Multiscale modeling of biological pattern formation.

In the past few decades, it has become increasingly popular and important to utilize mathematical models to understand how microscopic intercellular interactions lead to the macroscopic pattern formation ubiquitous in the biological world. Modeling methodologies come in a large variety and presently it is unclear what is their interrelationship and the assumptions implicit in their use. They can be broadly divided into three categories according to the spatial scale they purport to describe: the molecular, the cellular and the tissue scales. Most models address dynamics at the tissue-scale, few address the cellular scale and very few address the molecular scale. Of course there would be no dissent between models or at least the underlying assumptions would be known if they were all rigorously derived from a molecular level model, in which case the laws of physics and chemistry are very well known. However in practice this is not possible due to the immense complexity of the problem. A simpler approach is to derive models at a coarse scale from an intermediate scale model which has the special property of being based on biology and physics which are experimentally well studied. In this article we use such an approach to understand the assumptions inherent in the use of the most popular models, the tissue-level ones. Such models are found to invariably rely on the hidden assumption that statistical correlations between cells can be neglected. This often means that the predictions of these models are qualitatively correct but may fail in spatial regions where cell concentration is small, particularly if there are strong long-range correlations in cell movement. Such behavior can only be properly taken into account by cellular models. However such models unlike the tissue-level models are frequently not easily amenable to analysis, except when the number of interacting cells is small or when the interactions are weak, and thus are rather more suited for simulation. Hence it is our conclusion that the simultaneous theoretical and numerical analysis of models of the same biological system at different spatial scales provides a more robust method of understanding biological systems than the utilization of a single scale model. In particular this enables one to clearly separate nonphysical predictions stemming from model artifacts from those due to genuine physiological behavior.