We elaborate on the theory developed in the main paper, for those readers who are interested in greater detail. We first discuss the statistical properties including the possible ion configurations in the pore and the link between our free energy and the potential of mean force (PMF). Then we derive the equations governing ionic conductivity at small applied potentials (linear response), and consider three important examples to further illuminate understanding the central figure of the main paper (Figure 3.). Finally we briefly review the important parameters ∆μK,1−4 and ∆μNa,1−4 and place them in the context of other results.
CONTENTSDerivation of the statistical theory Definition of the system Configurations in the pore Derivation of the ensemble and its properties Derivation of the ionic conductivity Electrostatics and effects of ionic binding Effects of ion binding and polarisation Electrostatic interaction Novel insights from the theory Fundamental importance of structure Principal results of the main paper Proposed future work Conductivity Examples Example 1: One species, one ion, multiple binding sites Example 2: One species, two ions, two binding sites Example 3: Two species, multiple ions, multiple binding sites Relation to PMF Analysis of Parameters References DERIVATION OF THE STATISTICAL THEORY Definition of the systemThe system is described by Fig. 1 which is copied from the main text. It represents the selectivity filter (SF) of a biological channel. This pore (denoted by (c)) is thermally and diffusively coupled to the left (L) and right (R) bulk reservoirs (b) . Each bulk contains mixed solutions of S total ionic species where s ∈ 1, • • • , S. The system as a whole is characterized by the canonical ensemble with constant total particle number N s , volume V and temperature T .