# Potential energy and ATP synthesis

## Main Question or Discussion Point

Hello!
I am at a loss as to the cause of an energy potential from a concentration gradient.
Take for example the H+ concentration gradient across the innermembrane in mitochondria.
The greater concentration of H+ in the intermembrane space relative to the matrix results in an electrical potential and a chemical potential (acoording to wiki). I assume the chemical potential merely makes reference to the difference in concentrations. For example, an uncharged molecule with a higher concentration on one side of a membrane would have chemical potential, but not electrical potential.
If we were to place a partition through the middle of a container filled with water and placed an uncharged species A in to only one of the two compartments, the individual molecules would simply go about their regular business of moving about, bumping in to each other and the water molecules, and would diffuse randomly throughout the section. If an apprture was introduced. Slowly the molecules of A would pass through the aperture in a random, undirected manner. There is no purposive movement to occupy the whole container and to distribute themselves evenly throughout the container. It is simply through random movements and the resulting (approximately) even distribution is simply due to probability.
In this example there is a concentration gradient, but I do not see the potential to do work.
I was thinking of a gas in a container (not partitioned), with a turbine located near an aperture. The container holds a molecule B which is not present outside the container. If the container is at the same pressure as the outside, then opening the aperture will not drive the turbine, yet the molecules will diffuse out and become evenly spread throughout the container and the outside. If the pressure inside the container is greater than outside the container, opening the aperture would then result in the movement of the molecules out of the container and their greater and directed kinetic energy may be used to drive the turbine. In both examples, there is a concentration gradient, yet only trhe second can be used to do work.
Returning to the H+ difference in the mitochondria, I do not therefore see how the concentration gradient in and of itself can be used to do work.
The difference in concentration is used to do work, however; but where does the potential energy come from?
Potential energy, as far as I am aware, isn't an energy 'on it's own' (I apologise if this becomes a little obscure): you can have gravitational potential energy due to the force of gravity trying to pull two masses togeather. If this is impeded in someway, the objects have potential energy. But if one of the objects suddenly disappeared, the potential energy would also disappear: there has to be something to provide that potential energy.
The only 'source' I can think of for the H+ gradient is the elecrostatic attraction. The H+ are pulled/pushed through ATP Synthase by the force produced by the electrostatic attraction (present due to differences in charge across the innermembrane). If the electrostatic attraction was taken away, the molecules would simply diffuse down ther concentration gradient randomly, if possible, and this could not be used to do work (in this case generate ATP) unless there was a difference in pressure, as with the gas example, but in this case liquid pressure. By my reasoning at least.
Apologies if this is tortuous, but I wanted to explain my reasoning, as the explanations I have found do not seem to answer these issues.

Any help appreciated,
Many thanks,
Nobahar.

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If you put water into a U-shaped pipe with a semipermeable membrane in the middle and then pour a lot of NaCl in the right opening, the water will move from a lower concentration of ions to a larger concentrations of ions due to osmosis. This will raise the surface level of the water on the right side due to the movement of water. You can put a feather on the surface of the water in the right side of the tube and it will rise, illustrating an increase in potential energy.

H+ moves down the electrochemical gradient, that is, the gradient formed when taking into account both the concentration and electrostatic gradient.

If there is just a concentration gradient, single molecules would move randomly, but the net effect would be of molecules spreading from an area of high concentration to an area of low concentration of that particular molecule. This is because there are more ways for the molecules to spread out than there is for them to be all in one corner, so therefore, the probability of them being in such a 'disordered' state is higher. According to the second law of thermodynamics, this 'disorder' always increases in a isolated system.

This is easy to imagine if you ever had a friend who wears a lot of perfume or deodorant. It just spreads and you can smell it across the room after a while.

Hi Nobahar.
This question bothered me for a while too, until I read chapter 7 of Nelson's 'Biological Physics'. One way of looking at this, the abstract way, is to think about the number of ways in which the particles can be spread out in a smaller volume compared to a larger volume. The larger volume has more ways, and more disorder, and so there is an entropic force which can perform work, such as pushing against gravity in Mkorr's example.

However, the other way in which you can look at this problem is in terms of forces - transfers of momentum. You find it difficult to see how the random Brownian motion across the concentration gradient can give rise to this Newtonian force. The way in which the abstract 'disorder' forces above correspons to the actual mechanism is as follows:

Osmotic flow is a 'rectifier' of Brownian motion. Imagine a particle moving in a solvent. As a particle moves, it leaves a vacuum behind it, which 'sucks' more solvent in. Because of this viscous friction, solvent is pulled along with the particle. If there is no membrane and the system were in equillibrium, then the net effect of this would cancel out. However, if there is a membrane with particles on one side and not on the other, then particles bouncing off the membrane will drag solvent along with it as is moves away - because this transfer of momentum only occurs on one side of the membrane and not the other, there is a net transfer of momentum and work is done. As you can see, there is potential for the gradient to do mechanical work, such as rotating the F-ATPase and 'pinching' together ADP and Pi in such a way as to make the ATP state lower energy than the ADP + Pi state.

I hope this helps,
I

Thanks for the response Iainois. That’s interesting, although I do not full understand it.

This is the reasoning I have put together since my last post. It may be consistent with what you said, I do not know.

I shall use an exemplum to illustrate the main points of my ‘understanding’ of how a concentration gradient can provide energy to drive a process.

There at two compartments separated by a membrane. Imbedded in the membrane are some channels through which molecule A in an aqueous solution can pass. The membrane itself is not permeable to molecule A but is permeable to water, and so the channels act as the only passage for molecule A to the other compartment. This channel can acquire some of the kinetic energy of the molecules and utilise this to rotate. Depending on the compartment the molecules is passing into, the channel can rotate either clockwise or anticlockwise.

Molecules in the liquid or gas state in particular have lots of kinetic energy. I do not know how they acquired this, I presume there are a number of ways, but it isn’t important, what’s important is that it is not due to a force continuously acting on the molecule, as is the case with gravity turning an overshot or breastshot waterwheel, or electrostatic attraction. The molecules of A have acquired their kinetic energy, and I guess a better measure of which is their average kinetic energy.

When the system is in equilibrium (there are equal concentrations of molecule A in both compartments) and the average kinetic energy per molecule is the same in both compartments, then the channels will have no net rotation in any direction: the molecules will move with equal frequency in both directions and the channels will rotate both ways by equal amounts.

When the system is not in equilibrium but the average kinetic energy per molecule is equal for both compartments, then there is a concentration gradient – there are more molecules in one compartment than the other but there is no pressure difference between the compartments: I think the movement of water will counter any attempt to alter the pressure of one of the compartments, if not, assume no pressure difference, which I believe is the case in the mitochondria in cells. For probabilistic reasons, again I stress there is no force acting to move the molecules down there concentration gradient (I assume this is why it is referred to as a passive process), there will be a net flow of molecules from the compartment with a higher concentration to the compartment with a lower concentration due to the random movement of the molecules. This means there will be a net rotation of the channels in a given direction, depending on the net direction the molecules move in.
The ‘passive process’ and ‘no force acting to move the molecules down their concentration gradient’ I have taken to be akin to the way a gas can be used to perform work when rotating a wheel. Again a fictitious scenario and again there are two compartments, this time separated by a small channel the passage of which is obstructed by a paddle of a paddle wheel.

If the two compartments are of equal volume and have the same number of molecules and equal kinetic energy (equal pressure), then the frequency of the molecules colliding with the paddle are equal on both sides of the paddle, and there is no net force acting in either direction, and so the paddle doesn’t move and the wheel doesn’t rotate.

To achiever higher pressure in one compartment, more molecules are added but the average kinetic energy per molecules is kept constant. One compartment is now at a higher pressure and the molecules in that compartment will collide with the paddle with a higher frequency than those in the lower pressure compartment. There will be net force acting on the paddle from the higher pressure compartment, pushing in the lower pressure direction. If this force is sufficient, the paddle will move and the wheel will rotate. The net force acting to move the paddle in this instance is somewhat like the net force in an undershot waterwheel – there is a flow of gas molecules resulting in a concerted effort to turn the paddle. As the pressure of the two compartments equilibrate, then the paddle will cease to turn (it will probably stop turning prior to this as the net force acting on it diminishes).

If this explanation is accurate, then the work performed and the movement of molecules, essentially down their concentration gradient is due to the random movement of molecules. There is no force acting to pull or push molecules from a higher concentration to a lower concentration. It’s simply the random movement of the molecules and simply due to probability that the molecules will move from an area of high concentration to low concentration. I also assume in both cases that the average kinetic energy of the molecules will decrease. In the first example, as the channels rotate, they acquire some of the molecules kinetic energy, and therefore the molecules have decreasing kinetic energy as this process repeats. In the second example, I think the average kinetic energy per molecule of both compartments considered together after the paddle has moved is less than before it moved, suggesting some energy has been used to move the paddle. I assume other collisions to be elastic.

Okay, so, that is my interpretation for how a concentration gradient is a source of potential energy. The kinetic energy of molecules can be utilised to perform some work. When there is a concentration gradient, there is a net movement of molecules in the direction of lower concentration purely due to their random movements. This net movement can be used as a source of energy because the kinetic energy of the molecules can be utilised, and the concentration gradient is simply a way of ‘providing’ that energy in a directed manner. I get confused and doubt this reasoning when considering active transport. I do not understand why energy is required if the transporter is unidirectional. An individual molecule cannot look across the membrane and say “There are more on that side; I shall remain here unless energy is available”. My understanding is that energy is necessary to induce conformational changes in the carrier protein. This means that it functions more like a unidirectional gate than a channel. By my reasoning, a concentration gradient is of no importance for a unidirectional channel: if a channel could move a molecule in one direction only and acquire some of the kinetic energy to do work, then the concentration gradient is of no relevance. The carrier would move the molecules to the side of high concentration and extract some work in the process; it’s just that, unless a continual source of the molecules is available, then acquisition of work would cease. I think it is just that a unidirectional carrier can only be selective in terms of direction if it function more like a gate, with a binding site on one side and a shape change to move the molecule across, otherwise it could not distinguish between the direction from which a molecules is coming.

I shall stop here, there are probably some other important points I wanted to make, but I cannot think of them and I think this is getting a little verbose.

Any help appreciated,

And thank you if you took the time to read this,

Nobahar.

Hi Nobahar,

OK, firstly, you wondered where the kinetic energy comes from: it is the thermal energy. The higher the temperature, the more kinetic energy the molecules have and vice versa. Secondly, the ATPase in mitochondrial membranes is not a uni-directional gate. The only way you can get a uni-directional gate to work is by supplying energy to it. Otherwise, you end up with a Brownian ratchet (see http://en.wikipedia.org/wiki/Brownian_ratchet), which is physically impossible. Without doing work on a system, molecules will never move opposing a concentration gradient in a macroscopic sence (ie, one or two molecules might oppose the flow at any one time, but the net flow will never move up a gradient).

Now, the concept of entropy is one of the most powerful ideas I have ever heard because it is an abstract concept, very well understood in simple systems, which can be very useful in more complicated systems. Allow me to make a quick detour about entropy. If you have a canister of gas and open the valve, the random motion of the gas molecules will cause them to leave the canister - and the directional nature of this motion can be used to, eg, drive a paddle and produce energy. Similarly, without any physical forces acting, a lump of sugar will dissolve, or the milk in your coffee will diffuse and become evenly distributed. Now, there is no directional physical force driving the molecules from where they started to where they end up, only the random physical forces acting upon the molecules when they collide with one another - the net migration down the concentration gradient is just a consequence of their random Brownian motion. As the paddle example shows, energy can be extracted. What we can say is that the gas molecules in the canister correspond to a low entropy state (low disorder), and when they are distributed throughout the room, they are in a high entropy state (high disorder). Indeed, for ideal gasses we can calculate exactly how much entropy is in the system using the Sackur-Tetrode equation. We can say that in any spontaneous process, entropy increases - eg) you will see gas leaving a canister, but you will never see all the gas in a room spontaneously enter into a canister. Now, this is just a statement of empirical statistics. We can understand it by saying that it is highly unlikely that the positions and velocities of the atoms of gas in the room will result in the gas spontaneously accumulating in the canister, but it is incredibly likely that the molecules in a canister which has just been opened This isn't a physical force in the sense of gravity or electromagnetic, but there IS a force moving the paddle. This is the entropic force and it can be calucalted in a number of systems. And, as you alluded to, when the paddle turns, it removes some of the kinetic energy of the gas molecules, reducing their temperature.

Now, imagine these molecules in solution, on both sides of a membrane. Now we have to contend with interactions between the solvent and the solute. However, don't fret, because the solvent-solute interactions is the same on both sides of the membrane. In fact, we can just about ignore it. Now you will see that there is a difference in concentration on either side of the membrane causes an entropic force and, as the concentrations moves towards equillibrium, energy can be extracted to do work.

Finally, ATPases can and do work in reverse. Ions across a membrane can be used to synthesise ATP, but ATP can also be used to push ions against the gradient. It all depends on the magnitude of the gradient and the concentration of ATP!

Anyway, that was a little rushed because I have to go.

Feel free to ask any other questions.
I