Why do our epiphyseal plates close up in our late teens or early twenties?
During Endochondral ossification  chondrocytes in the plate are rapidly dividing, newer daughter cells stack facing the epiphysis while the older cells are pushed towards the diaphysis. As the older chondrocytes degenerate, osteoblasts ossify the remains to form new bone. 

In puberty increasing levels of estrogen, in both females and males, leads to increased apoptosis of chondrocytes in the epiphyseal plate. Depletion of chondrocytes due to apoptosis leads to less ossification and growth slows down and later stops when the entire cartilage have become replaced by bone, leaving only a thin epiphyseal scar which later disappears. Once the adult stage is reached, maximum height is reached.

Your question addresses the genetics of this process however it is more so regulated by the endocrine system. There has not been a single gene assigned to growth plates, or its regulation, and I find it unlikely that there ever will be. 

Rather, endochondrial ossification takes place because of hundreds of genes interacting with each other. If you start to look at genes involved in apoptosis and the estrogen receptor pathway, you will begin to appreciate more the complexity behind the physiology.

I have a hunch you were looking for a gene, that with age just shuts off, or turns on. Recent evidence has show this does in fact happen as methylation may be a candidate mechanism of mediating not only environmental, but also genetic effects on age-related phenotypes.

http://journals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1002629