Cilia

Clia and flagella are proteic extrusions of cellular surfaces, preposed to move the extra cellular fluids and/or to move the cell itself within its environment.

This figure shows a cross section of a cilium next to a longitudinal section.


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The motion of an individual cilium superficially resembles that of an oar, in that it sweeps through the medium with a power stroke that propels the cell in the opposite direction. Each power stroke and return stroke involves perhaps thousands of chemical reactions. There may be dozens of strokes per second, and perhaps thousands of cilia.


The motion of cilia and flagella involves at least two aspects of interest for theoretical physics. Namely first, a random supply of enegy -- coming from a biochemical reaction -- is converted in ordered motion. Second, cilia and flagella are able to sustain wave motion without damping in a viscous fluid.
  • Ciliar motion and macroscopic fluid current. We model ciliar motion as resulting by rectification of brownian processes. We can prove the existence of a macroscopic fluid current which is compatible with a stationary state for the cilia motion. We write a system of Fokker--Plank equations with two distinct potentials. Coupling with thermal noise is considered. Rate transitions from active and passive phases of motions are determined by biochemical reactions. Cilia--cilia interactions are considered (mean field). The macroscopic current is studied from numerical solutions.
In M.Cosentino Lagomarsino,B.Bassetti,P.Jona" "Rowers" coupled hydrodynamically. Modeling possible mechanisms for the cooperation of Cilia. " Eur. Phys. J. B 26, 81--88 (2002) (see PS version)



Fig. 1

During the active phase -- the POWER stroke -- the cilium moves in the quartic potentials W. In the passive phase -- RECOVERY stroke -- the potential V is parabolic. Both potentials are asymmetric, due to the self--generated current in the fluid. Permanence times in the potentials are different.


Fig. 2

Active current as a function of the asymmetry parameter f in the potentials. The intersection J=f represents the self--generated current.


Coordinated motion of cilia (the metachronal wave).

We employ a model system, called rowers, as a generic physical framework to define the problem of the coordinated motion of cilia as a far from equilibrium process.

Rowers are active (two-state) oscillators interacting solely through forces of hydrodynamic origin. We consider the case of fully deterministic dynamics, find analytical solutions of the equation of motion in the long wavelength (continuum) limit, and investigate numerically the short wavelength limit.

We prove the existence of metachronal waves below a characteristic wavelength. Such waves are unstable and become stable only if the sign of the coupling is reversed. We also find that with normal hydrodynamic interaction the metachronal pattern has the form of stable trains of traveling wave packets sustained by the onset of anti-coordinated beating of consecutive rowers. In M.Cosentino Lagomarsino,P.Jona,B.Bassetti " Metachronal wave and hydrodynamic interaction for deterministic switching rowers " Phys.Rev. E (2003) physics/0301061


Metachronal pattern in the large wavelength limit. In the continuum limit of the equation of motion for the rowers we prove that wave solutions f(x) exists and study their stability. There is an upper critical wavelength for metachronal solutions whose wavelength is large compared to the spacing between rowers (2 units).

The stability analysis gives one line (zero measure) of stability in the plane of model parameters (A,B), with runaway hyperbolic trajectories around it.

Metachronal solutions from numerical simulations for linear arrays of many (50-250) rowers interacting hydrodynamically: here the configuration as a function of the rower index.The dashed line shows the actual configuration, the solid line outlines the shape of the wave packet.

Time evolution of the 40th rower of the previous array.

Short wavelengths To analyze the system at short wavelengths we run numerical simulation of linear array of a few rowers with periodic boundary conditions.

For three rowers we analyzed the results using Poincare' maps.

If the potentials are linear the system may be either chaotic (fig.1) or quasi-periodic (fig.2), which means that the motion does not result in a travelling wave.

fig.1
fig.2

If, instead, the potentials are not linear there is no chaos and attractors appear in the configuration space section (fig.3) and (fig.4). Thus, the curvature k of the potential plays a role in determining the organization properties of the system.

fig.3
fig.4

The relaxation time diverges with power law as the curvature of the potential k goes to zero .

Also, the relaxation time diverges with power law as the strenght of hydrodynamics hydrodynamic interaction (alpha) goes to zero.