Problem 10 - Biochemistry: Concepts and Connections

Question

Calculate the equilibrium membrane potentials to be expected across a membrane at \(37{ }^{\circ} \mathrm{C}\), with a \(\mathrm{NaCl}\) concentration of \(0.10 \mathrm{M}\) on the "right side" and \(0.01 \mathrm{M}\) on the "left side", given the following conditions. In each case, state which side is \((+)\) and which is \((-)\). (a) Membrane permeable only to \(\mathrm{Na}^{+}\) (b) Membrane permeable only to \(\mathrm{Cl}^{-}\) (c) Membrane equally permeable to both ions.

Step-by-Step Solution

Answer

(a) + Right, 61.5 mV (b) + Left, -61.5 mV (c) Neutral, 0 mV

Step 1: Identify the Nernst Equation

The Nernst equation is used to calculate the equilibrium potential for a specific ion. It is given by the formula: \[ E = \frac{RT}{zF} \ln \left( \frac{[\text{ion outside}]}{[\text{ion inside}]} \right), \] where \( E \) is the equilibrium potential, \( R \) is the universal gas constant (8.314 J/mol·K), \( T \) is the temperature in Kelvin, \( z \) is the ion charge, and \( F \) is Faraday's constant (96485 C/mol). At \(37{ }^{\circ} \mathrm{C}\), which is 310 K, the equation can be approximated as: \[ E = \frac{61.5}{z} \log \left( \frac{[\text{ion outside}]}{[\text{ion inside}]} \right). \]

Step 2: Calculate Equilibrium Potential for Na+

For part (a), the membrane is permeable only to \( \mathrm{Na}^{+} \). The concentration on the right side (outside) is \(0.10\, \mathrm{M}\), and the concentration on the left side (inside) is \(0.01\, \mathrm{M}\). Using the Nernst equation: \[ E_{\mathrm{Na}^+} = \frac{61.5}{1} \log \left( \frac{0.10}{0.01} \right). \] Calculate: \[ E_{\mathrm{Na}^+} = 61.5 \log(10) = 61.5 \times 1 = 61.5\, \text{mV}. \] The right side will be more positive, so it is the \((+)\) side.

Step 3: Calculate Equilibrium Potential for Cl−

For part (b), the membrane is permeable only to \( \mathrm{Cl}^{-} \). Keep in mind that \( \mathrm{Cl}^{-} \) has a charge of \(-1\). Using the Nernst equation: \[ E_{\mathrm{Cl}^-} = \frac{61.5}{-1} \log \left( \frac{0.10}{0.01} \right). \] Calculate: \[ E_{\mathrm{Cl}^-} = -61.5 \log(10) = -61.5 \times 1 = -61.5\, \text{mV}. \] The left side will be more positive, so it is the \((+)\) side.

Step 4: Calculate Membrane Potential When Permeable to Both Na+ and Cl−

For part (c), if the membrane is equally permeable to both \( \mathrm{Na}^{+} \) and \( \mathrm{Cl}^{-} \), the Goldman equation is used. It takes into account multiple ion permeabilities, but since they are equal, the potential will average out: \[ E = \frac{61.5}{1} \log \left( \frac{[\text{Na outside}] + [\text{Cl inside}]}{[\text{Na inside}] + [\text{Cl outside}]} \right) = \frac{61.5}{2}(61.5 + (-61.5)) = 0. \] Therefore, no side is more positive or negative.

Key Concepts

Equilibrium Potential

The concept of equilibrium potential is essential in understanding how ions move across a membrane to achieve balance in electrical and chemical gradients. Equilibrium potential refers to the potential difference across a cell membrane that exactly opposes the net diffusion of a specific ion through the membrane. This happens when the forces of chemical gradient and electrical gradient for that ion reach a point of balance.

It is calculated by the Nernst equation, which combines temperature and concentration differences to find the potential needed to counteract the flow of ions. For a positive ion like sodium (Na^+), the equilibrium potential is positive if more inflow is required to balance the chemical gradient. Conversely, for a negative ion like chloride (Cl^-), the equilibrium potential often comes up as negative because movement out of the cell leads to greater electrical neutrality.

• The equilibrium potential is essential for understanding neuronal excitability and muscle contraction.
• It provides insights into how cells maintain resting membrane potential.

This knowledge aids in deciphering mechanisms like action potentials, where shifts in equilibrium lead to electrical signals across neurons.

Membrane Permeability

Membrane permeability determines the ease with which ions can cross the cell membrane. It is a critical factor affecting the cell's electrical potential and overall function. Permeability varies depending on the type of membrane and the specific ion involved.

This property underlies the Nernst equation calculations, which assume that only one type of ion is permeable at a time. In reality, many ions can be permeable simultaneously, but the degree of permeability influences the equilibrium potential significantly.

• A membrane highly permeable to Na^+ will develop a potential according to the sodium concentration gradient.
• Conversely, a membrane more permeable to Cl^- will rely on the chloride ion distribution.

Furthermore, when a membrane is equally permeable to multiple ions, such as sodium and chloride, the calculation of potential becomes more complex, as each ion contributes to the overall membrane potential. An understanding of permeability is crucial for physiology and biochemistry, especially in scenarios involving cellular signaling and osmoregulation.

Goldman Equation

The Goldman equation provides a more comprehensive understanding of membrane potential compared to the Nernst equation, especially when considering multiple ion types. It accounts for the varying permeabilities of ions like sodium (Na^+), potassium (K^+), and chloride (Cl^-), all at once. Unlike the Nernst equation, which is only applicable when the membrane is permeable to a single ion, the Goldman equation is useful when multiple ions affect the membrane potential. It calculates an average of all the equilibrium potentials, weighted by their respective permeabilities.

• When dealing with multiple ions, the overall potential depends on both the concentrations and the relative permeabilities of each ion species.
• This approach provides a realistic depiction of the cell's actual membrane potential, taking into account the influence of several ions.

In situations where the membrane is equally permeable to sodium and chloride, the Goldman equation reflects a neutral potential, indicating no overall driving force favoring either side of the membrane. This level of analysis is critical when assessing physiological functions such as nerve impulse transmission, where multipoint ion influences are at play.