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The application of mathematics to biological systems. Mathematical biology spans all levels of biological organization and biological function, from the configuration of biological macromolecules to the entire ecosphere over the course of evolutionary time.
The influence of physics on mathematical biology has been twofold. On the one hand, organisms simply are material systems, and presumably can be analyzed in the same terms as any other material system. Reductionism, the theory that biological processes find their resolution in the particularities of physics, finds its practical embodiment in biophysics. Thus, one of the roots of mathematical biology is what was originally called mathematical biophysics. On the other hand, other early investigations in mathematical biology, such as population dynamics (mathematical ecology), exploited the form of such analyses, such as using differential rate equations, but they expressed their analyses in strictly biological terms. Such approaches were guided by analogy with mathematical physics rather than by reduction to physics and so rest on the form rather than the substance of physics. See Biophysics
Both of these approaches are important, especially since organisms possess characteristics that have no obvious counterpart in inorganic systems. As a result, mathematical biology has acquired an independent and unique character. In several important cases, these characteristics have required a reconsideration of physics itself, as in the impact of open systems on classical thermodynamics.
Surrogacy and models
The idea that something can be learned about a system by studying a different system, or surrogate, is central to all science. The relation between a system and its surrogates is embodied in the concept of a model. The basic idea of mathematical biology is that an appropriate formal or mathematical system may similarly be used as a surrogate for a biological system. The use of mathematical models offers possibilities that transcend what can be done on the basis of observation and experiment alone.
For example, morphological differences between related species can be made to disappear by means of relatively simple coordinate transformations of the space in which the forms are embedded. Surrogacy explicitly becomes a matter of intertransformability, or similarity, and what is true for morphology also holds true for other functional relationships that are characteristic of organisms, whether they be chemical, physical, or evolutionary. These assertions of surrogacy and modeling can be restated: closely related implies similar. This is a nontrivial assertion: “closely related” is a metric relation pertaining to genotypes, whereas “similar” is an equivalence relation based on phenotypes. It is the similarity relation between phenotypes that provides the basis for surrogacy. Thus the question immediately arises: given a genotype, how far can it be varied or changed or mutated, and still preserve similarity?
Such questions fall mathematically into the province of stability theory, particularly structural stability. Under very general conditions, there exist many genomes that are unstable (bifurcation points) in the sense that however high a degree of metric approximation is chosen, the associated phenotypes may be dissimilar, that is, not intertransformable. That observation by R. Thom provides the basis for his theory of catastrophes and demonstrates the complexity of the surrogacy relationship. The fundamental importance of such ideas for phenomena of development, for evolution (particularly for macroevolution), and for the extrapolation of data from one species to another, or the relation between health and disease, is evident.
A closely related group of ideas that are characteristic of mathematical biology may be described as metaphoric. One example of a metaphoric approach is the study of brain activities through the application of the properties of neural networks, that is, networks of interconnected boolean (binary-state) switches. Appropriately configured switching networks are known to exhibit behaviors that are analogous to those that characterize the brain, such as learning, memory, and discrimination. That is, networks of neuronlike units can automatically manifest brainlike behaviors and can be regarded as metaphorical brains. Such boolean neural nets also underlie digital computation, a relationship which is explored in the hybrid area of artificial intelligence. The same mathematical formulation of switching networks arises in genetic and developmental phenomena, such as the concept of operon, and in other physiological systems, such as the immune system.
Another important example of metaphor in biology is morphogenesis, or pattern generation, through the coupling of chemical reactions with physical diffusion. Chemical reactions tend to make systems heterogeneous, diffusion tends to smooth them out, and combining the two can lead to highly complex behaviors. Since reactions and diffusions typically occur together in biological systems, exploring the general properties of such systems can illuminate pattern generation in general.
Such ideas turn out to be closely related to those of bifurcation and catastrophe and have a profound impact on physics itself, since they are inherently associated with systems that are thermodynamically open and hence completely outside the realm of classical thermodynamics. The behavior of such open systems can be infinitely more complicated than those that are commonly explored in physics. Open systems may possess large numbers of stable and unstable steady states of various types, as well as more complicated oscillatory steady-state behaviors (limit cycles) and still more general behaviors collectively called chaotic. Changes in initial conditions or in environmental circumstances can result in dramatic switching (bifurcations) between these modes of behavior. See Chaos
Perhaps the biotechnology that has affected everyone most directly is medicine. Medicine can be regarded as a branch of control theory, geared to the maintenance or restoration of a state of health. It is unique in that the systems needed for control are themselves control systems that are far more intricate and complex than any that can be fabricated. In addition to the light it sheds on the processes needed for control, mathematical biology is indispensable for designing the controls themselves and for assessing their costs, benefits, safety, and efficacy.
In general, the object of any theory of control is to produce an algorithm, or protocol, that will achieve optimal results. Mathematical biology allows one to relate systems of different characters through the exploitation of their mathematical commonalities. Biology has many optimal designs and optimal controls, which are the products of biological evolution through natural selection. The design of optimal therapies in medicine is analogous to the generation of optimal organisms. Thus the mathematical theory appropriate for analyzing one discipline of biology, such as evolution, itself becomes transmuted into a theory of control in an entirely different realm. The same holds true for other biotechnologies, such as the efficient exploitation of biological populations.