Electrical processes through a membrane - active transport

[MMS]

Homework Statement

[/B]
I am given the configuration below of a membrane and it is said that Na+ ions are transported actively through transporters from the inside to the outside.
The J_Na_a seen below is the active current density of the Na ions and it is constant.
Vm is the potential difference on the membrane.
M is a neutral macromolecule that cannot be transported (n_M mol of it).
Both the sodium and chloride ions can be transported with the conductivities stated in the figure below.

Using the quantities given (everything I've mentioned and in the figure), I need provide equations that satisfy:
A:
1. Electo-neutrality
2. Osmotic balance
3. Chloride ion equilibrium
4. Sodium ion equilibrium

B:
Draw an equivalent circuit that describes this.

C:
Vm0 is the resting potential. Find the inner concentrations, the resting potential and the volume V as a function of the outer concentrations, n_M, the conductivites and the active current density.

Homework Equations

The Attempt at a Solution

This is what I've done so far:

A:
1. For electro-neutrality I required outside and inside that the concentrations be equal

2. For osmotic balance, equal concentrations

3. For each ion equilibrium, the potential across the membrane equal to the Nernst potential of each of the ions

However I am quite unsure of 3. The fact that there's this macromolecule is bothering me and making me think that taking the Nernst potential of each ions equal to the Vm is wrong.

Can anybody tell me if this is right please?Thanks in advance.

 

[nate808]

Thank you for your post. I would like to offer some guidance and suggestions for your problem.

A:
1. For electro-neutrality, you are correct that the concentrations of ions inside and outside the membrane should be equal. This is because for a system to be electrically neutral, the positive and negative charges must balance each other out. Therefore, the total number of positive charges (sodium ions) must be equal to the total number of negative charges (chloride ions) in both the inside and outside compartments.
2. For osmotic balance, you are also correct that the concentrations of ions should be equal inside and outside the membrane. However, in addition to this, you must also take into account the presence of the macromolecule (M). This macromolecule cannot be transported and therefore, it will contribute to the osmotic pressure inside the membrane. This means that the total concentration of ions and macromolecules must be equal on both sides of the membrane for osmotic balance to be satisfied.
3. For chloride ion equilibrium, you can use the Nernst equation to determine the equilibrium potential for chloride ions (E_Cl). This equation takes into account the concentration gradient of chloride ions across the membrane. However, as you correctly pointed out, the presence of the macromolecule may affect the equilibrium potential. Therefore, you may need to modify the Nernst equation to take into account the osmotic pressure contribution of the macromolecule.
4. Similarly, for sodium ion equilibrium, you can use the Nernst equation to determine the equilibrium potential for sodium ions (E_Na). However, as mentioned before, the presence of the macromolecule may affect the equilibrium potential and you may need to modify the Nernst equation accordingly.

B:
To draw an equivalent circuit, you can represent the membrane as a capacitor (representing the lipid bilayer) in parallel with two resistors (representing the conductivities of sodium and chloride ions). The presence of the macromolecule can be represented as a resistor in series with the capacitor. The active current density can be represented as a current source in parallel with the capacitor.

C:
To find the inner concentrations, you can use the equations you have derived for electro-neutrality and osmotic balance. The resting potential (Vm0) can be determined by setting the net current across the membrane to zero. This will give you an equation