Mathematical models of C-type and N-type inactivating heteromeric voltage gated potassium channels

Voltage gated potassium channels can be composed of either four identical, or different, pore-forming protein subunits. While the voltage gated channels with identical subunits have been extensively studied both physiologically and mathematically, those with multiple subunit types, termed heteromeric channels, have not been. Here we construct, and explore the predictive outputs of, mechanistic models for heteromeric voltage gated potassium channels that possess either N-type or C-type inactivation kinetics. For both types of inactivation, we first build Markov models of four identical pore-forming inactivating subunits. Combining this with previous results regarding non-inactivating heteromeric channels, we are able to define models for heteromeric channels containing both non-inactivating and inactivating subunits of any ratio. We simulate each model through three unique voltage clamp protocols to identify steady state properties. In doing so, we generate predictions about the impact of adding additional inactivating subunits on a total channel's kinetics. We show that while N-type inactivating subunits appear to have a non-linear impact on the level of inactivation the channel experiences, the effect of C-type inactivating subunits is almost linear. Finally, to combat the computational issues of working with a large number of state variables we define model reductions for both types of heteromeric channels. For the N-type heteromers we derive a quasi-steady-state approximation and indicate where the approximation is appropriate. With the C-type heteromers we are able to write an explicit model reduction bringing models of greater than 10 dimensions down to 2.