Understanding Cellular Electrophysiology: Ion Distribution, The Goldman Equation, and Resting Membrane Potential
Cellular life depends critically on maintaining distinct internal environments separate from the external surroundings. A fundamental aspect of this is the unequal distribution of ions across the cell membrane, which generates electrical potentials essential for diverse cellular functions, from nerve impulse transmission to muscle contraction and nutrient transport. This guide outlines the typical ion gradients across the cell membrane, introduces the Goldman-Hodgkin-Katz (GHK) equation, and explains how this equation helps clarify the generation of the resting membrane potential (RMP).
Normal Distribution of Key Ions Across the Cell Membrane
The cell membrane acts as a selective barrier, actively and passively controlling the passage of ions. This selective permeability, combined with active transport mechanisms, results in significant concentration differences for key ions between the intracellular fluid (cytosol) and the extracellular fluid. The major ions involved in cellular electrophysiology are Sodium (Na+), Potassium (K+), Calcium (Ca2+), and Chloride (Cl-). While exact concentrations vary slightly depending on cell type and species, a typical mammalian cell exhibits the following general distribution:
- Sodium Ions (Na+):
- Extracellular Concentration: High (typically around 140-150 mM)
- Intracellular Concentration: Low (typically around 5-15 mM)
- Concentration Gradient: Steep, with Na+ concentration significantly higher outside the cell than inside. This gradient represents substantial potential energy.
- Maintenance: Primarily maintained by the Sodium-Potassium Pump (Na+/K+-ATPase), an active transporter that pumps 3 Na+ ions out of the cell for every 2 K+ ions pumped in, consuming ATP. This pump works against the concentration gradient for both ions and is crucial for establishing and maintaining the Na+ gradient.
- Potassium Ions (K+):
- Extracellular Concentration: Low (typically around 4-5 mM)
- Intracellular Concentration: High (typically around 140-150 mM)
- Concentration Gradient: Steep, with K+ concentration significantly higher inside the cell than outside.
- Maintenance: Primarily maintained by the Sodium-Potassium Pump (Na+/K+-ATPase), which actively transports K+ into the cell. The membrane also has ‘leak’ channels that are relatively permeable to K+, meaning there is a passive outward movement of K+ down its concentration gradient, which the pump counteracts.
- Calcium Ions (Ca2+):
- Extracellular Concentration: High (typically around 1-2 mM)
- Intracellular Concentration: Extremely Low (typically around 50-100 nM – note the nanomolar scale, a difference of over 10,000 times compared to outside).
- Concentration Gradient: Extremely Steep, with Ca2+ concentration orders of magnitude higher outside the cell than in the cytoplasm.
- Maintenance: Achieved through a variety of mechanisms: Plasma Membrane Ca2+-ATPases (pumps Ca2+ out), Na+/Ca2+ exchangers (use the Na+ gradient to extrude Ca2+), and uptake into intracellular stores like the endoplasmic/sarcoplasmic reticulum via SERCA pumps. This low intracellular Ca2+ concentration is critical for Ca2+ acting as a signaling molecule.
- Chloride Ions (Cl-):
- Extracellular Concentration: High (typically around 100-120 mM)
- Intracellular Concentration: Low (typically around 5-15 mM)
- Concentration Gradient: High, with Cl- concentration significantly higher outside than inside.
- Maintenance: The Cl- distribution is often more complex and can vary significantly between cell types. Passive distribution is influenced by the membrane potential (negative inside repels Cl-). However, active transport mechanisms like Na+-K+-2Cl- cotransporters (NKCC) bring Cl- in, and K+-Cl- cotransporters (KCC) extrude Cl-. The net effect often contributes to maintaining the outward chloride gradient, although its exact intracellular concentration is significantly influenced by the cell’s resting potential.
These unequal ion distributions represent stored electrochemical potential energy, ready to be harnessed for cellular work when ion channels in the membrane open, allowing ions to flow down their electrochemical gradients.
Introducing Membrane Potential and Equilibrium Potential
Because ions carry electrical charge, their unequal distribution across the membrane creates an electrical potential difference. This potential difference is known as the membrane potential (Vm), defined as the electrical potential inside the cell relative to the outside (which is arbitrarily set to zero). In a resting cell, the membrane potential is typically negative, around -70 to -90 mV in many excitable cells.
For a single ion species, if the membrane were only permeable to that ion, the ion would flow down its concentration gradient until the electrical force opposing its movement exactly balanced the chemical force driving it. The membrane potential at which this electrochemical equilibrium is reached for a single ion is called the equilibrium potential (Eion) for that ion. It can be calculated using the Nernst equation:
Eion = (RT / zF) * ln ([Ion]out / [Ion]in)
Where:
- R is the ideal gas constant
- T is the absolute temperature (in Kelvin)
- z is the valence (charge) of the ion (+1 for Na+, K+; +2 for Ca2+; -1 for Cl-)
- F is Faraday’s constant
- [Ion]out and [Ion]in are the extracellular and intracellular concentrations of the ion, respectively.
At typical body temperature (37°C), the Nernst equation simplifies for monovalent ions to:
Eion ≈ 61.5 mV * log10 ([Ion]out / [Ion]in) (for z = +1, e.g., Na+, K+) Eion ≈ -61.5 mV * log10 ([Ion]out / [Ion]in) (for z = -1, e.g., Cl-) Eion ≈ 30.7 mV * log10 ([Ion]out / [Ion]in) (for z = +2, e.g., Ca2+)
Using the typical concentrations from Step 1, we can estimate the equilibrium potentials for the key ions:
- EK+ ≈ 61.5 mV * log10 (4.5 / 150) ≈ -90 mV
- ENa+ ≈ 61.5 mV * log10 (145 / 10) ≈ +65 mV
- ECl- ≈ -61.5 mV * log10 (110 / 8) ≈ -70 mV (Note: Cl- distribution is complex, its resting potential is often close to RMP).
- ECa2+ ≈ 30.7 mV * log10 (1.5e-3 / 100e-6) ≈ +120 mV (Using mM for outside, µM * 1000 for inside to get approximate mM).
Crucially, the resting membrane potential of a cell (typically -70 mV) is generally not equal to the equilibrium potential of any single ion. This indicates that the resting membrane potential is determined by the simultaneous flow of multiple ions across the membrane.
Physiological Basis of the Goldman (GHK) Equation
The Nernst equation is a powerful tool for understanding the equilibrium potential of a single ion. However, biological membranes at rest are typically permeable to multiple ion species simultaneously, albeit to varying degrees. To calculate the membrane potential under conditions where multiple ions contribute, we need a more comprehensive equation. This is where the Goldman-Hodgkin-Katz (GHK) voltage equation comes in.
The GHK equation extends the Nernst concept by considering the contribution of multiple ions (typically Na+, K+, and Cl-) and, most importantly, their relative permeabilities (P) across the membrane. Permeability reflects the ease with which an ion can cross the membrane, primarily determined by the number of open ion channels specific to that ion.
The simplified GHK equation for a membrane permeable to Na+, K+, and Cl- is:
Vm = (RT / F) * ln ((PK*[K+]out + PNa*[Na+]out + PCl*[Cl-]in) / (PK*[K+]in + PNa*[Na+]in + PCl*[Cl-]out))
Where:
- Vm is the membrane potential
- R, T, F are constants as in the Nernst equation
- PNa, PK, PCl are the permeabilities of the membrane to Na+, K+, and Cl-, respectively.
- [Ion]out/[Ion]in are the extracellular/intracellular concentrations from Step 1.
The physiological basis of the GHK equation lies in the following:
- Multiple Ion Contributions: It acknowledges that the total electrical current across the membrane is the sum of the currents carried by all permeant ions.
- Electrochemical Driving Force: For each ion, the equation considers its electrochemical driving force, which is the difference between the actual membrane potential (Vm) and its equilibrium potential (Eion). This driving force determines the direction of net ion movement.
- Permeability (P) as the Key Factor: The critical advance of the GHK equation is the inclusion of permeability terms. The rate at which an ion flows across the membrane is not just dependent on the driving force but also on how easily it can permeate the membrane barriers. Permeability is proportional to the number of open channels specific to that ion. A higher permeability means a given driving force will result in a larger ion flux.
- Weighted Average: The GHK equation essentially calculates a weighted average of the equilibrium potentials of the contributing ions, with the weighting factor being the relative permeability of the membrane to each ion. If the membrane is highly permeable to K+ but less permeable to Na+, the membrane potential will be closer to EK+ than to ENa+.
The GHK equation calculates the membrane potential at which the net flow of charge across the membrane is zero. This is a steady state, not a true thermodynamic equilibrium, because ions are still flowing down their gradients (as long as Vm ≠ Eion), but the pump is continuously working to counteract this leakage and maintain the gradients.
Role of the Goldman Equation in Generation of RMP
The Resting Membrane Potential (RMP) is defined as the stable, negative membrane potential maintained by a cell when it is not actively signaling or undergoing major electrical changes. The GHK equation is the fundamental tool for understanding how the RMP is generated and why it has the value it does.
The role of the GHK equation in RMP generation can be explained as follows:
- Dominant Permeability at Rest: In most resting cells, the membrane is significantly more permeable to K+ than to Na+ or Cl-. This resting K+ permeability is primarily due to specific types of non-gated K+ channels known as “leak” channels, which are typically open at rest. The permeability to Na+ at rest is much lower due to a smaller number of open sodium channels. Similarly, resting permeability to Ca2+ is extremely low. The permeability to Cl- can vary but often contributes to the RMP value, sometimes following the potential established by K+ and Na+.
- Weighted Influence: According to the GHK equation, the membrane potential will be most strongly influenced by the ion with the highest relative permeability. Since PK is much greater than PNa and PCl (in many resting cells, PK : PNa : PCl might be roughly 1 : 0.05 : 0.1 or similar ratios depending on cell type), the RMP will be pulled closer to the equilibrium potential of K+ (EK+ ≈ -90 mV).
- Net Zero Current at RMP: The RMP is calculated by the GHK equation as the potential at which the sum of the individual ion currents (driven by the electrochemical gradient and scaled by permeability) is zero:
- At rest, K+ tends to leak out of the cell down its steep concentration gradient through leak channels, creating an outward positive current. The electrical force (negative inside) opposes this, but the concentration gradient is stronger until the membrane potential becomes very negative (approaching EK+).
- Na+ tends to leak into the cell down its steep electrochemical gradient (both concentration and electrical forces drive it inward) through the few resting Na+ channels. This creates an inward positive current.
- Cl- movement depends on its ECl- relative to Vm. If ECl- is close to Vm, there’s little net movement. If Vm is more positive than ECl-, Cl- leaks in; if more negative, Cl- leaks out.
The GHK equation finds the Vm where the outward positive K+ current is exactly balanced by the inward positive Na+ current (and sometimes contributions from Cl- and other minor ions). This net zero ionic current defines the stable RMP.
- Dependence on Gradients and Permeabilities: The GHK equation explicitly shows that the RMP value is directly dependent on:
- The concentration gradients of Na+, K+, and Cl- across the membrane.
- The relative permeabilities of the membrane to these ions (PNa, PK, PCl).
- Role of the Na+/K+ Pump (Indirect but Essential): While the GHK equation calculates the RMP based on existing gradients and permeabilities, it is crucial to remember that the Na+/K+-ATPase pump is ultimately responsible for maintaining the concentration gradients of Na+ and K+ over time. Ion leakage down their gradients, even at rest, would eventually dissipate these gradients if not actively counteracted by the pump. Thus, the pump indirectly but fundamentally contributes to the RMP by ensuring the necessary concentration differences are in place for the ion flows described by the GHK equation to occur and establish the potential. The pump itself also contributes a small electrogenic potential because it moves 3 positive charges out for every 2 positive charges in, making the inside slightly more negative. However, the primary determination of RMP value comes from the passive ion permeabilities described by GHK, acting on the gradients maintained by the pump.
In summary, the Goldman-Hodgkin-Katz equation provides the mathematical framework for understanding how the simultaneous diffusion of multiple ions, each driven by its electrochemical gradient and restricted by the membrane’s specific permeability to that ion, results in the stable, negative resting membrane potential characteristic of excitable and many non-excitable cells. The dominance of K+ permeability through leak channels at rest means the RMP is close to the K+ equilibrium potential, while the small but significant Na+ permeability prevents it from reaching EK+, resulting in a typical RMP of around -70 mV.
