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We have discussed earlier in Chapter 4, that Langmuir adsorption is an important physical property and can be observed in the lter. To consider this property we need to x the value of the xed charge, and so we consider the suitable xed

nf = −2.5. At this nf the state space reduces to our optimal transport regime, and so we can try to calculate the adsorption isotherms for either excited K+ and

Na+state. If we subtract the ground state from_hn

Ki, and renormalise by dividing by three, we can dene the K+ _{isotherm as,}

ΘK =

xKe∆˜µK/kT

1 +xKe∆˜µK/kT +xN ae∆˜µN a−kTln(3)/kT

. (5.71)

This resembles the Langmuir isotherm for mixed species solutions as introduced earlier in equation (4.27). The equivalent isotherm for Na+ can be written as,

ΘN a =

xN ae∆˜µN a−kTln(3)/kT

1 +xKe∆˜µK/kT +xN ae∆˜µN a−kTln(3)/kT

These functions are not exactly equivalent to equation (4.27), because these de- scribe a non-ideal electrolyte solution. However if we neglect the concentration dependence in the excess chemical potential, the occupancy can exactly be de- scribed by a Langmuir isotherm, (see gure 5.11). Including this Debye-Hückel term as given by (5.24), slightly shifts the form of the adsorption isotherm. How- ever, it remains a saturating function vs. the bulk concentration.

5.4 Summary

In this chapter we have derived a multi-species statistical theory that is applicable to narrow ion channels coupled to mixed-species particle reservoirs. This involved: a review of the literature to understand the properties of charged particles in bulk solutions; a directed look at the application of this theory to ion channels; and nally the derivation of the main theory. This derivation proves that the selectivity between alike-charged ions is purely a result of the chemical interactions, as expected because of the shared valence of the ions.

Our derivation of the grand canonical ensemble for narrow channels with multiple binding sites and mixed-species bulk solutions, involved:

ˆ The derivation of the Gibbs free energy equation (5.39), which is important because it describes the energy state of the lter. This equation takes ac- count of all interactions in the system, including the bulk ideal and non-ideal interactions, the ideal term in the lter, the electrostatic interaction with the xed charge and further non-ideal interactions via the excess chemical poten- tial. This produces energy spectra for the system that are parabolic vs. Qf, with xed energy barriers due to dierences of the excess chemical potentials. ˆ The GCE probability distribution function and its partition function, which was derived from standard techniques. This equation describes the occu- pancy and statistical properties of the system as a function of the energy

interactions. The mean number of particles produces a selective staircase function vs. Qf, where the midpoint of steps in hnKi occur at degeneracies between the energy levels, andhnN aiwas∼40times smaller as a result of the dierence in free energy spectra. The multi-species adsorption isotherm, has been derived within the optimal transport regime of KcsA. This results in a highly selective saturating occupancy function vs. the bulk concentration.

We have also derived the generalised Einstein relation at linear response, thereby relating the conductivity through the lter to the uctuations in particle number. The equation for K+ _{conductivity resulted in a sequence of diusion limited peaks}

that maximise at the degeneracy between neighbouring energy levels. This occurs exactly at the midpoint in the occupancy steps. This property conrms Coulomb blockade within the lter for K+_{. The Na}+ _{conduction meanwhile was selectively}

blocked by K+ resulting in a staircase _{∼ 200} times smaller than the K+ peaks,

occurring due to the continual uctuation between excited K+ _{and Na}+ _states.

Conduction of multi-species must include explicitly the interaction between dif- ferent species within the lter and it has been shown that Fickian diusion fails to fully describe this phenomenon in many examples [226, 227, 228]. Interaction between ionic species in biological channels has been proposed by two mechanisms: either as a drag exerted on the conducting ion from the other species [229, 230] or from a physical electrostatic exclusion as is included here [105]. To include these terms explicitly a Maxwell-Stefan diusion theory needs to be developed. This theory replaces the Fickian uxes with linear combinations of chemical potential gradients for all species [231]:

xi RT∇ηi =− X i= j6=i xjji−xijj cTDij , (5.73)

where all terms are as previously dened with R being the molar gas constant, cT being the total concentration of the solution andDij being the Maxwell-Stefan diusivity between species. It remains an active area for future research to explain

the conduction of the disfavoured species.

Finally, we have derived the Eisenman selectivity relation directly from our free energy spectra and condition for optimal conductivity. Inserting the corresponding value of nf into the free energy barrier for Na+ resulted in equation (5.70). In solutions with identical numbers of K+ and Na+ ions, and neglecting the minor

inuence from mixing, the energy barrier is solely described by the dehydration energy dierence between species.

By deriving a theory that includes mixed-species solutions, there is the opportunity for direct investigation of the equilibrium occupancy, selectivity and conductivity properties of ion channels under equilibrium physiological conditions.

6. Transition rates

6.1 Introduction

To move beyond investigating quasi-equilibrium behaviour we can introduce a kinetic model. Fundamental to such models are the transition rates, which we will briey review and derive. In application to ion channels it is common to choose Eyring or Arrhenius transition rates within transition state theory (TST) but this is often criticised and leaves many unanswered questions [117, 232, 124, 125, 126] . We will introduce the Grand-Canonical-Monte-Carlo transition rates used in BD simulations [82, 83, 101], derive the transition rates using our the GCE derived in this thesis (see Chapter 5), demonstrate their suitability for use in a self-consistent kinetic model.