https://orcid.org/0000-0002-0998-0995
Figures
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.
Citation: Beenish, Samreen M, Alshammari FS (2025) Lie symmetry approach to the dynamical behavior and conservation laws of actin filament electrical models. PLoS One 20(9): e0331243. https://doi.org/10.1371/journal.pone.0331243
Editor: Angelo Marcelo Tusset, Federal University of Technology - Parana, BRAZIL
Received: April 15, 2025; Accepted: August 12, 2025; Published: September 9, 2025
Copyright: © 2025 Beenish et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: No datasets were generated or analyzed during the current study.
Funding: This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2503).
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
The study of NLPDEs and their corresponding solutions has increasingly drawn focus as a pivotal and demanding area in both pure and applied mathematics. These equations play a crucial role in understanding nonlinear physical phenomena, with applications spanning engineering and natural sciences. Exact or analytical solutions are highly valued for their ability to reveal the intrinsic properties of nonlinear systems [1–4]. Given their broad relevance in nonlinear sciences, interest in studying NLPDEs has grown substantially. Various mathematical techniques, including the modified generalized exponential rational function method [5], the unifed method [6], the group analysis [7,8], the Lie isomorphism method [9], have been employed to obtain solitary wave solutions [10–12]. The cytoskeleton is an essential component of all living cells, consisting of three filamentous structures: actin-based microfilaments, intermediate filaments, and tubulin-based microtubules. These networks, interconnected by specific proteins, regulate cellular processes such as migration, division, and intracellular transport. actin-based microfilaments facilitate cell migration and remodeling at the leading edge, while tubulin-based microtubules anchor chromosomes and mediate cell division. Additionally, molecular motors, as protein complexes, facilitate directional transport along tubulin-based microtubules and actin-based microfilaments [13,14]. In the analysis of F-actin surrounded by a saline solution see Fig 1, it is important to determine the resistance, inductance, and capacitance of the entire filament. This requires obtaining effective values using the appropriate properties of addition. As shown in Fig 2, both parallel and series components contribute to the total resistance, the capacitance follows a parallel-addition property, and the inductance is strictly a series contribution. For a filament consisting of m monomers, the effective resistance (), inductance (S), and capacitance (
) are given by [15]:
Where and
, satisfying the relation
. It is important to note that we have assigned
,
, Sp = S, and
. Consequently, for a
segment of the actin filament, we obtain
In this part, we establish an electrical model of the actin filament using inductive, capacitive, and resistive components, building on the concepts outlined earlier. Essentially, Kirchhoff’s laws are applied to the segment of the effective electrical circuit corresponding to a single monomer, N, which is coupled to neighboring monomers see Fig 2. These Bjerrum ions have been shown to generate a time-dependent current as they move along helical paths, contributing to the inductance denoted as S. Due to viscosity, a resistive component is also anticipated in these currents, which is included in series with S and labeled as see Fig 2. In parallel with these components, another resistance [16,17], referred to as
, exists between the Bjerrum ions and the filament surface. A capacitance
is placed in series with this resistance, and it is assumed that the charge on this capacitor exhibits a nonlinear relationship with voltage, similar to the charge-voltage behavior of a reverse-biased pn-junction diode as described by Ma et al. [1]. Consequently, for the mth monomer, we assume [18,19]:
where α is anticipated to be small. Applying Kirchhoff’s laws and referring to Fig 2, if Km represents the current passing through the inductance S and resistor , while Km−1 is the current flowing along EF, then the current through HF must be
. For the BC segment of the mth monomer:
where wm and wm + 1 represent the voltages across EG and CJ, respectively, as illustrated in Fig 2. Likewise, if the voltage across the capacitor is Wm − W0, where W0 denotes the capacitor’s bias voltage, then we obtain:
The current through section FH equals the rate of change of charge , giving:
From Eq (3), it follows that:
and
From Eq (5), it follows that:
Thus, using Eqs (6)–(8), we obtain:
Substituting Eq (4) into Eq (9) gives:
Applying a continuum approximation with Wm = W and a Taylor expansion in β gives:
At this point, all derivatives up to the fourth order are considered. Using Eq (11), we get:
Applying Eq (12), and Eq (11), we rewrite Eq (10) as:
Assuming time variations are small relative to the constant background voltage, we consider the nonlinear capacitance term as second order. With time derivatives of order ε, nonlinear voltage terms of order , and β of order ε, we retain terms up to
in Eq (13) to derive the final expression [20]:
Eq (14) presents W as the dependent variable, while τ and are the independent variables. Specifically, τ corresponds to the temporal variable, and
denotes the spatial variable. J. A. Tuszynski et al. [20] studied Eq (14), solving it using the maximum propagation velocity wave expressed in Jacobi elliptic functions. In this work, we analyze Eq (14) from different perspectives: Firstly, we conduct a symmetry analysis of Eq (14), derive its symmetry group, and obtain solutions via the tanh method. Additionally, we visualize the solution’s behavior using Mathematica. Secondly, we examine the dynamical behavior through bifurcation and chaos analysis. Thirdly, we investigate chaotic behavior using methods such as time series analysis, poincare maps, power spectrum, return maps, fractal dimension, and Lyapunov exponents. Each tool has distinct significance in chaos detection:
- Time analysis: Investigates the progression of a system’s state variables across temporal dimensions, discerning patterns, trends, and anomalies that signify chaotic behavior.
- Poincare map: Represents a discrete framework of a continuous dynamical system, encapsulating its intersections with a lower-dimensional subspace, thereby elucidating periodicity or chaotic trajectories.
- Power spectrum: Examines the frequency constituents of a time series to differentiate between regular and chaotic dynamics, chaotic systems manifest broad, continuous spectra as opposed to distinct peaks.
- Return map: Illustrates the correlation between consecutive values of a variable, facilitating the visualization of attractors, periodicity, and the manifestation of chaos through complex structures.
- Lyapunov exponent: Quantifies the sensitivity of trajectories to initial conditions, with positive values denoting exponential divergence and affirming chaotic behavior.
- Fractal dimension: The fractal dimension quantifies the complexity of chaotic attractors, reflecting the scaling behavior of system intricacies. A non-integer value indicates self-similarity and irregular dynamics.
- Lastly, we highlight three further aspects of our study. we perform a sensitivity analysis of Eq (14). We investigate the conservation laws of Eq (14). We analyze the stability of Eq (14).
Lie point symmetry plays a crucial role in various scientific fields, especially in integrable systems with infinitely many symmetries. The Lie symmetry analysis method [21] is recognized as an effective approach for obtaining analytical solutions to NLPDEs. Additionally, it is instrumental in deriving conservation laws, which are essential for studying nonlinear physical phenomena [22]. A significant recent contribution to nonlinear physics can be found in [23]. Conservation laws mathematically represent the principle that a specific physical quantity remains constant during the evolution of a physical system. They also aid in refining mathematical methods to establish the existence and uniqueness of solutions. Various techniques are available for formulating conservation laws for differential equations (DEs) [24,25]. The well-known Noether theorem [26] links Lie point symmetries to conservation laws, while Ibragimov [27] introduced a new conservation law theorem. Conservation laws, symmetry analysis, bifurcation, and chaos analysis have become prominent research topics in recent studies. Tariq Mahmood et al. [28] investigated symmetry analysis, conservation laws, and bifurcation analysis using the Klein–Gordon equation. The graphical abstract has been depicted in Fig 3.
2 Lie point symmetry analysis
A one-parameter Lie group of infinitesimal transformations is presented [29,30]:
Here, ,
, and
are infinitesimal generators, and the symmetry of Eq (14) follows from the symmetry conditions of the vector field (15):
are complete operators concerning
and τ.
Theorem 1: Eq (14) possesses a two-dimensional Lie algebra spanned by two generators [31]:
Using Theorem 1 along with the related ODEs and initial conditions:
We obtain one-parameter groups using
:
This leads to the following result.
Theorem 2: If is a solution of Eq (14), another solution is given by [32]:
Using the Lie bracket definition , the commutators are given by:
Theorem 3: A two-dimensional Lie symmetry algebra is generated by
in Theorem 1. Next, we determine the adjoint representation of the vector fields using the Lie series [32]:
Applying system (21), we obtain:
Theorem 4: A one-parameter optimal system [33] for the Lie algebra is formed by the operators
with an arbitrary constant Ψ.
3 Exact solutions via symmetry reductions
This section focuses on analyzing symmetry reductions and exact solutions for Eq (14).
3.1 The operator 
Utilizing the generator , one derives
Setting and inserting Eq (24) into Eq (14), we obtain the ODE:
We solve Eq (25) in maple and obtain the following solution:
Substituting Eq (26) into Eq (24) yields the solution of Eq (14):
3.2 The operator 
Based on the generator , we have :
With , substituting Eq (28) into Eq (14) yields the following ODE:
The obtained solution for Eq (29) is:
Substituting Eq (30) into Eq (28) yields the solution of Eq (14):
3.3 The operator 
From , it follows that:
Taking , substituting Eq (32) into Eq (14) leads to the ODE:
Integrating Eq (33) with respect to , we obtain the following ODE:
3.4 Exploring wave solution behaviors
In this part, We aim to derive solitary wave solutions for Eq (14) using the tanh method. A detailed explanation of the method is available in [34]. The balancing number between the higher-order derivative and the nonlinear terms
and
to determine the value from Eq (34). We apply the balancing procedure to the ODE in Eq (34) and obtain
. Consequently, from
, we get:
By inserting Eq (35) into Eq (34), we derive a system of algebraic equations. Solving these equations using Maple, we obtain:
3.5 Graphical representation
This section presents graphical representations of the obtained solutions. By selecting appropriate parameter values, we derive both 2D and 3D visualizations of Eq (36) with a2 = 0.56 and . As shown in Figs 4 and 5, the solution exhibits a kink profile for
, while for
, the corresponding representations in Figs 6 and 7 reveal an anti-kink-shaped solution. Kink and anti-kink solitons play a vital role in nonlinear wave dynamics due to their stability and energy-preserving nature. They model signal transmission in optical fibers, energy transport in plasmas, and biological processes like DNA dynamics. In condensed matter physics, they describe domain walls in ferromagnets and phase transitions in superconductors. Additionally, they appear in fluid dynamics and cosmology, representing shock waves and topological defects. Their robustness makes them essential in various scientific and engineering applications [35–38].
4 Bifurcation analysis of Eq (14)
Bifurcation occurs in dynamical systems when slight changes in system parameters or conditions lead to a qualitative shift in behavior. As parameters surpass critical thresholds, known as bifurcation points, the system undergoes transformations such as the emergence or disappearance of stable fixed points, limit cycles, or chaotic dynamics. These transitions may cause the system to shift between stability and instability or give rise to new dynamic patterns. Eq (34) written as a dynamical system is [39]:
System (37) has the following fixed points along the axis :
The Jacobian matrix for system (37) is obtained through the following computations [40]:
Remark: The point is a saddle if
, a center if
, and a cusp if
. The nature of system (37) at critical points is determined accordingly [41]. The detailed results of the global phase portraits are discussed in Tables 1–2 and illustrated in Figs 8–19.
5 Hamilton analysis of Eq (14)
Using Hamilton’s equations from classical mechanics, the system’s behavior can be effectively analyzed and understood [42]
A system is called Hamiltonian if a function exists that satisfies:
The Hamiltonian function of the system is denoted by [43].
Definition: For the dynamical system (41) to be classified as Hamiltonian, specific conditions must be met:
Eq (37) qualifies as a Hamiltonian dynamical system since it meets the required conditions for the state equations:
The Hamiltonian corresponding to system (37) is given by [44]:
where z1 is a real parameter.
6 Investigation of chaotic behavior
Understanding the intrinsic dynamics of Eq (34) is crucial. This section presents a detailed examination of Eq (37)’s response to noise-induced perturbations and its display of chaotic behavior. The analysis follows the proposed model [45]:
To analyze chaotic dynamics, we will evaluate noise sensitivity using two different approaches [46]. Our investigation centers on the parameters , representing amplitude, and
, indicating frequency. By varying these parameters, we aim to assess their impact on the system’s chaotic behavior [47,48]. The influence of the applied force on dynamical system (37) is analyzed using various chaos detection methods outlined in the introduction, including phase portraits, Poincaré maps, power spectra, fractal dimensions, return maps, time series, and Lyapunov exponents. These tools are employed to illustrate chaotic behavior in the perturbed system, with graphical representations provided. Identification of chaos via 2D phase portraits for system (45) at
,
,
, and varying
behavioue such as:
- When
: The phase portrait demonstrates a complex nested architecture characterized by elaborate loops, signifying quasi-periodic dynamics as shown in Fig 20.
- When
: The trajectory reveals oscillatory behavior accompanied by periodic fluctuations, implying regular dynamical characteristics as shown in Fig 21.
- When
: The heightened complexity manifested in the loops indicates a potential transition towards chaotic dynamics as shown in Fig 22.
- When
: The phase portrait reveals irregular and intersecting trajectories, which serve as a definitive indicator of chaotic motion as shown in Fig 23.
- When
: The structure continues to exhibit intricate features with prominent folding, further substantiating chaotic dynamics as shown in Fig 24.
- When
: The trajectory, characterized by its density and entanglement, corroborates the presence of pronounced chaotic behavior within the system as shown in Fig 25.
- Time Series: Figs 26 and 27 presents the time series analysis of system (45) under quasi-periodic feedback, with
,
in shown Fig 26, and
,
in Fig 27.
- Poincare Maps: System (45) shows periodic behavior at
, while at
and 0.55, it transitions to chaotic behavior as shown in Figs 28–31.
- Lyapunov Exponents: The Lyapunov Exponents simulations of the proposed chaotic system indicate one negative and one positive exponent, confirming its chaotic nature, as shown in Fig (32). The computed LEs are h1 = + 0.060156, and h2 = −0.060156. Given the positivity of the first and negativity of the second exponent, the Kaplan-Yorke dimension is determined as follows:
(46)
As we can see,is greater than one, which indicates that the system exhibits chaotic behavior. The dynamical system denoted as (45) with parameters
,
, and
demonstrates chaotic behavior, as substantiated by the diagnostics illustrated in Fig 32.
- Power Spectrum The expansive and continuous spectrum reveals a diverse array of excited frequencies, devoid of discrete peaks. This characteristic is indicative of chaotic systems, wherein trajectories exhibit periodicity and a high sensitivity to initial conditions as shown in Fig 33.
- Return Map: The plotted points exhibit a fractal-like, irregular configuration as opposed to forming a closed curve or distinct points. This observation corroborates the existence of a strange attractor, which is a defining feature of chaos as shown in Fig 34.
- Fractal Dimension: The non-integer value associated with the fractal dimension provides further evidence of chaotic dynamics, as it encapsulates the system’s intricate, self-similar geometry within phase space as shown in Fig 35.
7 Conservation laws of Eq (14)
Bluman [49] developed a systematic method for generating important conserved vectors. Their approach entails identifying multiplier of specific orders for a given differential equation, which are subsequently used to determine the associated fluxes:
shows the Euler operator:
where , and
represent the total derivative operators.
By analyzing Eq (47) and conducting further standard computations, we obtain the following multiplier:
When , the corresponding multiplier is
, yielding the conservation laws:
When , the corresponding multiplier is
, yielding the conservation laws :
8 Sensitivity analysis of Eq (14)
In this section, we perform sensitivity analyses on the following dynamical system using two distinct initial conditions:
We analyze the system’s behavior across multiple scenarios, each illustrated in distinct figures. Fig 36 presents the sensitivity analysis for the parameter values: , a2 = 0.24, R1 = 0.38,
, and R2 = −0.04. The initial conditions for the red trajectory are set as
, while for the blue trajectory, they are
. Fig 37 depicts the sensitivity analysis under different parameter values:
, a2 = 0.24, R1 = 0.38,
, and R2 = 0.04. The initial conditions for the red trajectory are
, whereas for the blue trajectory, they are
. Fig 38 illustrates the sensitivity analysis for another parameter set:
, a2 = −0.24, R1 = 0.38,
, and R2 = −0.04. The red trajectory starts from
, while the blue trajectory originates from
. Notably, even slight variations in the initial parameters result in significant deviations in the system’s behavior, demonstrating a high degree of sensitivity in the examined model.
9 Stability analysis of Eq (14)
Examine the perturbed solution expressed as:
It is evident that any constant Y1 serves as a steady-state solution of Eq (14). Substituting Eq (54) into Eq (14) yields:
Linearizing Eq (55) results in:
Assume that Eq (56) admits a solution of the form [50]:
where k0 represents the dimensionless wave number, substituting Eq (57) into Eq (56) and solving for , we obtain:
According to the stability analysis, the sign of determines the stability of the system. If
, the perturbation grows exponentially, indicating an unstable system. If
, the perturbation decays, leading to a stable system.
10 Conclusion
The governing equations of the model are derived using Kirchhoff’s laws applied to the LRC resonant circuit, representing an actin monomer within a filament. By taking the continuum limit, we obtained second-order partial differential equations describing the spatio-temporal voltage distribution. This study aims to conduct the follwong results:
- First, we perform the Lie point symmetry analysis of the considered model and obtain a two-dimensional Lie algebra. It is observed that these Lie symmetries preserve an abelian algebraic structure. Using the identified symmetries, we construct the corresponding symmetry group.
- Next, we derive exact solutions through symmetry reduction by combining the symmetries to reduce the original PDE into an ODE. The tanh method is applied to obtain the soliton solution. To visualize these solutions, we utilized Mathematica to generate 3D and 2D plots as shown in Figs 4–7, illustrating kink and anti-kink solitons. The tanh method simplifies nonlinear PDEs into solvable algebraic forms, making it computationally efficient. It is widely applicable for constructing exact soliton and wave solutions in nonlinear systems.
- Further, we apply the Galilean transformation to the ODE and convert it into a dynamical system. The resulting system is analyzed using bifurcation analysis. We discuss all parameter cases where the system exhibits stable and unstable behavior, as illustrated in Figs 8–19. Finally, we derive the Hamiltonian corresponding to the dynamical system.
- We apply an external forcing term to the dynamical system to study its chaotic behavior. This chaotic nature was demonstrated through time series plots, phase portraits, return maps, power spectrum, Poincaré maps, fractal dimensions, and Lyapunov exponents, as shown in Figs 20–35.
- Sensitivity analysis was carried out using the Runge-Kutta method, as depicted in Figs 36–38.
- Additionally, conservation laws were derived through the multiplier method.
- Finally, a stability analysis was performed, indicating that the system remains stable when
; otherwise, it becomes unstable.
The limitation of bifurcation and chaos analysis lies in their current applicability mainly to second-order ODEs and integrable systems. In future work, this model can be explored numerically using advanced techniques such as the Finite Element Method, Finite Volume Method, Spectral Method, Lattice Boltzmann Method, Pseudo-Spectral Method, and the Lump solution approach. These directions open up possibilities for extending the analysis to more complex nonlinear PDEs, offering a broader scope for future research.
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