Water bridging in protein crystals. Protein molecules can crystallize. A puzzling observation is that two protein molecules in the crystal are sometimes separated by a spacer water molecule. However, if proteins are attracted by a dipolar electrostatic interaction, the system should be most stable when the proteins are as close together as possible.
To model this, consider each protein molecule to be spherical with a radius $R_{o}$, as shown in Figure 21.19. Each protein has a net dipole moment due to two charges, $+q$ and $-q$ respectively, each a distance $d$ from the center of the protein, and collinear.
(a) Calculate the electrostatic potential for the proteinprotein contact pair without the water, with the charges distributed as shown in Figure $21.19 .$ Assume the system is in a vacuum.
(b) Now, calculate the electrostatic potential for the water-bridged protein-protein pair shown in Figure $21.20$. To do this, first recognize that the water molecule also has a dipole moment. Assume that water has a charge of $-q_{w}$ at the center of a sphere of radius $R_{w}$, and a charge of $+q_{w}$ at a distance $\left|R_{w}-d_{w}\right|$ from the center, as shown in Figure $21.21$. Suppose that in the crystal, the water dipole is collinear with the dipoles of the proteins shown in Figure $21.20$.
(c) Assume $q=2$ e in both systems in (a) and (b). Assume that $q_{w}=1 \mathrm{e}, d=5 \mathrm{~A}, d_{w}=1.2 \mathrm{~A}, R_{o}=10 \mathrm{~A}$ and $R_{w}=5 \mathrm{~A}$. Which system, protein contacting protein or proteins contacting a central water, is the most stable?
(d) Replace the vacuum surroundings of the proteinwater-protein system by water surroundings, and assume a standard Coulombic treatment of the medium. What is the difference in total interaction energy in this case?