The gel and PCR calculator

You will write three functions for the work of a molecular biology laboratory: the size of a band in a gel, the pieces of a cut ring, and the copies in a PCR.

The idea. In a gel, the logarithm of the size falls in a straight line with the distance. A ring cut at $n$ sites gives $n$ pieces. A PCR multiplies the copies by $1 + E$ in each cycle.

Your tasks.

1. Finish size_from_gel. It takes a ladder, as a dictionary of sizes and distances, and the distance of a band. It fits the line of $\log_{10}(\text{size})$ against the distance by least squares, and gives the size.
2. Finish ring_pieces. It takes the length of the ring and a list of cut sites, and gives the sizes of the pieces, from the largest to the smallest.
3. Finish pcr_copies. It gives the number of copies after some cycles, from some starting molecules, with an efficiency between 0 and 1.