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Excitons dynamic in a three-stranded \(\alpha \)-helix protein chains with diagonal and off-diagonal couplings: effects of strong long-range interactions

Abstract

Exciton dynamics through a three-stranded \(\alpha \)-helix protein chain in the presence of inter-spines and diagonal and off-diagonal couplings as well as long-range interactions are investigated. Its is shown that the behavior of the system is governed by three-coupled modified continuous nonlinear Schrödinger equations. Performing a modulational instability analysis, it appears that both the off-diagonal coupling and the long-range interaction reduce the instability regions and the amplitude of the growth rate. More analytical insights of the system dynamics are obtained by constructing its solutions with the F-expansion method. Many families of solutions are unveiled among them are bright and dark solitary waves, Jacobian elliptic function solutions and hyperbolic function solutions to name just a few. Intensive numerical simulations carried corroborate the analytical solutions found with a good accuracy. In addition, our numerical findings prove that the long-range interactions increase the energy of the waves propagating through the protein chain. From a biological point of view, the model used in the current work better describes the energy transport in protein chains.

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Correspondence to R. Y. Ondoua.

Appendix: coefficients of Eq. (21)

Appendix: coefficients of Eq. (21)

$$\begin{aligned} \theta _{5}&=m_{{55}}-m_{{22}}+m_{{44}}+m_{{66}}-m_{{33}}-m_{{11}} \\ \theta _{4}&= m_{{11}}m_{{44}}+m_{{11}}m_{{66}}-m_{{22}}m_{{33}}+m_{{22}}m_{{55}}-m_ {{11}}m_{{33}} \\&\quad +m_{{11}}m_{{55}}+m_{{33}}m_{{66}}-{m_{{14}}}^{2}+m_{{22 }}m_{{44}}\\&\quad -m_{{22}}m_{{11}}-{m_{{25}}}^{2}+m_{{33}}m_{{44}}-m_{{55}}m_{{66}}-{m_{{36}}}^{2} \\&\quad +m_{{22}}m_{{66}} -m_{{44}}m_{{55}}-m_{{44}}m_{{66}}+m_{{33}}m_{{55}}, \\ \theta _{3}&=m_{{22}}m_{{33}}m_{{66}}+m_{{22}}m_{{33}}m_{{55}}+m_{{22}}m_{{11}}m_{{ 44}}-m_{{22}}m_{{44}}m_{{66}}\\&\quad +m_{{22}}m_{{11}}m_{{55}}-m_{{33}}m_{{44} }m_{{55}}\\&\quad +m_{{44}}m_{{55}}m_{{66}}-{m_{{25}}}^{2}m_{{11}} -{m_{{14}}}^{2}m_{{22}}+{m_{{25}}}^{2}m_{{44}}+{m_{{25}}}^{2}m_{{66}} \\&\quad +{m_{{14}}}^{2 }m_{{55}}+{m_{{14}}}^{2}m_{{66}}-{m_{{14}}}^{2}m_{{33}}\\&\quad +{m_{{36}}}^{2}m_{{55}}+{m_{{36}}}^{2}m_{{44}}-m_{{22}}{m_{{36}}}^{2}-{m_{{25}}}^{2}m _{{33}}\\&\quad -m_{{11}}{m_{{36}}}^{2}-m_{{11}}m_{{55}}m_{{66}}-m_{{11}}m_{{44 }}m_{{66}}\\&\quad -m_{{33}}m_{{44}}m_{{66}}+m_{{11}}m_{{33}}m_{{66}}-m_{{11}}m _{{44}}m_{{55}}+m_{{11}}m_{{33}}m_{{44}}\\&\quad +m_{{11}}m_{{33}}m_{{55}}-m_{{ 22}}m_{{55}}m_{{66}}\\&\quad -m_{{22}}m_{{11}}m_{{33}}+m_{{22}}m_{{11}}m_{{66}} -m_{{33}}m_{{55}}m_{{66}}-m_{{22}}m_{{44}}m_{{55}}+m_{{22}}m_{{33}}m_{ {44}}, \\ \theta _{2}&=-m_{{11}}m_{{33}}m_{{44}}m_{{55}}+m_{{22}}m_{{11}}m_{{33}}m_{{44}}+m_{ {22}}m_{{11}}m_{{33}}m_{{55}} \\&\quad -m_{{22}}m_{{11}}m_{{44}}m_{{66}}-m_{{22} }m_{{11}}m_{{55}}m_{{66}}\\&\quad +m_{{33}}m_{{44}}m_{{55}}m_{{66}}+m_{{11}}m_{ {44}}m_{{55}}m_{{66}}-m_{{22}}m_{{11}}m_{{44}}m_{{55}} \\&\quad -m_{{22}}m_{{33} }m_{{55}}m_{{66}}-m_{{22}}m_{{33}}m_{{44}}m_{{66}}\\&\quad +m_{{22}}m_{{44}}m_{ {55}}m_{{66}}+{m_{{25}}}^{2}m_{{11}}m_{{66}}-{m_{{25}}}^{2}m_{{33}}m_{ {11}}+{m_{{25}}}^{2}m_{{33}}m_{{44}} \\&\quad -{m_{{14}}}^{2}m_{{22}}m_{{33}}+{m _{{12}}}^{2}m_{{22}}m_{{44}}\\&\quad -2\,m_{{25}}m_{{33}}{m_{{23}}}^{2}+{m_{{25 }}}^{2}m_{{44}}m_{{11}}-{m_{{25}}}^{2}m_{{44}}m_{{66}}+{m_{{13}}}^{2}m _{{33}}m_{{44}} \\&\quad +{m_{{25}}}^{2}m_{{33}}m_{{66}}+{m_{{12}}}^{2}m_{{44}}m _{{55}}\\&\quad -2\,{m_{{12}}}^{2}m_{{44}}m_{{25}}-2\,{m_{{12}}}^{2}m_{{14}}m_{ {55}}+m_{{11}}{m_{{36}}}^{2}m_{{55}}-2\,{m_{{12}}}^{2}m_{{22}}m_{{14}} \\&\quad -{m_{{14}}}^{2}m_{{55}}m_{{66}}+{m_{{14}}}^{2}m_{{22}}m_{{55}}\\&\quad +{m_{{14 }}}^{2}m_{{22}}m_{{66}}+{m_{{14}}}^{2}m_{{33}}m_{{55}}+m_{{22}}m_{{11} }m_{{33}}m_{{66}} \\&\quad -m_{{22}}m_{{33}}m_{{44}}m_{{55}}-2\,{m_{{13}}}^{2}m_ {{33}}m_{{14}}\\&\quad +{m_{{13}}}^{2}m_{{33}}m_ {{11}}+4\,{m_{{12}}}^{2}m_{{14}}m_{{25}}+m_{{22}}m_{{33}}{m_{{23}}}^{2 }-{m_{{36}}}^{2}m_{{44}}m_{{55}} \\&\quad +m_{{22}}{m_{{23}}}^{2}m_{{66}}+{m_{{ 12}}}^{2}m_{{22}}m_{{11}}\\&\quad +m_{{11}}{m_{{36}}}^{2}m_{{44}}-2\,m_{{22}}{m _{{23}}}^{2}m_{{36}}+{m_{{12}}}^{2}m_{{11}}m_{{55}}-2\,{m_{{12}}}^{2}m _{{11}}m_{{25}} \\&\quad +m_{{22}}{m_{{36}}}^{2}m_{{55}}-2\,{m_{{13}}}^{2}m_{{36 }}m_{{44}}\\&\quad +{m_{{13}}}^{2}m_{{11}}m_{{66}}+4\,{m_{{13}}}^{2}m_{{36}}m_{ {14}}-2\,{m_{{13}}}^{2}m_{{36}}m_{{11}} \\&\quad -2\,{m_{{13}}}^{2}m_{{14}}m_{{ 66}}-m_{{11}}m_{{33}}m_{{44}}m_{{66}}\\&\quad +4\,{m_{{23}}}^{2}m_{{36}}m_{{25} }+{m_{{13}}}^{2}m_{{44}}m_{{66}}-2\,{m_{{23}}}^{2}m_{{36}}m_{{55}}-2\, {m_{{23}}}^{2}m_{{25}}m_{{66}} \\&\quad +{m_{{23}}}^{2}m_{{55}}m_{{66}}+m_{{22}} {m_{{36}}}^{2}m_{{44}}\\&\quad -m_{{22}}m_{{11}}{m_{{36}}}^{2}-m_{{11}}m_{{33}} m_{{55}}m_{{66}}-{m_{{14}}}^{2}{m_{{36}}}^{2} \\&\quad -{m_{{25}}}^{2}{m_{{36}}} ^{2}+{m_{{14}}}^{2}m_{{33}}m_{{66}}\\&\quad -{m_{{25}}}^{2}{m_{{14}}}^{2}+{m_{{23}}}^{2}m_{{33}}m_{{55}}, \\ \theta _{1}&=4\,{m_{{12}}}^{2}m_{{33}}m_{{14}}m_{{25}}-{m_{{12}}}^{2}m_{{44}}m_{{55 }}m_{{66}} \\&\quad +2\,{m_{{12}}}^{2}m_{{44}}m_{{25}}m_{{66}}+2\,{m_{{12}}}^{2} m_{{14}}m_{{55}}m_{{66}}\\&\quad -4\,{m_{{12}}}^{2}m_{{14}}m_{{25}}m_{{66}}-2\, m_{{25}}m_{{33}}{m_{{23}}}^{2}m_{{11}}+2\,m_{{25}}m_{{33}}m_{{44}}{m_{ {23}}}^{2} \\&\quad -{m_{{25}}}^{2}m_{{33}}{m_{{14}}}^{2}+{m_{{25}}}^{2}{m_{{36} }}^{2}m_{{44}}\\&\quad -{m_{{14}}}^{2}m_{{22}}{m_{{36}}}^{2}-2\,{m_{{12}}}^{2}m _{{33}}m_{{44}}m_{{25}}+{m_{{12}}}^{2}m_{{22}}m_{{33}}m_{{44}} \\&\quad +{m_{{14 }}}^{2}m_{{22}}m_{{33}}m_{{55}}-{m_{{14}}}^{2}m_{{22}}m_{{55}}m_{{66}} \\&\quad -{m_{{25}}}^{2}m_{{33}}m_{{44}}m_{{66}}+{m_{{25}}}^{2}m_{{33}}m_{{11}} m_{{66}} \\&\quad +{m_{{25}}}^{2}m_{{33}}m_{{44}}m_{{11}}-{m_{{25}}}^{2}m_{{44}} m_{{11}}m_{{66}}\\&\quad -m_{{22}}m_{{ 11}}m_{{33}}m_{{55}}m_{{66}}-m_{{22}}m_{{11}}m_{{33}}m_{{44}}m_{{66}}- m_{{22}}m_{{11}}m_{{33}}m_{{44}}m_{{55}} \\&\quad +m_{{22}}m_{{11}}m_{{44}}m_{{ 55}}m_{{66}}\\&\quad +m_{{22}}m_{{33}}m_{{44}}m_{{55}}m_{{66}}+{m_{{12}}}^{2}m_ {{33}}m_{{44}}m_{{55}} \\&\quad +m_{{22}}m_{{33}}{m_{{23}}}^{2}m_{{11}}-m_{{22}} m_{{33}}m_{{44}}{m_{{23}}}^{2}\\&\quad -{m_{{23}}}^{2}m_{{33}}m_{{44}}m_{{55}}+2\,{m_{{13}}}^{2}m_{{33}}m_{{14 }}m_{{55}} \\&\quad -m_{{22}}{m_{{23}}}^{2}m_{{44}}m_{{66}}-2\,m_{{22}}{m_{{23}} }^{2}m_{{11}}m_{{36}}\\&\quad +2\,m_{{22 }}{m_{{23}}}^{2}m_{{44}}m_{{36}}-2\,{m_{{12}}}^{2}m_{{33}}m_{{14}}m_{{ 55}}+{m_{{12}}}^{2}m_{{33}}m_{{11}}m_{{55}} \\&\quad +2\,{m_{{23}}}^{2}m_{{36}}m _{{44}}m_{{55}}\\&\quad -4\,{m_{{23}}}^{2}m_{{36}}m_{{44}}m_{{25}}-{m_{{13}}}^{ 2}m_{{44}}m_{{55}}m_{{66}}+2\,{m_{{13}}}^{2}m_{{14}}m_{{55}}m_{{66}} \\&\quad -{ m_{{13}}}^{2}m_{{11}}m_{{55}}m_{{66}} \\&\quad -4\,{m_{{13}}}^{2}m_{{36}}m_{{14} }m_{{55}}+2\,{m_{{13}}}^{2}m_{{36}}m_{{11}}m_{{55}} \\&\quad +2\,{m_{{13}}}^{2}m _{{36}}m_{{44}}m_{{55}}+{m_{{13}}}^{2}m_{{22}}m_{{44}}m_{{66}}\\&\quad -{m_{{13 }}}^{2}m_{{33}}m_{{11}}m_{{55}} -{m_{{13}}}^{2}m_{{33}}m_{{44}}m_{{55}} -2\,{m_{{13}}}^{2}m_{{22}}m_{{33}}m_{{14}} \\&\quad +{m_{{13}}}^{2}m_{{22}}m_{{ 33}}m_{{11}}\\&\quad +2\,{m_{{12}}}^{2}m _{{22}}m_{{14}}m_{{66}}-{m_{{12}}}^{2}m_{{22}}m_{{11}}m_{{66}}-{m_{{12 }}}^{2}m_{{22}}m_{{44}}m_{{66}} \\&\quad -2\,{m_{{12}}}^{2}m_{{22}}m_{{33}}m_{{ 14}}\\&\quad -2\,{m_{{12}}}^{2}m_{{33}}m_{{11}}m_{{25}}-{m_{{12}}}^{2}m_{{11}}m _{{55}}m_{{66}} \\&\quad +2\,{m_{{12}}}^{2}m_{{11}}m_{{25}}m_{{66}}-m_{{11}}{m_{ {36}}}^{2}m_{{44}}m_{{55}}\\&\quad +m_{{ 22}}m_{{11}}{m_{{36}}}^{2}m_{{44}}-m_{{22}}{m_{{36}}}^{2}m_{{44}}m_{{ 55}}+{m_{{23}}}^{2}m_{{11}}m_{{55}}m_{{66}} \\&\quad -2\,{m_{{23}}}^{2}m_{{11}}m _{{25}}m_{{66}}\\&\quad +2\,{m_{{23}}}^{ 2}m_{{44}}m_{{25}}m_{{66}}-2\,{m_{{23}}}^{2}m_{{36}}m_{{55}}m_{{11}}+4 \,{m_{{23}}}^{2}m_{{36}}m_{{25}}m_{{11}} \\&\quad +{m_{{14}}}^{2}m_{{22}}m_{{33} }m_{{66}}\\&\quad +{m_{{12}}}^{2}m_{{22}}m_{{33}}m_{{11}}-2\,{m_{{13}}}^{2}m_{{ 22}}m_{{14}}m_{{66}}+{m_{{13}}}^{2}m_{{22}}m_{{11}}m_{{66}} \\&\quad +4\,{m_{{13 }}}^{2}m_{{22}}m_{{36}}m_{{14}}\\&\quad -2\,{m_{{13}}}^{2}m_{{22}}m_{{36}}m_{{ 11}}-2\,{m_{{13}}}^{2}m_{{22}}m_{{36}}m_{{44}}-{m_{{14}}}^{2}m_{{33}}m _{{55}}m_{{66}} \\&\quad +{m_{{14}}}^{2}{m_{{36}}}^{2}m_{{55}}+{m_{{25}}}^{2}{m_ {{14}}}^{2}m_{{66}}\\&\quad -{m_{{25}}}^{2}{m_{{36}}}^{2}m_{{11}}-{m_{{23}}}^{2}m_{{44}}m_{{55}}m_{{66}} +m_{{22}}m_{{11}}{m_{{36}}}^{2}m_{{55}} \\&\quad +{m_{{13}}}^{2}m_{{22}}m_{{33}}m_{{44}} +m_{{22}}{m_{{23}}}^{2}m_{{11}}m_{{66}}\\&\quad +{m_{{23}}}^{2}m_{{33}}m_{{55}}m_{{11}}+m_{{11}}m_{{33}}m_{{44}}m_{{55}}m_{{66}}, \\ \theta _{0}&=-m_{{22}}{m_{{23}}}^{2}m_{{11}}m_{{44}}m_{{66}}-4\,m_{{12}}m_{{33}}m_{ {14}}m_{{55}}m_{{23}}m_{{13}} +8\,m_{{12}}m_{{33}}m_{{14}}m_{{25}}m_{{ 23}}m_{{13}}\\&\quad -2\,{m_{{36}}}^{2}{m_{{12}}}^{2}m_{{44}}m_{{25}}+2\,m_{{23 }}m_{{13}}m_{{44}}m_{{55}}m_{{12}}m_{{66}} \\&\quad +{m_{{36}}}^{2}{m_{{12}}}^{2 }m_{{44}}m_{{55}}-2\,{m_{{36}}}^{2}{m_{{12}}}^{2}m_{{14}}m_{{55}}\\&\quad -4\,m _{{23}}m_{{13}}m_{{44}}m_{{25}}m_{{12}}m_{{66}}-m_{{22}}m_{{33}}m_{{44 }}{m_{{23}}}^{2}m_{{11}}-{m_{{23}}}^{2}m_{{33}}m_{{44}}m_{{55}}m_{{11} } \\&\quad +4\,{m_{{36}}}^{2}{m_{{12}}}^{2}m_{{14}}m_{{25}}\\&\quad +{m_{{36}}}^{2}{m_{{ 12}}}^{2}m_{{11}}m_{{55}}-2\,{m_{{36}}}^{2}{m_{{12}}}^{2}m_{{11}}m_{{ 25}}-4\,m_{{23}}m_{{13}}m_{{14}}m_{{55}}m_{{12}}m_{{66}} \\&\quad -2\,{m_{{25}}} ^{2}{m_{{13}}}^{2}m_{{14}}m_{{66}}\\&\quad -4\,m_{{36}}m_{{12}}m_{{23}}m_{{13}} m_{{44}}m_{{55}}+8\,m_{{36}}m_{{12}}m_{{23}}m_{{13}}m_{{44}}m_{{25}} \\&\quad +8 \,m_{{36}}m_{{12}}m_{{23}}m_{{13}}m_{{14}}m_{{55}}\\&\quad -16\,m_{{36}}m_{{12} }m_{{23}}m_{{13}}m_{{14}}m_{{25}}+2\,m_{{12}}m_{{22}}m_{{33}}m_{{44}}m _{{23}}m_{{13}}-2\,m_{{22}}{m_{{23}}}^{2}{m_{{14}}}^{2}m_{{36}}\\&\quad +{m_{{23}}}^{2}m_{{33}}{m_{{14}} }^{2}m_{{55}}+8\,m_{{23}}m_{{13}}m_{{14}}m_{{25}}m_{{12}}m_{{66}} \\&\quad -4\,{ m_{{13}}}^{2}m_{{22}}m_{{36}}m_{{14}}m_{{55}}\\&\quad +2\,m_{{12}}m_{{22}}m_{{ 23}}m_{{13}}m_{{11}}m_{{66}}-4\,m_{{12}}m_{{22}}m_{{13}}m_{{11}}m_{{23 }}m_{{36}}+2\,{m_{{12}}}^{2}m_{{22}}m_{{33}}m_{{14}}m_{{66}}\\&\quad +{m_{{25}} }^{2}m_{{33}}{m_{{14}}}^{2}m_{{66}}-2\,{m_{{25}}}^{2}m_{{33}}m_{{14}}{ m_{{13}}}^{2}+{m_{{25}}}^{2}m_{{33}}m_{{11}}{m_{{13}}}^{2} \\&\quad +{m_{{25}}}^ {2}m_{{33}}m_{{44}}{m_{{13}}}^{2}\\&\quad -2\,{m_{{25}}}^{2}{m_{{13}}}^{2}m_{{ 11}}m_{{36}}+{m_{{25}}}^{2}{m_{{13}}}^{2}m_{{11}}m_{{66}}+{m_{{25}}}^{ 2}{m_{{36}}}^{2}m_{{44}}m_{{11}} \\&\quad +m_{{22}}m_{{33}}{m_{{14}}}^{2}{m_{{23 }}}^{2}\\&\quad -{m_{{12}}}^{2}m_{{33}}m_{{44}}m_{{55}}m_{{66}}+2\,{m_{{12}}}^{ 2}m_{{33}}m_{{44}}m_{{25}}m_{{66}} \\&\quad +2\,{m_{{12}}}^{2}m_{{33}}m_{{14}}m_ {{55}}m_{{66}}\\&\quad -{m_{{ 12}}}^{2}m_{{33}}m_{{11}}m_{{55}}m_{{66}}+2\,{m_{{12}}}^{2}m_{{33}}m_{ {11}}m_{{25}}m_{{66}}-{m_{{12}}}^{2}m_{{22}}m_{{33}}m_{{11}}m_{{66}} \\&\quad -{ m_{{12}}}^{2}m_{{22}}m_{{33}}m_{{44}}m_{{66}}\\&\quad +2\,m_{{12}}m_{{33}}m_{{ 44}}m_{{55}}m_{{23}}m_{{13}}+2\,{m_{{13}}}^{2}m_{{22}}m_{{36}}m_{{11}} m_{{55}} \\&\quad +2\,{m_{{13}}}^{2}m_{{22}}m_{{36}}m_{{44}}m_{{55}}\\&\quad -{m_{{14}}}^ {2}m_{{22}}m_{{33}}m_{{55}}m_{{66}}-{m_{{25}}}^{2}m_{{33}}m_{{44}}m_{{ 11}}m_{{66}}+2\,m_{{25}}m_{{33}}m_{{44}}{m_{{23}}}^{2}m_{{11}} \\&\quad -2\,{m_{ {12}}}^{2}m_{{22}}{m_{{36}}}^{2}m_{{14}}\\&\quad +{m_{{12}}}^{2}m_{{22}}{m_{{36 }}}^{2}m_{{11}}+{m_{{12}}}^{2}m_{{22}}{m_{{36}}}^{2}m_{{44}}+2\,{m_{{ 13}}}^{2}m_{{22}}m_{{33}}m_{{14}}m_{{55}} \\&\quad -{m_{{13}}}^{2}m_{{22}}m_{{33 }}m_{{11}}m_{{55}}\\&\quad -{m_{{13}}}^{2}m_{{22}}m_{{33}}m_{{44}}m_{{55}}-{m_{ {25}}}^{2}{m_{{36}}}^{2}{m_{{14}}}^{2} \\&\quad -4\,m_{{12}}m_{{22}}m_{{13}}m_{{ 44}}m_{{23}}m_{{36}}\\&\quad +8\,m_{{12}}m_{{22}}m_{{13}}m_{{14}}m_{{23}}m_{{36}}-4\,m_{{12}}m_{{ 22}}m_{{13}}m_{{14}}m_{{23}}m_{{66}} \\&\quad -4\,m_{{12}}m_{{22}}m_{{33}}m_{{14 }}m_{{23}}m_{{13}}\\&\quad +2\,m_{{12}}m_{{22}}m_{{33}}m_{{11}}m_{{23}}m_{{13}} -4\,m_{{12}}m_{{33}}m_{{44}}m_{{25}}m_{{23}}m_{{13}} \\&\quad +{m_{{23}}}^{2}{m_ {{14}}}^{2}m_{{55}}m_{{66}}\\&\quad -{m_{{23}}}^{2}m_{{11}}m_{{44}}m_{{55}}m_{{66}}+2\,{m_{{23}}}^{2}m _{{11}}m_{{44}}m_{{25}}m_{{66}} \\&\quad +2\,{m_{{23}}}^{2}m_{{36}}m_{{44}}m_{{ 55}}m_{{11}}\\&\quad -{m_{{13 }}}^{2}m_{{22}}m_{{44}}m_{{55}}m_{{66}}+2\,{m_{{13}}}^{2}m_{{22}}m_{{ 14}}m_{{55}}m_{{66}} \\&\quad -{m_{{13}}}^{2}m_{{22}}m_{{11}}m_{{55}}m_{{66}} \\&\quad +m_{{22}}m_{{11}}m_{{ 33}}m_{{44}}m_{{55}}m_{{66}}-2\,{m_{{23}}}^{2}{m_{{14}}}^{2}m_{{25}}m_ {{66}}-2\,{m_{{23}}}^{2}m_{{36}}{m_{{14}}}^{2}m_{{55}} \\&\quad +4\,{m_{{23}}}^{ 2}m_{{36}}{m_{{14}}}^{2}m_{{25}}\\&\quad + {m_{{14}}}^{2}m_{{22}}{m_{{36}}}^{2}m _{{55}}-2\,m_{{25}}m_{{33}}{m_{{14}}}^{2}{m_{{23}}}^{2}-2\,{m_{{25}}}^ {2}{m_{{13}}}^{2}m_{{44}}m_{{36}} \\&\quad +{m_{{25}}}^{2}{m_{{13}}}^{2}m_{{44}} m_{{66}}\\&\quad +4\,{m_{{25}}}^{2}{m_{{13}}}^{2}m_{{14}}m_{{36}}+2\,m_{{12}}m_ {{33}}m_{{11}}m_{{55}}m_{{23}}m_{{13}} \\&\quad -4\,m_{{12}}m_{{33}}m_{{11}}m_{{ 25}}m_{{23}}m_{{13}}\\&\quad +2\,m_{{23}}m_{{13}}m_{{11}}m_{{55}}m_{{12}}m_{{66 }}-4\,m_{{23}}m_{{13}}m_{{11}}m_{{25}}m_{{12}}m_{{66}} \\&\quad -4\,m_{{36}}m_{{ 12}}m_{{23}}m_{{13}}m_{{11}}m_{{55}}\\&\quad +8\,m_{{36}}m_{{12}}m_{{23}}m_{{13 }}m_{{11}}m_{{25}}-4\,{m_{{23}}}^{2}m_{{36}}m_{{44}}m_{{25}}m_{{11}} \\&\quad -m_{{22}}m_{{11}}{m_{{36}}}^{2}m_{{44}}m_{{55}} \\&\quad +2\,m_{{22}}{m_{{23}}}^{2}m_{{11}}m_{{44}}m_{{36}}+2\,m_{{12}}m_{{22}}m_{{13}}m_{{44}}m_{{23}}m_{{66 }} \\&\quad -4\,{m_{{12}}}^{2}m_{{33}}m_{{14}}m_{{25}}m_{{66}}\\&\quad +m_{{22}}{m_{{23}}}^{2}{m_{{14}}}^{2}m_{{66}}. \end{aligned}$$

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Ondoua, R.Y., Mimshe Fewu, J.C., Belobo Belobo, D. et al. Excitons dynamic in a three-stranded \(\alpha \)-helix protein chains with diagonal and off-diagonal couplings: effects of strong long-range interactions. Eur. Phys. J. Plus 136, 274 (2021). https://doi.org/10.1140/epjp/s13360-021-01279-1

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  • DOI: https://doi.org/10.1140/epjp/s13360-021-01279-1