Membrane Potential: Zero Permeability Contribution | Goldman-Hodgkin-Katz

[skandy]

In membrane potential determination of a cell, the Goldman Hodgkin Katz equation says the contribution of an ion to diffusion potential is dependent on its membrane permeability.
So in case an ion, one that has a zero permeability, is present outside the cell, using the equation , one will get contribution to diffusion potential as zero but thinking rationally, the charge on the ion must contribute to the electrical potential across the membrane.
Though I see that the equation is meant for diffusion potential, I see that the same equation is being used to find the resting membrane potential.

Here my questions are
1. Are membrane potential and net diffusion potential the same?
2. Isn't there any contribution to membrane Potential by an ion with zero permeability? If so how? Is there a different equation for it? Or will its presence influence the membrane permeability for other ions and thus have an effect?

 

[atyy]

Usually we assume that the bulk of the solution starts off neutral, eg. the solution could be A⁺B^- both inside and outside the cell. Let's suppose AB has a high concentration inside the cell and a low concentration outside. If there is a concentration difference across an impermeable membrane, then since no ions can move, both sides will stay neutral, and there will be no membrane potential difference.

If the membrane is permeable to A and impermeable to B, then A will try to diffuse down the concentration gradient from the inside of the cell to the outside of the cell. This will cause a net positive charge inside the cell and a net negative charge outside the cell. This charge excess will try to couteract the diffusion of A, since negatively charged A will be attracted back to the now positive environment inside the cell. The steady membrane potential occurs when the excess positive charge caused inside the cell exactly balances the tendency of A to diffuse out of the cell according to the concentration gradient.

Together, these two pictures are why we only put (explicitly) the permeable ions into the equation. You can also see by this reasoning that if A and B were both permeable, then both would diffuse together down the concentration gradient, and there would be no electric charge separation.

One approximation we usually make is that in principle the diffusion of A from the inside to the outside changes the concentration in the cell. But we ignore this because it takes relatively few ions to move across the cell membrane to cause a big membrane potential difference, and these few ions don't change the concentrations inside and outside by much.

These approximations can break down, but they illustrate why impermeable ions don't have to be explicitly considered in most approximations.

 

[Andy Resnick]

<snip>
2. Isn't there any contribution to membrane Potential by an ion with zero permeability?<snip>

Consider soluble proteins- they have lots of negative charge but cannot leave the cell. This leads to movement of *water* across the membrane (Donnan equilibrium). Small ions will move down their concentration gradient until a final equilibrium concentration and membrane potential is reached for each semipermeable ion species (GHZ equations) and water will move until the osmotic gradient and hydrostatic gradient are equalized.

As a specific example, consider the extracellular space to initially have 150mM of NaCl and the cytosol to have [Na+] = 150 mM, [Cl-] = 0 mM, and [protein] = 1mM = 150 mEq (each protein molecule has 150 negative charges). Initially, Cl moves down the concentration gradient into the cell, which draws additional Na into maintain electroneutrality. Final equilibrium is reached when the extracellular space has [Na] = [Cl] = 100mM and the cytosol has [Na] = 200 mM, [Cl] = 50 mM, the membrane potential is -18.4 mV (cell is negative) and the hydrostatic pressure jump is 967 mmHg (cell is positive).

 

[somasimple]

How is it possible to reach a final state where [Na] = [Cl] = 100mM when we know that the cell volume is negligible regarding the volume of the external milieu?
Your explanation may work with same volumes, right?
How is it possible to have a potential when electroneutrality (that states equality of charges) is preserved?

references :
Biological membranes
Foundations of cellular neurophysiology
https://www.physicsforums.com/showpost.php?p=3635686&postcount=2

 

[SW VandeCarr]

How is it possible to reach a final state where [Na] = [Cl] = 100mM when we know that the cell volume is negligible regarding the volume of the external milieu?
Your explanation may work with same volumes, right?
How is it possible to have a potential when electroneutrality (that states equality of charges) is preserved?

references :
Biological membranes
Foundations of cellular neurophysiology
https://www.physicsforums.com/showpost.php?p=3635686&postcount=2

It's not really dependent on volumes. The intracellular compartment has a net negative charge relative to the extracellular compartment. This is due in part to the net negative charge of intracellular proteins which cannot diffuse across cell membranes. In addition the sodium-potassium ATP dependent pumps actively maintain low intracellular sodium vs potassium concentrations while the relative concentration of these cations is reversed in the extracellular compartment. By actively pumping sodium out of the cell, this helps to maintain a net negative intracellular charge because potassium can passively diffuse out the cells (following the concentration gradient) which tends to lower the overall cation concentration inside the cell.

 

[somasimple]

It contradicts how Donnan equilibrium is computed. Working with concentrations may lose the dimensional aspect of the phenomenon.

 

[SW VandeCarr]

It contradicts how Donnan equilibrium is computed.

Yes it does. The ATP dependent sodium-potassium pump maintains a disequilibrium.

http://www.asianscientist.com/books/wp-content/uploads/2013/06/4987_chap1_1.pdf

 

[somasimple]

Yes it does. The ATP dependent sodium-potassium pump maintains a disequilibrium.

The second sentence does not change the computation of the equilibrium neither the involved volumes.

 

[SW VandeCarr]

The second sentence does not change the computation of the equilibrium neither the involved volumes.

So what's your point? Are you interested in how cells work or how a model works?

"Ion transporters are divided into pumps and exchangers, but in all cases the duty of the
transporter is to move specific ions against their electrochemical gradients in order to
maintain a non-equilibrium steady state, such as the resting membrane potential." (Section 2.4 of the link in post 7.)

 

[somasimple]

Are you interested in how cells work or how a model works.

Both. A model may describe how a cell works.

 

[SW VandeCarr]

Both. A model may describe how a cell works.

Yes, but not the Donnan equilibrium model.

 

[somasimple]

Yes, but not the Donnan equilibrium model.

So, are you saying that the second response describes "how a cell works" with a model that does not describe "how a cell works"?
BTW, the first reply is based upon a " common" cell model. This one is far from the second.

 

[SW VandeCarr]

So, are you saying that the second response describes "how a cell works" with a model that does not describe "how a cell works"?
BTW, the first reply is based upon a " common" cell model. This one is far from the second.

I can only speak for my own responses. I described basically how the cell works in this regard in post 5. If you have some disagreement with this description, specifically point it out based on acceptable science. Otherwise I would advise that you cease this argumentative and seemingly pointless line of discussion.

 

[Andy Resnick]

How is it possible to reach a final state where [Na] = [Cl] = 100mM when we know that the cell volume is negligible regarding the volume of the external milieu?
Your explanation may work with same volumes, right?
How is it possible to have a potential when electroneutrality (that states equality of charges) is preserved?

I'm not entirely sure what you are asking.

 

[SW VandeCarr]

I'm not entirely sure what you are asking.

It seems somasimple doesn't understand that there is no final electrochemical equilibrium state with regard to the animal cell except with cell death. The living cell functions in a non-equilibrium steady state between the intracellular and extracellular compartments as I've tried to explain. If the sodium-potassium pumps were poisoned and failed to function, sodium and water would diffuse in, potassium would diffuse out, and the cells of the body would expand and probably lyse. If one wants a purely passive diffusion model, that's a question for physics, not biology.

 

[somasimple]

I'm not entirely sure what you are asking.

As a specific example, consider the extracellular space to initially have 150mM of NaCl and the cytosol to have [Na+] = 150 mM, [Cl-] = 0 mM, and [protein] = 1mM = 150 mEq (each protein molecule has 150 negative charges). Initially, Cl moves down the concentration gradient into the cell, which draws additional Na into maintain electroneutrality. Final equilibrium is reached when the extracellular space has [Na] = [Cl] = 100mM and the cytosol has [Na] = 200 mM, [Cl] = 50 mM

Things that are clearly stated may be easily understood.

1/ In your example, the extracellular space is 150mM of NaCl at start.
2/ In your example, the extracellular space is 100mM of NaCl at End.
3/ You have diluted the extracellular content/volume with the content/volume of the cell.
4/ There is a problem of scale/volume in your explanation because the concentration of the extracellular space/volume can't be changed/modified by a concentration of a volume that is million and million time smaller.

 

[Andy Resnick]

Things that are clearly stated may be easily understood.

1/ In your example, the extracellular space is 150mM of NaCl at start.
2/ In your example, the extracellular space is 100mM of NaCl at End.
3/ You have diluted the extracellular content/volume with the content/volume of the cell.
4/ There is a problem of scale/volume in your explanation because the concentration of the extracellular space/volume can't be changed/modified by a concentration of a volume that is million and million time smaller.

I see your error(s)- in my example, ions are transported into the cell; this is not the same thing as the transport of water into the extracellular space. Also, in normal physiological systems, there is not a single cell in an infinite reservoir; for example an epithelial cell layer develops directed ion transport and forms a physical/functional barrier between two extracellular compartments; osmotic differences across the cell membrane as well as transepithelial are balanced by electrochemical differences and not hydraulic gradients. These gradients are maintained as long as the cell/tissue is alive and hydrolyzing ATP.

Now, consider a single cell organism: yeast, bacteria, algae, etc. These organisms have additional structures to resist osmotic pressure- a rigid cell wall, for example. Putting an isolated mammalian cell into a hypo- or hyperosmotic solution results in cell swelling/dehydration and cell death.

Does this help?

 

[Andy Resnick]

It seems somasimple doesn't understand that there is no final electrochemical equilibrium state with regard to the animal cell except with cell death. <snip>. If one wants a purely passive diffusion model, that's a question for physics, not biology.

Agreed.

 

[somasimple]

Firstly I must thank you since you brought a bit of osmosis in the model. It is sure the movement of the solvent (water) is yet underestimated.

I see your error(s)- in my example, ions are transported into the cell; this is not the same thing as the transport of water into the extracellular space.

Thanks again to point out a mistake I made but it does not change the problem at all.
Effectively, you may dilute a solution by adding some solvent and there will be a change in its volume.

You may, also, dilute a solution by removing/transporting some quantity of its content, as in your example. The external volume remains unchanged but I'm quite sure there's not enough room in a cell to move such a quantity of ions. The internal concentration will reach some summits, alas unknown, in physiological models or alive ones.
I'm sure you'll find another faulty point in this argument...

Also, in normal physiological systems, there is not a single cell in an infinite reservoir; for example an epithelial cell layer develops directed ion transport and forms a physical/functional barrier between two extracellular compartments; osmotic differences across the cell membrane as well as transepithelial are balanced by electrochemical differences and not hydraulic gradients. These gradients are maintained as long as the cell/tissue is alive and hydrolyzing ATP.

Also, I'm glad you introduce a more physiological model but epithelium is a very specialized cells tissue that is in direct contact with environment. It has functions and properties that are far from our simple cell model.
Also, the cell models described in books are constructed/based on a relation 1 to 1.
A cell tissue brings a new relation: 1 to n and some properties that worked with the previous relation may not work anymore.
Thus, you can't put the blame on me since I was not aware of such a model AND this model is not the subject of this thread.

Now, consider a single cell organism: yeast, bacteria, algae, etc. These organisms have additional structures to resist osmotic pressure- a rigid cell wall, for example. Putting an isolated mammalian cell into a hypo- or hyperosmotic solution results in cell swelling/dehydration and cell death.

Does this help?

I think it does not help since cell volume variations exist and are the proof of a perfect functioning and a fate of life. Cells may swell without a chance of death.
Here is a very common example: neurons.
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1458751/
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3201844/
Unfortunately, water movement (and all concentrations changes that happen with such changes) are not integrated, yet, in a more physiological cell model.

 

[Andy Resnick]

It is sure the movement of the solvent (water) is yet underestimated.
<snip>

The family of water channels (aquaporins) are well-studied- Peter Agre received a Nobel for discovering them. Water permeability of tissue is also fairly well-measured.

Maybe I don't understand your questions? Complaining about a simplified model presented in a textbook is hardly sporting...

 

[somasimple]

The family of water channels (aquaporins) are well-studied- Peter Agre received a Nobel for discovering them. Water permeability of tissue is also fairly well-measured.

Well, Andrew,
I think I'm already aware of aquaporins =>
Something stated at Physics forums in 2010
https://www.physicsforums.com/showpost.php?p=2736847&postcount=40
But if aquaporins are found in astrocytes, their presence is not yet defined in neurons.
http://chemse.oxfordjournals.org/content/33/5/481.full.pdf

AQP4 is expressed in the supporting and basal cells but not in the neuronal sensory cells (Ablimit et al. 2006). All of these observations show that, in the nervous tissues, AQP4 is localized not in nerve cells per se but in supporting cells that are situated in close contact with nerve cells.

http://www.ncbi.nlm.nih.gov/pubmed/21602841
Also, I may understand why you remain silent about ions transport. Silence is often a kind of reply.