Diploma thesis – Martin Reichelsdorfer
Martin Reichelsdorfer
On Brownian motion of asymmetric particles – An application as molecular motor
finished 2010-05
supervised by Klaus Mecke
Abstract
Asymmetric Brownian particles are able to conduct directed average motion under non-equilibrium conditions. This can be used for the construction of molecular motors. The present thesis is concerned with the extension and analysis of a biologically inspired model. Due to the often quite abstract formulation, the considerations are also relevant in the broader context of general Brownian motion of asymmetric objects.
The model consists of an asymmetric two-dimensional body, which moves along a rail through a two-dimensional, infinite ideal gas. By temporarily slowing it down at stopping sites, the motor or the corresponding statistical ensemble, respectively, is pushed away from equilibrium. The example of a parabolic potential with exponentially decreasing spring constant shows that these binding sites can be modelled by time-dependent potential wells. Similar to the cooling of a gas by expansion, this release process leads to a narrowing of the velocity distribution. Particularly during the subsequent relaxation the random motion is biased until the equilibrium distribution is established again. A singular release of the motor is found to be rather inefficient in the considered case, in the sense that the width of the distribution of positions is always profoundly larger than the drift of its mean value. By periodically distributing the binding sites, however, an effective rectification of the motion can be achieved, since the equilibration of the system is prevented and a stationary non-equilibrium state establishes instead.
Probability P(X) to find the motor at position X after different (dimensionless) times. The model consists of only three possible (dimensionless) velocities (-1, 0 and 1) and periodic stopping sites, where the motor’s velocity is instantly set to zero. The asymmetry of the particle is reflected by appropriate asymmetric transition rates betweenn the three states.
As the model can only be solved approximately due to its complexity, a series of reduced models is developed. These systems exhibit the same characteristic features as the more general model, but are easier to access analytically. In particular, a system composed of only three discrete velocity levels can be regarded as a minimal model of an asymmetric Brownian particle in one dimension. It turns out that the motion of asymmetric and symmetric particles differ not only in non-equilibrium (as might have been anticipated from prior observations), but also in equilibrium. Moreover, augmenting the model with discrete position states allows for the already mentioned investigation of repeated stopping sites. Hereat, qualitative accordances to experimental data from a monomeric kinesin KIF1A motor are observed. To test, whether the results can also be reproduced quantitatively could be the subject of further studies.
Singular release from a parabolic potential well, pushing the motor away from thermal equilibrium, and subsequent relaxation. As long as in non-equilibrium, the particle has a preferred direction of motion.
Comparatively large thermal fluctuations are the driving force of the motor, wherefore new methods have to be applied for its thermodynamic description. In the framework of so-called stochastic thermodynamics, quantities like energy, work, heat or even entropy may be attributed to individual phase space trajectories. Hence, they are now to be understood as stochastic variables with wide-stretching probability distributions. With respect to these distributions a multitude of relations, called fluctuation theorems, can be found, which often constitute a more detailed version of common thermodynamic statements like the second law, and which have already been applied in the analysis of experimental data (e.g. in the determination of free energies). The validity of these theorems for the present model can be confirmed by direct calculation.
Asymmetric Brownian particles are able to conduct directed average motion under non-equilibrium conditions. This can be used for the construction of molecular motors. The present thesis is concerned with the extension and analysis of a biologically inspired model. Due to the often quite abstract formulation, the considerations are also relevant in the broader context of general Brownian motion of asymmetric objects.
The model consists of an asymmetric two-dimensional body, which moves along a rail through a two-dimensional, infinite ideal gas. By temporarily slowing it down at stopping sites, the motor or the corresponding statistical ensemble, respectively, is pushed away from equilibrium. The example of a parabolic potential with exponentially decreasing spring constant shows that these binding sites can be modelled by time-dependent potential wells. Similar to the cooling of a gas by expansion, this release process leads to a narrowing of the velocity distribution. Particularly during the subsequent relaxation the random motion is biased until the equilibrium distribution is established again. A singular release of the motor is found to be rather inefficient in the considered case, in the sense that the width of the distribution of positions is always profoundly larger than the drift of its mean value. By periodically distributing the binding sites, however, an effective rectification of the motion can be achieved, since the equilibration of the system is prevented and a stationary non-equilibrium state establishes instead.
Probability P(X) to find the motor at position X after different (dimensionless) times. The model consists of only three possible (dimensionless) velocities (-1, 0 and 1) and periodic stopping sites, where the motor’s velocity is instantly set to zero. The asymmetry of the particle is reflected by appropriate asymmetric transition rates betweenn the three states.
As the model can only be solved approximately due to its complexity, a series of reduced models is developed. These systems exhibit the same characteristic features as the more general model, but are easier to access analytically. In particular, a system composed of only three discrete velocity levels can be regarded as a minimal model of an asymmetric Brownian particle in one dimension. It turns out that the motion of asymmetric and symmetric particles differ not only in non-equilibrium (as might have been anticipated from prior observations), but also in equilibrium. Moreover, augmenting the model with discrete position states allows for the already mentioned investigation of repeated stopping sites. Hereat, qualitative accordances to experimental data from a monomeric kinesin KIF1A motor are observed. To test, whether the results can also be reproduced quantitatively could be the subject of further studies.
Singular release from a parabolic potential well, pushing the motor away from thermal equilibrium, and subsequent relaxation. As long as in non-equilibrium, the particle has a preferred direction of motion.
Comparatively large thermal fluctuations are the driving force of the motor, wherefore new methods have to be applied for its thermodynamic description. In the framework of so-called stochastic thermodynamics, quantities like energy, work, heat or even entropy may be attributed to individual phase space trajectories. Hence, they are now to be understood as stochastic variables with wide-stretching probability distributions. With respect to these distributions a multitude of relations, called fluctuation theorems, can be found, which often constitute a more detailed version of common thermodynamic statements like the second law, and which have already been applied in the analysis of experimental data (e.g. in the determination of free energies). The validity of these theorems for the present model can be confirmed by direct calculation.