Question

Rate Enhancement by Urease The enzyme urease enhances the rate of urea hydrolysis at \(\mathrm{pH} 8.0\) and \(20{ }^{\circ} \mathrm{C}\) by a factor of \(10^{14}\). Suppose that a given quantity of urease can completely hydrolyze a given quantity of urea in \(5.0 \mathrm{~min}\) at \(20^{\circ} \mathrm{C}\) and \(\mathrm{pH} 8.0\). How long would it take for this amount of urea to be hydrolyzed under the same conditions in the absence of urease? Assume that both reactions take place in sterile systems so that bacteria cannot attack the urea.

Short answer

Without urease, the reaction would take \(5.0 \times 10^{14}\) minutes.

Step-by-step solution

Understanding the Problem

We are asked to determine how long a reaction would take without the enzyme urease, given that with the enzyme, it takes 5.0 minutes and the rate enhancement factor by urease is \(10^{14}\).

Defining the Variables

Let \( t_{ ext{without}} \) be the time required to hydrolyze urea without urease. The time with urease, \( t_{ ext{with}} \), is given as 5.0 minutes, and the rate enhancement factor is \(10^{14}\).

Setting Up the Relationship

The rate enhancement factor is defined as the rate of the reaction with the enzyme divided by the rate of the reaction without the enzyme. Therefore, \( \text{Rate Enhancement Factor} = \frac{t_{ ext{without}}}{t_{ ext{with}}} = 10^{14}\).

Solving for Time Without Urease

The time without urease, \( t_{ ext{without}} \), can be found using the equation \( t_{ ext{without}} = t_{ ext{with}} \times 10^{14} \). Substitute 5.0 minutes for \( t_{ ext{with}} \): \( t_{ ext{without}} = 5.0 \times 10^{14} \).

Calculating the Time

Calculate \( t_{ ext{without}} \) using the equation from the previous step: \( t_{ ext{without}} = 5.0 \times 10^{14} \). This gives \( t_{ ext{without}} = 5.0 \times 10^{14} \text{ minutes} \).

Key concepts

Understanding Urease

Urease is a fascinating enzyme that plays a crucial role in biochemistry. It acts as a catalyst in the hydrolysis of urea, a process in which urea is converted into ammonia and carbon dioxide. This transformation is particularly important in the nitrogen cycle. Urease, found commonly in soil bacteria, plants, and certain fungi, helps in nitrogen metabolism by breaking down urea found in the environment.
This enzyme is notable for its ability to significantly speed up the reaction, which is otherwise extremely slow without it. In this context, urease enhances the rate of urea hydrolysis by a tremendous factor of \(10^{14}\). This means that the reaction is \(10^{14}\) times faster when urease is present, highlighting its efficiency as a biocatalyst.
The study and understanding of urease have provided insights into enzyme kinetics, a field that explores the rates of enzymatic reactions, giving us deeper knowledge into how enzymes function and their applications in biotechnology.

Exploring Rate Enhancement

Rate enhancement refers to how much faster a reaction occurs in the presence of an enzyme compared to without it. In the case of urease helping in the hydrolysis of urea, the rate enhancement is astronomical: \(10^{14}\) times faster. This illustrates how enzymes can dramatically influence the pace of biochemical reactions.
To understand rate enhancement, think of it as the factor that amplifies the reaction speed. For example, if a reaction normally takes a billion years, an enzyme with a rate enhancement factor of \(10^{14}\) could complete it in just a fraction of a second.
Enzymes lower the activation energy needed for reactions to proceed. This allows reactants to convert more readily into products within biological systems. By comparing the rates with and without an enzyme, we can calculate this enhancement factor, which provides valuable insight into the enzyme's efficiency and effectiveness.

Urea Hydrolysis Explained

Urea hydrolysis is a chemical process where urea is broken down to produce ammonia and carbon dioxide. The reaction is represented as: \[ \text{(NH}_2\text{)_2CO + H}_2\text{O} \rightarrow 2\text{NH}_3 + \text{CO}_2 \].
This biochemical process is crucial for nitrogen recycling in ecosystems as it aids in converting nitrogen waste into usable forms. Without enzymes like urease, urea hydrolysis would proceed extremely slowly, making it less effective in natural nitrogen cycles.
Urease catalyzes this process, facilitating life-sustaining transformations. Without urease or in its absence, the rate of urea hydrolysis drops dramatically, making its study essential for understanding both biological systems and potential applications in agriculture and medicine.

Determining Reaction Rate

The reaction rate is a measurement of how fast a reaction occurs. In enzyme kinetics, the reaction rate can be significantly influenced by the presence of an enzyme. Urease, for example, greatly increases the reaction rate of urea hydrolysis. This makes it a textbook case for studying how enzymes boost reaction rates.
In our exercise scenario, the reaction rate with urease allows urea hydrolysis to complete in just 5 minutes. Without urease, the same process would take a hugely extended period to finish, as the rate enhancement factor is \(10^{14}\). By calculating the different rates, students can learn how enzyme kinetics work and how essential enzymes are in facilitating rapid biochemical reactions.
Understanding these concepts helps in fields like biotechnology and pharmacology, where enzymes are used to catalyze various processes efficiently. Knowing how to determine and apply reaction rates opens doors to developing new technologies and solutions in science and industry.