The fluctuating dynamics of a filament embedded in a spherical membrane with applications to microrheology

Many of the cell membrane’s vital functions, including molecular transport and signal transduction, are achieved by self-organization of its lipids and proteins. These structural variations coincide with changes in membrane mechanics. Microrheology has been extensively used for studying the mechanics of biological materials. Yet, the theoretical frameworks of microrheology need to be modified for applications to 2D membranes. Here, we outline the theoretical framework that relates the linear viscoelastic properties of a freely suspended and supported spherical membrane to the fluctuating dynamics of a semiflexible filament embedded in that membrane. By using filaments as probes one can simultaneously measure the membrane mechanics over multiple length scales. We begin by deriving the response function that computes the time-dependent filament shape, specifically the time-dependent amplitudes of different filament wavelengths, as a function of the external forces on the filament as well as membrane viscoelasticity, and the mechanics of 3D domains that surround the membrane. We, then, use Fluctuation-Dissipation theorem to express these shape amplitudes in terms of thermal forces and the response function. We show that the confined spherical geometry and the presence of rigid boundaries in supported membranes give rise to unique dynamical features in the relaxation behavior of the fluctuating filament. Finally, we show how the filament’s shape can be used to compute membrane rheology by considering examples where the membrane is modeled as a Maxwell fluid, Kelvin-Void solid and a power-law viscoelastic fluid.