Maybe it would help to think about two different size scales. On the scale of a single molecule of the antiporter, the transmembrane proton and sodium gradients can directly be put to use by the protein as it exchanges a proton on one side of the membrane for a Na+ ion on the other. The protein can bind to the ions. The protein molecule has no way of directly interacting with the transmembrane voltage. However, on the size scale of the whole mitochondrion, the proton gradient interacts with the transmembrane voltage. So, indirectly, the antiporter interacts with the transmembrane voltage.

Adam: I understand but that doesn't really what the question asked. Ok, so if you consider the whole mitochondrion, then the driving force for H+/Na+ should be both electrical potential and the concentration gradient. But in the lecture note, it said that the driving force is only the concentration but not the electrical potential (the attraction of the relatively negative charge in the matrix) because 1 Na+ is exchanged for 1 H+ so its neutral, but it doesnt seem like that in my opinion! 

"For example, if one side of the membrane is 4 volts, the other is 2 volts then because its an electroneutral reaction, the voltage stays that same but there still is a difference in the potential between two sides."

In this part of "vectorial tranlslocation" voltage is a difference in electric potential between two points on opposite sides of a membrane. so it is not clear what you meant. YEs, the antiporter works because of difference in [H+] (delta pH).   you can prove it by using nigericin+K+ and inhibit this exchange.

If you prefer to talk about Williams  "localized potentials"  and protonic microcircuits - yes you can involve this, but do you think it would make students understand the topic better?

Basically, my example is: if one side is 4 volts, and the other side is 2 volts, then when it translocate 1 Na+ out, 1 H+ in, then basically the voltage stays the same: 4 on one side, 2 on the other side which means the driving force for the this translocation must include the electropotential as well. But my lecturer said the driving force for this H+/Na+ is only the difference in concentration which means the difference in [H+] between two sides drives this translocation and the reason there is no electropotential gets involved is because we exchange 1Na+ for 1 H+ so its neutral. But this doesnt make sense because of the reason in my example!

I thought my question was clear??

voltage is a difference in electric potential between two points on opposite sides of a membrane

Im sure you know about the chemiosmotic theory, right? So there are two factors that contribute to this: first is the concentration between two sides, and secondly is the voltage difference.

For example, the symporter H+/Lactose using the difference in the H+ concentration between two membranes to drive the import of Lactose. Also, the protein use the free energy that is drawn from the voltage difference between 2 membrane (on the cytoplasm, its more negative relative to the periplasm or cell exterior). And my lecturer said that since Lactose is neutral, it doesnt affect this electric potential (I have nothing to say about this).

But then another example was considered is the Na+/H+ antiporter. My lecturer said this only uses the difference between H+ conc. to drive the in 1H+ for 1 Na+ out but not the difference in potential. It was explained because Na+ is equally charged like H+ so there wasnt a net change in potential. But that doesnt make sense to me because if, just if, there was a difference in potential between two membrane before, then when there is no net change in potential, this original difference in potential should stay the same and it will still act as a driving force for the protein to work. And in fact, there is a difference in H+ electrical potential since the matrix is more alkaline than the intermembrane space.

Is it clear now?

Perhaps a good way for you to investigate this conundrum is to go back to the original research literature on the antiporter. Find out whether experiments have been done using purified antiporter reconstituted into proteoliposomes in which the transmembrane accumulation of Na+ was accomplished either by setting up a transmembrane pH or a transmembrane potential, each in the absence of the other.

Here is an article to get you started:

http://www.jbc.org/content/266/10/6518.long

Another thought addressing a comment of yours: If the transmembrane potential is not reduced as a result of transmembrane ion transport by the antiporter (in an experimental situation in which the potential is not constantly regenerated), then how can it act as a driving force for transport? That would be perpetual motion. Energy in some form must be consumed in order to move ions against a concentration gradient.

Look for experiments in the literature that examine this question of driving force. That is the way to answer your question decisively.

voltage is a difference in electric potential between two points on opposite sides of a membrane.

It was explained because Na+ is equally charged like H+ so there wasnt a net change in potential...  across the membrane.

Maybe I may add a comment to this discussion. First, I think that the question should be considered in term of thermodynamics and fluxes must be distinguished from equilibria.

In fact the transporter is effectively neutral and is only sensitive to concentration gradients: its action results in a decrease in both Na+ and H+ concentrations gradients across the inner mitochondrial membrane (that’s where the energy is coming from) until both gradients equilibrate, but at lower values. However, this occurs across the mitochondrial membrane where the proton motive force is continuously maintained by respiratory chain proton pumping and results in a ∆µH+ with two components : ∆pH and membrane potential. Since ∆pH is decreased by the Na+/H+ antiporter, we do not observe a decrease but instead an increase in membrane potential. This is what has been constantly observed with nigericin, an artificial K+/H+ antiporter.

I hope this may help.

Philippe: You mentioned that there is a difference in pH between two sides of the mitochondrial membrane. I agree that this means the concentration gradient will be one of the forces that drive the action of this transporter. But since H+ is an ion, the difference in concentration will create a difference in electrical charge between two sides of the membrane as well, which means the side that is higher in pH will be more negative relative to the side that is lower in pH. So wouldn't this an other force that drives the action of the transporter as well, along with concentration gradient?

If H+ were the only ion having a concentration gradient across the membrane, then there would be a charge difference across the membrane, too. However, there are many different ions that can have concentration gradients across the membrane. It is therefore possible to have a delta pH without a voltage if another ion gradient cancels out the voltage of the proton gradient. This is, in fact, at least one function of the H+/ion antiporters.

You’re right, the side with higher pH (matrix side) is also negative (negative inside membrane potential), that’s why ∆µH+ has two components (∆pH and membrane potential), both coming from proton pumping by respiratory complexes. However, you are talking about an "antiporter" which associates the movement of H+ with an opposite movement of Na+, the total effect is therefore neutral. You cannot separate both transports, if the transporter was only transporting H+ it would be an uncoupler and would totally collapse ∆µH+.

Adam: Can you confirm for me if the concentration of other cations in the matrix of mitochondria add up to cancel out the effect of the proton? I dont think it is because for the Calcium transporter, the driving force is the difference in potential between two sides and Ca2+ ions are attracted to the negative side of the membrane.

Philippe: No, thats the problem. The fact that its an antiporter doesnt mean there is no difference in potential between two sides. Its an antiporter, the only thing we can say is that there will be no net charge change (1Na+ exchange for 1H+), so if there is a difference in potential between 2 sides, then this difference will stay the same and will NOT be affected by the activity of the antiporter (as its neutral in charge). So if there is a difference in potential created by H+ ions (and in fact there is this difference) then this should be the driving force as well!!

Anh.  Have you ever looked at the Peter Stewart/John Kellum view of Acid Base.  There are some papers and a recent book titled, "Stewart's textbook of Acid-Base 2nd edition" As par tof their acid-base work they actual question the accepted NHe exchanger explanation.  The basic summary is that the Na+ cation moves, but they indicate no studies that actually show proton transported across a membrane in the opposite direction.(ie no radio-labelled H+ tracked)  In lieu they hypothesize that to maintain electric neutrality, in addition to the Na+ cation moving, perhaps there is a weak anion that also moves with it in the same direction.  Their view of acid base will also bend your mind if you haven't heard of it yet