Problem 12 - Biochemistry: Concepts and Connections

Question

(a) A mitochondrion can be modeled as a cylinder \(1.5 \mu \mathrm{m}\) (microns) in length and \(0.6 \mu \mathrm{m}\) in diameter. Calculate the number of citrate synthase enzyme molecules contained in a single mitochondrion, assuming the mítochondrial citrate synthase concentration is \(1 \mu \mathrm{M}\). (b) Is this a reasonable answer? Assume that a single citrate synthase molecule is roughly spherical with a diameter of \(100 \mathrm{~A}(10 \mathrm{~nm})\). Show your reasoning.

Step-by-Step Solution

Answer

The mitochondrion contains about 256 citrate synthase molecules, which is reasonable given molecular sizes and packing.

Step 1: Calculate the Volume of the Mitochondrion

To find the volume of the mitochondrial cylinder, use the formula for the volume of a cylinder: \( V = \pi r^2 h \). Here, the radius \( r \) is half of the diameter, \( 0.6 \mu m / 2 = 0.3 \mu m \), and the height \( h \) is the length of the mitochondrion, \( 1.5 \mu m \). Thus, the volume: \( V = \pi (0.3 \mu m)^2 (1.5 \mu m) \approx 0.424 \mu m^3 \) or \( 0.424 \times 10^{-15} L \).

Step 2: Calculate the Number of Molecules

Using the concentration of citrate synthase \( 1 \mu M = 1 mol/m^3 \), we calculate the number of molecules using Avogadro's number \( 6.022 \times 10^{23} \) molecules/mol. The number of molecules is given by: \( \text{Number of molecules} = 1 \times 10^{-6} \times 0.424 \times 10^{-15} \times 6.022 \times 10^{23} \approx 256 \) molecules.

Step 3: Assess Reasonability

A citrate synthase molecule is roughly spherical with a diameter of \( 100 \mathrm{~A} = 10 \mathrm{~nm} = 0.01 \mu m \). Calculate the volume of a single molecule with \( V = \frac{4}{3} \pi r^3 \), where \( r = 0.005 \mu m \), resulting in \( V \approx 5.24 \times 10^{-7} \mu m^3 \). This implies very small volumes, and having \( 256 \) molecules spread in the mitochondrion's volume seems reasonable when considering molecular packing.

Key Concepts

Citrate Synthase

Citrate synthase is a critical enzyme in the mitochondrial matrix, playing a vital role in the citric acid cycle (also known as the Krebs cycle). This enzyme catalyzes the first step of the cycle, where acetyl-CoA is combined with oxaloacetate to form citrate. One important characteristic of citrate synthase is its specificity; it ensures that only citrate is formed, initiating energy production within the mitochondria. The concentration of citrate synthase within mitochondria can influence the overall rate of the citric acid cycle. Understanding its function is key for comprehending how cells produce energy efficiently. Since each mitochondrion can hold a certain concentration of citrate synthase, calculating the number of enzyme molecules helps in visualizing efficiency on a cellular level.

Mitochondrial Volume

Mitochondria, often referred to as the powerhouses of the cell, have a distinct cylindrical shape that can be modeled to assess their volume. For a cylinder, the volume is calculated using the formula: \[ V = \pi r^2 h \]where \( r \) is the radius and \( h \) is the height (length) of the cylinder. In this problem, the mitochondrion is modeled with a length of 1.5 micrometers and a diameter of 0.6 micrometers. Therefore, the radius is half of the diameter, i.e., 0.3 micrometers. Substituting these dimensions into the volume formula allows us to find that the volume is approximately \( 0.424 \mu m^3 \), or \( 0.424 \times 10^{-15} \) liters when converted to more standard units. Calculating mitochondrial volume helps in estimating how many molecules, such as citrate synthase, can fit within.

Enzyme Concentration

Enzyme concentration in mitochondria is fundamental for determining how effectively biochemical reactions occur. Concentration is typically measured in molarity, which stands for moles per liter (mol/L). In this exercise, the given concentration is 1 micromolar (\( 1 \mu M \)), translating into \( 1 \times 10^{-6} \) moles per liter. This indicates a sparse but adequate number of enzyme molecules within the mitochondrion for catalytic activity. Enzyme concentration directly affects the rate at which reactions can proceed. Having an appropriate concentration ensures that metabolic pathways like the citric acid cycle can operate smoothly without interruption. In practical terms, calculating the concentration helps in understanding the number of reactive molecules available in the mitochondrial environment, impacting energy production efficiency.

Avogadro's Number

Avogadro's number, a fundamental constant in chemistry, is essential for converting measurements from moles to molecules. It is defined as \( 6.022 \times 10^{23} \) molecules per mole. This remarkable constant allows chemists to bridge the gap between macroscopic measurements and microscopic reactions.In the context of mitochondrial biochemistry, Avogadro's number helps us determine how many molecules of citrate synthase are present by multiplying it with the concentration of the enzyme and the volume of the mitochondrion. For example, using the concentration \( 1 \mu M \), the volume \( 0.424 \times 10^{-15} \) liters, and Avogadro's number, we find approximately 256 molecules of citrate synthase in a mitochondrion.Avogadro's number makes it feasible to delve into the minute molecular world by translating the chemistry into quantifiable and tangible data. It's a cornerstone in understanding and calculating biochemical interactions at a molecular level.