Science 20 December 1996:
Vol. 274. no. 5295, pp. 2039 - 2040
DOI: 10.1126/science.274.5295.2039
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Perspectives
Biologists Put on Mathematical Glasses

Torbjörn Fagerström, Peter Jagers, Peter Schuster, Eörs Szathmary

T. Fagerström is in the Department of Theoretical Ecology, S-223 62 Lund, Sweden. 
E-mail: torbjorn.fagerstrom@teorekol.lu.se. P. Jagers is in the Department of 
Mathematics, Chalmers and Gothenburg Universities, S-412 96 Gothenburg, Sweden. 
E-mail: jagers@math.chalmers.se. P. Schuster is at the Institut fur Theoretische 
Chemie und Strahlenchemie, Universität Wien, A-1090 Wien, Austria. E-mail: 
peter.schuster@tbi.univie.ac.at. E. Szathmary is at the Collegium Budapest, 1014 
Budapest, Hungary. E-mail: szathmary@colbud.hu
Nothing has shaped the development of thought in physics more than Galileo's 
statement: "The book of Nature is written in the language of mathematics" (1). 

Physicists take it for granted that the important questions have answers that can 
be cast into mathematical formulas. Ever since Newton, physics and mathematics 
have lived in a fruitful symbiosis, with a great deal of cross-fertilization to 
the benefit of both disciplines. Physics, as we phrase it nowadays, is concerned 
mainly with the development of a comprehensive and unifying mathematical theory 
of Nature. The physicists are approaching this goal by experimentation, 
abstraction, and generalization.

Biology differs from physics in that it has an indispensable historical 
component. This was stressed already by Ernst Haeckel and most properly phrased 
by Theodosius Dobzhansky in his statement that "nothing in biology makes sense 
except in the light of evolution" (2). For this and other reasons, much of 
biology has traditionally described the overwhelming diversity and unique 
variability of the living world and has used a scientific methodology based on 
observation, description, and classification. Although generalization and 
abstraction, where feasible, was always aimed for, mathematics was usually not 
used in this process. For example, last century's greatest naturalist, Charles 
Darwin, laid down his great theory of evolution and the origin of species without 
making use of a single equation.
 
This century has witnessed an increased use of mathematics in many fields of 
biology, prominent examples being cell kinetics, population ecology, 
epidemiology, and population genetics. 
In the case of population genetics, the 
formal approach was introduced already last century by Gregor Mendel, who applied 
his knowledge of probability theory to the problem of inheritance, and the 
mathematics of heredity was then taken up and successfully developed by Fisher, 
Haldane, and Wright. Indeed, the neo-Darwinian synthesis that emerged by the 
middle of this century owes much of its success to the abstractions, 
generalizations, and formalizations of theoretical population ecology and 
theoretical population genetics, and it may well be considered to be the first 
case of a reasonably mature, proper theoretical biology.


What, then, is theoretical biology about?
 Can problems be identified in biology 
that would benefit from current mathematical approaches?
 Is there an obvious need 
for the development of new mathematics?

 These questions were addressed at a 
meeting in Sweden organized recently by the European Science Foundation.

As to the first question, a mere heap of mathematical models in biology does not 
constitute theoretical biology. It is rather a discipline, aiming at a coherent 
body of concepts and a family of models, in which passage from one concept to 
another, or from one model to another, must follow a regularized pathway. Adding 
to theoretical biology can occasionally mean little or no mathematical work, at 
least in terms of "numbers" and "solutions," but the conceptual contribution must 
then be significant. Waddington's epigenetic landscape is such an example. But 
these concepts must ultimately turn out to be mathematizable. Mathematical 
biology, on the other hand, deals with questions where a solid conceptual 
framework already exists: it constitutes an extension, refinement, and 
elaboration of established, simple models. It also incorporates the rigorous 
analysis of mathematical structures applied in biology.

Mathematics inspired by physics is largely based on symmetry considerations, but 
symmetry never played the same fundamental role in biology as it does in physics: 
there are no quantum numbers of life. Nevertheless, theoretical biology has so 
far largely borrowed its methods from physics; in particular, the use of 
differential equations--this brilliant product of the Newtonian world-view--and 
the techniques for analyzing their dynamical properties have dominated much of 
theoretical biology. One must acknowledge, however, that biology is only partly 
Newtonian.
 It is individuals who make up populations, at all levels from 
molecules and tumors to whole species, and we must not forget that variation 
occurs at all these levels in the most conspicuous manner. It is not without 
reason that Boltzmann remarked that  Darwin's work was an intuitive "statistical 
mechanics of populations."

Modern probability theory offers possibilities to describe individual behavior, 
even in situations that exhibit much individual variation, and even when these 
variations do not follow the standard distributions of elementary statistics. 
From these individual properties, characteristics of the whole, like rates of 
growth, evolution, or extinction, might then be deduced. 
A more rigorous 
application of probability theory to biology, in terms of individuals with 
varying behavior, rather than in the streams or fluxes of classical physics, 
should be close to biological thinking. This may even inspire developments in 
mathematics and eventually turn out to be relevant for modern physics, which long 
ago left the tradition of thinking in terms of continua in favor of systems of 
interacting particles, as discrete as are the individuals of biological 
populations.

In addition, biology should also stimulate 
the development of new branches of mathematics, tailored to the specific needs of theoretical biology.
 Most urgent 
is the current explosion of data produced in the various branches of biology. 
Leading the pack in this regard are genome sequencing and the molecular genetics 
of development
. Similarly, as taxonomists and ecologists gather more information 
on Earth's biodiversity, databases that are already large will continue to grow. 
This data explosion requires theory to be advanced because "no new principle will 
declare itself from below a heap of facts," as Sir Peter Medawar stated 
precisely.
 To this field of urgent need for theory one can add a list of unique 
biological phenomena, like reproduction, selection, adaptation, symbiosis and 
"arms races," that await to be cast in a rigorous theoretical framework. Advances 
by mathematicians on these topics should be welcomed, but in a critical spirit, 
by the theoretical biology community.

In trying to meet the increasing need for conceptualization, formalization, and 
abstraction, biology should borrow some of the virtues of physics. At the same 
time biologists must neither deny nor forget their heritage, nor fall into the 
traps of shortlived fads. 

Several latecomers among the concepts of theoretical 
physics are key concepts in biology as well. This applies to network dynamics, 
which is relevant to the analysis of metabolism or the immune system; to 
self-organization, which is what developmental biology is partly about; and to 
the emergence of new properties from synergy of interactions, which is presumably 
why multicellularity arose eons ago.

All these topics were discussed at the recent meeting in Sweden, together with an 
outlook to the open questions of biology.

 Why and how do information and 
complexity increase in evolution?
 What causes Nature to build more complex things 
in a hierarchical manner with principles that are recurrent at all levels?


 Genes, 
for example, are integrated into genomes, cells into multicellular organisms, and 
individuals into societies. 
Recurrence of the same principle gives rise to ever 
higher forms of complex life in an apparently open-ended evolutionary process. 

"Organization"--the operation of control systems in specific canalizing 
structures--has been an integrative concept in many areas of biology, but 
theoretical biology has so far had limited success in describing it in formal 
terms. 
Related to this problem is the ongoing occupation of evolutionary 
biologists with equilibrium situations and microevolution: usually, the applied 
dynamic (if explicit at all) is a closed one.
 Major transitions in 
evolution--such as

the origin of life,
the emergence of eukaryotic cells, 
and 
the origin of the human capacity for language,
 to name but a few--could not be 
farther away from an equilibrium.  
Also, they cannot be described satisfactorily 
by established models of microevolution.
 What is needed is an open-ended model, 
in which evolutionary novelties (or, rather, representations thereof) can 
continue to arise indefinitely. This model must be related to a currently 
nonexisting theory of variation, which in turn must be related to the theory of 
organization of objects, the evolution of which we would like to describe.

 A theory of development (ontogenesis) is still missing.

The origin of language is an even more formidable problem. Attempts to solve it 
must be based on ingredients in the theory of evolution, neurobiology, and formal 
linguistics. Because important insights have been mathematized in all these 
fields already, their joint application to this problem will have a strong 
mathematical element as well. But how much of this can be achieved by the 
standard methods is an open question.

Our ultimate goal must be a unifying theory of biology emerging from the 
forthcoming synthesis of three great disciplines: molecular, developmental, and 
evolutionary biology. Such a concept will provide the theoretical basis for a 
biology of the future with its own tools and methods, some coming from 
mathematics, some from computer science, and others, perhaps, from somewhere 
else. We suspect that an enormously exciting period lies ahead.

References

   1. S. Drake, Discoveries and Opinions of Galileo (Doubleday-Anchor, New York, 
1957).
   2. T. Dobzhansky, Am. Biol. Teach. 35, 125 (1973).


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