At its best—which is usually when the model building is done collaboratively, drawing on the particular insights and strengths of both mathematical and chemical scientists—this diagnostic approach is extremely valuable, leading to both theoretical and numerical understanding of the model, which can then return both quantitative information and conceptual understanding about the original physical problem. Indeed, carefully designed models can suggest and explain properties that are counterintuitive or unexpected to the problem posers themselves. An example is found in macroscopic models of supercoiled DNA, in which higher buckling catastrophes, as extensions to elasticity theory, were recently found. Furthermore, analysis of mathematical models can indicate directions for additional physical and numerical experimentation, as well as for extensions to the model. A potentially negative effect of mathematical abstraction, however, is that the theoretical formulation may lose its relevance to the original application that motivated it.
Beyond the "cultural" differences just described, attempts to build collaborations between mathematicians and chemists encounter boundaries imposed within most universities as well as within other structures in which disciplinary divisions are strong. These borders are especially difficult to cross early in a scientist's career, ironically when the potential for interdisciplinary work may be greatest because of the appeal of "new frontiers." Institutional practices influence the style of work that is valued (and hence often pursued) in each discipline, as well as the level of regular interaction and communication. Since many computational chemists and mathematicians are faculty members at research universities, both fields are affected by the value system of academia, in which recognition (promotion and tenure) requires a record of individual accomplishment judged as outstanding by one's peers. During the process of creating a portfolio of publications, any perceived dilution of a faculty member's personal contribution through collaborations may be seen as undesirable. This discouragement of collaborative work early in one's career applies to both mathematics and chemistry, although chemists have a strong countervailing tradition of working in groups.
For mathematicians, the potential career damage of collaboration rises when it involves work in a field seen as peripheral to mathematics. In some instances, interdisciplinary work may be regarded by one's mathematical colleagues as "not real mathematics" or as less valuable than traditional mathematics. Most academic mathematicians would agree that it is difficult to obtain accurate and convincing evaluations of "interdisciplinary" work (meaning work that involves significant contributions from other sciences) and research in nontraditional areas of mathematics. In this connection, there is a recent report on the recognition and reward system in the mathematical sciences (Joint Policy Board for Mathematics, 1994). Such issues are particularly worrying for junior mathematicians, since it would be unusual for nonmathematicians to be asked for help in a tenure or promotion evaluation; mathematics departments might well be reluctant to rely on outsiders for judgments and decisions viewed as a departmental prerogative. Another related issue is the value attached to work in which an existing body of mathematics is applied to another scientific problem area; even if the impact is great and the work represents a significant scientific advance, it is not ''new mathematics,'' and hence may be accorded little weight in an evaluation of research contributions.
Because of the tendency to preserve and protect departmental boundaries, mathematics departments are ill-equipped to cope with questions that inevitably arise if mathematicians become seriously interested in interdisciplinary problems. For example, if a young mathematician is hired as a