Calculating Ensemble Averaged Descriptions of Protein Rigidity without Sampling

Previous works have demonstrated that protein rigidity is related to thermodynamic stability, especially under conditions that favor formation of native structure. Mechanical network rigidity properties of a single conformation are efficiently calculated using the integer body-bar Pebble Game (PG) algorithm. However, thermodynamic properties require averaging over many samples from the ensemble of accessible conformations to accurately account for fluctuations in network topology. We have developed a mean field Virtual Pebble Game (VPG) that represents the ensemble of networks by a single effective network. That is, all possible number of distance constraints (or bars) that can form between a pair of rigid bodies is replaced by the average number. The resulting effective network is viewed as having weighted edges, where the weight of an edge quantifies its capacity to absorb degrees of freedom. The VPG is interpreted as a flow problem on this effective network, which eliminates the need to sample. Across a nonredundant dataset of 272 protein structures, we apply the VPG to proteins for the first time. Our results show numerically and visually that the rigidity characterizations of the VPG accurately reflect the ensemble averaged properties. This result positions the VPG as an efficient alternative to understand the mechanical role that chemical interactions play in maintaining protein stability

Changes in Lysozyme Flexibility upon Mutation Are Frequent, Large and Long-Ranged

We investigate changes in human c-type lysozyme flexibility upon mutation via a Distance Constraint Model, which gives a statistical mechanical treatment of network rigidity. Specifically, two dynamical metrics are tracked. Changes in flexibility index quantify differences within backbone flexibility, whereas changes in the cooperativity correlation quantify differences within pairwise mechanical couplings. Regardless of metric, the same general conclusions are drawn. That is, small structural perturbations introduced by single point mutations have a frequent and pronounced affect on lysozyme flexibility that can extend over long distances. Specifically, an appreciable change occurs in backbone flexibility for 48% of the residues, and a change in cooperativity occurs in 42% of residue pairs. The average distance from mutation to a site with a change in flexibility is 17–20 Å. Interestingly, the frequency and scale of the changes within single point mutant structures are generally larger than those observed in the hen egg white lysozyme (HEWL) ortholog, which shares 61% sequence identity with human lysozyme. For example, point mutations often lead to substantial flexibility increases within the β-subdomain, which is consistent with experimental results indicating that it is the nucleation site for amyloid formation. However, β-subdomain flexibility within the human and HEWL orthologs is more similar despite the lowered sequence identity. These results suggest compensating mutations in HEWL reestablish desired properties

Mechanics of Chemo-Mechanical Stimuli Responsive Soft Polymers

Responsive materials, often obtained by designing the molecular structure of polymers or gels, are able to respond with detectable physical changes to external stimuli of various nature, ranging from chemical (such as pH), temperature, light radiation, mechanical stress, etc. In this paper we propose a micromechanical model, rooted in the statistical approach to the network conformation of polymeric materials, to predict the mechanical response of polymers with embedded responsive molecules. The model makes use of the so-called chains distribution function, aimed at providing the current state of the network’s chains. A univocal relation between the distribution function and the mechanical state of the material is established, so the knowledge of the evolution of such a function with the applied deformation or other external stimuli allows to get the macroscopic response of the material. Finally, the case of responsive molecules inserted into the network as crosslinkers is considered. The responsiveness of the molecules is assumed to depend either on the mechanical and/or chemical stimuli coming from the external environment. The cases involving molecules sensible to chemical stimuli, always imply the presence of a solvent carrying the triggering chemical agent, and thus require to consider the swelling phenomenon induced by the fluid absorbed by the polymer network. The theoretical framework of the micromechanical model is illustrated and some examples are finally presented and discussed