Question

A purified protein is in a Hepes ( \(N\)-(2-hydroxyethyl)piperazine- \(N^{\prime}\)-(2-ethanesulfonic acid)) buffer at pH 7 with \(500 \mathrm{~mm} \mathrm{NaCl}\). A dialysis membrane tube holds a \(1 \mathrm{~mL}\) sample of the protein solution. The sample in the dialysis membrane floats in a beaker containing \(1 \mathrm{~L}\) of the same Hepes buffer, but with \(0 \mathrm{~mm} \mathrm{NaCl}\), for dialysis. Small molecules and ions (such as \(\mathrm{Na}^{+}, \mathrm{Cl}^{-}\), and Hepes) can diffuse across the dialysis membrane, but the protein cannot. a. Calculate the concentration of \(\mathrm{NaCl}\) in the protein sample, once the dialysis has come to equilibrium. Assume that no volume changes occur in the sample during the dialysis. b. Calculate the final \(\mathrm{NaCl}\) concentration in the protein sample after dialysis in \(250 \mathrm{~mL}\) of the same Hepes buffer, with \(0 \mathrm{~mm} \mathrm{NaCl}\), twice in succession.

Short answer

a. After dialysis, the NaCl concentration is approximately 0.4995 mM. b. After two 250 mL dialyses, the concentration is approximately 7.93 nM.

Step-by-step solution

Understanding Initial Conditions

Initially, we have a protein sample containing 1 mL of Hepes buffer with 500 mM NaCl inside a dialysis membrane. This sample is placed in a beaker with 1 L (1000 mL) of Hepes buffer that contains no NaCl. Our goal is to find the concentration of NaCl in the protein sample after equilibrium is reached.

Analyzing the Equilibrium Condition

At equilibrium, the concentration of NaCl outside the dialysis membrane (the beaker) and inside (the protein sample) will be the same, as NaCl can freely diffuse across the membrane. The total number of moles of NaCl will be conserved.

Calculating Moles of NaCl Initialy Present

Originally, the protein sample contains \(500 \text{ mM} \times 1 \text{ mL} = 0.5 \text{ mmol} \) of NaCl. There's no NaCl in the initial 1 L of surrounding buffer.

Calculating Final Concentration After Equilibrium

The total volume at equilibrium is 1001 mL because the 1 mL and the 1000 mL are combined. Therefore, the final concentration \(C_f\) of NaCl is given by the equation \( C_f = \frac{0.5 \text{ mmol}}{1001 \text{ mL}}\). This results in \( C_f \approx 0.4995 \text{ mM} \).

Setting Up Conditions for the 250 mL Dialysis

To complete part b, after the first dialysis in 1000 mL buffer at equilibrium, the concentration was 0.4995 mM. Now, imagine placing the 1 mL protein sample in 250 mL of Hepes buffer without NaCl. Analyzing this scenario follows similar logic.

First 250 mL Dialysis Step

Consider the total initial moles of NaCl: \(0.4995 \text{ mM} \times 1 \text{ mL} = 0.4995 \text{ µmol} \). The total final volume is 251 mL (because 1 mL of protein sample plus 250 mL of buffer). Thus, \( C_{f1} = \frac{0.4995 \text{ µmol}}{251 \text{ mL}} \approx 1.99 \text{ µM} \).

Second 250 mL Dialysis Step

Repeat the process using the concentration from the first dialysis and the same volume of buffer. The concentration of NaCl before the second dialysis is \(1.99 \text{ µM} \times 1 \text{ mL} = 1.99 \text{ nmol} \). The final concentration is \( C_{f2} = \frac{1.99 \text{ nmol}}{251 \text{ mL}} \approx 7.93 \text{ nM} \).

Key concepts

Dialysis Process

Dialysis is a separation technique often used in biochemistry to remove unwanted small molecules from a protein solution. The method relies on a semi-permeable membrane, which allows specific particles and molecules to pass through while retaining others. In our case, the dialysis bag contains a protein solution with NaCl and is immersed in a large volume of buffer with no NaCl.
Here's how it works:

• The small molecules like Na⁺ and Cl^- ions can pass through the membrane into the buffer solution.
• The larger protein molecules are retained inside the membrane.
• After some time, equilibrium is reached, equalizing the concentration of NaCl inside and outside the membrane.

Dialysis is crucial in experiments where you need to purify proteins without altering their properties by removing salts or other small contaminants.

Equilibrium Concentration

Equilibrium concentration refers to the state where the concentrations of solutes on either side of a dialysis membrane have stabilized. For NaCl in our scenario, its diffusion through the membrane into the large buffer volume occurs until the concentration of NaCl inside the dialysis bag equals the concentration in the surrounding solution.
When calculating equilibrium, several steps are involved:

• Initially measure the moles of NaCl within the dialysis bag.
• The total volume after the membrane and buffer mix is the sum of both volumes (e.g., 1 mL + 1000 mL gives 1001 mL total volume).
• The equilibrium concentration is determined by dividing the initial moles of NaCl by the total volume.

Once equilibrium is reached, the NaCl concentration before dialysis was 500 mM. After the process, concentration adjusts to around 0.4995 mM due to its distribution in a larger buffer volume.

Molecular Diffusion

Molecular diffusion is the movement of molecules from an area of higher concentration to an area of lower concentration until equilibrium is achieved. This process is fundamental in the dialysis, where NaCl diffuses through the membrane until its concentration balances on both sides.
Diffusion relies on:

• The concentration gradient – molecules naturally move from crowded areas to less crowded ones.
• The permeability of the membrane – only certain molecules can cross.

In dialysis, the protein molecules are too large to diffuse through the membrane, while smaller ions like Na⁺ and Cl^- migrate freely. This allows the NaCl concentration to decrease inside the dialysis bag as they move into the buffer, showing how molecular diffusion purifies the sample from unwanted small molecules.

Protein Purification

Protein purification is a vital technique in biochemistry, used for isolating proteins from a complex mixture to study their properties or prepare them for other experiments. Dialysis often forms a step in this multi-stage process.
This technique focuses on:

• Removing impurities such as salts, reducing the risk of protein denaturation.
• Maintaining protein functionality by ensuring they remain in a suited environment without unwanted substances.

Dialysis helps remove small, soluble impurities, one key step in protein purification pathways. Post-dialysis, the protein solution should be closer to pure, free of extraneous molecules that might interfere with subsequent analysis or applications in research.
Overall, these methods form part of a larger effort to ensure proteins are isolated in the most natural and functional state possible, ready for further scientific exploration.