Question

Suppose that the cell shown in Figure \(12.12 a\) is permeable to \(\mathrm{Na}^{+}\), \(\mathrm{K}^{+}\), and \(\mathrm{Cl}^{-}\)but not to \(\mathrm{A}^{-}\). In the absence of ion pumps, \(\mathrm{Na}^{+}\)and \(\mathrm{Cl}^{-}\)will diffuse in, the cell will become out of osmotic balance, and water will enter. Explain why a cell cannot be in thermodynamic equilibrium and also be in osmotic balance, unless there is a nonpermeating ion in the extracellular fluid (as would be the case if \(\mathrm{Na}^{+}\)were nonpermeating). What does this suggest about the evolutionary origin of the \(\mathrm{Na}^{+}-\mathrm{K}^{+}\)exchange pump?

Short answer

Given the cell is permeable to \(\mathrm{Na}^{+}\), \(\mathrm{K}^{+}\), and \(\mathrm{Cl}^{-}\), and non-permeable to \(\mathrm{A}^{-}\), a cell cannot be in both thermodynamic equilibrium and osmotic balance because in absence of a non-permeating ion, the concentrations will equalize across the membrane disrupting the osmotic balance. This suggests the origin of the \(\mathrm{Na}^{+}-\mathrm{K}^{+}\) pump could be due to evolutionary pressure to ensure osmotic balance and cell survival.

Step-by-step solution

Understand the Baseline Assumption

Thermodynamic equilibrium occurs when a system reaches a state of maximum entropy, exhibiting no changes in pressure, temperature, or concentration. On the other hand, osmotic balance requires different concentrations on either side of the cell membrane to facilitate the movement of molecules across it. Hence, a cell in thermodynamic equilibrium would therefore not expect any net movement of molecules without a specific pump to enforce the osmotic balance while maintaining the equilibrium.

Reasoning Out the Requirement

Given that the cell is at equilibrium, it means that there are no net movement of ions in or out of the cell, hence no gradient is maintained. Now, in the absence of a non-permeating ion outside the cell, all ions will eventually move until equilibrium is reached. This means no gradient is maintained, thus no osmosis occurs and the cell is out of osmotic balance.

Inferring the Evolutionary Origin

If \(\mathrm{Na}^{+}\) were non-permeating (like \(\mathrm{A}^{-}\)), it would remain in the extracellular fluid and maintain osmotic balance by preventing excessive water inflow. However, it's permeable and without a pump it would eventually equalize concentrations across the cell membrane, upsetting the osmotic balance. The \(\mathrm{Na}^{+}-\mathrm{K}^{+}\) exchange pump actively moves \(\mathrm{Na}^{+}\) out and \(\mathrm{K}^{+}\) into the cell, ultimately maintaining osmosis and cellular functions, suggesting an evolutionary necessity of the pump.

Key concepts

Thermodynamic Equilibrium

Thermodynamic equilibrium is a state where a system is perfectly balanced, resulting in no net change in entropy over time. Entropy, often referred to as the degree of disorder, reaches its maximum in equilibrium, indicating that energy is uniformly distributed, and no further energy exchanges are viable.
In a cellular context, achieving thermodynamic equilibrium means that ionic concentrations inside and outside the cell become equal over time.
This state eradicates any gradient that would allow ion or molecule movement through the cell membrane, crucial for dynamic processes.

• Stable pressure and temperature
• Equal concentrations across the membrane
• No energy transfer or net ion movement

Thus, in thermodynamic equilibrium, there is no spontaneous chemical energy available to support essential cellular functions, highlighting the importance of cellular mechanisms that actively maintain concentration gradients.

Osmotic Balance

Osmotic balance is crucial for maintaining cell integrity and function. Osmosis is the process where water moves across a semipermeable membrane to equalize solute concentrations on both sides.
For proper cell function, a gradient must exist so that water moves to balance solute concentrations without disrupting cell volume.

• Different solute concentrations inside and outside the cell
• Water movement to equalize concentrations
• Prevents cell swelling or shrinking

Without osmotic balance, cells either swell and burst due to water influx (lysis) or shrink from excessive water efflux (crenation). Cells require structures and mechanisms, such as impermeable ions, to maintain this critical balance, making it clear how delicate and important osmoregulation is in cellular health.

Ion Permeability

Ion permeability refers to the ability of ions to move across the cell membrane, which is selectively permeable. This selectivity is vital for cellular homeostasis and signaling.
Some ions easily cross the membrane while others do not, depending on their charge, size, and solubility.
Membrane proteins often assist in this process, allowing controlled ion flow essential for maintaining concentration gradients.

• Selective movement of ions
• Influences electrical potential and signaling
• Maintains homeostatic balance

Without control over ion permeability, cells wouldn't be able to regulate their internal environment, which could disrupt important functions like nutrient absorption, pH regulation, and electrical signaling.

Na+-K+ Exchange Pump

The Na+-K+ exchange pump is a fundamental cellular mechanism necessary for maintaining ion gradients across the cell membrane. This active transport process uses ATP to move 3 Na+ ions out of the cell and 2 K+ ions into the cell against their concentration gradients.
By doing this, the pump ensures that Na+ remains more concentrated outside, and K+ more concentrated inside, aiding in cell volume regulation and electrical activity.

• Uses energy (ATP) for ion transport
• Maintains concentration gradients crucial for osmotic balance
• Supports functions like nerve impulse transmission

Evolutionarily, the emergence of this pump highlights its essential role in managing osmotic pressure and sustaining cell integrity, demonstrating how cells evolved mechanisms to thrive despite permeable membranes.