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Metachronal waves in the flagellar beating of Volvox and their hydrodynamic origin.

Brumley DR, Polin M, Pedley TJ, Goldstein RE - J R Soc Interface (2015)

Bottom Line: A minimal model of hydrodynamically coupled oscillators can reproduce semi-quantitatively the characteristics of the average metachronal dynamics, and the emergence of defects.Our results suggest that metachronal coordination follows from deformations in the oscillators' limit cycles induced by hydrodynamic stresses, and that defects result from sufficiently steep local biases in the oscillators' intrinsic frequencies.Additionally, we find that random variations in the intrinsic rotor frequencies increase the robustness of the average properties of the emergent metachronal waves.

View Article: PubMed Central - PubMed

Affiliation: Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA.

ABSTRACT
Groups of eukaryotic cilia and flagella are capable of coordinating their beating over large scales, routinely exhibiting collective dynamics in the form of metachronal waves. The origin of this behavior--possibly influenced by both mechanical interactions and direct biological regulation--is poorly understood, in large part due to a lack of quantitative experimental studies. Here we characterize in detail flagellar coordination on the surface of the multicellular alga Volvox carteri, an emerging model organism for flagellar dynamics. Our studies reveal for the first time that the average metachronal coordination observed is punctuated by periodic phase defects during which synchrony is partial and limited to specific groups of cells. A minimal model of hydrodynamically coupled oscillators can reproduce semi-quantitatively the characteristics of the average metachronal dynamics, and the emergence of defects. We systematically study the model's behaviour by assessing the effect of changing intrinsic rotor characteristics, including oscillator stiffness and the nature of their internal driving force, as well as their geometric properties and spatial arrangement. Our results suggest that metachronal coordination follows from deformations in the oscillators' limit cycles induced by hydrodynamic stresses, and that defects result from sufficiently steep local biases in the oscillators' intrinsic frequencies. Additionally, we find that random variations in the intrinsic rotor frequencies increase the robustness of the average properties of the emergent metachronal waves.

No MeSH data available.


Related in: MedlinePlus

Modelling flagella. (a) Tip trajectory over 10 beats of a flagellum of a Volvox somatic cell. (b) Rotor as a model flagellum: a sphere of radius a elastically bound to a circular trajectory of radius r0 (dashed red) perpendicular to a no-slip plane, driven by a tangential force in the ϕ-direction. (c) Intrinsic trajectory of an isolated rotor. Steady-state limit cycle of the sphere above the no-slip wall at z = 0, for Λ = 0.1, 1, 2, 5, 10, ∞. Perturbations from the circular trajectory have been magnified by a factor of 100 so that the shape is clearly visible. The evolution of the (d) radius and (e) geometric phase through one beating period is also shown.
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RSIF20141358F3: Modelling flagella. (a) Tip trajectory over 10 beats of a flagellum of a Volvox somatic cell. (b) Rotor as a model flagellum: a sphere of radius a elastically bound to a circular trajectory of radius r0 (dashed red) perpendicular to a no-slip plane, driven by a tangential force in the ϕ-direction. (c) Intrinsic trajectory of an isolated rotor. Steady-state limit cycle of the sphere above the no-slip wall at z = 0, for Λ = 0.1, 1, 2, 5, 10, ∞. Perturbations from the circular trajectory have been magnified by a factor of 100 so that the shape is clearly visible. The evolution of the (d) radius and (e) geometric phase through one beating period is also shown.

Mentions: The surface-mounted somatic cells of Volvox are a few tens of micrometres apart, and their flagella are therefore more nearly in the weak-coupling limit than the cilia of historically studied organisms such as Paramecium. It is thus appropriate to model the fluid disturbance produced by their operation as a multi-pole expansion [45], of which we will only keep the mode with the slowest spatial decay—the Stokeslet—representing the effect of a point force. This flow is analogous to the far field of a rigid sphere pulled through the fluid. It has recently been shown that representing a Volvox flagellum as a single Stokeslet provides an accurate representation of its flow field down to distances of approximately 10 µm, smaller than those typically separating cells within colonies (approx. 20 µm) [4]. High-speed tracking of the flagellar waveform combined with resistive force theory also confirms that the distributed forces associated with the motion can be well represented by a single point force which periodically traverses a closed loop [4]. Inspired by the trajectories of flagellar tips in Volvox, and following an approach similar to others [27–29], a beating flagellum will thus be modelled as a small sphere of radius a elastically bound to a circular trajectory of radius r0 by radial and transversal springs of stiffnesses λ and η, respectively, and driven by a tangential force of magnitude fdrive (figure 3). The position of the sphere is given by x = x0 + s(ζ, r, ϕ), where s = (r sin(ϕ), ζ, r cos(ϕ)). The prescribed trajectory, defined by (ζ = 0, r = r0), is perpendicular to a no-slip plane at z = 0 representing the surface of Volvox, and its centre x0 is at a distance d from the plane. The proximity of the no-slip boundary causes an asymmetry in the sphere's motion which induces a net flow, thus mimicking power and recovery strokes of real flagella.Figure 3.


Metachronal waves in the flagellar beating of Volvox and their hydrodynamic origin.

Brumley DR, Polin M, Pedley TJ, Goldstein RE - J R Soc Interface (2015)

Modelling flagella. (a) Tip trajectory over 10 beats of a flagellum of a Volvox somatic cell. (b) Rotor as a model flagellum: a sphere of radius a elastically bound to a circular trajectory of radius r0 (dashed red) perpendicular to a no-slip plane, driven by a tangential force in the ϕ-direction. (c) Intrinsic trajectory of an isolated rotor. Steady-state limit cycle of the sphere above the no-slip wall at z = 0, for Λ = 0.1, 1, 2, 5, 10, ∞. Perturbations from the circular trajectory have been magnified by a factor of 100 so that the shape is clearly visible. The evolution of the (d) radius and (e) geometric phase through one beating period is also shown.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4528573&req=5

RSIF20141358F3: Modelling flagella. (a) Tip trajectory over 10 beats of a flagellum of a Volvox somatic cell. (b) Rotor as a model flagellum: a sphere of radius a elastically bound to a circular trajectory of radius r0 (dashed red) perpendicular to a no-slip plane, driven by a tangential force in the ϕ-direction. (c) Intrinsic trajectory of an isolated rotor. Steady-state limit cycle of the sphere above the no-slip wall at z = 0, for Λ = 0.1, 1, 2, 5, 10, ∞. Perturbations from the circular trajectory have been magnified by a factor of 100 so that the shape is clearly visible. The evolution of the (d) radius and (e) geometric phase through one beating period is also shown.
Mentions: The surface-mounted somatic cells of Volvox are a few tens of micrometres apart, and their flagella are therefore more nearly in the weak-coupling limit than the cilia of historically studied organisms such as Paramecium. It is thus appropriate to model the fluid disturbance produced by their operation as a multi-pole expansion [45], of which we will only keep the mode with the slowest spatial decay—the Stokeslet—representing the effect of a point force. This flow is analogous to the far field of a rigid sphere pulled through the fluid. It has recently been shown that representing a Volvox flagellum as a single Stokeslet provides an accurate representation of its flow field down to distances of approximately 10 µm, smaller than those typically separating cells within colonies (approx. 20 µm) [4]. High-speed tracking of the flagellar waveform combined with resistive force theory also confirms that the distributed forces associated with the motion can be well represented by a single point force which periodically traverses a closed loop [4]. Inspired by the trajectories of flagellar tips in Volvox, and following an approach similar to others [27–29], a beating flagellum will thus be modelled as a small sphere of radius a elastically bound to a circular trajectory of radius r0 by radial and transversal springs of stiffnesses λ and η, respectively, and driven by a tangential force of magnitude fdrive (figure 3). The position of the sphere is given by x = x0 + s(ζ, r, ϕ), where s = (r sin(ϕ), ζ, r cos(ϕ)). The prescribed trajectory, defined by (ζ = 0, r = r0), is perpendicular to a no-slip plane at z = 0 representing the surface of Volvox, and its centre x0 is at a distance d from the plane. The proximity of the no-slip boundary causes an asymmetry in the sphere's motion which induces a net flow, thus mimicking power and recovery strokes of real flagella.Figure 3.

Bottom Line: A minimal model of hydrodynamically coupled oscillators can reproduce semi-quantitatively the characteristics of the average metachronal dynamics, and the emergence of defects.Our results suggest that metachronal coordination follows from deformations in the oscillators' limit cycles induced by hydrodynamic stresses, and that defects result from sufficiently steep local biases in the oscillators' intrinsic frequencies.Additionally, we find that random variations in the intrinsic rotor frequencies increase the robustness of the average properties of the emergent metachronal waves.

View Article: PubMed Central - PubMed

Affiliation: Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA.

ABSTRACT
Groups of eukaryotic cilia and flagella are capable of coordinating their beating over large scales, routinely exhibiting collective dynamics in the form of metachronal waves. The origin of this behavior--possibly influenced by both mechanical interactions and direct biological regulation--is poorly understood, in large part due to a lack of quantitative experimental studies. Here we characterize in detail flagellar coordination on the surface of the multicellular alga Volvox carteri, an emerging model organism for flagellar dynamics. Our studies reveal for the first time that the average metachronal coordination observed is punctuated by periodic phase defects during which synchrony is partial and limited to specific groups of cells. A minimal model of hydrodynamically coupled oscillators can reproduce semi-quantitatively the characteristics of the average metachronal dynamics, and the emergence of defects. We systematically study the model's behaviour by assessing the effect of changing intrinsic rotor characteristics, including oscillator stiffness and the nature of their internal driving force, as well as their geometric properties and spatial arrangement. Our results suggest that metachronal coordination follows from deformations in the oscillators' limit cycles induced by hydrodynamic stresses, and that defects result from sufficiently steep local biases in the oscillators' intrinsic frequencies. Additionally, we find that random variations in the intrinsic rotor frequencies increase the robustness of the average properties of the emergent metachronal waves.

No MeSH data available.


Related in: MedlinePlus