^{1,2,a)}, Julian Shillcock

^{1,3}and Reinhard Lipowsky

^{1,b)}

### Abstract

Actin polymerization is coupled to the hydrolysis of adenosine triphosphate (ATP) into adenosine diphosphate (ADP) and inorganic phosphate . Therefore, each protomer within an actin filament can attain three different nucleotide states corresponding to bound ATP, , and ADP. These protomer states form spatial patterns on the growing (or shrinking) filaments. Using Brownian dynamics simulations, the growth behavior of long filaments is studied, together with the associated protomer patterns, as a function of ATP-actin monomer concentration, , within the surrounding solution. For concentrations close to the critical concentration , the filaments undergo treadmilling, i.e., they grow at the barbed and shrink at the pointed end, which leads to directed translational motion of the whole filament. The corresponding nonequilibrium states are characterized by several global fluxes and by spatial density and flux profiles along the filaments. We focus on a certain set of transition rates as deduced from *in vitro* experiments and find that the associated treadmilling (or turnover) rate is about 0.08 monomers per second.

Kunkun Guo thanks Professor Dong Qiu and Xin Li for stimulating discussions and acknowledges financial support by BNLMS and NSFC (Grant Nos. 10947177 and 21004018). Julian Shillcock was supported by the Danish National Research Foundation via MEMPHYS.

I. INTRODUCTION

II. THEORETICAL APPROACH

A. Basic aspects of coarse-grained description

B. Transition rates and filament dynamics

C. Detailed balance in the absence of hydrolysis

D. Simulation parameters

E. Rescaling procedure

III. RESULTS AND DISCUSSION

A. Growth rates and concentration regimes

B. Treadmilling rate: Single filament simulations

C. Treadmilling rate: On-off statistics at filament ends

D. State probabilities of terminal protomers

E. Phosphate release from terminal protomers

F. Steady state protomer patterns

G. Flux balance relations

IV. CONCLUSIONS

### Key Topics

- Polymers
- 37.0
- Polymerization
- 16.0
- Nucleotides
- 7.0
- Diffusion
- 6.0
- Brownian dynamics
- 5.0

## Figures

Schematic view of the different processes that determine the time evolution of actin polymerization and ATP hydrolysis. Actin monomers and protomers with a bound ATP, , and ADP molecule are denoted by , and , respectively. The two conical regions at the barbed (right) and pointed (left) end of the filament have linear dimensions and . These regions represent capture zones for the diffusing monomers and determine the attachment rates and for these monomers at the two ends. The protomers of the filament are transformed into protomers with the ATP cleavage rate , the protomers are changed into protomers with the release rate . The two terminal protomers at the two filament ends can also detach from the filament as illustrated for a protomer at the barbed end and for a protomer at the pointed end. Since both terminal protomers can attain three different nucleotide states, one has to distinguish six detachment rates, see text.

Schematic view of the different processes that determine the time evolution of actin polymerization and ATP hydrolysis. Actin monomers and protomers with a bound ATP, , and ADP molecule are denoted by , and , respectively. The two conical regions at the barbed (right) and pointed (left) end of the filament have linear dimensions and . These regions represent capture zones for the diffusing monomers and determine the attachment rates and for these monomers at the two ends. The protomers of the filament are transformed into protomers with the ATP cleavage rate , the protomers are changed into protomers with the release rate . The two terminal protomers at the two filament ends can also detach from the filament as illustrated for a protomer at the barbed end and for a protomer at the pointed end. Since both terminal protomers can attain three different nucleotide states, one has to distinguish six detachment rates, see text.

The overall growth rate as a function of free actin concentration . The two different sets of data have been obtained by measuring the average filament length (black squares) and the average waiting times between successive attachment and detachment events (red stars) as explained in the text. The transition rates are the same as in Table I.

The overall growth rate as a function of free actin concentration . The two different sets of data have been obtained by measuring the average filament length (black squares) and the average waiting times between successive attachment and detachment events (red stars) as explained in the text. The transition rates are the same as in Table I.

Filament length and positions of barbed and pointed end as functions of simulation time . The length is measured in units of protomer number (left -axis), the end positions are measured in units of the basic length scale (right -axis). The filament grows parallel to the long side of the simulation box, which has an extension of , the longitudinal extension of one protomer is . The transition rates are the same as in Table I, the free actin concentration is close to the critical concentration .

Filament length and positions of barbed and pointed end as functions of simulation time . The length is measured in units of protomer number (left -axis), the end positions are measured in units of the basic length scale (right -axis). The filament grows parallel to the long side of the simulation box, which has an extension of , the longitudinal extension of one protomer is . The transition rates are the same as in Table I, the free actin concentration is close to the critical concentration .

Normalized histograms for time intervals between two successive monomer attachments and two successive protomer detachments at the (a) barbed and (b) pointed end. The parameters are the same as in Fig. 3.

Normalized histograms for time intervals between two successive monomer attachments and two successive protomer detachments at the (a) barbed and (b) pointed end. The parameters are the same as in Fig. 3.

The three probabilities with , and for the terminal protomer at the (a) barbed and (b) pointed end as a function of free actin concentration . The transition rates are the same as in Table I.

The three probabilities with , and for the terminal protomer at the (a) barbed and (b) pointed end as a function of free actin concentration . The transition rates are the same as in Table I.

Overall growth rate as a function of free actin concentration for three different values of the rate , which governs the detachment of protomers from the barbed end. The other transition rates have the same values as in Table I.

Overall growth rate as a function of free actin concentration for three different values of the rate , which governs the detachment of protomers from the barbed end. The other transition rates have the same values as in Table I.

Overall growth rate as a function of free actin concentration for three different choices of the release rates and , which govern the internal and terminal protomers, respectively. The values of all other transition rates are in Table I.

Overall growth rate as a function of free actin concentration for three different choices of the release rates and , which govern the internal and terminal protomers, respectively. The values of all other transition rates are in Table I.

The density profiles (left) and (right) for , and protomers as a function of the spatial coordinate where is measured in units of the projected protomer size. The barbed end is located at , the pointed end at or . The values of the transition rates are given in Table I; the free actin concentration is .

The density profiles (left) and (right) for , and protomers as a function of the spatial coordinate where is measured in units of the projected protomer size. The barbed end is located at , the pointed end at or . The values of the transition rates are given in Table I; the free actin concentration is .

## Tables

Numerical values for the different transition rates as used in the simulations: Attachment rate constants for monomers; detachment rates for , , and protomers; ATP cleavage rate ; and release rate . All rates are given in units of 1/s, all rate constants in units of . The superscripts ba and po refer to the barbed and pointed end, respectively. Except for the detachment rate , all rates have the same numerical values as those in Table I of Ref. 23. The rate is determined by detailed balance in the absence of ATP hydrolysis.

Numerical values for the different transition rates as used in the simulations: Attachment rate constants for monomers; detachment rates for , , and protomers; ATP cleavage rate ; and release rate . All rates are given in units of 1/s, all rate constants in units of . The superscripts ba and po refer to the barbed and pointed end, respectively. Except for the detachment rate , all rates have the same numerical values as those in Table I of Ref. 23. The rate is determined by detailed balance in the absence of ATP hydrolysis.

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