_{1}

We use the Brownian dynamics with hydrodynamic interactions simulation in order to describe the movement of an elastically coupled dimer Brownian motor in a ratchet potential. The only external forces considered in our system were the load, the random thermal noise and an unbiased thermal fluctuation. We observe differences in the dynamic behaviour if hydrodynamic interactions are considered as compared with the case without them. In conclusion, hydrodynamic interactions influence substantially the dynamics of a ratchet dimer Brownian motor; consequently they have to be considered in any theory where the molecular motors are in a liquid medium.

Brownian motors are small physical micro- or even nano-machines that operate far from thermal equilibrium by extracting the energy from both thermal and non-equilibrium fluctuations in order to generate work against external loads. They present the physical analogue of bio-molecular motors that also work out of equilibrium to direct intracellular transport and to control motion in cells. In such bio-molecular motors, proteins such as kinesins, myosins and dyneins, move unidirectionally on one-dimensional “tracks” while hydrolysing adenosine triphosphate (ATP). These molecular motors are powered by a ratchet mechanism [

In the present work, we use the Brownian dynamics with hydrodynamic interactions simulation in order to describe the movement of an elastically coupled dimer Brownian motor in a ratchet potential. In Section 2, 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 Section 3, we describe the formalism given by Ermak and McCammon (1978) [

We consider an elastically coupled dimer in 3 dimensions in an asymmetrical potential (ratchet) in the

The corresponding force on the particles produced by the ratchet potential is given by:

In the former equations

We define

i, j, k are unit vectors in the direction of the cartesian axis, with

Then the modulus of the harmonic force is,

where

Then the components of the harmonic force are,

The corresponding components of the harmonic force on each dimer particle are,

Then the forces acting on the dimer particles are:

The load force,

Consider a system of N spherical interacting Brownian particles suspended in a hydrodynamic medium, the displacement of particle

where the superscript “0” indicates that the variable is to be evaluated at the beginning of the time step.

In our case of a dimer in three dimensions the tensor

A first basic quantity of interest, in our case, is the average center of mass velocity in the

where

Another quantity of central interest will be the effective diffusion coefficient,

where

The competition between the drift

here

We performed the simulation in dimensionless units. Distance is in units of the separation distance

The average velocity of a molecular motor is a function of the load force resisting the motor’s advancement. One of the characteristic of a molecular motor is the load force-velocity curve. In

In the range

In

as a function of time in the long time limit, i.e., after transients due to initial conditions have died out. The effective diffusion coefficient is giving by the slope of the linear fitting,

In

In

In

time, in the long-time limit, for a given load force,

case with hydrodynamic interactions greater than the case without them.

In

In

A similar result was found by Houtman et al. [

In conclusion, hydrodynamic interactions influence substantially the dynamics of a ratchet dimer Brownian motor; consequently they have to be considered in any theory where the molecular motors are in a liquid medium.

José A.Fornés, (2015) Hydrodynamic Interactions Introduce Differences in the Behaviour of a Ratchet Dimer Brownian Motor. Journal of Biomaterials and Nanobiotechnology,06,81-90. doi: 10.4236/jbnb.2015.62008