Aspartate residues, specifically the β-carboxylate group, play an essential role in the active sites of many enzymes. To determine how this group functions at physiological pH, let's explore its behavior regarding acidity and nucleophilicity.

The pKa of the β-carboxylate group in aspartate is 3.86. The pKa value provides insight into the protonation state of a group at a given pH. If the pH is above the pKa, the group is largely deprotonated, and at physiological pH (around 7.4), the β-carboxylate group will be deprotonated.

1. Deprotonation and function as a conjugate base:
   • At a physiological pH of roughly 7.4, which is above the pKa of 3.86, the β-carboxylate group of aspartate will exist primarily in its deprotonated form, COO^-.
   • This deprotonated form is the conjugate base of the carboxylic acid.
2. Nucleophilic Properties:
   • A nucleophile is an electron-rich species that donates electrons to an electron-deficient center.
   • The negatively charged COO^- group can act as a nucleophile due to its electron density.

Given these explanations, the β-carboxylate group of aspartate, at physiological pH, acts as a nucleophile and a conjugate base. This matches the correct answer: a nucleophile and a conjugate base.

Conclusion: Aspartate, with its deprotonated β-carboxylate group at physiological pH, efficiently plays its role in enzyme active sites as both a nucleophile due to its negative charge and as a conjugate base from its acid form.