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THE ROLE OF MODELS IN THEORETICAL BIOLOGY 209

1.    True model. Ali necessary criteria of similarity, including geometrie factors, are invariant, and pertinent data obtained from the model can be extrapolated to prototype by predictable scaling factors.

2.    Adequate model. Not all similarity criteria are invariant, but some measured data can be extrapolated to prototype on the basis of rational analysis.

3.    Distorted model. Some of the pertinent similarity criteria have been deliberately distorted (usually for practical reasons of model construc-tion) and some data from the model can be extrapolated to prototype following analysis.

4.    Dissimilar model. The model does not look like the prototype and may not be geometrically similar, but some data from the model can be extrapolated because other pertinent similarity criteria are invariant.

To this listing may be added an additional category, not generally of much value in engineering work.

5.    Qualitative model. The model does not allow prediction of any data for the prototype, in spite of the fact that the two systems share in common one or morę criteria of similarity and/or dimensional constants, but has conceptual utility.

Examples of all these types of models of biological systems are known, although most published reports do not make such distinctions and do not include sufficient information on validity of model data to be certain to what extent the model was adeąuate.

B. Modeling and Theoretical Biology

It would probably not be an exaggeration to state that modeling and simulation represent the most important single tools of theoretical biology. In modern science the mere verbal statement of a “theory” is no longer considered adeąuate and mathematical constructs of various kinds are expected. These may be in the form of conventional mathematical eąuations or presented as algorithms or axiomatic structures; all these representations may be encompassed by Computer simulation.

If it is granted that theoretical biology does now, or will shortly, ful-fill these scientific reąuirements, modeling analysis may be construed as a means of compariDg “theories” or specific symbolic representations of systems with the goal of discovering their similarities. Altematively, one may say the goal is to find isomorphisms between various descrip-tions of biological systems.

Although many scientists associate modeling theory mainly with the laboratory or workshop fabrication of smali models, the important progress in the field has been largely along theoretical mathematical