^{1}and Jean-Paul Ryckaert

^{1,a)}

### Abstract

To study the compressional forces exerted by a bundle of living stiff filaments pressing on a surface, akin to the case of an actin bundle in filopodia structures, we have performed particulate molecular dynamics simulations of a grafted bundle of parallel living (self-assembling) filaments, in chemical equilibrium with a solution of their constitutive monomers. Equilibrium is established as these filaments, grafted at one end to a wall of the simulation box, grow at their chemically active free end, and encounter the opposite confining wall of the simulation box. Further growth of filaments requires bending and thus energy, which automatically limit the populations of longer filaments. The resulting filament sizes distribution and the force exerted by the bundle on the obstacle are analyzed for different grafting densities and different sub- or supercritical conditions, these properties being compared with the predictions of the corresponding ideal confined bundle model. In this analysis, non-ideal effects due to interactions between filaments and confinement effects are singled out. For all state points considered at the same temperature and at the same gap width between the two surfaces, the force per filament exerted on the opposite wall appears to be a function of a rescaled free monomer density . This quantity can be estimated directly from the characteristic length of the exponential filament size distribution P observed in the size domain where these grafted filaments are not in direct contact with the wall. We also analyze the dynamics of the filament contour length fluctuations in terms of effective polymerization (U) and depolymerization (W) rates, where again it is possible to disentangle non-ideal and confinement effects.

The authors wish to thank M. Baus, G. Ciccotti, J.-F. Joanny, P. B. S. Kumar, D. Lacoste, and C. Pierleoni for useful discussions about the present work. They warmly thank G. Destrée for invaluable technical help. S. Ramachandran acknowledges financial help from the BRIC (Bureau des Relations Internationales et de Coopération) of the Université Libre de Bruxelles. J.-P. R. thanks P.B.S. Kumar for hosting him at IIT Madras where this work was completed.

I. FORCE GENERATION BY A BUNDLE OF PARALLEL LIVING FILAMENTS: INTRODUCTION

II. THERMODYNAMIC DESCRIPTION OF THE CONFINED BUNDLE

A. The general approach

B. The ideal bundle properties

III. THE SPECIFIC MODEL WITH INTERMOLECULAR INTERACTIONS FOR THE CONFINED SELF-ASSEMBLING GRAFTED BUNDLE

A. Model Hamiltonian for a grafted and confined single filament and single living filament properties

B. Model Hamiltonian for a bundle of self-assembled filaments and free monomers in interaction and the corresponding reactive canonical ensemble

C. Computer simulation experiments: Choice of parameters and list of experiments

IV. SIMULATION RESULTS

A. Distribution of filament lengths and free monomer distribution

B. Data on normal pressure/compressive force and free monomers chemical potential

C. Filament size kinetics

V. DISCUSSION AND PERSPECTIVES

## Figures

Simulation box with N f living filaments (shown in red) anchored normal to the solid wall on the left, in chemical equilibrium with free monomers (shown in green). Single monomer end-filament (de)polymerization takes place continuously for a prescribed total number N t of monomers. The filament growth (in supercritical conditions) is obstructed by the second solid wall on the right which exerts a normal equilibrium force on the bundle. The snapshot shown (note that periodic boundary conditions apply in lateral directions) is extracted from the N f = 32, N t = 500 case within the IIa experiments series (see Table I ).

Simulation box with N f living filaments (shown in red) anchored normal to the solid wall on the left, in chemical equilibrium with free monomers (shown in green). Single monomer end-filament (de)polymerization takes place continuously for a prescribed total number N t of monomers. The filament growth (in supercritical conditions) is obstructed by the second solid wall on the right which exerts a normal equilibrium force on the bundle. The snapshot shown (note that periodic boundary conditions apply in lateral directions) is extracted from the N f = 32, N t = 500 case within the IIa experiments series (see Table I ).

Distribution of filament sizes obtained in the simulation series IIa of experiments at surface density σ f d 2 = 0.222 for N t values 230 (circles), 300 (squares), 370 (lozenges), 437 (triangles pointing up), 500 (triangles pointing left), and 550 (triangles pointing down). Continuous lines have been drawn for better data visualization. (Inset) The ratios P i + 1/P i are plotted versus i (for i < z) for the experiments N t = 437, 500, 550 to which corresponds, in the same order, an increasing fitting value indicated by an horizontal line (see text).

Distribution of filament sizes obtained in the simulation series IIa of experiments at surface density σ f d 2 = 0.222 for N t values 230 (circles), 300 (squares), 370 (lozenges), 437 (triangles pointing up), 500 (triangles pointing left), and 550 (triangles pointing down). Continuous lines have been drawn for better data visualization. (Inset) The ratios P i + 1/P i are plotted versus i (for i < z) for the experiments N t = 437, 500, 550 to which corresponds, in the same order, an increasing fitting value indicated by an horizontal line (see text).

Free monomer density as a function of x/d from the anchoring wall at (x = 0) to the obstacle wall at (x/d = 16) in the simulation series IIa of experiments at surface density σ f d 2 = 0.2222, for N t values 230 (black circles), 370 (green lozenges), 550 (violet triangles). Corresponding values of are indicated as a continuous horizontal line of same color except for the highest density (for N t = 550) lying outside the shown density window.

Free monomer density as a function of x/d from the anchoring wall at (x = 0) to the obstacle wall at (x/d = 16) in the simulation series IIa of experiments at surface density σ f d 2 = 0.2222, for N t values 230 (black circles), 370 (green lozenges), 550 (violet triangles). Corresponding values of are indicated as a continuous horizontal line of same color except for the highest density (for N t = 550) lying outside the shown density window.

Osmotic force per filament f N as a function of for simulation data corresponding to various surface densities σ f d 2 = 0.125 (experiment I, blue lozenges), 0.222 (red filled squares for experiment IIa and red open squares for experiment IIb), and 0.320 (experiment III, black circles) compared to the ideal solution prediction of the force per filament as a function of based on the filament-wall microscopic model used in the bundle simulations.

Osmotic force per filament f N as a function of for simulation data corresponding to various surface densities σ f d 2 = 0.125 (experiment I, blue lozenges), 0.222 (red filled squares for experiment IIa and red open squares for experiment IIb), and 0.320 (experiment III, black circles) compared to the ideal solution prediction of the force per filament as a function of based on the filament-wall microscopic model used in the bundle simulations.

Osmotic force per filament f N as a function of for simulation data corresponding to various surface densities σ f d 2 = 0.125 (experiment I, blue lozenges), 0.222 (red filled squares for experiment IIa and red open squares for experiment IIb), and 0.320 (experiment III, black circles) compared to the ideal solution prediction of the force per filament as a function of based on the filament-wall microscopic model used in the bundle simulations.

Local reduced force (red dashed curve) and local reduced average force (green continuous curve) exerted by the right wall located at gap distance L/d = 16 on one filament anchored normally at the left wall and growing towards the right, as a function of , for the filament model (with l p = 250d) and for the filament-wall interaction model which are used in bundle simulations. The black dashed-dotted curve shows predicted by Eq. (1) . (Inset) Same local reduced force for a right wall position varying between L/d = 15.5 and L/d = 16.5. The three curves (from bottom upwards) correspond to free monomer reduced number densities , , and .

Local reduced force (red dashed curve) and local reduced average force (green continuous curve) exerted by the right wall located at gap distance L/d = 16 on one filament anchored normally at the left wall and growing towards the right, as a function of , for the filament model (with l p = 250d) and for the filament-wall interaction model which are used in bundle simulations. The black dashed-dotted curve shows predicted by Eq. (1) . (Inset) Same local reduced force for a right wall position varying between L/d = 15.5 and L/d = 16.5. The three curves (from bottom upwards) correspond to free monomer reduced number densities , , and .

Polymerization rate U i and depolymerization rate W i (in units ) for filaments of various sizes i in the series of experiments I, N t = 450 (circles), 500 (squares), 525 (lozenges), and 550 (triangles). Only points known with reasonable statistics (corresponding to significant P i values) are indicated, with filled symbols for U i and empty symbols for W i . The error bars being set to one σ (estimated from four independent runs per experiment). Continuous lines are shown to facilitate data observation. Estimated bulk constant values U 0 (see text) are indicated by horizontal dashed lines. The inset shows the polymerization rates for all values of i in their rescaled form U i /U 0.

Polymerization rate U i and depolymerization rate W i (in units ) for filaments of various sizes i in the series of experiments I, N t = 450 (circles), 500 (squares), 525 (lozenges), and 550 (triangles). Only points known with reasonable statistics (corresponding to significant P i values) are indicated, with filled symbols for U i and empty symbols for W i . The error bars being set to one σ (estimated from four independent runs per experiment). Continuous lines are shown to facilitate data observation. Estimated bulk constant values U 0 (see text) are indicated by horizontal dashed lines. The inset shows the polymerization rates for all values of i in their rescaled form U i /U 0.

## Tables

List of simulation experiments performed at k B T = 1, L = 16d, l p = 250d, and , regrouped into four experiment series in which only the total number of monomers N t is changing. The transverse area is A = H 2, the brush surface density is σ f = N f /A, the equilibrium constant K 0 (see Eq. (31) ) is based on different values of , namely, 8.04211 (I, IIb), 7.34894 (IIa), and 6.61497 (III).

List of simulation experiments performed at k B T = 1, L = 16d, l p = 250d, and , regrouped into four experiment series in which only the total number of monomers N t is changing. The transverse area is A = H 2, the brush surface density is σ f = N f /A, the equilibrium constant K 0 (see Eq. (31) ) is based on different values of , namely, 8.04211 (I, IIb), 7.34894 (IIa), and 6.61497 (III).

Equilibrium data on a brush of N f = 32 filaments pressing against a fixed wall at density σ f d 2 = 0.2222 (experiment IIa with K 0 d −3 = 39.0698). N t is the total number of monomers in the volume V/d 3 = 2304 which also contains 1440 MPCD solvent particles. ⟨i⟩ is the average size of a filament expressed in the number of monomers, including those which initiate the filament at the left wall. The two next columns provide data on the filament size fluctuations, namely, their amplitude and characteristic relaxation time τ i . ⟨X i ⟩ is the averaged projection of the end-to-end vector of the filaments on the normal to the walls (x axis). ⟨U⟩ and ⟨W⟩ are the average (de)polymerization rates per free filament end. Note that times and frequencies are made dimensionless by using the time unit u t defined in Sec. III C and unmentioned errors are of one unit on the last digit indicated.

Equilibrium data on a brush of N f = 32 filaments pressing against a fixed wall at density σ f d 2 = 0.2222 (experiment IIa with K 0 d −3 = 39.0698). N t is the total number of monomers in the volume V/d 3 = 2304 which also contains 1440 MPCD solvent particles. ⟨i⟩ is the average size of a filament expressed in the number of monomers, including those which initiate the filament at the left wall. The two next columns provide data on the filament size fluctuations, namely, their amplitude and characteristic relaxation time τ i . ⟨X i ⟩ is the averaged projection of the end-to-end vector of the filaments on the normal to the walls (x axis). ⟨U⟩ and ⟨W⟩ are the average (de)polymerization rates per free filament end. Note that times and frequencies are made dimensionless by using the time unit u t defined in Sec. III C and unmentioned errors are of one unit on the last digit indicated.

Equilibrium data on a brush of N f = 32 filaments pressing against a fixed wall at density σ f d 2 = 0.125 (experiment I). N t is the total number of monomers in the volume V/d 3 = 4096 which also contains 2560 MPCD solvent particles. ⟨ρ1⟩ is the average free monomer density in the total volume V. The quantity ⟨E/(N 1 + 1)⟩ is the average expression in the argument of the logarithm term used in Widom-like formula Eq. (40) , which then leads to reported in next column. The two next columns provide the reduced free monomer concentration defined as (with K 0/d 3 = 78.13968 in the present case) and the effective reduced density obtained by an exponential fitting of the P i simulation data. The next column provides an estimate of the apparently i independent ratio r = f i − 1/f i evaluated as the ratio . and are, respectively, the contributions of the brush and the free monomers to the pressure exerted on the right wall, p ∞ is the pressure in the pure monomer solution in chemical equilibrium with the brush. The last column provides the reduced force per filament computed as . Note that the limit for buckling for an independent filament is , so that the N = 600 case needs to be considered with care as the effective .

Equilibrium data on a brush of N f = 32 filaments pressing against a fixed wall at density σ f d 2 = 0.125 (experiment I). N t is the total number of monomers in the volume V/d 3 = 4096 which also contains 2560 MPCD solvent particles. ⟨ρ1⟩ is the average free monomer density in the total volume V. The quantity ⟨E/(N 1 + 1)⟩ is the average expression in the argument of the logarithm term used in Widom-like formula Eq. (40) , which then leads to reported in next column. The two next columns provide the reduced free monomer concentration defined as (with K 0/d 3 = 78.13968 in the present case) and the effective reduced density obtained by an exponential fitting of the P i simulation data. The next column provides an estimate of the apparently i independent ratio r = f i − 1/f i evaluated as the ratio . and are, respectively, the contributions of the brush and the free monomers to the pressure exerted on the right wall, p ∞ is the pressure in the pure monomer solution in chemical equilibrium with the brush. The last column provides the reduced force per filament computed as . Note that the limit for buckling for an independent filament is , so that the N = 600 case needs to be considered with care as the effective .

Equilibrium data on a brush of N f = 32 filaments pressing against a fixed wall at density σ f d 2 = 0.2222 (experiment II). The first set of data coined as experiment IIa and the second set of data coined as experiment IIb differ only by a different choice of (see Table I ) leading to different K 0 values indicated in the extra second column. N t is the total number of monomers in the volume V/d 3 = 2304 which also contains 1440 MPCD solvent particles. See caption of similar Table III for explanations on the nature of the shown data and for the last caution sentence which applies here also for the N = 551 case in experiment IIb.

Equilibrium data on a brush of N f = 32 filaments pressing against a fixed wall at density σ f d 2 = 0.2222 (experiment II). The first set of data coined as experiment IIa and the second set of data coined as experiment IIb differ only by a different choice of (see Table I ) leading to different K 0 values indicated in the extra second column. N t is the total number of monomers in the volume V/d 3 = 2304 which also contains 1440 MPCD solvent particles. See caption of similar Table III for explanations on the nature of the shown data and for the last caution sentence which applies here also for the N = 551 case in experiment IIb.

Equilibrium data on a brush of N f = 32 filaments pressing against a fixed wall at density σ f d 2 = 0.32 (experiment III). N t is the total number of monomers in the volume V/d 3 = 1600 which also contains 1000 MPCD solvent particles. See caption of similar Table III for explanations on the nature of the shown data. Note that K 0/d 3 = 18.75352 in the present case.

Equilibrium data on a brush of N f = 32 filaments pressing against a fixed wall at density σ f d 2 = 0.32 (experiment III). N t is the total number of monomers in the volume V/d 3 = 1600 which also contains 1000 MPCD solvent particles. See caption of similar Table III for explanations on the nature of the shown data. Note that K 0/d 3 = 18.75352 in the present case.

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