Polymer uncrossing and knotting in protein folding, and their role in minimal folding pathways

Optimizing Epitope Conformational Ensembles Using α-Synuclein Cyclic Peptide "Glycindel" Scaffolds: A Customized Immunogen Method for Generating Oligomer-Selective Antibodies for Parkinson's Disease

Effectively presenting epitopes on immunogens, in order to raise conformationally selective antibodies through active immunization, is a central problem in treating protein misfolding diseases, particularly neurodegenerative diseases such as Alzheimer's disease or Parkinson's disease. We seek to selectively target conformations enriched in toxic, oligomeric propagating species while sparing the healthy forms of the protein that are often more abundant. To this end, we computationally modeled scaffolded epitopes in cyclic peptides by inserting/deleting a variable number of flanking glycines ("glycindels") to best mimic a misfolding-specific conformation of an epitope of α-synuclein enriched in the oligomer ensemble, as characterized by a region most readily disordered and solvent-exposed in a stressed, partially denatured protofibril. We screen and rank the cyclic peptide scaffolds of α-synuclein in silico based on their ensemble overlap properties with the fibril, oligomer-model and isolated monomer ensembles. We present experimental data of seeded aggregation that support nucleation rates consistent with computationally predicted cyclic peptide conformational similarity. We also introduce a method for screening against structured off-pathway targets in the human proteome by selecting scaffolds with minimal conformational similarity between their epitope and the same solvent-exposed primary sequence in structured human proteins. Different cyclic peptide scaffolds with variable numbers of glycines are predicted computationally to have markedly different conformational ensembles. Ensemble comparison and overlap were quantified by the Jensen-Shannon divergence and a new measure introduced here, the embedding depth, which determines the extent to which a given ensemble is subsumed by another ensemble and which may be a more useful measure in developing immunogens that confer conformational selectivity to an antibody.

Topology of polymer chains under nanoscale confinement

Spatial confinement limits the conformational space accessible to biomolecules but the implications for bimolecular topology are not yet known. Folded linear biopolymers can be seen as molecular circuits formed by intramolecular contacts. The pairwise arrangement of intra-chain contacts can be categorized as parallel, series or cross, and has been identified as a topological property. Using molecular dynamics simulations, we determine the contact order distributions and topological circuits of short semi-flexible linear and ring polymer chains with a persistence length of l_p under a spherical confinement of radius R_c. At low values of l_p/R_c, the entropy of the linear chain leads to the formation of independent contacts along the chain and accordingly, increases the fraction of series topology with respect to other topologies. However, at high l_p/R_c, the fraction of cross and parallel topologies are enhanced in the chain topological circuits with cross becoming predominant. At an intermediate confining regime, we identify a critical value of l_p/R_c, at which all topological states have equal probability. Confinement thus equalizes the probability of more complex cross and parallel topologies to the level of the more simple, non-cooperative series topology. Moreover, our topology analysis reveals distinct behaviours for ring- and linear polymers under weak confinement; however, we find no difference between ring- and linear polymers under strong confinement. Under weak confinement, ring polymers adopt parallel and series topologies with equal likelihood, while linear polymers show a higher tendency for series arrangement. The radial distribution analysis of the topology reveals a non-uniform effect of confinement on the topology of polymer chains, thereby imposing more pronounced effects on the core region than on the confinement surface. Additionally, our results reveal that over a wide range of confining radii, loops arranged in parallel and cross topologies have nearly the same contact orders. Such degeneracy implies that the kinetics and transition rates between the topological states cannot be solely explained by contact order. We expect these findings to be of general importance in understanding chaperone assisted protein folding, chromosome architecture, and the evolution of molecular folds.

Minimum action principle and shape dynamics

In this paper, we propose a new method for computing a distance between two shapes embedded in three-dimensional space. Instead of comparing directly the geometric properties of the two shapes, we measure the cost of deforming one of the two shapes into the other. The deformation is computed as the geodesic between the two shapes in the space of shapes. The geodesic is found as a minimizer of the Onsager-Machlup action, based on an elastic energy for shapes that we define. Its length is set to be the integral of the action along that path; it defines an intrinsic quasi-metric on the space of shapes. We illustrate applications of our method to geometric morphometrics using three datasets representing bones and teeth of primates. Experiments on these datasets show that the variational quasi-metric we have introduced performs remarkably well both in shape recognition and in identifying evolutionary patterns, with success rates similar to, and in some cases better than, those obtained by expert observers.

Circuit topology of self-interacting chains: implications for folding and unfolding dynamics

Understanding the relationship between molecular structure and folding is a central problem in disciplines ranging from biology to polymer physics and DNA origami. Topology can be a powerful tool to address this question. For a folded linear chain, the arrangement of intra-chain contacts is a topological property because rearranging the contacts requires discontinuous deformations. Conversely, the topology is preserved when continuously stretching the chain while maintaining the contact arrangement. Here we investigate how the folding and unfolding of linear chains with binary contacts is guided by the topology of contact arrangements. We formalize the topology by describing the relations between any two contacts in the structure, which for a linear chain can either be in parallel, in series, or crossing each other. We show that even when other determinants of folding rate such as contact order and size are kept constant, this 'circuit' topology determines folding kinetics. In particular, we find that the folding rate increases with the fractions of parallel and crossed relations. Moreover, we show how circuit topology constrains the conformational phase space explored during folding and unfolding: the number of forbidden unfolding transitions is found to increase with the fraction of parallel relations and to decrease with the fraction of series relations. Finally, we find that circuit topology influences whether distinct intermediate states are present, with crossed contacts being the key factor. The approach presented here can be more generally applied to questions on molecular dynamics, evolutionary biology, molecular engineering, and single-molecule biophysics.

Pierced Lasso Bundles are a new class of knot-like motifs

A four-helix bundle is a well-characterized motif often used as a target for designed pharmaceutical therapeutics and nutritional supplements. Recently, we discovered a new structural complexity within this motif created by a disulphide bridge in the long-chain helical bundle cytokine leptin. When oxidized, leptin contains a disulphide bridge creating a covalent-loop through which part of the polypeptide chain is threaded (as seen in knotted proteins). We explored whether other proteins contain a similar intriguing knot-like structure as in leptin and discovered 11 structurally homologous proteins in the PDB. We call this new helical family class the Pierced Lasso Bundle (PLB) and the knot-like threaded structural motif a Pierced Lasso (PL). In the current study, we use structure-based simulation to investigate the threading/folding mechanisms for all the PLBs along with three unthreaded homologs as the covalent loop (or lasso) in leptin is important in folding dynamics and activity. We find that the presence of a small covalent loop leads to a mechanism where structural elements slipknot to thread through the covalent loop. Larger loops use a piercing mechanism where the free terminal plugs through the covalent loop. Remarkably, the position of the loop as well as its size influences the native state dynamics, which can impact receptor binding and biological activity. This previously unrecognized complexity of knot-like proteins within the helical bundle family comprises a completely new class within the knot family, and the hidden complexity we unraveled in the PLBs is expected to be found in other protein structures outside the four-helix bundles. The insights gained here provide critical new elements for future investigation of this emerging class of proteins, where function and the energetic landscape can be controlled by hidden topology, and should be take into account in ab initio predictions of newly identified protein targets.