I am not a chemist, but a neuroscientist, so bear with me. I have struggled with this problem for over a week now and have realized it's a chemical question. I'll try to explain it so no neuroscience background is needed.
I am trying to fit a model to experimental data. The data describes how a certain channel opens and closes in response to a change in membrane potential of the cell the channel sits in. This is captured in two datasets:
- The fraction of open channels at equilibrium for different values of the membrane potential (an example of a plot of this can be found here. This fraction I call G/G$_{max}$
- The time constant describing the time it takes for the fraction to come to its equilibrium. This time constant I call $\tau_{G/G_{max}}$
It is thought that for each open channel, 4 gates have to be in the 'open' state. That is, it is thought that:
G/G$_{max}$ = n$_{\infty}^4$
The gate n, too, can be described by its fraction of open-state/total. The rates of n moving between these two states are described by $\alpha_n$ and $\beta_n$. Once there are four gates in the open-state, the channel is instantly open. If I try to describe this in a chemical reaction, I would guess something like:
$$\ce{C_n <=>[\alpha_n][\beta_n] O_n}$$
where $C_n$ and $O_n$ depict 'closed' and 'open' n-gates respectively. And
$$\ce{4\cdot O_n <=>[instant][instant] O_{ch} }$$, where $O_{ch}$ describes the whole channel being open.
Now is the data I've got for G/G$_{max}$(V) and $\tau_{G/G_{max}}$ (this is the only thing the experimentalists can measure). However, to model this behavior, I need $\alpha$(V) and $\beta$(V). I already know that
$n_{\infty} = (G/G_{max})^{1/4}$
but I cannot figure out how $\tau_n$ and $\tau_{G/G_{max}}$ are related to each other. Even though the reaction 4O --> open channel is instant, the shape of approaching the equilibrium differs and thus the point that they cross 1-1/e would as well. n will reach its equilibrium with a single exponential, while G/G$_{max}$ reaches it with an exponential to the fourth power.
I think the answer should be quite simple, and should only depend on the 4'th power.
I hope my question is clear, but if there are more details that will help solve this I am more than happy to clarify certain points.
Edit: Here is a figure that shows how the 'measured' time constant and the one for the differential variable n differ. Also, it shows that there is not a linear relationship.
Just to be clear about what I did:
The differential system describing this model looks like: G/G$_{max}$ = n$^4$ with $\frac{dn}{dt} = \alpha_n(V)(1 - n_{\infty}(V)) - \beta(V) n_{\infty}(V)$
As one of the answers pointed out (sort of), a solution to this system with initial condition V1 is:
$n(t) = n_{\infty}(V) \cdot e^{-(\alpha_n(V) + \beta_n(V))} + n_{\infty}(V) \cdot \Big(1 - e^{-(\alpha_n(V) + \beta_n(V))}\Big)$
and thus
$G/G_{max}(t) = n(t)^4$
To find the $\tau$ of either of these traces (the time constant being the time it takes to reach $1-1/e \cdot$(final value) (see link), I plotted:
$\frac{n - min(n)}{max(n) - min(n)}$
and
$\frac{n^4 - min(n^4)}{max(n^4) - min(n^4)}$