Introduction to Natural Science
CAL Lab
Modeling Cell Growth

Introduction:
The cells in a culture reproduce by splitting in half forming two new daughter cells in a process called fission. As the population of cells grows more cells will be available to divide. Provided there are no barriers to cell division such as limited nutrients or space, or some type of predation the rate of growth of cells will increase in directly proportion to the number of cells present. If N is the number of cells and t is the time then the rate of change in N is

where the proportionality constant  is called the growth constant or the intrinsic growth rate of the cell culture. Since the cells are dividing continuously and with a constant growth rate the number of cells present as a function of time can be described by the exponential function

where  is the initial number of cells.

The time it takes for the population to double is called the generation time g or the doubling time. The generation time g is related to the growth constant  by the following relation

.

Eventually the nutrient supply will be exhausted and the cell culture will no longer grow at an exponential rate. The time period during which the cell culture grows exponentially is called the exponential phase or log phase by biologists.

In this lab we will model cell growth using tacks to represent cells

Procedure:
Take 20 tacks and place them in the petri dish. Count out another 60 tacks and keep them at the side — these are the nutrient supply for the cells in the petri dish. Shake the dish horizontally then count the number of tacks which point up. These are the tacks which are ready to divide and will do so provided sufficient nutrients are available to them. If sufficient nutrients are available add one tack for each tack which is pointing up in the petri dish. If there insufficient nutrients for all the tacks which are pointing up add what is available and remove those tacks that were not fed. These cells have died and should be put back in the tack box. The maximum number of nutrient tacks which are available each feeding period is eight. This process represents one growth cycle. Repeat this process for at least 20 cycles. Record the total number of tacks in the petri dish as a function of the number of cycles. Due to statistical variation you may want to repeat the entire experiment several times and find the average number of tacks for each cycle.

Plot average number of tacks vs number of cycles. Use logs to find the growth constant , the generation number g, and the time period that the cells are in the log phase.