# Elliptic Davydov solitons in $\alpha$-helix protein chain with exciton-
exciton and exciton-phonon couplings

Nkeh Oma Nfor  Corresponding Author, Email<EMAIL_ADDRESS>Department of
Physics, Higher Teacher Training College Bambili, The University of Bamenda,
P. O. Box 39, Bambili-Cameroon.  Michael Nana Jipdi  Department of Physics,
Higher Teacher Training College Bambili, The University of Bamenda, P. O. Box
39, Bambili-Cameroon.

###### Abstract

We consider the Davydov model of $\alpha$-helix protein chain with both
exciton-exciton and exciton-phonon couplings and investigate on the evolution
of elliptic solitons. In the discrete regime of the adiabatic limit, we
analytically and numerically show that modulational instability induces the
self-localization of energy in the $\alpha$-helix protein chain. By
incorporating the continuous limit approximations, various nonlinear periodic
modes are traced; strongly suggesting that the energy of the ATP hydrolysis is
locally distributed over the $\alpha$-helix protein chain. It is generally
found that the exciton-exciton coupling induces the inhomogeneity in the
protein chain, which greatly enhances energy localization that is physically
manifested as nonlinear periodic modes. Results of numerical simulations
clearly depicts the evolution of these nonlinear periodic modes in the highly
discrete, nonlinear, and coupled system that governs the dynamics of
$\alpha$-helix protein chain.

Davydov; elliptic solitons; modulational instability; $\alpha$-helix protein
chain.

##  I Introduction

The solitary wave concept as a localized and dynamic entity in nonlinear media
plays a crucial role in modern science Peyrard2006 . This concept has greatly
permeated the biophysical field because the first identification of biomedical
soliton was made in $\alpha$-helix protein chains Davydov1973 . The Davydov
soliton results from a nonlinear coupling between the vibrational excitation
and a deformation in the protein chain, triggered by the amide-I vibrational
energy Scott1992 ; Davydov1985 ; Christiansen1990 . This vibrational energy
emanates from the hydrolysis of adenosine triphosphate (ATP). In fact, an ATP
molecule binds to a specific site on the protein, reacts with water, and under
normal physiological conditions releases the appropriate amount of energy for
the metabolic activities of the cell. Concretely, Davydov mainly assumed that
the amide-I molecules absorbed the energy released by the ATP and channelled
it along the protein chain via quantum robust self-trapping excitations known
as solitons. In this study, we mainly explore the quantum mechanical approach
in dealing with the amide-I vibration, while classical mechanics is used in
the lattice dynamics.

Despite the remarkable successes recorded by the Davydov model, there still
exists some challenges like the exciton-phonon bound states constraint to
travel only at subsonic velocity Scoot1982 , and the thermal instability of
Davydov soliton at physiological temperatures Cottingham1989 . The issue
related to the velocity has been appropriately addressed by the correction of
the linear hydrogen bond potential which link peptides. For instance, the
energy of hydrogen bond that links peptide groups was modelled by the cubic
nonlinear potential Davydov1985 ; Davydov1983 ; Davydov1984 , with such
modification giving an opportunity for the introduction of supersonic soliton
velocity in the continuum limit approximation. To address the latter case,
numerical analysis by Forner Forner1991  predicted the stability of Davydov
soliton at physiological temperatures provided the nonlinear coupling constant
is increased. Consequently this study seeks to explore additional mechanism
for energy localization in the $\alpha$-helix protein chain, probably
triggered by increase in nonlinearity. It should be noted that in the original
model proposed by Davydov, anharmonicity is introduced in the protein chain
only by the nonlinear coupling between excitons and the acoustic phonon field
which describes the harmonic oscillations of the molecules along the chain.
However, the effects of two exciton states in both the discrete and continuum
levels in $\alpha$-helix protein chain greatly suggests that such dynamics
induces additional nonlinearity in the chain Latha2012 ; Merlin2013 . The
relatively large electric dipoles of the amide-I excitons further complement
the idea of incorporating the exciton-exciton interaction in the protein chain
Xiao1997 ; Nji2018 . It is important to note that in Ref. Xiao1997 , the
exciton-exciton interaction is the driving force behind energy localization
via the process of modulational instability in the $\alpha$-helix protein
chain. The self-trapped amide-I vibrational energy may propagate along the
chain as a result of the hydrolysis of ATP molecules.

A good number of research activities has been focused on identifying spatially
localized soliton-type excitons in the $\alpha$-helix protein chain
Davydov1973 ; Scott1992 ; Davydov1985 ; Christiansen1990 , with little focus
on spatial periodic excitons. Such spatial periodic wave trains may be
considered as a superposition of individual localized exciton modes. From
experimental and theoretical standpoints, nonlinear periodic wavetrains are
ubiquitous in nonlinear optics Rodrigues2018 ; Defi2015 ; Amrani2011 ;
Haboucha2008 , neural networks Villacorta2013 ; Nfor2018 , liquid films
Noumana2018 , among a plethora of nonlinear systems. Furthermore for long-
range intermolecular interactions, Mvogo $et.\,al$ Mvogo2013 ; Mvogo2014
showed the existence of elliptic-type solutions in $\alpha$-helix protein
chain when taking into account inter spine coupling, while Nji $et.\,al$
Nji2018  equally investigated on the propagation of periodic soliton
wavetrains which is characterised by the $dn$ function, when dealing with a
longitudinal interacting amide-I exciton chains of $\alpha$-helix proteins.
These previous studies on the propagation of elliptic solitons in various
nonlinear media among many others, serves like a big motivation behind the
current investigation. In fact, this study seeks to elucidate on the dynamics
of elliptic solitons in $\alpha$-helix protein chain, experiencing both
exciton-exciton and exciton-phonon nonlinear coupling forces.

In sec. II, we consider a model Hamiltonian of $\alpha$-helix protein chain
which is inherently described by spins. The Hamiltonian is eventually
transformed to the classical Davydov Hamiltonian which exclusively deals with
boson operators, with the aid of the Dayson-Maleev transformation. Such
transformation gives the perfect opportunity to incorporate the nonlinear
exciton-exciton coupling. By exploring the quantum and classical mechanical
approaches, we basically derived the original equations obtained by Davydov,
with the addition of the nonlinear force term introduced by the exciton-
exciton interactions. We carry out the adiabatic approximation in sec. III to
obtain the discrete nonlinear Schr$\ddot{o}$dinger amplitude equation. It is
shown that the exciton-exciton interactions enhance energy localization in the
chain via the process of modulational instability. The various regions of the
observation of localized modes in the $(q,\,Q)$ plane are equally identified.
We numerically investigate the evolution of plane waves in the system to
complement the predictions from the modulational instability analysis.
Analytic solution of the discrete amplitude equation are equally obtained via
the Jacobian elliptic method. We further explore the continuum limit
approximation in sec. IV to obtain the inhomogeneous nonlinear
Schr$\ddot{o}$dinger equation; with the inhomogeneity induced by the exciton-
exciton coupling. In the absence of the exciton-exciton coupling, we observe
periodic trains of excitons that signifies how energy of the ATP hydrolysis is
uniformly distributed over the protein chain. However when the soliton
velocity exceeds that of the phonon, the system supports the propagation of
dark solitons which is characterized by the $sn$ function. Solutions of the
inhomogeneous nonlinear Schr$\ddot{o}$dinger equation generally depicts
periodic exciton modes, which is characterized by the elliptic $cn$ function.
We numerically solve the discrete and coupled equations, in order to
investigate on the evolution of the elliptic solitons in the protein chain. We
finally conclude in sec. V by highlighting the important results obtained, and
articulating on some brighter perspectives.

##  II model equations of the coupled system

As a model, we consider a hydrogen-bonded chain of $\alpha$-helix protein made
up of N-peptide groups which is characterised by two levels. The amide-I
exciton is completely absent in the ground state and created in the excited
state; which eventually spreads to neighbouring molecules with the aid of the
hopping parameter $J$. The Hamiltonian of each two-level molecule described by
a spin one-half operator $s_{n}$ is given by Dicke1954 ; Xiao1998

$\displaystyle H=\hbar\omega_{0}\sum_{n}s_{n}^{z}$
$\displaystyle-\frac{1}{2}J\sum_{n,\delta}(s_{n}^{+}s_{n+\delta}^{-}+s_{n}^{-}s_{n+\delta}^{+})$
$\displaystyle-D\sum_{n,\delta}(s_{n}^{z}+$
$\displaystyle\frac{1}{2})(s_{n+\delta}^{z}+\frac{1}{2})$
$\displaystyle+\chi\sum_{n}(U_{n+1}-$ $\displaystyle
U_{n})(s_{n}^{z}+\frac{1}{2})+H_{ph},$ (1)

where

$H_{ph}=\sum_{n}\Big{(}\frac{P_{n}^{2}}{2M}+\frac{1}{2}K(U_{n+1}-U_{n})^{2}\Big{)}.$
(2)

In Eq. (II), the operators $s_{n}^{+}$ and $s_{n}^{-}$ indicate excitation and
de-excitation, respectively, between two levels with an excitation energy
$\hbar\omega_{0}$ at the $n$th site, $\delta$ runs over the nearest neighbors
of $n$, and

$[s_{i}^{z},s_{j}^{\pm}]=\pm
s_{i}^{\pm}\delta_{ij},\,\,\,\,[s_{i}^{+},s_{j}^{-}]=2s_{i}^{z}\delta_{ij}.$
(3)

Concretely in Eq. (II), the first term corresponds to the energy of the non-
interacting molecules, the second term describes propagation of the excitation
associated with the transition dipole moment $d$ and the third term comes from
electrostatic interaction between two amide-I excitons that incorporates the
static dipole moment $\mu$. The fourth term is the exciton phonon interaction
Hamiltonian, with $\chi$ being the exciton-phonon coupling constant which is
generally obtained from the temperature dependence of the vibrational $C$=$O$
modes, observed in infrared spectroscopy experiments. Moreover, the constants
$J=2d^{2}/4\pi\epsilon R^{3}$ and $D=2\mu^{2}/4\pi\epsilon R^{3}$, where both
the transition and static dipole moments are parallel to the chain direction,
$R$ is the nearest neighbor spacing. It should be noted that $J$ and $D$ have
different numerical values. $J$ is determined by the transition dipole moment
of the amide-I oscillator and the term with $J$ describes the transfer of
excitations through the resonance interaction between a molecule in an excited
dipole state with an identical non-excited molecule while $D$ is determined by
the static electric moment of the amide-I oscillator and describes the
electrostatic interactions between two excited molecules, i.e., the
electrostatic interaction between two excitons. The value of $J$ can be
derived from the experimental data of infrared spectra and it was found that
$J=1.55$ x $10^{-22}J$ Nevskay1976 . A rough estimate of the static dipole
moment $\mu$ can be made from the charge distribution of the static $C$=$O$
bond, which is of the same order of magnitude as $d$. Practically, the value
of $D$ is difficult to calculate because of the effects of polar $N$-$H$ bonds
and solvent, hence we always assume that $D\approx J$.

For the phonon Hamiltonian $H_{ph}$ given in Eq. (2), $P_{n}$ and $u_{n}$
respectively denote the momentum and position operators for longitudinal
displacement of the peptide group of site $n$. $M$ is the mass of the peptide
group and $K$ is the spring constant of the hydrogen bond.

The Hamiltonian (II) exclusively deals with spins, explaining why it is in
terms of Pauli operators. This is contrary to the classical Davydov
Hamiltonian that deals with boson operators. However, the spin operators in
Eq. (II) can be transformed to the Davydov Hamiltonian with the aid of Dyson-
Maleev transformation and boson commutation relations to obtain Xiao1998

$\displaystyle H=-\frac{1}{2}N\omega_{0}+\omega_{0}\sum_{n}a_{n}^{+}a_{n}$
$\displaystyle-\frac{1}{2}J\sum_{n,\delta}(a_{n}^{+}a_{n+\delta}+a_{n}a_{n+\delta}^{+})$
$\displaystyle-D\sum_{n,\delta}a_{n}^{+}a_{n}a_{n+\delta}^{+}a_{n+\delta}+\chi\sum_{n}$
$\displaystyle(U_{n+1}-U_{n})a_{n}^{+}a_{n}+H_{ph},$ (4)

where the term with four boson operators is omitted, because it describes the
simultaneous excitations of two amide-I excitons at one molecule.

Since we are interested with multi-exciton states, it is therefore natural to
consider excitons as coherent states whose basis states can be represented in
terms of Glauber coherent states Glauber1958

$|\Psi\big{>}=|\left\\{\alpha_{n}\right\\}\big{>}=\prod_{n}|\alpha_{n}\big{>},$
(5)

as the ansatz for the eigenstates of $H$. The coherent states satisfy the
following equation,

$a_{n}|\alpha_{n}\big{>}=\alpha_{n}|\alpha_{n}\big{>},$ (6)

where $\alpha_{n}$ is a complex eigenvalue. Taking $\big{<}\Psi|H|\Psi\big{>}$
as the usual classical Hamiltonian function, the time evolution of
$\alpha_{n}$ proceeds according to the time-dependent Schr$\ddot{o}$dinger
equation

$\displaystyle i\hbar\frac{\partial\alpha_{n}}{\partial
t}=\hbar\omega_{0}\alpha_{n}-J(\alpha_{n+1}+\alpha_{n-1}$ $\displaystyle)$
$\displaystyle-D(|\alpha_{n+1}|^{2}+|\alpha_{n-1}|^{2})\alpha_{n}+\chi(U_{n+1}$
$\displaystyle-U_{n})\alpha_{n}.$ (7)

While the dynamics of the classical lattice coordinates $U_{n}$, is governed
by

$M\frac{\partial^{2}U_{n}}{\partial
t^{2}}=K(U_{n+1}+U_{n-1}-2U_{n})+\chi(|\alpha_{n}|^{2}-|\alpha_{n-1}|^{2}).$
(8)

By using the transformation
$\alpha_{n}=\phi_{n}exp[-i/\hbar(\hbar\omega_{0}-2J)t]$, Eqs. (II) and (8) can
be rewritten as

$\displaystyle i\hbar\frac{\partial\phi_{n}}{\partial t}=$
$\displaystyle-J(\phi_{n+1}+\phi_{n-1}-2\phi_{n})-D(|\phi_{n+1}|^{2}+$
$\displaystyle|\phi_{n-1}|^{2})\phi_{n}$
$\displaystyle+\chi(U_{n+1}-U_{n})\phi_{n},$ (9a) $\displaystyle
M\frac{\partial^{2}U_{n}}{\partial t^{2}}=K($ $\displaystyle
U_{n+1}+U_{n-1}-2U_{n})+\chi(|\phi_{n}|^{2}-|\phi_{n-1}|^{2}).$ (9b)

In the absence of the exciton-exciton coupling (i.e. $D=0$), Eq. (9) models
the original equation obtained by Davydov, which describe the time evolution
of amide-I vibrational energy coupled to displacements of the hydrogen-bonded
chain of peptide groups. Furthermore, the distribution of the amide-I energy
over the individual peptide groups of the chain is determined by the quantity
$|\phi_{n}(t)|^{2}$.

##  III Modulational instability and discrete periodic excitons in the
adiabatic limit

Basically, the principle behind adiabatic approximation is to minimise the
delay necessary for the entire lattice to readjust to perturbations induced by
a $C$=$O$ bond excitations. Consequently in this adiabatic approximation (i.e.
$\dot{u}_{n}=0$), Eq. (9b) becomes

$U_{n+1}-U_{n}=-(\frac{\chi}{K})|\phi_{n}|^{2},$ (10)

reducing Eq. (9a) to

$\displaystyle i\frac{\partial\phi_{n}}{\partial
t}+\beta(\phi_{n+1}+\phi_{n-1}$
$\displaystyle-2\phi_{n})+\gamma_{1}(|\phi_{n+1}|^{2}+|\phi_{n-1}|^{2})\phi_{n}$
$\displaystyle+\,\,\,\,\,\gamma_{2}|\phi_{n}|^{2}\phi_{n}=0$ , (11)

where $\beta=J/\hbar,\,\gamma_{1}=D/\hbar,\,\gamma_{2}=\chi^{2}/\hbar K$.
Equation (III) is the discrete non-linear Schr$\ddot{o}$dinger (DNLS)
amplitude equation in which both the exciton-exciton and exciton-phonon
coupling constants provides the necessary non-linear force terms in the
adiabatic limit.

We now investigate the modulational instability (MI) analysis of the discrete
amplitude equation (III), in order to establish the necessary conditions for
energy localization in the $\alpha$-helix protein chain. It is important to
note that Eq. (III) has exact plane wave solutions in the form
$\phi_{n}(t)=\phi_{0}\,exp[i(qn-\omega t)]$, provided the relation between the
wavenumber $q$, the angular frequency $\omega$, and the amplitude $\phi_{0}$
is given as

$\omega=4\beta\,sin^{2}\left(\frac{q}{2}\right)-\Big{[}2\gamma_{1}+\gamma_{2}\Big{]}|\phi_{0}|^{2}.$
(12)

The MI of amplitude and phase of this plane wave solution is achieved by
introducing small perturbations $\theta_{n}(t),\,\,\theta^{\prime}_{n}(t)$, to
the amplitude and phase respectively, so that one can linearize the equation
of the envelope of the carrier wave. Therefore we look for a solution of the
form $\phi_{n}(t)=\Big{[}\phi_{0}+\theta_{n}(t)\Big{]}\,exp[i(qn-\omega
t+\theta^{\prime}_{n}(t))]$, where $\theta_{n}(t),\,\,\theta^{\prime}_{n}(t)$
are assumed to be small perturbation parameters of the system with frequency
$\Omega$ and wave number $Q$. Introducing this perturbation solution in (III),
using (12) and after linearization with respect to constant amplitudes, one
obtains the following dispersion relation

$\Big{[}\Omega-2\beta\,sin(Q)sin(q)\Big{]}^{2}=\Big{[}4\beta\,sin^{2}(Q/2)cos(q)\Big{]}\Big{[}4\beta\,sin^{2}(Q/2)$
$\times
cos(q)-2(\gamma_{2}-4\gamma_{1})|\phi_{0}|^{2}-12\gamma_{1}sin^{2}(Q/2)|\phi_{0}|^{2}\Big{]}.$
(13)

Equation (13) determines the condition for the stability of a plane wave with
wavenumber $q$ in the $\alpha$-helix protein chain. If the RHS of Eq. (13) is
negative, then the perturbation grows exponentially and MI is possible only if
the initial amplitude $|\phi_{0}|$ exceeds the threshold amplitude
$|\phi_{0,cr}|$ defined as follows

$|\phi_{0}|^{2}>|\phi_{0,cr}|^{2}=\frac{2\beta\,sin^{2}(Q/2)cos(q)}{(\gamma_{2}-4\gamma_{1})+6\gamma_{1}sin^{2}(Q/2)},$
(14)

with this instability region constraint to

$\Big{[}(\gamma_{2}-4\gamma_{1})+6\gamma_{1}sin^{2}(Q/2)\Big{]}cos(q)>0.$ (15)

For $\gamma_{1}>0,\,\gamma_{2}>0,\,cos(q)>0$, MI is only observed when the
strength of the exciton-phonon coupling far exceeds that of the exciton-
exciton coupling (.i.e $\gamma_{2}>4\gamma_{1}$). For $\gamma_{2}=0$, a plane
wave will be unstable provided $sin^{2}(Q/2)>2/3$, irrespective of the value
of $\gamma_{1}.$ Finally in the absence of exciton-exciton coupling (.i.e
$\gamma_{1}=0$), we obtain the exact results by Yuri Kivshar in the study of
localized modes in a chain with nonlinear on-site potential s26 .

Figure 1 shows the profile of the MI conditions on the $(q,\,Q)$ plane for
$cos(q)>0$ (i.e. $0\leq q<\pi/2$). We maintain the value of J to be $1.55$ x
$10^{-22}J$ (.i.e $\beta=1.47$ x $10^{12}s^{-1}$), as derived from the
experimental data of infrared spectra Nevskay1976 , and set
$|\phi_{0}|^{2}=1.00$ without loss of generality. The thick blue regions
generally depicts areas of plane wave stability, while the yellowish areas
signifies regions of unstable plane waves; strongly indicating the existence
of discrete solitons.

Figure 1: (color online) Regions of MI in the $(q,\,Q)$ plane indicated by the
yellow area(s) and according to Eq. (13). This is for parameters $\beta=1.47$
x $10^{12}s^{-1}$, and $|\phi_{0}|^{2}=1.00$. (a) $\gamma_{1}=3.16$ x
$10^{12}s^{-1}$, $\gamma_{2}=6.20$ x $10^{11}s^{-1}$, (b) $\gamma_{1}=1.50$ x
$10^{11}s^{-1}$, $\gamma_{2}=6.20$ x $10^{11}s^{-1}$, (c) $\gamma_{1}=3.16$ x
$10^{12}s^{-1}$, $\gamma_{2}=0.00$, (d) $\gamma_{1}=0.00$, $\gamma_{2}=6.20$ x
$10^{11}s^{-1}$.

In fact, Fig. 1(a) clearly shows the regions of MI on the $(q,\,Q)$ plane with
the protein parameters of the system obtained from Ref. Scott1992 . We have
that the exciton-exciton coupling constant, $D\approx 3.33$ x $10^{-22}J$, the
exciton-phonon coupling constant $\chi=62pN$, and the spring constant
$K=58.8N/m$; which effectively leads to the values $\gamma_{1}=3.16$ x
$10^{12}s^{-1}$ and $\gamma_{2}=6.20$ x $10^{11}s^{-1}$. We observe that
discrete solitons are easily traced in the protein chain for perturbation wave
numbers $0.50\leq Q\leq 1.90$ and $Q\geq 2.30$. An increase in the wave number
$q$ in these regions, destroys the solitons by favoring the stability of the
plane waves. Figure 1(b) gives the stability/instability regions in the
$(q,\,Q)$ plane for $\gamma_{1}=1.50$ x $10^{11}s^{-1}$, with other parameter
values maintained. Note that the decrease in the $\gamma_{1}$ value is
temperature dependent, because variations of the orientations of the peptide
groups are mainly due to the rotation around the single bonds connected to the
$C_{\alpha}$ atoms. Since we are interested in one degree of freedom, the
exciton-exciton coupling constant, $D$, is now inextricably linked to
temperature by the relation Xiao1998 ; Bergethon1990

$D(T)=\frac{2}{k_{B}T}\Big{(}\frac{1}{4\pi\epsilon_{0}}\frac{2\mu^{2}}{R^{3}}\Big{)}^{2},$
(16)

where $k_{B}$ is the the Boltzmann constant. An increase in temperature
clearly constraint the identification of discrete bright solitons in a single
region where $Q\geq 1.70$ as in Fig. 1(b); which is more prominent with small
values of wave number $q$. In the absence of exciton-phonon coupling (i.e.
$\gamma_{2}=0$), the stability/instability curve is given in Fig. 1(c); with
similar profile to that in Fig. 1(a). However, the intensity of the yellowish
region in Fig. 1(c) supersedes that in Fig. 1(a), greatly suggesting that the
exciton-phonon coupling favors plane wave stability. Lastly for
$\gamma_{1}=0$, we obtain the stability/instability curve in Fig. 1(d);
clearly showing that soliton-like modes in the protein chain can only be
observed in a single region. An increase in temperature generally leads to a
decrease in $\gamma_{1}$, hence enhancing the existence of Davydov soliton
Xiao1998 , as reflected in Figs. 1(b,d).

We now perform numerical simulations of the amplitude Eq. (III), in order to
give a comprehensive account on the evolution of plane waves and energy
localization in the $\alpha-$helix protein chain under small perturbations.
Consequently, we consider the initial condition

$\phi_{n}(t=0)=\phi_{0}[1+\Lambda_{0}\,cos(Qn)]cos(qn),$ (17)

with periodic boundary conditions $\phi_{0}=\phi_{N},\,\phi_{N+1}=\phi_{1}$,
where $N=200.$

The regions of MI shown in Fig. 1 may induce the formation of localized modes.
Consequently we now use the Jacobian elliptic function analytical method
Dai2006 ; Liu2001 ; Tiofack2007 , to show the existence of this kind of
elliptic Davydov solitons in the $\alpha$-helix protein chain. Before applying
this analytical method to Eq. (III), it is important for us to consider the
following transformations

$\phi_{n}=e^{i\eta_{n}}\rho_{n}(\xi_{n}),\,\eta_{n}=pn+\omega
t+\zeta,\,\xi_{n}=kn+ct+\nu.$ (18)

Upon substitution of the ansatz (18) into Eq. (III), and, separating the real
and imaginary parts yield:

$\displaystyle-(\omega+2\beta)\rho_{n}$
$\displaystyle+\gamma_{1}(\rho_{n+1}^{2}+\rho_{n-1}^{2})\rho_{n}+\gamma_{2}\rho_{n}^{3}$
$\displaystyle+\,\,\,\,\,\,\beta($
$\displaystyle\rho_{n+1}+\rho_{n-1})cos(p)=0,$ (19a) $\displaystyle
c\rho^{\prime}_{n}+\beta($ $\displaystyle\rho_{n+1}-\rho_{n-1})sin(p)=0.$
(19b)

We use the following series expansion as a solution of Eqs. (19):

$\rho_{n}(\xi_{n})=a_{0}+a_{1}\,sn(\xi_{n}),$ (20)

where $sn(\xi_{n})\equiv sn(\xi_{n},m)$, and $m$ is the modulus of Jacobian
elliptic functions which is constraint by $0\leq m\leq 1$. The constants
$a_{0},\,a_{1}$ are to be determined, and using the identity

$sn(A+B)=\frac{sn(A)cn(B)dn(B)+sn(B)cn(A)dn(A)}{1-m^{2}sn^{2}(A)sn^{2}(B)},$
(21)

we obtain

$\rho_{n+1}(\xi_{n})=a_{0}+a_{1}\Big{[}\frac{sn(\xi_{n})cn(k)dn(k)+sn(k)cn(\xi_{n})dn(\xi_{n})}{1-m^{2}sn^{2}(\xi_{n})sn^{2}(k)}\Big{]},$
(22)
$\rho_{n-1}(\xi_{n})=a_{0}+a_{1}\Big{[}\frac{sn(\xi_{n})cn(k)dn(k)+sn(k)cn(\xi_{n})dn(\xi_{n})}{1-m^{2}sn^{2}(\xi_{n})sn^{2}(k)}\Big{]}.$
(23)

We further substitute the expansions (20), (22), (23) into Eqs. (19); clearing
the denominator and setting the coefficients of all powers like
$sn^{i}(\xi_{n})\,(i=\,0,\,1,\,2,\,3)$ to zero. After some long calculations,
this leads us to the following solutions

$\omega\,\,=\frac{-4m^{2}\beta\gamma_{1}cos(p)sn^{4}(k)cn(k)dn(k)}{[2\gamma_{1}+\gamma_{2}-2\gamma_{1}(m^{2}+2)sn^{2}(k)+4m^{2}\gamma_{1}sn^{4}(k)]}$
$\displaystyle+$ $\displaystyle\,\,2\beta[cos(p)cn(k)dn(k)-1],$ (24a)
$\displaystyle c\,\,\,$ $\displaystyle=-2\beta sin(p)sn(k),$ (24b)
$\displaystyle a_{0}$ $\displaystyle=0,$ (24c) $\displaystyle a_{1}^{2}$
$\displaystyle=\frac{-2m^{2}\beta
cos(p)sn^{2}(k)cn(k)dn(k)}{[2\gamma_{1}+\gamma_{2}-2\gamma_{1}(m^{2}+2)sn^{2}(k)+4m^{2}\gamma_{1}sn^{4}(k)]}.$
(24d)

The constants $p,\,k,\,\zeta,\,\nu$, are given arbitrary values in order to
obtain the most appropriate discrete solution (18), as shown in Fig. 2. As the
values of the modulus $m$ increases from $0.60$ to $1.00$, the periodic dark
excitons in the protein chain degenerates into a single dark soliton in Fig.
2(d).

Figure 2:  (color online) Magnitude of spatial profile of discrete excitons in
the $\alpha-$helix protein chain according to solution (18). The parameters
are given in Eqs. (24), with values
$p=\pi,\,k=0.3,\,t=0.4,\,\zeta=0.2,\,\nu=0.00,\,\beta=1.47$ x $10^{12}s^{-1}$,
$\gamma_{1}=3.16$ x $10^{12}s^{-1}$, and $\gamma_{2}=6.20$ x $10^{11}s^{-1}$.
This is for $(a)\,m=0.60\,(b)\,m=0.70\,(c)\,m=0.98\,(d)\,m=1.00\,.$

##  IV Excitons in the coupled system

###  IV.1 Analytic solutions in the continuum limit approximation

In order to effectively comprehend the amide-I $\alpha$-helix protein chain
dynamics in the exciton-exciton and exciton-phonon couplings regime, it is
incumbent on us to solve the discrete coupled system (9). However, it is
cumbersome to solve Eqs. (9) as such due to its high non-linearity and
discreteness. Therefore it is quite useful to explore the continuum limit
approximation in the low-temperature and long-wavelength limit. Consequently,
we carry out a continuous-limit approximation by introducing the spatial
variable $x=\varepsilon na$, where $a$ is the lattice spacing and
$\varepsilon<<1$ is a perturbation parameter which physically accounts for the
interaction of the system with its quantum environment. This leads to

$\displaystyle\phi_{n\pm 1}(t)$ $\displaystyle=\phi(x,t)\pm\varepsilon
a\frac{\partial\phi(x,t)}{\partial
x}+\frac{\varepsilon^{2}a^{2}}{2!}\frac{\partial^{2}\phi(x,t)}{\partial
x^{2}}+\bigcirc(\varepsilon^{3}),$ (25a) $\displaystyle U_{n\pm 1}(t)$
$\displaystyle=U(x,t)\pm\varepsilon a\frac{\partial U(x,t)}{\partial
x}+\frac{\varepsilon^{2}a^{2}}{2!}\frac{\partial^{2}U(x,t)}{\partial
x^{2}}+\bigcirc(\varepsilon^{3}).$ (25b)

Also in the weak exciton-phonon coupling regime, we perturb $\chi$ to the
order $\varepsilon$ and transform Eqs. (9) to

$\displaystyle
i\hbar\phi_{t}+J\varepsilon^{2}a^{2}\phi_{xx}+2D|\phi|^{2}\phi=-\varepsilon^{2}D$
$\displaystyle\big{[}(\phi\phi_{x}^{*})_{x}+|\phi_{x}|^{2}\big{]}\phi\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle+\chi\varepsilon^{2}aU_{x}\phi\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(26a) $\displaystyle
MU_{tt}-K\varepsilon^{2}a^{2}U_{xx}=\chi\varepsilon^{2}a|\phi|_{x}^{2}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(26b)

We shall seek travelling wave solutions of Eqs. (26) in the form of
excitations that propagate along the chain with a velocity $v_{0}$ such that

$U(x,t)=U(x-\varepsilon v_{0}t).$ (27)

Inserting Eq. (27) in Eq. (26b), we get

$U_{x}=\frac{-\chi|\phi|^{2}}{Ka\big{(}1-v_{0}^{2}/v_{0s}^{2}\big{)}},$ (28)

where $v_{0s}=\sqrt{\frac{Ka^{2}}{M}}$ is the velocity of sound which is
greater than the velocity of excitations in the chain i.e. $v_{0s}>v_{0}$.
Substituting Eq. (28) into Eq. (26a), we get

$i\phi_{t}+P\phi_{xx}+Q|\phi|^{2}\phi+R\big{[}(\phi\phi_{x}^{*})_{x}+|\phi_{x}|^{2}\big{]}\phi=0,$
(29)

where the coefficients are given by

$\displaystyle P$ $\displaystyle=\frac{J\varepsilon^{2}a^{2}}{\hbar},$ (30a)
$\displaystyle Q$
$\displaystyle=\frac{2D}{\hbar}+\frac{\varepsilon^{2}\chi^{2}}{\hbar
K\big{(}1-v_{0}^{2}/v_{0s}^{2}\big{)}},$ (30b) $\displaystyle R$
$\displaystyle=\frac{\varepsilon^{2}D}{\hbar}.$ (30c)

Equation (29) is the inhomogeneous NLS equation, which governs the amide-I
$\alpha$-helix protein chain dynamics in the continuous limit approximation,
with the inhomogeneity being induced by the exciton-exciton coupling. The form
of the envelope soliton solution of Eq. (29) is given by

$\phi(x,t)=\phi_{0}\,u(X)\,exp[i(\psi(X)+\omega t)],$ (31)

where

$X=X(x,t)=x-ct.$

$u$ and $\psi$ are real functions of $X$, while $\phi_{0},\,\omega,\,c$ are
real constants with $c$ being the speed while $\omega$ is the frequency.
Inserting the ansatz (31) into Eq. (29) and separating the real and imaginary
parts gives

$\displaystyle P(u^{\prime\prime}-u\psi^{\prime 2})+$
$\displaystyle\phi_{0}^{2}Qu^{3}+(c\psi^{\prime}-\omega)u+\phi_{0}^{2}R(2u^{\prime
2}+uu^{\prime\prime}+$ $\displaystyle\psi^{\prime
2}u^{2})u\,\,\,\,\,\,\,=\,\,\,\,\,\,$ $\displaystyle 0,$ (32a) $\displaystyle
P(2u^{\prime}\psi^{\prime}+u\psi^{\prime\prime})$ $\displaystyle-
cu^{\prime}-\phi_{0}^{2}R(\psi^{\prime\prime}u^{2}+2uu^{\prime}\psi^{\prime})u=0.$
(32b)

At this stage, we assume that

$\psi^{\prime}=bu^{2}+\sigma,$ (33)

where $b$ and $\sigma$ are real constants. Upon substituting (33) into Eq.
(32) and neglecting terms of order $\bigcirc(u^{4})$ and above yields

$\displaystyle P(u^{\prime\prime}-\sigma^{2}u-2\sigma bu^{3})+$
$\displaystyle\phi_{0}^{2}Qu^{3}+c(bu^{2}+\sigma)u-\omega u+$
$\displaystyle\phi_{0}^{2}R(2u^{\prime 2}+uu^{\prime\prime}+\sigma^{2}u^{2})u$
$\displaystyle=\,\,\,\,\,\,\,\,\,\,0,$ (34a)
$\displaystyle(2P\sigma-c)u^{\prime}+(4bP-2\phi_{0}^{2}$ $\displaystyle
R\sigma)u^{\prime}u^{2}=0.$ (34b)

Since $u^{\prime}\neq 0$, $u^{\prime}u^{2}\neq 0$ in Eq. (34b), we
respectively obtain $c=2P\sigma,\,\,\phi_{0}^{2}R=2bP/\sigma$ and transform
Eq. (34a) to

$u^{\prime\prime}=\Gamma_{1}u-\Gamma_{2}u^{3}-\Gamma_{3}(u^{2}u^{\prime})^{\prime},$
(35)

where

$\Gamma_{1}=\frac{(\omega-P\sigma^{2})}{P},\,\Gamma_{2}=\frac{\phi_{0}^{2}}{P}(R\sigma^{2}+Q),\,\Gamma_{3}=\frac{\phi_{0}^{2}R}{P}.$
(36)

In the absence of the of exciton-exciton coupling (i.e.
$D=0,\,\Gamma_{3}=0,\,R=0,\,Q=\varepsilon^{2}\chi^{2}/\hbar
K\big{(}1-\frac{v_{0}^{2}}{v_{0s}^{2}}\big{)}$, Eq. (35) possesses solution of
the form Vyas2006 ; Hioe1999 ; Hioe2003

$u(X)=A\,dn(BX,\,m)+C,$ (37)

where $A,\,B$ and $C$ are parameters to be determined, with $m$ being the
modulus $(0\leqslant m\leqslant 1)$ of the elliptic $dn$ function. By
substituting (37) into Eq. (35) and equating coefficients of terms with the
same powers of $dn(BX,\,k)$ to have

$A=\sqrt{\frac{2(\omega-P\sigma^{2})}{\phi_{0}^{2}Q(2-m^{2})}},\,B=\sqrt{\frac{\omega-P\sigma^{2}}{P(2-m^{2})}},\,C=0.$
(38)

Figure 3 is the $dn$ function which depicts the train of spatial periodic
excitons in the amide-I $\alpha$-helix protein chain.

Figure 3:  (color online) Spatial periodic excitons which is solution (37).
The parameters are
$\phi_{0}=\sqrt{2},\,P=1.00,\,Q=1.00,\,\sigma=1.0,\,\omega=2.00$. This is for
$(a)\,m=0.60\,(b)\,m=0.70\,(c)\,m=0.98\,(d)\,m=1.00\,.$

The non localized solutions depicted in Fig. 3(a-c) shows that the energy of
the ATP hydrolysis is locally distributed over the the $\alpha$-helix protein
chain. This energy may be self-localized to form a single bright soliton on
one side of protein chain for $m=1.0$ (i.e. it is as a result of superposition
of continuous plane waves oscillating at a constant frequency shift), as shown
in Fig. 3(d). However from experimental standpoint, the observation of such
localized nature of energy transfer in a protein chain is very rare. In fact,
$m$ may be considered as a pulse dispersion parameter with the periodic
excitons mainly conveying coded information being processed in the amide-I
$\alpha$-helix protein chain in the spatial domain.

The Cherenkov-like effect initially demonstrated by Cherenkov in 1934
Landau1960 , may equally be identified in the no exciton-exciton (i.e. $D=0$)
coupling regime; provided the soliton velocity exceeds that of the phonon
(.i.e $v_{0}>v_{0s}$ as in ref Mesquita1998 ). Consequently, we equally obtain
dark profile soliton of Eq. (35) given by

$u(X)=\sqrt{\frac{2m^{2}(P\sigma^{2}-\omega)}{\phi_{0}^{2}|Q|(1+m^{2})}}\,sn\Big{[}\sqrt{\frac{(P\sigma^{2}-\omega)}{P(1+m^{2})}}X,\,m\Big{]},$
(39)

provided the frequency $\omega<P\sigma^{2}$. This leads to the propagation of
the elliptic $sn$ exciton in the $\alpha$-helix protein chain as shown in Fig.
4; with the profile of this periodic excitons degenerating to a kink soliton
for $m=1.00$ as shown in Fig. 4(d). Solution (39) manifest the same structural
features as the spatial profile of discrete excitons in the adiabatic limit,
as dictated by Eq. (20).

Figure 4:  (color online) Profile of Jacobian $sn$ function along the
$\alpha$-helix protein chain according to solution (39). The parameters are
$\phi_{0}=1.00,\,P=1.00,\,Q=-1.00,\,\sigma=2.00,\,\omega=3.00$. This is for
$(a)\,m=0.60\,(b)\,m=0.70,(c)\,m=0.98\,(d)\,m=1.00\,.$

For $\Gamma_{3}\neq 0,$ traveling wave solutions of Eq. (35) can be obtained
by using the following series expansion Vyas2006 ; Hioe1999 ; Hioe2003

$u(X)=c_{0}+c_{1}\,cn(X)+c_{2}\,cn^{2}(X),$ (40)

where $cn(X)\equiv cn(X,m)$ and $c_{0},\,c_{1},\,c_{2}$ are real constants to
be determined. Since our goal is to look for spatial periodic solutions, we
substitute our proposed solution (40) into Eq. (35), and equating the
coefficients of equal powers of $cn(X)$ to have:

$\displaystyle c_{0}^{2}$
$\displaystyle=\frac{2m^{2}-1}{4\Gamma_{2}+(2m^{2}-1)\Gamma_{3}},$ (41a)
$\displaystyle c_{1}^{2}$
$\displaystyle=\frac{4\Gamma_{1}\Gamma_{2}+(2m^{2}-1)\big{[}\Gamma_{1}\Gamma_{3}-\Gamma_{2}\big{]}}{2(1-m^{2})\big{[}4\Gamma_{2}\Gamma_{3}+(2m^{2}-1)\Gamma_{3}^{2}\big{]}},$
(41b) $\displaystyle c_{2}^{2}$ $\displaystyle=0.$ (41c)

Figure (5) shows the spatial profile of elliptic solitons in the
$\alpha-$helix protein chain as the modulus $m$ is varied. On the other hand,
the intensity profile of this spatial periodic soliton is given in Fig. (6).

Figure 5:  (color online) Spatial periodic solitons in the protein chain as
governed by solution (40). The parameters are
$\phi_{0}=\sqrt{2},\,P=1.00,\,Q=1.00,\,\sigma=1.00,\,R=3.00,\,\omega=5.00,\,\Gamma_{1}=4.00,\,\Gamma_{2}=8.00,\,\Gamma_{3}=6.00$.
This is for $(a)\,m=0.70\,(b)\,m=0.90,(c)\,m=0.98\,(d)\,m=0.99\,.$ Figure 6:
(color online) Intensity of spatial periodic solitons in the protein chain as
governed by solution (31). The parameters are
$\phi_{0}=\sqrt{2},\,P=1.00,\,Q=1.00,\,\sigma=1.00,\,R=3.00,\,\omega=5.00,\,\Gamma_{1}=4.00,\,\Gamma_{2}=8.00,\,\Gamma_{3}=6.00$.
This is for $(a)\,m=0.70\,(b)\,m=0.90,(c)\,m=0.98\,(d)\,m=0.99\,.$

We equally plot the spatial profile of the displacement of the peptide groups
given by solution (28) in Fig. (7).

Figure 7:  (color online) Spatial profile of the displacement of the peptide
groups given by solution (28). The parameters are
$\phi_{0}=\sqrt{2},\,P=1.00,\,Q=1.00,\,\sigma=1.00,\,R=3.00,\,\omega=5.00,\,\Gamma_{1}=4.00,\,\Gamma_{2}=8.00,\,\Gamma_{3}=6.00,\,a=1.00,\,v_{0}^{2}=0.90v_{0s}^{2},\,K=58.8Nm^{-1},$
and the exciton-phonon coupling constant $\chi=62pN.$ This is for
$(a)\,m=0.70\,(b)\,m=0.90,(c)\,m=0.98\,(d)\,m=0.99\,.$

###  IV.2 Numerical analysis

A comprehensive study of Eqs. (9) is quite cumbersome, because of the high
non-linearity and discreteness that characterizes such an equation. In order
to effectively comprehend the amide-I $\alpha$-helix protein chain dynamics in
the exciton-exciton and exciton-phonon couplings regime, it is incumbent on us
to numerically solve the discrete coupled system (9). We now consider the
following approximation $U_{n+1}-U_{n}\approx\varepsilon a\partial U/\partial
x$, in order to obtain the following approximate initial conditions

$\displaystyle\phi_{n}(t=0)$
$\displaystyle\approx\phi_{0}u_{n}\,exp\Big{[}-i\varepsilon
a\big{[}bu_{n}^{2}+\sigma\big{]}\Big{]},$ (42a) $\displaystyle U_{n}(t=0)$
$\displaystyle\approx\frac{\varepsilon\chi|\phi_{0}|^{2}|u_{n}|^{2}}{K\big{(}1-v_{0}^{2}/v_{0s}^{2}\big{)}},$
(42b)

for the numerical integration of Eqs. (9). Without loss of generality, we set
$\varepsilon=a=1$ and consider the following three initial conditions:

  1. 1.

For $D=0$, the $dn$ soliton solution (37) gives:

$u_{n}=\sqrt{\frac{2(\omega-P\sigma^{2})}{\phi_{0}^{2}Q(2-m^{2})}}\,dn\Big{[}\sqrt{\frac{\omega-P\sigma^{2}}{P(2-m^{2})}}n,\,m\Big{]}.$
(43)

  2. 2.

For $D=0$, the $sn$ soliton solution (39) gives:

$u_{n}=\sqrt{\frac{2m^{2}(P\sigma^{2}-\omega)}{\phi_{0}^{2}|Q|(1+m^{2})}}\,sn\Big{[}\sqrt{\frac{(P\sigma^{2}-\omega)}{P(1+m^{2})}}n,\,m\Big{]},$
(44)

provided the frequency $\omega<P\sigma^{2}$.

  3. 3.

For $D\neq 0$, the $cn$ soliton solution (40) gives:

$u_{n}=\sqrt{\frac{2m^{2}-1}{4\Gamma_{2}+(2m^{2}-1)\Gamma_{3}}}\,\,\,\,\,+$
$\sqrt{\frac{4\Gamma_{1}\Gamma_{2}+(2m^{2}-1)\big{[}\Gamma_{1}\Gamma_{3}-\Gamma_{2}\big{]}}{2(1-m^{2})\big{[}4\Gamma_{2}\Gamma_{3}+(2m^{2}-1)\Gamma_{3}^{2}\big{]}}}cn(n,m).$
(45)

##  V Discussion and Conclusion

It is clearly an indisputable fact that proteins are the principal organs used
in the generation of energy for the metabolic activities of the cell and by
extension to maintain life activities. Consequently there is a growing
interest in the field of bioenergetics, because of the quest to fully
understand the mechanism of generation and transfer of energy in protein
chains. Davydov argued that the energy released by ATP hydrolysis is stored in
the amide-I vibration through nonlinear interactions, that traps the energy in
the form of a robust entity called solitons. The soliton concept is a crucial
idea that should be rigorously considered by biophysicists. Even though it can
not explain every aspect of protein chain dynamics, it however serve like a
guide for future experimental investigations. In this study, we considered a
model Hamiltonian of $\alpha$-helix protein chain which deals with spins, and
later transformed it to the classical Davydov Hamiltonian to exclusively
handle boson operators. The quantum mechanical approach was then explored to
treat the amide-I excitations, while the displacements of longitudinal sound
wave along the hydrogen-bonded chain was treated classically. In the adiabatic
limit approximation, we have demonstrated that the mechanism of the formation
of the localized modes in the protein chain is inextricably linked to the
modulational instability phenomena. We further explored all the nonlinearity
in the protein chain, in order to enhance the self-trapping mechanism of the
amide-I vibrational energy. Such mechanism has been shown to provide a
possible solution to the problem of the thermal stability of the Davydov
soliton Xiao1998 . By applying the continuum limit approximation to our
coupled system, the inhomogeneous NLS equation was obtained. It was generally
observed that the inhomogeneity induced by the exciton-exciton coupling,
favors energy localization in the chain as elaborated by the results of
numerical simulations.

In perspective; firstly we intend to use the soliton concept of $\alpha$-helix
protein chain to explain a possible mechanism of general anaesthesia. This
will be based on the assumption that the binding of an anaesthetic molecule to
a protein interferes with soliton propagation. To validate this idea from a
theoretical standpoint, we need to calculate the effect of anaesthetic binding
on soliton propagation. Secondly, the interaction of a system with its
environment is given by the dissipative effect in quantum system, which has
attracted many interests in the last decades. Such dissipative effects can be
effectively investigated by adding new term the inhomogeneous NLS equation. A
remarkable result of this present study is that the exciton-exciton coupling
enhances self-trapping, which accounts for many aspects of protein dynamics.

## Ackowledgments

N. Oma Nfor appreciates the enriching discussions with members of the LaRAMaNS
research group.

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