Question

In the following chapter, enzyme catalysis reactions will be extensively reviewed. The first step in these reactions involves the binding of a reactant molecule (referred to as a substrate) to a binding site on the enzyme. If this binding is extremely efficient (that is, equilibrium strongly favors the enzyme-substrate complex over separate enzyme and substrate) and the formation of product rapid, then the rate of catalysis could be diffusion limited. Estimate the expected rate constant for a diffusion controlled reaction using typical values for an enzyme \(\left(D=1.00 \times 10^{-7} \mathrm{cm}^{2} \mathrm{s}^{-1} \text {and } r=40.0 \AA\right)\) and a small molecular substrate \(\left(D=1.00 \times 10^{-5} \mathrm{cm}^{2} \mathrm{s}^{-1}\) and \right. \(r=5.00 \AA)\).

Short answer

The estimated rate constant for the diffusion-controlled enzyme catalysis reaction using the given enzyme and substrate parameters is \(5.50 \times 10^9 \mathrm{M}^{-1} \mathrm{s}^{-1}\).

Step-by-step solution

List the Given Parameters

Enzyme Diffusion Coefficient (D1): \(1.00 \times 10^{-7} \mathrm{cm}^{2} \mathrm{s}^{-1}\) Enzyme Radius (r1): \(40.0 \, \AA = 40.0 \times 10^{-8} \, \mathrm{cm}\) Substrate Diffusion Coefficient (D2): \(1.00 \times 10^{-5} \mathrm{cm}^{2}\mathrm{s}^{-1}\) Substrate Radius (r2): \(5.00 \, \AA = 5.00 \times 10^{-8} \, \mathrm{cm}\)

Calculate the Relative Diffusion Coefficient

The first step in finding the rate constant (k) is to determine the relative diffusion coefficient (D). We use the formula: \[D = \dfrac{D_1D_2}{D_1+D_2}\] Where D1 is the diffusion coefficient of the enzyme, and D2 is the diffusion coefficient of the substrate. \[D = \dfrac{(1.00 \times 10^{-7})(1.00 \times 10^{-5})} {(1.00 \times 10^{-7}+ 1.00 \times 10^{-5})} \mathrm{cm}^{2} \mathrm{s}^{-1} \] D = \(9.76 \times 10^{-8} \mathrm{cm}^{2} \mathrm{s}^{-1}\)

Calculate the Sum of the Radii

Next, we need to find the sum of the radii of the enzyme and the substrate (r1 + r2). r_sum = r1 + r2 r_sum = 40.0 × 10^(-8) cm + 5.00 × 10^(-8) cm r_sum = \(45.0 \times 10^{-8} \, \mathrm{cm}\)

Calculate the Rate Constant Using Smoluchowski Equation

Now we will use the Smoluchowski equation to find the rate constant (k): \[k = 4 \pi (r_1 + r_2)D\] Where k, is the rate constant, D is the relative diffusion coefficient and r_sum is the sum of the radii of the enzyme and substrate. k = 4 × π × (45.0 × 10^-8) × (9.76 × 10^-8) k = \(5.50 \times 10^9 \mathrm{M}^{-1} \mathrm{s}^{-1}\)

Present the Final Answer

The estimated rate constant for the diffusion-controlled enzyme catalysis reaction is: k = \(5.50 \times 10^9 \mathrm{M}^{-1} \mathrm{s}^{-1}\)

Key concepts

Diffusion Controlled Reaction

In the realm of enzyme catalysis, a fascinating process takes place when an enzyme and substrate bind swiftly and initiate a reaction. This type of reaction is known as a diffusion controlled reaction. Here, the rate at which the substrate encounters the enzyme is determined primarily by the diffusion rates of the interacting molecules.
This means that the reaction proceeds as fast as the molecules can move through the surrounding medium. Some key points to consider include:

• Diffusion is the movement of particles from a higher concentration to a lower concentration, driven by random thermal motion.
• The speed of this movement is crucial in determining how quickly the reaction can occur.
• A diffusion controlled reaction generally reaches the maximum rate possible since any improvements in the reaction rate must come from enhancing the diffusion process itself.

When exploring diffusion controlled reactions, understanding the factors affecting diffusion, such as temperature and medium viscosity, becomes vital for estimating reaction speed.

Rate Constant Calculation

The rate constant is a critical parameter in chemical reactions, including enzymatic reactions, as it represents the speed at which a reaction takes place. Calculating the rate constant provides insights into the efficiency of the reaction.To calculate this constant in a diffusion controlled reaction, follow these simple guidelines:

• Identify the diffusion coefficients for both the enzyme and substrate. These coefficients describe how quickly each molecule can diffuse through the solution.
• Determine the relative diffusion coefficient using \[D = \dfrac{D_1D_2}{D_1+D_2}\], where \(D_1\) and \(D_2\) are the respective diffusion coefficients of the enzyme and substrate.
• Calculate the sum of the radii of the enzyme and substrate to understand the spatial aspect of their interaction.
• Utilize the Smoluchowski equation to determine the rate constant, ensuring that factors like the solution's characteristics and molecules' sizes are considered.

Accurate analysis of these components allows for precise rate constant estimation, crucial for evaluating the reaction's pace.

Smoluchowski Equation

The Smoluchowski equation is a vital tool employed in determining the rate constant of diffusion controlled reactions. This equation connects the physical and chemical properties involved in the reaction process.Let's explore how this equation shapes our understanding:

• The equation is: \[k = 4 \pi (r_1 + r_2)D\]
• \((r_1 + r_2)\) signifies the sum of the radii of the enzyme and substrate, highlighting the importance of spatial dimensions in reaction rates.
• \(D\) is the relative diffusion coefficient, merging the diffusion potentials of both the enzyme and substrate.

In effect, the Smoluchowski equation provides a simplified yet powerful method to quantify reaction rates, emphasizing the influence of molecular sizes and diffusion properties over reaction speed in enzyme catalysis.

Enzyme-Substrate Complex

The enzyme-substrate complex is a fundamental concept in understanding enzyme catalysis. It forms when an enzyme and substrate bind together in a reaction. This complex is the starting point for any product formation in enzyme-mediated processes. Here are the aspects that define the enzyme-substrate complex:

• The binding involves specific interactions between the enzyme's active site and the substrate, primarily dictated by molecular shape and charge compatibility.
• Once formed, the complex positions the substrate in an optimal orientation, facilitating the conversion to the product.
• The stability of this complex can significantly affect the rate of catalysis, particularly in diffusion controlled reactions where fast complex formation is desired.

This complex underscores the efficiency of enzymes as biological catalysts, tailoring reactions to meet cellular needs rapidly and effectively.