T-World 6: Problem 4 - Voltage-dependence of sodium-potassium pump

 At one point, I noticed how EADs are quite a bit harder to evoke in the Shannon-Bers model than in ToR-ORd. I.e., when I ported key currents (I_CaL, I_NaL, I_Kr, I_Ks) to the newly created model built closer to the Bers/Grandi framework, it would still not generate EADs as readily in the right conditions as models like ToR-ORd. The difference was not huge, but it was noticeable. In a separate investigation, when I was trying to build an understanding of differences in each current in either framework, I noticed how relatively different is the voltage-dependence – near-linear in ToR-ORd, but sublinear in the Shannon model. And there I had a spark of thinking that clarified why the Shannon-like models could be naturally less prone to EAD formation, arising from the sodium-potassium pump differences. Let’s go over this in more detail.

Below is shown a comparison of the linear voltage-dependence used in T-World, versus the sublinear voltage-dependence of the sodium-potassium pump used in the Shannon model and its successors.

Here, fnak in Shannon was scaled down to 89% to reflect the fact that when using this scaling, one gets a highly comparable behaviour of the cell to the one with a linear dependence. As seen below, the action potential (top left), calcium transient (top right), and sodium concentration after 100 beats are very similar between such models.

 

Now, the bottom right plot shows the sodium potassium pump current, and a difference is much more substantial there. You can check how the shape of the current corresponds to the voltage-dependence shown in the previous plot. During early plateau, above ca. 0 mV, the sublinear (Shannon-based) pump’s current is lower than that of the linear – and that’s exactly what you’d expect based on the comparison of the voltage dependency curve in the previous plot. On the other hand, below ca. 0 mV, the Shannon (red) curve in the voltage dependency plot is higher, and that is why the pump current is higher too – until the diastolic potential is reached, where the models meet again.

Early afterdepolarisations with the L-type calcium current in T-World tend to take-off around – 10 mV. As you can see from the top left panel, this happens around 200 ms. And looking back at the NaK pump on the bottom right panel of the above figure, that is where the sublinear-based pump produces a markedly higher current. I.e., there is more repolarising current at this point of simulation, and that is why an aspiring early afterdepolarisation faces stronger opposition and is less likely to develop. (In reality, EADs happen in models with reduced repolarization reserve, starting much later than at 200 ms; this plot merely illustrates that when the EAD-friendly membrane potential is reached, INaK is higher in the sublinear model.)

With regards to experimental evidence on linear/sublinear voltage-dependence, the classical study by Nakao & Gatsby of 1989 shows the following (Figure 4C):

Different voltage-dependence curves represent three levels of intracellular sodium ([Na]_pip – what is in the pipette controls the intracellular concentration). What can be seen there is that the voltage-dependence is clearly sublinear at 50 mM sodium. However, at the physiological level of 8 mM, it is pretty linear, certainly up to ca. 40 mV, i.e., for the range of membrane potentials relevant for a cardiac action potential. The rest of the paper by Nakao and Gadsby mostly uses 50 mM intracellular sodium to get results in other figures, and that is probably why this condition was mainly used to develop models. This data at 50 mM were also used to develop the Luo-Rudy model of 1994 , where the formulation in the Shannon model comes from (see Figure 9 of the Luo-Rudy model, also carried out at intracellular sodium of 50 mM). This point aside, I think that one take-home message I took from my years making models is that the Luo-Rudy model is INSANE. So many things done right, even with limited information of the time, so much complexity that made sense. It really is a monumental piece of work.

For some time, I wanted to use the ORd/ToR-ORd model of the sodium potassium pump, but that became untenable when I noticed in another project that the pump’s representation in those models is quite problematic with regards to extracellular potassium changes. Briefly, the model doesn’t show much of a reduced pump rate with hypokalaemia. This issue is quite stealthy, and that is why I missed it until relatively recently – it does not show when the same unphysiologically high sodium concentration is used as in its reference study; i.e., the model matches the underlying data, it’s just those data are not in physiological condition. However, when physiological sodium levels are used, this issue appears and can cause problems in certain domains of application – additional plots and discussion are around the Supplementary Figure S3 in the T-World paper.

In the end, we used an approach closer to the Bers/Grandi model of sodium-potassium pump, but one with a linear dependence on membrane potential. The moment we switched from sublinear to linear and adjusted pump rate to maintain reasonable sodium and potassium levels, EADs started appearing more readily.

Altogether, this section illustrates just how much experimental conditions matter, and a model working perfectly in unphysiological conditions may be surprisingly problematic in physiological ones.