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Nonlinear amplitude dynamics in flagellar beating

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ABSTRACT

The physical basis of flagellar and ciliary beating is a major problem in biology which is still far from completely understood. The fundamental cytoskeleton structure of cilia and flagella is the axoneme, a cylindrical array of microtubule doublets connected by passive cross-linkers and dynein motor proteins. The complex interplay of these elements leads to the generation of self-organized bending waves. Although many mathematical models have been proposed to understand this process, few attempts have been made to assess the role of dyneins on the nonlinear nature of the axoneme. Here, we investigate the nonlinear dynamics of flagella by considering an axonemal sliding control mechanism for dynein activity. This approach unveils the nonlinear selection of the oscillation amplitudes, which are typically either missed or prescribed in mathematical models. The explicit set of nonlinear equations are derived and solved numerically. Our analysis reveals the spatio-temporal dynamics of dynein populations and flagellum shape for different regimes of motor activity, medium viscosity and flagellum elasticity. Unstable modes saturate via the coupling of dynein kinetics and flagellum shape without the need of invoking a nonlinear axonemal response. Hence, our work reveals a novel mechanism for the saturation of unstable modes in axonemal beating.

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Principal component analysis of flagellar beating. (a) Covariance matrix for μa=4300 (left) and μa=5600 (right). We can identify characteristic wavelengths λ,λ′ from negative long-range correlations in C. Note λ′>λ and the number of local maxima decreases when μa is increased. (b) (Left) Two principal shape modes v1,v2 (solid and dashed lines, respectively), corresponding to the two maximum eigenvalues of the covariance matrix in figure 4a (left). (Right) The flagellar shape at time t=733 ms (thick solid line) is reconstructed (white line) by a superposition of the two principal shape modes v1,v2 in figure 4b (left) fitting the scores B1,B2. (c) Flagellar dynamics in a reduced two-dimensional shape space for μa=4300 (left) and μa=5600 (right). Elliptic limit cycles are rescaled to better appreciate the distortion owing to the nonlinear terms. Sp=10, μ=50, η=0.14, ζ=0.4.
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RSOS160698F4: Principal component analysis of flagellar beating. (a) Covariance matrix for μa=4300 (left) and μa=5600 (right). We can identify characteristic wavelengths λ,λ′ from negative long-range correlations in C. Note λ′>λ and the number of local maxima decreases when μa is increased. (b) (Left) Two principal shape modes v1,v2 (solid and dashed lines, respectively), corresponding to the two maximum eigenvalues of the covariance matrix in figure 4a (left). (Right) The flagellar shape at time t=733 ms (thick solid line) is reconstructed (white line) by a superposition of the two principal shape modes v1,v2 in figure 4b (left) fitting the scores B1,B2. (c) Flagellar dynamics in a reduced two-dimensional shape space for μa=4300 (left) and μa=5600 (right). Elliptic limit cycles are rescaled to better appreciate the distortion owing to the nonlinear terms. Sp=10, μ=50, η=0.14, ζ=0.4.

Mentions: In this section, we study the obtained nonlinear solutions using principal component analysis [26,27]. This technique treats the flagellar shapes as multi-feature datasets, which can be projected to a lower dimensional space characterized by principal shape modes. Here we analyse the numerically resolved data following reference [26] to study sperm flagella. We discretize our flagella data with time-points ti, i=1,…,p and M=4×103 intervals corresponding to points sj=( j−1)/M, j=1,…,r along the flagellum, with r=M+1. We construct a measurement matrix Φ of size p×r for the tangent angle where Φij=ϕ(sj,ti). This matrix represents a kymograph of the flagellar beat. We define the r×r covariance matrix as , where with all rows equal to , where is the mean tangent angle at si. The covariance matrix C is shown for μa=4300 (figure 4a, left) and μa=5600 (figure 4a, right). In figure 4a (left), we find negative correlation between tangent angles that are a distance λ/2 apart. Hence, a characteristic wavelength λ can be identified in the system, which manifests as a long-range correlation in the matrix C. On the other hand, strong positive correlations around the main diagonals correspond to short-ranged correlations mainly owing to the bending stiffness of the bundle [26]. The number of local maxima along the diagonals in C decreases from μa=4300 to μa=5600, and at the same time λ′>λ. Hence, an increase in motor activity slightly increases the characteristic wavelength while decreasing the number of local maxima, which is related to the characteristic wavenumber. Employing an eigenvalue decomposition of the covariance matrix, we can obtain the eigenvectors v1,…,vr and their corresponding eigenvalues d1,…,dr, such that Cvj=djvj. Without loss of generality, we can sort the eigenvalues in descending order d1≥⋯≥dr. We find that the first two eigenvalues capture >99% variance of the data. This fact indicates that our flagellar waves can be suitably described in a two-dimensional shape space, because they can be regarded as single-frequency oscillators. Each flagellar shape Φi=[ϕ(s1,ti),…,ϕ(sr,ti)] can be expressed now as a linear combination of the eigenvectors vk [26]:5.1Φi=ϕ¯+∑k=1rBk(ti)vk,where Bk are the shape scores computed by a linear least-square fit. In figure 4b (left), the two first eigenvectors v1,v2 are shown for μa=4300. In figure 4b (right), the flagellar shape at a certain time (thick solid line) is reconstructed (white line) by using a superposition of the two principal shape modes v1,v2 (solid and dashed lines, respectively) and fitting the scores B1,B2. Finally, in figure 4c, we show the shape space trajectories beginning with small amplitude eigenmode solutions. While close to the bifurcation the limit cycle is elliptic (figure 4c, left), far from the bifurcation the limit cycle becomes distorted (figure 4c, right). Elliptic limit cycles were also found experimentally for bull sperm flagella [26]. Hence, as found in §5, motor activity in the nonlinear regime significantly affects the shape of the flagellum when compared with the linear solutions, which only provide good estimates sufficiently close to the Hopf bifurcation.Figure 4


Nonlinear amplitude dynamics in flagellar beating
Principal component analysis of flagellar beating. (a) Covariance matrix for μa=4300 (left) and μa=5600 (right). We can identify characteristic wavelengths λ,λ′ from negative long-range correlations in C. Note λ′>λ and the number of local maxima decreases when μa is increased. (b) (Left) Two principal shape modes v1,v2 (solid and dashed lines, respectively), corresponding to the two maximum eigenvalues of the covariance matrix in figure 4a (left). (Right) The flagellar shape at time t=733 ms (thick solid line) is reconstructed (white line) by a superposition of the two principal shape modes v1,v2 in figure 4b (left) fitting the scores B1,B2. (c) Flagellar dynamics in a reduced two-dimensional shape space for μa=4300 (left) and μa=5600 (right). Elliptic limit cycles are rescaled to better appreciate the distortion owing to the nonlinear terms. Sp=10, μ=50, η=0.14, ζ=0.4.
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getmorefigures.php?uid=PMC5383814&req=5

RSOS160698F4: Principal component analysis of flagellar beating. (a) Covariance matrix for μa=4300 (left) and μa=5600 (right). We can identify characteristic wavelengths λ,λ′ from negative long-range correlations in C. Note λ′>λ and the number of local maxima decreases when μa is increased. (b) (Left) Two principal shape modes v1,v2 (solid and dashed lines, respectively), corresponding to the two maximum eigenvalues of the covariance matrix in figure 4a (left). (Right) The flagellar shape at time t=733 ms (thick solid line) is reconstructed (white line) by a superposition of the two principal shape modes v1,v2 in figure 4b (left) fitting the scores B1,B2. (c) Flagellar dynamics in a reduced two-dimensional shape space for μa=4300 (left) and μa=5600 (right). Elliptic limit cycles are rescaled to better appreciate the distortion owing to the nonlinear terms. Sp=10, μ=50, η=0.14, ζ=0.4.
Mentions: In this section, we study the obtained nonlinear solutions using principal component analysis [26,27]. This technique treats the flagellar shapes as multi-feature datasets, which can be projected to a lower dimensional space characterized by principal shape modes. Here we analyse the numerically resolved data following reference [26] to study sperm flagella. We discretize our flagella data with time-points ti, i=1,…,p and M=4×103 intervals corresponding to points sj=( j−1)/M, j=1,…,r along the flagellum, with r=M+1. We construct a measurement matrix Φ of size p×r for the tangent angle where Φij=ϕ(sj,ti). This matrix represents a kymograph of the flagellar beat. We define the r×r covariance matrix as , where with all rows equal to , where is the mean tangent angle at si. The covariance matrix C is shown for μa=4300 (figure 4a, left) and μa=5600 (figure 4a, right). In figure 4a (left), we find negative correlation between tangent angles that are a distance λ/2 apart. Hence, a characteristic wavelength λ can be identified in the system, which manifests as a long-range correlation in the matrix C. On the other hand, strong positive correlations around the main diagonals correspond to short-ranged correlations mainly owing to the bending stiffness of the bundle [26]. The number of local maxima along the diagonals in C decreases from μa=4300 to μa=5600, and at the same time λ′>λ. Hence, an increase in motor activity slightly increases the characteristic wavelength while decreasing the number of local maxima, which is related to the characteristic wavenumber. Employing an eigenvalue decomposition of the covariance matrix, we can obtain the eigenvectors v1,…,vr and their corresponding eigenvalues d1,…,dr, such that Cvj=djvj. Without loss of generality, we can sort the eigenvalues in descending order d1≥⋯≥dr. We find that the first two eigenvalues capture >99% variance of the data. This fact indicates that our flagellar waves can be suitably described in a two-dimensional shape space, because they can be regarded as single-frequency oscillators. Each flagellar shape Φi=[ϕ(s1,ti),…,ϕ(sr,ti)] can be expressed now as a linear combination of the eigenvectors vk [26]:5.1Φi=ϕ¯+∑k=1rBk(ti)vk,where Bk are the shape scores computed by a linear least-square fit. In figure 4b (left), the two first eigenvectors v1,v2 are shown for μa=4300. In figure 4b (right), the flagellar shape at a certain time (thick solid line) is reconstructed (white line) by using a superposition of the two principal shape modes v1,v2 (solid and dashed lines, respectively) and fitting the scores B1,B2. Finally, in figure 4c, we show the shape space trajectories beginning with small amplitude eigenmode solutions. While close to the bifurcation the limit cycle is elliptic (figure 4c, left), far from the bifurcation the limit cycle becomes distorted (figure 4c, right). Elliptic limit cycles were also found experimentally for bull sperm flagella [26]. Hence, as found in §5, motor activity in the nonlinear regime significantly affects the shape of the flagellum when compared with the linear solutions, which only provide good estimates sufficiently close to the Hopf bifurcation.Figure 4

View Article: PubMed Central - PubMed

ABSTRACT

The physical basis of flagellar and ciliary beating is a major problem in biology which is still far from completely understood. The fundamental cytoskeleton structure of cilia and flagella is the axoneme, a cylindrical array of microtubule doublets connected by passive cross-linkers and dynein motor proteins. The complex interplay of these elements leads to the generation of self-organized bending waves. Although many mathematical models have been proposed to understand this process, few attempts have been made to assess the role of dyneins on the nonlinear nature of the axoneme. Here, we investigate the nonlinear dynamics of flagella by considering an axonemal sliding control mechanism for dynein activity. This approach unveils the nonlinear selection of the oscillation amplitudes, which are typically either missed or prescribed in mathematical models. The explicit set of nonlinear equations are derived and solved numerically. Our analysis reveals the spatio-temporal dynamics of dynein populations and flagellum shape for different regimes of motor activity, medium viscosity and flagellum elasticity. Unstable modes saturate via the coupling of dynein kinetics and flagellum shape without the need of invoking a nonlinear axonemal response. Hence, our work reveals a novel mechanism for the saturation of unstable modes in axonemal beating.

No MeSH data available.


Related in: MedlinePlus