Topology and structural self-organization in folded proteins

Exploring Structural Flexibility and Stability of α-Synuclein by the Landau-Ginzburg-Wilson Approach

α-Synuclein (αS) is the principal protein component of the Lewy body and Lewy neurite deposits that are found in the brains of the victims of one of the most prevalent neurodegenerative disorders, Parkinson's disease. αS can be qualified as a chameleon protein because of the large number of different conformations that it is able to adopt: it is disordered under physiological conditions in solution, in equilibrium with a minor α-helical tetrameric form in the cytoplasm, and is α-helical when bound to a cell membrane. Also, in vitro, αS forms polymorphic amyloid fibrils with unique arrangements of cross-β-sheet motifs. Therefore, it is of interest to elucidate the origins of the structural flexibility of αS and what makes αS stable in different conformations. We address these questions here by analyzing the experimental structures of the micelle-bound, tetrameric, and fibrillar αS in terms of a kink (heteroclinic standing wave solution) of a generalized discrete nonlinear Schrödinger equation. It is illustrated that without molecular dynamics simulations the kinks are capable of identifying the key residues causing structural flexibility of αS. Also, the stability of the experimental structures of αS is investigated by simulating heating/cooling trajectories using the Glauber algorithm. The findings are consistent with experiments.

Local topology and bifurcation hot-spots in proteins with SARS-CoV-2 spike protein as an example

Novel topological methods are introduced to protein research. The aim is to identify hot-spot sites where a bifurcation can alter the local topology of the protein backbone. Since the shape of a protein is intimately related to its biological function, a substitution that causes a bifurcation should have an enhanced capacity to change the protein's function. The methodology applies to any protein but it is developed with the SARS-CoV-2 spike protein as a timely example. First, topological criteria are introduced to identify and classify potential bifurcation hot-spot sites along the protein backbone. Then, the expected outcome of asubstitution, if it occurs, is estimated for a general class of hot-spots, using a comparative analysis of the surrounding backbone segments. The analysis combines the statistics of structurally commensurate amino acid fragments in the Protein Data Bank with general stereochemical considerations. It is observed that the notorious D614G substitution of the spike protein is a good example of a bifurcation hot-spot. A number of topologically similar examples are then analyzed in detail, some of them are even better candidates for a bifurcation hot-spot than D614G. The local topology of the more recently observed N501Y substitution is also inspected, and it is found that this site is proximal to a different kind of local topology changing bifurcation.

Obtaining Tertiary Protein Structures by the ab Initio Interpretation of Small Angle X-ray Scattering Data

Small angle X-ray scattering (SAXS) is an important tool for investigating the structure of proteins in solution. We present a novel ab initio method representing polypeptide chains as discrete curves used to derive a meaningful three-dimensional model from only the primary sequence and SAXS data. High resolution structures were used to generate probability density functions for each common secondary structural element found in proteins, which are used to place realistic restraints on the model curve’s geometry. This is coupled with a novel explicit hydration shell model in order to derive physically meaningful three-dimensional models by optimizing against experimental SAXS data. The efficacy of this model is verified on an established benchmark protein set, and then it is used to predict the lysozyme structure using only its primary sequence and SAXS data. The method is used to generate a biologically plausible model of the coiled-coil component of the human synaptonemal complex central element protein.

Braiding topology and the energy landscape of chromosome organization proteins

Assemblies of structural maintenance of chromosomes (SMC) proteins and kleisin subunits are essential to chromosome organization and segregation across all kingdoms of life. While structural data exist for parts of the SMC-kleisin complexes, complete structures of the entire complexes have yet to be determined, making mechanistic studies difficult. Using an integrative approach that combines crystallographic structural information about the globular subdomains, along with coevolutionary information and an energy landscape optimized force field (AWSEM), we predict atomic-scale structures for several tripartite SMC-kleisin complexes, including prokaryotic condensin, eukaryotic cohesin, and eukaryotic condensin. The molecular dynamics simulations of the SMC-kleisin protein complexes suggest that these complexes exist as a broad conformational ensemble that is made up of different topological isomers. The simulations suggest a critical role for the SMC coiled-coil regions, where the coils intertwine with various linking numbers. The twist and writhe of these braided coils are coupled with the motion of the SMC head domains, suggesting that the complexes may function as topological motors. Opening, closing, and translation along the DNA of the SMC-kleisin protein complexes would allow these motors to couple to the topology of DNA when DNA is entwined with the braided coils.

Topological Indices of Proteins

Protein molecules can be approximated by discrete polygonal chains of amino acids. Standard topological tools can be applied to the smoothening of the polygons to introduce a topological classification of folded states of proteins, for example, using the self-linking number of the corresponding framed curves. In this paper we extend this classification to the discrete version, taking advantage of the “randomness” of such curves. Known definitions of the self-linking number apply to non-singular framings: for example, the Frenet framing cannot be used if the curve has inflection points. However, in the discrete proteins the special points are naturally resolved. Consequently, a separate integer topological characteristics can be introduced, which takes into account the intrinsic features of the special points. This works well for the proteins in our analysis, for which we compute integer topological indices associated with the singularities of the Frenet framing. We show how a version of the Calugareanu’s theorem is satisfied for the associated self-linking number of a discrete curve. Since the singularities of the Frenet framing correspond to the structural motifs of proteins, we propose topological indices as a technical tool for the description of the folding dynamics of proteins.

Bloch spin waves and emergent structure in protein folding with HIV envelope glycoprotein as an example

We inquire how structure emerges during the process of protein folding. For this we scrutinize collective many-atom motions during all-atom molecular dynamics simulations. We introduce, develop, and employ various topological techniques, in combination with analytic tools that we deduce from the concept of integrable models and structure of discrete nonlinear Schrödinger equation. The example we consider is an α-helical subunit of the HIV envelope glycoprotein gp41. The helical structure is stable when the subunit is part of the biological oligomer. But in isolation, the helix becomes unstable, and the monomer starts deforming. We follow the process computationally. We interpret the evolving structure both in terms of a backbone based Heisenberg spin chain and in terms of a side chain based XY spin chain. We find that in both cases the formation of protein supersecondary structure is akin the formation of a topological Bloch domain wall along a spin chain. During the process we identify three individual Bloch walls and we show that each of them can be modelled with a precision of tenths to several angstroms in terms of a soliton solution to a discrete nonlinear Schrödinger equation.

Clustering and percolation in protein loop structures

Background

High precision protein loop modelling remains a challenge, both in template based and template independent approaches to protein structure prediction.

Method

We introduce the concepts of protein loop clustering and percolation, to develop a quantitative approach to systematically classify the modular building blocks of loops in crystallographic folded proteins. These fragments are all different parameterisations of a unique kink solution to a generalised discrete nonlinear Schrödinger (DNLS) equation. Accordingly, the fragments are also local energy minima of the ensuing energy function.

Results

We show how the loop fragments cover practically all ultrahigh resolution crystallographic protein structures in Protein Data Bank (PDB), with a 0.2 Ångström root-mean-square (RMS) precision. We find that no more than 12 different loop fragments are needed, to describe around 38 % of ultrahigh resolution loops in PDB. But there is also a large number of loop fragments that are either unique, or very rare, and examples of unique fragments are found even in the structure of a myoglobin.

Conclusions

Protein loops are built in a modular fashion. The loops are composed of fragments that can be modelled by the kink of the DNLS equation. The majority of loop fragments are also common, which are shared by many proteins. These common fragments are probably important for supporting the overall protein conformation. But there are also several fragments that are either unique to a given protein, or very rare. Such fragments are probably related to the function of the protein. Furthermore, we have found that the amino acid sequence does not determine the structure in a unique fashion. There are many examples of loop fragments with an identical amino acid sequence, but with a very different structure.

Electronic supplementary material

The online version of this article (doi:10.1186/s12900-015-0049-x) contains supplementary material, which is available to authorized users.

Peierls-Nabarro barrier and protein loop propagation

When a self-localized quasiparticle excitation propagates along a discrete one-dimensional lattice, it becomes subject to a dissipation that converts the kinetic energy into lattice vibrations. Eventually the kinetic energy no longer enables the excitation to cross over the minimum energy barrier between neighboring sites, and the excitation becomes localized within a lattice cell. In the case of a protein, the lattice structure consists of the C(α) backbone. The self-localized quasiparticle excitation is the elemental building block of loops. It can be modeled by a kink that solves a variant of the discrete nonlinear Schrödinger equation. We study the propagation of such a kink in the case of the protein G related albumin-binding domain, using the united residue coarse-grained molecular-dynamics force field. We estimate the height of the energy barriers that the kink needs to cross over in order to propagate along the backbone lattice. We analyze how these barriers give rise to both stresses and reliefs, which control the kink movement. For this, we deform a natively folded protein structure by parallel translating the kink along the backbone away from its native position. We release the transposed kink, and we follow how it propagates along the backbone toward the native location. We observe that the dissipative forces that are exerted on the kink by the various energy barriers have a pivotal role in determining how a protein folds toward its native state.

Kinks, loops, and protein folding, with protein A as an example

The dynamics and energetics of formation of loops in the 46-residue N-terminal fragment of the B-domain of staphylococcal protein A has been studied. Numerical simulations have been performed using coarse-grained molecular dynamics with the united-residue (UNRES) force field. The results have been analyzed in terms of a kink (heteroclinic standing wave solution) of a generalized discrete nonlinear Schrödinger (DNLS) equation. In the case of proteins, the DNLS equation arises from a C(α)-trace-based energy function. Three individual kink profiles were identified in the experimental three-α-helix structure of protein A, in the range of the Glu16-Asn29, Leu20-Asn29, and Gln33-Asn44 residues, respectively; these correspond to two loops in the native structure. UNRES simulations were started from the full right-handed α-helix to obtain a clear picture of kink formation, which would otherwise be blurred by helix formation. All three kinks emerged during coarse-grained simulations. It was found that the formation of each is accompanied by a local free energy increase; this is expressed as the change of UNRES energy which has the physical sense of the potential of mean force of a polypeptide chain. The increase is about 7 kcal/mol. This value can thus be considered as the free energy barrier to kink formation in full α-helical segments of polypeptide chains. During the simulations, the kinks emerge, disappear, propagate, and annihilate each other many times. It was found that the formation of a kink is initiated by an abrupt change in the orientation of a pair of consecutive side chains in the loop region. This resembles the formation of a Bloch wall along a spin chain, where the C(α) backbone corresponds to the chain, and the amino acid side chains are interpreted as the spin variables. This observation suggests that nearest-neighbor side chain-side chain interactions are responsible for initiation of loop formation. It was also found that the individual kinks are reflected as clear peaks in the principal modes of the analyzed trajectory of protein A, the shapes of which resemble the directional derivatives of the kinks along the chain. These observations suggest that the kinks of the DNLS equation determine the functionally important motions of proteins.