Question

Many biochemical reactions that occur in cells require relatively high concentrations of potassium ion \(\left(\mathrm{K}^{+}\right) .\) The concentration of \(\mathrm{K}^{+}\) in muscle cells is about \(0.15 \mathrm{M}\). The concentration of \(\mathrm{K}^{+}\) in blood plasma is about \(0.0050 \mathrm{M}\). The high internal concentration in cells is maintained by pumping \(\mathrm{K}^{+}\) from the plasma. How much work must be done to transport 1.0 mole of \(\mathrm{K}^{+}\) from the blood to the inside of a muscle cell at \(37^{\circ} \mathrm{C}\), normal body temperature? When 1.0 mole of \(\mathrm{K}^{+}\) is transferred from blood to the cells, do any other ions have to be transported? Why or why not?

Short answer

The work required to transport 1.0 mole of K⁺ ions from blood plasma into the muscle cell is approximately -6860 J. Other ions, specifically sodium ions (Na⁺), need to be transported along with K⁺ ions to maintain electrical neutrality within the cell. This occurs through the sodium-potassium pump (Na⁺/K⁺-ATPase), an active transport system that helps regulate the balance of ions and maintain overall electrical neutrality in cells.

Step-by-step solution

Find the Gibbs free energy change

The Gibbs free energy change can be found using the formula: ΔG = -nRT ln(Q) where ΔG is the change in Gibbs free energy, n is the number of moles of K⁺, R is the gas constant (8.314 J/mol K), T is the temperature in Kelvin, and Q is the reaction quotient. First, we need to convert the given temperature in Celsius to Kelvin: T(K) = 37°C + 273.15 = 310.15 K

Calculate the reaction quotient (Q)

The reaction quotient (Q) can be calculated as: Q = [K⁺]_inside / [K⁺]_outside where [K⁺]_inside is the concentration of K⁺ inside the muscle cell, and [K⁺]_outside is the concentration of K⁺ in blood plasma. Given concentrations are: [K⁺]_inside = 0.15 M [K⁺]_outside = 0.0050 M We can now plug the values and find Q: Q = 0.15 / 0.0050 = 30

Calculate the work required

Now, we can use the given values to calculate the work required: ΔG = -nRT ln(Q) ΔG = -(1.0 mol)(8.314 J/mol K)(310.15 K) ln(30) ΔG ≈ -6860 J The work required to transport 1.0 mole of K⁺ ions from blood plasma into the muscle cell is approximately -6860 J.

Discuss if any other ions need to be transported

Yes, other ions need to be transported along with K⁺ ions. The reason is that cells need to maintain electrical neutrality. When 1.0 mole of positively charged K⁺ ions is transported into the muscle cell, the same amount of a charge must be moved out of the cell to maintain the overall charge balance in the cell. Typically, sodium ions (Na⁺) are moved out of the cell in exchange for the K⁺ ions being pumped in. This process is commonly known as the sodium-potassium pump (Na⁺/K⁺-ATPase). This active transport system allows cells to maintain a proper balance of ions and overall electrical neutrality.

Key concepts

Potassium Ion Concentration

The potassium ion (K⁺) concentration within cells plays a critical role in maintaining cellular functions such as electrical signaling in neurons and muscle contraction. In muscle cells, a high concentration of K⁺ is essential, with typical values around 0.15 M compared to just 0.0050 M in blood plasma. This gradient is crucial for various physiological processes and is actively regulated by cellular mechanisms.

To understand the importance of maintaining this ion concentration gradient, let’s consider the cellular environment as a busy marketplace. In the market, certain items (or ions) need to be in greater abundance on one side to ensure the town (or cell) operates smoothly. The K⁺ ions, much like a popular product, are in higher demand within the cell than outside in the blood plasma, akin to a storage facility. This difference in concentration is not just a happenstance but a carefully regulated balance that allows cells to function optimally.

Sodium-Potassium Pump

The sodium-potassium pump, scientifically referred to as Na⁺/K⁺-ATPase, is a vital cellular mechanism that uses energy to move ions across the cell membrane against their concentration gradients. For every three sodium ions (Na⁺) pumped out, two potassium ions (K⁺) are pumped into the cell. This mechanism functions much like a gate that controls the entry and exit of these ions, and it's fueled by ATP, the cell’s energy currency.

Picture a bustling cafe where coffee (Na⁺) needs to leave the shop for every two cups of tea (K⁺) that enter. This exchange is not equal, reflecting the actual 3:2 ratio that the sodium-potassium pump works with to maintain cellular homeostasis. This pump is fundamental because it helps in setting the stage for electrical signal passage, nutrient transport, and overall cellular health.

Gibbs Free Energy Equation

The Gibbs free energy equation, given by ΔG = -nRT ln(Q), allows us to calculate the amount of energy change during a process. In this equation, ΔG represents the Gibbs free energy change, n is the number of moles, R is the gas constant, T is the temperature in Kelvin, and Q is the reaction quotient, which represents the ratio of the concentrations of products to reactants.

Imagine you are assessing whether a business venture (a biochemical reaction) is profitable. The Gibbs free energy is like the profit (or loss) from this venture. If ΔG is negative, the process (or reaction) can occur spontaneously, just as a business makes a profit without any extra investment. In the given exercise, the negative ΔG signifies that the work required to move K⁺ from the blood to the inside of a cell is energy favorable under standard conditions.

Cellular Ion Transport

Cellular ion transport involves moving ions across the cell membrane, requiring specialized proteins such as ion channels and pumps. This transport is essential for maintaining the electrolyte and fluid balance within cells and throughout the body.

Consider a diligent mailroom in a large office. The mailroom workers (ion channels and pumps) are responsible for sorting and delivering packages (ions) to various departments (cellular compartments). These workers need to recognize and move the packages to the right places, despite a constant influx of items coming in and going out. In biology, maintaining the proper distribution of ions like K⁺ and Na⁺ is paramount for physiological functions such as muscle contraction, nerve impulse generation, and osmoregulation. The transport process, especially when against the concentration gradient, consumes energy but ensures the cell has the right ingredients for optimal functionality.