I am using this model to estimate the number of different ways a protein containing $25$ amino acids can fold.
Each bead is unique but can magnetically attach to any other. They are arranged on the string in a unique order, but for simplicity assume that the string can stretch as much as needed.
I figured out that, just considering one bead attaching to every other bead (i.e. just 1 "fold" of the protein), over each bead in the string, there are overall $300$ unique ways the string can fold $(24+23+22+21+...+0)$.
Obviously the problem gets much more complex when multiple folds are considered.
An estimate, even just to an order of magnitude, is welcome, given sound mathematical reasoning.
How can a reasonable estimate (or even exact figure) be reached?
If I understand correctly, there are $25$ beads in total, and hence $\tbinom{25}{2}=\tfrac{25\times24}{2}=300$ connections that can be made in total. The beads are attached in a string in a unique order, so these are $24$ connections that are fixed. This leaves $276$ available connections between beads. Each connection is either made or not made, yielding a whopping total of $$2^{276}\approx1.1214168\times10^{83},$$ configurations.