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Mathematical models of biological systems
Reproducttve Toxwology, Vol 3, p 79, 1989
0890-6238/89 $3 00 + 00 Copyright © 1989 Pergamon Press plc
Pnnted in the U S A
• Editorial
M A T H E M A T I C A L MODELS OF BIOLOGICAL SYSTEMS
In this issue of Reproductive Toxtcology, Kfirka and Jelfnek offer an explanation for the apparent stabilIty of the incidence of birth defects in humans. Their explanation takes the form of a mathematical model that relates dose of teratogen to the frequency of malformed offspring. Since many re~'ders may be unfamiliar with the rationale underlying Kfirka and Jelinek's work and as a consequence may dismiss their effort as irrelevant to empirical issues, it is perhaps worthwhile to take a moment to reflect on the role that such models play in understanding biological phenomena For most experimentalists, the primary apphcation of mathematical models to biological problems hes with the testing of theory. General-purpose statistical models, frequently consisting of simple additive relationships, are among the most familiar. These models, although not always explicitly stated, are often useful in distinguishing among competing theories when applied to data from well-designed experiments. Quantitative predictmn from theory generally requires a more explicit model-building effort with close attentmn to the definition of parameters and a clear statement of assumptions. Models designed for prediction can be improved with more precise parameter estimates and sometimes by an increase in model complexity through the inclusion of more parameters and interactions among them. These model-using activities are often thought of as part of a process m understanding a biological phenomenon, beginning with identification of relevant sources of variation and ending with estimating their sizes and Interrelationships. So where does the work of Kfirka and Jelinek fit into this paradigm of scientific methodology? It doesn't. Their purpose in
examining mathematical models zs to find one whose behavior mimics the phenomenon of interest - - in this case, a stable frequency of birth defects in the face of an increase in environmental insult. This IS a fundamentally different use of mathematical modelling than discussed above. Here the object of Interest IS the behavior of the model itself. The intellectual challenge is to construct the most parsimonious model possible, one that possesses a minimal number of parameters yet retains the behavior characterizing the phenomenon. This constitutes an attempt to distinguish what can happen from what cannot happen as a consequence of a given set of assumptions. Modelling efforts such as this, however, are among the easiest for researchers immersed in empirical detail to dismiss as irrelevant. This is because the strength of studying models for their behavioral properties hes with their abstract conceptuahzatlon, which is a significant hurdle to testing such models in the manner alluded to above. For example, Kfirka and Jelinek formulate their model in terms of a "critical period" dunng ontogeny and invoke the notion of "swiftness of reaction" to a teratogen. These are difficulties that will arise for any general theory formulated in mathematical abstraction. As a consequence, the search for experimental systems to test such theory will not be an easy one Nevertheless, if progress in scientific understanding can be characterized as a series of reconceptualizatlons of a phenomenon, then abstract modelling efforts must play a pivotal role RICHARD
W.
MORRIS
Analytwal Scwnces, Inc. Durham, North Carolina
79
0890-6238/89 $3 00 + 00 Copyright © 1989 Pergamon Press plc
Pnnted in the U S A
• Editorial
M A T H E M A T I C A L MODELS OF BIOLOGICAL SYSTEMS
In this issue of Reproductive Toxtcology, Kfirka and Jelfnek offer an explanation for the apparent stabilIty of the incidence of birth defects in humans. Their explanation takes the form of a mathematical model that relates dose of teratogen to the frequency of malformed offspring. Since many re~'ders may be unfamiliar with the rationale underlying Kfirka and Jelinek's work and as a consequence may dismiss their effort as irrelevant to empirical issues, it is perhaps worthwhile to take a moment to reflect on the role that such models play in understanding biological phenomena For most experimentalists, the primary apphcation of mathematical models to biological problems hes with the testing of theory. General-purpose statistical models, frequently consisting of simple additive relationships, are among the most familiar. These models, although not always explicitly stated, are often useful in distinguishing among competing theories when applied to data from well-designed experiments. Quantitative predictmn from theory generally requires a more explicit model-building effort with close attentmn to the definition of parameters and a clear statement of assumptions. Models designed for prediction can be improved with more precise parameter estimates and sometimes by an increase in model complexity through the inclusion of more parameters and interactions among them. These model-using activities are often thought of as part of a process m understanding a biological phenomenon, beginning with identification of relevant sources of variation and ending with estimating their sizes and Interrelationships. So where does the work of Kfirka and Jelinek fit into this paradigm of scientific methodology? It doesn't. Their purpose in
examining mathematical models zs to find one whose behavior mimics the phenomenon of interest - - in this case, a stable frequency of birth defects in the face of an increase in environmental insult. This IS a fundamentally different use of mathematical modelling than discussed above. Here the object of Interest IS the behavior of the model itself. The intellectual challenge is to construct the most parsimonious model possible, one that possesses a minimal number of parameters yet retains the behavior characterizing the phenomenon. This constitutes an attempt to distinguish what can happen from what cannot happen as a consequence of a given set of assumptions. Modelling efforts such as this, however, are among the easiest for researchers immersed in empirical detail to dismiss as irrelevant. This is because the strength of studying models for their behavioral properties hes with their abstract conceptuahzatlon, which is a significant hurdle to testing such models in the manner alluded to above. For example, Kfirka and Jelinek formulate their model in terms of a "critical period" dunng ontogeny and invoke the notion of "swiftness of reaction" to a teratogen. These are difficulties that will arise for any general theory formulated in mathematical abstraction. As a consequence, the search for experimental systems to test such theory will not be an easy one Nevertheless, if progress in scientific understanding can be characterized as a series of reconceptualizatlons of a phenomenon, then abstract modelling efforts must play a pivotal role RICHARD
W.
MORRIS
Analytwal Scwnces, Inc. Durham, North Carolina
79