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Volume with time is simply the change in length with

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volume with time is simply the change in length with time times a constant prefactor; then consider the change in protein concentration as a function of the change in the total volume into which the original N proteins are diluted. (b) We can repeat a calculation like that given above using a discrete language. Imagine that before cell division, the number of copies of a given transcription factor in the cell is N . In particular, for every cell doubling, the number of proteins is reduced by a factor of 2. Using such a picture, write a formula for the average number of proteins per cell as a function of the number of cell divisions and relate this result to that obtained in part (a). Furthermore, by using the fact that 2 = exp (ln 2), reconcile the discrete and continuous pictures precisely. (c) Interestingly, the model used in part (b) opens the door to one of the most important themes in physics, namely, that of fluctuations. In particular, as the cells divide from one generation to the next, each daughter does not really get N/ 2 copies of the protein since the dilution effect is a stochastic process. Rather the partitioning of the N proteins into daughter cells during division follows the binomial distribution. Analyzing these fluctuations can actually lead to a quantification of the number of copies of a protein in a cell. In this part of the problem, work out the expected fluctuations after each division by noting that the fluctuations can be written as q < ( N 1 - N 2 ) 2 > , where N 1 and N 2
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