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

Abstract

In biological settings membranes typically interact locally with other membranes or the extracellular matrix in the exterior, as well as with internal cellular structures such as the cytoskeleton. Characterization of the dynamic properties of such interactions presents a difficult task. Significant progress has been achieved through simulations and experiments, yet analytical progress in modelling 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, where the elasticity of the membrane's attachment to the cytoskeleton could be measured.

Comparison of thermal and athermal dynamics of the cell membrane slope fluctuations in the presence and absence of Latrunculin-B

“…The best fit with a straight line having 0 slope yields a mean value of −1.27 ± 0.15 which is in accordance with the with the theoretical value of −1 (equation ( 21)). However, the higher value of exponent could be a result of the finite size of the particle [46,47]. The slope fluctuations show a consistent inverse law behaviour in both the cases.…”

Nuclear Stiffness Decreases with Disruption of the Extracellular Matrix in Living Tissues

“…Generally, AFM data is used to extract material properties of biological materials by fitting the AFM indentation curve to the Hertz theory of elasticity to extract Young's modulus. [ 30 ] However, the native ECM exerts an isotropic pre‐stress tension through cell‐matrix attachments which oppose the applied force from the AFM probe during membrane deformation, limiting the applicability of typical force‐indentation models for elasticity. [ 31 ] Furthermore, an accurate derivation of elastic modulus for a material requires a careful estimation of the AFM tip geometry.…”

Long-range correlations in pinned athermal networks

“…These results are also relevant in biological networks, where contractile forces are important in many processes such as wound healing and the motion of cytoskeletons mediated by active internal forces [12,13,44]. Our analysis can be generalized to incorporate transverse forces [10,11,45], as well as different periodic backgrounds, in two as well as three dimensions. Our techniques can also be used to study the response of athermal networks in the presence of disorder, such as in disordered crystals [9,46].…”

“…Such systems are robust to thermal agitations and differ from thermal systems in their response to external perturbations, as well as fluctuations in positions and forces [9]. The behaviour of athermal systems driven by local active forces arise in various contexts in biology and physics [10][11][12][13], and can give rise to extreme outof-equilibrium behaviour. Recently, dense active systems composed of athermal particles have been used to model diverse phenomena such as animal flocking and transport through channels [14].…”