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Energy transport mechanism in the form of proton soliton in a one-dimensional hydrogen-bonded polypeptide chain

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Abstract

The dynamics of protons in a one-dimensional hydrogen-bonded (HB) polypeptide chain (PC) is investigated theoretically. A new Hamiltonian is formulated with the inclusion of higher-order molecular interactions between peptide groups (PGs). The wave function of the excitation state of a single particle is replaced by a new wave function of a two-quanta quasi-coherent state. The dynamics is governed by a higher-order nonlinear Schrödinger equation and the energy transport is performed by the proton soliton. A nonlinear multiple-scale perturbation analysis has been performed and the evolution of soliton parameters such as velocity and amplitude is explored numerically. The proton soliton is thermally stable and very robust against these perturbations. The energy transport by the proton soliton is more appropriate to understand the mechanism of energy transfer in biological processes such as muscle contraction, DNA replication, and neuro-electric pulse transfer on biomembranes.

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Acknowledgments

L.K. gratefully acknowledges financial support from UGC, India, in the form of a Research Award, NBHM, India, in the form of a major research project, DAE-BRNS, India, in the form of a Young Scientist Research Award, and ICTP, Italy, for providing support under regular associateship scheme. N.A. acknowledges financial support from Central University of Tamilnadu.

Author information

Authors and Affiliations

  1. Department of Physics, School of Basic and Applied Sciences, Central University of Tamilnadu, Thiruvarur, 610 101, Tamilnadu, India

    L. Kavitha & N. Ayyappan

  2. The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

    L. Kavitha

  3. Department of Physics, Periyar University, Salem, 636 011, Tamilnadu, India

    R. Priya & S. Jayanthi

  4. Center for Nanoscience and Nanotechnology, Periyar University, Salem, 636 011, Tamilnadu, India

    D. Gopi

  5. Department of Chemistry, Periyar University, Salem, 636 011, Tamilnadu, India

    D. Gopi

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  1. L. Kavitha
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  3. N. Ayyappan
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Correspondence to L. Kavitha.

Appendix

Appendix

$$\begin{array}{@{}rcl@{}} \hat{\phi}_{1}&=&\delta_{1}{S}{\Delta} T{\Delta}+\frac{\delta_{2}}{2\eta}\left( 3{\Delta} {S}{\Delta} T{\Delta}+T{\Delta}\sinh{\Delta}\right)-\frac{\delta_{2}}{\eta}{S}{\Delta} -\alpha\eta^{5} \left[ \left( \frac{49}{10}\right){S}{\Delta} T^{2}{\Delta}-\left( \frac{14}{5}\right)\cosh(2{\Delta}){S}^{3}\right.\\ &&\times{\Delta} T^{4}{\Delta} - \left( \frac{56}{5} \right){S}^{3}{\Delta} T^{4}{\Delta} + \left( \frac{21}{2}\right){\Delta} {S}^{7}{\Delta} T{\Delta} - \left( \frac{63}{4}\right){\Delta} {S}^{5}{\Delta} T{\Delta} -\left( \frac{21}{10}\right){S}^{5}{\Delta} T{\Delta} + \left( \frac{49}{20} \right){S}^{3}{\Delta} T^{2}{\Delta}\\ &&+ \left( \frac{49}{10} \right){S}{\Delta} T{\Delta} + \left( \frac{21}{5} \right){S}{\Delta} T^{6}+\left( \frac{21}{5}\right){S}{\Delta} T{\Delta}{\Delta}- \left( \frac{36}{35}\right)\cosh(2{\Delta}){S}^{3}{\Delta} T^{6}{\Delta} + \left( \frac{216}{35}\right){S}^{3}{\Delta} T^{6}{\Delta} \\ && -\left( \frac{87}{35}\right){S}{\Delta} T^{2}{\Delta}+ \left( \frac{27}{4}\right){\Delta} {S}^{9}{\Delta} T{\Delta} - 18{\Delta} {S}^{7}{\Delta} T{\Delta} + \left( \frac{27}{2} \right){\Delta} {S}^{5}{\Delta} T{\Delta} - \left( \frac{27}{28} \right){S}^{7}{\Delta} T^{2}{\Delta} + \left( \frac{171}{70} \right)\\ && {S}^{5}{\Delta} T^{2}{\Delta} -\left( \frac{87}{70}\right){S}^{3}{\Delta} T^{2} {\Delta}- \left( \frac{87}{35}\right){S}{\Delta} T{\Delta} - \left( \frac{18}{7}\right){S}{\Delta} T^{8}{\Delta} - \left( \frac{18}{7} \right){S}{\Delta} T{\Delta} + \left( \frac{5}{2} \right){S}^{3}{\Delta} T^{2}{\Delta} \\ && +10{S}{\Delta} T{\Delta}-\left( \frac{15}{8} \right){S}{\Delta} T^{2}{\Delta}+\left( \frac{45}{16} \right){\Delta} {S}^{5}{\Delta} T{\Delta} - \left( \frac{45}{8} \right){S}^{3}{\Delta} T^{2}{\Delta} - \left( \frac{15}{8} \right){S}{\Delta} T{\Delta} - \left( \frac{5}{4} \right){S}{\Delta} T^{4}{\Delta} \\ && - \left( \frac{5}{4} \right){S}{\Delta} T{\Delta} + \left( \frac{88}{21}\right){S}{\Delta} T^{2}{\Delta} - \left( \frac{55}{7} \right){S}^{7}{\Delta} T^{2}{\Delta} -\left( \frac{55}{7} \right){S}^{7}{\Delta} T^{2}{\Delta}+\left( \frac{11}{7} \right){S}^{5}{\Delta} T^{2}{\Delta}+\left( \frac{44}{21}\right){S}^{3}{\Delta}\\ && T^{2}{\Delta} +\left( \frac{88}{21}\right){S}{\Delta} T{\Delta}+ \left( \frac{44}{21}\right){\Delta} {S}{\Delta} T^{2}{\Delta}+\left( \frac{55}{48} \right){S}^{9}{\Delta} T{\Delta} -\left( \frac{275}{126} \right){S}^{7}{\Delta} T{\Delta}+\left( \frac{11}{56} \right){S}^{5}{\Delta} T{\Delta} + \left( \frac{11}{21} \right)\\ &&{S}^{3}{\Delta} T{\Delta} +\left( \frac{11}{14} \right){\Delta} {S}^{5}{\Delta} T^{2}{\Delta} + \left( \frac{11}{14}\right)\cosh(2{\Delta}){S}^{3}{\Delta} T^{6}{\Delta}+\left( \frac{22}{21}\right){\Delta}{S}^{3}{\Delta} T^{2}{\Delta} +\left( \frac{11}{14} \right){S}^{3}{\Delta} T^{6}{\Delta} \\ && +\left( \frac{36}{35}\right){S}^{3}{\Delta} T^{6}{\Delta}+\left( \frac{6}{35} \right) \cosh(2{\Delta}){S}^{3}{\Delta} T^{6}{\Delta} - \left( \frac{29}{70} \right){S}{\Delta} T^{2}{\Delta} - 3{\Delta} {S}^{7}{\Delta} T{\Delta} +\left( \frac{9}{8} \right){\Delta}{S}^{9}{\Delta} T{\Delta}\\ && +\left( \frac{9}{4} \right){\Delta} {S}^{5}{\Delta} T{\Delta} - \left( \frac{9}{56} \right){S}^{7}{\Delta} T^{2}{\Delta} +\left( \frac{57}{140}\right){S}^{5}{\Delta} T^{2}{\Delta} - \left( \frac{29}{140} \right){S}^{3}{\Delta} T^{2}{\Delta} - \left( \frac{29}{70} \right){S}{\Delta} T{\Delta} \\ \end{array} $$
$$\begin{array}{@{}rcl@{}} && -\left( \frac{3}{7}\right){S}{\Delta} T^{8}{\Delta} -\left( \frac{3}{7} \right){S}{\Delta} T{\Delta} - \left( \frac{16}{7} \right){S}{\Delta} T^{2}{\Delta} +\left( \frac{30}{7}\right){S}^{7}{\Delta} T^{2}{\Delta} - \left( \frac{6}{7} \right){S}^{5}{\Delta} T^{2}{\Delta} - \left( \frac{8}{7} \right){S}^{3}{\Delta} T^{2}{\Delta} \\ && -\left( \frac{16}{7} \right){S}{\Delta} T{\Delta} +\left( \frac{10}{7} \right){S}{\Delta} T^{2}{\Delta} + \left( \frac{45}{8} \right){\Delta} {S}^{9}{\Delta} T{\Delta} -\left( \frac{15}{2} \right){\Delta} {S}^{7}{\Delta} T{\Delta} - \left( \frac{45}{56} \right){S}^{7}{\Delta} T^{2}{\Delta} - \left( \frac{10}{7} \right) \\ &&{S}{\Delta} T^{2}{\Delta} + \left( \frac{45}{8} \right){\Delta} {S}^{9}{\Delta} T{\Delta} - \left( \frac{15}{2} \right){\Delta} {S}^{7}{\Delta} T{\Delta} - \left( \frac{45}{56} \right){S}^{7}{\Delta} T^{2}{\Delta} +\left( \frac{15}{28}\right){S}^{5}{\Delta} T^{2}{\Delta} + \left( \frac{5}{7} \right){S}^{3}{\Delta} T^{2}{\Delta} \\ && + \left( \frac{10}{7}\right){S}{\Delta} T{\Delta} +\left( \frac{3}{7}\right)\cosh(2{\Delta}){S}^{3}{\Delta} T^{6}{\Delta} + \left( \frac{18}{7} \right){S}^{3}{\Delta} T^{6}{\Delta} +\left( \frac{184}{35}\right){\Delta} T^{2}{\Delta} -\left( \frac{69}{7} \right){S}^{7}{\Delta} T^{2}{\Delta}\\ && +\left( \frac{69}{35} \right){S}^{5}{\Delta} T^{2}{\Delta} +\left( \frac{92}{35} \right){S}^{3}{\Delta} T^{2}{\Delta} + \left( \frac{184}{105}\right){S}{\Delta} T{\Delta} - \left( \frac{23}{7}\right){S}{\Delta} T^{2}{\Delta} -\left( \frac{207}{4} \right){\Delta} {S}^{9}{\Delta} T{\Delta} \\ && +\left( \frac{69}{4} \right){\Delta} {S}^{7}{\Delta} T{\Delta}+\left( \frac{207}{112} \right){S}^{7}{\Delta} T^{2}{\Delta} -\left( \frac{69}{56} \right){S}^{5}{\Delta} T^{2}{\Delta} - \left( \frac{69}{42} \right){S}^{3}{\Delta} T^{2}{\Delta} - \left( \frac{23}{7} \right){S}{\Delta} T{\Delta} \\ && -\left( \frac{69}{70} \right)\cosh(2{\Delta}){S}^{3}{\Delta} T^{6}{\Delta} + \left( \frac{9}{70} \right)\cosh(2{\Delta}){S}^{3}{\Delta} T^{6}{\Delta} -\left( \frac{207}{35} \right){S}^{3}{\Delta} T^{6}{\Delta} + \left( \frac{27}{35} \right){S}^{3}{\Delta} T^{6}{\Delta} \\&& - \left( \frac{87}{280} \right){S}{\Delta} T^{2}{\Delta} +\left( \frac{27}{32} \right){\Delta} {S}^{9}{\Delta} T{\Delta} -\left( \frac{9}{4}\right){\Delta} {S}^{7}{\Delta} T{\Delta}+\left( \frac{27}{16} \right){\Delta} {S}^{5}{\Delta} T{\Delta} -\left( \frac{27}{224} \right){S}^{7}{\Delta} T^{2}{\Delta} \\ &&\left. +\left( \frac{1071}{560} \right){S}^{5}{\Delta} T^{2}{\Delta}-\left( \frac{1071}{1680} \right){S}{\Delta} T{\Delta} -\left( \frac{87}{280} \right){S}{\Delta} T{\Delta} -\left( \frac{9}{28} \right){S}{\Delta} T^{8}{\Delta}-\left( \frac{9}{28} \right){S}{\Delta} T{\Delta} \right] \\ && -\left( \frac{b_{3}\varepsilon^{4}}{12}-1\right)\eta^{3} \left[ \left( \frac{28}{3}\right){S}{\Delta} T^{4}{\Delta} + \left( \frac{28}{3} \right){S}{\Delta} T{\Delta} - 14{S}{\Delta} T^{2}{\Delta} -\left( \frac{21}{2} \right){\Delta} {S}^{5}{\Delta} T{\Delta} +21{\Delta} {S}^{3}{\Delta} T{\Delta}\right.\\ && + \left( \frac{14}{4} \right){S}^{3}{\Delta} T{\Delta} - 14{S}{\Delta} T{\Delta} - 14{\Delta} {S}{\Delta} T{\Delta} + \left( \frac{56}{3} \right){S}{\Delta} T^{2}{\Delta} - \left( \frac{14}{3} \right){S}^{3}{\Delta} T^{2}{\Delta} + \left( \frac{56}{3} \right){S}{\Delta} T{\Delta} \\ && +6{S}{\Delta} T^{5}{\Delta} + 6{S}{\Delta} T{\Delta} + 23{S}{\Delta} T^{2}{\Delta} - 6{\Delta} {S}^{7}{\Delta} T{\Delta} +18{\Delta}{S}^{5}{\Delta} T{\Delta} - 18{\Delta} {S}^{3}{\Delta} T{\Delta} + \left( \frac{6}{5} \right){S}^{5}{\Delta} T{\Delta} \\ && - \left( \frac{33}{5}\right){S}^{3}{\Delta} T^{2}{\Delta} + \left( \frac{46}{5}\right){S}{\Delta} T{\Delta} + 12{\Delta} {S}{\Delta} T{\Delta} - \left( \frac{52}{5} \right){S}{\Delta} T^{2}{\Delta} - \left( \frac{12}{5} \right){S}^{5}{\Delta} T{\Delta} + \left( \frac{44}{5} \right){S}^{3}{\Delta} T^{2}{\Delta}\\ && -\left( \frac{92}{5} \right){S}{\Delta} T{\Delta} -5{S}{\Delta} T^{2}{\Delta} - 5{S}{\Delta} T{\Delta} + \left( \frac{15}{4} \right){S}{\Delta} T^{2}{\Delta} - \left( \frac{15}{4} \right){\Delta} {S}^{3}{\Delta} T{\Delta} + \left( \frac{15}{2} \right){S}{\Delta} T{\Delta} + \left( \frac{5}{2}\right) \\ &&\left.{\Delta} {S}{\Delta} T{\Delta}- \left( \frac{5}{2}\right){S}{\Delta} T^{2}{\Delta} - \left( \frac{5}{2} \right){S}{\Delta} T{\Delta}\right] -\eta\left( 6\xi^{2}+b_{3}\varepsilon^{2}-\frac{b_{3}\varepsilon^{4}\xi^{2}}{2}\right) \left[ \left( \frac{1}{3} \right){S}{\Delta} T^{4}{\Delta} +\left( \frac{1}{3} \right){S}{\Delta} T{\Delta} \right.\\ &&-\left( \frac{1}{2}\right){S}{\Delta} T^{2}{\Delta} - \left( \frac{3}{8} \right){\Delta} {S}^{5}{\Delta} T{\Delta} + \left( \frac{3}{4} \right){\Delta} {S}^{3}{\Delta} T{\Delta} +\left( \frac{1}{8}\right){S}^{3}{\Delta} T{\Delta} - \left( \frac{1}{2} \right){S}{\Delta} T{\Delta} - \left( \frac{1}{2} \right){\Delta} {S}{\Delta} T{\Delta}\\ &&\left. +\left( \frac{2}{3} \right){S}{\Delta} T^{2}{\Delta} - \left( \frac{1}{6} \right){S}^{3}{\Delta} T^{2}{\Delta} - \left( \frac{2}{3}\right){S}{\Delta} T{\Delta} \right] -\left( 6\xi^{2}+b_{3}\varepsilon^{2}-\frac{b_{3}\varepsilon^{4}\xi^{2}}{2}\right)\frac{1}{\eta} \left[ \left( -\frac{1}{2}\right){S}^{3}{\Delta} T^{2}{\Delta}\right.\\ && -\left( \frac{2}{3}\right){S}{\Delta} T^{2}{\Delta} -\left( \frac{2}{3}\right){S}{\Delta} T{\Delta} + \left( \frac{1}{4}\right){S}{\Delta} T^{2}{\Delta} - \left( \frac{3}{8}\right){\Delta} {S}^{5}{\Delta} T{\Delta} +\left( \frac{1}{8}\right){S}^{3}{\Delta} T^{2}{\Delta} +\left( \frac{1}{4}\right){S}{\Delta} T{\Delta}\\ &&\left. +\left( \frac{1}{3}\right){S}{\Delta} T^{4}{\Delta} +\left( \frac{1}{3}\right){S}{\Delta} T{\Delta} \right] - \eta\left( 12\alpha\xi^{2}-20\right) \left[\left( \frac{1}{15}\right)\cos(2{\Delta}){S}^{3}{\Delta} T^{4}{\Delta} +\left( \frac{4}{15}\right){S}^{3}{\Delta} T^{4}{\Delta} \right.\\ && - \left( \frac{7}{60}\right){S}{\Delta} T^{2}{\Delta} - \left( \frac{1}{4}\right){\Delta} {S}^{7}{\Delta} T{\Delta} +\left( \frac{3}{8}\right){\Delta} {S}^{5}{\Delta} T{\Delta}+\left( \frac{1}{60}\right){S}^{5}{\Delta} T{\Delta} -\left( \frac{7}{120}\right){S}^{3}{\Delta} T^{2}{\Delta} \\ && -\left( \frac{7}{60}\right){S}{\Delta} T{\Delta} - \left( \frac{1}{10}\right){S}{\Delta} T^{6}{\Delta} - \left( \frac{1}{10}\right){S}{\Delta} T{\Delta} -\left( \frac{8}{15}\right){S}{\Delta} T{\Delta} -\left( \frac{1}{5}\right){S}^{5}{\Delta} T^{2}{\Delta} - \left( \frac{4}{15}\right){S}^{3}{\Delta} T^{2}{\Delta} \\ && \left.-\left( \frac{8}{15}\right){S}{\Delta} T{\Delta} \right] - \left( 6\alpha\xi^{2}-10\right) \left[\left( \frac{4}{15}\right){S}{\Delta} T^{2}{\Delta} -\left( \frac{1}{2}\right){\Delta} {S}^{7}{\Delta} T{\Delta}+\left( \frac{1}{10}\right){S}^{5}{\Delta} T^{2}{\Delta} + \left( \frac{2}{15}\right){S}^{3}{\Delta} T^{2}{\Delta} \right.\\ && \left.+ \left( \frac{4}{15}\right){S}{\Delta} T{\Delta} + \left( \frac{1}{15}\right)\cos(2{\Delta}){S}^{3}{\Delta} T^{4}{\Delta} + \left( \frac{4}{5}\right){S}^{3}{\Delta} T^{4}{\Delta}\right] -\alpha\eta^{2} \left[ \left( \frac{176}{7}\right){S}{\Delta} T^{2}{\Delta} + \left( \frac{11}{7}\right){S}^{7}{\Delta} T^{2}{\Delta} \right.\\ && + \left( \frac{66}{7}\right){S}^{5}{\Delta} T^{2}{\Delta} +\left( \frac{88}{7}\right){S}^{3}{\Delta} T^{2}{\Delta} + \left( \frac{176}{7}\right){S}{\Delta} T{\Delta} - \left( \frac{33}{7}\right){S}{\Delta} T^{2}{\Delta} +\left( \frac{165}{16}\right){\Delta} {S}^{9}{\Delta} T{\Delta}-\left( \frac{165}{112}\right) \end{array} $$
$$\begin{array}{@{}rcl@{}} &&{S}^{7}{\Delta} T^{2}{\Delta} - \left( \frac{99}{56}\right){S}^{5} {\Delta} T^{2}{\Delta} -\left( \frac{33}{14}\right){S}^{3}{\Delta} T^{2}{\Delta} - \left( \frac{33}{7}\right){S}{\Delta} T{\Delta}- \left( \frac{44}{21}\right){S}{\Delta} T^{2}{\Delta} + \left( \frac{55}{14}\right){S}^{7}{\Delta} T^{2}{\Delta} \\ && - \left( \frac{11}{14}\right){S}^{5}{\Delta} T{\Delta} - \left( \frac{22}{21}\right){S}^{3}{\Delta} T^{2}{\Delta} - \left( \frac{44}{21}\right){S}{\Delta} T{\Delta} + \left( \frac{24}{35}\right){S}{\Delta} T^{2}{\Delta}- \left( \frac{9}{7}\right){S}^{7}{\Delta} T^{2}{\Delta} + \left( \frac{9}{35}\right){S}^{5}\\ && {\Delta} T^{2}{\Delta}+ \left( \frac{24}{35}\right){S}{\Delta} T{\Delta} + \left( \frac{3}{7}\right){S}{\Delta} T^{2}{\Delta} +\left( \frac{27}{16}\right){\Delta} {S}^{9}{\Delta} T{\Delta} - \left( \frac{9}{4}\right){\Delta} {S}^{7}{\Delta} T{\Delta} - \left( \frac{27}{112}\right){S}^{7}{\Delta} T^{2}{\Delta} \\ &&\left. + \left( \frac{9}{56}\right){S}^{5}{\Delta} T^{2}{\Delta} + \left( \frac{3}{14}\right){S}^{3}{\Delta} T^{2}{\Delta} + \left( \frac{3}{7}\right){S}{\Delta} T{\Delta} +\left( \frac{9}{70}\right)\cos(2{\Delta}){S}^{3}{\Delta} T^{6}{\Delta}+\left( \frac{27}{35}\right){S}^{3}{\Delta} T^{6}{\Delta} \right] \\ && -\eta^{2} \left[\left( \frac{5}{2}\right){S}^{5}{\Delta} T{\Delta} +\left( \frac{5}{2}\right){S}{\Delta} T{\Delta}+ 2{\Delta} {S}{\Delta} T^{2}{\Delta} -\left( \frac{3}{4}\right){S}^{5}{\Delta} T{\Delta} + \left( \frac{1}{2}\right){S}^{3}{\Delta} T{\Delta} - 2{\log}\left( \cosh{\Delta}\right){S}{\Delta} \right.\\ &&\left. T{\Delta}- 3{\Delta} {S}^{5}{\Delta} T^{2}{\Delta} - {\Delta} {S}^{3}{\Delta} T^{2}{\Delta} + \left( \frac{5}{4}\right){S}{\Delta} T^{5}{\Delta} +\left( \frac{5}{4}\right){S}{\Delta} T{\Delta} \right] -\eta^{3}\left( 12\alpha\xi^{2}-30\right) \left[\left( \frac{1}{15}\right)\cos(2{\Delta})\right.\\ &&{S}^{3}{\Delta} T^{4}{\Delta}+\left( \frac{4}{15}\right){S}^{3}{\Delta} T^{4}{\Delta} + \left( \frac{7}{60}\right){S}{\Delta} T^{2}{\Delta} + \left( \frac{1}{4}\right){S}^{7}{\Delta} T{\Delta} -\left( \frac{3}{8}\right){\Delta} {S}^{5}{\Delta} T{\Delta} - \left( \frac{1}{20}\right){S}^{5}{\Delta} T^{2}{\Delta} \\ &&\left. + \left( \frac{7}{120}\right){S}^{3}{\Delta} T^{2}{\Delta} +\left( \frac{7}{60}\right){S}{\Delta} T{\Delta} + \left( \frac{1}{10}\right){S}{\Delta} T^{6}{\Delta} + \left( \frac{1}{10}\right){S}{\Delta} T{\Delta}\right] -\frac{\alpha}{\eta} \left[\left( \frac{72}{35}\right){S}{\Delta} T^{2}{\Delta} + \left( \frac{9}{14}\right){S}^{7} \right.\\ &&{\Delta} T^{2}{\Delta}+\left( \frac{27}{35}\right){S}^{5}{\Delta} T^{2}{\Delta} +\left( \frac{36}{35}\right){S}^{3}{\Delta} T^{2}{\Delta} + \left( \frac{72}{35}\right){S}{\Delta} T{\Delta} - \left( \frac{27}{70}\right){S}{\Delta} T^{2}{\Delta} +\left( \frac{27}{32}\right){\Delta} {S}^{9}{\Delta} T{\Delta} \\ && - \left( \frac{27}{224}\right){S}^{7}{\Delta} T^{2}{\Delta} - \left( \frac{81}{560}\right){S}^{5}{\Delta} T^{2}{\Delta} - \left( \frac{27}{140}\right){S}^{3}{\Delta} T^{2}{\Delta} - \left( \frac{27}{70}\right){S}{\Delta} T{\Delta} - \left( \frac{12}{70}\right){S}{\Delta} T^{2}{\Delta} \\ && \left.+ \left( \frac{9}{28}\right){S}^{7}{\Delta} T^{2}{\Delta} - \left( \frac{9}{140}\right){S}^{5}{\Delta} T^{2}{\Delta} - \left( \frac{3}{35}\right){S}^{3}{\Delta} T^{2}{\Delta} - \left( \frac{6}{35}\right){S}{\Delta} T{\Delta} \right] -\eta\left( 20\xi^{2}-\frac{7}{2}\alpha\xi^{4}-24b_{2} \right.\\ &&\left.+\frac{48}{\varepsilon^{2}}\right)\left[\left( \frac{1}{3}\right){S}^{3}{\Delta} T^{2}{\Delta} + \left( \frac{2}{3}\right){S}{\Delta} T^{2}{\Delta} + \left( \frac{2}{3}\right){S}{\Delta} T{\Delta} + \left( \frac{1}{4}\right){S}{\Delta} T^{2}{\Delta} -\left( \frac{3}{8}\right) {\Delta} {S}^{5}{\Delta} T{\Delta} + 8{S}^{3}{\Delta} T^{2}{\Delta}\right.\\ &&\left.-\frac{12\omega_{0}}{\varepsilon^{2}} \right) \left( {S}{\Delta} T^{2}{\Delta}+{S}{\Delta} T{\Delta} \right)+ \left( \frac{3}{2}\xi_{T}({\Theta}-{\Theta}_{0}) -3b_{1}+\frac{3b_{3}\varepsilon^{2}\xi^{2}}{2}+\frac{b_{3}\varepsilon^{4}\xi^{4}}{8} +\frac{18\omega_{0}}{\varepsilon^{4}} \right) \\ &&\left.\left( {\Delta} {S}{\Delta} T{\Delta} - {S}{\Delta} T^{2}{\Delta} - {S}{\Delta} T{\Delta} \right) \right] - \xi^{4}\eta^{2} \left[ \left( \frac{3}{4}\right){S}{\Delta} T{\Delta} Tan^{-1}\left( T(\frac{\Delta}{2})\right) + \left( \frac{1}{4}\right) {S}^{4}{\Delta} T^{2}{\Delta}\right.\\ && + \left( \frac{3}{8}\right){S}^{2}{\Delta} T^{2}{\Delta} + \left( \frac{3}{4}\right){S}{\Delta} T{\Delta} Tan^{-1}\left( \frac{1}{2}\right) + \left( \frac{3}{10}\right){\Delta} {S}^{6}{\Delta} T{\Delta} - \left( \frac{9}{40}\right){S}{\Delta} T{\Delta} Tan^{-1}\left( T\left( \frac{\Delta}{2}\right)\right)\\ && -\left( \frac{3}{40}\right){S}^{4}{\Delta} T^{2}{\Delta} - \left( \frac{9}{80}\right){S}^{2}{\Delta} T^{2}{\Delta} - \left( \frac{9}{40}\right){S}{\Delta} T{\Delta} Tan^{-1}\left( \frac{1}{2}\right) - \left( \frac{1}{8}\right){S}{\Delta} T{\Delta} Tan^{-1}\left( T\left( \frac{\Delta}{2}\right)\right)\\ &&\left. + \left( \frac{1}{8}\right){S}^{4}{\Delta} T^{2}{\Delta} - \left( \frac{1}{16}\right){S}^{2}{\Delta} T^{2}{\Delta} - \left( \frac{1}{8}\right){S}{\Delta} T{\Delta} Tan^{-1}\left( \frac{1}{2}\right) \right] - \left( 40\alpha\xi^{2}-60\right)\eta^{3} \left[ \left( \frac{8}{15}\right){S}{\Delta} T^{2}{\Delta}\right.\\ && + \left( \frac{1}{5}\right){S}^{5}{\Delta} T^{2}{\Delta} + \left( \frac{4}{15}\right){S}^{3}{\Delta} T^{2}{\Delta} + \left( \frac{8}{15}\right){S}{\Delta} T{\Delta} + \left( \frac{2}{15}\right){S}{\Delta} T^{2}{\Delta} - \left( \frac{1}{4}\right){\Delta} {S}^{7}{\Delta} T{\Delta}\\ && \left.+ \left( \frac{1}{20}\right){S}^{5}{\Delta} T^{2}{\Delta} + \left( \frac{1}{15}\right){S}^{3}{\Delta} T^{2}{\Delta} + \left( \frac{2}{15}\right){S}{\Delta} T{\Delta} + \left( \frac{2}{15}\right){S}^{3}{\Delta} T^{4}{\Delta} + \left( \frac{1}{30}\right)\cos(2{\Delta}){S}^{3}{\Delta} T^{4}{\Delta} \right]\\ && -\alpha\xi^{4} \left[ \left( \frac{1}{4}\right){S}^{2}{\Delta} T^{2}{\Delta} + \left( \frac{1}{4}\right){\Delta} {S}^{4}{\Delta} T{\Delta} \right] + \left[\frac{161}{6}\alpha\eta^{5} +\frac{35}{3}\eta^{3}+\frac{20}{3}\alpha\xi^{2}\eta^{3}+2\alpha\xi^{2}-\frac{10}{3} \right] {S}^{7}{\Delta}\\ && - \left[-\frac{139}{4}\alpha\eta^{5}+\frac{5}{2}\eta^{3}+\frac{15}{4}\eta^{5} - \left( \frac{3}{2}\xi^{2}+\frac{b_{3}\varepsilon^{2}}{4}-\frac{b_{3}\varepsilon^{4}\xi^{2}}{8} \right)\frac{1}{\eta^{2}} -\left( 5\xi^{2}-\frac{7}{8}\alpha\eta^{4} -6b_{2}+\frac{12}{\varepsilon^{2}} \right)\eta\right] {S}^{5}{\Delta}\\ && - \left[\frac{103}{2}\alpha\eta^{5} +\frac{55}{8}\alpha\eta^{2} +\frac{9\alpha}{16\eta} \right]{S}^{9}{\Delta} + \left[\frac{3\xi^{2}}{2}+\frac{b_{3}\varepsilon^{2}}{4} -\frac{b_{3}\varepsilon^{4}\xi^{2}}{8} \right] \eta {S}{\Delta} T^{4}{\Delta} -\left( \frac{b_{3}\varepsilon^{4}}{3}-4 \right) \eta^{3}{S}{\Delta} T^{6}{\Delta}\\ && - \left( \frac{5}{2}-\frac{5}{24}b_{3}\varepsilon^{4} \right) \eta^{3} {S}^{3}{\Delta} T{\Delta} -\frac{2}{3}\eta^{2}\cos(2{\Delta}){S}^{3}{\Delta} T^{3}{\Delta} - \frac{8}{3}\eta^{2}{S}^{3}{\Delta} T^{3}{\Delta} -\left[{\vphantom{0\frac{2}{4}^{y}}}\xi_{T}({\Theta}-{\Theta}_{0}) -2b_{1}\right. \end{array} $$
$$\begin{array}{@{}rcl@{}} &&\left.+b_{3}\varepsilon^{2}\xi^{2}+\frac{b_{3}\varepsilon^{4}\xi^{4}}{12}+\frac{12\omega_{0}}{\varepsilon^{2}} \right] \eta {S}^{2}{\Delta} -\frac{\xi^{4}\eta^{2}{S}^{6}{\Delta}}{4} -\frac{\alpha\varepsilon^{4}{S}^{4}{\Delta}}{6} + \left( \frac{7b_{3}\varepsilon^{4}}{12} -7 \right)\frac{1}{\eta^{2}}{S}{\Delta} + \left[6\xi^{2}+b_{3}\varepsilon^{2}\right.\\ &&\left. -\frac{b_{3}\varepsilon^{4}\xi^{2}}{2} \right]\frac{\eta^{2}}{4}{S}{\Delta} - \left( \frac{b_{3}\varepsilon^{4}\eta^{5}}{3}-4\eta^{5} \right){S}{\Delta} + \left[\frac{75}{4}\alpha\eta^{5}-\frac{25}{2}\eta^{3}-3\alpha\xi^{2}-10\alpha\xi^{2}\eta^{3}+5 \right]{\Delta} {S}^{7}{\Delta} T{\Delta}\\ && +\left[\frac{15}{4}\eta^{3}+\frac{45}{8}\eta^{5}-\frac{849}{16}\alpha\eta^{5}-\frac{\xi^{2}}{\eta^{2}}-\frac{3}{8}b_{3}\varepsilon^{2}-\frac{3}{16}\varepsilon^{4}\xi^{2}-\frac{15\xi^{2}}{2\eta} -\frac{21\alpha\eta\xi^{4}}{16} -9b_{2}\eta+\frac{18\eta}{\varepsilon^{2}} \right] {\Delta} {S}^{5}{\Delta} T{\Delta}\\ && + \left[\frac{165}{16}\alpha\eta^{2} +\frac{27\alpha}{32\eta}+\frac{309}{2}\alpha\eta^{5} \right]{\Delta} {S}^{9}{\Delta} T{\Delta} -\eta\left[\frac{9}{4}\xi^{2}+\frac{3b_{3}\varepsilon^{2}}{8} -\frac{3b_{3}\varepsilon^{4}\xi^{2}}{16} \right]{\Delta} {S}{\Delta} T^{5}{\Delta} + \left( \frac{b_{3}\varepsilon^{4}}{2}-6 \right)\\ && \eta^{3}{S}{\Delta} T^{7}{\Delta}+\left[\frac{15\eta^{3}}{4}-\frac{5\eta^{3}b_{3}\varepsilon^{4}}{16} \right] {\Delta} {S}^{3}{\Delta} T{\Delta} + 4\eta^{2}{S}^{3}{\Delta} T^{4}{\Delta} +\eta^{2}{\Delta}\cos(2{\Delta}){S}^{3}{\Delta} T^{4}{\Delta} - \frac{3}{2\eta}\left[{\vphantom{0\frac{2}{5}}}\xi_{T} \right.\\ &&\left.({\Theta}-{\Theta}_{0})-2b_{1}+b_{3}\varepsilon^{2}\xi^{2}+\frac{b_{3}\varepsilon^{4}\xi^{4}}{12}+\frac{12\omega_{0}}{\varepsilon^{2}} \right]{\Delta} {S}^{2}{\Delta} T{\Delta} +\frac{3}{10}\xi^{4}\eta^{2}{\Delta} {S}^{6}{\Delta} T{\Delta}+\frac{1}{4}\alpha\varepsilon^{4}{\Delta} {S}^{4}{\Delta}\\ && T{\Delta}+\left[\frac{3}{2\eta}\left( 7-\frac{7}{12}b_{3}\varepsilon^{4} \right)-\frac{3\eta}{8} \left( 6\xi^{2}+b_{3}\varepsilon^{2}-\frac{b_{3}\varepsilon^{4}\xi^{2}}{2} \right) +\frac{3}{2}\left( b_{3}\varepsilon^{4}\eta^{3}-4\eta^{3} \right) \right] {\Delta} {S}{\Delta} T{\Delta}\\ && +\left[-\frac{125}{12}\alpha\eta^{5} - \frac{35}{6}\eta^{3}-\alpha\xi^{2}+\frac{5}{3} -\frac{5}{3}\alpha\xi^{2}\eta^{3} \right] {S}^{6}{\Delta} T{\Delta} s{\Delta}+ \left[\frac{5}{4}\eta^{3} - \frac{41}{2}\alpha\eta^{5} - \left( \frac{3}{4}\xi^{2}+\frac{b_{3}\varepsilon^{2}}{8} \right.\right.\\ &&\left.\left. -\frac{b_{3}\varepsilon^{4}\xi^{2}}{16} \right)\frac{1}{\eta^{2}} - \left( \frac{5}{2}\xi^{2} -\frac{7}{16}\alpha\xi^{4}-3b_{2} +\frac{6}{\varepsilon^{2}} \right)\eta \right]{S}^{4}{\Delta} T{\Delta} s{\Delta} + \left[\frac{103}{2}\alpha\eta^{5}+\frac{55}{16} \alpha\eta^{2} +\frac{9\alpha}{32\eta} \right]\\ && {S}^{8}{\Delta} T{\Delta} s{\Delta}- \eta\left[\frac{3\xi^{2}}{4} +\frac{b_{3}\varepsilon^{2}}{8} -\frac{b_{3}\varepsilon^{4}\xi^{2}}{16} \right]T^{5}{\Delta} s{\Delta} +\left[\frac{b_{3}\varepsilon^{4}}{6}-2 \right] \eta^{3}T^{7}{\Delta} s{\Delta} +\left[\frac{5}{4} -\frac{5b_{3}\varepsilon^{4}}{48} \right]\\ && \eta^{3}{S}^{2}{\Delta} T{\Delta} s{\Delta}+ \frac{4}{3}\eta^{2}{S}^{2}{\Delta} T^{4}{\Delta} s{\Delta} +\frac{\eta^{2}}{3}\cos(2{\Delta}){S}^{2}{\Delta} T^{4}{\Delta} s{\Delta} - \left[\xi_{T}({\Theta}-{\Theta}_{0})-2b_{1} +b_{3}\varepsilon^{2}\xi^{2} \right.\\ &&\left.+\frac{b_{3}\varepsilon^{4}\xi^{4}}{12}+\frac{12\omega_{0}}{\varepsilon^{2}} \right]\frac{1}{2\eta}{S}{\Delta} T{\Delta} s{\Delta} + \frac{\xi^{4}\eta^{2}}{10}{S}^{5}{\Delta} T{\Delta} s{\Delta} + \frac{\alpha}{2}\varepsilon^{4}{S}^{3}{\Delta} T{\Delta} s{\Delta} + \left[\left( 7-\frac{7b_{3}\varepsilon^{4}}{12}\right)\right.\\ &&\left. \frac{1}{2\eta^{2}} + \frac{b_{3}\varepsilon^{4}\eta^{3}}{6}-2\eta^{3} -\frac{\eta}{8}\left( 6\xi^{2}+b_{3}\varepsilon^{2} -\frac{b_{3}\varepsilon^{4}\xi^{2}}{2}\right) \right]T{\Delta} s{\Delta}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{array} $$
(A1)
$$\begin{array}{@{}rcl@{}} \hat{\psi}_{1}&=&\left[\left( \frac{34}{3}\right)\xi\eta^{2}-\left( \frac{14}{3}\right)\alpha\xi\eta^{2}-\left( \frac{118}{45}\right)\alpha\xi\eta^{4}-2\xi^{3}+3\varepsilon^{4}\xi^{3}+\left( \frac{1}{3}\right)b_{3}\varepsilon^{4}\xi\eta^{3} +\left( \frac{5}{18}\right)b_{3} \right.\\ &&\left.\xi\varepsilon^{4}\eta^{2}+\left( \frac{8}{3}\right)\alpha\xi^{3}\eta^{2}-2b_{3}\varepsilon^{2}\xi+\frac{\theta-\theta_{0}}{2} \right]{S}{\Delta} T{\Delta} +\left[\delta_{3}-18\xi\eta^{2}-14\alpha\xi\eta^{2}\right.\\ &&\left.+\left( \frac{3}{2}\right)b_{3}\varepsilon^{4}\xi\eta^{2} +\left( \frac{227}{15}\right)\alpha\xi\eta^{4}+8\alpha\xi^{3}\eta^{2} + \left( \frac{2}{3}\right)b_{3}\varepsilon^{4}\xi^{3}-2\xi^{3}-2b_{3}\varepsilon^{2}\xi +\frac{(\theta-\theta_{0})_{T}}{2} \right] \\ &&{S}{\Delta} + \left[\left( \frac{\delta_{4}}{2\eta}\right) + \left( \frac{1}{3}\right)b_{3}\varepsilon^{4}\xi - \left( \frac{7}{2}\right)\xi\eta^{2}+4\xi^{3}+2b_{3}\varepsilon^{2}\xi -\left( \frac{1}{3}\right)b_{3}\varepsilon^{4}\xi^{3} -\left( \frac{1}{24}\right)b_{3}\varepsilon^{4}\xi\eta^{2}\right. \\ &&\left.{\vphantom{0\frac{2}{4}}}-(\theta-\theta_{0})_{T} \right] {\Delta}~{S}{\Delta} + \left[ \left( \frac{23}{3}\right)\xi\eta^{2}- 3b_{3}\varepsilon^{4}\xi\eta^{2} +\left( \frac{14}{3}\right)\alpha\xi\eta^{2}-2\alpha\xi^{3}\eta^{2}-2\alpha\xi\eta^{4} \right.\\ &&\left.-\left( \frac{5}{24}\right)b_{3}\varepsilon^{4}\eta^{2} \right] {\Delta}~{S}^{5}{\Delta} +\left[2\xi\eta^{2}+2\alpha\xi\eta^{2} - \left( \frac{1}{6}\right)b_{3}\varepsilon^{4}\xi -\xi^{3}-b_{3}\varepsilon^{2} - \left( \frac{1}{2}\right)b_{3}\varepsilon^{4}\xi^{3} \right.\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} &&\left.+\left( \frac{1}{2}\right)b_{3}\varepsilon^{2}\xi-\frac{(\theta-\theta_{0})_{T}}{3} \right] {\Delta}~{S}^{3}{\Delta}+\left[ \left( \frac{-13}{3}\right) \xi\eta^{2} +\left( \frac{1}{3}\right)\alpha\xi\eta^{2}-\left( \frac{5}{36}\right)\varepsilon^{4}\xi\eta^{2} +\left( \frac{4}{3}\right)\right.\\ &&\left.\alpha\xi^{3}\eta^{2}-\left( \frac{59}{45}\right)\alpha\xi\eta^{4} \right] {S}^{3}{\Delta} T{\Delta} +\left[ \left( \frac{-20}{3}\right)\xi\eta^{2}- \left( \frac{5}{9}\right)b_{3}\varepsilon^{4}\xi\eta^{2}+\left( \frac{16}{3}\right)\alpha\xi^{3}\eta^{2} \right] {S}{\Delta} T{\Delta}\\ && -\left( \frac{112}{15}\right)\alpha\xi\eta^{4} {S}^{3}{\Delta} T^{3}{\Delta} \left( \frac{14}{3}\right){\Delta} {S}{\Delta}+ \left( \frac{\eta_{T}}{\eta^{2}}\right)~{\Delta} {S}{\Delta} T{\Delta} - \left( \frac{4}{3}\right)\alpha\xi\eta^{4} \cos(2{\Delta}) {S}\\ &&{\Delta} T^{3}{\Delta}-\frac{4}{5}\alpha\xi\eta^{4} {S}{\Delta} T{\Delta} + \left[ \frac{\delta_{4}}{2\eta} - \left( \frac{1}{2}\right)b_{3}\varepsilon^{4}\xi\eta^{2} +\frac{\xi\eta^{2}}{2} \right] s{\Delta} -\frac{26}{15}\alpha\xi\eta^{4} {S}{\Delta} T{\Delta}\\ &&+2\alpha\xi\eta^{4} {S}{\Delta} +3\alpha\xi\eta^{4} {S}{\Delta} T^{4}{\Delta}+\frac{\eta_{T}}{2\eta^{2}}{\Delta}^{2} {S}{\Delta}+ \left[ \frac{\xi\eta^{2}}{2}-\left( \frac{1}{24}\right)b_{3}\varepsilon^{4}\xi\eta^{2} \right] {\Delta} {S}{\Delta} T^{6}{\Delta}\\ &&-\alpha\xi\eta^{4} {S}^{4}{\Delta} +\left[ \left( \frac{-5}{2}\right) \xi\eta^{2}-\left( \frac{5}{8}\right)b_{3}\varepsilon^{4}\eta^{2}+\alpha\xi\eta^{4} +2\alpha\xi^{3}\eta^{2} \right] {S}^{4}{\Delta} s{\Delta} +\left[\left( \frac{-1}{24}\right)b_{3}\varepsilon^{4} \right.\\ &&\left.\xi\eta^{2}+\left( \frac{1}{2}\right)\xi\eta^{2} \right] \tanh^{6}{\Delta}s{\Delta} -\left( \frac{7}{3}\right)\alpha\xi\eta^{4} {S}{\Delta} s{\Delta} +\left[ \xi^{3}+\left( \frac{1}{2}\right)b_{3}\varepsilon^{2}\xi -\left( \frac{1}{12}\right){S}^{2}{\Delta} s{\Delta}\right]. \end{array} $$
(A2)

where, SΔ=sechΔ, \(\mathrm {T}{\Delta }=\tanh {\Delta }\) and \(\mathrm {s}{\Delta }=\sinh {\Delta }\)

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Kavitha, L., Priya, R., Ayyappan, N. et al. Energy transport mechanism in the form of proton soliton in a one-dimensional hydrogen-bonded polypeptide chain. J Biol Phys 42, 9–31 (2016). https://doi.org/10.1007/s10867-015-9389-9

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