The quaternary structure stability of proteins is typically studied under conditions that accelerate their aggregation/unfolding processes on convenient laboratory time scales. Such conditions include high temperature or pressure, chaotrope-mediated unfolding, or low or high pH. These approaches have the limitation of being nonphysiological and that the concentration of the protein in solution is changing as the reactions proceed. We describe a methodology to define the quaternary structure stability of the amyloidogenic homotetrameric protein transthyretin (TTR) under physiological conditions. This methodology expands from a described approach based on the measurement of the rate of subunit exchange of TTR with a tandem flag-tagged (FT₂) TTR counterpart. We demonstrate that subunit exchange of TTR with FT₂·TTR can be analyzed and quantified using a semi-native polyacrylamide gel electrophoresis technique. In addition, we biophysically characterized two FT₂·TTR variants derived from wild-type and the amyloidogenic variant Val122Ile TTR, both of which are associated with cardiac amyloid deposition late in life. The FT₂·TTR variants have similar amyloidogenic potential and similar thermodynamic and kinetic stabilities compared to those of their nontagged counterparts. We utilized the methodology to study the potential of the small molecule SOM0226, a repurposed drug under clinical development for the prevention and treatment of the TTR amyloidoses, to stabilize TTR. The results enabled us to characterize the binding energetics of SOM0226 to TTR. The described technique is well-suited to study the quaternary structure of other human aggregation-prone proteins under physiological conditions.