lipid molecule PIP2 nositol 4,5-bisphosphate) i membranes. Its philic head group has a greater he an important constituent of eukaryotic cell negative charge (typically -4e under physiological pH) than most of the other phospholipids present in eukaryotic cell membranes. PIP2 makes up only a small fraction of the membrane (typically 1% by mole fraction), but in spite of this low concentration, it is known to form clusters with multiple PIP2 molecules that are thought to be important in cell signaling. It is an area of active research to understand the basis of this cluster formation. Recent research Cz. Li, et al., Biophys. J., 97, 155-163 (2009 provides support for an electrostatic mechanism for this clustering, in which Ca ons provide an attractive 2+ nteraction holding together the PIP2 molecules. The detailed structure of how Ca ons and PIP2 molecules are arranged in these clusters is not yet known. In this problem we consider a highly simplified model simply to give a feel for how these interactions work. 2+ ions. They are all placed on a grid where each grid A simplified model of such a cluster consists of five PIP molecules, each with a headgroup charge of -4e, and four Ca square has sides of length s, as shown in the figure below. PIP 2+