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6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions
motion of particles and macromolecules and may find application in a wide variety
of fields, including particle separation and the design of molecular motors and
pumps. Zimmermann and Seifert [14] studied the efficiencies of a molecular motor
for a generic hybrid model applied to the F1-ATPase they obtained good quantitative
agreement with the experimental data. Pinkoviezky and Gov [15] motivated by
the observed pulses of backward-moving myosin-X in the filopodia structure, they
model interacting molecular motors with an internal degree of freedom, introducing
a novel modification to the approximation scheme.
In the present work we use the Brownian dynamics with hydrodynamic interactions simulation in order to describe the movement of a elastically coupled dimer
Brownian motor in a ratchet potential. In section entitled “The Model” we describe
the forces acting on an oscillating dimer in a ratchet potential with a load force
and an external unbiased fluctuation, which acts simultaneously on two particles.
In the section entitled “Brownian dynamics with hydrodynamic interactions” we
describe the formalism given by Ermak and McCammon (1978) [16], which couples
the forces described in “The Model” section and thermal noise with the diffusion
tensor. A striking feature of fluid mechanics in the viscously dominated regime,
or equivalently, at low Reynolds number, is the long range of hydrodynamic
interactions. For example, the Stokeslet, or flow field induced by a point force,
falls off inversely with distance. Hydrodynamic interactions have been considered
by several authors to explain different phenomena: Kemps and Bhattacharjee, [17],
used a particle tracking model for colloid transport near planar surfaces covered
with spherical asperities This model provides a preliminary step in investigating how
geometrically tractable asperities alter the undisturbed flow field and hydrodynamic
interactions between the particle and the substrate. Hydrodynamic interactions allow
in average for directed motion of a three-sphere system, the spheres are connected
by two identical active linker arms. Each linker arm contains molecular motors
and elastic elements and can oscillate spontaneously, see [18]. Microorganisms
are often subjected to swimming in close proximity to each other as well as
other boundaries. The resulting hydrodynamicn interactions may have puzzling
effects on their swimming speed, trajectory, and power dissipation. These effects
were investigated by Ramia et al. [19], each microorganism consisted of a sphere
propelled by a rotating helix. It was found that only a small increase (less than
10%) results in the mean swimming speed of an organism swimming near and
parallel to another identical organism. In a later paper, Kim and Powers [20],
focus on hydrodynamic interactions by considering two rotating rigid helices. They
suppose the helices were driven by stationary motors, and obtained complementary
results to those of Ramia et al., since the hydrodynamic interactions between their
helices were stronger. Fornés [21], showed that hydrodynamic interactions induce
movement against an external load in a ratchet dimer Brownian motor. In the
present paper we show that hydrodynamic interactions introduce differences in the
behaviour of a ratchet dimer Brownian motor as compared without them. We report
differences in the effective diffusion coefficient, D eff , mean vc x component of the
mass center velocity and spatial cross correlations in x direction.
6 Ratchet Dimer Brownian Motor with Hydrodynamic Interactions
motion of particles and macromolecules and may find application in a wide variety
of fields, including particle separation and the design of molecular motors and
pumps. Zimmermann and Seifert [14] studied the efficiencies of a molecular motor
for a generic hybrid model applied to the F1-ATPase they obtained good quantitative
agreement with the experimental data. Pinkoviezky and Gov [15] motivated by
the observed pulses of backward-moving myosin-X in the filopodia structure, they
model interacting molecular motors with an internal degree of freedom, introducing
a novel modification to the approximation scheme.
In the present work we use the Brownian dynamics with hydrodynamic interactions simulation in order to describe the movement of a elastically coupled dimer
Brownian motor in a ratchet potential. In section entitled “The Model” we describe
the forces acting on an oscillating dimer in a ratchet potential with a load force
and an external unbiased fluctuation, which acts simultaneously on two particles.
In the section entitled “Brownian dynamics with hydrodynamic interactions” we
describe the formalism given by Ermak and McCammon (1978) [16], which couples
the forces described in “The Model” section and thermal noise with the diffusion
tensor. A striking feature of fluid mechanics in the viscously dominated regime,
or equivalently, at low Reynolds number, is the long range of hydrodynamic
interactions. For example, the Stokeslet, or flow field induced by a point force,
falls off inversely with distance. Hydrodynamic interactions have been considered
by several authors to explain different phenomena: Kemps and Bhattacharjee, [17],
used a particle tracking model for colloid transport near planar surfaces covered
with spherical asperities This model provides a preliminary step in investigating how
geometrically tractable asperities alter the undisturbed flow field and hydrodynamic
interactions between the particle and the substrate. Hydrodynamic interactions allow
in average for directed motion of a three-sphere system, the spheres are connected
by two identical active linker arms. Each linker arm contains molecular motors
and elastic elements and can oscillate spontaneously, see [18]. Microorganisms
are often subjected to swimming in close proximity to each other as well as
other boundaries. The resulting hydrodynamicn interactions may have puzzling
effects on their swimming speed, trajectory, and power dissipation. These effects
were investigated by Ramia et al. [19], each microorganism consisted of a sphere
propelled by a rotating helix. It was found that only a small increase (less than
10%) results in the mean swimming speed of an organism swimming near and
parallel to another identical organism. In a later paper, Kim and Powers [20],
focus on hydrodynamic interactions by considering two rotating rigid helices. They
suppose the helices were driven by stationary motors, and obtained complementary
results to those of Ramia et al., since the hydrodynamic interactions between their
helices were stronger. Fornés [21], showed that hydrodynamic interactions induce
movement against an external load in a ratchet dimer Brownian motor. In the
present paper we show that hydrodynamic interactions introduce differences in the
behaviour of a ratchet dimer Brownian motor as compared without them. We report
differences in the effective diffusion coefficient, D eff , mean vc x component of the
mass center velocity and spatial cross correlations in x direction.