Question

Energetics of the \(\mathrm{Na}^{+} \mathbf{K}^{+}\)ATPase For a typical vertebrate cell with a membrane potential of \(-0.070 \mathrm{~V}\) (inside negative), what is the free-energy change for transporting 1 mol of \(\mathrm{Na}^{+}\) from the cell into the blood at \(37^{\circ} \mathrm{C}\) ? Assume the \(\mathrm{Na}^{+}\) concentration is 12 mm inside the cell and 145 mm in blood plasma.

Short answer

The free-energy change for transporting 1 mol of \( \mathrm{Na}^{+} \) into the blood is 9,838 J/mol.

Step-by-step solution

Understand the Nernst Equation

The Nernst equation is used to determine the equilibrium potential for a given ion, and it can also relate the membrane potential to ion concentrations on either side of the membrane. This equation is fundamental in calculating the free-energy change across the membrane.

Write Down the Nernst Equation

The Nernst equation is given by: \[ E = \frac{RT}{zF} \ln \frac{[\text{ion outside}]}{[\text{ion inside}]} \]where- \( R \) is the ideal gas constant \( (8.314 \text{ J/mol}\cdot\text{K}) \),- \( T \) is the temperature in Kelvin,- \( z \) is the charge of the ion (for \( \mathrm{Na^+} \), \( z = +1 \)),- \( F \) is Faraday's constant \( (96485 \text{ C/mol}) \),- \( [\text{ion outside}] \) and \( [\text{ion inside}] \) are the concentrations outside and inside the cell.

Convert Temperature to Kelvin

The temperature needs to be converted to Kelvin for use in the Nernst equation. \[ T = 37^\circ \text{C} = 37 + 273.15 = 310.15 \text{ K} \]

Calculate the Nernst Potential

Substitute the values into the Nernst equation:\[ E = \frac{(8.314)(310.15)}{(1)(96485)} \ln \frac{145}{12} \approx 0.067 \text{ V} \]

Write Formula for Free-Energy Change

The free-energy change for moving 1 mol of ions across the membrane is given by the equation:\[ \Delta G = RT \ln \frac{[\text{ion outside}]}{[\text{ion inside}]} + zF\Delta\Psi \]where \( \Delta\Psi \) is the transmembrane potential (\(-0.070\ \text{V}\)).

Calculate the Chemical Work

Calculate the chemical work portion using the Nernst equation:\[ \Delta G_{\text{chemical}} = (8.314)(310.15) \ln \frac{145}{12} \approx 16,593 \text{ J/mol} \]

Calculate the Electrical Work

Calculate the electrical work:\[ \Delta G_{\text{electric}} = (+1)(96485)(-0.070) \approx -6,755 \text{ J/mol} \]

Calculate the Total Free-Energy Change

Add the chemical work and the electrical work to find the total free-energy change:\[ \Delta G = 16,593 + (-6,755) = 9,838 \text{ J/mol} \]

Interpret the Result

The positive sign of the free-energy change indicates that the transport of \( \mathrm{Na^+} \) from the cell to the blood is non-spontaneous and requires energy input.

Key concepts

Nernst Equation

The Nernst Equation plays a pivotal role in bioenergetics, specifically in understanding ion movement across cell membranes. This equation helps us calculate the equilibrium potential for an ion. The beauty of the Nernst Equation is that it takes into account the concentration gradient of ions. This gradient is the driving force for ions to move across membranes. The equation is represented as: \[ E = \frac{RT}{zF} \ln \frac{[\text{ion outside}]}{[\text{ion inside}]} \]Here's what the symbols mean:

• R: The ideal gas constant, valued at 8.314 J/mol·K.
• T: Absolute temperature in Kelvin, crucial for thermal consideration in biological contexts.
• z: Ionic charge, which influences how strongly ions interact with the electric field across the membrane.
• F: Faraday's constant, a factor in converting between chemical and electrical energy.

By using this equation, we can determine how the concentration of ions affects the membrane potential, which is essential for understanding nerve impulse transmission and muscle contraction.

Membrane Potential

A membrane potential refers to the electrical potential difference across a cell's plasma membrane. It's an essential concept for bioenergetics because it influences how ions and molecules move across the membrane. Essentially, the membrane potential is created by differences in ion concentrations inside and outside the cell. This difference creates an electrical gradient, working alongside the concentration gradients established by ion channels and pumps. For example, if we have a high concentration of sodium ions outside the cell compared to inside, an electrical potential is formed across the membrane. The typical resting membrane potential for a vertebrate cell is about -70 mV, with the inside being negative relative to the outside. This is crucial for processes like:

• Signal transmission in neurons.
• Muscle contraction.
• Transport of substances across the membrane.

The interplay of the chemical and electrical gradients drives vital cellular processes.

Free-Energy Change

The concept of free-energy change, denoted as \( \Delta G \), is crucial for understanding how energy is utilized in cellular processes. In the context of ion transport, \( \Delta G \) determines whether the movement of ions is spontaneous or requires energy input. The equation for free-energy change is:\[ \Delta G = RT \ln \frac{[\text{ion outside}]}{[\text{ion inside}]} + zF\Delta\Psi \]Here, \( \Delta \Psi \) represents the transmembrane potential. It's important to understand that a positive \( \Delta G \) indicates a process is non-spontaneous, meaning it won't occur without external energy. This is key in active transport where energy is required, such as the pumping of sodium ions out of a cell. Conversely, a negative \( \Delta G \) means the process can happen spontaneously, driven by existing gradients and potentials. Understanding \( \Delta G \) helps us gauge the energetics of cellular functions and how cells maintain homeostasis.

Ion Transport

Ion transport is a fundamental concept in cellular biology that involves the movement of ions across cell membranes. This process is vital for maintaining essential functions such as nerve signal transmission, muscle contraction, and osmoregulation. There are two main types of ion transport: passive and active.

• Passive Transport: Ions move down their concentration gradient through channels without requiring energy.
• Active Transport: Ions are moved against their concentration gradients using energy, often from ATP hydrolysis.

One critical aspect of ion transport is that it enables cells to maintain ionic gradients crucial for survival. These gradients are foundational for the generation of electrical signals in excitable cells such as neurons and muscle cells. The dynamic nature of ion transport allows cells to respond and adapt to changing environments, maintain their internal conditions, and perform biological functions effectively.

Sodium-Potassium Pump

The sodium-potassium pump is an essential component of ion transport and plays a crucial role in maintaining cell homeostasis. This pump actively transports sodium and potassium ions across the cell membrane, balancing their concentrations. Here’s a simple breakdown of how it works:

• The pump moves 3 sodium ions out of the cell, which is crucial in maintaining the low internal sodium concentration.
• Simultaneously, it brings 2 potassium ions into the cell, which aids in maintaining high internal potassium levels.
• This process uses the energy from ATP, as it goes against the sodium and potassium concentration gradients.

The sodium-potassium pump is not just about balancing ion concentrations. It also helps in regulating cell volume, providing the sodium gradient for secondary active transport, and maintaining the resting membrane potential. Understanding its operation is vital for grasping how cells manage energy and maintain vital functions.