The variational modulus density theory explains mechanical responses of cell membranes and membrane crosslinkers.

Both classical mechanics and quantum mechanics explain Brownian motion. However, it remains unclear whether they are compatible with each other, as the physical and mathematical identity of the wavefunction in quantum mechanics has been elusive. In this work, a continuum theory using grammars in classical mechanics modeling, but potentially compatible with the quantum wavefunction, is introduced. The theory explains the confined Brownian motion of cell membrane inclusions interacting with extracellular matrices or cytoskeletons via elastic molecular crosslinkers. This crosslinker theory is integrated into the Canham-Helfrich-Evans model for fluid membranes. Calculations, based on a finite element method for the combined theory, reproduced measured data from adhesion molecular machineries and cell membranes. Overall, by providing physical and mathematical interpretations of the quantum wavefunction, the presented theoretical model provides improved capabilities for the realistic simulation of cell membranes and membrane linker proteins.