The membrane proximal region (MPR, residues 649–683) and transmembrane domain (TMD, residues 684–705) of the gp41 subunit of HIV-1’s envelope protein are highly conserved and are important in viral mucosal transmission, virus attachment and membrane fusion with target cells. Several structures of the trimeric membrane proximal external region (residues 662–683) of MPR have been reported at the atomic level; however, the atomic structure of the TMD still remains unknown. To elucidate the structure of both MPR and TMD, we expressed the region spanning both domains, MPR-TM (residues 649–705), in Escherichia coli as a fusion protein with maltose binding protein (MBP). MPR-TM was initially fused to the C-terminus of MBP via a 42 aa-long linker containing a TEV protease recognition site (MBP-linker-MPR-TM).
Biophysical characterization indicated that the purified MBP-linker-MPR-TM protein was a monodisperse and stable candidate for crystallization. However, crystals of the MBP-linker-MPR-TM protein could not be obtained in extensive crystallization screens. It is possible that the 42 residue-long linker between MBP and MPR-TM was interfering with crystal formation. To test this hypothesis, the 42 residue-long linker was replaced with three alanine residues. The fusion protein, MBP-AAA-MPR-TM, was similarly purified and characterized. Significantly, both the MBP-linker-MPR-TM and MBP-AAA-MPR-TM proteins strongly interacted with broadly neutralizing monoclonal antibodies 2F5 and 4E10. With epitopes accessible to the broadly neutralizing antibodies, these MBP/MPR-TM recombinant proteins may be in immunologically relevant conformations that mimic a pre-hairpin intermediate of gp41.
The coffee cup system can be simplified and modeled by a cart-and-pendulum system. Bazzi et al. and Maurice et al. present two different cart-and-pendulum systems to represent the coffee cup system [1],[2]. The purpose of this project was to build upon these systems and to gain a better understanding of the coffee cup system and to determine where chaos existed within the system. The honors thesis team first worked with their senior design group to develop a mathematical model for the cart-and-pendulum system based on the Bazzi and Maurice papers [1],[2]. This system was analyzed and then built upon by the honors thesis team to build a cart-and-two-pendulum model to represent the coffee cup system more accurately.
Analysis of the single pendulum model showed that there exists a low frequency region where the pendulum and the cart remain in phase with each other and a high frequency region where the cart and pendulum have a π phase difference between them. The transition point of the low and high frequency region is determined by the resonant frequency of the pendulum. The analysis of the two-pendulum system also confirmed this result and revealed that differences in length between the pendulum cause the pendulums to transition to the high frequency regions at separate frequency. The pendulums have different resonance frequencies and transition into the high frequency region based on their own resonant frequency. This causes a range of frequencies where the pendulums are out of phase from each other. After both pendulums have transitioned, they remain in phase with each other and out of phase from the cart.
However, if the length of the pendulum is decreased too much, the system starts to exhibit chaotic behavior. The short pendulum starts to act in a chaotic manner and the phase relationship between the pendulums and the carts is no longer maintained. Since the pendulum length represents the distance between the particle of coffee and the top of the cup, this implies that coffee near the top of the cup would cause the system to act chaotically. Further analysis would be needed to determine the reason why the length affects the system in this way.