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# Modeling

Recently, many biologists try to grasp biological systems behavior with computational and mathematical approach. But, when we make a model, in its process, we are apt to lose our sight about what kind things we want to grasp with those approach. So this time firstly we set our position to be clear.

## Problem

We considered the accuracy of Timer vector. It is supposed that when the length of repetitive sequence is short, the length of DNA cut off in each replication varies from cell to cell, and as a result, the fluctuation affects on the dynamics of the population, and the accuracy of our timer is reduced. We wondered if we make repetitive sequence of lacI binding site longer, the fluctuation among population is reduced and the accuracy increase. To confirm this, we simulated the time development of cell population.

## Assumption

We assume following conditions.

1. The length decreased in each DNA replication varies from 15 to 185 bps in uniform randomness.
2. There is no interaction among cells and the life duration only depends on the behavior of Timer vector.
3. Each division occurs at the same time duration.

## Numerical calculation

We run below program in mathematica for four cases, and plot each result.

## Result

horizontal axis=>Number of division.

vertical axis=>Population.

## Conclusion

As the result shows, by making repetitive sequences longer, the fluctuation becomes smaller and the time-developing behavior of cell population is averaged. Then we concluded our assumption is certain.