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Methods of local topology are introduced to the field of protein physics. This is achieved by explaining how the folding and unfolding processes of a globular protein alter the local topology of the protein's C-alpha backbone through conformational bifurcations. The mathematical formulation builds on the concept of Arnol'd's perestroikas, by extending it 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 Schroedinger 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., Comment: 18 pages, 11 figures

Cats use the connection governing parallel transport in the space of shapes to land safely on their feet. Here we argue that this connection also explains the impressive performance of molecular motors by enabling molecules to evade conclusions of Feynman's ratchet-and-pawl analysis. We first demonstrate, using simple molecular models, how directed rotational motion can emerge from shape changes even without angular momentum. We then computationally design knotted polyalanine molecules and show how their shape space connection organizes individual atom thermal vibrations into collective rotational motion, independently of angular momentum. Our simulations show that rotational motion arises effortlessly even in ambient water, making the molecule an effective theory time crystal. Our findings have potential for practical molecular motor design and engineering and can be verified through high-precision nuclear magnetic resonance measurements., Comment: 3 figures 2 movies

A Hamiltonian time crystal can emerge when a Noether symmetry is subject to a condition that prevents the energy minimum from being a critical point of the Hamiltonian. A somewhat trivial example is the Schr\"odinger equation of a harmonic oscillator. The Noether charge for its particle number coincides with the square norm of the wave function, and the energy eigenvalue is a Lagrange multiplier for the condition that the wave function is properly normalized. A more elaborate example is the Gross-Pitaevskii equation that models vortices in a cold atom Bose-Einstein condensate. In an oblate, essentially two dimensional harmonic trap the energy minimum is a topologically protected timecrystalline vortex that rotates around the trap center. Additional examples are constructed using coarse grained Hamiltonian models of closed molecular chains. When knotted, the topology of a chain can support a time crystal. As a physical example, high precision all-atom molecular dynamics is used to analyze an isolated cyclopropane molecule. The simulation reveals that the molecular D$_{3h}$ symmetry becomes spontaneously broken. When the molecule is observed with sufficiently long stroboscopic time steps it appears to rotate like a simple Hamiltonian time crystal. When the length of the stroboscopic time step is decreased the rotational motion becomes increasingly ratcheting and eventually it resembles the back-and-forth oscillations of Sisyphus dynamics. The stroboscopic rotation is entirely due to atomic level oscillatory shape changes, so that cyclopropane is an example of a molecule that can rotate without angular momentum. Finally, the article is concluded with a personal recollection how Frank's and Betsy's Stockholm journey started., Comment: Based on talk presented at the FrankFest 2021

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 Poincar\'e 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., Comment: 18 pages, 4 figures, 2 figure tables; Discussion of analogy with the Kosterlitz-Thouless transition; Additionnal discussion of the fixed point structures (animations are available in ancillary files)

Vortices in a Bose-Einstein condensate are modelled as spontaneously symmetry breaking minimum energy solutions of the time dependent Gross-Pitaevskii equation, using the method of constrained optimization. In a non-rotating axially symmetric trap, the core of a single vortex precesses around the trap center and, at the same time, the phase of its wave function shifts at a constant rate. The precession velocity, the speed of phase shift, and the distance between the vortex core and the trap center, depend continuously on the value of the conserved angular momentum that is carried by the entire condensate. In the case of a symmetric pair of identical vortices, the precession engages an emergent gauge field in their relative coordinate, with a flux that is equal to the ratio between the precession and shift velocities., Comment: improved version; Replaced with a version in print in JHEP; Supplemental Material as ancillary file; 21 pages, 5 figures

A deformable body can rotate even with no angular momentum, simply by changing its shape. A good example is a falling cat, how it maneuvers in air to land on its feet. Here a first principles molecular level example of the phenomenon is presented. For this the thermal vibrations of individual atoms in an isolated cyclopropane molecule are simulated in vacuum and at ultralow internal temperature values, and the ensuing molecular motion is followed stroboscopically. It is observed that in the limit of long stroboscopic time steps the vibrations combine into an apparent uniform rotation of the entire molecule even in the absence of angular momentum. This large time scale rotational motion is then modeled in an effective theory approach, in terms of timecrystalline Hamiltonian dynamics. The phenomenon is a temperature sensitive measurable. As such it has potential applications that range from models of autonomous molecular motors to development of molecular level detector, sensor and control technologies., Comment: 3 figures, 3 movies

We focus on a Hamiltonian system with a continuous symmetry, and dynamics that takes place on a presymplectic manifold. We explain how the symmetry can become spontaneously broken by a time crystal, that we define as the minimum of the available mechanical free energy that is simultaneously a time dependent solution of Hamilton's equation. The mathematical description of such a timecrystalline spontaneous symmetry breaking builds on concepts of equivariant Morse theory in the space of Hamiltonian flows. As an example we analyze a general family of timecrystalline Hamiltonians that is designed to model polygonal, piecewise linear closed strings. The vertices correspond to the locations of pointlike interaction centers; the string is akin a chain of atoms, that are joined together by covalent bonds, modeled by the links of the string. We argue that the timecrystalline character of the string can be affected by its topology. For this we show that a knotty string is usually more timecrystalline than a string with no self-entanglement. We also reveal a relation between phase space topology and the occurrence of timecrystalline dynamics. For this we show that in the case of three point particles, the presence of a time crystal can relate to a Dirac monopole that resides in the phase space. Our results propose that physical examples of Hamiltonian time crystals can be realized in terms of closed, knotted molecular rings., Comment: 4 figures

Classical Hamiltonian systems with conserved charges and those with constraints often describe dynamics on a pre-symplectic manifold. Here we show that a pre-symplectic manifold is also the proper stage to describe autonomous energy conserving Hamiltonian time crystals. We explain how the occurrence of a time crystal relates to the wider concept of spontaneously broken symmetries; in the case of a time crystal, the symmetry breaking takes place in a dynamical context. We then analyze in detail two examples of time crystalline Hamiltonian dynamics. The first example is a piecewise linear closed string, with dynamics determined by a Lie-Poisson bracket and Hamiltonian that relates to membrane stability. We explain how the Lie-Poisson brackets descents to a time crystalline pre-symplectic bracket, and we show that the Hamiltonian dynamics supports two phases; in one phase we have a time crystal and in the other phase time crystals are absent. The second example is a discrete Hamiltonian variant of the Q-ball Lagrangian of time dependent non-topological solitons. We explain how a Q-ball becomes a time crystal, and we construct examples of time crystalline Q-balls., Comment: 17 pages 4 figures

A time crystal is a time dependent physical system that does not reach a standstill, even in state of minimum energy. Here we show that the stability of a time crystal can be enhanced by its topology. For this we simulate time crystals made of chainlike ensembles of mutually interacting point particles. When we tie the chain into a knot we find that its timecrystalline qualities improve. The theoretical models we consider are widely used in coarse grained descriptions of linear polymers. Thus we expect that physical realizations of time crystals can be found in terms of knotted molecules., Comment: 1 figure

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 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., Comment: 17 pages, many figures

We inquire to what extent protein peptide plane and side chain dynamics can be reconstructed from knowledge of C-alpha dynamics. Due to lack of experimental data we analyze all atom molecular dynamics trajectories from Anton supercomputer, and for clarity we limit our attention to the peptide plane O atoms and side chain C-beta atoms. We try and reconstruct their dynamics using four different approaches. Three of these are the publicly available reconstruction programs Pulchra, Remo Scwrl4. The fourth, Statistical Method, builds entirely on statistical analysis of Protein Data Bank (PDB) structures. All four methods place the O and C-beta atoms accurately along the Anton trajectories. However, the Statistical Method performs best. The results suggest that under physiological conditions, the all atom dynamics is slaved to that of C-alpha atoms. The results can help improve all atom force fields, and advance reconstruction and refinement methods for reduced protein structures. The results provide impetus for development of effective coarse grained force fields in terms of reduced coordinates., Comment: 24 figures

In applications of mechanics, including quantum mechanics, we often consider complex systems, where complete solutions of the underlying "fundamental" equations is both impractical and unnecessary to describe appropriate observations accurately. For example, practical chemistry, including even precision first-principles quantum chemistry, is never concerned with the behavior of the subnuclear quarks and gluons. Instead, we often focus on a few key variables, and construct a so-called effective theory for those. Such effective theories can become complicated and non-local, even for fairly simple systems. But in many circumstances, when there is a separation of scales, we can treat the reduced set of variables as a conventional dynamical system in its own right, governed by an energy conserving Lagrangian or Hamiltonian, in a useful approximation. The structure of that emergent description can display qualitatively new features, notably including reduced dimensionality, manifested through unconventional Poisson brackets. Here we discuss the physical meaning and consequences of such truncated dynamics. We propose physically realizable toy models of molecular rings, wherein time crystals emerge at the classical level. We propose that such behavior occurs in the effective theory of highly diamagnetic aromatic ring molecules, and could be widespread., Comment: 4 figures

A protein is traditionally visualised as a piecewise linear discrete curve, and its geometry is conventionally characterised by the extrinsically determined Ramachandran angles. However, a protein backbone has also two independent intrinsic geometric structures, due to the peptide planes and the side chains. Here we adapt and develop modern 3D virtual reality techniques to scrutinize the atomic geometry along a protein backbone, in the vicinity of a peptide plane. For this we compare backbone geometry-based (extrinsic) and structure-based (intrinsic) coordinate systems, and as an example we inspect the trans and cis peptide planes. We reveal systematics in the way how a cis peptide plane deforms the neighbouring atomic geometry, and we develop a virtual reality based visual methodology that can identify the presence of a cis peptide plane from the arrangement of atoms in its vicinity. Our approach can easily detect exceptionally placed atoms in crystallographic structures. Thus it can be employed as a powerful visual refinement tool which is applicable also in the case when resolution of the protein structure is limited and whenever refinement is needed. As concrete examples we identify a number of crystallographic protein structures in Protein Data Bank (PDB) that display exceptional atomic positions around their cis peptide planes., Comment: 25 figures

Chiral heteropolymers such as larger globular proteins can simultaneously support multiple length scales. The interplay between different scales brings about conformational diversity, and governs the structure of the energy landscape. Multiple scales produces also complex dynamics, which in the case of proteins sustains live matter. However, thus far no clear understanding exist, how to distinguish the various scales that determine the structure and dynamics of a complex protein. Here we propose a systematic method to identify the scales in chiral heteropolymers such as a protein. For this we introduce a novel order parameter, that not only reveals the scales but also probes the phase structure. In particular, we argue that a chiral heteropolymer can simultaneously display traits of several different phases, contingent on the length scale at which it is scrutinized. Our approach builds on a variant of Kadanoff's block-spin transformation that we employ to coarse grain piecewise linear chains such as the C$\alpha$ backbone of a protein. We derive analytically and then verify numerically a number of properties that the order parameter can display. We demonstrate how, in the case of crystallographic protein structures in Protein Data Bank, the order parameter reveals the presence of different length scales, and we propose that a relation must exist between the scales, phases, and the complexity of folding pathways., Comment: 22 figures

We propose to combine a mean field approach with all atom molecular dynamics 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 modelling 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 C-alpha 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 C-alpha profile with ultra high 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 analyse the thermal stability of this Glauber attractor using room temperature molecular dynamics. We identify and scrutinise 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 alpha-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., Comment: 17 figures

We propose to develop mean field theory in combination with Glauber algorithm, 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 analysing the Engrailed homeodomain protein as an example. We demonstrate how to interpret S/WAXS data with a good precision and over an extended temperature range. We explain experimentally observed phenomena in terms of protein phase structure, and we make predictions for future experiments how the scattering data behaves at different ambient temperature values. We conclude that a combination of mean field theory with Glauber algorithm has the potential to develop into a highly accurate, computationally effective and predictive tool for analysing S/WAXS data. Finally, we compare our results with those obtained previously in an all-atom molecular dynamics simulation., Comment: 10 figures

The Landau-Ginzburg-Wilson paradigm is applied to model the low-temperature crystallographic C$\alpha$ 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 $\alpha$-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 characterise 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., Comment: 18 pages 16 figures

We inquire how structure emerges during the process of protein folding. For this we scrutinise col- lective 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 Schroedinger equation. The example we consider is an alpha-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 super-secondary 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 very high precision in terms of a soliton solution to a discrete nonlinear Schroedinger equation., Comment: 20 pages 29 figures

This is a set of lectures that I presented at the Les Houches 2014 Summer School "Topological Aspects in Condensed Matter Physics". The lectures are an introduction to physics of proteins. To physicists, and by a physicist. My lectures at les Houches were also celebration of the anniversary of Schroedinger's 1944 lectures, and for that reason I decided to share my title with his book.

A commonly recurring problem in structural protein studies, is the determination of all heavy atom positions from the knowledge of the central alpha-carbon coordinates. We employ advances in virtual reality to address the problem. The outcome is a 3D visualisation based technique where all the heavy backbone and side chain atoms are treated on equal footing, in terms of the C-alpha coordinates. Each heavy atom can be visualised on the surfaces of the different two-spheres, that are centered at the other heavy backbone and side chain atoms. In particular, the rotamers are visible as clusters which display strong dependence on the underlying backbone secondary structure. Our method easily detects those atoms in a crystallographic protein structure which have been been likely misplaced. Our approach forms a basis for the development of a new generation, visualisation based side chain construction, validation and refinement tools. The heavy atom positions are identified in a manner which accounts for the secondary structure environment, leading to improved accuracy over existing methods.

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 NMR 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 alpha-helical conformations that might cause the transition to amyloidosis, by nucleation.

It's been said that "mathematics is biology's next microscope, only better; biology is mathematics' next physics, only better". Here we aim for something even better. We try to combine mathematical physics and biology into a picoscope of life. For this we merge techniques which have been introduced and developed in modern mathematical physics, largely by Ludvig Faddeev to describe objects such as solitons and Higgs and to explain phenomena such as anomalies in gauge fields. We propose a synthesis that can help to resolve the protein folding problem, one of the most important conundrums in all of science. We apply the concept of gauge invariance to scrutinize the extrinsic geometry of strings in three dimensional space. We evoke general principles of symmetry in combination with Wilsonian universality and derive an essentially unique Landau-Ginzburg energy that describes the dynamics of a generic string-like configuration in the far infrared. We observe that the energy supports topological solitons, that pertain to an anomaly in the manner how a string is framed around its inflection points. We explain how the solitons operate as modular building blocks from which folded proteins are composed. We describe crystallographic protein structures by multi-solitons with experimental precision, and investigate the non-equilibrium dynamics of proteins under varying temperature. We simulate the folding process of a protein at in vivo speed and with close to pico-scale accuracy using a standard laptop computer: With pico-biology as mathematical physics' next pursuit, things can only get better., Comment: A section on thermostatting has been reformulated

This paper deals with various interrelations between strings and surfaces in three dimensional ambient space, two dimensional integrable models and two dimensional and four dimensional decomposed SU(2) Yang-Mills theories. Initially, a spinor version of the Frenet equation is introduced in order to describe the differential geometry of static three dimensional string-like structures. Then its relation to the structure of the su(2) Lie algebra valued Maurer-Cartan one-form is presented; while by introducing time evolution of the string a Lax pair is obtained, as an integrability condition. In addition, it is show how the Lax pair of the integrable nonlinear Schroedinger equation becomes embedded into the Lax pair of the time extended spinor Frenet equation and it is described how a spinor based projection operator formalism can be used to construct the conserved quantities, in the case of the nonlinear Schroedinger equation. Then the Lax pair structure of the time extended spinor Frenet equation is related to properties of flat connections in a two dimensional decomposed SU(2) Yang-Mills theory. In addition, the connection between the decomposed Yang-Mills and the Gauss-Godazzi equation that describes surfaces in three dimensional ambient space is presented. In that context the relation between isothermic surfaces and integrable models is discussed. Finally, the utility of the Cartan approach to differential geometry is considered. In particular, the similarities between the Cartan formalism and the structure of both two dimensional and four dimensional decomposed SU(2) Yang-Mills theories are discussed, while the description of two dimensional integrable models as embedded structures in the four dimensional decomposed SU(2) Yang-Mills theory are presented.

Vortex rings are ubiquitous in fluids, with smoke rings being a familiar example. The interaction of multiple vortex rings produces complex dynamical behaviour, such as the leapfrogging motion first analysed by Helmholtz more than a century and a half ago. Here we report on numerical investigations of vortex ring dynamics in a different setting from fluids, namely, as solutions of the Landau-Lifshitz equation that models the evolution of the local magnetization in a ferromagnetic medium. We present the results of the first study on the dynamics of interacting magnetic vortex rings and provide a novel link between fluids and magnetism, by showing that a range of phenomena familiar in fluids are reproduced in ferromagnets. This includes the leapfrogging motion of a pair of vortex rings and evidence for the chaotic dynamics of a trio of rings., Comment: 18 pages, 9 figures. Version to appear in Nonlinearity

The conformational complexity of linear polymers far exceeds that of point-like atoms and molecules. Polymers can bend, twist, even become knotted. Thus they may also display a much richer phase structure than point particles. But it is not very easy to characterize the phase of a polymer. Essentially, the only attribute is the radius of gyration. The way how it changes when the degree of polymerization becomes different, and how it evolves when the ambient temperature and solvent properties change, discloses the phase of the polymer. Moreover, in any finite length chain there are corrections to scaling, that complicate the detailed analysis of the phase structure. Here we introduce a quantity that we call the folding angle, a novel tool to identify and scrutinize the phases of polymers. We argue for a mean-field relationship between its values and those of the scaling exponent in the radius of gyration. But unlike in the case of the radius of gyration, the value of the folding angle can be evaluated from a single structure. As an example we estimate the value of the folding angle in the case of crystallographic {\alpha}-helical protein structures in the Protein Data Bank (PDB). We also show how the value can be numerically computed using a theoretical model of {\alpha}-helical chiral homopolymers., Comment: 2 figures

The theory of string-like continuous curves and discrete chains have numerous important physical applications. Here we develop a general geometrical approach, to systematically derive Hamiltonian energy functions for these objects. In the case of continuous curves, we demand that the energy function must be invariant under local frame rotations, and it should also transform covariantly under reparametrizations of the curve. This leads us to consider energy functions that are constructed from the conserved quantities in the hierarchy of the integrable nonlinear Schr\"odinger equation (NLSE). We point out the existence of a Weyl transformation that we utilize to introduce a dual hierarchy to the standard NLSE hierarchy. We propose that the dual hierarchy is also integrable, and we confirm this to the first non-trivial order. In the discrete case the requirement of reparametrization invariance is void. But the demand of invariance under local frame rotations prevails, and we utilize it to introduce a discrete variant of the Zakharov-Shabat recursion relation. We use this relation to derive frame independent quantities that we propose are the essentially unique and as such natural candidates for constructing energy functions for piecewise linear polygonal chains. We also investigate the discrete version of the Weyl duality transformation. We confirm that in the continuum limit the discrete energy functions go over to their continuum counterparts, including the perfect derivative contributions.

We construct an energy function that describes the crystallographic structure of spermwhale myoglobin backbone. As a model in our construction, we use the Protein Data Bank entry 1ABS that has been measured at liquid helium temperature. Consequently the thermal B-factor fluctuations are very small, which is an advantage in our construction. The energy function that we utilize resembles that of the discrete non-linear Schrodinger equation. Likewise, ours supports solitons as local minimum energy configurations. We describe the 1ABS backbone in terms of solitons with a precision that deviates from 1ABS by an average root-mean-square distance, which is less than the experimentally observed Debye-Waller B-factor fluctuation distance. We then subject the multisoliton solution to extensive numerical heating and cooling experiments, over a very wide range of temperatures. We concentrate in particular to temperatures above 300K and below the theta-point unfolding temperature, which is around 348K. We confirm that the behavior of the multisoliton is fully consistent with Anfinsen's principle, up to very high temperatures. We observe that the structure responds to an increase of temperature consistently in a very similar manner. This enables us to characterize the onset of thermally induced conformational changes in terms of three distinct backbone ligand gates. One of the gates is made of the helix F and the helix E. This is a pathway that is presumed to have a major role in ligand migration between the heme and the exterior. The two other gates are chosen similarly, when open they provide a direct access route for a ligand to reach the heme. We find that out of the three gates we investigate, the one which is formed by helices B and G is the most sensitive one to thermally induced conformational changes. Our approach provides a novel perspective to the important problem of ligand migration., Comment: 24 figures

The enterobacteria lambda phage is a paradigm temperate bacteriophage. Its lysogenic and lytic life cycles echo competition between the DNA binding $\lambda$-repressor (CI) and CRO proteins. Here we scrutinize the structure, stability and folding pathways of the $\lambda$-repressor protein, that controls the transition from the lysogenic to the lytic state. We first investigate the super-secondary helix-loop-helix composition of its backbone. We use a discrete Frenet framing to resolve the backbone spectrum in terms of bond and torsion angles. Instead of four, there appears to be seven individual loops. We model the putative loops using an explicit soliton Ansatz. It is based on the standard soliton profile of the continuum nonlinear Schr\"odinger equation. The accuracy of the Ansatz far exceeds the B-factor fluctuation distance accuracy of the experimentally determined protein configuration. We then investigate the folding pathways and dynamics of the $\lambda$-repressor protein. We introduce a coarse-grained energy function to model the backbone in terms of the C$_\alpha$ atoms and the side-chains in terms of the relative orientation of the C$_\beta$ atoms. We describe the folding dynamics in terms of relaxation dynamics, and find that the folded configuration can be reached from a very generic initial configuration. We conclude that folding is dominated by the temporal ordering of soliton formation. In particular, the third soliton should appear before the first and second. Otherwise, the DNA binding turn does not acquire its correct structure. We confirm the stability of the folded configuration by repeated heating and cooling simulations.

Folded proteins have a modular assembly. They are constructed from regular secondary structures like alpha-helices and beta-strands that are joined together by loops. Here we develop a visualization technique that is adapted to describe this modular structure. In complement to the widely employed Ramachandran plot that is based on toroidal geometry, our approach utilizes the geometry of a two-sphere. Unlike the more conventional approaches that only describea given peptide unit, ours is capable of describing the entire backbone environment including the neighboring peptide units. It maps the positions of each atom to the surface of the two-sphere exactly how these atoms are seen by an observer who is located at the position of the central C-alpha atom. At each level of side-chain atoms we observe a strong correlation between the positioning of the atom and the underlying local secondary structure with very little if any variation between the different amino acids. As a concrete example we analyze the left-handed helix region of non-glycyl amino acids. This region corresponds to an isolated and highly localized residue independent sector in the direction of the C-beta carbons on the two-sphere. We show that the residue independent localization extends to C-gamma and C-delta carbons, and to side-chain oxygen and nitrogen atoms in the case of asparagine and aspartic acid. When we extend the analysis to the side-chain atoms of the neighboring residues, we observe that left-handed beta-turns display a regular and largely amino acid independent structure that can extend to seven consecutive residues. This collective pattern is duu to the presence of a backbone soliton. We show how one can use our visualization techniques to analyze and classify the different solitons in terms of selection rules that we describe in detail., Comment: arXiv admin note: substantial text overlap with arXiv:1104.2261

We investigate the fine structure of the sp3 hybridized covalent bond geometry that governs the tetrahedral architecture around the central C$_\alpha$ carbon of a protein backbone, and for this we develop new visualization techniques to analyze high resolution X-ray structures in Protein Data Bank. We observe that there is a correlation between the deformations of the ideal tetrahedral symmetry and the local secondary structure of the protein. We propose a universal coarse grained energy function to describe the ensuing side-chain geometry in terms of the C$_\beta$ carbon orientations. The energy function can model the side-chain geometry with a sub-atomic precision. As an example we construct the C$_\alpha$-C$_\beta$ structure of HP35 chicken villin headpiece. We obtain a configuration that deviates less than 0.4 \.A in root-mean-square distance from the experimental X-ray structure.

Protein collapse can be viewed as a dynamical phase transition, during which new scales and collective variables become excited while the old ones recede and fade away. This causes formidable computational bottle-necks in approaches that are based on atomic scale scrutiny. Here we consider an effective dynamical Landau theory to model the folding process at biologically relevant time and distance scales. We reach both a substantial decrease in the execution time and improvement in the accuracy of the final configuration, in comparison to more conventional approaches. As an example we inspect the collapse of HP35 chicken villin headpiece subdomain, where there are detailed molecular dynamics simulations to compare with. We start from a structureless, unbend and untwisted initial configuration. In less than one second of wall-clock time on a single processor personal computer we consistently reach the native state with 0.5 Angstrom root mean square distance (RMSD) precision. We confirm that our folding pathways are indeed akin those obtained in recent atomic level molecular dynamics simulations. We conclude that our approach appears to have the potential for a computationally economical method to accurately understand theoretical aspects of protein collapse., Comment: 5 pages 2 figs

The emergence of biochemical activities in a protein seem to commence with the onset of atomic mean-square displacements along the protein lattice. The ensuing protein dynamical transition has been discussed extensively in the literature, and often with conflicting conclusions. Here we clarify the phenomenon by establishing a deep connection between the dynamical transition and the pseudogap state where high-temperature superconductivity comes to its end. For this we first show how to endow proteins with an order parameter akin the quasiparticle wave function in superconductors. We then present universality arguments to claim that the protein dynamical transition takes place in tandem with a pseudogap transmutation. We confirm that available experimental data fully supports our proposal., Comment: 4 pages one figure

Structural classification shows that the number of different protein folds is surprisingly small. It also appears that proteins are built in a modular fashion, from a relatively small number of components. Here we propose to identify the modular building blocks of proteins with the dark soliton solution of a generalized discrete nonlinear Schrodinger equation. For this we show that practically all protein loops can be obtained simply by scaling the size and by joining together a number of copies of the soliton, one after another. The soliton has only two loop specific parameters and we identify their possible values in Protein Data Bank. We show that with a collection of 200 sets of parameters, each determining a soliton profile that describes a different short loop, we cover over 90% of all proteins with experimental accuracy. We also present two examples that describe how the loop library can be employed both to model and to analyze the structure of folded proteins., Comment: 7 pages 6 figs

Alzheimer's disease causes severe neurodegeneration in the brain that leads to a certain death. The defining factor is the formation of extracellular senile amyloid plaques in the brain. However, therapeutic approaches to remove them have not been effective in humans, and so our understanding of the cause of Alzheimer's disease remains incomplete. Here we investigate physical processes that might relate to its onset. Instead of the extracellular amyloid, we scrutinize the intracellular domain of its precursor protein. We argue for a phenomenon that has never before been discussed in the context of polymer physics: Like ice and water together, the intracellular domain of the amyloid precursor protein forms a state of phase coexistence with another protein. This leads to an inherent instability that could well be among the missing pieces in the puzzle of Alzheimer's disease., Comment: 4 figures

Both symmetry and organized breaking of symmetry have a pivotal r\^ole in our understanding of structure and pattern formation in physical systems, including the origin of mass in the Universe and the chiral structure of biological macromolecules. Here we report on a new symmetry breaking phenomenon that takes place in all biologically active proteins, thus this symmetry breaking relates to the inception of life. The unbroken symmetry determines the covalent bond geometry of a sp3 hybridized carbon atom. It dictates the tetrahedral architecture of atoms around the central carbon of an amino acid. Here we show that in a biologically active protein this symmetry becomes broken. Moreover, we show that the pattern of symmetry breaking is in a direct correspondence with the local secondary structure of the folded protein., Comment: 5 pages 3 figs

We inspect the geometry of proteins by identifying their backbones as framed polygons. We find that the left-handed helix region of the Ramachandran map for non-glycyl residues corresponds to an isolated and highly localized sector in the orientation of the $C_\beta$ carbons, when viewed in a Frenet frame that is centered at the corresponding $C_\alpha$ carbons. We show that this localization in the orientation persists to $C_\gamma$ and $C_\delta$ carbons. Furthermore, when we extend our analysis to the neighboring residues we conclude that the left-handed helix region reflects a very regular and apparently residue independent collective interplay of at least seven consecutive amino acids., Comment: 13 figures

The lambda phage is a paradigm temperate bacteriophage. Its lysogenic and lytic life cycles echo competition between the DNA binding CI and CRO proteins. Here we address the Physics of this transition in terms of an energy function that portrays the backbone as a multi-soliton configuration. The precision of the individual solitons far exceeds the B-factor accuracy of the experimentally determined protein conformations giving us confidence to conclude that three of the four loops are each composites of two closely located solitons. The only exception is the repressive DNA binding turn, it is the sole single soliton configuration of the backbone. When we compare the solitons with the Protein Data Bank we find that the one preceding the DNA recognition helix is unique to the CI protein, prompting us to conclude that the lysogenic to lytic transition is due to a saddle-node bifurcation involving a soliton-antisoliton annihilation that removes the first loop., Comment: 5 pages 3 figures

We develop a transfer matrix formalism to visualize the framing of discrete piecewise linear curves in three dimensional space. Our approach is based on the concept of an intrinsically discrete curve, which enables us to more effectively describe curves that in the limit where the length of line segments vanishes approach fractal structures in lieu of continuous curves. We verify that in the case of differentiable curves the continuum limit of our discrete equation does reproduce the generalized Frenet equation. As an application we consider folded proteins, their Hausdorff dimension is known to be fractal. We explain how to employ the orientation of $C_\beta$ carbons of amino acids along a protein backbone to introduce a preferred framing along the backbone. By analyzing the experimentally resolved fold geometries in the Protein Data Bank we observe that this $C_\beta$ framing relates intimately to the discrete Frenet framing. We also explain how inflection points can be located in the loops, and clarify their distinctive r\^ole in determining the loop structure of foldel proteins., Comment: 14 pages 12 figures

Recently it has been pointed out that the "Faddeev-Niemi" equations that correspond to the Yang-Mills equations of motion for a decomposed gauge field, can have solutions that obey the standard Yang-Mills equations with a source term. Here we present a general class of such gauge field configurations., Comment: 4 pages; v2 typos corrected

A comparative classification scheme provides a good basis for several approaches to understand proteins, including prediction of relations between their structure and biological function. But it remains a challenge to combine a classification scheme that describes a protein starting from its well organized secondary structures and often involves direct human involvement, with an atomary level Physics based approach where a protein is fundamentally nothing more than an ensemble of mutually interacting carbon, hydrogen, oxygen and nitrogen atoms. In order to bridge these two complementary approaches to proteins, conceptually novel tools need to be introduced. Here we explain how the geometrical shape of entire folded proteins can be described analytically in terms of a single explicit elementary function that is familiar from nonlinear physical systems where it is known as the kink-soliton. Our approach enables the conversion of hierarchical structural information into a quantitative form that allows for a folded protein to be characterized in terms of a small number of global parameters that are in principle computable from atomary level considerations. As an example we describe in detail how the native fold of the myoglobin 1M6C emerges from a combination of kink-solitons with a very high atomary level accuracy. We also verify that our approach describes longer loops and loops connecting $\alpha$-helices with $\beta$-strands, with same overall accuracy., Comment: 3 figures

We introduce a novel generalization of the discrete nonlinear Schr\"odinger equation. It supports solitons that describe how proteins fold. As an example we scrutinize the villin headpiece HP35, an archetypal protein for testing both experimental and theoretical approaches to protein folding. Using explicit soliton profiles we construct its carbon backbone with an unprecedented accuracy., Comment: 5 pages 2 figures

We present a numerical Monte Carlo analysis of a continuos spin Ising chain that can describe the statistical proterties of folded proteins. We find that depending on the value of the Metropolis temperature, the model displays the three known nontrivial phases of polymers: At low temperatures the model is in a collapsed phase, at medium temperatures it is in a random walk phase, and at high temperatures it enters the self-avoiding random walk phase. By investigating the temperature dependence of the specific energy we confirm that the transition between the collapsed phase and the random walk phase is a phase transition, while the random walk phase and self-avoiding random walk phase are separated from each other by a cross-over transition. We also compare the predictions of the model to a phenomenological elastic energy formula, proposed by Huang and Lei to describe folded proteins., Comment: 12 pages, 23 figures, RevTeX 4.1

We consider the coupling between four dimensional Yang-Mills field and a three brane that fluctuates into higher dimensions. For this we interpret the Yang-Mills theory as a higher dimensional bulk gravity theory with dynamics that is governed by the Einstein action, and with a metric tensor constructed from the gauge field in a manner that displays the original gauge symmetry as an isometry. The brane moves in this higher dimensional space-time under the influence of its bulk gravity, with dynamics determined by the Nambu action. This introduces the desired interaction between the brane and the gauge field in a way that preserves the original gauge invariance as an isometry of the induced metric. After a prudent change of variables the result can be interpreted as a gauge invariant and massive vector field that propagates in the original space-time $\mathbb R^4$: The presence of the brane becomes entirely invisible, except for the mass., Comment: 9 pages

We inquire whether a resolution to the electroweak hierarchy problem could reside in symmetries that relate the bosonic Weinberg-Salam Lagrangian with a higher dimensional generally covariant theory. For this we consider a three-brane that moves under the influence of a seven dimensional pure Hilbert Einstein-like generally covariant theory. We introduce a change of variables that combines the conformal scale of the metric tensor with the brane fluctuations, so that the conformal scale becomes the modulus and the fluctuations become the angular field degrees of freedom of a polarly decomposed Higgs. When we assume that the four dimensional space-time background of the generally covariant theory is locally conformally flat and that the internal space is a squashed three-sphere, we arrive at one massless and three massive vector fields akin those in the Weinberg-Salam model and recover all the familiar ingredients of the symmetry broken bosonic Weinberg-Salam model, except that there is no bare Higgs mass. The Higgs mass is subject to dimensional censorship, its presence is forbidden by general covariance. This proposes that a resolution to the electroweak hierarchy problem might well reside in higher dimensions, in a Ward-Takahashi like identities for general covariance that relate a non-vanishing Higgs mass to dynamical breaking of general covariance. Moreover, the two electroweak gauge couplings are both determined by the squashing parameter of the internal three sphere and when we impose the condition that the vector boson masses must be in line with custodial symmetry we arrive at the classical level to the Weinberg angle sin^2 \theta_W = 0.296.

We propose that protein loops can be interpreted as topological domain-wall solitons. They interpolate between ground states that are the secondary structures like alpha-helices and beta-strands. Entire proteins can then be folded simply by assembling the solitons together, one after another. We present a simple theoretical model that realizes our proposal and apply it to a number of biologically active proteins including 1VII, 2RB8, 3EBX (Protein Data Bank codes). In all the examples that we have considered we are able to construct solitons that reproduce secondary structural motifs such as alpha-helix-loop-alpha-helix and beta-sheet-loop-beta-sheet with an overall root-mean-square-distance accuracy of around 0.7 Angstrom or less for the central alpha-carbons, i.e. within the limits of current experimental accuracy., Comment: 4 pages, 4 figures

Proteins can sometimes be knotted, and for many reasons the study of knotted proteins is rapidly becoming very important. For example, it has been proposed that a knot increases the stability of a protein. Knots may also alter enzymatic activities and enhance binding. Moreover, knotted proteins may even have some substantial biomedical significance in relation to illnesses such as Parkinson's disease. But to a large extent the biological role of knots remains a conundrum. In particular, there is no explanation why knotted proteins are so scarce. Here we argue that knots are relatively rare because they tend to cause swelling in proteins that are too short, and presently short proteins are over-represented in the Protein Data Bank (PDB). Using Monte Carlo simulations we predict that the figure-8 knot leads to the most compact protein configuration when the number of amino acids is in the range of 200-600. For the existence of the simplest knot, the trefoil, we estimate a theoretical upper bound of 300-400 amino acids, in line with the available PDB data., Comment: 13 pages 3 figures