Defining the Nernst Potential
The Nernst potential (or equilibrium potential, symbolized as Eion) for a specific ion is the theoretical membrane potential at which the net movement of that particular ion across the cell membrane is zero. This occurs despite the presence of a concentration gradient for the ion across the membrane.
Think of it as a point of balance. Ions tend to move from an area of high concentration to an area of low concentration (down their chemical gradient). However, as charged ions move across the membrane, they create an electrical potential difference (voltage). This electrical potential difference then exerts a force on the charged ions (the electrical gradient).
The Nernst potential is the specific membrane voltage at which these two forces – the chemical driving force and the electrical driving force – are exactly equal in magnitude and opposite in direction for a single, specific ion. At this potential, there is no net flow of the ion across the membrane, even if channels permeable to that ion are open. It’s a theoretical state of equilibrium for that ion alone.
It is important to distinguish the Nernst potential for a single ion from the resting membrane potential of a cell. The resting membrane potential is the actual electrical potential difference across the membrane of a cell at rest and is determined by the permeability of the membrane to multiple ions (primarily K+, Na+, and Cl-) and their respective equilibrium potentials. The Nernst potential for each ion represents the driving force and equilibrium point for that ion in isolation.
Physiological Basis of the Nernst Potential
The existence and relevance of the Nernst potential in biological systems are predicated on several key physiological properties of cell membranes:
- Ion Concentration Gradients: There are significant differences in ion concentrations between the intracellular fluid (cytoplasm) and the extracellular fluid. These gradients are actively maintained by energy-consuming pumps, most notably the Na+/K+-ATPase pump, which pumps 3 Na+ ions out for every 2 K+ ions pumped in, and also by other transporters and channels. For example, intracellular K+ concentration is much higher than extracellular K+, while intracellular Na+ and Ca++ concentrations are much lower than extracellular concentrations.
- Selective Membrane Permeability: The cell membrane itself is largely impermeable to ions due to its lipid bilayer structure. However, embedded within the membrane are various types of ion channels, which are proteins that form selective pores allowing specific ions (like K+, Na+, Ca++, Cl-) to pass through the membrane. The number and type of open channels determine the membrane’s permeability to different ions at any given time.
- Electrochemical Driving Force: The combination of the concentration gradient (chemical force) and the potential difference across the membrane (electrical force) creates an electrochemical driving force for each ion.
- The chemical force pushes ions down their concentration gradient.
- The electrical force pushes positive ions towards negative areas and negative ions towards positive areas.
The Nernst potential arises because of the interplay of these forces. Consider potassium ions (K+). Intracellular K+ is much higher than extracellular K+. If K+ channels are open, K+ will move out of the cell down its chemical gradient. As positive charge (K+) leaves the intracellular space and accumulates slightly on the outer surface of the membrane (leaving behind negative counter-ions like proteins inside), an electrical potential difference develops across the membrane, with the inside becoming negative relative to the outside. This developing negative potential inside the cell attracts the positive K+ ions and opposes their outward movement. As more K+ leaves, the inside negativity increases, and the electrical force pulling K+ back in strengthens. Eventually, a specific negative membrane potential is reached where the electrical force pulling K+ inward precisely balances the chemical force pushing K+ outward. At this potential, the net movement of K+ is zero – this is the Nernst potential for K+.
A similar process occurs for other ions, but the direction of movement and the resulting Nernst potential differ based on the ion’s charge and concentration gradient. For Na+, which is high outside and low inside, opening channels allows Na+ to move inward down its chemical gradient. This influx of positive charge makes the inside of the membrane positive relative to the outside, creating an electrical force that opposes further Na+ entry. The Nernst potential for Na+ is reached when this positive internal potential balances the inward chemical force, resulting in a positive Nernst potential.
Thus, the physiological basis is the existence of stable ion gradients maintained by pumps, combined with the presence of selective ion channels that allow controlled movement of ions, leading to the establishment of electrochemical equilibrium for individual ion species at specific membrane potentials (the Nernst potentials).
The Nernst Equation
The Nernst equation is the mathematical formula used to calculate the theoretical Nernst potential (equilibrium potential) for a specific ion at a given temperature, based on its concentrations on either side of the membrane.
The general form of the Nernst equation is:
Eion = (RT / zF) * ln([Ion]out / [Ion]in)
Where:
- Eion: The Nernst potential (equilibrium potential) for the specific ion (in volts).
- R: The ideal gas constant (8.314 J/(mol·K)).
- T: The absolute temperature (in Kelvin; typically assumed to be 310 K for mammalian body temperature, 37°C).
- z: The valence (charge) of the ion (e.g., +1 for Na+ and K+, +2 for Ca++, -1 for Cl-).
- F: Faraday’s constant (96,485 C/mol), the amount of electrical charge carried by one mole of univalent ions.
- ln: The natural logarithm.
- [Ion]out: The extracellular concentration of the ion (in mM or moles/L).
- [Ion]in: The intracellular concentration of the ion (in mM or moles/L).
In many physiological calculations, the equation is simplified by plugging in the constants R, T (at 310K), and F, and converting the natural logarithm (ln) to the base-10 logarithm (log10) using the conversion factor ln(x) = 2.303 * log10(x).
At a typical mammalian body temperature of 37°C (310 K), the simplified form often used is:
Eion ≈ (61.5 / z) * log10([Ion]out / [Ion]in)
Where Eion is now in millivolts (mV). The value 61.5 mV is approximately (2.303 * R * T / F) at 310K. For simplicity, some textbooks use 60 mV or 61 mV.
This simplified version highlights that the Nernst potential is directly proportional to the logarithm of the ratio of extracellular to intracellular concentration and inversely proportional to the valence of the ion.
Calculating Nernst Potentials for Key Ions (Na+ and K+)
Let’s calculate the approximate Nernst potentials for Sodium (Na+) and Potassium (K+) using typical mammalian physiological concentrations and the simplified Nernst equation (with 61.5 mV at 37°C).
Typical physiological concentrations are approximately:
- Sodium (Na+):
- [Na+]out ≈ 145 mM
- [Na+]in ≈ 15 mM
- Valence (z) = +1
- Potassium (K+):
- [K+]out ≈ 4 mM
- [K+]in ≈ 150 mM
- Valence (z) = +1
Let’s calculate:
Calculation for Potassium (EK):
EK = (61.5 / z) * log10([K+]out / [K+]in) EK = (61.5 / +1) * log10(4 mM / 150 mM) EK = 61.5 * log10(0.0267)
Now, calculate the log10(0.0267): log10(0.0267) ≈ -1.57
Substitute this back into the equation: EK ≈ 61.5 * (-1.57) EK ≈ -96.5 mV
So, the approximate Nernst potential for Potassium (EK) is around -97 mV. This negative value means that the electrical potential difference required to balance the outward chemical gradient for K+ (which pushes K+ out) is one where the inside of the cell is negative relative to the outside, pulling K+ back in.
Calculation for Sodium (ENa):
ENa = (61.5 / z) * log10([Na+]out / [Na+]in) ENa = (61.5 / +1) * log10(145 mM / 15 mM) ENa = 61.5 * log10(9.67)
Now, calculate the log10(9.67): log10(9.67) ≈ 0.985
Substitute this back into the equation: ENa ≈ 61.5 * 0.985 ENa ≈ +60.6 mV
So, the approximate Nernst potential for Sodium (ENa) is around +61 mV. This positive value means that the electrical potential difference required to balance the inward chemical gradient for Na+ (which pushes Na+ in) is one where the inside of the cell is positive relative to the outside, pushing positive Na+ ions back out.
These calculations show that EK is strongly negative, while ENa is strongly positive. This difference reflects the opposite concentration gradients and the forces required to equilibrate them. The actual resting membrane potential of neurons and muscle cells (typically around -70 mV) is much closer to EK than ENa, reflecting the higher permeability of the resting membrane to K+ compared to Na+.
Effects of Altering Ion Concentrations on the Equilibrium Potential for that Ion
The Nernst equation, Eion = (RT / zF) * ln([Ion]out / [Ion]in), clearly shows that the equilibrium potential for an ion is directly dependent on the ratio of its extracellular to intracellular concentration ([Ion]out / [Ion]in). Altering either the extracellular or intracellular concentration will change this ratio and, consequently, change the Nernst potential for that ion.
Let’s examine the effects of altering the extracellular concentration of K+, Na+, and Ca++:
- Altering Potassium (K+) Concentration:
- Increasing Extracellular K+ ([K+]out): If [K+]out increases (e.g., from 4 mM to 8 mM, a condition called hyperkalemia), the ratio [K+]out/[K+]in increases (e.g., 8/150 = 0.0533 vs 4/150 = 0.0267). In the simplified Nernst equation, log10(0.0533) ≈ -1.27 compared to log10(0.0267) ≈ -1.57. The resulting Nernst potential (EK) becomes less negative (e.g., 61.5 * -1.27 ≈ -78 mV vs 61.5 * -1.57 ≈ -97 mV). A less negative EK means the membrane potential required to stop K+ efflux is less negative. Since the resting membrane potential is heavily influenced by EK, an increase in [K+]out causes the resting membrane potential to depolarize (become less negative, closer to zero). This can make excitable cells more easily activated initially, but with substantial increases can lead to persistent depolarization, inactivation of voltage-gated Na+ channels, and loss of excitability (e.g., cardiac arrhythmias).
- Decreasing Extracellular K+ ([K+]out): If [K+]out decreases (e.g., from 4 mM to 2 mM, hypokalemia), the ratio [K+]out/[K+]in decreases (e.g., 2/150 = 0.0133 vs 4/150 = 0.0267). log10(0.0133) ≈ -1.88 compared to -1.57. The resulting Nernst potential (EK) becomes more negative (e.g., 61.5 * -1.88 ≈ -115 mV vs -97 mV). A more negative EK means the membrane potential required to stop K+ efflux is more negative. This hyperpolarizes the resting membrane potential (makes it more negative), making excitable cells less responsive to stimuli.
- Altering Sodium (Na+) Concentration:
- Increasing Extracellular Na+ ([Na+]out): If [Na+]out increases (hypernatremia), the ratio [Na+]out/[Na+]in increases. For example, if [Na+]out goes from 145 mM to 155 mM (with [Na+]in at 15 mM), the ratio changes from 9.67 to 10.33. log10(10.33) ≈ 1.01 vs log10(9.67) ≈ 0.985. The Nernst potential for Na+ (ENa) becomes slightly more positive (e.g., 61.5 * 1.01 ≈ +62 mV vs 61.5 * 0.985 ≈ +61 mV). While ENa becomes more positive, changes in [Na+]out have a relatively smaller effect on the resting membrane potential compared to changes in [K+]out because resting membrane permeability to Na+ is low. However, ENa determines the peak of the action potential overshoot in many excitable cells. A higher ENa can result in a slightly larger action potential amplitude.
- Decreasing Extracellular Na+ ([Na+]out): If [Na+]out decreases (hyponatremia), the ratio [Na+]out/[Na+]in decreases. For example, if [Na+]out goes from 145 mM to 135 mM, the ratio changes from 9.67 to 9. log10(9) ≈ 0.95 vs log10(9.67) ≈ 0.985. ENa becomes less positive (e.g., 61.5 * 0.95 ≈ +58.4 mV vs +61 mV). This directly affects the peak of the action potential, reducing its amplitude, which can impair signal transmission.
- Altering Calcium (Ca++) Concentration:
- Increasing Extracellular Ca++ ([Ca++]out): If [Ca++]out increases (hypercalcemia), the ratio [Ca++]out/[Ca++]in increases. Since Ca++ has a valence (z) of +2, the Nernst equation is ECa = (61.5 / 2) * log10([Ca++]out / [Ca++]in). A higher [Ca++]out ratio makes the log10 term more positive, increasing ECa (making it more positive). While ECa changes, extracellular Ca++ has a more significant physiological impact on membrane excitability by binding to proteins on the cell surface (including voltage-gated ion channels) and altering their gating properties. High extracellular Ca++ tends to stabilize the membrane, making it less excitable by shifting the voltage dependence of Na+ channels (e.g., making it harder for them to open).
- Decreasing Extracellular Ca++ ([Ca++]out): If [Ca++]out decreases (hypocalcemia), the ratio [Ca++]out/[Ca++]in decreases. This makes the log10 term less positive, decreasing ECa (making it less positive). More importantly, low extracellular Ca++ reduces the binding of Ca++ to membrane proteins, making voltage-gated Na+ channels more sensitive and prone to opening at more negative membrane potentials. This increases membrane excitability, leading to symptoms like muscle twitches (tetany).
In summary, the Nernst potential for any given ion is a direct function of the concentration gradient across the membrane for that ion. Changes in the extracellular concentration of ions, particularly K+, significantly alter the ratio used in the Nernst equation, thereby changing the calculated equilibrium potential and impacting the electrical behavior of the cell membrane, especially in excitable tissues.
Conclusion
The Nernst potential is a fundamental concept in cellular electrophysiology. It represents the theoretical membrane voltage at which the chemical driving force for a single ion across the membrane is perfectly balanced by the electrical driving force, resulting in zero net movement of that ion. Its physiological basis lies in the selective permeability of the cell membrane to ions and the active maintenance of ion concentration gradients. The Nernst equation provides the mathematical framework for calculating this potential based on ion concentrations and temperature. Calculations for key ions like Na+ and K+ reveal significantly different equilibrium potentials, reflecting their distinct gradients. Furthermore, alterations in extracellular ion concentrations directly impact their respective Nernst potentials, leading to profound consequences for cellular excitability and function. While the Nernst potential applies to individual ions, it serves as a critical building block for understanding the complex interplay of ion movements that determine the actual membrane potential of living cells.
