Electrostatics of ions inside the nanopores and trans-membrane channels

– A model of a finite cylindrical ion channel through a phospholipid membrane of width L separating two electrolyte reservoirs is studied. Analytical solution of the Poisson equation is obtained for an arbitrary distribution of ions inside the trans-membrane pore. The solution is asymptotically exact in the limit of large ionic strength of electrolyte on the two sides of membrane. However, even for physiological concentrations of electrolyte, the electrostatic barrier sizes found using the theory are in excellent agreement with the numerical solution of the Poisson equation. The analytical solution is used to calculate the electrostatic potential energy profiles for pores containing charged protein residues. Availability of a semi-exact interionic potential should greatly facilitate the study of ionic transport through nanopores and ion channels. Ion channels are water filled holes which facilitate exchange of electrolyte between the exterior and interior of a cell. Pores are formed by specific proteins embedded into the phospholipid membrane [1]. Depending on the conformation of the protein, the pore can be open or closed. When open, the protein is very specific to the kind of ions that it allows to pass through the channel [2,3]. In order to function properly the channel has to conduct thousands of ions in a period of few milliseconds. Considering that the channel passes through a phospholipid membrane which has a very low dielectric constant and is very narrow, producing high energetic penalties for ions entering the nonopores, it is fascinating to contemplate how Natures manages to perform this amazing task. In fact, as long ago as 1969, Parsegian observed that for an infinitely long cylindrical channel [4] of radius a = 3 Å, the electrostatic barrier is over 16kBT , which should completely suppress any ionic flow [5]. Later numerical work by Levitt [6], Jordan [7] and others demonstrated that for more realistic finite channels the barrier is dramatically reduced. For example, for a channel of length L = 25 Å and radius a = 3 Å, the barrier is about 6kBT , which although still quite large, should allow ionic conductivity. Recently the study of ion channels has expanded to other parts of applied physics. Water filled nanopores are introduced into silicon oxide films, polymer membranes, etc [8, 9]. In all of these cases the dielectric constant of the interior of a nanopore greatly exceeds that of the surrounding media. To quantitatively study the conductance of a nanopore one has three options: the all atom molecular dynamics simulation (MD) [3]; the Brownian dynamics simulation (BD) [10, 11] with implicit water treated as a uniform dielectric continuum; or the mean-field Poisson-Nernst-Planck theory (PNP) [12] c © EDP Sciences 2 EUROPHYSICS LETTERS which treats both water and ions implicitly. While clearly the most accurate, MD simulations are computationally very expensive [13]. Brownian dynamics is significantly faster than MD, but because of the dielectric discontinuities across the various interfaces a new solution of the Poisson equation is required for each new configuration of ions inside the pore. The simplest approach to study the ionic conduction is based on the PNP theory [12]. This combines the continuity equation with the Poisson equation and Ohm’s and Fick’s laws. PNP is intrinsically mean-field and is, therefore, bound to fail when ionic correlations become important. This has been well studied for its static version — the Poisson-Boltzmann equation, which is known to break down for aqueous electrolytes with multivalent ions and also for monovalent electrolytes in low dielectric solvents [14, 15]. For narrow channels, the cylindrical geometry, combined with the field confinement, results in a pseudo one dimensional potential of very long range [16,17]. Under these conditions the correlational effects dominate, and the mean-field approximation fails [14]. Indeed recent comparison between the BD and the PNP showed that PNP breaks down when the pore radius is smaller than about two Debye lengths [10, 11]. At the moment, therefore, it appears that a semi-continuum (implicit solvent) Brownian dynamics simulation is the best compromise between the cost and accuracy [13,18,19] for narrow pores. Unfortunately even this, simplified strategy demands a tremendous computational effort. The difficulty is that BD requires a new solution of the Poisson partial differential equation at each time step. This can be partially overcome by using lookup tables [11] and variational methods [20], but still requires a supercomputer. If the interaction potential between the ions inside the channel would be known, the simulation could proceed orders of magnitude faster. However, up to now the only exact solution to the Poisson equation in a cylindrical geometry was for the case of an infinitely long pore [4, 5, 16]. In this letter we shall provide another exact solution, but now for a finite trans-membrane channel. We shall work in the context of a primitive model of electrolyte and membrane. The membrane will be modeled as a uniform dielectric slab of width L located between z = 0 and z = L. The dielectric constant of the membrane and the channel forming protein is taken to be ǫp ≈ 2. On both sides of the membrane there is an electrolyte solution composed of point-like ions and characterized by the inverse Debye length κ. A channel is a cylindrical hole of radius a and length L filled with water. As is usual for continuum electrostatic models [13], we shall take the dielectric constant of water inside and outside the channel to be the same, ǫw ≈ 80. It is convenient to set up a cylindrical coordinate system (z, ρ, φ) with the origin located at the center of the channel at z = 0. Suppose that an ion is located at an arbitrary position x inside the channel. The electrostatic potential φ(z, ρ, φ;x) inside the channel and membrane satisfies the Laplace equation ∇φ = − 4πq ǫw δ(x − x) . (1) For z > L and z < 0, φ(x;x) satisfies the linearized Poisson-Boltzmann or the Debye-Hückel equation [14] ∇φ = κφ . (2) The inverse Debye length is related to the ionic strength I of electrolyte, ξ D = κ = √ 8πλBI , where λB = q /ǫkBT is the Bjerrum length and I = (αcα + αcα + 2c)/2. Here cα is the concentration of α : 1 valent electrolyte and c is the concentration of 1 : 1 electrolyte. All the usual boundary conditions must be enforced: the potential must vanish at infinity and be continuous across all the interfaces; the tangential component of the electric field and the normal component of the electric displacement must be continuous across all the interfaces. These boundary conditions guaranty the uniqueness of solution. Unfortunately, even this relatively simple geometry can not, in general, be solved exactly. We observe, however, that an exact solution is possible in the limit that κ → ∞. In this special case the system of differential equations becomes separable. Our, strategy then will be to solve exactly this asymptomatic problem and then extend the solution to finite values of Debye length. Yan Levin: ELECTROSTATICS OF IONS INSIDE THE NANOPORES AND TRANS-MEMBRANE CHANNELS3 We start by making the following fundamental observation. The condition κ → ∞ signifies that electrolyte perfectly screens any electric field — the Debye length is zero. This, combined with the boundary condition — electrostatic potential must vanishes at infinity — implies that in this limit φ(0, ρ, φ;x) = φ(L, ρ, φ;x) = 0, for any position x of an ion inside the pore. This is a dramatic simplification. Now it is no longer necessary to solve the Debye-Hückel equation, but only the Poisson equation with a perfect grounded conductor boundary conditions at z = 0 and z = L. To proceed we expand the δ(z − z) in eigenfunctions of the differential operator dψn dz2 + k nψn = 0 , (3) satisfying the perfect conductor boundary condition. The normalized eigenfunctions are ψn(z) =