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Bacteria are fascinating microorganisms that reproduce through a process known as binary fission. This means a single bacterium splits into two identical cells. In an ideal environment, free from constraints like nutrient shortage or competition, a bacterial population can grow exponentially. This is described by the equation \( N(t) = N_0 \cdot 2^{t/t_b} \), which calculates the number of bacteria \( N(t) \) at a given time \( t \). In this equation:

• \( N_0 \) is the initial number of bacteria, provided as the starting point of the population.
• \( t_b \) denotes the mean time between cell divisions—here it is 2 hours.

A shorter \( t_b \) means the population doubles more quickly, while a longer \( t_b \) would indicate slower growth. This exponential growth can result in explosive increases in population size over short periods, especially in ideal conditions where cell death (\( m=0 \)) is ignored.

When dealing with exponential growth, plotting on a semilogarithmic graph provides a clearer perspective. Normally, exponential functions curve upwards steeply on linear axes, making it hard to compare or predict values. However, using a semilogarithmic graph, where the y-axis (population size \( N(t) \)) is scaled logarithmically, exponential curves become straight lines. To sketch a semilogarithmic graph, you plot:

• \( t \) on the x-axis, representing time.
• The logarithm (log base 10) of the cell number on the y-axis. For example, for \( N(t) = 1000 \), you'd plot log(1000) about 3.

The relationship remains linear, allowing different populations to be compared easily regardless of their initial size. It helps better visualize growth and compare wild types and mutant strains by the slope of the lines. The steeper the slope, the faster the growth rate.

A mutation is a change in a bacterium's DNA, potentially affecting its division rate. In our scenario, the mutant bacterial strain divides two times slower than its wild type counterpart. The division rate slows down because the mutant's mean time between divisions \( t_b \) increases from 2 to 4 hours.This change can be reflected in the modified exponential growth formula:\[ N(t) = N_0 \cdot 2^{t/4} \]Where:

• The wild type remains at \( t_b = 2 \) hours for quicker division.
• The mutant bacterium grows slower, doubling every 4 hours.

Understanding mutation rates is key in biological studies, as they explain variations in growth rates. In practical terms, slower-growing strains may adapt differently to environments due to changes in selection pressures associated with mutations. Thus, mutation rates significantly influence ecological balances and evolutionary dynamics in microbial communities.