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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

Properties of metachronal waves. (a) Radial ur(r*, θ, t) and (b) tangential uθ(r*, θ, t) components of the flow field, measured at r* = 1.3 R. Phase defects are evidenced by white circles. (c) Correlation function C(Δθ, Δt) for a single representative Volvox colony, (d) its fitted correlation function  and the (e) corresponding error. The scale bar for (c–e) is the same. (f) Power spectrum of the autocorrelation function C(0, Δt) (for one Volvox colony), calculated for three distinct values of θ. (g) Average beat frequency as a function of polar angle θ for n = 60 different colonies (black) as well as the ensemble average (white dotted).
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RSIF20141358F2: Properties of metachronal waves. (a) Radial ur(r*, θ, t) and (b) tangential uθ(r*, θ, t) components of the flow field, measured at r* = 1.3 R. Phase defects are evidenced by white circles. (c) Correlation function C(Δθ, Δt) for a single representative Volvox colony, (d) its fitted correlation function and the (e) corresponding error. The scale bar for (c–e) is the same. (f) Power spectrum of the autocorrelation function C(0, Δt) (for one Volvox colony), calculated for three distinct values of θ. (g) Average beat frequency as a function of polar angle θ for n = 60 different colonies (black) as well as the ensemble average (white dotted).

Mentions: The fluid flow time-averaged over the whole duration t0 of the movie, (figure 1c), is described accurately by the modes of the ‘squirmer’ expansion of the flow field around a sphere [40–42] modified to take into account the net force exerted by the holding pipette. The results are in agreement with previous measurements on freely swimming colonies [43], and support the hypothesis that flagellar dynamics are the same in held and freely swimming colonies, a fact already established for the closely related unicellular species Chlamydomonas [34]. Subtracting the time average from the instantaneous flow field, highlights the flow's dependence on the local flagellar phase within the beating cycle. It is this phase that we wish to measure. Figure 2 shows representative kymographs, through the same time interval, for the radial and tangential components of u at r* = 1.3 R. This value of r*, which corresponds to the dashed circle in figure 1a, has been found empirically to maximize the kymographs' signals in most experiments.Figure 2.


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)

Properties of metachronal waves. (a) Radial ur(r*, θ, t) and (b) tangential uθ(r*, θ, t) components of the flow field, measured at r* = 1.3 R. Phase defects are evidenced by white circles. (c) Correlation function C(Δθ, Δt) for a single representative Volvox colony, (d) its fitted correlation function  and the (e) corresponding error. The scale bar for (c–e) is the same. (f) Power spectrum of the autocorrelation function C(0, Δt) (for one Volvox colony), calculated for three distinct values of θ. (g) Average beat frequency as a function of polar angle θ for n = 60 different colonies (black) as well as the ensemble average (white dotted).
© Copyright Policy - open-access
Related In: Results  -  Collection

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

RSIF20141358F2: Properties of metachronal waves. (a) Radial ur(r*, θ, t) and (b) tangential uθ(r*, θ, t) components of the flow field, measured at r* = 1.3 R. Phase defects are evidenced by white circles. (c) Correlation function C(Δθ, Δt) for a single representative Volvox colony, (d) its fitted correlation function and the (e) corresponding error. The scale bar for (c–e) is the same. (f) Power spectrum of the autocorrelation function C(0, Δt) (for one Volvox colony), calculated for three distinct values of θ. (g) Average beat frequency as a function of polar angle θ for n = 60 different colonies (black) as well as the ensemble average (white dotted).
Mentions: The fluid flow time-averaged over the whole duration t0 of the movie, (figure 1c), is described accurately by the modes of the ‘squirmer’ expansion of the flow field around a sphere [40–42] modified to take into account the net force exerted by the holding pipette. The results are in agreement with previous measurements on freely swimming colonies [43], and support the hypothesis that flagellar dynamics are the same in held and freely swimming colonies, a fact already established for the closely related unicellular species Chlamydomonas [34]. Subtracting the time average from the instantaneous flow field, highlights the flow's dependence on the local flagellar phase within the beating cycle. It is this phase that we wish to measure. Figure 2 shows representative kymographs, through the same time interval, for the radial and tangential components of u at r* = 1.3 R. This value of r*, which corresponds to the dashed circle in figure 1a, has been found empirically to maximize the kymographs' signals in most experiments.Figure 2.

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