to the quantitative sciences and technology and have changed the way we think about the living matter. Many scientists predict or see already in progress, the sixth revolution, which Ian Stewart1 calls the mathematical revolution and which should be similar to what happened in physics. There, the ability of mathematics to create relatively faithful images of real-world phenomena has been behind the successes of physics where mostly, instead of talking about real objects, we talk about their mathematical models. In principle, we should be able to do the same in biology the problem, however, is that biology is incredibly complex. The fundamental difficulty is the lack of fundamental principles in the field, such as the Newton laws in physics. The problems are further compounded by the diversity and specialisation, the interplay of levels and scales, the difficulties of experimentation due to the emergent behaviour coming from interactions of various levels of organisation that disallows studying the system through its isolated parts and the problems of feedback control. Despite these difficulties, mathematical biology has been growing at a fast rate, confirming the prediction about the incoming mathematical revolution in biology. The first and the oldest role of mathematics is providing a unified language for biological processes, complementing in this way experimental biology that directly deals with the collection and analysis of data. However, our ambitions go further. Mathematical biology not only should be the language of biology but, drawing on all fields of pure and applied mathematics as well as statistics and computer science, it should develop specific analytical tools tailor-made for the applications in natural sciences, facilitating thus a better understanding of biological phenomena and improving the predictive powers of theories based on observations and experiments. Conversely, mathematical biology should draw the inspiration from life sciences and use biological phenomena as the driving force to push the frontiers of mathematics. It is a subject of current debate whether such an interplay between mathematics and biology is possible. However, though mathematical thinking and mathematics were used in biology before, the scale of mutual interactions has enormously grown in the last couple of decades to the extent that in some disciplines the research has been determined by