If you lose the private key for some address that contains bitcoins, then you will never be able to spend those bitcoins. The private key is required in order to spend, and there is no known way to work backwards to discover the private key for an account.

If the private key is lost, the bitcoins become un-spendable.

Bitcoins are unspendable without the private key (as chytrik already stated). That being said, several wallets implement BIP032 which allows you to recover those private keys. It uses a "seed" (usually shown as several words) to initialize the wallet. Then the creation of the private keys is deterministic. With this seed you would be able to "recover" your private key and therefore the unspent BTC.