^{1}, Stefan Klumpp

^{2,3}, Reinhard Lipowsky

^{3}and Ulrich S. Schwarz

^{1,4,a)}

### Abstract

We use stochastic computer simulations to study the transport of a spherical cargo particle along a microtubule-like track on a planar substrate by several kinesin-like processive motors. Our newly developed adhesive motor dynamics algorithm combines the numerical integration of a Langevin equation for the motion of a sphere with kinetic rules for the molecular motors. The Langevin part includes diffusive motion, the action of the pulling motors, and hydrodynamic interactions between sphere and wall. The kinetic rules for the motors include binding to and unbinding from the filament as well as active motor steps. We find that the simulated mean transport length increases exponentially with the number of bound motors, in good agreement with earlier results. The number of motors in binding range to the motor track fluctuates in time with a Poissonian distribution, both for springs and cables being used as models for the linker mechanics. Cooperativity in the sense of equal load sharing only occurs for high values for viscosity and attachment time.

This work was supported by the Center for Modeling and Simulation in the Biosciences (BIOMS) and the Cluster of Excellence Cellular Networks at Heidelberg. S.K. was supported by a fellowship from Deutsche Forschungsgemeinschaft (Grant Nos. KL818/1-1 and KL818/1-2) and by the NSF through the Center for Theoretical Biological Physics (Grant No. PHY-0822283).

I. INTRODUCTION

II. MODEL AND COMPUTATIONAL METHODS

A. Bead dynamics

B. Motor dynamics

C. Bead versus motor friction

D. Vertical forces

E. Master equation approach

F. Mean run length for a spherical cargo particle

G. Computer simulations

III. RESULTS

A. Single motor simulations

B. Run length for several motors pulling

C. Distribution of motors in binding range

D. Escape rate distributions

E. Cargo transport against high viscousfriction

IV. DISCUSSION AND OUTLOOK

### Key Topics

- Viscosity
- 36.0
- Velocity measurement
- 24.0
- Friction
- 23.0
- Probability theory
- 13.0
- Motor proteins
- 12.0

## Figures

A single sphere of radius and surface separation from a planar wall. The translational coordinates of the sphere are given relative to a reference frame that is fixed to the wall. The sphere is pulled by one molecular motor that is attached to the surface at position measured with respect to the center of the sphere. The bead is subject to the motor force with -component . In addition, an external force acts parallel to the filament, typically arising from an optical trap. The force unbalance between and leads to the bead velocity . The motor with resting length is firmly attached to the bead and can bind to and unbind from a MT and moves with velocity . denotes the angle between the motor and the MT.

A single sphere of radius and surface separation from a planar wall. The translational coordinates of the sphere are given relative to a reference frame that is fixed to the wall. The sphere is pulled by one molecular motor that is attached to the surface at position measured with respect to the center of the sphere. The bead is subject to the motor force with -component . In addition, an external force acts parallel to the filament, typically arising from an optical trap. The force unbalance between and leads to the bead velocity . The motor with resting length is firmly attached to the bead and can bind to and unbind from a MT and moves with velocity . denotes the angle between the motor and the MT.

Force-velocity relation for a single motor: velocity as a function of load according to Eq. (10) with maximum velocity and stall force .

Force-velocity relation for a single motor: velocity as a function of load according to Eq. (10) with maximum velocity and stall force .

Illustration of the area fraction of the sphere (cut and placed besides the sphere) on which motor proteins can reach the MT (thin cylinder). depends in a geometrical fashion on the minimum distance between the sphere and the MT and on the resting length .

Illustration of the area fraction of the sphere (cut and placed besides the sphere) on which motor proteins can reach the MT (thin cylinder). depends in a geometrical fashion on the minimum distance between the sphere and the MT and on the resting length .

(a) Measured force-velocity relation of a single motor (with ) pulling a sphere of radius for three different viscosities . Shown is the relation according to Eq. (9), the actual measured force-velocity relation of the motor head and the bead center, respectively, and the theoretical prediction according to Eqs. (10) and (11). (b) The measured force-velocity relation for is shown where in Eq. (10) not but is used. The dotted line emphasizes the linear decrease of the velocity. The negative velocity of the bead at large results from thermal fluctuations. Fluctuations against walking direction increase the escape probability. In the case of escape they cannot be compensated by fluctuations in walking direction. (Numerical parameters: , number of runs .)

(a) Measured force-velocity relation of a single motor (with ) pulling a sphere of radius for three different viscosities . Shown is the relation according to Eq. (9), the actual measured force-velocity relation of the motor head and the bead center, respectively, and the theoretical prediction according to Eqs. (10) and (11). (b) The measured force-velocity relation for is shown where in Eq. (10) not but is used. The dotted line emphasizes the linear decrease of the velocity. The negative velocity of the bead at large results from thermal fluctuations. Fluctuations against walking direction increase the escape probability. In the case of escape they cannot be compensated by fluctuations in walking direction. (Numerical parameters: , number of runs .)

Mean run length of a bead pulled by a single motor as a function of an external force on the bead and for three different viscosities . The lines give the theoretical predictions according to Eq. (22) assuming an angle of 60° between the motor and the MT. For comparison also the theoretically predicted -curve for is shown (double dotted line).

Mean run length of a bead pulled by a single motor as a function of an external force on the bead and for three different viscosities . The lines give the theoretical predictions according to Eq. (22) assuming an angle of 60° between the motor and the MT. For comparison also the theoretically predicted -curve for is shown (double dotted line).

Distribution of run lengths in semilogarithmic scale for different values of motor coverage . The motor protein is modeled as a harmonic spring according to Eq. (5). (a) Resting length of the motor protein . (b) Resting length of the motor protein . (Numerical parameters: time step , number of simulation runs .)

Distribution of run lengths in semilogarithmic scale for different values of motor coverage . The motor protein is modeled as a harmonic spring according to Eq. (5). (a) Resting length of the motor protein . (b) Resting length of the motor protein . (Numerical parameters: time step , number of simulation runs .)

(a) Mean run length (data points with error bars) as a function of motors on the bead obtained from adhesive motor dynamics. The lines are fits of Eq. (19) with respect to the area fraction . (b) Mean number of bound motors (data points with error bars). The lines are the values obtained from the Poisson-averaged mean number of bound motors in Eq. (20) using for the fit value from (a). (Parameters: , , , , and .)

(a) Mean run length (data points with error bars) as a function of motors on the bead obtained from adhesive motor dynamics. The lines are fits of Eq. (19) with respect to the area fraction . (b) Mean number of bound motors (data points with error bars). The lines are the values obtained from the Poisson-averaged mean number of bound motors in Eq. (20) using for the fit value from (a). (Parameters: , , , , and .)

Combination of data from Figs. 7(a) and 7(b) for the resting lengths . is shown as a function of . For the dashed line (theory), Eqs. (19) and (20) were combined with a truncation of the sums at . (Parameters: , , , , and .)

Combination of data from Figs. 7(a) and 7(b) for the resting lengths . is shown as a function of . For the dashed line (theory), Eqs. (19) and (20) were combined with a truncation of the sums at . (Parameters: , , , , and .)

Histograms for the number of motors that are in binding range to the MT. The Symbols refer to simulation results for different values of the total number of motors on the sphere . Lines are Poisson distributions with mean value that is proportional to . (a) Resting length , spring model [Eq. (5)]. (b) Resting length , cable model [Eq. (6)]. (Parameters: , , , , .)

Histograms for the number of motors that are in binding range to the MT. The Symbols refer to simulation results for different values of the total number of motors on the sphere . Lines are Poisson distributions with mean value that is proportional to . (a) Resting length , spring model [Eq. (5)]. (b) Resting length , cable model [Eq. (6)]. (Parameters: , , , , .)

The relative frequencies of the number of motors that are in binding range to the MT during a single run are shown for six different sample runs. For the motor protein the cable model with resting length is used. (Parameters: , , , , and .)

The relative frequencies of the number of motors that are in binding range to the MT during a single run are shown for six different sample runs. For the motor protein the cable model with resting length is used. (Parameters: , , , , and .)

Measured probability distribution density for the escape rate given . The data were obtained for different values of . For the motor proteins the full spring model [Eq. (5)], with resting length was used. (Parameters: , , , , and .)

Measured probability distribution density for the escape rate given . The data were obtained for different values of . For the motor proteins the full spring model [Eq. (5)], with resting length was used. (Parameters: , , , , and .)

(a) Probability density of the cargo particle’s velocity that is obtained by averaging over a time interval of for different values of the viscosity . The inset shows the mean velocity as a function of the . (b) The mean velocity of the bead given that motors are simultaneously pulling is plotted as a function of (symbols). For comparison the theoretical expectation according to Eq. (13) is plotted, too (lines). (c) For high viscosity the conditional velocity distribution density given that a certain number of motors is pulling is plotted. (The full harmonic spring model was used; parameters: , , numerical time step , other parameters as in Table I.)

(a) Probability density of the cargo particle’s velocity that is obtained by averaging over a time interval of for different values of the viscosity . The inset shows the mean velocity as a function of the . (b) The mean velocity of the bead given that motors are simultaneously pulling is plotted as a function of (symbols). For comparison the theoretical expectation according to Eq. (13) is plotted, too (lines). (c) For high viscosity the conditional velocity distribution density given that a certain number of motors is pulling is plotted. (The full harmonic spring model was used; parameters: , , numerical time step , other parameters as in Table I.)

Frequencies of the motor forces in walking direction relative to the total load force on the cargo particle for different numbers of pulling motors. (a) Viscosity , escape rate . (b) , . (c) , . (d) , . For the other parameters the same values as in Fig. 12 were used.

Frequencies of the motor forces in walking direction relative to the total load force on the cargo particle for different numbers of pulling motors. (a) Viscosity , escape rate . (b) , . (c) , . (d) , . For the other parameters the same values as in Fig. 12 were used.

## Tables

Parameters used for adhesive motor dynamics. For ambient temperature, we used for viscosity (if not otherwise stated). If a range is given, then figure in boldface denotes the value used in the numerical simulations.

Parameters used for adhesive motor dynamics. For ambient temperature, we used for viscosity (if not otherwise stated). If a range is given, then figure in boldface denotes the value used in the numerical simulations.

Obtained fit values for the area fraction for different and the two applied polymer models. For comparison, the area fraction that is obtained from the measured mean distance is also displayed. is measured for fixed , the left boundary of the provided interval corresponds to the largest .

Obtained fit values for the area fraction for different and the two applied polymer models. For comparison, the area fraction that is obtained from the measured mean distance is also displayed. is measured for fixed , the left boundary of the provided interval corresponds to the largest .

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