Lie symmetry approach to the dynamical behavior and conservation laws of actin filament electrical models

Abstract

This research explores the dynamical properties and solutions of actin filaments, which serve as electrical conduits for ion transport along their lengths. Utilizing the Lie symmetry approach, we identify symmetry reductions that simplify the governing equation by lowering its dimensionality. This process leads to the formulation of a second-order differential equation, which, upon applying a Galilean transformation, is further converted into a system of first-order differential equations. Additionally, we investigate the bifurcation structure and sensitivity of the proposed dynamical system. When subjected to an external force, the system exhibits quasi-periodic behavior, which is detected using chaos analysis tools. Sensitivity analysis is also performed on the unperturbed system under varying initial conditions. Moreover, we establish the conservation laws associated with the equation and conduct a stability analysis of the model. Employing the tanh method, we derive exact solutions and visualize them through 3D and 2D graphical representations to gain deeper insights. These findings offer new perspectives on the studied equation and significantly contribute to the understanding of nonlinear wave dynamics.

Fig 1. F-actin is enveloped by water molecules and counterions.

Fig 2. A simplified circuit for the mth monomer shows current Jm flowing through inductance S and resistance ℜ1.

Fig 3. Chart of the implemented analysis.

Fig 4. Anti-kink soliton profile of W(ϑ,τ) from Eq (36) for positive wave speed using 3D plot.

Fig 5. Anti-kink soliton profile of W(ϑ,τ) from Eq (36) for positive wave speed using 2D plot.

Fig 6. Kink soliton profile of W(ϑ,τ) from Eq (36) for negative wave speed.

Fig 7. Kink soliton profile of W(ϑ,τ) from Eq (36) for negative wave speed.

Fig 8. Global phase portraits of the dynamical system (37) for positive 𝔉2 and 𝔉3, when 𝔉1 is absent.

Fig 9. Global phase portraits of the dynamical system (37) for positive 𝔉2 and 𝔉3, when 𝔉1 is positive.

Fig 10. Global phase portraits of the dynamical system (37) for positive 𝔉2 and 𝔉3, when 𝔉1 is negative.

Fig 11. Global phase portraits of the dynamical system (37) for negative 𝔉2 and 𝔉3, when 𝔉1 is absent.

Fig 12. Global phase portraits of the dynamical system (37) for negative 𝔉2 and 𝔉3, when 𝔉1 is positive.

Fig 13. Global phase portraits of the dynamical system (37) for negative 𝔉2 and 𝔉3, when 𝔉1 is negative.

Fig 14. Global phase portraits of the dynamical system (37) for 𝔉2>0 and 𝔉3<0, when 𝔉1 is absent.

Fig 15. Global phase portraits of the dynamical system (37) for 𝔉2>0 and 𝔉3<0, when 𝔉1 is positive.

Fig 16. Global phase portraits of the dynamical system (37) for 𝔉2>0 and 𝔉3<0, when 𝔉1 is negative.

Fig 17. Global phase portraits of the dynamical system (37) for 𝔉2<0 and 𝔉3>0, when 𝔉1 is absent.

Fig 18. Global phase portraits of the dynamical system (37) for 𝔉2<0 and 𝔉3>0, when 𝔉1 is positive.

Fig 19. Global phase portraits of the dynamical system (37) for 𝔉2<0 and 𝔉3>0, when 𝔉1 is negative.

Fig 20. Identification of quasi-periodic behaviour through 2D phase portrait analysis for dynamical system (45) at 𝔉1 = 0.85, 𝔉2 = -0.45, Δ=0.11, and κ=2.28.

Fig 21. Identification of quasi-periodic behaviour through 2D phase portrait analysis for dynamical system (45) at 𝔉1 = 0.85, 𝔉2=−0.45, Δ=0.33, and κ=2.28.

Fig 22. Identification of chaotic behaviour through 2D phase portrait analysis for dynamical system (45) at 𝔉1=0.85, 𝔉2 = -0.45, Δ=0.44, and κ=2.28.

Fig 23. Identification of chaotic behaviour through 2D phase portrait analysis for dynamical system (45) at 𝔉1 = 0.85, 𝔉2 = -0.45, Δ=0.55, and κ=2.28.

Fig 24. Identification of chaotic behaviour through 2D phase portrait analysis for dynamical system (45) at 𝔉1 = 0.85, 𝔉2 = -0.45, Δ=0.66, and κ=2.28.

Fig 25. Identification of quasi-periodic and chaotic behaviour through 2D phase portrait analysis for dynamical system (45) at 𝔉1 = 0.85, 𝔉2 = -0.45, Δ=0.77, and κ=2.28.

Fig 26. Chaotic behaviour in time analysis of system (45) for Δ=0.33 and κ=2.28 with 𝔉2 > 0, 𝔉3 > 0.

Fig 27. Chaotic behaviour in time analysis of system (45) for Δ=0.33 and κ=2.28 with 𝔉2 < 0, 𝔉3 < 0.

Fig 28. Identification of chaotic behaviour through poincaré map for dynamical system (45) at 𝔉1 = 0.85, Δ=0.11 , 𝔉2 = -0.45, and κ=2.28.

Fig 29. Identification of chaotic behaviour through poincaré map for dynamical system (45) at 𝔉1=0.85, Δ=0.33 , 𝔉2=−0.45, and κ=2.28.

Fig 30. Identification of chaotic behaviour through poincaré map for dynamical system (45) at 𝔉1=0.85, Δ=0.44 , 𝔉2=−0.45, and κ=2.28.

Fig 31. Identification of chaotic behaviour through poincaré map for dynamical system (45) at 𝔉1=0.85, Δ=0.77, 𝔉2=−0.45, and κ=2.28.

Fig 32. Detection of chaotic behavior through the Lyapunov exponent for dynamical system (45) with 𝔉1=0.85, 𝔉2=−0.45, and κ=2.28.

Fig 33. Detection of chaotic behavior through power spectrum for dynamical system (45) with 𝔉1=0.85, 𝔉2=−0.45, and κ=2.28.

Fig 34. Detection of chaotic behavior through return map for dynamical system (45) with 𝔉1=0.85, 𝔉2=−0.45, and κ=2.28.

Fig 35. Detection of chaotic behavior through fractal dimension for dynamical system (45) with 𝔉1=0.85, 𝔉2=−0.45, and κ=2.28.

Fig 36. Analysis of sensitivity in the dynamical system (53) using (0.45,0.03) and (0.12,0.03).

Fig 37. Analysis of sensitivity in the dynamical system (53) using (0.42,0.03) and (0.34,0.03).

Fig 38. Analysis of sensitivity in the dynamical system (53) using using (0.35,0.03) and (0.24,0.03).