ABSTRACT 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 singlemolecule experiments.
INTRODUCTION
Biological molecules have traditionally been studied in ensemble experiments, where large numbers are probed and analyzed simultaneously. Such experiments yield smoothly varying time and population averages, and can be rigorously interpreted using conventional thermodynamics and chemical kinetics. The picture that emerges represents mainly the states of highest probability within the population. Recently, however, researchers have begun to employ methods designed to study biomolecules one at a time. Such singlemolecule experiments have been used, for example, to investigate the elastic behavior of single DNA polymers, to unfold protein and RNA molecules, to measure the forces associated with molecular recognition, to investigate the kinematics and dynamics of single-motor molecules, and to follow the catalysis of individual enzymes. Single-molecule measurements often do not yield smooth, averaged values: individual molecules are found in states far from the mean of the population, and their instantaneous dynamics display chaotic and seemingly random behavior. The presence of large fluctuations affects the interpretation of data collected in single molecules. In particular, it raises the question, "what is the relationship between single-molecule measurements and thermodynamic data from ensemble measurements on the same systems?" Here we consider this question in the context of a common single-molecule experiment, measurement of a force vs. extension curve as a polymer molecule is stretched. We show that by replacing the familiar thermodynamic potentials (e.g., Gibbs free energy or Helmholtz free energy) with appropriate potentials of mean force, most but not all familiar thermodynamic concepts have close analogs on the single-molecule level. The primary difference is that although the choice of statistical ensemble is irrelevant for macroscopic systems, it can have a profound effect on single-molecule systems. Thus, in contrast to macroscopic thermodynamic systems, the results of singlemolecule experiments may depend on which variables are held fixed and which are allowed to fluctuate. Though these conclusions are reached by considering a particular experimental situation, the general ideas and conclusions hold for a wide variety of microscopic systems.
The mechanical extension of a single molecule
[Reference]
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[Author Affiliation]
David Keller,* David Swigon,^ and Carlos Bustamante^^(sec)(para)
*Department of Chemistry, University of New Mexico, Albuquerque, New Mexico; ^Department of Chemistry, Rutgers University, Rutgers, New Jersey; and ^^Howard Hughes Medical Institute; (sec)Department of Molecular and Cell Biology; and (para)Department of Physics, University of California at Berkeley, Berkeley, California
[Author Affiliation]
Submitted March 25, 2002, and accepted for publication July 29, 2002. Address reprint requests to David J. Keller, University of New Mexico, Clark 103, Albuquerque, NM 87131. Tel.: 505-277-3621/2060; Fax: 505277-2609; E-mail: dkeller@unm.edu.
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