The effect of fluctuations on the conductivity of ion channels is investigated. It is shown that modulation of the potential barrier at the selectivity site due to electrostatic amplification of charge fluctuations at the channel mouth exerts a leading-order effect on the channel conductivity. A Brownian dynamical model of ion motion in a channel is derived that takes into account both fluctuations at the channel mouth and vibrational modes of the wall. The charge fluctuations are modeled as a short noise flipping the height of the potential barrier. The wall fluctuations are introduced as a slow vibrational mode of protein motion that modulates ion conductance both stochastically and periodically. The model is used to estimate the contribution of the electrostatic amplification of charge fluctuations to the conductivity of ion channels.
Ion transport through the channels in cellular membranes underlies the electrical signal transduction and processing by living organisms. Accordingly ion channels, being natural nanotubes, control a vast range of biological functions in health and disease. The understanding of their structure-properties relationship is the subject of intensive, ever-growing, fundamental and applied research in biology, physics, and nanotechnology [1,2]. A central problem in studies of ion permeation through biological membrane channels is to understand how channels can be both highly selective between alike ions and yet still conduct millions of ions per second [3]. Indeed, selectivity between ions of the same charge implies that there exists a deep potential well for conducting type ions at the selectivity site of the channel. On the other hand such channels can pass up to 10 8 ions per second [4] corresponding to almost free diffusion.
Significant progress has been made towards an understanding of this problem over the last few decades. In particular, the molecular structure of the KcsA potassium channel [5] that discriminates between Na + and K + was determined by crystallographic analysis. Furthermore, by detecting the size of the structural fluctuations [6] and conformational changes [7], it has become possible to provide the experimental information needed for molecular modelling of the dynamical features of the observed selectivity and gating [8,9]. In particular, the minimum radius of the selectivity filter in KcsA is ∼0.85 Å, which is to be compared with 1.33 Å for the ionic radius of K + , suggesting that flexibility of the filter is coupled to ionic translocation [10]. It has therefore become apparent that fluctuations in the channel walls plays a fundamental role in maintaining high conductivity in highly selective channels (see also Elber [11]).
Another important source of modulation of the electrostatic potential identified in earlier research [12,13] relates to the interaction of the ion in the channel with charge fluctuations in the bath solutions. The effect of current fluctuations and noise on the channel entrance rates and on the channel conductivity was also considered in [14,15]. It becomes clear that fluctuations of the electrostatic potential within ion channels induced by various sources may provide a key to the solution of the central problems of permeation and selectivity. Models of such fluctuations have thus become one of the central topics of research on the permeability of ion channels. It is important to note that dynamical models of ion motion in the channel can also provide a link between studies of the permeability of open channels and channels gating. Notwithstanding recent advances, theoretical modeling of the dynamical features of ion channels is still in its infancy. In particular, little is yet known about the relative importance of the different dynamical mechanisms and sources of fluctuations in the ion channels.
In our earlier work we have started to develop a dynamical model [16,17,18,19,20,21] of ionic conductivity through open channels. It takes into account the coupling of ion motion to vibrations of the wall [16,17] and to charge fluctuations at the channel mouth [19,20]. Our goal is to derive a self-consistent model that allows for analytical estimation of the potential barrier at the selectivity site and for the effects of fluctuations on the conductivity of the channels. The starting point of our approach is a self-consistent quasi-analytical solution of the Poisson and Nernst-Planck equations in the channel, and in the bulk [21], allowing for accurate estimation of the current-voltage characteristics of ion channels [22] (see also [23,24,25,26,27]). The electrostatic channel potentials resulting from these estimates can be further used to estimate relative contribution to the channel conductivity from the different sources of fluctuations.
In this paper we introduce a model of ion permeation that takes into account dynamical effect of the charge fluctuations through the resultant shot noise, and we demonstrate that the latter has a leading-order effect on the transition probabilities. We show that the charge fluctuations at the channel mouth can be modeled as a flipping of the electrostatic potential at the selectivity site, which fluctuates between two maximum values at a rate corresponding to the random arrivals of ions at the channel mouth. The developed theoretical framework will allow us in the future to include into the model the modulation of the potential at the selectivity site due to hydration effects inside the channel.
A model of 3D Brownian dynamics simulation of ions in the bulk and inside the channel is described in Sec. 2. Using results of the 3D simulations in the bulk we present in Sec. 3 a reduced model of an ion moving in the channel and interacting with the wall vibrational modes and with charge fluctuations at the channel entrance. The model uses
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