## 10 Body-Hinge Frameworks
### Applications to Protein Structures  
The paper [126] presents a short summary of how biomolecules, including proteins, can be analysed using the geometry and combinatorics of body-hinge frameworks and the Molecular Theorem. Other helpful papers for this are [137; 138; 139; 140]. **Example 36**. : _Rings of atoms, particularly carbon rings, are important parts of many organic molecules (Figure 59). We start with an analysis of some simple counting for rings of 7, 6, and 5 molecules._  
_For a ring of 7 atoms with hinges around the ring, we have a body-hinge framework \(G=(B,H)\) with \(5|H|=35<36=6(7-1)=6(|B|-1)\). The ring is generically flexible, as are all longer rings. For a ring of 6, such as Cyclohexane (see Figure 59a), we have \(5|H|=30=6(6-1)=6(|B|-1)\). The body-hinge structure, and the molecular framework, will be generically isostatic. The figure shows added hydrogen atoms, so that each carbon is bonded to 4 other atoms and all the bonds are single bonds which allow rotation. For a ring of 5, such as in Proline (see Figure 59b), we have \(5|H|=25>24=6(5-1)=6(|B|-1)\). The body-hinge structure, and the molecular framework, will be redundant and globally rigid. Proline is one of the 21 amino acids that are the building blocks of proteins, and it plays a particular role in forming rigid substructures in a larger protein._  
_Figure 59c focuses on the core ring in a form chemists call the chair. Since the angles are all fixed, if we join 3 alternate atoms, we have an implied triangle, and the other 3 atoms form an additional triangle. This is now the edge skeleton of a convex octahedron which is isostatic in that geometry (by Cauchy's Theorem). Figure 59d is in the form the chemists call the boat. It is flexible with a full finite flex, due to the half-turn symmetry [138]. It takes some deformation of lengths and angles to switch between the chair and the boat. This is called an energy barrier in molecular modeling._  
In biomolecules, function depends on both having a shape and having some flexibility. So the rings of size \(>6\) are not common, and many rings of length 6 occur--sometimes linked together. **Example 37**. : _In'mad cow disease' a protein 'prion' switches shape to become too rigid--not able to be recycled, both building up as junk in the brain and offering a template for other copies of the protein to refold in the rigid form. The misfolded variant has more rigidity and aggregates by binding with other copies of the same protein along the beta sheets that then resist recycling. In cystic fibrosis, mutations in a gene cause the CFTR protein to become dysfunctional--essentially they become too floppy, so that the body recycles it before it can take a functional shape. Good functioning of proteins happen on the boundary of having a functioning shape and being able to make small changes in shape [126; 137]._There are extended fast programs, built from the Molecular Theorem and body-hinge models, to predict which parts of a protein are rigid and which are flexible [19; 20]. This software can analyze a biomolecule with 400,000 atoms in a minute--a quick and somewhat approximate prediction which is helpful and much faster than the many week molecular dynamics simulations. This information is valuable in drug design, because the drug may work by removing a functioning motion (as in HIV inhibitors) or even in deforming the protein so that some other part becomes active (allostery, or shape change at a distance). **Example 38**. : _Consider the initial drug treatment for HIV (Figure 60): the inhibitor reduces the flexibility of a critical functional motion of the protease, which is critical to the replication of the HIV virus by clipping one of the virus components. This drug shuts down the replication by rigidifying the functional motion._  
There is a rich and growing literature on applications of rigidity to drug design, and to validating protein models [141]. Figure 59: Two models of molecular rings: Cyclohexane (**a**) \(C_{6}H_{12}\) and Proline \(C_{5}H_{9}NO_{2}\) (**b**).