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Wrapping transition and wrapping-mediated interactions for discrete binding along an elastic filament: An exact solution
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10.1063/1.4757392
/content/aip/journal/jcp/137/14/10.1063/1.4757392
http://aip.metastore.ingenta.com/content/aip/journal/jcp/137/14/10.1063/1.4757392
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

A schematic drawing of the geometry as well as various quantities defined in the text for one and two cylinders wrapped on the Eulerian filament (shown in a symmetric mode). Specifically, for one cylinder (top) the angle at which the filament first touches the cylinder (advancing along the arc-length to the cylinder) is ψ and the angle at which it leaves is ψ. For two cylinders (bottom), separated by a distance , the contact angles α and α are as shown in the figure.

Image of FIG. 2.
FIG. 2.

Top: Schematic view of the wrapping of two cylinders in an antisymmetric mode where the wrapping angles for the two cylinders differ in sign. Bottom: The symmetric looped configuration (full line) of two wrapped cylinders corresponds to a negative value of the horizontal projected separation (i.e., cylinder 2 lies to the left of cylinder 1) and the symmetric extended configuration (dotted line) with positive (i.e., cylinder 2 lies to the right of cylinder 1).

Image of FIG. 3.
FIG. 3.

The wrapping transition of a single cylinder on an elastic filament. Exact solution for the average wrapping angle ratio is shown as a function of the dimensionless external tension = β for fixed μ = 1 and σ = 0.75.

Image of FIG. 4.
FIG. 4.

The wrapping transition as a function of the actual force and for temperatures = 210 K up to 390 K in steps of 30 K. The physical values for the parameters are given in the text. In both cases the curves move right as increases. Left panel: we have σ/μ = 0.15, σ < 0. The wrapping transition is relatively smooth and as increases it becomes stronger and moves to larger values of . Right panel: σ/μ = 0.35, σ > 0. The wrapping transition is much stronger and sharper as compared with the former case and, as predicted, the transition moves to larger values of as increases. It is this behavior in both cases which might be considered counter-intuitive.

Image of FIG. 5.
FIG. 5.

Dependence of the average horizontal separation ⟨ ⟩ (normalized to = + 2) on the external dimensionless tension = β for different sets of parameters in the case of constrained symmetric wrapping with α = α = π. Left panel: μ = 10 (high rigidity). Right panel: μ = 5 (low rigidity). Negative values of the average horizontal separation indicate the presence of a looped phase, i.e., the cylinder at larger distance along the elastic filament lies to the left of the other cylinder. Curves from left to right correspond to = for integer = 3, …, 8.

Image of FIG. 6.
FIG. 6.

Dependence of the average horizontal separation ⟨ ⟩ (normalized to = + 2) on the external dimensionless force = β for different sets of parameters in the case of constrained symmetric wrapping α = α = 3π/8. Left panel: μ = 10 (high rigidity). Right panel: μ = 5 (low rigidity). Clearly, for this set of parameters only the extended phase is allowed for large values of force regardless of rigidity. Curves from left to right correspond to = for integer = 3, …, 8.

Image of FIG. 7.
FIG. 7.

Mean projected separation, ⟨ ⟩ (blue curves), and mean wrapping angle, ⟨α⟩ (red curves), for unconstrained wrapping of elastic filament around two cylinders. Left panel: σ = 1. Right panel: σ = 10. Solid curves correspond to μ = 10 and dashed curves correspond to μ = 1. Note that the results have been normalized to maximum displacement, = + 2, or to maximum angle.

Image of FIG. 8.
FIG. 8.

Free energy as a function of the dimensionless external tension for μ = 10 and σ = 4.5. Left: Red solid curve corresponds to a single cylinder α = π, dashed curves correspond to antisymmetric double cylinder, α = −α = π. The lower (blue) curve corresponds to = 2π and the upper (black) curve is limit of increased separation. Right: Red solid curve corresponds to a single cylinder α = π, dashed curves correspond to symmetric double cylinder, α = α = π. The upper (blue) curve corresponds to = 2π and the lower (black) curve is limit of increased separation. Note the double cylinder (black) curve that coincides with the single cylinder (red) curve at free energy equal to zero gives the limit of infinite separation.

Image of FIG. 9.
FIG. 9.

Free energy as a function of the horizontal displacement projection ⟨ ⟩ is shown for the case of constrained wrapping with α = ±α = 5π/8, = 0.4, 1, 2, 3 (from lower to upper curve), μ = 50 and σ = 13.0. In the antisymmetric case (left) the effective interaction is attractive and in the symmetric case (right) it is repulsive.

Image of FIG. 10.
FIG. 10.

Left: The free energy as a function of the horizontal displacement projection ⟨ ⟩ (normalized to = + 2). Right: the required externally applied force, λ, as a function of the horizontal displacement projection ⟨ ⟩. Here we consider the case of constant wrapping with α = 5π/8 > π/2 and α = 3π/8 < π/2 for = 0.4, μ = 50, σ = 13.0, and = 4π.

Image of FIG. 11.
FIG. 11.

The free energy for the case of unconstrained wrapping angles. Left: The free energy is shown as a function of external force for a single cylinder and for two cylinders with symmetric wrapping. The upper dashed curve is for = π and the lower dashed curve corresponds to = 20 π, being effectively infinite. As one expects, for large separation and low force, the free energy of two cylinders is double that of a single cylinder (red curve). Right: The free energy as a function of projected horizontal distance , for = 0.01, 1, 2, 3 (from bottom to top). The free energy decreases with separation (entropic effect) and increases with force. In both cases μ = 1 and σ = 1.25.

Image of FIG. 12.
FIG. 12.

The required externally applied force, λ, as a function of the horizontal displacement projection ⟨ ⟩ (normalized to = + 2) is shown for the case of unconstrained wrapping with = 3, μ = 1, σ = 1.25, and = 2π.

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/content/aip/journal/jcp/137/14/10.1063/1.4757392
2012-10-10
2014-04-25
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Wrapping transition and wrapping-mediated interactions for discrete binding along an elastic filament: An exact solution
http://aip.metastore.ingenta.com/content/aip/journal/jcp/137/14/10.1063/1.4757392
10.1063/1.4757392
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