Poisson-Nernst-Planck-Fermi theory for modeling biological ion channels

Abstract

A Poisson-Nernst-Planck-Fermi (PNPF) theory is developed for studying ionic transport through biological ion channels. Our goal is to deal with the finite size of particle using a Fermi like distribution without calculating the forces between the particles, because they are both expensive and tricky to compute. We include the steric effect of ions and water molecules with nonuniform sizes and interstitial voids, the correlation effect of crowded ions with different valences, and the screening effect of water molecules in an inhomogeneous aqueous electrolyte. Including the finite volume of water and the voids between particles is an important new part of the theory presented here. Fermi like distributions of all particle species are derived from the volume exclusion of classical particles. Volume exclusion and the resulting saturation phenomena are especially important to describe the binding and permeation mechanisms of ions in a narrow channel pore. The Gibbs free energy of the Fermi distribution reduces to that of a Boltzmann distribution when these effects are not considered. The classical Gibbs entropy is extended to a new entropy form - called Gibbs-Fermi entropy - that describes mixing configurations of all finite size particles and voids in a thermodynamic system where microstates do not have equal probabilities. The PNPF model describes the dynamic flow of ions, water molecules, as well as voids with electric fields and protein charges. The model also provides a quantitative mean-field description of the charge/space competition mechanism of particles within the highly charged and crowded channel pore. The PNPF results are in good accord with experimental currents recorded in a 10(8)-fold range of Ca(2+) concentrations. The results illustrate the anomalous mole fraction effect, a signature of L-type calcium channels. Moreover, numerical results concerning water density, dielectric permittivity, void volume, and steric energy provide useful details to study a variety of physical mechanisms ranging from binding, to permeation, blocking, flexibility, and charge/space competition of the channel.

A GPU Poisson–Fermi solver for ion channel simulations

“…since it saturates [11], i.e., C i (r) < 1 v i for any arbitrary (or even infinite) electric potential φ(r) at any location r ∈ Ω s for all i = 1, · · · , K + 1 (ions and water), where β i = q i /k B T with q i being the charge on species i particles and q K+1 = 0, k B is the Boltzmann constant, T is an absolute temperature, v i = 4πa 3 i /3 with radius a i , and v 0 = K+1 i=1 v i /(K + 1) an average volume. The steric potential S trc (r) [9] is an entropic measure of crowding or emptiness at r with Γ(r) = 1 − K+1 i=1 v i C i (r) being a function of void volume fractions and Γ B = 1 − K+1 i=1 v i C B i a constant bulk volume fraction of voids when φ(r) = 0 that yields Γ(r) = Γ B , where C B i are constant bulk concentrations.…”

“…These effects and properties cannot be described by the classical Poisson-Boltzmann theory that consequently has been slowly modified and improved [18][19][20][21][22][23][24][25][26][27][28] for more than 100 years since the work of Gouy and Chapman [29,30]. It is shown in [11,17] However, in addition to the computational complexity of PB solvers for biophysical simulations, the PF model incurs more difficulties in numerical stability and convergence and is thus computationally more expensive than the PB model as described and illustrated in [9,13]. To reduce long execution times of PF solver on CPU, we propose here two GPU algorithms, one for linear algebraic system solver and the other for nonlinear PDE solver.…”

Poisson-Fermi modeling of ion activities in aqueous single and mixed electrolyte solutions at variable temperature

“…For an aqueous electrolyte solution with K species of ions, the Poisson-Fermi theory proposed in [18,21] treats all ions and water of any diameter as nonuniform hard spheres with interstitial voids between these spheres. The activity coefficient γ i of an ion of species i in the solution describes the deviation of the chemical potential of the ion from ideality (γ i = 1).…”

“…The factor v k /v 0 multiplying the steric potential function S trc (r) in Eq. (5) is a modification of the unity used in our previous work [19,21]. The steric energy…”

“…The steric potential is a mean-field approximation of Lennard-Jones (L-J) potentials that describe local variations of L-J distances (and thus empty voids) between any pair of particles. L-J potentials are highly oscillatory and extremely expensive and unstable to compute numerically [21]. Calculations that involve L-J potentials, or even truncated versions of L-J potentials must be extensively checked to be sure that results do not depend on irrelevant parameters.…”