How are fractions of bitcoin implemented and spent?

If the ability to spend bitcoins and therefore bitcoins depend on/correspond to private keys, how can fractions of coins exist? Can the key be split up? How are fractional transactions signed versus integer transactions? I am not asking what the subdivisions of bitcoin are or how it can be further subdivided.

The analogy of "coins" as individual objects that can be handled is misleading. The atomic unit in Bitcoin is a satoshi, but satoshis are not handled individually, rather in each transaction it is specified how many satoshis (as an integer variable) move from where to where. The total number of satoshis in the input must be at least as great as in the output, and all the inputs must be digitally signed by the owners.

The private key is what allows you to prove ownership of given outputs. The way it is used for signatures doesn't really depend on how many different outputs you have and what is the value of each.

There isn't even a meaningful concept of "a bitcoin" in the software, a bitcoin is just shorthand for 100,000,000 satoshis.