Deriving Equations for Two-State Ion Channel Kinetics

[thame]

I'm asked to derive two equations related to ion channel kinetics.

First, some background. These are two-state ion channels (open-closed) with the form shown below:

http://img9.imageshack.us/img9/2039/testsg5.gif

The first equation represents the open probability of the channel "P_open":

P_O = K_CO/(K_CO + K_OC)

From class, it seems that the equation is related to the Boltzmann equation, but I'm not sure how they're connected.

The second equation is of the time constant (tau) for these ion channels. That equation is:

t = 1/(K_CO + K_OC)


Is anyone familiar with these equations? My research so far has turned up a few options, though they vary subtly from the equation I received in class. For example, there is an alternative form of the P_open equation utilizing the Boltzmann constant where the K's are replaced with e^-U/k_BT

Any help at all would be really appreciated. Thanks Physics Forum!

 

[Ygggdrasil]

One way to derive these equations is to think about chemical kinetics in terms of activation energies. In this case, you would use the Boltzmann equation and the Arrhenius equation (it's better to look it up in a general chemistry text, but here's the wikipedia link http://en.wikipedia.org/wiki/Arrhenius_equation).

Another way is too use the kinetic equations. Using the rate constants give, you can relate the rates of change of closed and open channels in terms of the number of open and closed channels. Then you can solve for the number of open and closed channels at equilibrium by solving for the case where the rates of change of the channels are zero.

 

[thame]

Hey Ygggdrasil,

First, thanks for the help. I was able to clarify the problem a bit, it seems that the equations are (or are derived from) first-order differential equations.

I'm afraid I've only taken basic calculus, so I'm not entirely sure how to do this. My professor said that it was a mass action system of two-state kinetics (he's also recommended I use MAPLE).

Any ideas on where to go from that? Does equation (xi) on http://www.klab.caltech.edu/~stemmler/s4node2.html" help?