Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical modeling, and abstractions about living organisms to investigate the principles that govern the structure, development, and behavior of biological systems. It can be understood in contrast to experimental biology, which involves the conduction of experiments to obtain evidence in order to construct and test theories. The field is sometimes called mathematical biology or biomathematics to emphasize the mathematical aspect, or as theoretical biology to highlight the theoretical aspect. Theoretical biology focuses more on the development of theoretical principles for biology, while mathematical biology focuses on the application of mathematical tools to study biological systems. These terms often converge, for instance in the topics of Artificial Immune Systems or Amorphous Computation. Mathematical biology aims at developing mathematical representations and models of biological processes, using the techniques and tools of applied mathematics. It can be useful in both theoretical and practical research. Describing systems quantitatively allows for more precise predictions about those systems and the isolation and consistent analysis of features which might not be immediately obvious to an observer noting down qualitative features. Because of the complexity of living systems, theoretical biology employs several fields of mathematics, and has contributed to the development of new techniques.
History
Early history Mathematics has been used in biology as early as the 13th century, when Fibonacci used the famous Fibonacci series to describe a growing population of rabbits. In the 18th century, Daniel Bernoulli applied mathematics to describe the effect of smallpox on the human population. Thomas Malthus' 1789 essay on the growth of the human population was based on the concept of exponential growth. Pierre François Verhulst formulated the logistic growth model in 1836. Fritz Müller described the evolutionary benefits of what is now called Müllerian mimicry in 1879, in an account notable for being the first use of a mathematical argument in evolutionary ecology to show how powerful the effect of natural selection would be, unless one includes Malthus's discussion of the effects of population growth that influenced Charles Darwin: Malthus argued that growth would be exponential (he uses the word "geometric") while resources (the environment's carrying capacity) could only grow arithmetically. The term "theoretical biology" was first used as a monograph title by Johannes Reinke in 1901, and soon after by Jakob von Uexküll in 1920. One founding text is considered to be On Growth and Form (1917) by D'Arcy Thompson, and other early pioneers include Ronald Fisher, Hans Leo Przibram, Vito Volterra, Nicolas Rashevsky and Conrad Hal Waddington.
Recent growth
Interest in the field has grown rapidly from the 1960s onwards. Some reasons for this include:
The rapid growth of data-rich information sets, due to the genomics revolution, which are difficult to understand without the use of analytical tools Recent development of mathematical tools such as chaos theory to help understand complex, non-linear mechanisms in biology An increase in computing power, which facilitates calculations and simulations not previously possible An increasing interest in in silico experimentation due to ethical considerations, risk, unreliability and other complications involved in human and non-human animal research
Areas of research Several areas of specialized research in mathematical and theoretical biology as well as external links to related projects in various universities are concisely presented in the following subsections, including also a large number of appropriate validating references from a list of several thousands of published authors contributing to this field. Many of the included examples are characterised by highly complex, nonlinear mechanisms, as it is being increasingly recognised that such examples may be best understood through a combination of mathematical, logical, physical/chemical, molecular and computational models.
Abstract relational biology Abstract relational biology (ARB) is concerned with the study of general, relational models of complex biological systems, usually abstracting out specific morphological, or anatomical, structures. Some of the simplest models in ARB are the Metabolic-Replication, or (M,R)--systems introduced by Robert Rosen in 1957–1958 as abstract, relational models of cellular and organismal organization. Other approaches include the notion of autopoiesis developed by Maturana and Varela, Kauffman's Work-Constraints cycles, and more recently the notion of closure of constraints.
Algebraic biology Algebraic biology (also known as symbolic systems biology) applies the algebraic methods of symbolic computation to the study of biological problems, especially in genomics, proteomics, analysis of molecular structures and study of genes.
Complex systems biology An elaboration of systems biology to understand the more complex life processes was developed since 1970 in connection with molecular set theory, relational biology and algebraic biology.
Computer models and automata theory A monograph on this topic summarizes an extensive amount of published research in this area up to 1986, including subsections in the following areas: computer modeling in biology and medicine, arterial system models, neuron models, biochemical and oscillation networks, quantum automata, quantum computers in molecular biology and genetics, cancer modelling, neural nets, genetic networks, abstract categories in relational biology, metabolic-replication systems, category theory applications in biology and medicine, automata theory, cellular automata, tessellation models and complete self-reproduction, chaotic systems in organisms, relational biology and organismic theories. Modeling cell and molecular biology This area has received a boost due to the growing importance of molecular biology.
Mechanics of biological tissues Theoretical enzymology and enzyme kinetics Cancer modelling and simulation Modelling the movement of interacting cell populations Mathematical modelling of scar tissue formation Mathematical modelling of intracellular dynamics Mathematical modelling of the cell cycle Mathematical modelling of apoptosis Modelling physiological systems
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