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Estimating the dielectric constant of the channel protein and pore

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Abstract

When modelling biological ion channels using Brownian dynamics (BD) or Poisson–Nernst–Planck theory, the force encountered by permeant ions is calculated by solving Poisson’s equation. Two free parameters needed to solve this equation are the dielectric constant of water in the pore and the dielectric constant of the protein forming the channel. Although these values can in theory be deduced by various methods, they do not give a reliable answer when applied to channel-like geometries that contain charged particles. To determine the appropriate values of the dielectric constants, here we solve the inverse problem. Given the structure of the MthK channel, we attempt to determine the values of the protein and pore dielectric constants that minimize the discrepancies between the experimentally-determined current–voltage curve and the curve obtained from BD simulations. Two different methods have been applied to determine these values. First, we use all possible pairs of the pore dielectric constant of water, ranging from 20 to 80 in steps of 10, and the protein dielectric constant of 2–10 in steps of 2, and compare the simulated results with the experimental values. We find that the best agreement is obtained with experiment when a protein dielectric constant of 2 and a pore water dielectric constant of 60 is used. Second, we employ a learning-based stochastic optimization algorithm to pick out the optimum combination of the two dielectric constants. From the algorithm we obtain an optimum value of 2 for the protein dielectric constant and 64 for the pore dielectric constant.

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Acknowledgments

This work was supported by grants from the National Health & Medical Research Council of Australia. Calculations were performed on the SGI Altix cluster at the APAC National Facility.

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Correspondence to Taira Vora.

Appendix

Appendix

The algorithm for the adaptive controlled BD simulations is presented here:

Algorithm 1: Stochastic Search Adaptive BD Algorithm for Dielectric Estimation

  • Step 0 : (Initialisation) At batch-time n = 0, initialise state of the algorithm by selecting a dielectric constant θ 0, a combination of ɛw and ɛp, as the optimal dielectric estimate of ion channel pore and protein randomly and with uniform probability.

  • Step 1 : (sampling and exploration) at a particular batch n, we conduct independent simulations on all experimental conditions Λ, and evaluate C n (θ n ) according to Eq. 5. We then generate an alternative candidate \(\tilde{\theta}_n\) and conduct Λ independent simulations on the new candidate \(\tilde{\theta}_n,\) and evaluate \(C_n(\tilde{\theta}_n).\)

  • Step 2 : (conditional acceptance test) the candidate with the lower C n (θ) value is the better fit to the experimental results. It is the better dielectric estimate for batch n. It is then selected as the optimal candidate for run n + 1 and another θ (choice of ɛw and ɛp) is randomly chosen as the comparison candidate.

  • Step 3 : update π n , which is a counter for the number of times each set of candidates has been a better estimate of the experimental results.

  • Step 4 : update estimate of ɛw and ɛp. Go to Step 1.

The function π n (θ) of π n generated in Step 3 of the above algorithm is merely a normalised counter for how many times the algorithm has visited any particular candidate of dielectric constants. In particular,

$$ \pi_n(\theta) =\frac {{\hbox{No. of times algorithm visits a particular set of }\varepsilon_{\rm w}\,\hbox{and} \,\varepsilon_{\rm p} \hbox{ in batches 1 to }n}}{n} $$
(6)

is the occupation probability of state θ. We see that for sufficiently large n, π n (θ *) > π n (θ), meaning that the algorithm spends more time at the optimal shape θ * than at any other shape θ ∈ Θ. As a consequence θ * n , which is the optimal shape on which the algorithm has spent maximum time until time n, converges to the optimal shape θ * with probability one. The attraction capability of this algorithm is proved by Andradottir (1999).

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Ng, J.A., Vora, T., Krishnamurthy, V. et al. Estimating the dielectric constant of the channel protein and pore. Eur Biophys J 37, 213–222 (2008). https://doi.org/10.1007/s00249-007-0218-3

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  • DOI: https://doi.org/10.1007/s00249-007-0218-3

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