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Nernst potential (2013)

Page history last edited by Bill Dorland 6 years ago

Prerequisites:

 

Membranes can act like "batteries", insofar as there can be electric potential differences (voltages) maintained across them. This is important to many biochemical processes, including the electrical signals in neurons.  In this page, we'll look deeper at the physical mechanism that is responsible for creating this potential difference.

 

Here's what's going on:

 

There are ions on both sides of the membrane, and each side has a roughly equal number of positive and negative ions.  If there is a higher salt concentration (i.e. more positive ions and more negative ions) on one side of a membrane than the other, and ions can pass through the membrane, then as you know, they'll pass from high to low concentrations.  We looked at the mechanism for this when we studied diffusion:  this movement from high to low concentrations is not caused by any physical force, but is a probabilistic phenomenon that results from random motion.  Each ion moves randomly, so each ion close to the membrane has an equal probability of crossing the membrane, but since one side of the membrane starts with more ions, there are more opportunities for ions on that side to cross over to the other side.  We can also think about this in terms of entropy:  having an equal concentration of ions on each side of the membrane maximizes the number of possible ways to arrange them.

 

Now let's say the membrane is semipermeable, as many biological membranes are.  Let's say that there is NaCl on both sides (but not the same concentration on both sides), and that the (positive) sodium ions can pass through the membrane but the (negative) chloride ions can't.  Then positive charges will go across the membrane, from the high-concentration side to the low-concentration side, while the negative charges will stay put.  Thus, as a result of the concentration difference, a net electric current flows across the membrane!  It's as if there's an electric potential there!  Even though the individual charges are just moving randomly (and there is no actual potential difference), the net effect looks like a current flowing as the result of a potential difference.

 

What would it take to stop this current?  As charges move across the membrane, one side of the membrane becomes slightly positive and the other becomes slightly negative, and these unbalanced charges create an electric field pointing in the other direction, so that the charges feel a force in the other direction.  If the magnitude of this electric field is just right, then the charges will reach equilibrium, with entropic effects (i.e. the diffusion resulting from the concentration difference) balanced out by forces.  We've seen several examples of this sort of effect already: screening (where random thermal motion of ions in solution is balanced by electrostatic attraction) and the isothermal atmosphere (where random thermal motion of air molecules is balanced by gravity).

 

The number of ions that need to move across the membrane for the system to reach equilibrium is very small, so the change in the concentrations is negligible.  This makes the math (below) much easier, unlike in the case of Debye screening (where we skipped the details of the math).

 

If we can figure out how much applied potential difference it would take to keep the charges in equilibrium, that will tell us the Nernst potential across this membrane.  (Similarly, when we put something on a scale to measure its weight, we're not directly measuring the gravitational pull of the Earth on the object!  Instead, we're measuring how much upward force we have to exert on the object to balance out the Earth's pull and keep it in equilibrium.  But it still works.)

 

Without any applied potential difference, the equilibrium state would be equal concentrations on both sides of the membrane.  What potential difference would it take to make the concentration c1 on one side and c2 on the other side?  Recall the Boltzmann distribution!  If there is an energy difference between the two sides of the membrane, then ions have a different probability to end up on each side of the membrane (when the system reaches equilibrium), and the ratio of these probabilities is the same as the ratio of the concentrations.

 

The Boltzmann factor tells us that the ratio

 

Formula

 

where ΔU is the energy difference, and that's equal to qΔV, where q is the charge of each ion, and ΔV is the potential difference across the membrane.  We can use this to solve for the Nernst potential, ΔV:

 

Formula

 

Therefore: 

Formula

 

This gives us the voltage across a membrane, if we know the concentrations on each side.  Here, q is the charge of a single ion, but this is sometimes written as ze, where z is the valence of the ion (e.g., for Na+, z = 1), and e is the elementary charge.

 

Just to check that this equation makes sense: If c1 = c2, then c1/c2 = 1, so ln (c1/c2) = 0, so there's no potential difference.  That's what we expect:  if there's no concentration difference, then there's no voltage.  As the concentration difference gets larger, this fraction gets larger, so the potential difference also gets larger.

 

Here, we were mainly paying attention to the magnitude of the voltage difference, and not the sign.  This was on purpose:  though it's possible to figure out the sign of the potential difference from the equation, it's better (and less error-prone) to think it through qualitatively and determine which sign makes sense.

 

In this example, where the positive ions are the ones that can move through the membrane, positive current flows from high concentration to low concentration, so the concentration gradient acts as if the high-concentration side of the membrane is at a higher potential.  This effective potential difference is the "Nernst potential". As the current flows, positive charge builds up on the low-concentration side (and therefore the high-concentration side becomes more negatively charged), which means there is an electric potential difference, and the electric potential is higher on the low-concentration side.  A net current continues to flow until the electric potential balances out the Nernst potential, and the system reaches equilibrium.

 

But in another situation, where the membrane is permeable to negative ions but not to positive ions, negative ions would flow from higher to lower concentration, so since the direction of the current is the direction that positive charge flows, this means that the current flows from the lower-concentration side to the higher-concentration side!  Therefore, in that case, the concentration gradient acts as if the lower-concentration side of the membrane is at a higher potential.   Again, this is the Nernst potential.  Therefore, a higher electric potential builds up on the high-concentration side.

 

It's worthwhile to think this through each time!

 

 Workout: Nernst potential

 

Ben Dreyfus 3/1/2012 and 3/23/2016

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