Three-dimensional simulation of biological ion channels under mechanical, thermal and fluid forces

In this article we address the three-dimensional modeling and simulation of biological ion channels using a continuum-based approach. Our multi-physics formulation self-consistently combines, to the best of our knowledge for the first time, ion electrodiffusion, channel fluid motion, thermal self-heating and mechanical deformation. The resulting system of nonlinearly coupled partial differential equations in conservation form is discretized using the Galerkin Finite Element Method. The validation of the proposed computational model is carried out with the simulation of two ion nanochannels. The first is a voltage operated channel with K⁺ and Na⁺ ions, and the second is the sodium-potassium pump in which also chlorine (Cl⁻) and bicarbonate (HCO₃⁻) ions are considered. In the first case study, we investigate the coupling between electrochemical and fluid-dynamical effects. Then, we enrich the modeling picture by investigating the influence of a thermal gradient. Finally, we add a mechanical stress responsible for channel deformation and investigate its effect on the functional response of the channel. Results show that fluid and thermal fields have no influence in absence of mechanical deformation whereas ion distributions and channel functional response are significantly modified if mechanical stress is included in the model. These predictions agree with biophysical conjectures on the importance of protein conformation in the modulation of channel electrochemical properties. In the second case study, we exploit our multiphysical mathematical formulation to investigate the effect of permanent surface charge on the function of the sodium-potassium pump. In particular, we consider the biophysical application in which the pump actively participates to the process of aqueous humor production across the transepithelial membrane in the ciliary body of the eye. In this study we are motivated by the fact that several data are available to verify the accuracy of model predictions and that the role of surface charge has not yet been mathematically analyzed in the specific context at hand. Results show that the model is able to predict in a very accurate manner the correct aqueous humor flow direction, the magnitude of aqueous velocity and the experimentally measured transepithelial membrane potential only if the (negative) surface permanent charge is larger than a limiting value, whereas if the charge is below this value, fluid inversion occurs.