From Feynman's ratchet to time crystalline molecular motors.

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.

Long-Time Dynamics of Selected Molecular-Motor Components Using a Physics-Based Coarse-Grained Approach.

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.

Statistical model for factors correlating with COVID-19 deaths.

Background: The COVID-19 pandemic has caused major disruption in societies globally. Our aim is to understand, what factors were associated with the impact of the pandemic on death rates. This will help countries to better prepare for and respond in future pandemics. Methods: We modeled with a linear mixed effect model the impact of COVID-19 with the dependent variable "Daily mortality change" (DMC) with country features variables and intervention (containment measurement) data. We tested both country characteristics consisting of demographic, societal, health related, healthcare system specific, environmental and cultural feature as well as COVID-19 specific response in the form of social distancing interventions. Results: A statistically significant country feature was Geert Hofstede's masculinity, i.e., the extent to which the use of force is endorsed socially, correlating positively with a higher DMC. The effects of different interventions were stronger that those of country features, particularly cancelling public events, controlling international travel and closing workplaces. Conclusion: Social distancing interventions and the country feature: Geert Hofstede's masculinity dimension had a significant impact on COVID-19 mortality change. However other country features, such as development and population health did not show significance. Thus, the crises responders and scholars could revisit the concept and understanding of preparedness for and response to pandemics.

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-ß-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.

Investigation of Phosphorylation-Induced Folding of an Intrinsically Disordered Protein by Coarse-Grained Molecular Dynamics.

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-BP218-62) 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 Cαs of selected pairs of residues. The key residues involved in the folding of the pT37pT46 4E-BP218-62 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-BP218-62 is analyzed in terms of a kink (heteroclinic standing wave solution) of a generalized discrete nonlinear Schrödinger 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-BP218-62.

On topology and knotty entanglement in protein folding.

We investigate aspects of topology in protein folding. For this we numerically simulate the temperature driven folding and unfolding of the slipknotted archaeal virus protein AFV3-109. Due to knottiness the (un)folding is a topological process, it engages the entire backbone in a collective fashion. Accordingly we introduce a topological approach to model the process. Our simulations reveal that the (un)folding of AFV3-109 slipknot proceeds through a folding intermediate that has the topology of a trefoil knot. We observe that the final slipknot causes a slight swelling of the folded AFV3-109 structure. We disclose the relative stability of the strands and helices during both the folding and unfolding processes. We confirm results from previous studies that pointed out that it can be very demanding to simulate the formation of knotty self-entanglement, and we explain how the problems are circumvented: The slipknotted AFV3-109 protein is a very slow folder with a topologically demanding pathway, which needs to be properly accounted for in a simulation description. When we either increase the relative stiffness of bending, or when we decrease the speed of ambient cooling, the rate of slipknot formation rapidly increases.

Autonomous topological time crystals and knotty molecular motors.

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.

New Insights into Folding, Misfolding, and Nonfolding Dynamics of a WW Domain.

Intermediate states in protein folding are associated with formation of amyloid fibrils, which are responsible for a number of neurodegenerative diseases. Therefore, prevention of the aggregation of folding intermediates is one of the most important problems to overcome. Recently, we studied the origins and prevention of formation of intermediate states with the example of the Formin binding protein 28 (FBP28) WW domain. We demonstrated that the replacement of Leu26 by Asp26 or Trp26 (in ∼15% of the folding trajectories) can alter the folding scenario from three-state folding, a major folding scenario for the FBP28 WW domain (WT) and its mutants, toward two-state or downhill folding at temperatures below the melting point. Here, for a better understanding of the physics of the formation/elimination of intermediates, (i) the dynamics and energetics of formation of ß-strands in folding, misfolding, and nonfolding trajectories of these mutants (L26D and L26W) is investigated; (ii) the experimental structures of WT, L26D, and L26W are analyzed in terms of a kink (heteroclinic standing wave solution) of a generalized discrete nonlinear Schrödinger equation. We show that the formation of each ß-strand in folding trajectories is accompanied by the emergence of kinks in internal coordinate space as well as a decrease in local free energy. In particular, the decrease in downhill folding trajectory is ∼7 kcal/mol, while it varies between 31 and 48 kcal/mol for the three-state folding trajectory. The kink analyses of the experimental structures give new insights into formation of intermediates, which may become a useful tool for preventing aggregation.

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.

Protein tertiary structure and the myoglobin phase diagram.

We develop an effective theory approach to investigate the phase properties of globular proteins. Instead of interactions between individual atoms or localized interaction centers, the approach builds directly on the tertiary structure of a protein. As an example we construct the phase diagram of (apo)myoglobin with temperature (T) and acidity (pH) as the thermodynamical variables. We describe how myoglobin unfolds from the native folded state to a random coil when temperature and acidity increase. We confirm the presence of two molten globule folding intermediates, and we predict an abrupt transition between the two when acidity changes. When temperature further increases we find that the abrupt transition line between the two molten globule states terminates at a tricritical point, where the helical structures fade away. Our results also suggest that the ligand entry and exit is driven by large scale collective motions that destabilize the myoglobin F-helix.

Can the geometry of all-atom protein trajectories be reconstructed from the knowledge of Cα time evolution? A study of peptide plane O and side chain Cß atoms.

We inquire to what extent can the geometry of protein peptide plane and side chain atoms be reconstructed from the knowledge of Cα time evolution. Due to the lack of experimental data, we analyze all atom molecular dynamics trajectories from the Anton supercomputer, and for clarity, we limit our attention to the peptide plane O atoms and side chain Cß atoms. We reconstruct their positions using four different approaches. Three of these are the publicly available reconstruction programs Pulchra, Remo, and Scwrl4. The fourth, Statistical Method, builds entirely on the statistical analysis of Protein Data Bank structures. All four methods place the O and Cß atoms accurately along the Anton trajectories; the Statistical Method gives results that are closest to the Anton data. The results suggest that when a protein moves under physiological conditions, its all atom structures can be reconstructed with high accuracy from the knowledge of the Cα atom positions. This can help to better understand and improve all atom force fields, and advance reconstruction and refinement methods for reduced protein structures. The results provide impetus for the development of effective coarse grained force fields in terms of reduced coordinates.

Study of correlations between protein peptide plane dynamics and side chain dynamics.

Protein dynamics is pivotal to biological processes. However, experiments are very demanding and difficult to perform, and all-atom molecular dynamics simulations can still not provide all the answers. This motivates us to analyze protein dynamics in terms of different reduced coordinate representations. We then need to resolve how to reconstruct the full all-atom dynamics from its coarse grained approximation. Accordingly we scrutinize all-atom molecular dynamics trajectories in terms of crystallographic Protein Data Bank (PDB) structures, and inquire to what extent is it possible to predict the dynamics of side chain Cß atoms in terms of the static properties of backbone Cα and O atoms. Here we find that simulated Cß dynamics at near physiological conditions can be reconstructed with very high precision, using the knowledge of the crystallographic backbone Cα and O positions. The precision we can reach with our PDB-based Statistical Method reconstruction exceeds that of popular all-atom reconstruction methods such as Remo and Pulchra, and is fully comparable with the precision of the highly elaborate Scwrl4 all-atom reconstruction method that we have enhanced with the knowledge of the backbone Cα and O atom positions. We then conclude that in a dynamical protein that moves around at physiological conditions, the relative positions of its Cß atoms with respect to the backbone Cα and O atoms, deviate very little from their relative positions in static crystallographic PDB structures. This proposes that the dynamics of a biologically active protein could remain subject to very similar, stringent stereochemical constraints that dictate the structure of a folded crystallographic protein. Thus, our results provide a strong impetus to the development of coarse grained techniques that are based on reduced coordinate representations.

Multistage modeling of protein dynamics with monomeric Myc oncoprotein as an example.

We propose to combine a mean-field approach with all-atom molecular dynamics (MD) into a multistage algorithm that can model protein folding and dynamics over very long time periods yet with atomic-level precision. As an example, we investigate an isolated monomeric Myc oncoprotein that has been implicated in carcinomas including those in colon, breast, and lungs. Under physiological conditions a monomeric Myc is presumed to be an example of intrinsically disordered proteins that pose a serious challenge to existing modeling techniques. We argue that a room-temperature monomeric Myc is in a dynamical state, it oscillates between different conformations that we identify. For this we adopt the Cα backbone of Myc in a crystallographic heteromer as an initial ansatz for the monomeric structure. We construct a multisoliton of the pertinent Landau free energy to describe the Cα profile with ultrahigh precision. We use Glauber dynamics to resolve how the multisoliton responds to repeated increases and decreases in ambient temperature. We confirm that the initial structure is unstable in isolation. We reveal a highly degenerate ground-state landscape, an attractive set towards which Glauber dynamics converges in the limit of vanishing ambient temperature. We analyze the thermal stability of this Glauber attractor using room-temperature molecular dynamics. We identify and scrutinize a particularly stable subset in which the two helical segments of the original multisoliton align in parallel next to each other. During the MD time evolution of a representative structure from this subset, we observe intermittent quasiparticle oscillations along the C-terminal α helix, some of which resemble a translating Davydov's Amide-I soliton. We propose that the presence of oscillatory motion is in line with the expected intrinsically disordered character of Myc.

Solution x-ray scattering and structure formation in protein dynamics.

We propose a computationally effective approach that builds on Landau mean-field theory in combination with modern nonequilibrium statistical mechanics to model and interpret protein dynamics and structure formation in small- to wide-angle x-ray scattering (S/WAXS) experiments. We develop the methodology by analyzing experimental data in the case of Engrailed homeodomain protein as an example. We demonstrate how to interpret S/WAXS data qualitatively with a good precision and over an extended temperature range. We explain experimental observations in terms of protein phase structure, and we make predictions for future experiments and for how to analyze data at different ambient temperature values. We conclude that the approach we propose has the potential to become a highly accurate, computationally effective, and predictive tool for analyzing S/WAXS data. For this, we compare our results with those obtained previously in an all-atom molecular dynamics simulation.

Conformational landscape of an amyloid intra-cellular domain and Landau-Ginzburg-Wilson paradigm in protein dynamics.

The Landau-Ginzburg-Wilson paradigm is proposed as a framework, to investigate the conformational landscape of intrinsically unstructured proteins. A universal Cα-trace Landau free energy is deduced from general symmetry considerations, with the ensuing all-atom structure modeled using publicly available reconstruction programs Pulchra and Scwrl. As an example, the conformational stability of an amyloid precursor protein intra-cellular domain (AICD) is inspected; the reference conformation is the crystallographic structure with code 3DXC in Protein Data Bank (PDB) that describes a heterodimer of AICD and a nuclear multi-domain adaptor protein Fe65. Those conformations of AICD that correspond to local or near-local minima of the Landau free energy are identified. For this, the response of the original 3DXC conformation to variations in the ambient temperature is investigated, using the Glauber algorithm. The conclusion is that in isolation the AICD conformation in 3DXC must be unstable. A family of degenerate conformations that minimise the Landau free energy is identified, and it is proposed that the native state of an isolated AICD is a superposition of these conformations. The results are fully in line with the presumed intrinsically unstructured character of isolated AICD and should provide a basis for a systematic analysis of AICD structure in future NMR experiments.

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.

Thermal unfolding of myoglobin in the Landau-Ginzburg-Wilson approach.

The Landau-Ginzburg-Wilson paradigm is applied to model the low-temperature crystallographic Cα backbone structure of sperm whale myoglobin. The Glauber protocol is employed to simulate its response to an increase in ambient temperature. The myoglobin is found to unfold from its native state by a succession of α-helical intermediates, fully in line with the observed folding and unfolding patterns in denaturation experiments. In particular, a molten globule intermediate is identified with experimentally correct attributes. A detailed, experimentally testable contact map is constructed to characterize the specifics of the unfolding pathway, including the formation of long-range interactions. The results reveal how the unfolding process of a protein is driven by the interplay between, and a successive melting of, its modular secondary structure components.

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.

Aspects of structural landscape of human islet amyloid polypeptide.

The human islet amyloid polypeptide (hIAPP) co-operates with insulin to maintain glycemic balance. It also constitutes the amyloid plaques that aggregate in the pancreas of type-II diabetic patients. We have performed extensive in silico investigations to analyse the structural landscape of monomeric hIAPP, which is presumed to be intrinsically disordered. For this, we construct from first principles a highly predictive energy function that describes a monomeric hIAPP observed in a nuclear magnetic resonance experiment, as a local energy minimum. We subject our theoretical model of hIAPP to repeated heating and cooling simulations, back and forth between a high temperature regime where the conformation resembles a random walker and a low temperature limit where no thermal motions prevail. We find that the final low temperature conformations display a high level of degeneracy, in a manner which is fully in line with the presumed intrinsically disordered character of hIAPP. In particular, we identify an isolated family of α-helical conformations that might cause the transition to amyloidosis, by nucleation.