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### Abstract

We discuss motions of an elastic N × M membrane model whose constituents can bind reversibly with strength ɛ to adhesive sites of a flat substrate. One of the edges of the membrane (“front”) is driven in one direction at rate constant *p* by *N* stochastically treadmilling short parallel lines (“cortex”). The main conclusions derived from Monte Carlo studies of this model are the following: (a) Since the polymerizing cortex pushes only the leading edge of the membrane, the major part of the membranes is dragged behind. Therefore, the locomotion of the membrane can be described by frictional sliding processes which are asymmetrically distributed between front and rear of the membrane. A signature of this asymmetry is the difference between the life times of adhesion bonds at front and rear, τ_{1} and τ_{ M }, respectively, where τ_{1} ≫ τ_{ M }. (b) There are four characteristic times for the membrane motion: The first time, *T* _{0} ∼ τ_{ M } ∼ *e* ^{ aɛ}, is the resting time where the displacement of the membrane is practically zero. The second time, *T* _{ p } ∼ τ_{1} ∼ *M*, is the friction time which characterizes the time between two consecutive ruptures of adhesion bonds at the front, and which signalizes the onset of drift (“protrusion”) at the leading edge. The third time, *T* _{ r } ∼ *M* ^{γ(ɛ)} (γ > 1), characterizes the “retraction” of the trailing edge, which is the retarded response to the pulling leading edge. The fourth time, *T* _{ L } ∼ *M* ^{2}, is the growth time for fluctuation of the end-to-end distance. (c) The separation of time scales, *T* _{ r }/*T* _{ p } ∼ *M* ^{γ(ɛ) − 1}, leads to stretched fluctuations of the end-to-end distance, which are considered as stochastic cycles of protrusion and retraction on the time scale of *T* _{ L }. (d) The drift velocity v obeys anomalous scaling, , where *f* (*z*) ∼ *const*. for small drag *pM* ≪ 1, and *f* (*z*) ∼ *z* ^{−γ(ɛ)} for *pM* ≫ 1, which implies . These results may also turn out to be useful for the (more difficult) problem of understanding the protrusion-retraction cycle of crawling biological cells. We compare our model and our results to previous two-particle theories for membrane protrusion and to known stochastic friction models.

I. INTRODUCTION

II. MODEL AND SIMULATION METHOD

A. Surface model

B. Adhesion

C. The cortex

D. Remarks and implications

E. Simulation protocol

III. RESULTS

A. Membrane stretching

B. Frictional sliding

C. Velocity

D. Growth of end-to-end distance

E. Protrusion-retraction cycles

IV. DISCUSSIONS

A. The model: Limitations and comparisons

B. Comparisons to friction models

C. Conjectures on cell crawling

### Key Topics

- Adhesion
- 28.0
- Cell membranes
- 24.0
- Polymerization
- 24.0
- Cell adhesion
- 23.0
- Cell migration
- 14.0

## Figures

(Upper) Cartoon of a side view of the model. Adhesive substrate (triangles), membrane (black circles), integrin bonds (blue lines), plus end of filament (green circle). (Lower) Snapshot of top view for *M* = *N* = 10: red lines are integrin bonds, which connect the membrane (red dot) to adhesion sites (blue crosses). The green line represents the plus ends of the filaments (“cortex”).

(Upper) Cartoon of a side view of the model. Adhesive substrate (triangles), membrane (black circles), integrin bonds (blue lines), plus end of filament (green circle). (Lower) Snapshot of top view for *M* = *N* = 10: red lines are integrin bonds, which connect the membrane (red dot) to adhesion sites (blue crosses). The green line represents the plus ends of the filaments (“cortex”).

Average strain *L* _{ k } along a membrane coordinate *k*/*M* for *M* = 640, and various polymerization parameters *p* and adhesion strengths ɛ. (Inset) *L* _{ k } *versus* *k*/*M* for ɛ = 1, *p* = 1.0 and two different membrane sizes, *M* = 40, 2560.

Average strain *L* _{ k } along a membrane coordinate *k*/*M* for *M* = 640, and various polymerization parameters *p* and adhesion strengths ɛ. (Inset) *L* _{ k } *versus* *k*/*M* for ɛ = 1, *p* = 1.0 and two different membrane sizes, *M* = 40, 2560.

Average strain *F* _{ k } of integrin-substrate bonds along a membrane (*M* = 640) for various rate parameters *p*, and adhesion strengths ɛ.

Average strain *F* _{ k } of integrin-substrate bonds along a membrane (*M* = 640) for various rate parameters *p*, and adhesion strengths ɛ.

Stretching of membranes measured by the scaled end-to-end distance *L*/*M* as function of drag velocity *pM* for polymerization parameters 0.001 ⩽ *p* ⩽ 1, adhesion strengths ɛ = 0, 1, 2, 4, 6, and membrane sizes 20 ⩽ *M* ⩽ 2560.

Stretching of membranes measured by the scaled end-to-end distance *L*/*M* as function of drag velocity *pM* for polymerization parameters 0.001 ⩽ *p* ⩽ 1, adhesion strengths ɛ = 0, 1, 2, 4, 6, and membrane sizes 20 ⩽ *M* ⩽ 2560.

RMSD *dy*(*t*) of an adherent membrane (*M* = 160, ɛ = 6, *p* = 0.1).

RMSD *dy*(*t*) of an adherent membrane (*M* = 160, ɛ = 6, *p* = 0.1).

Location of integrin bonds, *S* _{ k }(*t*), at membrane site *k* as function of time for *k* = 1 (LE), *k* = *M* (TE), and *k* = *M*/2 (parameters: *M* = 160, ɛ = 6, *p* = 0.1).

Location of integrin bonds, *S* _{ k }(*t*), at membrane site *k* as function of time for *k* = 1 (LE), *k* = *M* (TE), and *k* = *M*/2 (parameters: *M* = 160, ɛ = 6, *p* = 0.1).

Average life time, τ_{ k }, of integrin bonds at membrane row *k*.

Average life time, τ_{ k }, of integrin bonds at membrane row *k*.

Integrin life times, τ_{1} and τ_{ M } as function of adhesion strength ɛ (parameters: *p* = 1.0, 0.1, 0.01, 0.001, and *M* = 160).

Integrin life times, τ_{1} and τ_{ M } as function of adhesion strength ɛ (parameters: *p* = 1.0, 0.1, 0.01, 0.001, and *M* = 160).

Log-log plot of scaled drift velocities v of membranes as function of drag *pM*. Parameters: 0.001 ⩽ *p* ⩽ 1; 20 ⩽ *M* ⩽ 2560; ɛ = 0, 1, 2. The broken line is . The inset shows a semi-log plot of the data for ɛ = 0 in order to illustrate the inflection point of the curve.

Log-log plot of scaled drift velocities v of membranes as function of drag *pM*. Parameters: 0.001 ⩽ *p* ⩽ 1; 20 ⩽ *M* ⩽ 2560; ɛ = 0, 1, 2. The broken line is . The inset shows a semi-log plot of the data for ɛ = 0 in order to illustrate the inflection point of the curve.

Finite size crossover scaling for RMSD of *L*(*t*). Parameters: *p* = 0.1; *M* = 40, 80, 160, 320; ɛ = 0, 6. (Upper inset) Exponent α as function of ɛ. (Lower inset) *dL*(*t*) *versus* *t*/*M* for ɛ = 6.

Finite size crossover scaling for RMSD of *L*(*t*). Parameters: *p* = 0.1; *M* = 40, 80, 160, 320; ɛ = 0, 6. (Upper inset) Exponent α as function of ɛ. (Lower inset) *dL*(*t*) *versus* *t*/*M* for ɛ = 6.

Scaled RMSD *dy*(*t*) for leading edge (LE) and trailing edge (TE) *versus* scaled time *t*/*M* and *t*/*M* ^{γ}, respectively. (Inset): Exponent γ as function of ɛ (parameters: *p* = 0.1; ɛ = 0, 6; *M* = 40, 80, 160, 320).

Scaled RMSD *dy*(*t*) for leading edge (LE) and trailing edge (TE) *versus* scaled time *t*/*M* and *t*/*M* ^{γ}, respectively. (Inset): Exponent γ as function of ɛ (parameters: *p* = 0.1; ɛ = 0, 6; *M* = 40, 80, 160, 320).

End-to-end distance, *L*(*t*) of the membrane (parameters: *M* = 160, *p* = 0.1).

End-to-end distance, *L*(*t*) of the membrane (parameters: *M* = 160, *p* = 0.1).

Scaled drift velocity v of membranes as function of *p* ^{1/γ} *M*. Parameters: 0.001 ⩽ *p* ⩽ 1; 20 ⩽ *M* ⩽ 2560; ɛ = 0, 1, 2, 4, 6. The exponent γ used for each ɛ = 0, 1, 2, 4, 6 is γ = 1, 1.05, 1.15, 1.25, 1.3, respectively.

Scaled drift velocity v of membranes as function of *p* ^{1/γ} *M*. Parameters: 0.001 ⩽ *p* ⩽ 1; 20 ⩽ *M* ⩽ 2560; ɛ = 0, 1, 2, 4, 6. The exponent γ used for each ɛ = 0, 1, 2, 4, 6 is γ = 1, 1.05, 1.15, 1.25, 1.3, respectively.

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