The Need for a Multiscale Modeling Framework

ences is that of a “field,” which, together with appropriate mathematical operations on fields, is used to express the physical conservation laws of nature, such as conservation of mass, momentum, charge, etc. Fields were first introduced by the great 19th-century British scientist Michael Faraday to represent the intensity and direction of the force experienced by a small charged particle in the presence of magnetic dipoles and electric charges throughout a three-dimensional space. The greatest success of 19thand early 20th-century physics was the discovery that most observable phenomena could be described by differential equations operating on these fields: Maxwell’s field equations for electricity and magnetism, Einstein’s field equation for gravity, the Yang-Mills field equations for subatomic particles, and the field equations of continuum mechanics were derived from Newton’s laws of conservation of mass and conservation of momentum. There is an interesting analogy with molecular biology here: just as the classical field theory world of physics was overshadowed in the early 20th century by quantum theory and particle physics, the integrative systems-level physiology of the first half of the 20th century gave way to the reductionist molecular biology of the second half. In both cases the challenge now is to reconcile the continuum/systems view with the particle/molecular view. For physicists, this may be achievable by describing their equations in a higher-dimensional space via superstring theory. For mathematical biologists, it is likely to be achieved via multiscale modeling. Many of the great accomplishments of electrical and mechanical engineering, respectively, in the 20th century were built (for the former), on the sound theoretical basis of Maxwell’s field theory and quantum mechanics and (for the latter) on continuum mechanics. In both cases, however, the theoretical field theory models had to be supplemented with empirical descriptions of material behavior that hide molecular or atomic detail behind approximate “constitutive laws.” For elec-