Question

The hydrolysis of pyrophosphate to orthophosphate is important in driving forward biosynthetic reactions such as the synthesis of DNA. This hydrolytic reaction is catalyzed in Escherichia coli by a pyrophosphatase that has a mass of \(120 \mathrm{kDa}\) and consists of six identical subunits. For this enzyme, a unit of activity is defined as the amount of enzyme that hydrolyzes \(10 \mu \mathrm{mol}\) of pyrophosphate in 15 minutes at \(37^{\circ} \mathrm{C}\) under standard assay conditions. The purified enzyme has a \(V_{\max }\) of 2800 units per milligram of enzyme. (a) How many moles of substrate are hydrolyzed per second per milligram of enzyme when the substrate concentration is much greater than \(K_{M} ?\) (b) How many moles of active sites are there in \(1 \mathrm{mg}\) of enzyme? Assume that each subunit has one active site. (c) What is the turnover number of the enzyme? Compare this value with others mentioned in this chapter.

Short answer

(a) \(3.11 \times 10^{-5} \, \text{mol s}^{-1 mg}^{-1}\), (b) \(3 \times 10^{-7} \, \text{mol active sites}\), (c) \(k_{cat} = 103.67 \, \text{s}^{-1}\).

Step-by-step solution

Define Variables and Constants

Before starting the calculation, let's gather the information given:- Molecular weight of the enzyme is 120 kDa.- The enzyme consists of six identical subunits, so the molecular weight of each subunit is \(\frac{120}{6} = 20\) kDa.- \(V_{\text{max}}\) is 2800 units per mg of enzyme.- 1 unit of enzyme activity hydrolyzes \(10 \mu M\) pyrophosphate in 15 minutes.

Convert Units of Enzyme Activity

A unit of enzyme activity corresponds to hydrolysis of \(10 \mu \text{mol} = 10 \times 10^{-6}\) mol pyrophosphate per 15 minutes. Convert the unit activity to moles of substrate hydrolyzed per second:\[\frac{10 \times 10^{-6}}{15 \times 60}\, \text{mol s}^{-1} = \frac{10 \times 10^{-6}}{900}\, \text{mol s}^{-1} \approx 1.11 \times 10^{-8} \, \text{mol s}^{-1}\, \text{per unit}\]

Calculate Moles of Substrate Hydrolyzed per Second per Milligram

With \(V_{\text{max}}\) of 2800 units per mg, calculate moles of substrate hydrolyzed per second per mg:\[2800\, \text{units mg}^{-1} \times 1.11 \times 10^{-8} \, \text{mol s}^{-1} \approx 3.11 \times 10^{-5} \, \text{mol s}^{-1}\, \text{mg}^{-1}\]

Determine Number of Active Sites in 1 mg Enzyme

1 subunit has a molecular weight of 20 kDa = 20,000 Da. Converting to grams, this is \(20,000 \times 10^{-3} \text{g mol}^{-1}\).Calculate moles of enzyme in 1 mg:\[\frac{1\, \text{mg}}{20,000\, \text{g mol}^{-1}} = \frac{1 \times 10^{-3}}{20}\, \text{mol} = 5 \times 10^{-8} \, \text{mol}\]Thus, there are \(6 \times 5 \times 10^{-8} = 3 \times 10^{-7} \, \text{mol of active sites}\) in 1 mg enzyme.

Calculate Turnover Number (kcat)

The turnover number, \(k_{cat}\), is calculated using the formula: \(k_{cat} = \frac{V_{\text{max}}}{\text{Number of active sites}}\)With \(V_{\text{max}} = 3.11 \times 10^{-5} \, \text{mol s}^{-1} \text{mg}^{-1}\) and number of active sites = \(3 \times 10^{-7} \, \text{mol}\):\[k_{cat} = \frac{3.11 \times 10^{-5}}{3 \times 10^{-7}} = 103.67 \, \text{s}^{-1}\]This means each active site releases about 103 substrate molecules per second at full occupancy.

Compare Turnover Number

In reference materials, typical \(k_{cat}\) values for enzymes range from 1 to 10,000 s⁻¹. The calculated \(k_{cat}\) of 103.67 s⁻¹ indicates this enzyme is moderately efficient, reflecting typical values for enzymes in biosynthetic pathways.

Key concepts

Pyrophosphatase

Pyrophosphatase enzymes play a critical role in the conversion of pyrophosphate into orthophosphate, an essential reaction that drives many biosynthetic pathways forward. These enzymes are crucial in cellular metabolism, especially in species like *Escherichia coli*. They help facilitate the synthesis of vital components like DNA by breaking down pyrophosphate.

This hydrolytic reaction is highly favorable and usually irreversible under physiological conditions, providing a substantial energetic push for endergonic reactions such as nucleic acid synthesis. In the context of *E. coli*, the pyrophosphatase has a molecular mass of 120 kDa and is made up of six identical subunits. This multi-subunit structure ensures efficient catalysis by providing multiple active sites within a single enzyme molecule. Thus, activating or regulating these enzymes can significantly influence cellular functions, as their catalytic actions are intimately linked with the energy balance and metabolic flux of the cell.

Turnover Number

The turnover number, or catalytic constant ( k_{cat} ), is an essential characteristic of enzyme kinetics. It describes the number of substrate molecules that an enzyme can convert into product per second when operating at full capacity. For the pyrophosphatase enzyme analyzed, the calculated turnover number is approximately 103.67 s⁻¹.

This number implies that each active site on the enzyme can catalyze the conversion of 103 substrate molecules per second. This rate is quite typical for enzymes involved in biosynthetic pathways, reflecting their necessity for moderate speed and efficiency in metabolic processes.

Turnover numbers can vary widely among different enzymes, often ranging from a few to many thousands per second, underscoring the diverse requirements and mechanisms of enzymes in different biological contexts.

Active Sites

Active sites are specific regions within an enzyme where substrate molecules bind and undergo a chemical reaction. For the pyrophosphatase enzyme discussed here, each of the six identical subunits contains one active site.

This structural aspect means that in each 120 kDa molecule, there are six active sites available to interact with the substrate. The number of moles of active sites in 1 mg of enzyme has been calculated to be around 3 \( \times \) 10⁻⁷ mol. The efficiency of the enzyme reaction is thus determined not only by the individual active sites' catalytic capabilities but also by their structural conformation and accessibility.

In enzymes with multiple subunits like pyrophosphatase, the arrangement and characteristics of active sites can greatly influence their functions and the effectiveness of enzymatic reactions.

Biosynthetic Reactions

Biosynthetic reactions are the pathways through which complex molecules are synthesized from simpler ones, forming the building blocks of life. These reactions are often driven by the energy provided by the hydrolysis of pyrophosphate, making pyrophosphatases key facilitators in metabolic pathways.

Through this hydrolytic action, pyrophosphatase helps maintain the necessary concentration of orthophosphate required for DNA, RNA, and protein synthesis, among other tasks. This central role in biosynthesis emphasizes the importance of enzyme kinetics in understanding how cellular processes are regulated and maintained.

Regulation of these pathways can occur at different levels, including enzyme concentration, activity modification by inhibitors or activators, and substrate availability. Understanding these processes helps in appreciating how cellular economies efficiently manage resources and respond to variations in metabolic needs and environmental conditions.