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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 proteins, for example, using the self-linking number of the corresponding framed curves. In this paper we add new details to the standard classification. 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. Meanwhile 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. For large number of proteins we compute integer topological indices associated with the singularities of the Frenet framing. We show how a version of the Calugareanus 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.
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We perform theoretical studies of stretching of 20 proteins with knots within a coarse grained model. The knots ends are found to jump to well defined sequential locations that are associated with sharp turns whereas in homopolymers they diffuse around and eventually slide off. The waiting times of the jumps are increasingly stochastic as the temperature is raised. Larger knots do not return to their native locations when a protein is released after stretching.
Experiments indicate that unbinding rates of proteins from DNA can depend on the concentration of proteins in nearby solution. Here we present a theory of multi-step replacement of DNA-bound proteins by solution-phase proteins. For four different kinetic scenarios we calculate the depen- dence of protein unbinding and replacement rates on solution protein concentration. We find (1) strong effects of progressive rezipping of the solution-phase protein onto DNA sites liberated by unzipping of the originally bound protein; (2) that a model in which solution-phase proteins bind non-specifically to DNA can describe experiments on exchanges between the non specific DNA- binding proteins Fis-Fis and Fis-HU; (3) that a binding specific model describes experiments on the exchange of CueR proteins on specific binding sites.
Intrinsically disordered proteins (IDPs) do not possess well-defined three-dimensional structures in solution under physiological conditions. We develop all-atom, united-atom, and coarse-grained Langevin dynamics simulations for the IDP alpha-synuclein that include geometric, attractive hydrophobic, and screened electrostatic interactions and are calibrated to the inter-residue separations measured in recent smFRET experiments. We find that alpha-synuclein is disordered with conformational statistics that are intermediate between random walk and collapsed globule behavior. An advantage of calibrated molecular simulations over constraint methods is that physical forces act on all residues, not only on residue pairs that are monitored experimentally, and these simulations can be used to study oligomerization and aggregation of multiple alpha-synuclein proteins that may precede amyloid formation.
Problems of search and recognition appear over different scales in biological systems. In this review we focus on the challenges posed by interactions between proteins, in particular transcription factors, and DNA and possible mechanisms which allow for a fast and selective target location. Initially we argue that DNA-binding proteins can be classified, broadly, into three distinct classes which we illustrate using experimental data. Each class calls for a different search process and we discuss the possible application of different search mechanisms proposed over the years to each class. The main thrust of this review is a new mechanism which is based on barrier discrimination. We introduce the model and analyze in detail its consequences. It is shown that this mechanism applies to all classes of transcription factors and can lead to a fast and specific search. Moreover, it is shown that the mechanism has interesting transient features which allow for stability at the target despite rapid binding and unbinding of the transcription factor from the target.
The ability to consistently distinguish real protein structures from computationally generated model decoys is not yet a solved problem. One route to distinguish real protein structures from decoys is to delineate the important physical features that specify a real protein. For example, it has long been appreciated that the hydrophobic cores of proteins contribute significantly to their stability. As a dataset of decoys to compare with real protein structures, we studied submissions to the bi-annual CASP competition (specifically CASP11, 12, and 13), in which researchers attempt to predict the structure of a protein only knowing its amino acid sequence. Our analysis reveals that many of the submissions possess cores that do not recapitulate the features that define real proteins. In particular, the model structures appear more densely packed (because of energetically unfavorable atomic overlaps), contain too few residues in the core, and have improper distributions of hydrophobic residues throughout the structure. Based on these observations, we developed a deep learning method, which incorporates key physical features of protein cores, to predict how well a computational model recapitulates the real protein structure without knowledge of the structure of the target sequence. By identifying the important features of protein structure, our method is able to rank decoys from the CASP competitions equally well, if not better than, state-of-the-art methods that incorporate many additional features.
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