No. They use different curves.
For each system the public and private keys of the same key pair are not independent. You choose a private key and that determines what your public key will be.
(Good thing it doesn't work in reverse. If you could derive someone's private key from their public key, then we wouldn't have an asymmetric algorithm.)
You can derive a private key from an old public key. (Wrong!) Or a private key from an old private key. But you cannot derive a public key directly from anything other than its corresponding private key.
Alternatively Alice can sign, using her old private key, a messages that informs others of her new public keys. When Bob wants to communicate with her, he can check the authenticity of that message using Alice's old public key. (Assuming the old private key has not been compromised.)
(Note that you should be using separate keys for signing and encrypting.)
As for Alice's private key, I recommend generating new (random) private keys. If her old private key were compromised then it would mean all new deterministically derived private keys would automatically be compromised as well.