^{1}, Priscilla Hardas

^{1}, Sanoop Ramachandran

^{1}and Jean-Paul Ryckaert

^{1,a)}

### Abstract

We propose a hybrid molecular dynamics/multi-particle collision dynamics model to simulate a set of self-assembled semiflexible filaments and free monomers. Further, we introduce a Monte Carlo scheme to deal with single monomer addition (polymerization) or removal (depolymerization), satisfying the detailed balance condition within a proper statistical mechanical framework. This model of filaments, based on the wormlike chain, aims to represent equilibrium polymers with distinct reaction rates at both ends, such as self-assembled adenosine diphosphate-actin filaments in the absence of adenosine triphosphate (ATP) hydrolysis and other proteins. We report the distribution of filament lengths and the corresponding dynamical fluctuations on an equilibrium trajectory. Potential generalizations of this method to include irreversible steps like ATP-actin hydrolysis are discussed.

The authors are grateful to Marc Baus, Ray Kapral, P. B. Sunil Kumar, and Jean-Louis Martiel for illuminating discussions on various aspects of the work. We thank Georges Destree at the ULB/VUB Centre de Calcul for efficient programming support. S.R. wishes to acknowledge the BRIC (Bureau des Relations Internationales et de la Coopération) of the ULB, for financial support.

I. INTRODUCTION

II. GENERAL MESOSCOPIC MODEL OF A SET OF INTERACTING LIVING FILAMENTS AND FREE MONOMERS IN A THERMAL BATH

A. The solute

B. The solvent bath

C. Modeling of (de)polymerization steps

D. Simulation ensemble including chemical reactions

III. THEORETICAL FRAMEWORK

A. The filament length distribution and equilibrium constants

1. Ideal solution

2. Effects of interactions

B. Mean field rate equations and rate constants

1. Effective (de)polymerization rate constant

2. Mean field filament length dynamics

IV. ADOPTED MICROSCOPIC PARAMETERS AND LIST OF SIMULATION EXPERIMENTS

V. SIMULATION RESULTS

A. Equilibrium distributions

B. State point dependence of rates and related equilibrium constant

C. Dynamic fluctuations of filament length

VI. DISCUSSION

A. Methodological aspects

B. Parametrization of our generic biofilament model to F-actin

VII. CONCLUSION

### Key Topics

- Polymers
- 154.0
- Polymerization
- 32.0
- Equilibrium constants
- 26.0
- Solvents
- 24.0
- Statistical mechanics models
- 23.0

## Figures

A schematic showing the model filament with beads of size σ and bond length *l* _{0}. The angle between successive bonds are denoted by θ.

A schematic showing the model filament with beads of size σ and bond length *l* _{0}. The angle between successive bonds are denoted by θ.

State *I*: The polymerized state where the reacting monomer shown in dark blue is connected to the filament through intramolecular interactions and is, necessarily, located in a low intramolecular energy volume resulting from the intersection of a conic volume (in green) and the spherical layer (red) which, respectively, limit the bending and the stretching energies (see main text). The dashed blue circles represent the volume *V* _{s} and the light green particles represent free monomers.

State *I*: The polymerized state where the reacting monomer shown in dark blue is connected to the filament through intramolecular interactions and is, necessarily, located in a low intramolecular energy volume resulting from the intersection of a conic volume (in green) and the spherical layer (red) which, respectively, limit the bending and the stretching energies (see main text). The dashed blue circles represent the volume *V* _{s} and the light green particles represent free monomers.

State *J*: The depolymerized state where the reacting monomer shown in blue is free but is necessarily located within the spherical layer shown with dashed blue lines whose minimum radius limits the excluded volume pair interaction energy with the last monomer of the target filament end (see main text).

State *J*: The depolymerized state where the reacting monomer shown in blue is free but is necessarily located within the spherical layer shown with dashed blue lines whose minimum radius limits the excluded volume pair interaction energy with the last monomer of the target filament end (see main text).

is plotted vs with (fixed) after numerical evaluation and inversion of Eq. (18). The solid line is for *z* = 18, the dashed line for *z* = 40 and the dashed-dotted line for *z* = 72. It is seen that when *z* → ∞, as becomes large. The green (light) line at is a guide for the eye. Such curves have only a universal character for ideal solution conditions as the reduction factor 1/*K* ^{0} is only a function of temperature (see text).

is plotted vs with (fixed) after numerical evaluation and inversion of Eq. (18). The solid line is for *z* = 18, the dashed line for *z* = 40 and the dashed-dotted line for *z* = 72. It is seen that when *z* → ∞, as becomes large. The green (light) line at is a guide for the eye. Such curves have only a universal character for ideal solution conditions as the reduction factor 1/*K* ^{0} is only a function of temperature (see text).

Theoretical bounded filament distributions for subcritical () and supercritical () monomer densities shown as *P* ^{eq}(*i*) = ρ_{ i }/ρ_{ f } vs *i*. The solid, dotted, dashed, and dashed-dotted lines are for and 1.67, respectively, in the conditions of the plot for *z* = 18 in Fig. 4. The solid horizontal line corresponds to the case.

Theoretical bounded filament distributions for subcritical () and supercritical () monomer densities shown as *P* ^{eq}(*i*) = ρ_{ i }/ρ_{ f } vs *i*. The solid, dotted, dashed, and dashed-dotted lines are for and 1.67, respectively, in the conditions of the plot for *z* = 18 in Fig. 4. The solid horizontal line corresponds to the case.

Plot of *P* ^{eq}(*i*) = ρ_{ i }/ρ_{ f } vs *i* for two different values, namely subcritical experiment G2 (black circles) and supercritical experiment G4 (black squares). The solid black lines are the corresponding fits. The two dashed straight lines indicate the expected distributions in ideal solution conditions for the same state point (*N* _{ t }, *N* _{ f }, *V*, *T*).

Plot of *P* ^{eq}(*i*) = ρ_{ i }/ρ_{ f } vs *i* for two different values, namely subcritical experiment G2 (black circles) and supercritical experiment G4 (black squares). The solid black lines are the corresponding fits. The two dashed straight lines indicate the expected distributions in ideal solution conditions for the same state point (*N* _{ t }, *N* _{ f }, *V*, *T*).

The equilibrium constant calculated from the simulations. The open black circles are calculated from the free monomer density and the exponential filaments densities while the red stars are estimated from the ratio of the rates (1/ρ_{1})*U*/*W*. Within the error bars, the agreement between the two estimates is excellent. The solid red line is the ratio (1/ρ_{1})*f*(ρ_{1})/*g*(ρ_{1}) estimated from the fits. The dotted line represents the equilibrium constant for the ideal case.

The equilibrium constant calculated from the simulations. The open black circles are calculated from the free monomer density and the exponential filaments densities while the red stars are estimated from the ratio of the rates (1/ρ_{1})*U*/*W*. Within the error bars, the agreement between the two estimates is excellent. The solid red line is the ratio (1/ρ_{1})*f*(ρ_{1})/*g*(ρ_{1}) estimated from the fits. The dotted line represents the equilibrium constant for the ideal case.

Polymerization rate *U* (circles) and depolymerization rate *W* (squares) vs ρ_{1} for all experiments. The solid lines are linear (*W*) and quadratic (*U*) fitting curves discussed in the text. In the inset, the global growth rate *J* = *U* − *W* is indicated (diamonds). The solid line showing *J* = 0 is only for reference.

Polymerization rate *U* (circles) and depolymerization rate *W* (squares) vs ρ_{1} for all experiments. The solid lines are linear (*W*) and quadratic (*U*) fitting curves discussed in the text. In the inset, the global growth rate *J* = *U* − *W* is indicated (diamonds). The solid line showing *J* = 0 is only for reference.

*C*(*t*) vs *t* for experiments G1 (squares), G3 (circles), and G5 (diamonds). The solid curves represent the theoretical expressions of *C*(*t*) based on Eqs. (51) and (52) where the conditional probabilities are computed by solving according to standard methods, the mean field population dynamics given by Eqs. (50), using the *U* and *W* actual tabulated values of the same state point.

*C*(*t*) vs *t* for experiments G1 (squares), G3 (circles), and G5 (diamonds). The solid curves represent the theoretical expressions of *C*(*t*) based on Eqs. (51) and (52) where the conditional probabilities are computed by solving according to standard methods, the mean field population dynamics given by Eqs. (50), using the *U* and *W* actual tabulated values of the same state point.

〈Δ*N*(*t*)〉_{ N(0) = i } vs *t* at short times for different values of . The lines refer (from bottom) to an average over filaments *i* = 8, 9, and 10 in experiment G1, an average over filaments *i* = 9, 10, and 11 in experiment G3 and an average over filaments *i* = 11, 12, and 13 in experiment G5.

〈Δ*N*(*t*)〉_{ N(0) = i } vs *t* at short times for different values of . The lines refer (from bottom) to an average over filaments *i* = 8, 9, and 10 in experiment G1, an average over filaments *i* = 9, 10, and 11 in experiment G3 and an average over filaments *i* = 11, 12, and 13 in experiment G5.

## Tables

Table with simulation results for different values of *N* _{ t }. The particle packing fraction is represented by η. We fix *N* _{ f } = 80 in all the experiments with their size limited between 3 and *z* = 18. Here is the reduced monomer density obtained from the slope of the logarithm of the equilibrium filament length distribution, and is the measured single monomer density. For comparison, we also tabulate ρ_{1} numerically calculated from Eq. (18). The polymerization (*U*) and depolymerization (*W*) rates are calculated directly by counting.

Table with simulation results for different values of *N* _{ t }. The particle packing fraction is represented by η. We fix *N* _{ f } = 80 in all the experiments with their size limited between 3 and *z* = 18. Here is the reduced monomer density obtained from the slope of the logarithm of the equilibrium filament length distribution, and is the measured single monomer density. For comparison, we also tabulate ρ_{1} numerically calculated from Eq. (18). The polymerization (*U*) and depolymerization (*W*) rates are calculated directly by counting.

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