Excitations of Solitary Waves along Alpha-Helical Protein Molecules

Abstract

A protein molecular structure consists of connected alpha-helices and/or beta-strands. The alpha-helix is a right-handed helix with peptide bonds located on the inner, and the side chains extending outward. It is stabilized by hydrogen bonds formed between the amino group and carbonyl groups of every fourth peptide bond. This study applies the Jacobian elliptic function method, a powerful tool for constructing nonlinear solutions, to investigate the nonlinear excitations in the form of solitary waves along alpha helical protein molecules.

Keywords

Soliton, protein, Jacobi-elliptic function method, symbolic computation

1. Introduction

A fundamental problem in biophysics involves understanding the storage and transport of energy within protein chains. Proteins are polymers made up of 20 different amino acids arranged in a specific sequence known as the primary structure. The actual three-dimensional structure of a protein is highly complex. The primary structure refers to the covalent backbone of the polypeptide chain, which includes the sequence of amino acid residues. One of the essential secondary structures of proteins is the alpha-helix, a periodic helical structure characterized by hydrogen bonds between amino acids along the helical axis.

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The energy released during processes such as the hydrolysis of adenosine triphosphate (ATP) is crucial for biological processes. According to Davydov's hypothesis, this energy, stored in the amide-I vibration of the C-O bond of the peptide group, can be transferred along the polypeptide chain in the form of a soliton. A soliton is a localized packet of energy that arises from nonlinear interactions of vibrational excitations and deformations in the protein structure.

Solitons are stable and can propagate over long distances without significant energy loss, making them potential mechanisms for energy transfer in proteins.

2. Dynamical Equation and Exact Solution using Jacobi Elliptic Function Method

We derive the Cubic-Quintic Nonlinear Schrödinger Equation (CQNSE) to describe the distribution of excitations in protein chains. The CQNSE is given by:

_iħ∂ψ/∂t + J₁∂²ψ/∂x² - λψ + k₁G|ψ|²ψ + k₂G²|ψ|⁴ψ = 0

Where:

• λ = ε + T + U - D - 2J₁
• J₁ = 2d²R^-3
• χ = βDR
• k₁ = χ²/M
• k₂ = χ³γ/2RM²

We seek a traveling wave solution ψ(x, t) = u(ξ) exp[i(k₀x - ω₀t)], where ξ = (x - ct), and obtain the ordinary differential equation:

d²u(ξ)/dξ² + M₁du(ξ)/dξ + M₂u(ξ) + M₃u³(ξ) + M₄u⁵(ξ) = 0

Here, M₁ = 2k₀, M₂ = k₀² - λJ₁, M₃ = k₁J₁G², M₄ = k₂J₁G².

We apply the Jacobian elliptic function method to solve this equation. Using the ansatz u(ξ) = a₀ + Σ(a_isnⁱ + b_isn^-i), we obtain the real and imaginary parts of the solution. The real part is:

The imaginary part:

Solving the above equations using symbolic computation, we find:

a₁ = -[(M₃a₃⁰ + M₂a₀ + M₆a₀ + M₄a₅⁰)/(M₁ + M₅)cn(ξ)dn(ξ)]

b₁ = 0

Substituting these values into Eq. (3), we obtain the exact solitary wave solution as:

u(ξ) = a₀ - [(M₂

Where:

• a₀ represents the amplitude of the anti-solitonic structure.
• M₁, M₂, M₃, M₄, M₅, and M₆ are defined in the equations above.
• sn(ξ), cn(ξ), and dn(ξ) are Jacobian-elliptic functions.

Using the equation (7), we can describe the exact solitary wave solution in the form of an anti-solitonic structure.

We have conducted simulations using this solution with various parameter values. When increasing the parameter a₀ from 1 x 10^-10 to 1.2 x 10^-10, the amplitude of the anti-solitonic structure increases. Additionally, varying the coefficients of the velocity parameter (c) from 0.1 to 0.7 results in the broadening of the middle region of the anti-solitonic structure.

3. Conclusions

In this study, we applied the Jacobian elliptic function method to find exact solitary wave solutions for the dynamics of alpha-helical protein molecules. This method proved to be a powerful tool for solving the Cubic-Quintic Nonlinear Schrödinger Equation (CQNSE), providing a new anti-solitonic structure solution. The results suggest that this approach can be applied to solve similar dynamical equations in various scientific fields.

The anti-solitonic structure obtained in our simulations shows that solitons may play a role in efficiently transferring energy within proteins. This differs from the traditional assumption that energy dissipates into the environment during folding processes in proteins. Solitons offer a mechanism for localized energy transfer over long distances, which could have important implications for biological processes.

In conclusion, this study contributes to our understanding of the dynamics of alpha-helical protein molecules and the potential role of solitons in energy transfer within biological systems.