Criticality and supradiffusion in biological membranes: The effect of transverse multiplicative fluctuations

We suggest that the cytoskeleton in contact with the inner surface of biological membranes in cells exhibiting tensegrity, may be considered as a system near critical conditions. This feature will influence the dynamical processes, such as diffusion, associated with the membrane’s fluctuations induced by the surrounding medium. In this work we analyze a model for the diffusion of particles attached to the membrane due to the transverse membrane fluctuations when the surrounding fluid is near a critical state. We describe these fluctuations by a multiplicative Langevin equation with colored noise which accounts for the rheological nature of the medium. From the associated Fokker–Planck equation we calculate analytically the mean square displacement (MSD) and the dynamic structure factor (DSF) of the particles. In the limit of additive white noise, it is well known that the MSD and the DSF exhibit sub-diffusive behavior with a scaling MSD∼t2/3 and DSF∼exp(ΓkAWt)2/3. In contrast, we show that for the case of external fluctuations arising from criticality and modeled by an Ornstein–Uhlenbeck multiplicative noise, the behavior of these quantities becomes supradiffusive, with scalings MSD∼t5/3 and DSF∼exp(ΓkMCt)5/3. We suggest that this supradiffusive behavior might be of importance for the biological functions of the cell and compare our work with other approaches which also predict the same transition from a sub-diffusive to a supra-diffusive regime.