Electric discharge of electrocytes: Modelling, analysis and simulation.

In this paper, we investigate the electric discharge of electrocytes by extending our previous work on the generation of electric potential. We first give a complete formulation of a single cell unit consisting of an electrocyte and a resistor, based on a Poisson-Nernst-Planck (PNP) system with various membrane currents as interfacial conditions for the electrocyte and a Maxwell's model for the resistor. Our previous work can be treated as a special case with an infinite resistor (or open circuit). Using asymptotic analysis, we simplify our PNP system and reduce it to an ordinary differential equation (ODE) based model. Unlike the case of an infinite resistor, our numerical simulations of the new model reveal several distinct features. A finite current is generated, which leads to non-constant electric potentials in the bulk of intracellular and extracellular regions. Furthermore, the current induces an additional action potential (AP) at the non-innervated membrane, contrary to the case of an open circuit where an AP is generated only at the innervated membrane. The voltage drop inside the electrocyte is caused by an internal resistance due to mobile ions. We show that our single cell model can be used as the basis for a system with stacked electrocytes and the total current during the discharge of an electric eel can be estimated by using our model.

Electric potential generation of electrocytes: Modelling, analysis, and computation.

In this paper, we developed a one-dimensional model for electric potential generation of electrocytes in electric eels. The model is based on the Poisson-Nernst-Planck system for ion transport coupled with membrane fluxes including the Hodgkin-Huxley type. Using asymptotic analysis, we derived a simplified zero-dimensional model, which we denote as the membrane model in this paper, as a leading order approximation. Our analysis provides justification for the assumption in membrane models that electric potential is constant in the intracellular space. This is essential to explain the superposition of two membrane potentials that leads to a significant transcellular potential. Numerical simulations are also carried out to support our analytical findings.

Selectivity of the KcsA potassium channel: Analysis and computation.

Ion channels regulate the flux of ions through cell membranes and play significant roles in many physiological functions. Most of the existing literature focuses on computational approaches based on molecular dynamics simulation or numerical solution of the modified Poisson-Nernst-Planck (PNP) system. In this paper, we present an analytical and computational study of a mathematical model of the KcsA potassium channel, including the effects of ion size (Bikerman model) and solvation energy (Born model). Under equilibrium conditions, we obtain an analytical solution of our modified PNP system, which is used to explain selectivity of KcsA of various ions (K^{+}, Na^{+}, Cl^{-}, Ca^{2+}, and Ba^{2+}) due to negative permanent charges inside the filter region and the effect of ion sizes. Our results show that K^{+} is always selected over Na^{+}, as smaller Na^{+} ions have larger solvation energy. As the amount of negative charges in the filter exceeds a critical value, divalent ions (Ca^{2+} and Ba^{2+}) can enter the filter region and block the KcsA channel. For the nonequilibrium cases, due to difficulties associated with a pure analytical or numerical approach, we use a hybrid analytical-numerical method to solve the modified PNP system. Our predictions of selectivity of KcsA channels and saturation phenomenon of the current-voltage (I-V) curve agree with experimental observations.

Electroneutral models for dynamic Poisson-Nernst-Planck systems.

The Poisson-Nernst-Planck (PNP) system is a standard model for describing ion transport. In many applications, e.g., ions in biological tissues, the presence of thin boundary layers poses both modeling and computational challenges. In this paper, we derive simplified electroneutral (EN) models where the thin boundary layers are replaced by effective boundary conditions. There are two major advantages of EN models. First, it is much cheaper to solve them numerically. Second, EN models are easier to deal with compared to the original PNP system; therefore, it would also be easier to derive macroscopic models for cellular structures using EN models. Even though the approach used here is applicable to higher-dimensional cases, this paper mainly focuses on the one-dimensional system, including the general multi-ion case. Using systematic asymptotic analysis, we derive a variety of effective boundary conditions directly applicable to the EN system for the bulk region. This EN system can be solved directly and efficiently without computing the solution in the boundary layer. The derivation is based on matched asymptotics, and the key idea is to bring back higher-order contributions into the effective boundary conditions. For Dirichlet boundary conditions, the higher-order terms can be neglected and the classical results (continuity of electrochemical potential) are recovered. For flux boundary conditions, higher-order terms account for the accumulation of ions in boundary layer and neglecting them leads to physically incorrect solutions. To validate the EN model, numerical computations are carried out for several examples. Our results show that solving the EN model is much more efficient than the original PNP system. Implemented with the Hodgkin-Huxley model, the computational time for solving the EN model is significantly reduced without sacrificing the accuracy of the solution due to the fact that it allows for relatively large mesh and time-step sizes.