Question

a. 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 \(0.0050 M\). The high internal concentration in cells is maintained by pumping \(\mathrm{K}^{+}\) from the plasma. How much work must be done to transport \(1.0 \mathrm{~mol} \mathrm{~K}^{+}\) from the blood to the inside of a muscle cell at \(37^{\circ} \mathrm{C}\) (normal body temperature)? b. When \(1.0 \mathrm{~mol} \mathrm{~K}^{+}\) is transferred from blood to the cells, do any other ions have to be transported? Why or why not? c. Cells use the hydrolysis of adenosine triphosphate, abbreviated ATP, as a source of energy. Symbolically, this reaction can be represented as $$\operatorname{ATP}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{ADP}(a q)+\mathrm{H}_{2} \mathrm{PO}_{4}^{-}(a q)$$ where ADP represents adenosine diphosphate. For this reaction at \(37^{\circ} \mathrm{C}, K=1.7 \times 10^{5}\). How many moles of ATP must be hydrolyzed to provide the energy for the transport of \(1.0 \mathrm{~mol}\) \(\mathrm{K}^{+}\) ? Assume standard conditions for the ATP hydrolysis reaction.

Short answer

The work required to transport 1.0 mol of potassium ions from the blood plasma to muscle cells is approximately 8283 J/mol. Other ions, such as chloride ions, need to be transported with potassium ions to maintain the cell's electrical neutrality. Approximately 0.239 moles of ATP must be hydrolyzed to provide the energy needed for the transport of 1.0 mol of potassium ions.

Step-by-step solution

Part a: Calculate work for potassium ion transport

First, we need to calculate the work required for potassium ion transport by using the formula: Work = nRT ln(C2/C1) where n is the number of moles, R is the ideal gas constant (8.314 J/mol·K), T is the temperature in Kelvins (310.15 K), C1 is the initial concentration (0.0050 M), and C2 is the final concentration (0.15 M). Plugging the values into the equation, we get: Work = (1.0 mol)(8.314 J/mol·K)(310.15 K) ln(0.15 M / 0.0050 M) Now, we can simplify and calculate the final value. Work = (2584.24 J/mol) ln(30) ≈ 8283 J/mol The work required to transport 1.0 mol of potassium ions from blood plasma to muscle cells is approximately 8283 J/mol.

Part b: The necessity of other ion transportation

Yes, other ions need to be transported along with potassium ions. The main reason behind this is to maintain the electrical neutrality of the cell. As potassium ions are positively charged, they would create an electric charge imbalance inside the cell if not accompanied by negatively charged ions such as chloride ions (Cl-) or other factors contributing to maintaining the cell's electrical neutrality. This is essential for proper cell functioning and maintaining membrane potential.

Part c: Calculate the moles of ATP required for potassium ion transport

To calculate the number of moles of ATP needed for potassium ion transport, we first need to find the standard Gibbs free energy change (ΔG°) of ATP hydrolysis using the following formula: ΔG° = -RT ln(K) where R is the ideal gas constant (8.314 J/mol·K), T is the temperature in Kelvins (310.15 K), and K is the equilibrium constant (1.7 ×10^5). Plugging the values into the equation, we get: ΔG° = -(8.314 J/mol·K)(310.15 K) ln(1.7 × 10^5) ≈ -34,600 J/mol Now, we will divide the work required for transporting potassium ions by the energy yielded per mole of ATP hydrolyzed to get the moles of ATP needed: Moles of ATP = (Work required) / (Energy per mole of ATP hydrolyzed) Moles of ATP = (8283 J/mol) / (-34,600 J/mol) ≈ 0.239 mol Approximately 0.239 moles of ATP must be hydrolyzed to provide the energy needed for the transport of 1.0 mol of potassium ions.

Key concepts

ATP Hydrolysis

Adenosine triphosphate (ATP) is the energy currency of the cell, providing the power needed for many biological processes. ATP hydrolysis is the reaction wherein ATP is broken down into adenosine diphosphate (ADP) and inorganic phosphate (\r\( \mathrm{H}_2 \mathrm{PO}_4^- \)\r), releasing energy in the process. This reaction is crucial for cellular activities, including ion transport across membranes.

The energy released during ATP hydrolysis comes from breaking the high-energy phosphate bonds within the molecule. It is this released energy that is then harnessed to perform cellular work, such as the active transport of potassium ions against a concentration gradient, as seen in muscle cells.

Energy and Work

ATP hydrolysis releases energy that can be quantified as work required to perform a task within the cell. For example, transporting potassium ions requires a specific amount of work, which can be covered by the hydrolysis of ATP molecules. The conversion of the stored chemical energy in ATP into mechanical energy is a foundational concept in biochemistry and cellular biology.

Gibbs Free Energy

Gibbs free energy, denoted as \r\( \Delta G \)\r, is a thermodynamic property that measures the usable energy or the maximum reversible work that can be performed by a system at constant temperature and pressure. It is a central concept in understanding whether a reaction can occur spontaneously.

In the context of cellular processes, the Gibbs free energy change for ATP hydrolysis is negative, indicating the reaction is exergonic and releases energy spontaneously under standard conditions. The value of \r\( \Delta G \)\r for a reaction helps predict how much energy is available for cellular tasks—such as the active transport of ions across a membrane.

ATP Hydrolysis and Muscle Cells

The Gibbs free energy released during the hydrolysis of ATP is what provides the muscle cells with the necessary energy to import potassium ions against their concentration gradient. This flow of energy is crucial in powering many biochemical pathways.

Equilibrium Constant

The equilibrium constant, represented as \r\( K \)\r, is a dimensionless value that indicates the ratio of the concentrations of the products to the reactants of a reaction at equilibrium. In biochemistry, it is a measure of the tendency of a reaction to proceed to completion under standard conditions.

For the hydrolysis of ATP, the equilibrium constant is quite high (\r\( K = 1.7 \times 10^5 \)\r), suggesting that the reaction heavily favors the formation of products (ADP and \r\( \mathrm{H}_2 \mathrm{PO}_4^- \)\r). The large value of \r\( K \)\r reflects the high degree of ATP hydrolysis under cellular conditions, highlighting why ATP is such an effective molecule for energy storage and transfer in biological systems.

Implications for Cellular Work

An understanding of the equilibrium constant informs us about the efficiency of energy utilization during biochemical reactions. In the case of potassium ion transport, the ATP hydrolysis reaction's high equilibrium constant ensures that adequate energy is available for the active transport of ions, such as \r\( \mathrm{K}^{+} \)\r ions into muscle cells.

Membrane Potential

Membrane potential is the electric potential difference across a cell's plasma membrane. It is essential for the function of cells, particularly in nerve and muscle cells. The membrane potential arises due to the difference in ion concentrations inside and outside the cell, as well as the permeability of the cell membrane to those ions.

Maintaining a specific internal concentration of ions like potassium is crucial for cell function and contributes to the resting membrane potential. The active transport of potassium ions into muscle cells against their concentration gradient helps to establish and maintain this membrane potential.

Ion Transport and Electrical Neutrality

Cells actively transport ions like \r\( \mathrm{K}^{+} \)\r to maintain electrical neutrality and proper cell functioning. This electrochemical gradient, driven by ion pumps that require ATP hydrolysis, is fundamental for processes such as neurotransmission and muscle contraction. Ensuring a stable membrane potential is a continuous and energy-demanding task for the cell, reflecting the deep interconnection between biochemistry, cellular biology, and physiology.