Molecular transport through channels and pores: Effects of in-channel interactions and blocking
- *Medizinische Universitätsklinik 1, Josef Schneider Strasse 2, D-97080 Würzburg, Germany; and
- ‡Department of Physics, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931-1295
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Edited by Nicholas J. Turro, Columbia University, New York, NY, and approved June 5, 2006 (received for review March 3, 2006)
Abstract
Facilitated translocation of molecules through channels and pores is of fundamental importance for transmembrane transport in biological systems. Several such systems have specific binding sites inside the channel, but a clear understanding of how the interaction between channel and molecules affects the flow is still missing. We present a generic analytical treatment of the problem that relates molecular flow to the first passage time across and the number of particles inside the channel. Both quantities depend in different ways on the channel properties. For the idealized case of noninteracting molecules, we find an increased flow whenever there is a binding site in the channel, despite an increased first passage time. In the more realistic case that molecules may block the channel, we find an increase of flow only up to a certain threshold value of the binding strength and a dependence on the sign of the concentration gradient, i.e., asymmetric transport. The optimal binding strength in that case is analyzed. In all cases the reason for transport facilitation is an increased occupation probability of a particle inside the channel that overcomes any increase in the first passage time because of binding.
Footnotes
- †To whom correspondence should be addressed. E-mail: w.bauer{at}medizin.uni-wuerzburg.de
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Author contributions: W.R.B. and W.N. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.
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Conflict of interest statement: No conflicts declared.
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This paper was submitted directly (Track II) to the PNAS office.
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See Commentary on page 11431.
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↵ § We use here the simple to prove, but apparently not very well known, fact that if a/b = c/d, then a/b = (ma + nc)/(mb + nd) holds with arbitrary m, n (as long as mb + nd ≠ 0).It is important to note already at this point that for noninteracting particles, asymmetries in the channel potential are not important, because only the symmetrized part of n and τ contribute to the flow. The macroscopic Fick’s diffusion equation (Eq. 7) now explicitly relates steady-state flow to the concentration gradient, which acts as the thermodynamic driving force, and to the conductivity n/τ. As already mentioned above, the relevant FPT characterizing steady-state flow is the one considering reflecting boundary conditions at its starting point and not a conditional FPT that arises in exit splitting situations (see ref. 19). The only other relevant quantity is the number of molecules trapped by the channel, which is measured by the specific particle number n. The effect of the molecule–channel interaction on the flow depends on how this interaction affects the FPT and the number of molecules trapped. This effect is analyzed below.
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↵ ¶ In all our considerations, we assume no potential difference between the baths. However, our model can be readily adapted to a situation when Φ(0) = Φ1 ≠ Φ(L) = Φ2. Flow vanishes if the chemical potentials of the baths μi = Φi + ln(ci) are equal. This result implies equivalence of the diffusive conductivities, e −Φ1 n 1→2/τ1→2 = e −Φ2 n 2→1/τ2→1. Defining ñi→j = e −Φi ni →j as the potential corrected specific particle numbers, we can generalize Fick’s diffusion law in Eq. 7 to J = (ñ/τ)(e μ1 − e μ2)) with ñ = (ñ1→2 + ñ2→1)/2. Also, all other results of our work, in particular those for self-interacting particles, can be generalized to this situation by replacing specific particle numbers n and Δn by the corresponding potential corrected parameters, and concentrations ci by the activities e μi.
- Abbreviations:
- FPT,
- first passage time;
- MFPT,
- mean FPT;
- CMFPT,
- conditional MFPT.
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Freely available online through the PNAS open access option.
- © 2006 by The National Academy of Sciences of the USA
