The purpose of an action potential is to transmit signals along a cell membrane by influencing voltage-gated proteins. These changes in protein function are fundamental to life processes, enabling thinking, movement, and survival. To grasp the phenomenon of action potential propagation, it's essential to understand three key concepts: membrane conductance, capacitance, and resistance.
Conductance measures the movement of ions across cell membranes via channel proteins. Ions have varying permeabilities, largely determined by the number of available channel proteins and the driving force for each ion. The driving force, primarily voltage-based, plays a significant role. For instance, at RMP, if a channel opens that allows both Na+ and K+ ions to flow equally, Na+ will have a greater driving force and flow in more than K+. K+ flows less at RMP because it has a smaller driving force. We would say that at RMP, the conductance for Na+ is larger than the conductance for K+. However, if the membrane potential were to be a positive value (above 0), and we opened a channel permeable to both Na+ and K+, we would see the conductance for Na+ be much smaller than the conductance for K+.
Capacitance is a ratio of the amount of charge separated (positive from negative) to the amount of voltage generated when charges are separated (called electric potential). This ratio is basically describing how much charge we have seperated per Volt.
Lets attempt more description of the unit volt. The volt comes into play because charges are difficult to separate. Placing “like” charges together takes force or energy, since they repel each other (+/+ or -/-). The work needed to separate charges is generated by electrochemical gradients. For the chemical gradient, just think particles in areas of high concentration moving towards areas of low concentration. For the electrical gradient, think of two forces in play; like charges pushing each other away (push forces) and unlike charges trying to pull (pull forces) the opposite charge closer. The combination of the chemical and electrical gradients creates a very tense situation between the two layers of the membrane. Although tense, the composition of the membrane and the push/pull across it can reach a kind of stalemate (equilibrium or RMP) creating a stable enough situation that we can measure a consistent potential, and we express that potential in volts (technically millivolts). That “stable” situation is highly dependent upon conductance across the membrane. If conductance changes (opening a channel) the “tenseness” will unload and charges will move immediately (because of the voltage that exists across the memebrane). Thus, the resting membrane potential value in millivolts (ie., -70mV) represents the potential energy available to return the separated charges to each other if conductance (open channels) allows. The more volts (-70mV, -80mV, -90mV), the more potential energy available.
Now lets attempt some understanding of capacitance. Keeping with our analogy, the push/pull of charge is the potential energy of the millivolt, but how much each push (+/+ or -/-) or pull (+/-) contributes to voltage is defined by capacitance. For example, consider the following figure that shows a high capacitance situation vs a low capacitance situation.
Notice in the high capacitance region there are more units of charge separation compared to the low capacitance situation. Each charge in the high capacitance membrane is experiencing different push/pull forces compared to the low capacitance membrane. However, both membranes are still experiencing a -70mV potential. How can the lower capacitance membrane, with less charges, still be sitting at -70mV?
The membrane has polar regions that attract charges (phosphate heads) and non-polar regions that are insulated from charges (phospholipid tails). As stated, “like” charges are difficult to arrange together because of repulsion (push) forces. To accumulate more charges, like in a high capacitance situation, increasing the pull forces can offset the repulsive push forces. The increase in pull forces can be achieved by adding polar molecules to the non-polar region of the membrane. These polar molecules will contribute an attractive force towards the center (pull forces). This newly added attractive force can “absorb” some of the charge potential, making their push force weaker. The weaker push force means less charge repulsion, so more charges are separated before we achieve -70mV of RMP potential energy. Thus, even though the high capacitance membrane has more charges separated, much of each charge’s potential is being used up by the increased pull forces (represented by solid circles) making less potential per charge available. Whereas the charges in the low membrane capacitance have more potential per charge because there are less pull forces (dotted circles), thus less charges are needed to maintain -70mV because more of their charge potential is available.
Mathematically then, capacitance can be represented by two equations. The first equation represents the ratio of charge on the surface of the membrane per volt of the membrane potential. It is expressed as:
Equation 1: C = Q/V
Where Q represents the amount of charge separated, and V represents the voltage difference between the membrane surfaces. In the figure, both membranes have the same V, but Q is different, making the membrane with more Q (charge separation) have a higher capacitance.
The second equation explains how the addition of more polar molecules can increase the pull force, allowing for more charges to be accumulated along the membrane, thereby increasing capacitance.
Equation 2: C = Є x (A/d)
The Greek symbol epsilon (Є) represents what is called the permittivity of the capacitor. A technical definition of permittivity can be complex so we will take some liberties in the definition and try to ignore our cringing physicist friends. Thus, for our purposes, it is best to think of permittivity as the number of polar molecules found within the center portion of the membrane. The more polar a membrane, the higher the capacitance, and the more charge is separated per volt. The variable (d) represents the distance between charge separations (the distance between the external and internal surface of the membrane). In most cell membranes, this value never changes, however, in the case of an insulated axon (myelination), the distance of charge separation (d) can increase substantially. Also, myelin layers have very little protein, so the overall polarity of the “center” of our sandwich decreases (Є decreases). The variable (A) is a unit of area available for charge separation. If (A) stays the same for a unit of membrane measured, capacitance still decreases with myelin because of changes in the Є and d.
Resistance of the membrane or (Rm) is the unit of measurement used to describe the ability of something to oppose electrical current across a cell membrane. You can think of resistance as opposing conductance. When ion channels open and ions move down their gradients into or out of the cell, we would say that conductance has increased, but, only because membrane resistance (Rm) has decreased.
Lets imagine that a bunch of sodium rushed through an open sodium channel to the inside of the cell. Once inside the cell we can imagine that this new accumulation of positive charge could start to influence negative charges down the length of the axon (assuming the cell is a neuron). As negative charges found at the inner surface of the membrane experience this "influence", we might imagine that this could affect voltage gated ion channels in the membrane but further away from that intial positive charge. How far the influence of the positive charge can reach “down-stream” might be thought of as a "sphere of influence". If we were actual physicists we would probably start pontificating about electrical fields at this point, but since we lack pocket protectors for our pens, we will instead try to stay simpler and simply try to visualize what we mean by the term "sphere of influence”. A sphere of influence is simply the 3 dimensional space that we imagine the positive charge reaching after it enters the cytoplasm through a channel. This distance can be impacted by negatively charged proteins and other molecules in the cytoplasm. Anything that absorbs charge or impedes the "influence" downstream is called internal resistance (Ri). The "sphere of influence" must be strong enough and the Ri weak enough that voltage gated channels can be triggered down stream from the orginal source of entering sodium that we have been discussing. For a membrane to depolarize and trigger an action potential that propagates, the values of Rm and Ri are important.
When channels are opened and ions start flowing, based on their driving force across the membrane, a current is generated and can be referred to as the capacitative current. Capacitors can only gain or lose charge because this movement of charge is what causes current. At the resting membrane potential, the cell membrane capacitor is maximally charged, and when additional ion channels are opened (i.e. Na+ voltage-gated channels), the capacitor discharges, and the current moves toward zero (less charge separation). This loss of voltage is exponential, decreasing more rapidly with each passing millisecond. To describe this exponential loss in current across time, physicists use a unit measurement called the time constant. The time constant is a measure of how long it takes to depolarize a section of membrane. The time constant value is dependent upon both resistance of the membrane (Rm) and the membrane capacitance (Cm):
Equation 3: Time Constant = Rm x Cm
In other words, the time constant is directly proportional to the Rm and to the Cm. If capacitance of the membrane increases, the time constant will also increase. Stated another way, because there are more charges separated, it will take longer to depolarize the membrane. As the time constant increases, the cell depolarizes slower and the longer it takes to propagate an action potential. If Rm increases, charges will be impeded in t heir movement across the membrane. This increased resistance to current flow will cause a larger time constant. In a myelinated neuron, Rm increases and Cm decreases so the overall affect on a myelinated neuron for time constant (compared to a non myelinated neuron) is likely to be minimal.
Once an action potential is generated, the next obstacle is propagating it down the membrane. Since action potentials are the result of cations crossing the membrane and influencing protein conformations, how the sphere of influence is distributed once it crosses the membrane will determine the distance the action potential will propagate. There are three possible ways that a cationic charge (sphere of influence) can be affected once it crosses the membrane:
1. The sphere of influence could be neutralized (causing it to have less effect) by anionic charges on the inner surface of the membrane, as well as the anionic charges on polar molecules found in the cytoplasm; this is referred to as internal resistance (Ri).
2. The cations may flow back through the membrane, essentially leaking back out and taking their charge with them, thereby diminishing the positive charge; this is referred to as membrane resistance (Rm).
3. Whatever is left of a positive charge sphere of influence after losing some to Ri and Rm is able to influence down-stream proteins (triggering their opening) and to add to the propagation of the depolarizing effect.
As mentioned above, membrane voltage changes can occur as a function of time (the time constant). However, voltage changes can also occur as a function of space. To describe the effect of the charge and its distribution in space we use the unit of measurement called the length constant. For example, a graded membrane potential (what we have been calling a positive charge “sphere of influence”) will decay as it travels away from its site of origin. Importantly, the distance that this influence can go and maintain strength depends on the ratio of the membrane resistance to the internal resistance according to the following equation:
Equation 4: Length Constant = (Rm/Ri)1/2
When the ratio of Rm to Ri is high the effect of the charge is large and travels for a longer distance which is equivalent to a larger length constant. The time constant and length constant are most evident when talking about the axon of neurons, especially in reference to the effect of myelin.
Myelin decreases the effect of polarity (decreased Є) and increases d (equation 2), thereby decreasing capacitance. Because capacitance is inversely correlated with the distance between the charged membranes, myelin reduces the amount of stored charge thereby reducing the time constant (equation 3) at each node.
Less capacitance also means less ATP is needed to maintain an RMP which makes myelinated neurons more efficient. Finally, we only find myelin on large diameter neurons. Larger diameter actually decreases Ri because there is a decreased density of current impeding molecules downstream from entering positive charges.
Thus, myelin allows the membrane to depolarize quicker and to spread out further. It is important to recognize that myelin does not cover the whole axon; instead, it covers it in sections, leaving gaps of unmyelinated membranes called nodes of Ranvier. The nodes act to regenerate the action potential and myelinated sections act to increase the effect of the sphere of influence. This "jumping" of action potential depolarization events from node to node is called saltatory conduction. The end result is increased speed of action potential propagation dowon a myelinated neuron.
In summary, understanding membrane conductance, capacitance, and resistance is crucial for comprehending how action potentials propagate along cell membranes. These concepts also help us see the advantage the myelin gives action potential propogation. This all contributes to the healthy functioning of a nervous system and various physiological processes that we will discuss.
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