Conductance

In the development of a physical model to predict conductance through ion channels, Hille initially considered a cylindrical approximation with length L and cross-sectional area A, allowing the resistance R to be expressed as

\[R = \dfrac{1}{g} = \dfrac{\rho L}{A}\]

where \(\rho\) represents bulk resistivity. This simplistic model was subsequently refined with a more accurate representation of ion channels as a series of stacked cylinders, where the resistance accumulates. Considering ohmic principles and utilizing the HOLE software which measures cross-sectional areas A(z) along the channel axis z, the refined resistance model becomes

\[R_{HOLE} = \dfrac{1}{g_{HOLE}} = \sum_i \dfrac{\rho_{bulk} (z_i-z_{i-1})}{\pi r_i^2}\]

However, relying on bulk property resistivity becomes problematic, as conductivity \(\kappa=1/\rho\) depends on the diffusion coefficients of ions. The bulk conductivity \(\kappa_{bulk}\) of a KCl solution with concentration c is defined as

\[\kappa_{bulk} = \dfrac{c\cdot q_e^2\cdot(D_K+D_{CL})}{k_B T}\]

where c is the concentration of salts in water, \(q_e\) is the elementary charge, \(D_K\) and \(D_{CL}\) are diffusion coefficients of potassium and chloride ions, \(k_B\) is the Boltzmann constant, and T is temperature. To refine the model for ion channel conductance further, we introduce a conductivity model, expressing the conductivity \(\kappa(a,b)\) as a function of the radii a and b of ellipsoidal probe particles. For larger radii, the ion movement is relatively unconstrained, resulting in \(\kappa(a,b)\approx \kappa_{bulk}\), while narrower constrictions with smaller radii lead to reduced conductivity \(\kappa(a,b)<\kappa_{bulk}\). Hence, we can further adapt the model for channel resistance / conductance based on the PoreAnalyser profile to

\[R_{PA} = \dfrac{1}{g_{PA}} = \sum_i \dfrac{(z_i-z_{i-1})}{\kappa(a_i,b_i)\cdot\pi\cdot a_i\cdot b_i}\]

Find out more about the conductance model in our publication .