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THE ROLE OF MODELS IN THEORETICAL BIOLOGY 189

ratios of generation rates, concentration ratios, and absolute concentra-tions, as well as relational criteria pertaining to topology of the bio-chemical pathways.

A “chemical vat” type of cellular biochemistry model is described in a report by Morowitz et al. (1964). It is pointed out that nonlinear differential eąuations are reąuired for such simulations, which raises some difficult ąuestions about system stability. French and Fork (1961) simulated photosynthesis in an analog model based on five differential eąuations. Enzyme induction mechanisms, including deoxyribonucleic acid (DN A) and ribonucleic acid (RNA) effects, are modeled mathe-matically in reports of Roberts and associates (1961) and Lincoln and Parker (1965). Numerical similarity criteria are applicable to these studies.

Somewhat less obvious criteria are implicit in a report of Dayhoff and Heinmets (1963) on a phenomenological model of a complete but very simple celi. In this case, description of the model by sets of differential eąuations is impractical, so special heuristic methods must be used. Simple ratios of rates and concentrations, together with numerous relational invariants, would appear to apply to such models. Largely nonnumerical relational invariants also govem the model of gene-enzyme kinetics of Sugita and Fukuda (1963), based on a combination of Boolean logie and analog computation.

Numerical kinetics may be applied to cellular proliferation models. An extensive numerical representation of tissue proliferation and dif-ferentiation was described by Weiss and Kavanau (1957) and later implemented on a Computer by Kavanau (1960). Lifelike biomass growth curves were obtained for many organs and tissues. Similar prin-ciples are used by Perret (1960) to create a numerical model of a bacterial population growing in a closed vessel. Kinetic models of bonę marrow celi proliferation demonstrating many observed qualitative responses of the hemopoietic system are provided by Patt and Maloney (1964) and Lajtha et al. (1962). Stochastic numerical methods were applied to the same problem in a study by Till and associates (1964).

D. Numerical Genetics Simulation

Simulation of genetic phenomena represents another large class of kinetic models. The goveming eąuations are usually in the form of linear differential eąuations, and the principal variable may relate to relative numbers of particular animals (phenotypes or genotypes) in a population, probability of survival of a gene or genome, or percentage compo-