Amplification at Equilibrium: Structural and Thermodynamic Limitations, and Implementation
Amplifying weak molecular signals is essential in both natural and engineered biochemical systems. While most amplification schemes operate out of equilibrium, relying on kinetic barriers and fuel-driven cascades, it is also possible to amplify at thermodynamic equilibrium by shifting the energy landscape upon addition of an analyte. Equilibrium amplification is appealing because, in principle, it can remain indefinitely in the untriggered state. In this work, we establish fundamental structural and thermodynamic limits on equilibrium-based amplification. We first prove that dimerization networks--systems restricted to complexes of at most two monomers--are inherently incapable of equilibrium amplification. This no-go theorem explains the absence of amplification in prior undercomplementary "strand commutation" designs. We then show that allowing trimeric complexes breaks this barrier. We propose an isometric trimer-based amplifier whose output preserves the size of the input, enabling modular composition, and validate it experimentally, achieving an amplification factor close to the expected $2\times$. Finally, we derive universal thermodynamic bounds applicable to any equilibrium network regardless of complex size: the maximum amplification factor scales linearly with the free energy of interaction between the analyte and the amplifier components. For nucleic acid systems, this implies that the analyte length must grow linearly with the desired amplification factor, and that composing modular amplifiers yields diminishing returns for a fixed analyte. Together, these results delineate the structural and energetic boundaries of equilibrium amplification and rigorously justify the necessity of out-of-equilibrium approaches for achieving high gain.