Mathematical biology

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Frank Hoppensteadt (2007), Scholarpedia, 2(6):2877. doi:10.4249/scholarpedia.2877 revision #91463 [link to/cite this article]
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Curator: Frank Hoppensteadt

Mathematicians and biologists, including medical scientists, have a long history of working successfully together. Sophisticated mathematical results have been used in and have emerged from the life sciences. Examples are given by the development of stochastic processes and statistical methods to solve a variety of population problems in demography, ecology, genetics and epidemics, and most joint work between biologists, physicists, chemists and engineers involves synthesis and analysis of mathematical structures. Pythagoras, Aristotle, Fibonacci, Cardano, Bernoulli, Euler, Fourier, Laplace, Gauss, von Helmholtz, Riemann, Einstein, Thompson, Turing, Wiener, von Neumann, Thom, and Keller are names associated with both significant applications of mathematics to life science problems and significant developments in mathematics motivated by the life sciences.


Highly interdisciplinary

Mathematical biology is a highly interdisciplinary area that defies classification into the usual categories of mathematical research, although it has involved all areas of mathematics (real and complex analysis, integral and differential systems, metamathematics, algebra, geometry, number theory, topology, probability and statistics, as well as computer sciences). The area lies at the intersection of significant mathematical problems and fundamental questions in biology. The value of mathematics in biology comes partly from applications of statistics and calculus to quantifying life science phenomena, but more importantly from the sophisticated point of view it can bring to complicated real life systems by organizing information and identifying and studying emergent structures. Mathematical scientists, and many more from physics, chemistry, engineering, and medicine have developed and used mathematical methods in biology investigations. It is difficult to grasp the broad influence mathematics has had in biology.

Significant mathematical problems

There are significant problems that need the attention of mathematicians in almost all areas of life and medical sciences. While these are not necessarily the classical problems of mathematics, they can involve novel mathematical structures and interesting insights to known ones. For example, Rene Thom developed and used topology and singularity theory to investigate problems in developmental biology [R. Thom (2001)]. He took the approach of deriving and analyzing canonical mathematical models that capture certain fundamental aspects of developmental phenomena, which in return clarified understanding of the underlying biological processes. Other important examples are investigations of pattern formation in nature, which began with work by d'Arcy Thompson [Thompson (1917)], led to the introduction of appropriate mathematical models for reaction-diffusion systems [A.M.Turing (1936)], and continues today to help studies in many other fields (eg., see [J. D. Murray (2005)]). In addition, methods of stochastic processes have been developed and applied to a variety of problems in all areas of biology, but particularly in both population genetics and molecular genetics [W. Feller (1950), R. A. Fisher (1958), A. V. Skorokhod (2002), E. Lander (2004)]. There are many more sophisticated mathematical results that have contributed to, and benefited from, investigations in biology.

Important problems in biology

Biology is the study of life. The National Research Council [NAS (1989)] identified eleven major themes in biological research:

  1. Cell organization
  2. Ecology and ecosystems
  3. Evolution and diversity
  4. Genome organization and expression
  5. Growth and development
  6. Immune system, pathogens and host defenses
  7. Integrative approaches to organism function and disease
  8. Molecular structure and function
  9. Neurobiology and behavior
  10. New technology and industrial biotechnology
  11. Plant biology and agriculture

Mathematics is used in all of these areas. In addition to this list are numerous applications in medical sciences (eg, tomography and models of physiological systems), in social sciences (eg, demographics), in law (eg, forensics and the television series NUMB3RS), in engineering (eg, design and fabrication of instrumentation),..., the list is long. There have been dramatic developments in biology and in mathematical and computational sciences since 1989, including radiology (eg, fMRI and PET imaging), computational biology and bioinformatics (eg, structure and analysis of genomes), systems biology (eg, metabolic pathways in cells), and brain science (eg, derivation and study of canonical models of neuronal systems). These developments are yielding new understandings of diseases and epidemics, ecological systems, our bodies, and fundamental life processes. Mathematics, along with the sciences and engineering, has played important roles in almost all of these areas.

The longstanding interaction between mathematical methods and biology is illustrated by approaches to the question: How does a brain work? Mathematics has played a role in studies of the brain, ranging from early work of von Helmholtz [H. F. von Helmholtz (1863)] who sought an energy-like function to describe physical and chemical bases of brain dynamics, through work of S. Freud on catharsis [Freud 1895)], to Norbert Wiener on Cybernetics (the study of biological control mechanisms [N. Wiener (1961)]), to mathematical studies of the nervous system [1], and recently to work using mathematics to study consciousness [G. Edelmann (1987), R. Penrose (1989, 1994)].

Ordinary language models and mathematical models

What are barriers to collaborations between mathematicians and biologists? Rene Thom [R. Thom (1993)] points out that two principal approaches are taken toward modeling: ordinary language models, that are precise where data are known and suitably vague otherwise, and mathematical models that are reduced to minimal parts but still produce results consistent with experimental observations. Iteration between these two approaches has produced useful descriptions of biological phenomena and new insights to mathematics.

Among the impediments to joint projects is the imprecision of word models - one sees scientists disagreeing over the interpretation of data. Also, word models are specific to the language used to phrase them; for example, translations of models between English and German require great care, knowledge and precision. On the other hand, mathematical modelers often do not appreciate the nuances, subtlety and complexity of word models; and, biologists often are unfamiliar with mathematical methodologies. In addition, there are cultural differences in funding models; for example, how post-doctoral investigators are trained. Finally, the literature is often biased in one direction or another, thereby limiting access; for example, a book might define what a bacterium is, but not describe the fundamental theorem of calculus where it is used.

There are compelling reasons for pursuing word models and math models together. Obviously, new biological facts are needed from experiments, and the results must be communicated in accessible ways. Word models are powerful for this. On the other hand, mathematics has been useful in many important ways. First, there are many developments in both biology and in mathematics that have benefited by the iteration, not always in phase, between biologists and mathematicians. For example, irregular solutions of nonlinear difference equations arising in ecology models [W. E. Ricker (1954)] were eventually explained by mathematicians in subsequent work on chaos [eg, see J. Glieck (1987)]. Early developments in probability theory motivated by studies of Mendel's and Darwin's theories of genetics and evolution [R. A. Fisher (1958)] led to development of statistical methods that now form a major part of design and interpretation of most experiments. Second, the power of mathematics to synthesize models and analyze their solutions is important. For instance, the Kermack-McKendrick threshold theorem identifies a dimensionless parameter, namely (contact rate between susceptibles and infectives)/(removal rate of infectives), that is used to determine when a population is at risk for propagation of an epidemic disease [W. Kermack, A. G. McKendrick (1927), N. T. J. Bailey (1957), R. M. Anderson, R. M. May (1991)]. Similar problems arise in risk analysis. Third, a mix of analysis and geometry brings to biological problems methods that are used to study and visualize higher dimensional structures. For example, joint work on tomography by mathematicians and radiologists continues to be highly successful [F. Natterer (1986), R. J. Gardner (1995)]. Finally, (for this list at least) mathematical models make it possible to bring high level computer languages and digital computers to bear on biology problems, and they shape the foundations of the emerging area of computational biology. Large scale applications at present are in the design of drugs and in analysis of genome-phenome organization [2].

Theory and computer simulation

Like in physics and engineering, early mathematical models in the life sciences described phenomena over broad ranges of parameter values and widely disparate time and space scales. But, the complexity of a mathematical model had often been negotiated between its level of realism, what information is being sought, and what methods of analysis might be available to study it. Notable examples are Euler's renewal theory in demography, Kermack and McKendrick's epidemic thresholds, Volterra and Lotka's models of prey-predator cycles, and Fisher and Kolmogorov's models of the formation of genetic clines.

Over the past 30 years there has been a shift from mathematical analysis to computer simulation due mostly to improvements in computer power and accessibility. This shift has made it possible to include more information in models and still derive useful insights to them. Moreover, bootstrapping and data mining procedures have introduced new computer-based methodologies that work directly with observed databases. Computers have freed investigators to explore more detailed mathematical descriptions of life.

Although sophisticated high level computer languages, such as MATLAB/SIMULINK, MAPLE and MATHEMATICA, are available, someone using them must understand basic ideas of error propagation, size limitations on data, and have at least a basic understanding of why mathematics has been so surprisingly useful to understanding systematic behaviors. An important point is made by R.M. May [May(2004)]: 'It makes no sense to convey a beguiling sense of "reality" with irrelevant detail, when other equally important factors can only be guessed at'. Highly complex models are now routinely simulated, and then their solutions are studied using data mining and mathematical methodologies, but without informed judgment the outcomes can be meaningless.

Mathematics has in the past allowed humans to visualize things beyond their own physical senses - the extremes in physics now are quantum mechanics, subatomic particles and cosmology. We are beginning to find the words to address comparable problems in biology: How can a brain store and recall many more bits of information than there are neurons? How can a complex disease, such as malaria that involves human, insect and parasite populations, their genetics, immune responses, phenotypic expressions and life cycle dynamics, be disrupted and defeated? How is the complexity of our bodies (our organs, our bacteria, our immune systems, etc.) bound together to function with intention beyond any constituent parts?

The clear thinking of canonical mathematical structures, the uncovering of their behaviors through theory and simulation, and corroboration of outcomes with word models and experimental observations create new knowledge about life.

Further reading

This overview outlines some facets of mathematical biology, but Scholarpedia provides entry points to in-depth treatment of the many topics emerging in collaborations between biology and mathematics. This article is based partly on [F. C. Hoppensteadt (1995)], and listed below the references are additional references that are useful in creating curricula in mathematical biology.


  1. R. Thom (2001), Structural Stability and Morphogenesis, Westview Press; New Ed edition.
  2. D'Arcy W. Thompson (1917) On Growth and Form: The Complete Revised Edition (1992), Dover Publics., New York.
  3. A. M. Turing (1936), Proc. London Math. Soc, 5, 230–265.
  4. J. D. Murray (2005), Mathematical Biology, Springer-Verlag, New York.
  5. R. A. Fisher (1958), The Genetical Theory of Natural Selection, Dover Publications, New York.
  6. W. Feller (1950), Introduction to Probability Theory and its Applications, J. Wiley, New York.
  7. A. V. Skorokhod, F. C. Hoppensteadt, H. D. Salehi, (2002) Random Perturbation Methods, Springer-Verlag, New York.
  8. E. Lander (2004) (ed.), Annual Review of Genomics and Human Genetics 2004, Annual Reviews.
  9. NAS (1989) Opportunities in Biology, National Academy of Sciences Publ. Co.,Washington, DC.
  10. H. F. von Helmholtz (1863), On the sensations of tone as a physiological basis for the theory of music. (2nd ed.) Dover Publics, New York, 1954.
  11. S. Freud (1895), Project on a Scientific Psychology, in The Standard Edition of the Complete Psychological Works of Sigmund Freud, Vol I, pp 283-346. Hogarth Press, London, 1966.
  12. N. Wiener (1961), Cybernetics, (2nd ed.) MIT Press, Cambridge.
  13. G. Edelmann (1987), Neural Darwinism: The Theory of Neuronal Group Selection, Basic Books, New York.
  14. R. Penrose (1989, 1994), The Emperor's New Mind: Concerning Minds and the Laws of Physics, Oxford University Press. Shadows of the Mind: A Search for the Missing Science of Consciousness, Oxford University Press.
  15. R. Thom (1993), Biological organisation: Its definition and mathematisation, in Mathematics Applied to Biology and Medicine (J. Demongeot, V. Cappasso, eds.) Wuerz, Winnipeg.
  16. W. E. Ricker (1954), Stock and recruitment, J. Fish. Board of Canada, 14, 669–681.
  17. J. Glieck (1987), Chaos: Making a new Science, Viking, New York.
  18. W. Kermack, A. G. McKendrick (1927), A contribution to the theory of epidemics, Proc. Roy. Soc. A 115, 700–721.
  19. N. T. J. Bailey (1957), The Mathematical Theory of Infectious Diseases and its Applications, Charles Griffen, London.
  20. R. M. Anderson, R. M. May (1991), Infectious Diseases of Humans, Oxford University Press.
  21. F. Natterer (1986), Mathematics of Computerized Tomography, Wiley, New York.
  22. R. J. Gardner (1995), Geometric Tomography, AMS Notices 42, 422–429.
  23. R. M. May (2004) Uses and Abuses of Mathematics in Biology, Science, 6 February: Vol. 303. no. 5659, pp. 790 - 793.
  24. F. C. Hoppensteadt (1995), Getting Started in Mathematical Biology, AMS Notices.

Internal references

Background materials

Following are some of the resources used as background for this article:

  • Cambridge Studies in Mathematical Biology (1979-2000), Cambridge University Press, New York.
  • L. Edelstein-Keshet (2005), Mathematical models in biology, SIAM: Society for Industrial and Applied Mathematics; New Ed edition.
  • R. A. Fisher (1990), Statistical Methods, Experimental Design, and Scientific Inference: A Re-issue of Statistical Methods for Research Workers, The Design of Experiments, and Statistical Methods and Scientific Inference, Oxford University Press, USA.
  • S. Freud (1955), The Interpretation of Dreams, Basic Books, New York.
  • F. C. Hoppensteadt, C. S. Peskin (2000), Modeling and Simulation in Medicine and the Life Sciences (2nd ed), Springer-Verlag, New York.
  • Mathematical Aspects of Microbial Ecology (1991), Special Issue of Microbial Ecology, F. C. Hoppensteadt, D. A. Lauffenburger, P. Waltman (eds.) 22, 109–226.
  • C. G. Jung (1970), Forward to the I Ching, by R. Wilhelm and C. F. Bayner, Bollingen Fndn. Inc.
  • N.Keyfitz, W. Flieger (1991), World Population Growth and Aging: Demographic Trends in the Late Twentieth Century, U. Of Chicago Press; 2Rev Ed edition.
  • S. Kleene (1952), Metamathematics, Van Nostrand, Princeton.
  • B. B. Mandelbrodt (1982), The Fractal Geometry of Nature, Freeman, San Francisco.
  • J. Maynard Smith (1971), Mathematical Ideas in Biology, Camb. University Press, Cambridge.
  • W. S. McCulloch, W. Pitts (1943), A logical calculus of the ideas imminent in nervous activity, Math. Biophys. 5, 115–133.
  • S. Rubinow (1973), Mathematical Problems in the Biological Sciences, Soc for Industrial & Applied Math., Philadelphia.
  • E. Schroedinger (1946), What is Life? Cambridge U. Press; Reprint edition, 1992.
  • H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, 2003.
  • J. von Neumann (1958), The Computer and the Brain, Yale University Press, New Haven.
  • A. Winfree (1982), The Geometry of Biological Time, Springer-Verlag, New York.

See also

Dynamical Systems, Encyclopedia of Computational Neuroscience, Morphogenesis

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