A rigorous derivation and energetics of a wave equation with fractional damping

We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the water-air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally-damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy-dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally-damped wave equation with a time derivative of order 3/2

A rigorous derivation and energetics of a wave equation with fractional damping

We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the water–air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy–dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally damped wave equation with a time derivative of order 3/2

Linear Waves at Viscoelastic Interfaces Between Viscoelastic Media

We derive the general dispersion relation for interfacial waves along a planar viscoelastic boundary that separates two viscoelastic bulk media, including the e�ect of gravity. Our uni�ed theory contains Rayleigh waves, capillary-gravity- exural waves, Lucassen waves, bending waves in elastic plates, and the standard dispersion-free sound waves, as limiting cases. To illustrate our results, we consider waves at a viscoelastic interface immersed in water, and also at an air-water interface. We furthermore investigate pressure waves at a viscoelastic interface separating two identical viscoelastic bulk media, for which we consider both Kelvin-Voigt and Maxwell materials, as applicable to polymer gels and solutions. For all cases, we study how material properties determine the crossovers, scaling, and existence regimes, of the various interfacial waves. Since we include viscoelastic e�ects for all media involved, our theory allows to model waveguiding phenomena in biology, such as pressure pulses in axon membranes, which are possibly relevant for acoustic nerve pulse propagation phenomena

Software for Elastic Module Simulation of Laminin Network

This archive contains the software files used for the diluted laminin network simulations in "Basement Membrane Stiffness Determines Metastases Formation"

Basement membrane stiffness determines metastases formation

The basement membrane stiffness is shown to be a more dominant determinant than pore size in regulating cancer cell invasion, metastasis formation and patient survival. This stiffness is now known to be affected by the ratio of netrin-4 to laminin, with more netrin-4 leading to softer basement membranes. The basement membrane (BM) is a special type of extracellular matrix and presents the major barrier cancer cells have to overcome multiple times to form metastases. Here we show that BM stiffness is a major determinant of metastases formation in several tissues and identify netrin-4 (Net4) as a key regulator of BM stiffness. Mechanistically, our biophysical and functional analyses in combination with mathematical simulations show that Net4 softens the mechanical properties of native BMs by opening laminin node complexes, decreasing cancer cell potential to transmigrate this barrier despite creating bigger pores. Our results therefore reveal that BM stiffness is dominant over pore size, and that the mechanical properties of 'normal' BMs determine metastases formation and patient survival independent of cancer-mediated alterations. Thus, identifying individual Net4 protein levels within native BMs in major metastatic organs may have the potential to define patient survival even before tumour formation. The ratio of Net4 to laminin molecules determines BM stiffness, such that the more Net4, the softer the BM, thereby decreasing cancer cell invasion activity