Question

Two different proteins \(\mathrm{X}\) and \(\mathrm{Y}\) are dissolved in aqueous solution at \(37^{\circ} \mathrm{C}\). The proteins bind in a 1: 1 ratio to form XY. A solution that is initially \(1.00 \mathrm{mM}\) in each protein is allowed to reach equilibrium. At equilibrium, \(0.20 \mathrm{mM}\) of free \(\mathrm{X}\) and \(0.20 \mathrm{~m} M\) of free Y remain. What is \(K_{c}\) for the reaction?

Short answer

The equilibrium constant \(K_c\) for the reaction between proteins X and Y, which bind in a 1:1 ratio to form XY, is 20. This was calculated using the given initial and equilibrium concentrations and the formula for the equilibrium constant.

Step-by-step solution

Write down the reaction

The given reaction is: \[X + Y \rightleftharpoons XY\] The stoichiometric coefficients for X, Y, and XY are all 1.

Calculate the equilibrium concentrations

Use the initial concentrations and equilibrium concentrations of X and Y to determine the equilibrium concentration of XY. Initial concentrations: [X] = [Y] = 1.00 mM Equilibrium concentrations: [X]\(_{eq}\) = [Y]\(_{eq}\) = 0.20 mM The equilibrium concentration of XY can be calculated by comparing the difference in initial and equilibrium concentrations of X (or Y): [XY]\(_{eq}\) = 1.00 mM - 0.20 mM = 0.80 mM.

Write the formula for \(K_c\)

For this reaction, the formula for the equilibrium constant \(K_c\) can be written as: \[K_c = \frac{[\mathrm{XY}]_{eq}}{[\mathrm{X}]_{eq} [\mathrm{Y}]_{eq}}\]

Calculate \(K_c\)

Now we can use the equilibrium concentrations calculated in Step 2 to find the value of \(K_c\): \[K_c = \frac{0.80 \text{ mM}}{(0.20 \text{ mM})(0.20 \text{ mM})} = \frac{0.80}{0.04} = 20\] Therefore, the equilibrium constant \(K_c\) for the reaction is 20.

Key concepts

Proteins and Their Binding Interaction

Proteins are complex biological molecules that play critical roles in virtually all biological processes. They're made up of amino acids and can interact with other molecules, including other proteins, in specific ways. In our exercise, two proteins X and Y are involved in a binding interaction where they come together to form a new complex. This specific kind of binding, known as "protein-protein interaction," is critical for many cellular processes such as signal transduction, where proteins communicate and initiate cellular responses.

- Proteins bind in various ways, often driven by changes in the surrounding environment, such as temperature or concentration.
- The binding between proteins can be reversible, meaning that they can associate and dissociate under the right conditions.

Here, X and Y bind in a 1:1 ratio, forming a complex XY. Such specific interaction ensures that biological processes proceed in an orderly manner, allowing for precise cellular function.

Understanding Equilibrium Concentrations

Equilibrium concentrations refer to the concentrations of reactants and products in a chemical reaction once the reaction has reached a state of balance. At equilibrium, the rate of the forward reaction (forming products) equals the rate of the reverse reaction (reacting products back to reactants).

In our scenario, starting with 1.00 mM of each protein X and Y, the system reached equilibrium, and both X and Y had a remaining concentration of 0.20 mM. This condition provides an insight into the extent of the reaction progress.

- Initially, both proteins were at 1.00 mM, and the remaining 0.80 mM indicates the concentration used to form the complex XY.
- Knowing the equilibrium concentrations allows us to calculate the equilibrium constant, a pivotal aspect in predicting the feasibility and extent of reactions.

It's crucial to understand that equilibrium concentrations are static only under constant conditions without external influences such as temperature change.

The Role of Aqueous Solutions in Protein Reactions

Aqueous solutions, meaning substances dissolved in water, are essential in facilitating biological reactions, including protein binding. In cells, most biochemical reactions, including those involving proteins, take place in aqueous environments.

Water, being a polar solvent, is excellent for dissolving various ionic and polar molecules, which helps in stabilizing mobile ions and proteins as they interact.
- The polarity of water molecules allows them to surround charged species, effectively reducing internal friction and enabling smooth interactions, such as the formation of the protein complex XY.
- The temperature of the aqueous solution, here noted as 37°C, replicates normal human body temperature, creating a realistic and biologically relevant environment for proteins X and Y to bind.

Understanding the aqueous medium's supportive role is vital, as it significantly affects protein stability, solubility, and overall reaction dynamics.