# 1.4 The life of a molecule in a charged environment

## What is voltage?

Potential energy is a quantitative measurement of a system’s potential to do work. For example, if you hold a spring in a stretched position and keep it that way, one could say that the spring has high potential energy. It has the potential to violently compress back into its equilibrium length the second you let it go. When that occurs, one could say that the spring has realized its potential. Similarly, if you hold a ball high up in the air, it has high potential energy. It has the potential to fall and release its potential onto the world in the form of kinetic energy. The potential energy is realized when it is let go of.

Voltage is considered a form of potential energy. It refers to the ability of a system to create electrochemical work, even though it is currently not doing any at that moment. As you have likely already encountered, like charges repel while opposite charges attract. When a negative charge accelerates towards a positive one, it is doing electrochemical work. However, if one was to place a non-permeable barrier between a negative particle and a positive one, then they would be unable to move towards each other and produce this work. They would still have the potential to accelerate towards each other, but this potential would not be realized unless barrier was lifted, or otherwise made permeable to ionically charged particles.

Electrochemical potential is termed voltage. Neurons always have a voltage across a membrane, as the inside of a neuron is more negatively charged than the outside. These negative charges would like to accelerate outside of the neuron, but they cannot because the lipid membrane is not permeable to such charges (unless it contains ion channels which allow them to pass). Theoretically, if there was no membrane, then these ions would exchange until they reach their equilibrium potential.

Voltage is measured in Volts; which is a fancy way of saying charge per meter. It denotes how much charge difference there is across a non-permeable membrane, and how far away these charges are from each other. The reference frame for the unit of Volts is completely arbitrary; one can decide to set the zero point anywhere they choose. Classically, the outside of the neuron is set as the zero point, making the inside of the neuron -70mV. However, one can easily set the zero point as the inside of the neuron, making the outside +70mV. Alternatively, the zero point can be a hallway through the membrane, making the outside +35mV and the inside -35mV. The only thing that matters is the difference between two points, which in this case, is always 70mV (between the outside and inside). Where exactly you start counting from does not matter, so long as you are communicating with people who are using the same reference point as you are.

## What is current?

Current is simply the movement of charge over time. It is measured by Amperes, or how fast every charge moves per second. It is very analogous to velocity in that way. Voltage refers to the potential for creating current. When the barrier which creates voltage, i.e the lipid membrane, becomes permeable to certain ions, then these ions will suddenly become capable of moving towards opposite charges and away from like charges. That sudden movement is measured as a current. Thinking back to the example of a spring; if voltage is analogous to holding a spring in a stretched position, then current is analogous to the velocity at which the spring compresses back into its original size as soon as you let it go.

### Molecules in Charged Environments

Unlike chemical gradients, which are a derivative of the law of entropy, the electrostatic force is itself a fundamental law of nature. In fact, it is one of the three fundamental forces that explain all understood attributes of the physical universe (the other two being quantum forces). It causes like charges to repel and opposite charges to attract. Unlike uncharged environments, ions in charged environments do not move randomly, and would therefore not have an equal probability of moving in each possible direction. In such a system, there would be two factors to consider when predicting the net movement of molecules: chemical gradients and the electrostatic force.

Moving forward, it will be useful to think of chemical gradients as a “force”, moving ions from high to low concentrations. While previous discussion on entropy is more accurate, thinking of chemical gradients as a force can help make this next section more intuitively understood.

If two people push a block in the same direction, it will move down that direction with more acceleration than if only one person was pushing it. However, if two people are pushing a block from opposite directions, the block may end up with 0 net movement. If one of them is stronger than the other, then maybe the block will slowly crawl towards one direction. One can extend this idea to ions separated by a membrane, treating the chemical and electrical forces as “different people”.

In a membrane based environment with only one ion, and no other external charges, then ions will still move down their concentration gradient until equal concentrations are achieved on both sides in a dynamic equilibrium. Remember, dynamic equilibrium means ions are still being exchanged, just at a net of zero.

Now imagine that there are many negative ions on one side of the membrane, and these cannot permeate through the membrane. On the same side is a large concentration of positive ions, which are permeable to the membrane. The chemical gradient will “push” the positive ions towards the other side, but the electrostatic attraction to negative ions will “pull” them back. As a result, the equilibrium state of this system is not characterized by equal concentrations on both sides. Instead, there will always remain more positive ions in the side with the negative ions.

What if we continue to extend this idea? It is possible to completely nullify the “chemical force” if you create an equal but opposite electrostatic force. If one achieves this, then ion concentrations can remain stably asymmetric, such that at equilibrium, one side of the membrane has much higher concentration of a particular ion than the other side.

Not only is this possible, but chemists have derived an equation that calculates the exact voltage necessary to counteract any possible concentration gradient. We call this the “reversal potential”. Remember, voltage/potential is just a fancy way to say that one side of the membrane is more charged than the other.

The equation used to calculate reversal potential is termed the Nernst equation, however, this equation is only applicable to single ion systems. For example, let’s say there are roughly 20 potassium ions inside the cell for every one potassium ion outside the cell. The chemical “force” would like to push most of these potassium ions outside, until there are roughly 10.5 ions on each side. However, using the Nernst equation, a chemist has calculated that this system has a reversal potential of -90mV. If this chemist makes sure that there is 90mV more negative charge inside the cell than, then he/she can guarantee that there will be no net movement of potassium ions, and the 20:1 imbalance will remain stable. Chemists working on similar systems use batteries to apply a voltage differential across membranes, but in biology, it is usually impermeable molecules that set a charge differential. Since impermeable molecules cannot pass through the membrane, they only create an electrostatic force without being ever influenced by chemical gradients.

$E = \frac{RT}{zF} \; ln \left(\frac{[X]_{out}}{[X]_{in}} \right)$

Above is the Nernst equation. Both R and F are the universal gas constant and Faraday’s constant, respectively. T represents temperature, z represents the charge of the ion that makes up the system (i.e +1 for potassium), and X represents concentration of an ion, inside or outside the membrane.

There are three intuitive patterns that can be extracted from this equation. The first and most obvious is that the larger the concentration gradient (i.e the greater the difference between Xout and Xin), the larger the reversal potential necessary to counteract it.

Second: since z is a denominator, the more charged an ion, the smaller the reversal potential. This is because highly charged molecules are relatively more sensitive to the electrostatic force, and can thus counteract the chemical “force” perfectly with small voltage magnitudes.

Third and maybe least intuitive: high temperatures increase the voltage necessary to counteract the chemical force. For this to be understood, it is best to analyze chemical gradients through the lens of entropy again. Molecules move down their gradients because of their intrinsically chaotic movement patterns. Temperature makes molecules move even faster and more chaotically. You can think of this as “increasing” the strength of the chemical “force”, thus resulting in a need for higher reversal potentials to counteract it. Don’t worry if this last point is confusing; physiological temperature remains within a narrow enough range that, at least in neuroscience, you will never really have to worry about the effects of temperature on reversal potentials and chemical gradients.

### Reversal Potentials in the Wild

While it would be an interesting scientific finding, there do not exist any chemists in your head that change the voltages of neurons to achieve reversal potentials. In fact, there are many cases in which the membrane voltage is not equal to the reversal potential. This would mean that the ion permeable to the membrane is at disequilibrium. Ions will thus move across the membrane until equilibrium is achieved.

Driving force can be thought of as an invisible “force” that exists in an electrochemical system at disequilibrium. It “pushes” ions towards the direction that will eventually result in equilibrium. Driving force is calculated as Membrane Voltage minus Reversal Voltage. It is obvious from this equation that when membrane voltage equals the reversal potential, the system is at equilibrium, and driving force is 0. The larger the magnitude of the driving force, the farther the system is from equilibrium. This creates a greater “push” for ions to move towards equilibrium. The valence (i.e positive or negative) of driving force can be confusing, as it is influenced by the charge of the ions in the systems, and the direction they are moving. We will ignore valence for now and think only of the absolute value of driving force.

It is important to understand that these electrochemical systems are very dynamic. If a system is at disequilibrium, that means the membrane voltage is not equal to reversal potential. When ions consequentially move towards equilibrium, concentration gradients change. As a result, the voltage at equilibrium will not be equal to the reversal potential that was calculated when the system was in disequilibrium. In other textbooks, you might see the term reversal potential referred to as equilibrium potential. Note that this can be very confusing, and we purposefully avoided using the latter term in this textbook. Reversal potential is a much more accurate way to think about this.

To iron out this idea, let us take the previously elaborated potassium example. Under physiological conditions, where there are 20 ions inside the membrane for every 1 ion outside, potassium has a reversal potential of -90mV (calculated using Nernst). What if the membrane potential is -70mV, and no other ions are permeable to the membrane? Then there is not enough negative charge inside the membrane relative to outside to keep 20:1 ratio stalely. The lack of negative charge inside means that there is a driving force of 20mV, pushing potassium ions outside of the cell, decreasing the severity of the concentration gradient.

The very nature of pushing ions out results in a decrease in concentration gradient. This changes the calculated reversal potential (Nernst equation), making it smaller in magnitude (somewhere greater than -90mV but less than -70). At the same time, the movement of positive ions outside the cell results in less positive charge inside the cell. This changes the existing membrane potential, making it something less than -70mV but greater than -90mV. Eventually, equilibrium is ached when reversal potential equals membrane potential, and driving force is zero.

To summarize, the incorrect way of thinking about this is that when a system is in disequilibrium, the ions move and change membrane potential until it becomes equal to the reversal potential that was calculated at disequilibrium. This is a very common misconception because people use the work “equilibrium potential” instead of “reversal potential” commonly. The correct way of thinking about this is that when a system is in disequilibrium, ions move such that move both membrane potential and reversal potential dynamically change in value until they are equal. It is not only membrane potential’s value that changes when ions are moving, but both.