Question

If a colony of \(1.5\) billion \(E\). coli cells were infected with a single phage T4 and each lytic replication cycle of the phage produced 200 new phages, how many replication cycles would it take for T4 phages to overwhelm the entire bacterial colony? (Assume for the sake of simplicity that every phage completes its replication cycle in a different cell and that the bacteria themselves do not reproduce.)

Short answer

The value of n calculated in step 4 will be the number of replication cycles of phage T4 needed to overcome the bacterial colony. Always round up, because you can't have part of a cycle.

Step-by-step solution

Understand the Question and Define the Problem

We are given that one T4 phage can infect and replicate within an E. coli cell, each replication producing 200 new phages. We assume for simplicity's sake that each phage infects a new cell, and the E. coli don't reproduce. We need to find out how many replication cycles it will take for the phages to overwhelm the entire colony of 1.5 billion E. coli.

Express the Problem Mathematically

We can express the problem as the equation: \(1 \cdot 200^n = 1.5 \times 10^9\), where n is the number of replication cycles.

Solve for n

To solve for n, we should take the logarithm of both sides. The base of the logarithm should be 200, the number of new phages produced per replication cycle. \(\log_{200}(1.5 \times 10^9) = n\). The value of this operation would yield the number of replication cycles needed for the phages to outnumber the bacteria.

Calculate the value of n

Using a calculator or a mathematical software, compute the value of \(\log_{200}(1.5 \times 10^9)\) to find the value of n.

Key concepts

E. coli infection

When it comes to microbiology, understanding the interaction between bacteria and bacteriophages is fundamental. E. coli, or Escherichia coli, is a common bacterium found in the intestines of humans and animals. While most E. coli strains are harmless, some can cause serious food poisoning and infections.

E. coli infection begins when the bacteria enter areas of the body where they're not typically found, such as the urinary tract, blood, or tissues. Symptoms can range from mild to severe and may include stomach cramps, diarrhea, and vomiting. In microbiological studies, it is essential to understand the dynamics of such infections, especially in the context of bacteriophage infestations, which can greatly alter the fate of bacterial colonies.

Lytic replication

Lytic replication is a cycle where a bacteriophage, a virus that infects bacteria, commandeers a bacterium's machinery to replicate its genetic material. It is a key process in the world of microorganisms, with great importance in both environmental microbiology and biotechnology. After infection, the phage introduces its DNA into the host cell, E. coli in our context, and redirects the cell's resources to produce new phages.

As new phage particles are assembled, they eventually cause the bacterium to burst, or lyse, releasing the new phages to infect neighboring cells. This lytic cycle is constant and can be extremely quick, leading to exponential increases in phage numbers if unchecked. The outcome of this cycle is crucial since it can determine the survival or devastation of bacterial populations.

Logarithmic calculations in microbiology

Mathematics often comes into play in microbiology, especially when dealing with the exponential growth patterns seen in bacterial and viral populations. Logarithmic calculations are particularly useful in understanding and predicting these patterns.

For example, to predict how many lytic replication cycles it would take for bacteriophage T4 to overwhelm a colony of E. coli, logarithms offer a precise method to reverse the exponential equation. This allows scientists to determine the number of cycles needed to reach a certain number of phages from a single initial infected cell. A deep understanding of these calculations enables microbiologists to model infection dynamics and potentially control or target these microbial processes.

Bacteriophage T4

Bacteriophage T4 is a type of virus that specifically infects E. coli bacteria. It is a member of the 'T-even' phages, a highly studied group due to their complex structure and replication mechanisms. T4 is particularly known for its efficient and aggressive infection cycle.

The phage attaches to the surface of the E. coli and injects its DNA, taking over the cell's machinery to reproduce its parts and assemble new phage particles. Ultimately, the T4 phage's lytic replication cycle leads to the destruction of the host cell. The role of T4 in molecular biology is significant as well, being used in various genetic and biochemical research because of the precise way it interacts with E. coli during replication.