Series Editor
Marie-Christine Maurel
First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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Library of Congress Control Number: 2019943329
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ISBN 978-1-78630-483-4
Roger Buis, professor emeritus at the University of Toulouse (INP – Institut National Polytechnique) – already the author of a considerable amount of research into the biomathematics of growth1 – delivers here a true panorama of the relationships between biology and mathematics over time, and in particular over the course of the last century, accompanied by a series of profound epistemological thoughts, thereby creating a book of great rarity and value.
As we know, mathematics is ancient, just like the interest taken in living things. But the word “biology” only appeared at the beginning of the 19th Century and, whilst E. Kant has already confirmed – and as Roger Buis rightly reminds us – a piece of knowledge is scientific insofar as mathematics has been integrated into it, the explicit idea of applying mathematics to biology is found only with C. Bernard2 – one of the great references in the book.
In contrast with physics, biology resists mathematization, for understandable reasons: the variability of living things, their dependence on time and on the environment, diversity and the complexity of biological processes, the diffuse aspect of causality (sometimes circular) and the difficulty of mastering the operational conditions of experiments have made obtaining consistencies problematic. Hence, the overall appraisal which may be interpreted as disappointing: apart from some specific sectors (in particular genetics), few laws are proven in biology, and even fewer that express themselves in mathematical language.
Roger Buis comments on this, but, going further than the “epistemological obstacles” – coined by G. Bachelard – he takes up the challenge. Even though the description in vernacular language (important in both natural history and Husserlian phenomenology), will always remain the most important in biology, in this discipline we more readily use “modeling” than “demonstration”. Nevertheless, mathematics has transferable applications to biology: beyond the savings made by the move to symbols, using mathematical language is not simply using a “language”, but a true instrument of thinking, of a remarkable tool of intelligibility which, whilst allowing hypotheses to be clearly laid down, will verify the conclusions by the same amount. Because – let us not doubt it – in biology and elsewhere, the scientific approach is always hypothetico-deductive. Whilst certain preliminary conjectures are less significant here than in physics – for example, the choice of a reference frame (dominant, certainly, in factor analysis, but hardly relevant, in general, elsewhere in biology) – others like approximations or simplifications that we will cautiously allow ourselves to use (linearization, or even quasi-stationarity of certain processes) are essential in this and necessarily lead to significant consequences. At least an advantage is drawn from this: modeling allows controlled experimentation. Thus, the modification of a parameter in a model that is elsewhere structurally stable is going to be possible at will. Mathematics, as a result, does not only provide symbolisms. It also contributes concepts and operating modes that allow real life to be simulated3.
By examining history, we also realize that, to use the expression of Roger Buis, mathematics has “sculpted” biology. From the point of view of the continuous, geometry, since ancient times, has given rise to the consideration of symmetries and continuous transformations, implicitly presented by Aristotle with the ago-antagonistic couple of power and action. The first separate formalizations appeared from the medieval period onwards, with the famous Fibonacci sequence, which, founded on strong hypotheses, provided the first model of the growth of a population (as it happened, rabbits). Then, in the Classical epoch, there was the era of the first phyllotaxic “laws” relating to the growth speeds of stems or leaves of plants, as well as to the mechanics of wood and its constraints (G.-L. Buffon, L. Euler). Finally, during the 19th Century and especially during the 20th Century, calculation of probabilities was based on development of Mendelian genetics, then of the genetics of populations, and subsequently on statistical biometrics (R. Fisher). Whereas C. Bernard highlighted the stationarity of the interior environment of living organisms, principles of optimality, underpinned by the calculation of variations (seeking the extremums of a functionality), are going to become dominant in biology, in particular in plant biology. Then formalisms from system theory (L. von Bertalanffy) encouraged A. J. Lotka and V. Volterra to model the dynamics of interactions between species (prey–predator systems, parasitism) with differential equations. At around the same time, projective geometry or geometry of transformations of coordinates will allow the morphology, shape and growth of living things to be summarized. The large project of a universal morphology, inaugurated by J.W. Goethe on the subject of plants4, began to be mathematized by D’Arcy Thomson, whilst awaiting the development of differential topology. With A. Turing and his reaction–diffusion systems, the mechanics of gradient – of which R. Thom later made great use in his famous “theory of catastrophes” – began to be introduced into the theory of morphogenesis5. Soon, the theory of automatons by J. Von Neuman took its turn, and cybernetics by N. Wiener with his command theory and retroaction loops, which were used amongst others in the description of hormone mechanisms. In addition, Roger Buis does not leave aside the formal grammar from N. Chomsky, at the origin of L-systems by A. Lindenmayer (useful to formalize the growth of certain algae), networks from Petri, well-suited to the logical representation of certain plant morphogenesis, the direct or indirect input from quantum physicians (such as N. Bohr, M. Delbrück or E. Schrödinger) to molecular biology, the influence, also on this, of information technology, with the notion of “program”, of linguistics (R. Jakobson) and of the theory of information (C. Shannon) with the notion of “code”. He also collects in great detail all the inputs of structuralist thinking in mathematics which, from the theory of Eilenberg–Mac Lane categories to that of graphs and networks, have allowed a systematic and relational biology to develop, in which the notions of self-organization, emergence, complexity6, scale invariance, order and disorder have become dominant today, leading in fine to the construction of biomimetic automatons (artificial life) and the development of an entire bio-informatics approach relating to simulation.
Obviously, given the spontaneous interaction of the disciplines, the existence of these empirical developments does not constitute a justification in itself. They therefore deserve to be revisited and for us to ask of them: when and under what conditions mathematics is really productive in biology? What do we expect to gain from applying it? Which mathematics should we use, where and why? Roger Buis, in the last chapter of his book, broaches these questions with courage and answers them very precisely, underlining each time the benefit that mathematics brings to biologists. If not predicting, is it about describing or explaining? Do we aim for architectures or processes? In contrast to modern philosophy, which often restricts itself, in the manner of Heraclitus, to dwell on the influence of difference, science – Roger Buis demonstrates this forcefully and epistemologists can but approve it – has the objective of finding invariants. The important thing is not that something changes. The important thing is to consider what does not change within things that change – because it is an invariant only in its connections to transformations. And we find some in biology and in physiology, as well as in physics. F. Cuvier, É. Geoffroy Saint-Hilaire and E. Haeckel already explained some. Today, we see them in metabolic cycles, macromolecules (DNA and RNA), genetic code (to the nearest few exceptions) and cell theory (J. Monod). It remains that in biomathematics, they must be linked to time and space. Since then, the use of continuous, or discrete, formalisms, of spatialized models of random or kinetic regulation processes – models such as those that Roger Buis studies competently and in detail – will contribute to their appearance. In conclusion, the singularity of living things must not be seen as an obstacle, and even though precautions and a certain modesty is required – because the model is not reality, it is, at best, only an isomorphic representation7 – there is no doubt about the usefulness of mathematics of living things in a well-defined conceptual framework, and that it allows us to achieve the objective of all well-understood science: making sense of what we are studying.
This eloquent pledge by Roger Buis in favor of biomathematics – a defense and illustration of rational models that are available – is much more than a simple memento or a catalogue. It is a true epistemological and scientific reflection, precise and nuanced, nourished in wide-reaching culture and which overcomes fractures and controversies. From Aristotle to G. Canguilhem, great names of Western thinking are found, which means that philosophy, and even the honesty of man, is not out of place. Going much further: the structure, which also reinforces the convictions of the researcher, is suitable ground for germination of new ideas. Without attempting to take on the role of a know-it-all, we even have a desire to extend it.
To mention an initial fact here, which will speak to mathematicians, today we know that non-associative algebra models Mendelian genetics, providing what is known as “genetic algebra”8 or, more precisely, “gametic”9. Applied to hemoglobin, mathematics then intervenes not only as an instrument of description and explanation, but also as a forecasting instrument in research into the evolution of certain blood diseases and their oscillations. They thus allow calculation to be made of the stable states of an infected population, which corresponds to what is known in mathematics as “idempotents” of algebra, whose coefficients verify Hardy–Weinberg’s law10. Each time the law is satisfied, a stable state for the disease will have been identified. Another example could involve generalization of the notion of an invariant. Roger Buis often highlights the need for well-defined models whose applications are valid at a certain scale, although in biology there are also multi-scale processes. But we are also aware of phenomena of scale invariants, which in particular demonstrate fractal structures. Certain forms – in particular, plants – follow these models that computer science has brought to the forefront very precisely11. Finally, not only what were previously known as primary qualities (above all the form), but also the secondary qualities (e.g. the color12), are just as easily modeled mathematically, which thus leads to the idea of a kind of “algorithmic beauty”13 of nature. Serious ecology itself has for a long time been sliding from politics towards science, resolutely entering the world of modeling14. On balance, then mathematics no longer has the “severity” that Lautréamont lent it. Well understood, it is again gaining favor. But this is also already, more than anything else, what Roger Buis’s book does.
Daniel PARROCHIA
Honorary professor
University Jean Moulin-Lyon III
If in biology we wish to reach understanding of the laws of life, it is then necessary not only to observe and notice vital phenomena, but in addition the intensity relations in which they are related to each other must be set up numerically.
This application of mathematics to natural phenomena is the objective of all science, because the expression of the law of phenomena must always be mathematical.
C. Bernard, 18651
This opinion from the physiologist C. Bernard is emblematic of the relationship that biologists must construct between experimental practice and formulation of their observations. Biology, and in particular physiology, an experimental discipline par excellence, is thus expressly encouraged to make use of mathematics, in its very essence an abstract science. This demanding position caught between two such different practices does not come without its problems. First, despite numerous examples that prove a positive connection between these two disciplines, its very principle sometimes causes incomprehension that is difficult to remove. Let us clearly state that some biologists do not fully appreciate the true contribution that they could obtain from mathematics, restricting themselves to using it simply to analyze their observations statistically, a given law or a given recognized model. Using it more in this manner is a calculated exercise, an illustration, than as a means of extracting new important information about properties of the processes that they are studying.
Thus, establishing a connection between these two disciplines and the true benefit of combining them leaves some biologists in doubt about the practical advantage of attempting an in-depth approach to bringing them together. In their defense, it can be said that bringing them together in this way implies an “intellectual investment”, which is doubtlessly more rigorous, in any case more rigorous, than the introduction of concepts and methods from chemistry and physics into biology, disciplines with the common factor of being “sciences of nature” based on the observation and measurement of tangible phenomena. Since the situation involves both combining or confronting points of view with each other and a collaboration of capabilities, we are aware of the difficulties of establishing real operational contacts, always subjected to institutional separation in both teaching and in the organization of research laboratories. Despite undeniable progress (both psychological and sociological), which must be acknowledged, here we have a recurring subject of debate that merits an overview, whilst there is an accentuation of this general evolution in which many sectors of human activity are being “mathematized”. In light of this observation, it is necessary to specify that these relations are necessarily marked by what is intrinsic to each discipline in its nature and practice.
The field of mathematics is today a highly diverse body of knowledge in terms of objectives and methods. Its current state results, as we know, from a progressive extension of its various fields of study. Whilst, in Antiquity, arithmetic, Euclidean geometry and trigonometry formed its undeniable foundations, several other axioms were laid down afterwards, introducing other points of view that were particularly fruitful. As an example, we can reference the extension of the notion of number (ranging from natural whole to complex and quaternions), the move from arithmetic to algebra (notions of “group” and algebraic structures), the calculation of probabilities or even infinitesimal calculus (differential and integral, optimization), the diversity and extent of whose applications is well-known. Moreover, this evolution continues to the point that “mathematics is currently undergoing an extraordinary prosperity” in both qualitative and quantitative terms2.
We know that there is a link between this evolution and the constant approach by mathematicians to always refer to a given referential, which allows them to set up the underlying layers to their work by defining an appropriate system of coordinates. According to the problems encountered, the mathematician places themselves “like a good surveyor” in a given “space”, allowing them to abandon the classic Euclidian reference to work in another world, a very varied world, the specifications of which are based on the notions of vector space and topological space. A proliferation of this kind obviously raises questions for biologists from the moment that they know that the processes they are studying cannot necessarily be reduced to questions of distances and metrics. But they also, depending on the case, call on questions of vicinity and limits. We are able to see in this the consequence of a sort of inhomogeneity of its workspace, which already illustrates, as we will see, the diversity of the dynamics of deterministic systems whose behavior can be marked by jumps or bifurcations – the point at which biology and mathematics comfortably coincide.
Whilst this mundane remark about the sudden appearance of new paradigms is of course true for all disciplines, we can question the status of each of them and examine what the specificity of each field of scientific knowledge would be. This is true for both epistemological assumptions and methodology, independently so from everything that arises from interdisciplinarity3. The question can be raised particularly for biology if we base our judgment on the permanence of the discussion surrounding the originality of this discipline with a view to specifying if and how it is different from physical sciences whilst allowing it to take root more and more4.
In fact, these interdisciplinary relationships are often highly interlinked, in such a way that the connections that biology makes with mathematics are not always independent of those that are made with physics, where the influence of the latter lies in both the clearly more advanced formalization and a certain affinity with biology that itself has for a long time involved various physical notions at work in the different processes that it examines. Thus, there is a point of view known as “physicalistic” (or “mechanistic”) that is obvious and consistent in various forms with deducing the properties of biological things from the existence of underlying physical mechanisms (or physico-chemical). This undeniably marks the positions of principle that biology attempts to establish with a view to a formalized representation of the phenomena of living things. In the opposite sense, a strictly mathematical point of view only calls on notions or mathematical concepts without needing, at least temporarily, to attach or inject into this a solid interpretation. The question of whether this discrimination is schematic is something we are able to agree on, as attested to by the variations in vocabulary, where it has been possible to describe a single general type of approach as “biophysical” as easily as “biomathematical”5. By this term of “formalized representation”, we understand (although the term “representation” is itself a subject of debate for certain epistemologists, speaking more about “theorization”), a description of objects or biological processes – their specific characteristics and their properties – that take on a “particular” language, characterized by a specific coding and rules, where writing in literary or vernacular language only constitutes an initial approach to a description, an initial report on the study.
To begin, let us lay out the objective of this book a little, using the following simple problem: how to mathematically express one of the most-researched biological processes – the kinetics of growth. Whether this is, for example, the evolution of biomass of an in vitro culture of cells or demographic variations of a natural population in situ.
A first method of expressing this, in a purely empirical way, does not lay down any a priori hypothesis on the phenomenon in question. It is simply associated with operation of a statistical smoothing of data by a “neutral” equation, meaning without a connection to the underlying biological mechanism (cell division, reproduction, mortality, etc.). This is the very objective of classic methods of statistical biometrics whose principles we recall hereafter. In the present case, we resort to a polynomial equation, for which we will need to determine the degree n, ensuring correct adequacy. The objective of this kind of statistical regression is to express the relationship between an explained variable Y and an explanatory variable X (which in this case would be time) according to the following stochastic model:
This model6 only postulates the probabilistic hypotheses that define the random part ε and which correspond to the primary notion of parent population P or theoretical set from which we suppose that the sample (the measured Yi) was drawn. The distribution law of P and its parameters (the β above) must be estimated from the sample data. The difference between this theoretical model estimated in this way and the observed data represents the part known as “residual”; differences between the observed values and the values predicted by the model. Hypotheses about P and the quality of estimations are the basis that is required to allow an analysis of variance and decide on the value of n, according to the risk of chosen decision.
Another approach consists of writing one or several differential equations that express the type of presumed variation f that the speed of growth undergoes over time, as a function of the biomass or of the instantaneous number, written:
where P is a set of parameters to be identified. This model can be modified by adding a delay effect that expresses the action of a previous state g[y(t – τ)], which corresponds, for example, to a period of latency or of maturation. We are, therefore, led towards formulation of specific hypotheses (that it will be necessary to verify) about these functions f and g, and about the parameters P, whilst also basing it on what we know by simple empirical observation. Each model of this type is characterized by the nature of the hypotheses set down and by the type of observations to which we resort in order to found them (e.g. the shape of the growth curves with one or several points of inflection). A model of this type is therefore mixed in nature in the sense that the hypotheses that define it are not entirely “free”.
Finally, let us add in another way of proceeding that adopts an “axiomatic” position so named because it is not based on an experimental basis, but instead on a priori ideas that confer to it a character that is considered more theoretical or more abstract. For all this, it cannot be considered more “mathematized”. Application of the theory of automatons to report a growth or morphogenesis correctly illustrates this position of principle. In this case, we lay down as point of departure the vocabulary of the different possible states and the rules of transition (generation of a new cell, change of state). We will see that the difference with the differential formalism is not reduced just to the choice of continuous versus discrete, but affects the adopted hypotheses.
This type of discrimination between different types of models (equivalent grosso modo to opposing something purely formal with something that is partly empirical) of course allows us to add a little order to a varied set of approaches7. However, these distinctions are rather conventional, meaning by this that the essential consists instead of clearly resting on the three following points:
Moreover, this is not without its link to the traditional debate about the nature of mathematics themselves. These, as we know, are not independent of considerations of an experimental nature that have contributed to their own development in parallel with their fundamental axiomatics. The subject is thus rich with a variety of positions, leading to underlining the characteristics, and taking into account the historical and epistemological aspects of these connections.
These aspects will be addressed later concerning the essential “points of view” of the biomathematical panorama, which is still in the process of being established and justified. But we can agree that this isn’t the place to talk about developments and must refer back to more specialized studies when a given fundamental situation relating to philosophy of biology takes shape. At least it seems useful for us that this place allows the diversity of situations to appear by trying to highlight the pertinence and the particularities of representation that each of the considered approaches automatically offers. This difficult task is not in vain, as the course of biomathematical literature allows distinction to be made between what still comes from simple speculations (like a kind of expedition to develop) and what, given a true effectiveness, presents itself from now on as a true instrument (both technical and conceptual) that the biologist can use for the benefit of their own work. In any case, we believe that these conditions must define the complexity of these biological–mathematical relationships, and must thus seize the originality of different approaches, which it would be better to specify without exaggerating their differences, as then it will not lead to irreducibility. In a rather abrupt illustrative manner, we could say that in this exploration, there are numerous points of view, yet not all their sources can be easily used. Thus, we can contribute in this discussion to the reasoned expectation of biologists confronted by the continuous extension of the place of mathematics, itself highly varied, in practice of his discipline.