Statistical Mechanics of an Elastically Pinned Membrane: Equilibrium Dynamics and Power Spectrum

Abstract

In biological settings, membranes typically interact locally with other membranes: the extracellular matrix in the exterior or internal cellular structures such as the cytoskeleton, locally pinning the membrane. Characterizing the dynamical properties of such interactions presents a difficult task. Significant progress has been achieved through simulations and experiments, yet analytical progress in modeling pinned membranes has been impeded by the complexity of governing equations. Here, we circumvent these difficulties by calculating analytically the time-dependent Green's function of the operator governing the dynamics of an elastically pinned membrane in a hydrodynamic surrounding and subject to external forces. This enables us to calculate the equilibrium power spectral density for an overdamped membrane pinned by an elastic, permanently attached spring subject to thermal excitations. By considering the effects of the finite experimental resolution on the measured spectra, we show that the elasticity of the pinning can be extracted from the experimentally measured spectrum. Membrane fluctuations can thus be used as a tool to probe mechanical properties of the underlying structures. Such a tool may be particularly relevant in the context of cell mechanics, in which the elasticity of the membrane's attachment to the cytoskeleton could be measured.

Figure 1

Snapshot from the Langevin simulation of a locally pinned membrane fluctuating in a nonspecific potential (left). A sketch of the system is shown (right). The membrane is residing in a harmonic potential of strength γ at h₀ separation from a flat substrate and pinned by an elastic spring of stiffness λ and rest length l₀ positioned at r = 0. To see this figure in color, go online.

Figure 2

Dynamical properties of a membrane at the pinning site. Comparison of modeling (lines) and simulations (symbols) shows excellent agreement across the entire parameter range. (a) PSD of a free membrane (Eq. 36) (full line) and a pinned tensionless membrane (Eq. 38) is shown (dashed lines for $\lambda /{\lambda }_m^0$ = 1, 3, 10, increasing in the direction of the arrows). The high-frequency regime of the PSD is unaffected by the pinning, and the free-membrane behavior (ω^−1.667) is recovered. (b) The low-frequency limit of the PSD (Eq. 39) is shown as a function of the membrane tension σ for different pinning strengths ($\lambda /{\lambda }_m^0$ = 1, 3, 10). For large tensions, a σ^−1.5 dependence is recovered irrespective of λ. (c) The low-frequency limit is shown as a function of the pinning strength for different membrane tensions ($8\sigma /{\lambda }_m^0$ = 1, 3, 10). For large bond stiffness, a λ⁻² dependence is displayed. All curves are plotted for κ = 20 k_BT, γ = 3 × 10⁻⁷k_BT/nm⁴, and η = 1 mPas. To see this figure in color, go online.

Figure 3

Effect of the finite resolution (averaging over a circle of radius R and a time interval τ) on the PSD at the pinning (Eq. 43). Lines represent analytical results, and symbols represent the simulation data. Note that the numerical averaging of the simulation data was done on a rectangular grid, which introduces a miniature underestimation of the averaged PSD amplitude in comparison to the analytical results, which assume averaging over a perfect circle (for more detail, see Supporting Materials and Methods, Section II.B). (a) Averaging decreases the difference between the free and the pinned PSD, therefore reducing the effect of the pinning on the PSD. Pinning has no effect on the high-frequency part of the spectrum for both the averaged and the unaveraged spectrum. (b) The low-frequency part of the PSD is a nonmonotone function of the spatial averaging area. On the other hand, increasing the averaging area always attenuates the high-frequency components, which, in this case, fall as ∼ω⁻² instead of ∼ω^−1.667. (c) The low-frequency part of the PSD for three different pinning stiffness values is shown as a function of the averaging area. The low-frequency part of the free-membrane PSD monotonically decreases, whereas the pinned-membrane PSD shows nonmonotonic behavior. The position of the peak is related to the pinning correlation length. (d) Time averaging introduces oscillatory behavior of the PSD, for which only the envelope of the PSD is shown. The combined effect of the spatial and temporal averaging gives an ∼ω⁻⁴ behavior of the high-frequency regime. Parameters: κ = 20 k_BT, σ = 10⁻²⁰k_BT/nm², γ = 3 × 10⁻⁷k_BT/nm⁴, η = 1 mPas, and $\lambda /{\lambda }_m^0$ = 10 (except in c, where the values of $\lambda /{\lambda }_m^0$ are noted in the legend). To see this figure in color, go online.