Relating Single-Molecule Measurements to Thermodynamics

Measurements made on large ensembles of molecules are routinely interpreted using thermodynamics, but the normal rules of thermodynamics may not apply to measurements made on single molecules. Using a polymer stretching experiment as an example, it is shown that in the limit of a single, short molecule the outcome of experimental measurements may depend on which variables are held fixed and which are allowed to fluctuate. Thus an experiment in which the end-to-end distance of the polymer molecule is fixed and the tension fluctuates yields a different result than an experiment where the force is fixed and the end-to-end distance fluctuates. It is further shown that this difference is due to asymmetry in the distribution of end-to-end distances for a single molecule, and that the difference vanishes in the appropriate thermodynamic limit; that is, as the polymer molecule becomes long compared to its persistence length. Despite these differences, much of the thermodynamic formalism still applies on the single-molecule level if the thermodynamic free energies are replaced with appropriate potentials of mean force. The primary remaining differences are consequences of the fact that unlike the free energies, the potentials of mean force are not in general homogeneous functions of their variables. The basic thermodynamic concepts of an intensive or extensive quantity, and the thermodynamic relationships that follow from them, are therefore less useful for interpreting single-molecule experiments.