Journal of Biophysics

Volume 2016 (2016), Article ID 1657679, 10 pages

http://dx.doi.org/10.1155/2016/1657679

## Self-Optimized Biological Channels in Facilitating the Transmembrane Movement of Charged Molecules

^{1}Institute for Bio-Medical Physics, 109A Pasteur, 1st District, Ho Chi Minh City 710115, Vietnam^{2}VAST/Institute of Physics, 1 Mac Dinh Chi, 1st District, Ho Chi Minh City 710116, Vietnam

Received 13 October 2015; Revised 13 January 2016; Accepted 27 January 2016

Academic Editor: João A. R. G. Barbosa

Copyright © 2016 V. T. N. Huyen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider an anisotropically two-dimensional diffusion of a charged molecule (particle) through a large biological channel under an external voltage. The channel is modeled as a cylinder of three structure parameters: radius, length, and surface density of negative charges located at the channel interior-lining. These charges induce inside the channel a potential that plays a key role in controlling the particle current through the channel. It was shown that to facilitate the transmembrane particle movement the channel should be reasonably self-optimized so that its potential coincides with the resonant one, resulting in a large particle current across the channel. Observed facilitation appears to be an intrinsic property of biological channels, regardless of the external voltage or the particle concentration gradient. This facilitation is very selective in the sense that a channel of definite structure parameters can facilitate the transmembrane movement of only particles of proper valence at corresponding temperatures. Calculations also show that the modeled channel is nonohmic with the ion conductance which exhibits a resonance at the same channel potential as that identified in the current.

#### 1. Introduction

Biological channels are responsible for regulating the fluxes of ions and molecules (hereafter referred to as particles for short) across membranes and, therefore, are critically important for the cell functioning [1]. As well-known, these protein channels are very efficient in the sense that they support a very fast, selective, and robust across membrane transport, regardless of environment fluctuations [2]. Surprisingly, such privileged properties have been observed even in the case of large water-filled channels, where the particle transport does not involve the use of metabolic energy or conformational changes and was assumed to be simply diffusive [3]. Understanding the nature of this channel-facilitated particle movement (CFPM) is crucially important from the fundamental molecular biology as well as the application point of view (many modern drugs are developed in the way of using the ion-channels to enhance their efficiency; see, e.g., [4–6]).

Experimentally, there are accumulative data showing that the observed CFPM really resulted from some interaction between the moving particle and the channel interior-lining [7, 8]. Recent advancements of high-resolution current recording enable single-channel measurements that provide directly a living picture of how an individual channel functions and, therefore, shed light on the characteristics of channel current in dependence on different (channel and environment) parameters [7–9]. However, revealing exactly the nature of channel-particle interaction as well as the mechanism of CFPM is still very experimentally problematic due to the puzzled complexities related to both the channel structure and the measurement systems.

Theoretically, to describe the CFPM several models have been suggested. Considering the one-dimensional (1D) diffusion model with a position-dependent diffusion coefficient, Berezhkovskii et al. supposedly introduced a square potential well, spanning the whole channel length, that brings about a channel-particle interaction [10–13]. It was then shown that at a given solute concentration difference there exists an optimum potential well depth that can maximize the particle current, facilitating the channel function. In this model (i) the channel is assumed to be large enough so that all the effects related to the particle size can be omitted, (ii) a single-particle diffusion is considered, neglecting all many particle correlations, and, particularly, (iii) no realistic potential was assigned as the source for the square potential well introduced. Bauer and Nadler considered a similar 1D diffusion model with a square potential well that is however associated locally with only the particle bound temporarily inside the channel [14]. Using the macroscopic version of Fick’s equation, it was then demonstrated that a transport increase always occurs for any square potential wells. However, as already noted by the authors, the square potential well exploited in this model is also rather crude and a more realistic potential should be found [14]. From the very other point of view, Kolomeisky models the channel as a set of discrete binding sites arranged stochastically [15]. In such the discrete-state model the particles are assumed to hop along the binding sites in translocations across the channel and the optimum current may be achieved depending on the spatial distribution of binding-sites and the site-particle interactions [15, 16]. This model is so simple that the main dynamic properties of the problem can be calculated exactly. It was also demonstrated that the discrete-state model [15] and the continuum diffusion model [10] are closely related and can be effectively mapped into each other [17]. Nevertheless, like the square potential well in the continuum models [10, 14], the nature of the binding sites (a kind of channel-particle interaction) and the hopping mechanism of particles in the discrete-state model [15] still need to be identified.

Importantly, in all the models mentioned [10, 14, 15] the channel-particle interaction (which was expressed by a square potential well or a binding site) is generally viewed as the crucial condition for the transmembrane transport to be facilitated (see also [18]). Note again that in these models the particle motion is merely considered one-dimensional. Recently, Dettmer et al. have measured the diffusivity of spherical particles in closely confining, finite length channels [19]. Measurements demonstrated a strongly anisotropic diffusion in the channel interior: while the diffusion coefficient parallel to the channel axis remained constant throughout the entire channel interior, the perpendicular diffusion coefficient showed an almost linear decrease from the axis towards the channel wall. These observations put forward a need for the two-dimensional (2D) description with direction-dependent diffusion coefficients when studying the movement of particles inside a large channel. Furthermore, experimentally, the single-channel kinetics was extensively studied at different external voltages [20, 21]. And, the experimental sublinear current-voltage () characteristics reported in [22, 23] is often used as one of the basic requirements for theoretical models [24].

In the present paper we consider a 2D diffusive movement of particles through a large water-filled channel, taking into account an anisotropy of diffusion coefficients as observed in [19] and an influence of external voltage as discussed in [20, 24]. The channel is modeled as a cylinder characterized by three structure parameters: radius, length, and surface density of negative charges of channel interior-lining. The potential created by this charged interior-lining inside the channel is exactly calculated. It causes the “channel-particle interaction” that plays a key role in facilitating the transmembrane particle movement. Solving the 2D stochastic Langevin equation for the model suggested we systematically analyze the typically dynamical characteristics of particles such as the translocation probabilities, the translocation times, the currents, and the channel ion conductance under the influence of various factors: the channel-induced potential, the external voltage, or the difference in reservoir particle concentrations. It was particularly shown that to facilitate the transmembrane particle movement the channel should be reasonably self-optimized with appropriate structure parameters so that its potential coincides with the resonant one. In addition, this facilitation is very selective in the sense that a channel of definite structure parameters can facilitate the transmembrane movement of only particles of proper valence at corresponding temperatures. So, the model suggests that facilitating the transmembrane particle movement is an intrinsic property of biological channels. This property is independent of the external factors such as the external voltage or the bulk particle concentration gradient, though these factors may strongly influence the magnitude of various particle dynamical characteristics.

The paper is organized as follows. Section 2 introduces the 2D diffusion model for the problem under study, including the motion equation with an exact expression of the channel-induced potential, and describes the calculating method. Section 3 presents the main numerical results obtained. These results are discussed in great detail, showing the influence of various factors on the particle dynamical characteristics. A particular attention is given to the self-optimized property of the channels in facilitating the transmembrane particle movement. The paper concludes with a brief summary in Section 4.

#### 2. Model and Calculating Method

We consider a cylindrical channel of length and radius that connects the two reservoirs with particle concentrations and as schematically drawn in Figure 1(a). The channel interior-lining carries negative charges which are for simplicity assumed to be continuously and regularly distributed with a surface density . (The cation channels are believed to contain a net negative charge in the pore lining region of the protein [25]. In the case of potassium and gramicidin channels this is due to the partially charged carbonyl oxygens [1, 25].) These negative surface charges create an electrostatic potential which affects the movement of particles inside the channel. Particles are assumed to diffuse independently, neglecting any many-particle correlation. In addition, the diffusivity of a particle inside the channel is assumed to be anisotropic with the two different diffusion coefficients, (parallel with) and (perpendicular to the channel axis). Following [19], we assume that (i) is constant throughout the channel cylinder and and somewhat smaller than the diffusion coefficient in the bulk, with (we choose in the present work for definition) and (ii) linearly decreases as going from the channel axis (where the diffusion is isotropic) to the channel wall, .