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I came across this simple analytical expression of time constant of deactivation in Elenes et al., 2006. How does one derive this expression

$$\displaystyle\tau_\text{deactivation}\approx\frac{1}{D_+ + 2j_- + \frac{\alpha_2 2k_-}{\beta_2+2k_-}}$$

for the model given below.

           

An allosteric reaction mechanism for the muscle AChR. C, O, and D denote the closed, open, and desensitized conformations of the channel, whereas A denotes a molecule of ACh. For simplicity, only one of the (probably several) desensitized conformations is included in the model. Also, the two neurotransmitter binding sites are assumed to be functionally equivalent and independent and, therefore, the two possible monoliganded configurations are considered to be functionally indistinguishable. These simplifications have no consequences on the interpretation of our data. The red arrows indicate the rate constants that, according to Eq. 1, determine the kinetics of the macroscopic current decay upon stepping the concentration of ACh from saturating to zero (i.e., during channel deactivation).

The material is esoteric, but I am hoping someone will be able to answer. The same paper also mentions that deactivation time constant is the reciprocal of $\frac{\alpha_2 2k_-}{\beta_2+2k_-}$ and references Colquhoun and Hawkes, 1982. This reference will take quite a while to grasp completely.

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  • $\begingroup$ I just checked the linked paper and the paper that it refers to. Both of them do not explain how the expression was derived. No supplementary methods also. You may find the derivation (not the final one, still) in the 1981 paper but you need to spend some time on it and perhaps read a bit about continuous time Markov chains. This is certainly beyond my scope and if we don't get a reasonable answer from anyone else then we may migrate it to Mathematics or Physics and try our luck there. $\endgroup$ – WYSIWYG Jan 28 at 15:28
  • $\begingroup$ I'm trying to solve the system, but I'm having some troubles with the mathematics. On the meantime I recommend reading the chapters of Colquhoun and Hawkes in the book Single Channel Recording specially the Q matrix cookbook. These are a digest of their papers. $\endgroup$ – BPinto Feb 20 at 4:23

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