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| [[Image:alpha helix neg60 neg45 sideview.png|right|thumb|300px|Side view of an [[α-helix]] of [[alanine]] residues in [[atom]]ic detail. Protein [[α-helix|α-helices]] provide substrate for Davydov soliton creation and propagation.]]
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| '''Davydov soliton''' is a quantum [[quasiparticle]] representing an excitation propagating along the [[protein]] [[α-helix]] self-trapped [[amide|amide I]]. It is a solution of the Davydov [[Hamiltonian (quantum mechanics)|Hamiltonian]]. It is named for the Soviet and Ukrainian physicist [[Alexander Davydov]]. The Davydov model describes the interaction of the amide I [[vibration]]s with the [[hydrogen bond]]s that stabilize the [[α-helix]] of [[protein]]s. The elementary excitations within the α-helix are given by the [[phonon]]s which correspond to the deformational oscillations of the lattice, and the [[exciton]]s which describe the internal [[amide|amide I]] excitations of the [[peptide group]]s. Referring to the atomic structure of an α-helix region of protein the mechanism that creates the Davydov soliton ([[polaron]], [[exciton]]) can be described as follows: [[vibration]]al [[energy]] of the [[carbonyl|C=O]] [[stretching]] (or [[amide|amide I]]) [[oscillator]]s 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''.<ref name="Davydov1973">{{cite journal
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| | author = Davydov AS
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| | title = The theory of contraction of proteins under their excitation
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| | journal = Journal of Theoretical Biology
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| | volume = 38
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| | pages = 559–569
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| | year = 1973
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| | doi = 10.1016/0022-5193(73)90256-7
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| | pmid = 4266326
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| | issue = 3
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| }}</ref><ref name="Davydov1974">
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| {{cite journal
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| | author = Davydov AS
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| | title = Quantum theory of muscular contraction
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| | journal = Biophysics
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| | volume = 19
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| | pages = 684–691
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| | year = 1974
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| }}</ref><ref name="Davydov1977">
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| {{cite journal
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| | author = Davydov AS
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| | title = Solitons and energy transfer along protein molecules
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| | journal = Journal of Theoretical Biology
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| | volume = 66
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| | pages = 379–387
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| | year = 1977
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| | doi = 10.1016/0022-5193(77)90178-3
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| | pmid = 886872
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| | issue = 2
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| }}</ref> [[Soliton]]s in which the [[energy]] is distributed in a fashion preserving the [[helix|helical]] [[symmetry]] are dynamically unstable, and such [[symmetric]]al solitons once formed decay rapidly when they propagate. On the other hand, an [[symmetry|asymmetric]] soliton which [[Spontaneous symmetry breaking|spontaneously breaks the local translational and helical symmetries]] possesses the lowest energy and is a robust localized entity.<ref name="Brizhik2004">
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| {{cite journal
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| | author = Brizhik L, Eremko A, Piette B, Zakrzewski W
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| | title = Solitons in α-helical proteins
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| | journal = Physical Review E
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| | volume = 70
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| | pages = 031914, 1–16
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| | year = 2004
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| |bibcode = 2004PhRvE..70a1914K |doi = 10.1103/PhysRevE.70.011914
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| | arxiv=cond-mat/0402644}}</ref>
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| Davydov's [[Hamiltonian (quantum mechanics)|Hamiltonian]] is formally similar to the [[Polaron#Polaron_theory|Fröhlich-Holstein Hamiltonian]] for the interaction of electrons with a polarizable lattice. Thus the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the [[energy operator]] <math>\hat{H}</math> is
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| :<math>
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| \hat{H}=\hat{H}_{\textrm{qp}}+\hat{H}_{\textrm{ph}}+\hat{H}_{\textrm{int}}</math>
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| where <math>\hat{H}_{\textrm{qp}}</math> is the [[quasiparticle]] ([[exciton]]) [[Hamiltonian (quantum mechanics)|Hamiltonian]], which describes the motion of the amide I excitations between adjacent sites; <math>\hat{H}_{\textrm{ph}}</math> is the [[phonon]] [[Hamiltonian (quantum mechanics)|Hamiltonian]], which describes
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| the [[vibration]]s of the [[lattice model (physics)|lattice]]; and <math>\hat{H}_{\textrm{int}}</math> is the [[interaction]] [[Hamiltonian (quantum mechanics)|Hamiltonian]], which describes the interaction of the amide I excitation with the lattice.<ref name="Davydov1973"/><ref name="Davydov1974"/><ref name="Davydov1977"/>
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| The [[quasiparticle]] ([[exciton]]) [[Hamiltonian (quantum mechanics)|Hamiltonian]] <math>\hat{H}_{\textrm{qp}}</math> is:
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| :<math>\hat{H}_{\textrm{qp}}=</math> <math>\sum_{n,\alpha}E_{0}\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n,\alpha}</math> <math>-J\sum_{n,\alpha}\left(\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n+1,\alpha}+\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n-1,\alpha}\right)</math> <math>+L\sum_{n,\alpha}\left(\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n,\alpha+1}+\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n,\alpha-1}\right)</math>
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| where the index <math>n=1,2,\cdots,N</math> counts the peptide groups along the α-helix spine, the index <math>\alpha=1,2,3</math> counts each α-helix spine, <math>E_{0}=3.28\times10^{-20}</math> J is the energy of the amide I
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| vibration (CO stretching), <math>J=2.46\times10^{-22}</math> J is the [[dipole]]-[[dipole]] coupling energy between a particular amide I bond and those ahead and behind along the same spine, <math>L=1.55\times10^{-22}</math> J is the
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| dipole-dipole coupling energy between a particular amide I bond and those on adjacent spines in the same unit cell of the [[protein]] [[α-helix]], <math>\hat{A}_{n,\alpha}^{\dagger}</math> and <math>\hat{A}_{n,\alpha}</math> are respectively
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| the [[boson]] [[creation and annihilation operator]] for a quasiparticle at the [[peptide group]]. <math>n,\alpha</math><ref name="Scott1992">
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| {{cite journal
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| | author = Scott AS
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| | title = Davydov's soliton
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| | journal = Physics Reports
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| | volume = 217
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| | pages = 1–67
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| | year = 1992
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| | doi = 10.1016/0370-1573(92)90093-F
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| |bibcode = 1992PhR...217....1S }}</ref><ref name="Takeno1997">
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| {{cite journal
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| | author = Cruzeiro-Hansson L, Takeno S.
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| | year = 1997
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| | title = Davydov model: the quantum, mixed quantum-classical, and full classical systems
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| | journal=[[Physical Review E]]
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| | volume=56 | issue=1 | pages = 894–906
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| | bibcode = 1997PhRvE..56..894C
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| | doi= 10.1103/PhysRevE.56.894
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| }}</ref>
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| The [[phonon]] [[Hamiltonian (quantum mechanics)|Hamiltonian]] <math>\hat{H}_{\textrm{ph}}</math> is
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| :<math>
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| \hat{H}_{\textrm{ph}}=\frac{1}{2}\sum_{n,\alpha}\left[w(\hat{u}_{n+1,\alpha}-\hat{u}_{n,\alpha})^{2}+\frac{\hat{p}_{n,\alpha}^{2}}{M}\right]</math>
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| where <math>\hat{u}_{n,\alpha}</math> is the [[displacement operator]] from the equilibrium position of the [[peptide group]] <math>n,\alpha</math>, <math>\hat{p}_{n,\alpha}</math> is the [[momentum operator]] of the [[peptide group]] <math>n,\alpha</math>, M is the [[mass]] of each [[peptide group]], and <math>w=19.5</math> N m<math>^{-1}</math> is an [[Elasticity (physics)|effective elasticity coefficient]] of the lattice (the [[spring constant]] of a [[hydrogen bond]]).
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| Finally, the [[interaction]] [[Hamiltonian (quantum mechanics)|Hamiltonian]] <math>\hat{H}_{\textrm{int}}</math> is
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| :<math>
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| \hat{H}_{\textrm{int}}=\chi\sum_{n,\alpha}\left[(\hat{u}_{n+1,\alpha}-\hat{u}_{n,\alpha})\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n,\alpha}\right]</math>
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| where <math>\chi=-30</math> pN is an anharmonic parameter arising from the coupling between the [[quasiparticle]] (exciton) and the lattice displacements (phonon) and parameterizes the strength of the [[exciton]]-[[phonon]] [[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.<ref name="Cruzeiro2005">
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| {{cite journal
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| | author = Cruzeiro-Hansson L
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| | title = Influence of the nonlinearity and dipole strength on the amide I band of protein α-helices
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| | journal = The Journal of Chemical Physics
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| | volume = 123
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| | pages = 234909, 1–7
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| | year = 2005
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| | doi = 10.1063/1.2138705
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| | pmid = 16392951
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| | issue = 23
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| |bibcode = 2005JChPh.123w4909C
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| | url=http://link.aip.org/link/?JCPSA6/123/234909/1}}</ref>
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| 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]] that is (i) ''large'' so the continuum limit approximation is justified, (ii) ''acoustic'' because the self-localization arises from interactions
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| with acoustic modes of the lattice, and (iii) ''weakly coupled'' because the anharmonic energy is small compared with the phonon bandwidth.<ref name="Scott1992"/>
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| 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.<ref name="Scott1992"/> Supposing that the Davydov soliton is localized to 5 turns of the [[α-helix]] results in significant uncertainty in the [[velocity]] of the [[soliton]] <math>\Delta v=133</math> m/s, a fact that is obscured if one models the Davydov soliton as a classical object.
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| There are three possible fundamental approaches towards Davydov model:<ref name="Takeno1997"/><ref name="Cruzeiro1997">
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| {{cite journal
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| | author = Cruzeiro-Hansson L
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| | title = Short timescale energy transfer in proteins
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| | journal = Solphys '97 Proceedings
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| | volume =
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| | pages =
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| | year = 1997
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| }}</ref> (i) the [[Quantum mechanics|quantum theory]], in which both the amide I vibration ([[exciton]]s) and the lattice site motion ([[phonon]]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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| ==References==
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| {{reflist|2}}
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| {{particles}}
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| {{DEFAULTSORT:Davydov Soliton}}
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| [[Category:Quantum mechanics]]
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| [[Category:Biophysics]]
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| [[Category:Biological matter]]
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I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Distributing production is how he makes a residing. The preferred pastime for him and his kids is style and he'll be starting some thing else alongside with it. Her family life in Ohio.
Here is my web-site ... psychic readings online
