General Theory of the Single-File Multi-Particle Diffusion in Narrow Pores

1. Introduction

The narrow nanoscale pores, which conduct ions or other small molecules, constitute a significant challenge for physical modeling. The particles in very narrow nanoscale pores cannot pass by each other (single-file motion). The pore can also be long enough to accommodate more than one particle at the same time. The examples of such nanopores, which are very important for practical applications, are the ion channels of biological membranes [1-3] and carbon nanotubes [4, 5]. The macroscopic flux of the permeating particles through these objects is of great importance for practical applications.

Despite the fact that the narrow pores with multiple occupancy are known for many years, there is no universally accepted dedicated physical theory of these objects. The goal of this work is development of the general theory of the multi-particle single-file diffusion in non-equilibrium conditions based on the very basic principles of non-equilibrium statistical physics and the theory of probability.

2. Outline of the theory

The main difficulty of describing the system under study is that the number of particles in the channel is variable, while all conventional dynamic equations and relations of the statistical physics, which describe non-equilibrium systems, operate with a fixed number of particles. The idea of our theoretical development is to present the distribution function of our system as a series in discrete channel occupancies. The outline of the theoretical development is the following:

  1. The N-particle distribution function of the whole system  is constructed taking into account the properties of the reservoirs. Hereafter  are the coordinates of particles, t is the time.
  2. The probabilities wn of the occupancy states with n ions inside the channel are found (), where M is the maximal occupancy of the channel.
  3. The partial distribution functions  of m (m ≤ n) particles in the channel, which contains n particles are found. All quantities of the reservoirs except the external concentrations c1,2 and the membrane potential  vanish at this step after transition to the thermodynamic limit.
  4. It is shown that any property of the channel can be obtained as a series in channel occupancies. They depend on unknown n-particle distribution functions inside the channel .
  5. It is shown that the distribution functions  are the solution of the hierarchical set of partial differential equations, which is obtained from the Langevin equation of motion of the ions inside the channel.

3. The super ions

The general theory of the single-file multi-particle diffusion in narrow pores [6] can be greatly simplified in rather wide class of specific bell-like shapes of the single-ion energy profiles, which are often observed in real ion channels. In such potentials the ions move in highly concerted manner, which corresponds to the existence of narrow and deep groove in the energetic landscape. The motion of multiple ions can be reduced to the motion of single quasi-particle (the super-ion), which moves in one-dimensional effective potential along the groove. It is shown that effective potentials of the super-ions, which correspond to the conducting occupancies of the channel, are essentially flat. This explains the phenomenon of the barrier-less conduction in the channels with multiple occupancy in very elegant way. The approximation of the super-ions also reduces the computational complexity of the problem dramatically in comparison with the generic computational procedure.

4. Conclusion

We developed a general analytical framework which describes single-file diffusion of multiple strongly interacting particles in non-equilibrium conditions. The model takes into account the external potential action on the diffusing particles and the fluctuations of the number of particles due to their exchange with external reservoirs. The model is constructed in a bottom-up manner from the very basic principles of statistical physics and probability theory. It is shown that the problem can be reduced to a hierarchical system of elliptic partial differential equations of increasing dimensionality, which can be solved numerically. Our framework allows us to compute any macroscopic characteristics of the single-file multi-particle diffusion, including the current and the occupancy probabilities. It is shown that the occupancy probabilities and the current are rational functions of external concentrations.

The theory is tested on a model of the narrow pore inspired by the selectivity filter of biological ion channel. The macroscopic characteristics of the model channel are obtained in a wide range of parameters. Obtained data correlate very well with the data of earlier studies performed on the same model, which serves as a validation of our theoretical framework.

 References

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[2]     S. Berneche, B. Roux, Nature 414 (2001) 73.

[3]     B. Hille, W. Schwarz, J. Gen. Physiol. 72 (1978) 409.

[4]     G. Hummer, J.C. Rasaiah, J.P. Noworyta, Nature. 414(6860) (2001) 156.

[5]     U. Zimmerli, P. Koumoutsakos, Biophys. J. 94 (2008) 2546.

[6]     V.N. Kharkyanen, S.O. Yesylevskyy, Phys. Rev. E 80 (2009) 031118.