How does symmetry impact the flexibility of proteins?

Abstract

It is well known that (i) the flexibility and rigidity of proteins are central to their function, (ii) a number of oligomers with several copies of individual protein chains assemble with symmetry in the native state and (iii) added symmetry sometimes leads to added flexibility in structures. We observe that the most common symmetry classes of protein oligomers are also the symmetry classes that lead to increased flexibility in certain three-dimensional structures—and investigate the possible significance of this coincidence. This builds on the well-developed theory of generic rigidity of body–bar frameworks, which permits an analysis of the rigidity and flexibility of molecular structures such as proteins via fast combinatorial algorithms. In particular, we outline some very simple counting rules and possible algorithmic extensions that allow us to predict continuous symmetry-preserving motions in body–bar frameworks that possess non-trivial point-group symmetry. For simplicity, we focus on dimers, which typically assemble with twofold rotational axes, and often have allosteric function that requires motions to link distant sites on the two protein chains.

On the Symmetric Molecular Conjectures

“…Our computations of the configuration space again confirm that there is indeed a two-dimensional C-space for this molecular linkage. In addition, our computations show that each of the two 6-revolute rings of the structure maintains its half-turn symmetry at all times, which supports our expectation that stressed (over-constrained under symmetry) components of a structure (such as the symmetric 6-rings in the linkage) are more likely to maintain their symmetry than unstressed ones (such as the connecting links between the 6-rings) [14].…”

“…This fundamental result leads to the following necessary conditions for an S-regular body-bar (or molecular) framework to be rigid: Theorem 2. (Schulze, Whiteley, 2010 [15,13] [14]. While these algorithms provide sufficient conditions for the flexibility of a symmetric structure, an area of ongoing research is to whether they also provide necessary conditions for flexibility.…”

“…For each linkage, we show the initial configuration used for continuation, its symmetry group, the number of predicted motions according to the Tay and orbit counts, and the C-space topology inferred from the numerical analysis. We focus on linkages exhibiting the half-turn symmetry group C 2 and the dihedral group D 2 of order 4 (which is generated by two perpendicular half-turn axes), as these are the most common groups for molecular structures such as proteins [4,14].…”

Symmetry Perspectives on Some Auxetic Body-Bar Frameworks

“…Symmetry analyses have dealt with classes of system such as rotating rings of tetrahedra [8], [N ]-loops [9] toroidal deltahedra [10] and various mechanical toys and models [11][12][13]. Symmetry arguments have been used to derive conditions for the existence of isostatic frameworks [14] and to discuss the flexibility of protein molecules [15]. Attention from the mathematical, engineering and materials-science communities has now started to shift to periodic systems.…”

“…Attention from the mathematical, engineering and materials-science communities has now started to shift to periodic systems. Several different mathematical approaches towards the treatment of periodic systems are described in the proceedings of the 2012 conference on Rigidity of Periodic and Symmetric Structures in Nature and Engineering [15][16][17][18].…”