Question

The growth of a bacterial colony can be described as a first-order process, where the probability of cell division is linear with respect to time, expressed as NdN​=ζdt, where dN is the number of cells that divide in the time interval dt, and ζ is a constant.

Use the given expression to demonstrate that the number of cells in a colony is represented by N=N0​eζt, where N is the number of cell colonies, and N0​ is the number of colonies present at t=0.

Answer 1

Exponential growth of a bacterial colony can be mathematically proven using integration, showing that the population grows according to N=N0eζt. Real-world examples include a bacterial population in a flask with unlimited nutrients, growing exponentially to over 16 billion in 24 hours.

Step-by-step explanation:

The growth of a bacterial colony as a first-order process implies the exponential growth of the population. This growth is represented by the equation N=N0eζt, where N is the final number of bacterial colonies, N0 is the initial number of colonies present at t=0, t is the time, and ζ is the growth constant indicating the rate of cell division over time.

Let's prove this via integration. Starting from the rate of division dN/dt = ζN, we can rearrange to dN/N = ζdt.

Upon integrating both sides, ∫ dN/N = ∫ ζdt, we get lnN = ζt + C. If we solve for C using the initial condition N0 when t = 0, we find C = lnN0. Thus, the equation becomes lnN = ζt + lnN0 or ln(N/N0) = ζt. After exponentiating both sides, we arrive at N = N0e^ζt, which shows the exponential nature of bacterial growth when resources are not limiting.

One real-world example of bacterial growth is seen in a large flask with an unlimited supply of nutrients where a population of 1,000 bacteria can grow to over 16 billion in just 24 hours. This corresponds to the exponential growth model, correlating well with the aforementioned growth equation and demonstrating the concept of the increasing growth rate.