Inclusion of protein lysozyme molecules in the lipidic mono-olein cubic phase induces a transition from a $Pn\overline{3}m$ structure to an $Im\overline{3}m$ one. The small-angle x-ray scattering method with high-intensity synchrotron radiation enabled us to follow closely the transition depending on the conditions of lysozyme solutions. We show that concentrated lysozyme solutions induced the appearance of the $Im\overline{3}m$ structure coexisting with the $Pn\overline{3}m$ structure. From the relation between the lattice parameters of these two structures it is shown that they are related by the Bonnet transformation of the underlying triply periodic minimal surfaces. We found that the transition also occurred at lower lysozyme concentration when $\mathrm{NaCl}$ induced an attraction between lysozyme molecules. The origin of the transition was considered as a frustration in the cubic phase where lysozyme molecules were highly confined. A simple estimation of the frustration was given, which took into account the translational entropy of lysozyme molecules. At the highest concentration of lysozyme and $\mathrm{NaCl}$ the $Im\overline{3}m$ structure was found to disappear and left only the $Pn\overline{3}m$ structure. This was probably either due to the crystallization or phase separation of lysozyme solutions ongoing microscopically, which absorbed lysozyme molecules from channels of the cubic phase and thus removed the frustration.

• Received 24 January 2006

DOI:https://doi.org/10.1103/PhysRevE.73.061510