Mathematical modelling in science
Benoît Perthame wrote a beautiful article Modélisation mathématique dans les sciences published as one of a series of articles Aspects de la Science Mathématique. The reference for Perthame's article is Rayonnement du CNRS No 68 (2016), 12-14. We give below a translation of part of the article, omitting some more technical parts.
Benoît Perthame
Professor at Pierre-et-Marie-Curie University in Paris and currently director of the Jacques-Louis Lions Laboratory (UMR CNRS 7598, Inria), Benoît Perthame is a specialist in partial differential equations.
Invited plenary speaker at the International Congress of Mathematicians in Seoul in 2014, he also received the Blaise Pascal Medal from the European Academy of Sciences and the Grand Prix Inria from the Académie des Sciences (2015). Since the end of the 1990s, B Perthame has been interested in the role played by non-linear models in the life sciences. Self-organisation of cell populations, living tissue dynamics, evolution, neuroscience, are his areas of interest where mathematical formalisms are commonly used.
Biologists now have instruments that generate massive amounts of data: images, genomes and other signals of all types. They are therefore confronted with a new difficulty which consists in processing these data, cleaning them from measurement errors and representing them usefully. Mathematics first appears as a tool to achieve this goal. It does not matter what biophysical mechanisms produce these data, it is already a matter of visualising them effectively, just as experimental methods aim to predict observations in a context too complex for physical analysis.
Besides this use, another mathematical approach to the life sciences consists in modelling, that is to say in expressing in mathematical terms the functioning of a biological system. It corresponds to the need to make sense of observations, to explain their main mechanisms, to predict the behaviour of the system and thus to be able to decide and control. Two fine historical examples are well known. The first concerns the development of epidemics; Daniel Bernoulli describes the model and uses it to calculate the number of lives that would be saved by inoculation in the article at the Académie royale des sciences de Paris in 1760, "Essai d'une nouvelle analyse de la mortalité causée par la petite vérole et des avantages de l'inoculation pour la prévenir". ... The second example is due to L Euler. In 1767 he wrote the article entitled "A general investigation into the mortality and multiplication of the human race". He motivates this research by the need for a "general approach" to interpret disparate administrative records. This is always the case, a model is the best way to explain data.
Among the more recent modelling tools, Turing's instability is one of the most spectacular since it describes a universal principle of formation of "patterns" observed in nature. Contrary to intuition, diffusion, which in general mixes and homogenises, can destabilise a stable differential system. Alan Turing proposes this mechanism in his famous 1952 article. He gives a mathematical analysis, introduces the notion of morphogen and shows that it can explain segmentation during embryonic development. This mechanism will not be observed experimentally until much later for chemical reactions and the computation nowadays allows effective numerical simulations which are reproducible by all. The Turing mechanism, its general principles and its limitations, are still at work in the heart of discussions in different fields of biology. Even if it is challenged with current observational means, its strength is that it still serves as a basis for comparing other organisational mechanisms. On this subject and for a detailed discussion, one can consult the work of Jim Murray.
A few classic areas have long been approached using equations. In ecology, the Lotka-Volterra model explains why the resumption of fishing at the end of the First World War changed the proportion of small and large fish (prey/predators) observed in the markets of Trieste. Predictive mathematical modelling and computation are now used, for example, to forecast forest management, possibly taking into account models of the wood economy. This field has extended to the description of ecological invasions, by combining the movement of individuals and reproduction in favourable areas, opening the way to the phenomena of propagation waves, the theory of which was compared to the experiments by J G Skellam who, from 1951, observed the invasion of the muskrat in Central Europe. These questions of cooperation and competition between species, of genetic diversity, of speed of propagation of populations, are currently finding a new field of interest with taking account of climate change: will the species be able to keep up? This question gives rise not only to practical studies (calculating predictions with models) but also to studies of mathematical interest because a new formalism must be introduced.
Intense activity is also focused on the different models describing adaptation/selection of organisms, Darwinian evolution or epigenetics. How will a phenotypic trait take advantage in a population (of individuals, of cells)? Can there be co-existence of several populations with different traits? Can there be co-evolution of traits representing different phenotypes? How do statistical effects disappear in large populations? What are the comparative effects of adaptation or mutations? All these questions, which appear in fields ranging from ecology (competition between species, resistance to pesticides) to medical fields (resistance to drugs), lend themselves to different levels of modelling using probabilistic tools, game theory, partial differential equations etc. We can intuitively understand the mathematical interest since selection is represented in a very singular way as a single point or a few isolated points in a continuum of possible traits subject to the unpredictability of mutations.
The very vast field of collective behaviour also attracts a great deal of research and covers subjects ranging from molecules to animals, including cells and biological tissues. The aggregation example represents this topic quite well at the individual level; if we simply assume that the individuals move according to the position of their neighbours, we find a specific equations. In this family of equations we find the famous Keller-Segel system which describes chemotaxis and is applicable to many biomedical fields. Despite its derivation in 1971, this system continues to generate an enormous mathematical literature as the behaviour of its solutions is so rich. Since a biological tissue can be understood as a set of cells in interaction with each other and with the extracellular environment, we see the emergence of another mathematical theory, which is multi-scale analysis: what are the properties of a population of individuals when we know the behaviour of each individual? Geometry is also involved, for example in morphogenesis and tissue repair, much studied in several organisms, since the contraction of actomyosin structures plays a key role in the deformation of biological cells and tissues and possibly associated with movements by mean curvature.
Once the models have been established and validated by comparison with observations or experiments, they can lead to very concrete applications in medical fields, as shown in the article by O Saut in this issue.
These few examples show that many questions arising from the life sciences motivate current mathematical research and it is difficult to provide an exhaustive overview, especially as many teams of physicists have focused on living things and open up many perspectives with formalisation tools closer to the mathematical tradition. Currently many other fields of life sciences lend themselves to modelling and neurosciences are arguably among the most mathematised following the success of the Hodgkin-Huxley system to describe the propagation of the potential of action in neurons. A sign of maturity, these models now pose intrinsic questions to mathematics and give rise to new theoretical analyses entirely motivated by the need to understand formalism ... which can confuse biologists.