Influence of quadrupole–quadrupole-type interaction on the chaotic dynamics of $\alpha$-helical proteins

Abstract

By proposing a model Hamiltonian in the first quantised form we investigate the chaotic dynamics of $\alpha$-helical proteins by taking into account the quadrupole–quadrupole-type interaction. The dynamics is studied by deriving Hamilton’s equations of motion and by plotting the time-series evolution and phase-space trajectories. Chaotic trajectories are observed in the phase-space plots. The effect of the interaction parameters on the stability of proteins is also discussed.

Appendix

The Jacobian matrix elements are as follows:

$$\begin{aligned} a_{11}&=-\,0.0298p_{x}-6.7708p_{x} x+4.96p_{x} x a,\\ a_{12}&=-\,0.0298p_{y}-6.7708p_{y} x+4.96p_{y} x a, \\ a_{13}&=1.4408+3.38543b+0.08963 x\\&\quad +10.1562 x^{2}-1.24 a \\&\quad -2.48X b-7.4399x^{2} a+0.01234 y \\&\quad +3.3854y^{2}-2.48 X y^{2}-2.48Yb, \\ a_{14}&=0.01237x+0.0298 y+6.7708xy-4.96xya,\\ a_{15}&=a_{16}=0,\quad \\ a_{17}&=-1.24x -2.48x(p_{x}^{2}+p_{y}^{2}+x^{2}+y^{2}),\\ a_{18}&=-1.24x-2.48(p_{x}^{2}+p_{y}^{2}+x^{2}+y^{2}),\quad \\ a_{21}&=-\,0.0123p_{x}-6.77086p_{x} y+4.96p_{x}ya,\\ a_{22}&=-\,0.0123p_{y}-6.77086p_{y} y+4.96p_{y}ya, \\ a_{23}&=0.01237 x\!+\! 0.02987 y+6.7708 xy-4.96xya,\\ a_{24}&=1.4408+3.38543d+0.0298 x-2.48d \\&\quad -1.24a\!+\! 0.037128 y\!+\!10.1562 y^{2}\!-\!7.4399y^{2}a,\\ a_{25}&=a_{26}=0,\quad a_{27}=a_{28}-1.24y-2.48c, \\ a_{31}&=-1.4405-10.1562 p_{x}^{2}-3.38543f +1.24a\\&\quad +7.4399p_{x}^{2}a+2.48(Xg+Yf)-0.012376y, \\ a_{32}&=-6.7708p_{x}p_{y}+4.96ap_{x}p_{y}, \\ a_{33}&=0.02987p_{x}+6.7708p_{x} x-4.96ap_{x}y,\\ a_{34}&=0.012376p_{x}+6.67708p_{x}y-4.96p_{x}ya,\quad \\ a_{35}&=a_{36}=0,\\ a_{37}&=-1.24p_{x}-2.48p_{x}c,\quad \\ a_{38}&=-1.24p_{x}-2.48p_{x}c,\\ a_{41}&=-6.77086p_{x}p_{y}+4.96p_{x}p_{y}a,\\ a_{42}&=-1.44057-3.3854e-10.15629p_{y}^{2} \\&\quad -0.02987x+1.24a,\\ a_{43}&=0.02987p_{y}+6.7708p_{y}x-4.96p_{y}xa,\quad \\ a_{44}&=0.01234p_{y}+6.7708p_{y}y-4.96p_{y}ya,\\ a_{45}&=a_{46}=0,\quad a_{47}=-1.24p_{y}-2.48p_{y}c,\quad \\ a_{48}&=-1.24p_{y}-2.48p_{y}c,\\ a_{51}&=1.24p_{y}\!+\! 2.48p_{y}c,\quad a_{52}=-1.24x-2.48xc,\quad \\ a_{53}&=-1.24y-2.48yc,\\ a_{54}&=a_{55}=0,\quad a_{57}=-26,\quad a_{58}=0,\quad \\ a_{61}&=1.24p_{x}+2.48p_{x}c,\\ a_{62}&=1.24p_{y}\!+\! 2.48P_{y}c,\quad a_{63}=-1.24x-2.48xc,\quad \\ a_{64}&=-1.24y-2.48yc,\\ a_{65}&=a_{66}=a_{67}=0,\quad a_{68}=26,\quad \\ a_{71}&=a_{72}=a_{73}=a_{74}=0, \\ a_{75}&=a_{76}=5.2316 \times 10^{24},\quad a_{77}=a_{78}=0, \\ a_{81}&=a_{82}=a_{83}=a_{84}=0 \\&=a_{85}=a_{86}=a_{87}=a_{88}=0, \end{aligned}$$

where $X+Y=a$, $p_{x}^{2}+p_{y}^{2}=b$, $p_{x}^{2}+p_{y}^{2}+x^{2}+y^{2}=c$, $p_{x}^{2}+p_{y}^{2}+x^{2}=d$, $x^{2}+y^{2}+p_{x}^{2}=e$, $x^{2}+y^{2}+p_{y}^{2}=f$, $x^{2}+y^{2}=g$.