Davydov Soliton Research Articles

Exact solitary wave solutions for a coupled gKdV–Schrödinger system by a new ODE reduction method

AbstractA new method is developed for finding exact solitary wave solutions of a generalized Korteweg–de Vries equation with ‐power nonlinearity coupled to a linear Schrödinger equation arising in many different physical applications. This method yields 22 solution families, with . No solutions for were known previously in the literature. For , four of the solution families contain bright/dark Davydov solitons of the 1st and 2nd kind, obtained in recent literature by basic ansatz applied to the ordinary differential equation (ODE) system for traveling waves. All of the new solution families have interesting features, including bright/dark peaks with (up to) symmetric pairs of side peaks in the amplitude and a kink profile for the nonlinear part in the phase. The present method is fully systematic and involves several novel steps that reduce the traveling wave ODE system to a single nonlinear base ODE for which all polynomial solutions are found by symbolic computation. It is applicable more generally to other coupled nonlinear dispersive wave equations as well as to nonlinear ODE systems of generalized Hénon–Heiles form.

Knowns and unknowns in the Davydov model for energy transfer in proteins

The Davydov model for amide I propagation in hydrogen-bonded chains of proteins is revisited. The many similarities between the mixed quantum-classical dynamical equations and those that are derived from the full quantum Davydov model while applying the so-called D2 ansatz are highlighted. The transition from a minimum energy localized amide I state to a fully delocalized state is shown to operate in four phases, one of which is abrupt and the last of which is a fast but smooth change from a very broad yet localized state to a completely delocalized one. Exploration of the dynamical phase space at zero temperature includes the well-known soliton propagation as well as double and triple discrete breathers, and dispersion of initially localized states. The uncertainties related to the question of the thermal stability of the Davydov soliton are illustrated. A solution to the seemingly endless problem of the short radiative lifetime of the amide I excitations is proposed.

Nonlinear Wave Molecules for the Lakshmanan–Porsezian–Daniel Equation in Nonlinear Optics and Biology

AbstractThe nonlinear wave molecules of the Lakshmanan–Porsezian–Daniel (LPD) equation describing the propagation of ultrashort optical pulses through optical fibers and Davydov soliton in α‐helical proteins are investigated. Based on the analysis of characteristic lines, the breather molecules consisting of two, three, or even four different breather atoms are derived, and the synthesis modes of molecules by adjusting the values of the phase parameters are generated. The state conversion of the breather molecules is studied and a variety of converted wave molecules are produced. In particular, it is reported that the full conversion of the breather molecules does not exist in the LPD equation. The formation mechanisms of different types of nonlinear wave molecules through the nonlinear superposition are further explored and the influence of higher‐order effects on the breather molecules is discussed. Finally, the intricate interactions between the molecules and nonlinear waves are considered. It is found that the distances between atoms in the molecules as well as the shapes of the converted waves change after the interactions, and the essence of such shape‐changed interactions is revealed.

Several new analytical solutions for Davydov solitons in α-helix proteins

Various forms of rational solitons, breathers, rogue wave, periodic wave and multiwave for Davydov solitons in [Formula: see text]-helix proteins are described in this paper. We will figure out periodic wave, rogue wave, multiwave, [Formula: see text]-shaped solutions, homoclinic breathers, kink cross rational solutions, periodic cross kink, periodic cross rational solitons, generalized breather, Akhmediev breather and Kuznetsov–Ma breather. We will also discuss some interactions between them. At the end, we will provide the graphical representation of our newly discovered solutions.

Collective excitations in α-helical protein structures interacting with the water environment

ABSTRACT Low-frequency vibrational excitations of protein macromolecules in the terahertz frequency region are suggested to contribute to many biological processes such as enzymatic catalysis, intra-protein energy/charge transport, recognition, and allostery. To explain high effectiveness of these processes, two possible mechanisms of the long-lived excitation were proposed by H. Fröhlich and A.S. Davydov, which relate to either vibrational modes or solitary waves, respectively. In this paper, we developed a quantum dynamic model of vibrational excitation in α-helical proteins interacting with the aqueous environment. In the model, we distinguished three coupled subsystems, i.e., (i) a chain of hydrogen-bonded peptide groups (PGs), interacting with (ii) the subsystem of the side-chain residuals which in turn interact with (iii) the environment, surrounding water responsible for dissipation and fluctuation in the system. It was shown that the equation of motion for phonon variables of the PG chain can be transformed to nonlinear Schrodinger equation which admits bifurcation into the solution corresponding to the weak-damped vibrational modes (Fröhlich-type regime) and Davydov solitons. A bifurcation parameter is derived through the strength of phonon–phonon interaction between the side-chains and hydration-shell water molecules. As shown, the energy of these excited states is pumped through the interaction of the side-chains with fluctuating water environment of the proteins. The suggested mechanism of the collective vibrational mode excitation is discussed in connection with the recent experiments on the long-lived collective protein excitations in the terahertz frequency region and vibrational energy transport pathways in proteins.

Electrosoliton dynamics in a thermalized molecular chain

The possibility of the electrosoliton formation in α-helical proteins which represents a localized state of an extra electron bound with the deformation region of the α-helix arising due to the electron interaction with chain of peptide groups is investigated in a quasiclassical approximation. Two possible mechanisms of the formation of collective dynamic modes in the form of Frohlich collective mode and Davydov were previously suggested by the authors. In this paper, we developed a unified quantum-mechanics approach to describe conditions of the formation of the Frohlich vibronic state and Davydov in α-helical protein molecules interacting with the environment. The concept of soliton is used not only in the strict mathematical sense, i.e. in the case of completely integrable Hamiltonian systems, but also to describe dynamically stable, nonlinear collective structures. Davydov solitons are stable due to a small probability of the dissipation of its energy into thermal energy which provides a high efficiency of transport of energy, charges, and conformation changes in biosystems at a physiological temperature of 310 K. Electrosolitons can be formed if the value of electron–phonon interaction (EPI) parameter exceeds a certain threshold. One of the most important characteristics of the electrosoliton’s state is the coupling energy of a quasi-particle (exciton or electron) with molecular chain deformation, which also determines the stability. Dynamic equations describing the motion of a one-dimensional electrosoliton in the continuum approximation are a self-consistent system which includes the time-dependent Schrodinger equation with a deformation potential and an inhomogeneous linear wave equation for this potential. This system, known as the Zakharov system, has significance in physics and, generally, describes the nonlinear interaction of two physical subsystems: fast and slow. Zakharov equations have a well-known solution in the hyperbolic secant form, describing the envelope profile of the high-frequency vibrations of a fast subsystem, which can propagate with any subsonic velocity. The suggested mechanism of emergent of macroscopic dissipative structures in the form of electrosolitons in α-helical proteins is discussed in connection with recent experimental data on long-lived collective protein excitation in the terahertz frequency region.

Quantum tunneling of Davydov solitons through massive barriers

Protein clamps provide the cell with effective mechanisms for sensing of environmental changes and triggering adaptations that maintain homeostasis. The general physical mechanism behind protein clamping action, however, is poorly understood. Here, we explore the Davydov model for quantum transport of amide I energy, which is self-trapped in soliton states that propagate along the protein {\alpha}-helix spines and preserve their shape upon reflection from the {\alpha}-helix ends. We study computationally whether the Davydov solitons are able to reflect from massive barriers that model the presence of external protein clamps acting on a portion of the {\alpha}-helix, and characterize the range of barrier conditions for which the Davydov solitons are capable of tunneling through the barrier. The simulations showed that the variability of amino acid masses in proteins has a negligible effect on Davydov soliton dynamics, but a massive barrier that is a hundred times the mass of a single amino acid presents an obstacle for the soliton propagation. The greater width of Davydov solitons stabilizes the soliton upon reflection from such a massive barrier and increases the probability of tunneling through it, whereas the greater isotropy of the exciton-lattice coupling suppresses the tunneling efficiency by increasing the interaction time between the Davydov soliton and the barrier, thereby prolonging the tunneling time through the barrier and enhancing the probability for reflection from the latter. The results as presented, demonstrate the feasibility of Davydov solitons reflecting from or tunneling through massive barriers, and suggest a general physical mechanism underlying the action of protein clamps through local enhancement of an effective protein {\alpha}-helix mass.

On the quantum dynamics of Davydov solitons in protein [formula omitted]-helices

The transport of energy inside protein α-helices is studied by deriving a system of quantum equations of motion from the Davydov Hamiltonian with the use of the Schrödinger equation and the generalized Ehrenfest theorem. Numerically solving the system of quantum equations of motion for different initial distributions of the amide I energy over the peptide groups confirmed the generation of both moving or stationary Davydov solitons. In this simulation the soliton generation, propagation, and stability were found to be dependent on the symmetry of the exciton–phonon interaction Hamiltonian and the initial site of application of the exciton energy.

Коллективные возбуждения в альфа-спиральной молекуле белка

Continuous energy supply, a necessary condition for life, excites a state far from thermodynamic equilibrium, in particular coherent electric polar vibrations depending on water ordering in the cell. The collective, low-frequency vibrational excitations in protein macromolecules in the terahertz frequency region are suggested to orchestrate many protein processes such as enzymatic activity, electron/energy transport, conformation transitions and others. The most significant type of excitations is long-lived coherent vibronic modes and waves which are either spreading over large protein subdomains or being localised states. Two possible mechanisms of the formation of collective dynamic modes in the form of Frohlich collective mode and Davydov soliton were previously suggested. We developed a unified quantum-mechanics approach to describe conditions of the formation of Frohlich vibronic state and Davidov soliton in alphahelical protein molecules interacting with the environment. We distinguish three subsystems in the model, i.e., (i) oscillating peptide groups (PGs), interacting with (ii) the subsystem of side residuals of proteins, which in turn interacts with the environment (surrounding water), which is responsible for dissipation and fluctuation processes and modelled by a system of harmonic oscillators. It was shown that the equation of motion for dynamic variables of PG chain (phonon variables) can be transformed to the nonlinear Schrodinger equation for order parameters, which determines energy “pumping” due to protein interaction with a reservoir. Solution of the order parameter equation was shown to admit bifurcation into the solution corresponding to the formation of weak damped collective vibronic mode with growing amplitude. It was also shown that the solution corresponding to Davydov soliton can exist at certain boundary conditions in this bifurcation region. The suggested mechanism of emergent of macroscopic dissipative structures in the form of collective vibronic modes in proteins is discussed in connection with recent experimental data on long-lived collective protein excitation in the terahertz frequency region

Multi-hump bright solitons in a Schrödinger–mKdV system

We consider the problem of energy transport in a Davydov model along an anharmonic crystal medium obeying quartic longitudinal interactions corresponding to rigid interacting particles. The Zabusky and Kruskal unidirectional continuum limit of the original discrete equations reduces, in the long wave approximation, to a coupled system between the linear Schrödinger (LS) equation and the modified Korteweg–de Vries (mKdV) equation. Single- and two-hump bright soliton solutions for this LS–mKdV system are predicted to exist by variational means and numerically confirmed. The one-hump bright solitons are found to be the anharmonic supersonic analogue of the Davydov's solitons while the two-hump (in both components) bright solitons are found to be a novel type of soliton consisting of a two-soliton solution of mKdV trapped by the wave function associated to the LS equation. This two-hump soliton solution, as a two component solution, represents a new class of polaron solution to be contrasted with the two-soliton interaction phenomena from soliton theory, as revealed by a variational approach and direct numerical results for the two-soliton solution.

Optical solitons, conservation laws and modulation instability analysis for the modified nonlinear Schrödinger’s equation for Davydov solitons

In this paper, the optical solitons to the modified nonlinear Schrödinger’s equation for davydov solitons are investigate. The modified F-expansion method is the integration technique employed to achieve this task. This yielded a combined and other soliton solutions. The Lie point symmetry generators of a system of partial differential equations acquired by decomposing the equation into real and imaginary components are derived. We prove that the system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to construct a set of local conservation laws (Cls) for the system using the general Cls theorem presented by Ibragimov. Furthermore, the modulation instability (MI) is analyzed based on the standard linear-stability analysis and the MI gain spectrum is got. Numerical simulation of the obtained results are analyzed with interesting figures showing the physical meaning of the solutions.

Soliton solutions for Davydov solitons in α-helix proteins

The propagation equation for describing Davydov solitons in α-helix proteins has been investigated analytically. There are seven integration tools to extract analytical soliton solutions. They are the Ricatti equation expansion approach, ansatz scheme, improved extended tanh-equation method, the extend exp(−Ψ(τ))-expansion method, the extended Jacobi elliptic function expansion method, the extended trial equation method and the extended G'/G− expansion method.

The VES Hypothesis and Protein Conformational Changes

Abstract This work started as a (biased) review of the state of the art of the Davydov/Scott model for energy transfer in proteins via the propagation of amide I excitations and of how this initial quantum stage may lead to protein conformational changes. It is not a traditional review because certain results reported in the literature have been complemented by extra simulations which revealed, for instance, a new class of possible states for the amide I excitation, here designated as double discrete breathers. The issue of the thermal stability of the Davydov soliton is discussed, as well as the deeper question of how to simulate the non-equilibrium regime of a mixed quantum-classical system at finite temperature. While an exact answer to the latter question is not yet available, a specific formalism that is able to reproduce the correct canonical ensemble is described. Finally, the question of how a quantum amide I excitation can generate of the specific protein conformational changes known to be associated with function is also explored. Indeed, computer simulations indicate that a local action can lead to reproducible conformational changes. The paper ends with a discussion of some of the open questions that plague/stimulate this field and with a suggestion for an experiment to test the basic assumption of the Davydov/Scott model.

Oscillation of Davydov Solitons in Three Wells

In this work, we propose a model of oscillation of Davydov solitons in three wells. It can be used as a mathematical and physical frame in simulation of circle of some nonlinear oscillation of excitations via acupuncture system. The calculation shows that this sort of oscillation is possible if the initial rate of average occupational number of the quasi-particles in the wells is not equal to zero. One of oscillations arising relies on the initial rate of average occupational number of quasi-particles to be equal with each other within three wells. Then, the oscillation is not a kind of Josephson oscillation and has complicated frequency distributions. However, the total behavior of oscillation played is similar to three big solitons concentrated in three wells. In this sense, this model generally reveals a sort of oscillation mechanism of the acupuncture system how to work in the body, which allows us to understand the oscillation that may be one of fundamental natures in the acupuncture system.

Rogue-Wave Interaction of a Nonlinear Schrödinger Model for the Alpha Helical Protein

Abstract In this article, we investigate a fourth-order nonlinear Schrödinger equation, which governs the Davydov solitons in the alpha helical protein with higher-order effects. By virtue of the generalised Darboux transformation, higher-order rogue-wave solutions are derived. Propagation and interaction of the rogue waves are analysed: (i) Coefficients affect the existence time of the first-order rogue waves; (ii) coefficients affect the interaction time of the second- and third-order rogue waves; (iii) direction of the rogue-wave propagation remain unchanged after interaction.

Influence of electromagnetic field on soliton-mediated charge transport in biological systems

It is shown that electromagnetic fields affect dynamics of Davydov's solitons which provide charge transport processes in macromolecules during metabolism of the system. There is a resonant frequency of the field at which it can cause the transition of electrons from bound soliton states into delocalised states. Such decay of solitons reduces the effectiveness of charge transport, and, therefore, inhibits redox processes. Solitons radiate their own electromagnetic field of characteristic frequency determined by their average velocity. This self-radiated field leads to synchronization of soliton dynamics and charge transport processes, and is the source of the coherence in the system. Exposition of the system to the oscillating electromagnetic field of the frequency, which coincides with the eigen-frequency of solitons can enhance eigen-radiation of solitons, and, therefore, will enhance synchronization of charge transpor, stimulate the redox processes and increase coherence in the system. Electromagnetic oscillating field causes also ratchet phenomenon of solitons, i.e., drift of solitons in macromolecules in the presence of unbiased periodic field. Such additional drift enhances the charge transport processes. It is shown that temperature facilitates the ratchet drift. In particular, temperature fluctuations lead to the lowering of the critical value of the intensity and period of the field, above which the drift of solitons takes place. Moreover, there is a stochastic resonance in the soliton dynamics in external electromagnetic fields. This means, that there is some optimal temperature at which the drift of solitons is maximal.

A New Information Soliton to Resist Decaying of the Excitation Based on the External Field Interaction

In this work, we reveal possibility using information soliton to resist senescence of certain important bio-excitations (such as Davydov solitons) which play a fundamental role in information processing of life. For this goal, a type of external field interaction with original system is introduced. This field enables the total system to be described by a nonlinear Master equation. Then we found that the nonlinear term in the equation drives the initial excitation to evolve as a kind of information soliton asymptotically. It is this information soliton to resist decaying of the excitations. This provides a constructive way to prolong age of biological excitations by exerting an external field, which forms a basis used in medical devices or treatments.

Quantum interactive dualism: From Beck and Eccles tunneling model of exocytosis to molecular biology of SNARE zipping

We commence this review by outlining the challenges faced by physical theories of consciousness and briefly describe the two main approaches based on classical or quantum mechanics. Next, we provide a detailed exposition of the motivation, the theoretical construction and experimental falsification of the celebrated model due to Beck and Eccles concerning mind-brain interaction purported to operate at the sites of neurotransmitter release in the brain. Finally, we propose our own model of a vibrationally assisted quantum tunneling mechanism involving a Davydov soliton propagating along the hydrogen bonds in the protein four-α-helix bundle of the SNARE complex (soluble NSF attachment protein receptor; NSF, N-ethylmaleimide sensitive fusion proteins) that drives synaptic vesicle fusion. We also discuss the possible experimental tests that could falsify our model. Since erasure of consciousness by volatile anesthetics results from binding to the hydrophobic core of the SNARE four-α-helix bundle, our model is well suited to support quantum interactive dualism. Biomedical Reviews 2014; 25: 15-24.

THE CHECKOUT AND VERIFICATION OF THEORY OF BIO-ENERGY TRANSPORT IN THE PROTEIN MOLECULES

The phosphorylation and de-phosphorylation reactions in the cell, through which the bio-energy is released from ATP hydrolysis in biological systems, are described in this paper. Firstly, the bio-energy is accepted by the vibrational amides in protein molecules in virtue of resonant mechanism of frequency or energy, and can transport along the protein molecules in soliton, which is formed by self-trapping of vibrational quantum (or exciton), by means of dipole–dipole interaction among the neighboring peptide groups (or amino acids). Theory and properties of bio-energy transport were proposed and described by many researchers. We here reviewed mainly the theories and features of Davydov's and Pang's models. However these theoretical models including Davydov's and Pang's model were all established based on a periodic and uniform proteins, which are different from practically biological proteins molecules. Therefore, it is necessary to inspect and verify the validity of the theory of bio-energy transport in real biological protein molecules. These problems were extensively studied by a lot of researchers using different methods in past thirty years, a considerable number of research results were obtained. I review here the situation and progresses of study on this problem, in which we reviewed the correctness of the theory of bio-energy transport including Davydov's and Pang's model and its investigated progresses under influences of structural non-uniformity and disorder, side groups and imported impurities of protein chains as well as the thermal perturbation and damping of medium arising from the biological temperature of the systems. The structural non-uniformity arises from the disorder distribution of sequence of masses of amino acid residues, side groups and imported impurities, which results in the changes and fluctuations of the spring constant, dipole–dipole interaction, exciton–phonon coupling constant, diagonal disorder or ground state energy and chain–chain interaction of the molecular channels in the dynamic equations in different models. The influences of structural non-uniformity, side groups and imported impurities as well as the thermal perturbation and damping of medium on the bio-energy transport in the proteins with single chain and three chains were studied by different numerical simulation techniques and methods including the average Hamiltonian way of thermal perturbation, fourth-order Runge–Kutta method, Monte Carlo method, quantum perturbed way and thermodynamic and statistical method, and so on. In this review, the numerical simulation results of bio-energy transport in uniform protein molecules, the influence of structural non-uniformity on the bio-energy transport, the effects of system temperature on the bio-energy transport and the simultaneous effects of structural non-uniformity, damping and thermal perturbation of proteins on the bio-energy transport in single chains and α-helical molecules were included and studied, respectively. The results obtained from these studies and reviews suggest that Davydov's soliton is unstable, but Pang's soliton is stable at physiologic temperature 300 K under influences of structural non-uniformity and disorder, side groups, imported impurities and damping of medium. Thus we can conclude that the soliton in Pang's model is exactly a carrier of the bio-energy transport and Pang's theory is appropriate to α-helical protein molecules. Finally we provided a few of experimental evidences for real existence of the soliton and validity of the theory of bio-energy transport in proteins and stated further the applications of the theory in living systems.

Polaron dynamics in two-dimensional photon-echo spectroscopy of molecular rings

We have developed a new approach to the computation of third-order spectroscopic signals of molecular rings, by incorporating the Davydov soliton theory into the nonlinear response function formalism. The Davydov D1 and D Ansätze have been employed to treat the interactions between the excitons and the primary phonons, allowing for a full description of arbitrary exciton-phonon coupling strengths. As an illustration, we have simulated a series of optical 2D spectra for two models of molecular rings.