4.2.4.P9
The phospholipid molecule PIP2 (phosphatidylinositol 4,5-bisphosphate) is an important constituent of eukaryotic cell membranes. Its hydrophilic head group has a greater 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.
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Recent research* (Wang, Janmey, and coworkers) provides support for an electrostatic mechanism for this clustering, in which Ca2+ ions provide an attractive interaction holding together the PIP2 molecules. The detailed structure of how Ca2+ ions 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.
A simplified model of such a cluster consists of five PIP2 molecules, each with a headgroup charge of -4e, and four Ca2+ ions. They are all placed on a grid where each grid square has sides of length s, as shown in the figure at the right.
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In this problem we will explore whether the electric forces on the PIP2 molecules in this model cluster tends to pull it together or blow it apart.
(a) What is the net electric force on the central PIP2 molecule in the cluster coming from all the other charges in the cluster?
(b) To find out whether the electric forces pull the PIP2 molecules in the cluster together or blow them apart, we only have to consider one of the PIP2 molecules, say the one at the very top. Explain why.
(c) The total force on the top ion will look like a pure number times the expression kCe2/s2. Explain why this is true. Your explanation could involve a consideration of each force or use dimensional analysis (or you might come up with something else).
(d) Find the net force on the top ion exerted by the other ions by finding the number that multiplies the expression given in part (c). What direction does it point? Does this force push the top ion toward the center (tending to hold the cluster together) or push it away (tending to blow the cluster apart)? Explain your reasoning carefully, showing with diagrams which forces you included and how you added them.
* Z. Li, et al., Biophys. J., 97, 155-163 (2009).
Catherine Crouch, Julia Svoboda, and Joe Redish 11/19/11
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