Deriving Equations for Two-State Ion Channel Kinetics

• thame
In summary, the first equation relates the open probability of the channel to the equilibrium concentration of the channel open state, while the second equation relates the time constant for the channel to the equilibrium concentration of the channel in the open state.
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 "Popen":

PO = KCO/(KCO + KOC)

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/(KCO + KOC)

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 Popen equation utilizing the Boltzmann constant where the K's are replaced with e-U/kBT

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

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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.

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?

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1. What is biophysics?

Biophysics is a branch of science that combines principles and methods from physics and biology to study the physical processes and mechanisms that occur in living organisms. It focuses on understanding biological systems at a molecular and cellular level.

2. What are ion channels?

Ion channels are specialized pores or channels found in the cell membrane that allow the passage of ions, such as sodium, potassium, and calcium, into and out of the cell. They play a crucial role in maintaining the electrical and chemical balance of cells and are involved in various physiological processes, including nerve transmission and muscle contraction.

3. How do ion channels work?

Ion channels open and close in response to different stimuli, such as changes in membrane voltage, mechanical stress, or the presence of specific molecules. When they open, ions can flow through, creating an electrical current that can trigger cellular processes. The opening and closing of ion channels are regulated by specific proteins and can be fine-tuned by various factors.

4. What is the significance of ion channels in biophysics?

Ion channels are essential for maintaining the proper functioning of cells and tissues. They are involved in a wide range of physiological processes, including muscle contraction, hormone secretion, and nerve transmission. Studying ion channels can provide valuable insights into the mechanisms of diseases and can lead to the development of new treatments.

5. How are biophysics and ion channels related?

Biophysics is crucial in understanding the structure and function of ion channels. By using techniques such as electrophysiology, X-ray crystallography, and molecular modeling, biophysicists can study the physical properties of ion channels and how they interact with other molecules. This knowledge is vital in understanding the role of ion channels in various physiological processes and diseases.

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