Methods of local topology are introduced to the field of protein physics. This is done by interpreting the folding and unfolding processes of a globular protein in terms of conformational bifurcations that alter the local topology of the proteins Cα backbone. The mathematical formulation extends Arnold's perestroikas to piecewise linear chains using the discrete Frenet frame formalism. In the low-temperature folded phase, the backbone geometry generalizes the concept of a Peano curve, with its modular building blocks modeled by soliton solutions of a discretized nonlinear Schrödinger equation. The onset of thermal unfolding begins when perestroikas change the flattening and branch points that determine the centers of solitons. When temperature increases, the perestroikas cascade, which leads to a progressive disintegration of the modular structures. The folding and unfolding processes are quantitatively characterized by a correlation function that describes the evolution of perestroikas under temperature changes. The approach provides a comprehensive framework for understanding the Physics of protein folding and unfolding transitions, contributing to the broader field of protein structure and dynamics.

The last years have witnessed remarkable advances in our understanding of the emergence and consequences of topological constraints in biological and soft matter. Examples are abundant in relation to (bio)polymeric systems and range from the characterization of knots in single polymers and proteins to that of whole chromosomes and polymer melts. At the same time, considerable advances have been made in the description of the interplay between topological and physical properties in complex fluids, with the development of techniques that now allow researchers to control the formation of and interaction between defects in diverse classes of liquid crystals. Thanks to technological progress and the integration of experiments with increasingly sophisticated numerical simulations, topological biological and soft matter is a vibrant area of research attracting scientists from a broad range of disciplines. However, owing to the high degree of specialization of modern science, many results have remained confined to their own particular fields, with different jargon making it difficult for researchers to share ideas and work together towards a comprehensive view of the diverse phenomena at play. Compelled by these motivations, here we present a comprehensive overview of topological effects in systems ranging from DNA and genome organization to entangled proteins, polymeric materials, liquid crystals, and theoretical physics, with the intention of reducing the barriers between different fields of soft matter and biophysics. Particular care has been taken in providing a coherent formal introduction to the topological properties of polymers and of continuum materials and in highlighting the underlying common aspects concerning the emergence, characterization, and effects of topological objects in different systems. The second half of the review is dedicated to the presentation of the latest results in selected problems, specifically, the effects of topological constraints on the viscoelastic properties of polymeric materials; their relation with genome organization; a discussion on the emergence and possible effects of knots and other entanglements in proteins; the emergence and effects of topological defects and solitons in complex fluids. This review is dedicated to the memory of Marek Cieplak.

Cats have an instinctive ability to use the connection governing parallel transport in the space of shapes to land safely on their feet. Here, we argue that the concept of connection, which is extensively used in general relativity and other parts of theoretical physics, also explains the impressive performance of molecular motors by enabling molecules to evade the conclusions of Feynman's ratchet-and-pawl analysis. First, we demonstrate the emergence of directed rotational motion from shape changes, which is independent of angular momentum. Then, we computationally design knotted polyalanine molecules and demonstrate the organization of individual atom thermal vibrations into collective rotational motion, which is independent of angular momentum. The motion occurs effortlessly even in ambient water and can be further enhanced through spontaneous symmetry breaking, rendering the molecule an effective theory time crystal. Our findings can be experimentally verified via nuclear magnetic resonance measurements and hold practical potential for molecular motor design and engineering.

Molecular motors are essential for the movement and transportation of macromolecules in living organisms. Among them, rotatory motors are particularly efficient. In this study, we investigated the long-term dynamics of the designed left-handed alpha/alpha toroid (PDB: 4YY2), the RBM2 flagellum protein ring from Salmonella (PDB: 6SD5), and the V-type Na+-ATPase rotor in Enterococcus hirae (PDB: 2BL2) using microcanonical and canonical molecular dynamics simulations with the coarse-grained UNRES force field, including a lipid-membrane model, on a millisecond laboratory time scale. Our results demonstrate that rotational motion can occur with zero total angular momentum in the microcanonical regime and that thermal motions can be converted into net rotation in the canonical regime, as previously observed in simulations of smaller cyclic molecules. For 6SD5 and 2BL2, net rotation (with a ratcheting pattern) occurring only about the pivot of the respective system was observed in canonical simulations. The extent and direction of the rotation depended on the initial conditions. This result suggests that rotatory molecular motors can convert thermal oscillations into net rotational motion. The energy from ATP hydrolysis is required probably to set the direction and extent of rotation. Our findings highlight the importance of molecular-motor structures in facilitating movement and transportation within living organisms.

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 Schro''dinger 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.

A dilute gas of Bose-Einstein condensed atoms in a non-rotated and axially symmetric harmonic trap is modelled by the time dependent Gross-Pitaevskii equation. When the angular momentum carried by the condensate does not vanish, the minimum energy state describes vortices (or antivortices) that propagate around the trap center. The number of (anti)vortices increases with the angular momentum, and they repel each other to form Abrikosov lattices. Besides vortices and antivortices there are also stagnation points where the superflow vanishes; to our knowledge the stagnation points have not been analyzed previously, in the context of the Gross-Pitaevskii equation. The Poincare index formula states that the difference in the number of vortices and stagnation points can never change. When the number of stagnation points is small, they tend to aggregate into degenerate propagating structures. But when the number becomes sufficiently large, the stagnation points tend to pair up with the vortex cores, to propagate around the trap center in regular lattice arrangements. There is an analogy with the geometry of the Kosterlitz-Thouless transition, with the angular momentum of the condensate as the external control parameter instead of the temperature.

In this paper, a novel discrete algebra is presented which follows by combining the SU(2) Lie-Poisson bracket with the discrete Frenet equation. Physically, the construction describes a discrete piecewise linear string in R³. The starting point of our derivation is the discrete Frenet frame assigned at each vertex of the string. Then the link vector that connects the neighboring vertices is assigned the SU(2) Lie-Poisson bracket. Moreover, the same bracket defines the transfer matrices of the discrete Frenet equation which relates two neighboring frames along the string. The procedure extends in a self-similar manner to an infinite hierarchy of Poisson structures. As an example, the first descendant of the SU(2) Lie-Poisson structure is presented in detail. For this, the spinor representation of the discrete Frenet equation is employed, as it converts the brackets into a computationally more manageable form. The final result is a nonlinear, nontrivial, and novel Poisson structure that engages four neighboring vertices.

We show that topology is a very effective tool, to construct classical Hamiltonian time crystals. For this we numerically analyze a general class of time crystalline Hamiltonians that are designed to model the dynamics of molecular closed strings. We demonstrate how the time crystalline qualities of a closed string are greatly enhanced when the string becomes knotted. The Hamiltonians that we investigate include a generalized Kratky-Porod wormlike chain model in combination with long range Coulomb and Lennard-Jones interactions. Such energy functions are commonplace in coarse grained molecular modeling. Thus we expect that physical realizations of Hamiltonian time crystals can be constructed in terms of knotted ring molecules.

Apart from being the most common mechanism of regulating protein function and transmitting signals throughout the cell, phosphorylation has an ability to induce disorder-to-order transition in an intrinsically disordered protein. In particular, it was shown that folding of the intrinsically disordered protein, eIF4E-binding protein isoform 2 (4E-BP2), can be induced by multisite phosphorylation. Here, the principles that govern the folding of phosphorylated 4E-BP2 (pT37pT46 4E-BP2(18-62)) are investigated by analyzing canonical and replica exchange molecular dynamics trajectories, generated with the coarse-grained united-residue force field, in terms of local and global motions and the time dependence of formation of contacts between Cas of selected pairs of residues. The key residues involved in the folding of the pT37pT46 4E-BP2(18-62) are elucidated by this analysis. The correlations between local and global motions are identified. Moreover, for a better understanding of the physics of the formation of the folded state, the experimental structure of the pT37pT46 4E-BP2(18-62) is analyzed in terms of a kink (heteroclinic standing wave solution) of a generalized discrete nonlinear Schrodinger equation. It is shown that without molecular dynamics simulations the kinks are able to identify not only the phosphorylated sites of protein, the key players in folding, but also the reasons for the weak stability of the pT37pT46 4E-BP2(18-62).

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.