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Many biological processes involve transport and organization of inclusions in thin fluid interfaces. A key aspect of these assemblies is the active dissipative stresses applied from the inclusions to the fluid interface, resulting in long-range active interfacial flows. We study the effect of these active flows on the self-organization of rod-like inclusions in the interface. Specifically, we consider a di- lute suspension of Brownian rods of length L, embedded in a thin fluid interface of 2D viscosity ηm and surrounded on both sides with 3D fluid domains of viscosity ηf . The momentum transfer from the interfacial flows to the surrounding fluids occurs over length l0 = ηm/ηf , known as Saffman- Delbru ̈ck length. We use zeroth, first and second moments of Smoluchowski equation to obtain the conservation equations for concentration, polar order and nematic order fields, and use linear stability analysis and continuum simulations to study the dynamic variations of these fields as a function of L/l0, the ratio of active to thermal stresses, and the dimensionless self-propulsion velocity of the embedded particles. We find that at sufficiently large activities, the suspensions of active extensile stress (pusher) with no directed motion undergo a finite wavelength nematic ordering, with the length of the ordered domains decreasing with increasing L/l0. The ordering transition is hindered with further increases in L/l0. In contrast, the suspensions with active contractile stress (puller) remain uniform with variations of activity. We notice that the self-propulsion velocity results in significant concentration fluctuations and changes in the size of the order domains that depend on L/l0. Our re- search highlights the role of hydrodynamic interactions in the self-organization of active inclusions on biological interfaces. 

In studying the transport of inclusions in multiphase systems we are often interested in integrated quantities such as the net force and the net velocity of the inclusions. In the reciprocal theorem the known solution to the first and typically easier boundary value problem is used to compute the integrated quantities, such as the net force, in the second problem without the need to solve that problem. Here, we derive a reciprocal theorem for poro-viscoelastic (or biphasic) materials that are composed of a linear compressible solid phase, permeated by a viscous fluid. As an example, we analytically calculate the time-dependent net force on a rigid sphere in response to point forces applied to the elastic network and the Newtonian fluid phases of the biphasic material. We show that when the point force is applied to the fluid phase, the net force on the sphere evolves over time scales that are independent of the distance between the point force and the sphere; in comparison, when the point force is applied to the elastic phase, the time scale for force development increases quadratically with the distance, in line with the scaling of poroelastic relaxation time. Finally, we formulate and discuss how the reciprocal theorem can be applied to other areas, including (i) calculating the network slip on the sphere's surface, (ii) computing the leading-order effects of nonlinearities in the fluid and network forces and stresses, and (iii) calculating self-propulsion in biphasic systems. 

Many of the cell membrane's vital functions are achieved by the self-organization of the proteins and biopolymers embedded in it. The protein dynamics is in part determined by its drag. A large number of these proteins can polymerize to form filaments.In vitrostudies of protein–membrane interactions often involve using rigid beads coated with lipid bilayers, as a model for the cell membrane. Motivated by this, we use slender-body theory to compute the translational and rotational resistance of a single filamentous protein embedded in the outer layer of a supported bilayer membrane and surrounded on the exterior by a Newtonian fluid. We first consider the regime where the two layers are strongly coupled through their inter-leaflet friction. We find that the drag along the parallel direction grows linearly with the filament's length and quadratically with the length for the perpendicular and rotational drag coefficients. These findings are explained using scaling arguments and by analysing the velocity fields around the moving filament. We then present and discuss the qualitative differences between the drag of a filament moving in a freely suspended bilayer and a supported membrane as a function of the membrane's inter-leaflet friction. Finally, we briefly discuss how these findings can be used in experiments to determine membrane rheology. In summary, we present a formulation that allows computation of the effects of membrane properties (its curvature, viscosity and inter-leaflet friction), and the exterior and interior three-dimensional fluids’ depth and viscosity on the drag of a rod-like/filamentous protein, all in a unified theoretical framework. 

The cell cytoskeleton is a dynamic assembly of semi-flexible filaments and motor proteins. The cytoskeleton mechanics is a determining factor in many cellular processes, including cell division, cell motility and migration, mechanotransduction and intracellular transport. Mechanical properties of the cell, which are determined partly by its cytoskeleton, are also used as biomarkers for disease diagnosis and cell sorting. Experimental studies suggest that in whole cell scale, the cell cytoskeleton and its permeating cytosol may be modelled as a two-phase poro-viscoelastic (PVE) material composed of a viscoelastic (VE) network permeated by a viscous cytosol. We present the first general solution to this two-phase system in spherical coordinates, where we assume that both the fluid and network phases are in their linear response regime. Specifically, we use generalized linear incompressible and compressible VE constitutive equations to describe the stress in the fluid and network phases, respectively. We assume a constant permeability that couples the fluid and network displacements. We use these general solutions to study the motion of a rigid sphere moving under a constant force inside a two-phase system, composed of a linear elastic network and a Newtonian fluid. It is shown that the network compressibility introduces a slow relaxation of the sphere and non-monotonic network displacements with time along the direction of the applied force. Our results can be applied to particle-tracking microrheology to differentiate between PVE and VE materials, and to measure the fluid permeability as well as VE properties of the fluid and the network phases.