Replacing lysine with aspartic acid is really a change in the primary structure (the sequence of the amino acids - think in a chain). But because they are really different amino acids, the effect is much more profound and will affect the tertiary structure of the protein.
Lysine has a basic, positively charged side chain. Aspartic acid has a negatively charged carboxyl group for its side chain. So, they are two very different amino acids.
Since the tertiary structure of a protein is a result of the interactions of the various interactions of the amino acid side chains, you have to think about what a swap of a basic positive amino acid with a negatively charged amino acid could cause.
For example, if the lysine side chain interacted in ionic interactions (i.e. attraction to a negatively charged amino acid), if you swap it for aspartic acid which is negatively charged it will now repel the other amino acid's side chain and that would disrupt the tertiary structure of the protein. It would also likely cause disruption to the quaternary structure as well.
If this change was in an important part of the protein (e.g. the active site of an enzyme) then it would likely disrupt the proper functioning of this protein.
If you wanted to make the least amount of change to a protein by making a mutation to that lysine amino acid, you would choose other basic amino acids which are histidine and arginine.