Suppose $X$ and $Y$ are smooth affine surfaces over $\mathbb C$. Suppose there is a biholomorphism $f: X\to Y$. Does it follow that $X$ and $Y$ are isomorphic as affine surfaces (i.e. there exists an algebraic isomorphism $g: X\to Y$)?

What if we additionally know that $X$ and $Y$ are rational surfaces?