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In my studies I keep coming across the form of an equation that is used in many different mathematical models for voltage gated ion channels. The most general form I have found is in the 1977 paper Reconstruction of the action potential of ventricular myocardial fibres by Beeler and Reuter. Each ion channel has a gating variable $y$ whose differential equation depends on functions $\alpha$ and $\beta$, both of which take on the general form:

$$\{\alpha,\beta\}=\frac{C_1\exp[C_2(V_m+C_3)]+C_4(V_m+C_5)}{\exp[C_6(V_m+C_3)]+C_7}$$ where $V_m$ is the membrane potential and each constant is determined to have different values depending on the channel, system, etc.

My question is where does this form come from? In all of the papers I have come across they seem to just use this form with numbers chosen to fit their data without explanation as to why this form is actually used. The Beeler and Reuter paper cites the 1952 paper by Hodgkin and Huxley, but it seems like H&H use simpler forms of this equation primarily motivated by just fitting the data rather than illuminating any underlying mechanisms. Therefore, I do not understand where this form in the B&R paper comes from that so many papers later on use as well.

Does this form come from assumptions about the workings of the ion channels, if so, what are these assumptions, and how do they give rise to this form? Or is this just a general form found to fit many data sets fairly well, and if so why was this chosen over other functions that could probably do the same thing?

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As I understand from the paper, it is a phenomenological equation that allows them to fit the rates α and 𝛽 for all the ion channel particles in the model. So, as in H&H model, it have "loose" theoretical foundations. In page 183:

In order to simplify the reporting of the actual values used in this model, we have expressed the alphas and betas entirely in terms of a generalized function, with eight defining coefficients. Table 1C gives the equation, and the defining coefficients for al rate constants.

I have read papers using Eyring equation for a theoretical formulation of transition rates, based on thermodynamics.

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    $\begingroup$ This is partly true, in that in the HH model the theoretical foundations are at best loose, but I would not say they are non-existent: they refer to Goldman for example and all relate to Boltzmann equations. The choice of the generalized function in the Beeler and Reuter paper is mostly motivated by algebraic simplicity. That said, all are approximations to more modern biophysical models. $\endgroup$ – Bryan Krause Jun 11 at 17:11
  • $\begingroup$ Goldman equation refers to the ion reversal potential not to the ion channel dynamic. $\alpha(V)$ and $\beta(V)$ are not Boltzmann function in H&H model. As far as I know, they are just phenomenological. If you know any derivation for them please let me know. $\endgroup$ – heracho Jun 11 at 17:30
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    $\begingroup$ @heracho Bryan isn't referring to the Goldman equation. I think a paper by Goldman in 1944 is being referred to, which is cited by the HH paper when discussing a lose physical interpretation of their equations for $\alpha$ and $\beta$ $\endgroup$ – Aaron Stevens Jun 11 at 18:48
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    $\begingroup$ @BryanKrause Thanks for the hint. In H&H 1952: "First, it is one of the simplest which fits the experimental results and, secondly, it bears a close resemblance to the equation derived by Goldman (1943) for the movements of a charged particle in a constant field. Our equations can therefore be given a qualitative physical basis if it is supposed that the variation of $\alpha$ and $\beta$ with membrane potential arises from the effect of the electric field on the movement of a negatively charged particle..." $\endgroup$ – heracho Jun 12 at 15:42
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    $\begingroup$ @heracho Yeah, that's the sentence I was referring to; they also talk about how it doesn't quite fit Goldman (I think in the next sentence, or maybe next paragraph), because the voltages that they fit are not symmetrical. They basically punt and say "we don't know what exactly is going on" which is pretty fair given they are writing this in the 1950s (fair enough for a Nobel prize at least ;) ). I think if you add this comment discussion to your answer and summarize this comment discussion a bit it would be a good answer. $\endgroup$ – Bryan Krause Jun 12 at 16:05

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