^{1,a)}, Julian Shillcock

^{1,2}and Reinhard Lipowsky

^{1,b)}

### Abstract

Brownian dynamics simulations are used to study the dynamical process of self-assembly of actin monomers into long filaments containing up to 1000 actin protomers. In order to overcome the large separation of time scales between the diffusive motion of the free monomers and the relatively slow attachment and detachment processes at the two ends of the filaments, we introduce a novel rescaling procedure by which we speed all dynamical processes related to actin polymerization and depolymerization up by the same factor. In general, the actin protomers within a filament can attain three different states corresponding to a bound adenosine triphosphate (ATP), adenosine diphosphate with inorganic phosphate (ADP/P), and ADP molecule. The simplest situation that has been studied experimentally is provided by the polymerization of ADP-actin, for which all protomers are identical. This case is used to unravel certain relations between the filament’s physical properties and the model parameters such as the attachment rate constant and the size of the capture zone, the detachment rate and the probability of the detached event, as well as the growth rate and waiting times between two successive attachment/detachment events. When a single filament is allowed to grow in a bath of constant concentration of free ADP-actin monomers, its growth rate increases linearly with the free monomer concentration in quantitative agreement with *in vitro* experiments. The results also show that the waiting time is governed by exponential distributions and that the two ends of a filament undergo biased random walks. The filament length fluctuations are described by a length diffusion constant that is found to attain a constant value at low ADP-actin concentration and to increase linearly with this concentration. It is straightforward to apply our simulation code to more complex processes such as polymerization of ATP-actin coupled to ATP hydrolysis, force generation by filaments, formation of filament bundles, and filament-membrane interactions.

We thank Xin Li for the stimulating discussions and acknowledge support by Grant No. RGP0072 of the Human Frontier Science Project. MEMPHYS is supported by the Danish National Research Foundation.

I. INTRODUCTION

II. THEORETICAL DESCRIPTION

A. Algorithm for monomer attachment and detachment

B. Simulation parameters

C. Time scales involved in actin polymerization and depolymerization

D. Rescaling procedure

III. RESULTS

A. Definition of growth rate

B. Attachment rate constants

C. Binding times and attachment rates

D. Concentration dependence of growth rate

E. Length fluctuations and length diffusion constants

IV. CONCLUSIONS AND OUTLOOK

### Key Topics

- Polymers
- 127.0
- Diffusion
- 21.0
- Polymerization
- 19.0
- Self assembly
- 10.0
- Random walks
- 6.0

## Figures

Coarse-grained model of monomer and filament (a) actin monomer built up from three particles labeled by , , and and connected into an equilateral triangle. (b) These triangles are tied together into a filament by Hookean springs, which are between adjacent , adjacent , and adjacent particles as well as between all next-nearest neighbors particles within the filament (for clarity, only two next-nearest neighbor bonds are shown).

Coarse-grained model of monomer and filament (a) actin monomer built up from three particles labeled by , , and and connected into an equilateral triangle. (b) These triangles are tied together into a filament by Hookean springs, which are between adjacent , adjacent , and adjacent particles as well as between all next-nearest neighbors particles within the filament (for clarity, only two next-nearest neighbor bonds are shown).

Schematic illustration of the computational mechanism of ADP-filament growth by attachment and detachment of monomers to its ends. Single actin monomers with a bound ADP molecule are labeled with the letter “.” The conical regions at the filament’s ends are capture zones with independent radius and at the filament’s barbed and pointed ends, whose sizes control the rate of free monomer attachment events. Detachment of the terminal monomers at the filament’s barbed and pointed ends is controlled by two more independent parameters, and . Note that the rates of monomer attachment to the filament ends, and , are not parameters of the model but are determined by the capture zone size and the free monomer concentration.

Schematic illustration of the computational mechanism of ADP-filament growth by attachment and detachment of monomers to its ends. Single actin monomers with a bound ADP molecule are labeled with the letter “.” The conical regions at the filament’s ends are capture zones with independent radius and at the filament’s barbed and pointed ends, whose sizes control the rate of free monomer attachment events. Detachment of the terminal monomers at the filament’s barbed and pointed ends is controlled by two more independent parameters, and . Note that the rates of monomer attachment to the filament ends, and , are not parameters of the model but are determined by the capture zone size and the free monomer concentration.

Time evolution of actin protomer number (black solid line, left axis); positions (right axis) of barbed end (red solid line), pointed end (blue solid line), and the initial center (cyan solid line) in the filament vs simulation time step. The capture sizes are and and the detachment rates are and at the barbed and pointed ends, respectively.

Time evolution of actin protomer number (black solid line, left axis); positions (right axis) of barbed end (red solid line), pointed end (blue solid line), and the initial center (cyan solid line) in the filament vs simulation time step. The capture sizes are and and the detachment rates are and at the barbed and pointed ends, respectively.

Dependence of kinetic rate constants on monomer concentration for different values of the capture zone size , which is varied from (open squares) to (closed stars). In the inset, the mean kinetic rate constants at each capture zone are plotted logarithmically as a function of height ; the straight line has slope 3. In these simulations, we chose .

Dependence of kinetic rate constants on monomer concentration for different values of the capture zone size , which is varied from (open squares) to (closed stars). In the inset, the mean kinetic rate constants at each capture zone are plotted logarithmically as a function of height ; the straight line has slope 3. In these simulations, we chose .

(a) Normalized histograms for time intervals between successive monomer attachments at barbed and pointed ends. The width of the bins is 10 000 time steps, the probability of attachment is approximately exponentially distributed, . (b) The inverse mean waiting time , which is equal to the attachment rate as a function of monomer concentration at different capture zone sizes.

(a) Normalized histograms for time intervals between successive monomer attachments at barbed and pointed ends. The width of the bins is 10 000 time steps, the probability of attachment is approximately exponentially distributed, . (b) The inverse mean waiting time , which is equal to the attachment rate as a function of monomer concentration at different capture zone sizes.

Dependence of the ADP filament growth rate (left axis) and the inverse mean waiting time at the barbed end (right axis) on actin monomer concentrations . The intersection point of the square symbol curves with the horizontal dotted line defines the critical concentration . The parameters are the same as in Fig. 3.

Dependence of the ADP filament growth rate (left axis) and the inverse mean waiting time at the barbed end (right axis) on actin monomer concentrations . The intersection point of the square symbol curves with the horizontal dotted line defines the critical concentration . The parameters are the same as in Fig. 3.

The growth rate as a function of . (a) The experimental data are taken from Fig. 5 of Ref. 7 and the simulation data in Fig. 6 are expressed in physical units with the rescaling factors and . (b) The experimental data are taken from Fig. 3 of Ref. 5 and the simulation data in Fig. 6 are expressed in physical units with the rescaling factors and . Note that the critical (or equilibrium) concentration at which the growth rate vanishes is rather different for the two sets of experimental data reflecting the different ionic conditions.

The growth rate as a function of . (a) The experimental data are taken from Fig. 5 of Ref. 7 and the simulation data in Fig. 6 are expressed in physical units with the rescaling factors and . (b) The experimental data are taken from Fig. 3 of Ref. 5 and the simulation data in Fig. 6 are expressed in physical units with the rescaling factors and . Note that the critical (or equilibrium) concentration at which the growth rate vanishes is rather different for the two sets of experimental data reflecting the different ionic conditions.

Length diffusion constant (solid squares) as defined in Eq. (39) for ADP-actin polymerization. The solid (red) line represents the analytical expression (41). The parameters are the same as in Fig. 3.

Length diffusion constant (solid squares) as defined in Eq. (39) for ADP-actin polymerization. The solid (red) line represents the analytical expression (41). The parameters are the same as in Fig. 3.

The growth rate (solid squares) as a function of . The solid (red) line is a linear fit. The detachment rate and the capture zone size .

The growth rate (solid squares) as a function of . The solid (red) line is a linear fit. The detachment rate and the capture zone size .

The growth rate (solid squares) as a function of , . The solid (red) line is a linear fit. (a) The capture zone size . (b) The capture zone size .

The growth rate (solid squares) as a function of , . The solid (red) line is a linear fit. (a) The capture zone size . (b) The capture zone size .

## Tables

Attachment rate constants , detachment rates , and equilibrium (or critical) concentration for Mg-ADP-actin (Ref. 12) at barbed and pointed ends of actin filaments.

Attachment rate constants , detachment rates , and equilibrium (or critical) concentration for Mg-ADP-actin (Ref. 12) at barbed and pointed ends of actin filaments.

Rescaled attachment rate constants , detachment rates , and equilibrium (or critical) concentration as used in the simulations.

Rescaled attachment rate constants , detachment rates , and equilibrium (or critical) concentration as used in the simulations.

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