Davydov soliton - AbsoluteAstronomy.com

Davydov soliton is a quantum quasiparticle

Quasiparticle

In physics, quasiparticles are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in free space...

representing an excitation propagating along the protein

Protein

Proteins are biochemical compounds consisting of one or more polypeptides typically folded into a globular or fibrous form, facilitating a biological function. A polypeptide is a single linear polymer chain of amino acids bonded together by peptide bonds between the carboxyl and amino groups of...

α-helix self-trapped amide I

Amide

In chemistry, an amide is an organic compound that contains the functional group consisting of a carbonyl group linked to a nitrogen atom . The term refers both to a class of compounds and a functional group within those compounds. The term amide also refers to deprotonated form of ammonia or an...

. It is a solution of the Davydov Hamiltonian

Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

. It is named for the Soviet and Ukrainian physicist Alexander Davydov

Alexander Davydov

Alexander Sergeevich Davydov was a Soviet and Ukrainian physicist. Davydov graduated from Moscow State University in 1939. In 1963-1990 he was Director of Institute for Theoretical Physics of the Ukrainian Academy of Sciences ....

. The Davydov model describes the interaction of the amide I vibration

Vibration

Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...

s with the hydrogen bond

Hydrogen bond

A hydrogen bond is the attractive interaction of a hydrogen atom with an electronegative atom, such as nitrogen, oxygen or fluorine, that comes from another molecule or chemical group. The hydrogen must be covalently bonded to another electronegative atom to create the bond...

s that stabilize the α-helix of protein

Protein

s. The elementary excitations within the α-helix are given by the phonon

Phonon

In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...

s which correspond to the deformational oscillations of the lattice, and the exciton

Exciton

An exciton is a bound state of an electron and hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and some liquids...

s which describe the internal amide I

Amide

excitations of the peptide groups. Referring to the atomic structure of an α-helix region of protein the mechanism that creates the Davydov soliton (polaron

Polaron

A polaron is a quasiparticle composed of a charge and its accompanying polarization field. A slow moving electron in a dielectric crystal, interacting with lattice ions through long-range forces will permanently be surrounded by a region of lattice polarization and deformation caused by the moving...

, exciton

Exciton

) can be described as follows: vibration

Vibration

al energy

Energy

In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

of the C=O

Carbonyl

In organic chemistry, a carbonyl group is a functional group composed of a carbon atom double-bonded to an oxygen atom: C=O. It is common to several classes of organic compounds, as part of many larger functional groups....

stretching

Stretching

Stretching is a form of physical exercise in which a specific skeletal muscle is deliberately elongated, often by abduction from the torso, in order to improve the muscle's felt elasticity and reaffirm comfortable muscle tone. The result is a feeling of increased muscle control, flexibility and...

(or amide I

Amide

) oscillators that is localized on the α-helix acts through a phonon coupling effect to distort the structure of the α-helix, while the helical distortion reacts again through phonon coupling to trap the amide I oscillation energy and prevent its dispersion. This effect is called self-localization or self-trapping. Soliton

Soliton

In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...

s in which the energy

Energy

In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

is distributed in a fashion preserving the helical

Helix

A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...

symmetry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

are dynamically unstable, and such symmetrical solitons once formed decay rapidly when they propagate. On the other hand, an asymmetric

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

soliton which spontaneously breaks the local translational and helical symmetries

Spontaneous symmetry breaking

Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....

possesses the lowest energy and is a robust localized entity.

Davydov's Hamiltonian

Hamiltonian (quantum mechanics)

is formally similar to the Fröhlich-Holstein Hamiltonian for the interaction of electrons with a polarizable lattice. Thus the Hamiltonian

Hamiltonian (quantum mechanics)

of the energy operator

where

is the quasiparticle

Quasiparticle

(exciton

Exciton

) Hamiltonian

Hamiltonian (quantum mechanics)

, which describes the motion of the amide I excitations between adjacent sites;

is the phonon

Phonon

In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...

Hamiltonian

Hamiltonian (quantum mechanics)

, which describes
the vibration

Vibration

s of the lattice

Lattice model (physics)

In physics, a lattice model is a physical model that is defined on a lattice, as opposed to the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are...

; and

is the interaction

Interaction

Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect...

Hamiltonian

Hamiltonian (quantum mechanics)

, which describes the interaction of the amide I excitation with the lattice.

The quasiparticle

Quasiparticle

(exciton

Exciton

) Hamiltonian

Hamiltonian (quantum mechanics)

is:

where the index

counts the peptide groups along the α-helix spine, the index

counts each α-helix spine,

J is the energy of the amide I
vibration (CO stretching),

J is the dipole

Dipole

In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

-dipole

Dipole

coupling energy between a particular amide I bond and those ahead and behind along the same spine,

J is the
dipole-dipole coupling energy between a particular amide I bond and those on adjacent spines in the same unit cell of the protein

Protein

α-helix,

and

are respectively
the boson

Boson

In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

creation and annihilation operator for a quasiparticle at the peptide group.

The phonon

Phonon

In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...

Hamiltonian

Hamiltonian (quantum mechanics)

where

is the displacement operator from the equilibrium position of the peptide group

is the momentum operator

Momentum operator

In quantum mechanics, momentum is defined as an operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once...

of the peptide group

, M is the mass

Mass

Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

of each peptide group, and

N m

is an effective elasticity coefficient

Elasticity (physics)

In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....

of the lattice (the spring constant of a hydrogen bond

Hydrogen bond

).

Finally, the interaction

Interaction

Hamiltonian

Hamiltonian (quantum mechanics)

where

pN is an anharmonic parameter arising from the coupling between the quasiparticle

Quasiparticle

(exciton) and the lattice displacements (phonon) and parameterizes the strength of the exciton

Exciton

-phonon

Phonon

In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...

interaction

Interaction

. The value of this parameter for α-helix has been determined via comparison of the theoretically calculated absorption line shapes with the experimentally measured ones and should be considered as the best estimate up to date.

The mathematical techniques that are used to analyze Davydov's soliton are similar to some that have been developed in polaron theory. In this context the Davydov's soliton corresponds to a polaron

Polaron

that is (i) large so the continuum limit approximation is justified, (ii) acoustic because the self-localization arises from interactions
with acoustic modes of the lattice, and (iii) weakly coupled because the anharmonic energy is small compared with the phonon bandwidth.

The Davydov soliton is a quantum quasiparticle and it obeys Heisenberg's uncertainty principle. Thus any model that does not impose translational invariance is flawed by construction. Supposing that the Davydov soliton is localized to 5 turns of the α-helix results in significant uncertainty in the velocity

Velocity

In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

of the soliton

Soliton

m/s, a fact that is obscured if one models the Davydov soliton as a classical object.

There are three possible fundamental approaches towards Davydov model: (i) the quantum theory

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, in which both the amide I vibration (exciton

Exciton

s) and the lattice site motion (phonon

Phonon

In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...

s) are treated quantum mechanically; (ii) the mixed quantum-classical theory, in which the amide I vibration is treated quantum mechanically but the lattice is classical; and (iii) the classical theory, in which both the amide I and the lattice motions are treated classically.

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