Actually it DOES matter because you said that their nature was immutable.
When the Word took a human flesh, it took another nature, it did not change in nature.Jesus didn’t relinquish his Deity in order to become man, but retained his divine nature since he can never cease being God. This is where immutability comes in. At Jesus’ incarnation, there wasn’t a subtraction of his Deity, but an addition of humanity; he added, not subtracted, something to his divinity.
Again, you are mixing Jesus’ 2 natures with the Trinity.
Discuss Father, Word, Spirit and not Jesus’ second nature. Remember, John says the Word is God, and later on talks about the taking of an additional human nature. I repeat, Jesus eating, sleeping, dying have nothing to do with the Trinity.