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THE BOLE OF MODELS IN THEOBETICAL BIOLOGY 169

whole problem of symbolic conceptualization of biological relationships. In all of these works there is a lack of basie theory for determining when a model can provide new and valid quantitative information about a biological prototype.

It becomes elear that many different kinds of models may be created of living entities. McLeod and Osbom (1966) suggest that models are really most useful from the conceptual and not quantitative yiewpoints, insofar as they provide insight into operational strategy of organisms. The lever model by Borelli of the human skeleton has still not been brought to a point where data from models may actually be extrapolated with confidence, although much progress has been madę in this direction, but it served as a valuable starting point for objective understanding of certain aspects of bodily functions. Reviewing the purpose of Computer models in biology, Waterman (Waterman and Morowitz 1965, p. 402) remarks: “Hypotheses conceming poorly known or as yet experimentally unapproachable relationships can be tested by experiments carried out on the Computer itself.”

The subsequent sections of this chapter attempt to present an orderly framework for classification of biological models. They provide examples of yirtually all published reports dealing with biological models and simulation.

II. Mathematical Modeling Theory

In pure mathematics the concept of a “model” has a very specialized meaning, quite different from that of technology. Mathematical model-ing theory is based on highly specialized techniques such as abstract algebra, group theory, similarity transformations of differential equa-tions, applied dimensional analysis, inyariant properties of systems of differential equations, inyariant properties of automata, and system-atics of algorithmic languages. A detailed discussion of these yarious methods is beyond the scope of this chapter.

A. SlMILABITY CEITEBIA FOB PhYSICAL AnALOGS

It is easiest to understand quantitative modeling criteria by reference to physical analogs, in which similarity with a prototype is precisely defined by invariance of “dimensionless numbers" or “physical criteria of similarity.” There are numerous books, such as those of Langhaar (1951), Duncan (1953), and Birkhoff (1960) devoted to dimensional analysis and physical analog theory, whose technical aspects are dis-cussed inmore detail inother works of the author (Stahl, 1963a,b, 1967).