Just a quick note to tell you about our latest batch of papers in the “Heart By Numbers” special issue of Biophysical Journal which arose from a meeting in Berlin last year. They are all about I_{Kr}, the rapid-delayed rectifying potassium current carried by the hERG potassium ion channel, which is important in drug safety, as pharmaceutical drugs can block it and lead to dangerous disturbances in your heart’s rhythm. The papers are all open access and supported with open code and datasets (see our Github site).

First paper was led by Michael Clerx and is called “Four Ways to Fit an Ion Channel Model” we thought that a recipe or tutorial about how to make a Hodgkin-Huxley model for a voltage-gated ion channel was needed: why the protocols for ‘activation’, ‘deactivation’, ‘inactivation’ etc. are designed the way they are, how you can assemble data from them to fit time-constant and steady state curves, and which numerical schemes are sensible to use for doing this. In a lot of ways this paper is the partner for Kylie’s sinusoidal clamp paper that we published last year, which explains how people have typically done it, and then goes on to weigh up the pros and cons by trying the four different methods on the same dataset. The basic ways are:

- Method 1: Fitting the model’s analytic equations for steady-states and time constants (e.g. m
_{∞}and τ_{m}for a gating variable m) directly to experimentally-derived current-voltage (I-V) and time-constant-voltage (τ-V) curves. This is generally a bad idea if you have more than one gate (for IKr at least, their timescales of gating aren’t as independent as this concept really needs them to be – e.g. inactivation mucks up your measurement of activation). You can show the problem quite easily by running a model forwards with some assumed gating properties. You then simulate the experiment and its post-processing to emulate experimental measurement of steady state and time constant gating properties – you get back different ones than the underlying equations suggest! But this method remains widely used in the literature. - Method 2: one way round the problem above is simulating the experiments and then deriving I-V and τ-V curves by postprocessing simulated currents, and then fitting these to data. So you get a model whose parameters are consistent with the data. BUT the optimisation problem becomes very hard because of the postprocessing steps to derive I-V and τ-V curves introducing more jumps and discontinuities in the objective function. There can be ‘divide by small number’ effects in the postprocessing, or long time constants fitted to flattish data, that lead to a lot of error on certain summary curve data points too.
- Method 3: Simulating the traditional experiments and fitting directly to experimental current traces. This works surprisingly well, we had thought this optimisation would be difficult and need careful consideration of weighting traces and suchlike, but even a simple approach worked well, and better than Methods 1 and 2.
- Method 4: Kylie’s sinusoidal clamp method. Experiments are very short (hence easier than Method 3), fitting is very simple and reliable.

A few highlights to look out for in this one:

- A phase-plane plot method really helps understanding of the traditional protocol design, see Fig 1:

- Some tips and tricks for reliable parameter optimisation (including the role of parameter transforms, sensible bounds for voltage-dependent ion channel rate parameters, and the solver tolerance issue discussed on this blog before).

The second paper is by Chon Lok Lei, he created a new Method 4 discussed above, but instead of using the sinusoidal clamp, made a similar ‘*Staircase Protocol*‘ out of conventional steps and ramps which was able to run on a high-throughput 384 well Nanion SyncroPatch machine. This has some distinct advantages: Chon Lok got 124 good cell recordings, in one run of the machine, in about half an hour; compare this to manual patch where getting 10 nice stable cell recordings might take a week or more! So we hope this will let people create models of variants of this channel (e.g. mutations) and what happens to it under drug action much more easily than before. Some important points:

- Fitting the same model using the staircase method returned very similar parameter sets to our previous manual-patch sinusoidal study.
- There’s some nice mostly-automated quality control criteria used in addition to the usual series resistance, seal resistance and cell capacitance.
- There’s some work on leak current and drug-subtraction to isolate IKr that will be worth looking at if you want to repeat such escapades.
- We did 8 validation protocols alongside the staircase protocol that we used for fitting, and got some excellent predictions, for instance Fig. 2:
- A hierarchical statistical model allowed us to describe the variability that we saw fitting to data across all the wells.
- The paper includes the invention (we think!) of the ‘
*reversal ramp*‘ to estimate error in applied voltage clamp, a special bit of the protocol designed to estimate an artefact in the experiment (a bit like a leak step does). Due to the parallel nature of the experiment and shared solutions/temperature this allows an estimate of this error in each individual well as shown in Fig 10 of the paper (small, but as we’ll see, maybe now the main source of error…). - Some strong circumstantial evidence that the principal variability in kinetic parameters fitted to different cells is due to this voltage clamp error rather than extrinsic variability between cells (see Fig. 9 of the paper).

The third and final paper, again by Chon Lok, uses the Staircase Protocol method introduced above to examine the temperature dependence of the parameters we get back. In essence we repeat the experiment at five distinct temperatures, so we can plot out how they vary with temperature. We can then compare this with how they should ‘theoretically’ vary with temperature following either:

- The commonly-used Q10 formulation.
- The sometimes-used (what we called ‘typical’) Eyring formulation (not actually as per Wikipedia – a voltage-dependent term is added to the Wikipedia definition in previous ion channel modelling! See our paper for the typical definition).
- The never-used in ion channel modelling (as far as we know!) Generalised Eyring formulation.

The punchline is that there’s strong evidence that Q10s are insufficient as they don’t give you any temperature-dependence on the parameter that governs the voltage dependence of the rate. But a strong dependence certainly appears in our refitted models. The typically-used Eyring formulation does confer some temperature dependence to this parameter, but is actually no better as the temperature-dependence can sometimes be constrained to be in the wrong direction (increasing with temperature instead of decreasing or vice-versa). The more generalised Eyring formulation, previously used in some electrical engineering battery applications, appears to work pretty well. So some significant consequences for any work that is relying on previously-used formulations for temperature dependence of voltage-dependent rate parameters.

If we are right, and the temperature dependence doesn’t follow a Q10, then we expect Q10 estimates to be protocol-dependent, and indeed we can explain some “this paper’s Q10 estimate was higher than this paper’s” (but not all) using our full temperature model with previously used protocols from the literature.

Our punchline is that you probably need to do the experiments at 37°C to be confident you are predicting well there.

In the discussion there are some interesting questions raised about whether the model breaking the first-principles ‘Typical Eyring’ formulation is actually a signpost that the model has some shortcomings/discrepancy, would a better mathematical model follow the first principles trends? Or is the need for a Generalised Eyring relation just a sign that the ‘single energy barrier’ model that the first principles rate equations assume is too simple for a large protein complex which may shift its preferred conformations with temperature. Open question!

Anyway, hope you enjoy the papers: one, two, three; comments welcome below.

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